Hypoelliptic Laplacian and Orbital Integrals (AM-177) (Annals of Mathematics Studies)

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Author: Jean-Michel Bismut

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Annals of Mathematics Studies Number 177

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Hypoelliptic Laplacian and Orbital Integrals

Jean-Michel Bismut

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2011

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c 2011 by Princeton University Press Copyright Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Bismut, Jean-Michel. Hypoelliptic Laplacian and Orbital Integrals / Jean-Michel Bismut p. cm. Includes bibliographical references and index. ISBN-13: 978-0-691-15129-8 (alk. paper) ISBN-13: 978-0-691-15130-4 (pbk. : alk. paper) 1. Hypoelliptic equations. 2. Index theory and related fixed point theorems. British Library Cataloging-in-Publication Data is available This book has been composed in LATEX The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

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Est-ce à votre cocher, Monsieur, ou bien à votre cuisinier, que vous voulez parler? car je suis l’un et l’autre. `re, L’avare Molie

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Contents

Introduction 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10 0.11 0.12

1

The trace formula as a Lefschetz formula A short history of the hypoelliptic Laplacian The hypoelliptic Laplacian on a symmetric space The hypoelliptic Laplacian and its heat kernel Elliptic and hypoelliptic orbital integrals The limit as b → 0 The limit as b → +∞: an explicit formula for the orbital integrals The analysis of the hypoelliptic orbital integrals The heat kernel for bounded b and the Malliavin calculus The heat kernel for large b, Toponogov, and local index The hypoelliptic Laplacian and the wave equation The organization of the book

1. Clifford and Heisenberg algebras 1.1 1.2 1.3 1.4 1.5 1.6

The The The The The The

Clifford algebra of a real vector space Clifford algebra of V ⊕ V ∗ Heisenberg algebra Heisenberg algebra of V ⊕ V ∗ Clifford-Heisenberg algebra of V ⊕ V ∗ Clifford-Heisenberg algebra of V ⊕ V ∗ when V is Euclidean

2. The hypoelliptic Laplacian on X = G/K 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15

A pair (G, K) The flat connection on T X ⊕ N The Clifford algebras of g The flat connections on Λ· (T ∗ X ⊕ N ∗ ) The Casimir operator The form κg The Dirac operator of Kostant The Clifford-Heisenberg algebra of g ⊕ g∗ The operator Db The compression of the operator Db A formula for D2b The action of Db on quotients by K The operators LX and LX b The scaling of the form B The Bianchi identity

1 2 3 4 5 5 6 6 7 9 9 9 12 12 14 15 17 18 19 22 23 25 25 25 27 28 30 32 33 34 34 35 39 41 41

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viii 2.16 2.17 2.18

CONTENTS

A fundamental identity The canonical vector fields on X Lie derivatives and the operator LX b

3. The displacement function and the return map 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Convexity, the displacement function, and its critical set The norm of the canonical vector fields The subset X (γ) as a symmetric space The normal coordinate system on X based at X (γ) The return map along the minimizing geodesics in X (γ) The return map on Xb The connection form in the parallel transport trivialization Distances and pseudodistances on X and Xb The pseudodistance and Toponogov’s theorem The flat bundle (T X ⊕ N ) (γ)

4. Elliptic and hypoelliptic orbital integrals 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

An algebra of invariant kernels on X Orbital integrals Infinite dimensional orbital integrals The orbital integrals for the elliptic heat kernel of X The orbital supertraces for the hypoelliptic heat kernel A fundamental equality Another approach to the orbital integrals The locally symmetric space Z

5. Evaluation of supertraces for a model operator 5.1 5.2 5.3 5.4 5.5

The operator Pa,Y0k and the function Jγ Y0k A conjugate operator An evaluation of certain infinite dimensional traces Some formulas of linear algebra A formula for Jγ Y0k

6. A formula for semisimple orbital integrals 6.1 6.2 6.3

Orbital integrals for the heat kernel A formula for general orbital integrals The orbital integrals for the wave operator

7. An application to local index theory 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Characteristic forms on X The vector bundle of spinors on X and the Dirac operator The McKean-Singer formula on Z Orbital integrals and the index theorem A proof of (7.4.4) The case of complex symmetric spaces The case of an elliptic element The de Rham-Hodge operator The integrand of de Rham torsion

41 45 46 48 49 50 54 57 62 64 65 67 68 75 76 77 78 81 84 84 85 86 87 92 92 94 95 103 110 113 113 114 116 120 120 122 124 125 126 130 131 134 136

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ix

CONTENTS

8. The case where [k (γ) , p0 ] = 0 8.1 8.2 8.3

The case where G = K The case a 6= 0, [k (γ) , p0 ] = 0 The case where G = SL2 (R)

9. A proof of the main identity 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11

X Estimates on the heat kernel qb,t away from bia N k−1 A rescaling on the coordinates (f, Y ) A conjugation of the Clifford variables The norm of α A conjugation of the hypoelliptic Laplacian The limit of the rescaled heat kernel A proof of Theorem 6.1.1 A translation on the variable Y T X A coordinate system and a trivialization of the vector bundles X The asymptotics of the operator Pa,A,b,Y k as b → +∞ 0 A proof of Theorem 9.6.1

10. The action functional and the harmonic oscillator 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

A variational problem The Pontryagin maximum principle The variational problem on an Euclidean vector space Mehler’s formula The hypoelliptic heat kernel on an Euclidean vector space Orbital integrals on an Euclidean vector space Some computations involving Mehler’s formula The probabilistic interpretation of the harmonic oscillator

11. The analysis of the hypoelliptic Laplacian 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

The The The The The The The The

X AX b , Bb

on X scalar operators Littlewood-Paley decomposition along the fibres T X Littlewood-Paley decomposition on X Littlewood Paley decomposition on X X heat kernels for AX b , Bb scalar hypoelliptic operators on Xb scalar hypoelliptic operator on Xb with a quartic term heat kernel associated with the operator LX A,b

12. Rough estimates on the scalar heat kernel 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

The Malliavin calculus for the Brownian motion on X The probabilistic construction of exp −tBbX over X The operator BbX and the wave equation The Malliavin calculus for the operator BbX The tangent variational problem and integration by parts A uniform control of the integration by parts formula as b → 0 X Uniform rough estimates on rb,t for bounded b The limit as b → 0 The rough estimates as b → +∞

138 138 139 140 142 142 145 147 150 150 152 153 153 156 158 159 161 162 164 166 173 175 177 182 183 187 188 189 192 193 201 205 206 210 212 214 217 219 222 223 226 228 230 237

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x

CONTENTS

b 12.10 The heat kernel rX b,t on X X 12.11 The heat kernel rb,t on Xb

241 244

13. Refined estimates on the scalar heat kernel for bounded b 13.1 13.2 13.3

The Hessian of the distance function Bounds on the scalar heat kernel on X for bounded b Bounds on the scalar heat kernel on Xb for bounded b

X 14. The heat kernel qb,t for bounded b

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11

248 251 260 262

−tLX A

A probabilistic construction of exp The operator LX b and the wave equation Changing Y into −Y A probabilistic construction of exp −tLX0 A,b Estimating V· Estimating W· A proof of (4.5.3) when E is trivial A proof of the estimate (4.5.3) in the general case X0 for bounded b Rough estimates on the derivatives of qb,t The behavior of V· as b → 0 X0 as b → 0 The limit of qb,t

X for b large 15. The heat kernel qb,t

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10

248

263 263 264 265 266 267 268 270 274 280 287 290

rX b,t

over X Uniform estimates on the kernel The deviation from the geodesic flow for large b The scalar heat kernel on X away from Fγ = ia X (γ) Gaussian estimates for rX b near ia X (γ) The scalar heat kernel on Xb away from Fbγ = bia N k−1 Estimates on the scalar heat kernel on Xb near bia N k−1 A proof of Theorem 9.1.1 A proof of Theorem 9.1.3 A proof of Theorem 9.5.6 A proof of Theorem 9.11.1

291 292 294 299 299 306 310 311 312 313

Bibliography

317

Subject Index

323

Index of Notation

325

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Acknowledgments The author is indebted to Yves Benoist, Laurent Clozel, Patrick Gérard, Fran¸cois Labourie, Gilles Lebeau, Yves Le Jan, Werner M¨ uller, and Yuri Tschinkel for very helpful discussions and remarks. I am especially grateful to Gilles Lebeau for many heated, fruitful, if nonacademic, discussions. Without his support, this work would not have been completed. I am also indebted to Xiaonan Ma and Weiping Zhang for their critical comments. Finally, I thank Courant Institute and New York University for supporting my stay during the spring terms of 2007, 2008, and 2009.

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Introduction The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula.

0.1 The trace formula as a Lefschetz formula Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. Let X be a compact Riemannian manifold, and let ∆X be the corresponding Laplace-Beltrami For t > 0, consider the trace of the heat kernel operator. X X Tr exp t∆ /2 . If L is the Hilbert space of square-integrable functions 2 on X, Tr exp t∆X /2 is the trace of the ‘group element’ exp t∆X /2 acting on LX 2 . X Suppose that of an acyclic complex R on which ∆X L2 is the cohomology acts. Then Tr exp t∆X /2 can be viewed as the evaluation of a Lefschetz trace, so that cohomological methods can be applied to the evaluation of this trace. In our case, R will be the fibrewise de Rham complex of the total space Xb of a flat vector bundle over X, which contains T X as a subbundle. The Lefschetz fixed point formulas of Atiyah-Bott [ABo67, ABo68] provide a model for the evaluation of such cohomological traces. The McKean-Singer formula [McKS67] indicates that if R is a Hodge like Laplacian operator acting on R and commuting with ∆X , for any b > 0, X TrL2 exp t∆X /2 = Trs R exp t∆X /2 − tR /2b2 . (0.1) In (0.1), Trs is our notation for the supertrace. Note that the formula involves two parameters: t is a parameter in a Lie algebra, and 1/b2 is a genuine time parameter. For b → 0, the right-hand side of (0.1) obviously converges to the left-hand side. To establish the Atiyah-Bott formulas, the heat equation method of Gilkey [Gi73, Gi84] and Atiyah-Bott-Patodi [ABoP73] consists in making b → +∞ in (0.1), and to show that the local supertrace in the right-hand side of (0.1) localizes on the fixed point set of the isometry exp t∆X /2 , while exhibiting the nontrivial local cancellations anticipated by McKean-Singer [McKS67]. One should obtain formulas this way that are analogous to the fixed point formulas of [ABo67, ABo68]. The present book is an attempt to make sense of the above, in the case

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INTRODUCTION

where X is a compact locally symmetric space of noncompact type. In this case, the Selberg trace formula should be thought of as the evaluation of a Lefschetz trace. Contrary to what happens in Atiyah-Bott [ABo67, ABo68], X R 2 the operator LX b = ∆ /2 − /2b is not elliptic, but just hypoelliptic.

0.2 A short history of the hypoelliptic Laplacian Let us now give the proper rigorous background to the present work. Let X be a compact Riemannian manifold, let X , X ∗ be the total spaces of its tangent and cotangent bundle. In [B05], we introduced a deformation of the classical Hodge theory of X. The corresponding Laplacian LX b , b > 0 is a hypoelliptic operator acting over X ∗ . It is essentially the weighted sum of the harmonic oscillator along the fibre and of the generator of the geodesic flow. Arguments given in [B05] showed that as b → 0, the operator LX b should converge in the proper sense to the Hodge Laplacian X /2 of X via a collapsing mechanism, and that as b → +∞, LX b converges to the generator of the geodesic flow. The program outlined in [B05] was carried out in Bismut-Lebeau [BL08], at least for bounded values of b. In [BL08], it was shown that in a very precise X way, LX b converges to /2. A consequence of the results of [BL08] is that is trace given t > 0, exp −tLX b class, and that as b → 0,Xits trace converges X to the trace of exp −t /2 . The spectral theory of Lb was also studied in [BL08], as well as its local index theory. An important result of [BL08] is that if F is a flat vector bundle on X, the Ray-Singer metric on det H · (X, F ) one can attach to LX b coincides with the classical elliptic Ray-Singer metric [RS71, BZ92]. This paves the way to a possible proof using the hypoelliptic Laplacian of the Fried conjecture [Fri86, Fri88] concerning the relation of the Ray-Singer torsion to special values of the dynamical zeta function of the geodesic flow. In [B08a], we gave a deformation of the classical Dirac operator, the deformed Dirac operator acting over X , its square still being hypoelliptic, and having the same analytic structure as the operator LX b . Results similar to the ones in [BL08] were established in [B08a] for Quillen metrics. As a warm-up to the present book, if G is a compact connected semisimple Lie group with Lie algebra g, we produced in [B08b] a deformation of the Casimir operator of G to a hypoelliptic Laplacian over G × g acting on smooth sections of Λ· (g∗ ), and we showed that the supertrace of its heat kernel coincides with the trace of the scalar heat kernel of G. In particular, the spectrum of the Casimir operator is embedded as a fixed part of the spectrum of the hypoelliptic Laplacian. By making b → +∞, we recovered known formulas [Fr84] expressing the heat kernel of G as Poisson sums over its coroot lattice. The deformation of the Casimir operator in [B08b] was obtained via a deformation of the Dirac operator of Kostant [Ko76, Ko97], whose square coincides, up to a constant, with the Casimir operator.

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INTRODUCTION

3

The question arises of knowing whether, in the case of locally symmetric spaces of noncompact type, a deformation of the Casimir operator to a hypoelliptic operator is possible, which would have the same interpolation properties as before, and would produce a version of the Selberg trace formula. Such a construction would provide a justification for the formal considerations we made in section 0.1. The purpose of this book is to show that this question has a positive answer. In this book, the spectral side of the trace formula will be essentially ignored. Our main result, which is given in chapter 6, is an explicit local formula for certain semisimple orbital integrals in a reductive group. The Selberg trace formula expresses the trace of certain trace class operators as a sum of orbital integrals. What is done in the book is to evaluate these orbital integrals individually, by a method that is inspired by index theory.

0.3 The hypoelliptic Laplacian on a symmetric space First, we will explain the construction of the hypoelliptic Laplacian that is carried out in the present book. Let G be a reductive Lie group with Lie algebra g, let K be a maximal compact subgroup of G with Lie algebra k, let B be an invariant nondegenerate bilinear form on g, and let g = p ⊕ k be the Cartan decomposition of g. Let U (g) be the enveloping algebra. Let C g ∈ U (g) be the Casimir element. Set X = G/K. Then X is contractible. Moreover, if ρE : K → Aut (E) is a unitary representation of K, E descends to a Hermitian vector bundle F on X. Then p, k descend to vector bundles T X, N , T X being the tangent bundle of X. Moreover, T X ⊕ N can be canonically identified with the trivial vector bundle g over X, and so it is equipped with a canonical flat connection. Let π : X → X, π b : Xb → X be the total spaces of T X, T X ⊕ N . Let b c (g) be the Clifford algebra of (g, −B). Following Kostant [Ko76, bg ∈ b Ko97], in Definition 2.7.1, we introduce the Dirac operator D c (g)⊗U (g), whose square coincides, up to a constant, with the negative of the Casimir. b X,2 , we obtain the elliptic operator LX In Definition 2.13.1, from − 21 D ∞ acting on C (X, F ). Up to a constant, LX coincides with the action of 21 C g on C ∞ (X, F ), so that LX is an elliptic operator. b g descends to an operator D b g,X acting on Also, the Dirac operator D C ∞ Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . In Definition 2.9.1 and in section 2.12, for b > 0, we introduce the operator DX b that acts on the above vector space. It is given by the formula b g,X + ic Y N , Y T X + 1 DT X + E T X − iDN + iE N . (0.1) DX b =D b In (0.1), c denotes the natural action of the Clifford algebra of (T X ⊕ N, B) on Λ· (T ∗ X ⊕ N ∗ ). Also the operator DT X + E T X − iDN + iE N is some

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INTRODUCTION

version of the standard Dirac operator along the Euclidean fibre T X ⊕ N introduced by Witten [Wi82]. In Definition 2.13.1, we define the operator LX b , which is given by 1 b g,X,2 1 X,2 + Db . (0.2) LX b =− D 2 2 As an aside, we suggest the reader compare the operator that appears in the right-hand side of (0.1) with the right-hand side of (0.2). The following formula is established in section 2.13, LX b

N Λ· (T ∗ X⊕N ∗ ) 1 N T X 2 1 2 T X⊕N + |Y | − m − n + = Y ,Y + 2 −∆ 2 2b b2 ∞ ∗ · ∗ ∗ b 1 C (T X⊕N,π (Λ (T X⊕N )⊗F )) + ∇Y T X +b c ad Y T X b ! TX N E N − c ad Y + iθad Y − iρ Y . (0.3)

Let us just say that in (0.3), ∆T X⊕N is the Euclidean Laplacian along the fibre T X ⊕ N . Also observe that in LX b , the harmonic oscillator along the fibres of the Euclidean vector bundle T X ⊕ N appears with the factor 1/b2 . ∂ + LX By results of H¨ ormander [Hö67], the operator ∂t b is hypoelliptic. Its structure is very close to the structure of the hypoelliptic Laplacian in [B05, BL08, B08a]. By adding a trivial matrix operator A to LX , LX b , we obtain operators X X LA , LX A,b . In the context of the present book, the operator LA,b will be called a hypoelliptic Laplacian.

0.4 The hypoelliptic Laplacian and its heat kernel In chapter 11, the proper functional analytic machinery is developed, in order to obtain a chain of Sobolev spaces on which the hypoelliptic Laplacian acts as an unbounded operator. This is done by inspiring ourselves from our previous work with Lebeau [BL08, chapter 15], which is valid for the case where the base manifold is compact. Also regularizing properties of its resolvent and of its heat operator are obtained. The heat operator is shown to be given by a smooth kernel. A probabilistic method is also given to construct the heat operator on X or Xb. The fact that the functional analytic and probabilistic constructions coincide is proved using the Itô calculus. The probabilistic construction of the heat kernel is relatively easy, but does not give the refined properties on the resolvent that one obtains by the functional analytic machinery. In the remainder of the book, most of the hard analysis is done via the probabilistic construction of the heat kernel, while the functional analytic estimates do not play a significant role. This is because contrary to the situation in [BL08], where it was essential to obtain proper understanding

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INTRODUCTION

of the spectral properties of the hypoelliptic Laplacian, here, this aspect can be essentially disregarded.

0.5 Elliptic and hypoelliptic orbital integrals Let γ ∈ G be semisimple. In chapter 4, we introduce the heat operators X X exp −tLA , exp −tLA,b , and we define the corresponding orbital inteh i grals Tr[γ] exp −tLX . These orbital integrals are , Trs [γ] exp −tLX A A,b said to be respectively elliptic and hypoelliptic. As the notation suggests, the elliptic orbital integrals are generalized traces, while the hypoelliptic orbital integrals are generalized supertraces. While the existence of the elliptic orbital integrals follows from standard Gaussian estimates for the X on X, the existence of the orbital integrals for heatkernel for exp −tL A exp −tLX A,b

relies on a nontrivial estimate on the hypoelliptic heat ker-

nel ((x, Y ) , (x0 , Y 0 )). This estimate is stated in Theorem 4.5.2. It says that given > 0, M > 0, ≤ M , there exist C > 0, C 0 > 0 such that for 0 < b ≤ M, ≤ t ≤ M , and (x, Y ) , (x0 , Y 0 ) ∈ Xb, then X qb,t

X qb,t ((x, Y ) , (x0 , Y 0 )) ≤ C exp −C 0 d2 (x, x0 ) + |Y |2 + |Y 0 |2 .

(0.1)

0.6 The limit as b → 0 In Theorem 4.6.1, we prove that for any b > 0, t > 0, . Trs [γ] exp −tLX = Tr[γ] exp −tLX A A,b

(0.1)

Equation (0.1) is closely related to a corresponding identity established for ordinary traces over a compact Lie group G in [B08b]. The proof of (0.1) consists of two steps. The fact that the left-hand side of (0.1) does not depend on b > 0 is proved by a method very closely related to the proof of the McKean-Singer formula [McKS67] in index theory. It is in this sense that the book unifies index theory and the evaluation of orbital integrals. The proof of (0.1) is then reduced to showing that as b → 0, the lefthand side converges to the right-hand side. Proving this fact is obtained X by a nontrivial analysis of the heat kernel for exp −tLA,b . The uniform estimate (0.1) plays a crucial role in the proof. In section 0.8, we will give more details on the analytic arguments used in the book.

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INTRODUCTION

0.7 The limit as b → +∞: an explicit formula for the orbital integrals Our final formula is obtained by making b → +∞ in (0.1). Let dγ (x) = d (x, γx) be the displacement function associated with γ [BaGSc85], which is known to be a convex function. Let X (γ) ⊂ X be its critical set, which is a totally geodesic submanifold of X. Then X (γ) is the symmetric space associated with the centralizer Z (γ)h⊂ Gof γ. Asi b → +∞, the analysis of the [γ] X exp −tLA,b localizes near X (γ). More hypoelliptic orbital integral Trs h i , precisely, in chapter 9, to obtain the asymptotics of Trs [γ] exp −tLX A,b we choose x ∈ X (γ), and we take the suitable expansion of LX A,b near the geodesic connecting x and γx. This expansion involves a rescaling of coordinates, and also a corresponding Getzler rescaling [Ge86] of the Clifford variables b c. Ultimately, the limit operator is a hypoelliptic operator acting on p × g. The existence of a canonical flat connection over T X ⊕N plays a critical role in the computations. Combining the existence of this flat connection with the existence of the central Casimir operator introduces two major differences with respect to what was done in [B05, BL08, B08a]. After conjugation, we may assume that γ = ea k −1 , a ∈ p, k ∈ K, and Ad (k) a = a. Let k (γ) ⊂ k beh the Lie algebra i of the centralizer of γ in K. [γ] X Our explicit formula for Tr exp −tLA,b is stated in Theorem 6.1.1. It is given by an explicit integral over k (γ). In chapter 6, we show how to derive corresponding formulas for arbitrary kernels, which include the wave kernel. X This is all the more remarkable, since, contrary to LX A , LA,b does not have a wave kernel.

0.8 The analysis of the hypoelliptic orbital integrals In the analysis of the hypoelliptic orbital integrals, there is some overlap with the analysis of the hypoelliptic Laplacian in [BL08]. In [BL08], the Riemannian manifold X was assumed to be compact, and genuine traces or supertraces were considered. Here X is noncompact, and the orbital integrals that appear in (0.1) are defined using explicit properties of the corresponding heat kernels like the estimate in (0.1). Such estimates do not follow from [BL08]. In [BL08], the limit as b → 0 of hypoelliptic supertraces was studied by functional analytic methods involving semiclassical pseudodifferential operators. Chapter 17 in [BL08] is entirely devoted to this question. Since here X is noncompact, and since we deal explicitly with the kernels of the considered operators, the results of [BL08] cannot be used as such. Finally, the limit as b → +∞ involves questions that were not addressed in [BL08]. Again uniform estimates are needed.

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7

In the present book, these analytic questions are dealt with by a combination of probabilistic and analytic methods. In probability, we use the Itô calculus, and also the stochastic calculus of variations, or Malliavin calculus [M78, St81b, B81a, M97].

0.9 The heat kernel for bounded b and the Malliavin calculus Estimates like (0.1) are essentially obtained in three steps: 1. In chapter 12, we obtain rough estimates on scalar heat kernels associated with a scalar version of LX b . By rough estimates, we mean uniform bounds on the kernels and their derivatives of arbitrary order. Such bounds are obtained using the Malliavin calculus. Also we study the limit as b → 0 of the scalar hypoelliptic heat kernel. 2. In chapter 13, using the semigroup property of the scalar heat kernel combined with the rough bounds, we establish decay estimates similar to (0.1) for the scalar heat kernel. 3. In chapter 14, the estimates for the scalar heat kernel are transferred X to the kernel qb,t . We will briefly explain why probabilistic methods are relevant for step (1). Note that the geodesic flow on X is a differential equation. Also, by (0.3), LX A,b is a differential operator of order 1 in the variables in X, while being of order 2 in the variables in T X ⊕ N . To the scalar part of LX A,b , b one can associate a stochastic differential equation on X , which projects to a differential equation on X. This differential equation is a perturbation of the geodesic flow. The heat equation semigroup for the scalar part of LX A,b describes the probability law in Xb of the corresponding diffusion process at a given time t. The Malliavin calculus consists in exploiting the structure of the stochastic differential equation. More precisely, the properties of the heat kernel are obtained by using the fact that the scalar heat kernel is the image by the stochastic differential equation map Φ of a classical Brownian measure. Integration by parts on Wiener space can then be used to control the derivatives of the heat kernel. Estimates on heat kernels are ultimately obtained via the estimation of the Malliavin covariance matrix Φ0 Φ0∗ . For bounded b, estimating the covariance matrix is essentially equivalent to the proper uniform control of an action functional depending on b > 0. For b > 0, if xs , 0 ≤ s ≤ t is a smooth curve with values in X with fixed (x0 , x˙ 0 ) , (xt , x˙ t ), set Z 1 t 2 2 |x| ˙ + b4 |¨ x| ds. (0.1) Hb,t (x· ) = 2 0 This action functional was introduced in [B05] for smooth curves in X, and the corresponding variational problem was studied by Lebeau [L05]. Still

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INTRODUCTION

problems remained because of the possible nonsmoothness of the solution of the associated Hamilton-Jacobi equation. However, it turns out that the estimate of the Malliavin covariance matrix represents a tangent variational problem, which can be controlled by the solution of a related variational problem, where X is replaced by the Euclidean vector space p. This problem had precisely been studied by Lebeau in [L05] as a warm-up to a full understanding of the variational problem on X. This is why, prior to chapter 12, we devote chapter 10 to a detailed study of the above variational problem on an Euclidean vector space. The results of chapter 10 are used in chapter 12 to obtain a control of the integration by parts formula. Besides, when properly interpreted, chapter 10 can be viewed as an explicit verification of the soundness of our method of proof, when G is an Euclidean vector space. The fact that in this case, the intermediate steps can be made completely explicit is of special interest. In chapter 12, we also obtain the limit as b → 0 of the scalar hypoelliptic heat kernel. As explained before, when the base manifold is compact, a functional analytic version of this problem was solved in Bismut-Lebeau [BL08, chapter 17]. Here, this result is reobtained by probabilistic methods for the noncompact manifold X. In chapter 13, we obtain a Gaussian decay of the scalar heat kernel similar to (0.1) using the rough bounds in chapter 12, and also by exploiting the semigroup property. Probabilistic methods are still used, but they are more elementary than in chapter 12. One difficulty is to show that as b → 0, in spite of the fact that the energy of the underlying diffusion in X tends to +∞ as b → 0, this diffusion does not escape to infinity in X, the energy being absorbed by random fluctuations. Ultimately, the estimates follow easily from the rough bounds, and from Mehler’s formula for the heat kernel of the harmonic oscillator. In chapter 14, the transfer of the estimates for the scalar heat kernel to X estimates on qb,t is obtained using a matrix version of the Feynman-Kac formula. In the case where F is nontrivial, the symmetric space associated with the complexification KC of K plays an important role. Let us point out that many of our estimates are still valid in the case of variable curvature. In particular, the techniques developed in the present book can be used to give a different proof of some of results established in [BL08], with explicit estimates on the heat kernels. Finally, let us explain in more detail how we use our estimates. The uniform bounds on the heat kernels are needed to prove that the orbital integrals are well defined, and also to show that dominated convergence can be applied to the integrand defining the orbital integral when b → 0 and b → +∞. The bounds on the higher derivatives are needed when computing the limit of the orbital integrals. This is done by establishing uniform bounds over compact subsets on the kernels and their derivatives, and by proving that the heat kernels converge in a weak sense. Ultimately, we get pointwise convergence, which combined with the uniform bounds on the heat kernels, gives the convergence of the orbital integrals.

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9

0.10 The heat kernel for large b, Toponogov, and local index In chapter 15, the hypoelliptic heat kernel is studied as b → +∞. Note that by (0.3), for large b > 0, after rescaling Y ∈ T X ⊕ N , b )) 1 4 T X N 2 1 C ∞ (T X⊕N,π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F 2 LX Y ,Y + |Y | + ∇Y T X b ' b 2 2 +b c ad Y T X − c ad Y T X + iθad Y N − iρE Y N + O 1/b2 . (0.1) Equation (0.1) indicates that the diffusion associated with the scalar part of LX b tends to propagate along the geodesic flow. Still, because we have to control the corresponding heat kernel, we ultimately need to obtain a quantitative estimate on how much this diffusion differs from the geodesic flow. This question is dealt with in two steps. 1. Let ϕt |t∈R be the geodesic flow in X . Then γϕ−1 1 is a symplectic transformation of X , whose fixed point set Fγ is simply related to X (γ), the critical set of dγ . A purely geometric question, which is dealt with in the end of chapter 3, is to find how much the return map ϕ−1 1 γ differs from the identity away from Fγ . The corresponding quantitative estimates are obtained by using Toponogov’s theorem repeatedly. 2. In chapter 15, these estimates are combined with the rough bounds on X the heat kernel qb,t to obtain the proper uniform estimates for b large. Local index theoretic methods are used to control local cancellations in the supertrace of the heat kernel near Fγ as b → +∞. 0.11 The hypoelliptic Laplacian and the wave equation A crucial observation is made in sections 12.3 and 14.2, which relates the heat equation for the scalar version of LX b to the classical wave equation on X. It is shown that after averaging in the fibre variables, the heat equation on X or Xb descends to a nonlinear version of the wave equation on X. This observation is at the heart of some of the key probabilistic arguments used in chapters 12 and 14 to establish the uniform Gaussian decay in (0.1). More fundamentally, it is connected with the fact that the Hamiltonian differential equation of order 1 for the Hamiltonian flow on X descends to a differential equation of order 2 on X for the geodesics. In some sense, this descent argument propagates to the heat equation for the hypoelliptic Laplacian.

0.12 The organization of the book The book is divided into two parts. A first part, which includes chapters 1–9, contains the construction of the objects which are considered in the book,

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INTRODUCTION

the geometric results which are needed and their proof, the statement of the main results and their proofs. The analytic results which are needed in the proofs are themselves stated without proof. The detailed proof of the analytic results is deferred to a second part, which includes chapters 10–15. This book is organized as follows. In chapter 1, we recall general results on Clifford and Heisenberg algebras. b In chapter 2, we construct the hypoelliptic Laplacian LX b over X . In chapter 3, we establish various geometric results. If γ ∈ G is semisimple, we introduce the displacement function dγ , its critical set X (γ), and the coordinate system on X, which one obtains from the normals to X (γ). Also we study the return map ϕ−1 1 γ using Toponogov’s theorem. In chapter 4, we define the elliptic and hypoelliptic orbital integrals, and we establish identity (0.1). In chapter 5, if Y0k ∈ k (γ), we evaluate the heat kernel for a hypoelliptic operator Pa,Y0k acting on p ⊕ g, and we compute the supertrace Jγ Y0k of this heat kernel. In chapter 9, the operator Pa,Y0k will appear as a rescaled limit of LX b when b → +∞. This chapter can be read independently. In chapter 6, we state without proof our main result, which expresses the elliptic orbital integrals associated with the heat kernel as a Gaussian integral on k (γ). Also we show how to derive from this formula a corresponding formula for other kernels, which include the wave kernel. In chapter 7, we show the compatibility of our formula for the orbital integrals to the Atiyah-Singer index theorem [AS68a, AS68b] and to the Lefschetz fixed point formulas of Atiyah-Bott [ABo67, ABo68]. Also we recover results of Moscovici-Stanton [MoSt91] that are related to the evaluation of the Ray-Singer analytic torsion of locally symmetric spaces. In chapter 8, we evaluate explicitly the integrals over k (γ) when γ verifies a simple commutation relation, and we recover Selberg’s trace formula when G = SL2 (R). In chapter 9, we prove the formula that was stated in chapter 6. The proof relies on estimates established in the second part of the book. In chapter 10, we establish detailed results on the variational problem associated with the action Hb,t in (0.1) when X is an Euclidean vector space E, and we state various versions of Mehler’s formula. Also we establish key estimates on integrals involving the heat kernel of the harmonic oscillator. This chapter can be read independently. In chapter 11, given b > 0, we adapt the functional analytic methods of [BL08] to construct the resolvent and the heat kernel for the hypoelliptic Laplacian. In chapter 12, we obtain rough estimates for the heat kernel that is associated with a scalar hypoelliptic operator on X and on Xb. Such estimates are given for b bounded, and for b large. Also we study the limit of the heat kernel as b → 0. In chapter 13, we obtain the analogue of the uniform estimates in (0.1)

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INTRODUCTION

for the scalar heat kernels on X and on Xb. X In chapter 14, we establish estimates for the heat kernel qb,t for bounded b, which include (0.1). X Finally, in chapter 15, we establish the required estimates on qb,t for large b. As explained before, many ideas and techniques used in the present book have already been tested in our previous work with Lebeau [BL08]. Still the present book is largely self-contained, except for the analytic results of [BL08, chapter 15], whose use can in part be avoided. Some familiarity with [B08b] could be useful. Also we have tried to make the reading easier, by separating the results from their proofs, and geometric arguments from analytic arguments. Also we tried to justify the role of probabilistic arguments as much as possible. The length of the book can be partly explained by the fact that we rederive many of the technical results of [BL08] by different methods. If A is a Z2 -graded algebra, if a, b ∈ A, [a, b] will be our notation for the supercommutator of a, b, so that deg(a)deg(b)

[a, b] = ab − (−1)

ba.

(0.1)

Also, in most of the book, we will use Einstein summation conventions. Moreover, in the whole text, constants C will always be positive. If a constant depends on a parameter , it will often be written as C . Also positive constants will often be denoted using the same notation, although they may well be different. The results contained in this book have been announced in [B09b].

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Chapter One Clifford and Heisenberg algebras The purpose of this chapter is to recall various results on Clifford algebras and Heisenberg algebras. The results of this chapter will be used in our construction of the hypoelliptic Laplacian over a symmetric space. This chapter is organized as follows. In section 1.1, we introduce the Clifford algebra of a vector space V equipped with a symmetric bilinear form B. In section 1.2, we specialize the construction of the Clifford algebra to the case of V ⊕ V ∗ . In section 1.3, if (V, ω) is a symplectic vector space, we construct the associated Heisenberg algebra. In section 1.4, we specialize the construction of the Heisenberg algebra to the case of V ⊕ V ∗ . In section 1.5, we consider the combination of the Clifford and Heisenberg algebras for V ⊕ V ∗ , and we construct the complex Λ· (V ∗ ) ⊗ S · (V ∗ ) , d , which is the subcomplex of polynomial forms in the de Rham complex. Finally, in section 1.6, when V is equipped with a scalar product, this complex is related to a Witten complex over V .

1.1 The Clifford algebra of a real vector space Let V be a finite dimensional real vector space of dimension n, let B be a real-valued symmetric bilinear form on V . Let c (V ) be the Clifford algebra associated to (V, B). Namely, c (V ) is the algebra generated over R by 1, a ∈ V and the commutation relations for a, b ∈ V , ab + ba = −2B (a, b) .

(1.1.1)

We denote by b c (V ) the Clifford algebra associated to −B. Then c (V ) , b c (V ) are filtered by length, and their corresponding Gr· is just Λ· (V ). Also they are Z2 -graded algebras. If B is zero, then c (V ) , b c (V ) coincide with Λ· (V ). In the sequel, we assume that B is nondegenerate, so that V and V ∗ can be identified by the form B. Let ϕ : V → V ∗ denote the corresponding isomorphism, so that if a, b ∈ B, hϕa, bi = B (a, b) .

(1.1.2)

Let B ∗ be the corresponding bilinear form on V ∗ . Then B ∗ induces a nondegenerate symmetric bilinear form on Λ· (V ∗ ), which we still denote B ∗ .

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Clifford and Heisenberg algebras

If a ∈ V , let c (a) , b c (a) ∈ End (Λ· (V ∗ )) be given by c (a) = ϕa ∧ −ia ,

b c (a) = ϕa ∧ +ia .

(1.1.3)

Then c (a) and b c (a) are odd operators, which are respectively antisymmetric and symmetric with respect to B ∗ . If a, b ∈ V , then [c (a) , c (b)] = −2B (a, b) , [b c (a) , b c (b)] = 2B (a, b) ,

[c (a) , b c (b)] = 0. (1.1.4)

By (1.1.4), we find that Λ· (V ∗ ) is a c (V ) and a b c (V ) Clifford module. Also if a∈V,b c (a) is the negative of c (a) associated with −B. This is a discrepancy that is unavoidable, given the fact that the definition of the operator b c (a) is well established by tradition. If A lies in c (V ) or b c (V ), we denote by c (A) , b c (A) the corresponding endomorphisms of Λ· (V ∗ ) associated with the representation of the Clifford algebras c (V ) , b c (V ) defined in (1.1.3). deg A0 · ∗ The maps A ∈ c (V ) → c (A) 1 ∈ Λ (V ) , A0 ∈ b c (V ) → (−1) ( ) b c (A0 ) 1 ∈ Λ· (V ∗ ) identify c (V ) , b c (V ) as vector spaces to Λ· (V ∗ ). If α ∈ Λ· (V ∗ ), we will denote by c (α) , b c (α) the corresponding elements in c (V ) , b c (V ). There will be no risk of confusion with the above definition of c (A) , b c (A). ∗ ∗ Let e1 , . . . , en be a basis of V , and let e1 , . . . , en be the dual basis of V ∗ with respect to B, so that B ei , ej = δij . Note that if a ∈ V , a=

n X

B (e∗i , a) ei .

(1.1.5)

i=1

Let e1 , . . . , en be the basis of V ∗ which is dual to the basis e1 , . . . , en . Of course for 1 ≤ i ≤ n, ei = ϕe∗i . Take α ∈ Λp (V ∗ ), so that X α= α e i1 , . . . , e ip e i1 ∧ . . . ∧ e ip . (1.1.6) 1≤i1 0, let LX b act on C

Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) by the formula

1 b g,X,2 1 X,2 LX + Db . b =− D 2 2

(2.13.4)

By (2.12.19), we get LX b =

N Λ· (T ∗ X⊕N ∗ ) 1 1 N T X 2 2 Y ,Y + 2 −∆T X⊕N + |Y | − m − n + 2 2b b2 1 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) + ∇Y T X +b c ad Y T X b ! − c ad Y T X + iθad Y N − iρE Y N . (2.13.5)

The fibres of Λ· (T ∗ X ⊕ N ∗ ) ⊗ F are naturally equipped with a Hermitian product, which is denoted h i. Let h i be the Hermitian product on the space of smooth compactly supported sections of π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) over Xb, Z hs, s0 i = hs, s0 i dvXb . (2.13.6) b X X Lb is

Theorem 2.13.2. The operator ∂ + LX η. Moreover, ∂t b is hypoelliptic.

C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

Also 1b ∇Y T X

formally self-adjoint with respect to is formally skew-adjoint with respect

C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

1 to h i, and LX b − b ∇Y T X respect to h i.

is formally self-adjoint with

Proof. The first part of our theorem follows from Theorem 2.12.5 and from (2.13.4). The second part is a consequence of a general result by Hörmander [H¨ o67] on second order differential operators. C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) Clearly 1b ∇Y T X is formally skew-adjoint with respect to h i. The first line in the right-hand side of (2.13.5) is formally C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) self-adjoint with respect to h i. Except for 1b ∇Y T X , this is also the case for all terms in the right-hand side of (2.13.5). Indeed TX since ad Y exchanges T X and N , this is clear for b c ad Y T X and TX N c ad Y . Also since θad Y is antisymmetric with respect to the scalar product of Λ· (T ∗ X ⊕ N ∗ ) andpreserves T X and N , c θad Y N is antisymmetric, and so ic θad Y N is self-adjoint. Finally, iρE Y N is clearly self-adjoint. The proof of our theorem is completed. The operator LX b will be called a hypoelliptic Laplacian. By proceeding as in (2.12.20), we can rewrite (2.13.5) in the form N Λ· (T ∗ X⊕N ∗ ) 1 N T X 2 1 2 T X⊕N LX = Y , Y −∆ + |Y | + − m − n + b 2 2b2 b2 ! 1 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),f ∗,fˆ + ∇Y T X − c iθad Y N − iρE Y N . b (2.13.7)

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The hypoelliptic Laplacian on X = G/K

2.14 The scaling of the form B For t > 0, we denote with an extra subscript t the above operators associated with the form B/t. By (2.12.19), it is clear that · ∗ ∗ √ g,X Λ· (T ∗ X⊕N ∗ ) i /2 X −N Λ (T X⊕N ) /2 b + √ c Y N, Y TX tN Db,t t = tD t √ √ 1 1 i + tDT X + √ E T X − i tDN + √ E N . (2.14.1) b t t For a > 0, set Ka s (x, Y ) = s (x, aY ) . (2.14.2) By (2.14.1), we get · ∗ ∗ √ X Λ· (T ∗ X⊕N ∗ ) /2 X −N Λ (T X⊕N ) /2 √ Db,t t . (2.14.3) K√t tN K −1t = tD√ tb By (2.13.4), (2.14.3), we obtain K√t tN

Λ· (T ∗ X⊕N ∗ )

/2

−N LX b,t t

Λ· (T ∗ X⊕N ∗ )

/2

−1 √ . K√ = tLX tb t

(2.14.4)

2.15 The Bianchi identity The classical Bianchi identity says that h i X,2 DX = 0. b , Db

(2.15.1)

We will establish a refinement of the Bianchi identity along the lines of [B08b, Proposition 3.12]. Proposition 2.15.1. For any b > 0, the following identity holds: X X Db , Lb = 0. (2.15.2) b g,X,2 coProof. We start from the Bianchi identity (2.15.1). By (2.12.17), D 1 ∗ g g g incides with −C − 4 B (κ , κ ), so that h i g,X,2 b DX , D = 0. (2.15.3) b By (2.13.4), (2.15.1), and (2.15.3), we get (2.15.2).

2.16 A fundamental identity Put · ∗ ∗ 1 2 −∆T X⊕N + |Y | − m − n + N Λ (T X⊕N ) , 2 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) β = ∇Y T X +b c ad Y T X − c ad Y T X + iθad Y N − iρE Y N , 2 1 γ = Y N, Y TX . 2 α=

(2.16.1)

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Chapter 2

By (2.13.5), we get β α + + γ. (2.16.2) 2 b b We denote by H the fibrewise kernel of α, so that n o 2 H = exp − |Y | /2 ⊗ F. (2.16.3) Let H ⊥ be the orthogonal to H in L2 Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . LX b =

Note that β maps H into H ⊥ . Let α−1 be the inverse of α restricted to H ⊥ . Let P, P ⊥ be the orthogonal projections on H, H ⊥ . We embed ∗ · ∗ ∗ b π L2 (X, F ) into L2 X, b (Λ (T X ⊕ N ) ⊗ F ) via the isometric embed 2 ding s → π b∗ s exp − |Y | /2 /π (m+n)/4 . Now we establish an analogue of [B05, Theorem 3.14], [B08a, Theorem 1.14], and [B08b, Theorem 3.11], this last reference being especially relevant. Theorem 2.16.1. The following identity holds: P γ − βα−1 β P = LX .

(2.16.4)

Proof. We can rewrite the first equation in (2.12.19) in the form F 1 √ DX . b =E+ b 2

(2.16.5)

Using (2.13.4), and comparing (2.16.2) and (2.16.5), we get 1 b g,X,2 γ = E2 − D . (2.16.6) 2 Moreover, as we saw in section 1.5, H is just the kernel of F , so that P F = 0, F P = 0. From (2.16.6), we obtain 1 b X,2 P γ − βα−1 β P = P E 2 − EP ⊥ E − D P. (2.16.7) 2 α = F 2,

β = [E, F ] ,

Equivalently, 1 b X,2 2 P γ − βα−1 β P = (P EP ) − P D P. 2 By equation (2.10.2) in Proposition 2.10.1, we get P EP = 0.

(2.16.8)

(2.16.9)

Using (2.7.7), (2.13.3), (2.16.8), and (2.16.9), we get (2.16.4). Another proof for our theorem is to use the explicit formulas for α, β, γ in (2.16.1), and to get (2.16.4) by an explicit computation. A similar explicit computation has been done in the proof of [B08b, Theorem 3.11], to which the reader is referred. Since such a computation will be needed in chapter 14, we will give the main steps of this computation.

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Set R (Y ) = b c ad Y T X

− c ad Y T X + iθad Y N .

(2.16.10)

Let e1 , . . . , em be an orthonormal basis of T X, let em+1 , . . . , em+n be an orthonormal basis of N . By (1.1.9), (1.1.11), we get X

T X b c ad Y T X − c ad Y T X = Y , ei , ej 1≤i≤m m+1≤j≤m+n

ei ∧ ej − iei iej , (2.16.11) X

i Y N , ei , ej c (ei ) c (ej ) . c iθad Y N = − 4 1≤i,j≤m+n

Also by (1.1.3), we get c (ei ) =ei − iei , 1 ≤ i ≤ m,

(2.16.12)

i

− e − iei , m + 1 ≤ i ≤ m + n. Let P be the projection from Λ· (T ∗ X ⊕ N ∗ ) on Λ0 (T ∗ X ⊕ N ∗ ) = R. Set P⊥ = 1 − P.

(2.16.13)

Note that by (2.16.10)–(2.16.12), PR (Y ) P = 0. Definition 2.16.2. For Y ∈ T X ⊕ N , set −1 · ∗ ∗ S (Y ) = PR (Y ) 1 + N Λ (T X⊕N ) R (Y ) P.

(2.16.14)

(2.16.15)

Then S (Y ) ∈ R. Note that ad2 Y T X maps N into itself, and moreover,

TrN ad2 Y T X = − C k,p Y T X , Y T X . Proposition 2.16.3. The following identity holds: 1 1 N 2 T X Tr ad Y + TrT X⊕N −ad2 Y N P. S (Y ) = 3 24

(2.16.16)

(2.16.17)

Proof. Using (2.16.10), (2.16.11), we get PR Y T X

·

1 + N Λ (T

=

∗

X⊕N ∗ )

−1

m 1 X T X 2 R Y TX P = Y , ei P 3 i=1

m+n 1 1 X T X 2 Y , ei P = TrN ad2 Y T X P. (2.16.18) 3 i=m+1 3

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Chapter 2

Also using (2.16.10), (2.16.11), and (2.16.12), we obtain PR Y N

·

1 + N Λ (T

∗

X⊕N ∗ )

−1

m+n 1 X N 2 R YN P= Y , ei P 24 i=1

=

1 T X⊕N Tr −ad2 Y N P. (2.16.19) 24

Finally, because in the first equation in (2.16.11), 1 ≤ i ≤ m, m + 1 ≤ j ≤ m+n, and in the second equation, either 1 ≤ i, j ≤ m or m+1 ≤ i, j ≤ m+n, we get easily PR Y T X

·

1 + N Λ (T

= PR Y N

∗

X⊕N ∗ )

−1 ·

1 + N Λ (T

R YN P −1 ∗ ∗ X⊕N )

R Y T X P = 0. (2.16.20)

By (2.16.18)–(2.16.20), we get (2.16.17). Definition 2.16.4. Put δ=−

3 1 Tr C k,p − Tr C k,k . 16 48

(2.16.21)

Proposition 2.16.5. The following identity holds: P S (Y ) P = δ.

(2.16.22)

Proof. If u ∈ g, we have the easy identity 2

P hu, Y i P =

1 2 |u| . 2

(2.16.23)

By (2.16.16), (2.16.17) and (2.16.23), we get (2.16.22). Proposition 2.16.6. The following identity holds: −1 · ∗ ∗ 1 T X N 2 − R (Y ) 1 + N Λ (T X⊕N ) R (Y ) P P Y ,Y 2 1 = B ∗ (κg , κg ) P. (2.16.24) 8 Proof. Using (2.16.23), we get P

1 T X N 2 1 Y ,Y P = − Tr C k,p . 2 8

(2.16.25)

By (2.6.11) and (2.16.21), we obtain 1 1 δ + Tr C k,p = − B ∗ (κg , κg ) . 8 8

(2.16.26)

From (2.16.15), (2.16.22), (2.16.25), and (2.16.26), we get (2.16.24).

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Using again (2.16.23), we get 1 (2.16.27) P = − ∆H,X , 2 2 1 P ρE Y N P = C k,F . 2 2 Finally, note that the functions hu, Y i exp − |Y | /2 span the eigenspace 2 of the harmonic oscillator 21 −∆T X⊕N + |Y | − m − n that is associated with the eigenvalue 1. This explains why in (2.16.4), we may as well replace · ∗ ∗ α by 1 + N Λ (T X⊕N ) . By combining (2.13.2), (2.13.3), (2.16.1), (2.16.11), (2.16.24), and (2.16.27), we get C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),2 P γ − βα−1 β P = −P ∇Y T X P −1 1 T X N 2 Λ· (T ∗ X⊕N ∗ ) − R (Y ) 1 + N R (Y ) P +P Y ,Y 2 2 1 1 1 + P ρE Y N P = − ∆H,X + C k,F + B ∗ (κg , κg ) = LX . (2.16.28) 2 2 8 This completes the computational proof of Theorem 2.16.1. C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),2

−P ∇Y T X

2.17 The canonical vector fields on X Recall that the curvature Ω of the canonical connection on the K-principal bundle p : G → G/K is given by (2.1.10). Let RT X be the curvature of the connection ∇T X . If a, b, c ∈ p, RT X is just the equivariant representation of the map a, b, c ∈ p → − [[a, b] , c] ∈ p. Observe that if a, b, c, d ∈ g, then B (− [[a, b] , c] , d) = −B ([a, b] , [c, d]) .

(2.17.1)

B (− [[a, b] , b] , a) = B ([a, b] , [a, b]) .

(2.17.2)

In particular, Since B is negative on k, from (2.17.2), by taking a, b ∈ p, we deduce that X has nonpositive curvature. Since X has nonpositive curvature, given a base point, the exponential map Tx X → X is a covering. Since X is simply connected, this map is one to one. If a ∈ g, let ∇a,r denote differentiation on G with respect to the rightinvariant vector field associated with a. The operator ∇a,r descends to an operator La acting on C ∞ (X, F ). The operator La is just the infinitesimal version of the action of G on C ∞ (X, F ). Let aT X be the vector field on X associated with a. Then La is a Lie derivative operator acting on C ∞ (X, F ) that is associated with aT X . We can write La in the form F La = ∇F aT X − µ (a) .

(2.17.3)

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Chapter 2

In (2.17.3), µF (a) is a skew-adjoint section of End (F ). Let us now give a more precise version of (2.17.3). If g ∈ G, we can split ad g −1 a ∈ g according to the splitting g = p ⊕ k, in the form k p (2.17.4) Ad g −1 a = Ad g −1 a + Ad g −1 a . p k Tautologically, the functions Ad g −1 a , Ad g −1 a define sections aT X , aN of T X, N over X, the section aT X being just the above vector field. Under the identification T X ⊕ N ' g given in (2.2.1), aT X , aN are precisely the components of a ∈ g with respect to this splitting, so that a = aT X + aN .

(2.17.5) k Also ρE Ad g −1 a descends to a skew-adjoint section of End (F ), E which will be denoted ρ aN . From the above considerations, we get µF (a) = ρE aN . (2.17.6)

By (2.17.3) and (2.17.6), we obtain E La = ∇F aN . aT X − ρ

(2.17.7)

Moreover, it is well-known that ∇F µF (a) + iaT X RF = 0.

(2.17.8)

Note that (2.17.8) follows easily from the above considerations on µF (a). On the other hand, a is a flat section of T X ⊕ N ' g, i.e., if A ∈ T X, T X⊕N,f ∇A a = 0.

From (2.2.2) and (2.17.9), we get ∇TAX aT X + A, aN = 0,

N TX ∇N = 0. A a + A, a

(2.17.9)

(2.17.10)

When La acts on sections on T X, it is just the Lie derivative operator LaT X . Classically, LaT X = ∇TaTXX − ∇T· X aT X . Comparing with (2.17.3), (2.17.6), we get µT X (a) = ρT X aN = ∇T· X aT X .

(2.17.11)

(2.17.12)

Of course the last equality in (2.17.12) also follows from (2.17.10).

2.18 Lie derivatives and the operator LX b If e ∈ T X ⊕ N , let ∇Ve denote the corresponding differentiation operator along the fibre T X ⊕ N , and LV[e,Y ] denotes the fibrewise Lie derivative operator associated with the fibrewise vector field [e, Y ].

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Take a ∈ g. The action of the Lie derivative operator La on C ∞ (X, F ) is given by (2.17.7). When replacing E by C ∞ (g, Λ· (g∗ ) ⊗ E), by (2.12.13) and (2.17.7), the action of La on C ∞ (X, C ∞ (T X ⊕ N, π ∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ))) is given by C ∞ (T X⊕N,π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

La = ∇aT X

+ LV[aN ,Y ] − ρE aN .

(2.18.1)

In (2.18.1), the fibrewise Lie derivative operator LV[aN ,Y ] is given by (2.12.4), i.e., LV[aN ,Y ] = ∇V[aN ,Y ] − (c + b c) ad aN . (2.18.2) Definition 2.18.1. Set X LX a,b = Lb + La .

(2.18.3)

In the sequel, we use the notation in (2.17.5). Theorem 2.18.2. The following identity holds: LX a,b

N Λ· (T ∗ X⊕N ∗ ) 1 N T X 2 1 2 T X⊕N + 2 −∆ Y ,Y + |Y | − m − n + = 2 2b b2 1 C ∞ (T X⊕N,π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),f ∗,fb + ∇Y T X +baT X −b c (bad (a)) b ! N N TX E N N − c iθad Y + bad a − a − ρ iY + ba + ∇V[aN ,Y ] . (2.18.4)

Moreover, the operator

LX a,b

commutes with Z (a).

Proof. Equation (2.18.4) follows from (2.4.5), (2.13.7), (2.18.1), and (2.18.2). X Since LX b commutes with G, and La commutes with Z (a), La,b commutes with Z (a).

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Chapter Three The displacement function and the return map The purpose of this chapter is to study the displacement function dγ on X that is associated with a semisimple element γ ∈ G. If ϕt , t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X (γ) ⊂ X of dγ can be easily related to the fixed point set Fγ ⊂ X of the symplectic transformation γ −1 ϕ1 of X . We study the nondegeneracy of γ −1 ϕ1 − 1 along Fγ . More fundamentally, we give important quantitative estimates on how much ϕ1/2 differs from ϕ−1/2 γ away from Fγ . These quantitative estimates are based on Toponogov’s theorem [BaGSc85]. They will play an essential role in chapter 15, where we establish the results needed to study the hypoelliptic orbital integrals as b → +∞, which are shown to concentrate in the proper way near a natural lift Fbγ of Fγ in Xb. This chapter is organized as follows. In section 3.1, if γ ∈ G is semisimple, we recall key properties of the displacement function dγ (x) = d (x, γx) on X along the lines of Ballmann-Gromov-Schroeder [BaGSc85]. In particular dγ is a convex function, its critical set X (γ) is introduced, and the factorization of γ as a product of commuting hyperbolic and elliptic elements is obtained. By conjugation, we will write γ in the form γ = ea k −1 , a ∈ p, k ∈ K, Ad (k) a = a. 2 In section 3.2, when a ∈ p, we show that the function ψa = aT X /2 is a Morse-Bott function, whose critical set is the submanifold X (ea ). In section 3.3, we show that X (γ) is the symmetric space associated with the centralizer Z (γ) of γ. In section 3.4, we show that d2γ is a Morse-Bott function. Also we describe the normal bundle NX(γ)/X , and the corresponding normal coordinate system that identifies X to the total space of NX(γ)/X . In section 3.5, we describe the obvious analogue of the return map along the geodesics in X (γ) that connect x, γx, x ∈ X (γ). We also interpret X (γ) as the fixed point set of a symplectic transformation of the cotangent bundle of X associated with the geodesic flow. In section 3.6, we consider the obvious lift of the return map to the total space Xb of T X ⊕ N . In section 3.7, we evaluate the connection form on the principal bundle p : G → X = G/K using the parallel transport trivialization. In section 3.8, we define a family of distances and pseudodistances on X and Xb. In section 3.9, we establish important estimates involving the flow ϕt |t∈R , which are based on Toponogov’s theorem.

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The displacement function and the return map

Finally, in section 3.10, we construct a flat subbundle (T X ⊕ N ) (γ) of T X ⊕ N . This subbundle will be used in chapter 9 when applying local index methods to hypoelliptic orbital integrals as b → +∞. Reading sections 3.6–3.10 can be deferred to the point where the corresponding results are used in the text.

3.1 Convexity, the displacement function, and its critical set If γ ∈ G, we denote by Z (γ) ⊂ G the centralizer of γ, and by z (γ) its Lie algebra. Then z (γ) = {f ∈ g, Ad (γ) f = f } .

(3.1.1)

Let Z 0 (γ) be the connected component of the identity in Z (γ). If a ∈ g, let Z (a) ⊂ G be the stabilizer of a, and let z (a) be its Lie algebra. Then z (a) = ker ad (a) .

(3.1.2)

Definition 3.1.1. A function h : X → R is said to be convex if for any geodesic t ∈ R → xt ∈ X with constant speed, the function t ∈ R → h (xt ) ∈ R is convex. A subset Y ⊂ X is said to be convex if when y, y 0 ∈ Y , the geodesic connecting y and y 0 is included in Y . Let d be the Riemannian distance in X. If γ ∈ G, the displacement function dγ is given by dγ (x) = d (x, γx) .

(3.1.3)

By [BaGSc85, §6.1], the function dγ is convex on X. Also if g ∈ G, dC(g)γ (gx) = dγ (x) .

(3.1.4)

By (3.1.4), the function dγ is Z (γ)-invariant. Moreover, dθγ (θx) = dγ (x) .

(3.1.5)

By [E96, 2.19.21], γ ∈ G is said to be semisimple if the function dγ has a minimum value in X. Let X (γ) ⊂ X be the subset of X where dγ takes its minimum value mγ . By [BaGSc85, p. 78], X (γ) is a closed convex subset of X. Since dγ is convex and is smooth on X \ X (γ), by [BaGSc85, 1.2], dγ has no critical points on X \ X (γ). Moreover, the action of Z (γ) on X preserves X (γ). By (3.1.5), if γ is semisimple, θγ is semisimple, and moreover, X (θγ) = θX (γ) ,

mθγ = mγ .

(3.1.6)

The element γ ∈ G is said to be elliptic if γ is semisimple and mγ = 0, i.e., if γ fixes some point in X. In this case, X (γ) is just the set of fixed points of γ. Equivalently γ is elliptic if and only if it is conjugate in G to an element of K. Finally, γ is said to be hyperbolic if it is conjugated in G to ea , a ∈ p.

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Chapter 3

Elliptic or hyperbolic elements are semisimple. By [E96, Theorem 2.19.23], γ is semisimple if and only if there exist h, e ∈ G, with h hyperbolic and e elliptic such that γ = he = eh. Also e, h are uniquely determined. Finally, Z (γ) = Z (h) ∩ Z (e) .

(3.1.7)

If g ∈ G is such that pg = x, then pγg = γx. If t ∈ [0, 1] → xt ∈ X is the geodesic connecting x and γx, let g· be the horizontal lift of x· in G such that g0 = g. We define the parallel transport element k ∈ K by the formula g1 = γgk.

(3.1.8)

Theorem 3.1.2. Let γ ∈ G be semisimple. If g ∈ G, x = pg ∈ X, then x ∈ X (γ) if and only if there exist a ∈ p, k ∈ K such that Ad (k) a = a, and moreover, γ = C (g) ea k −1 . (3.1.9) Also C (g) ea ∈ G, C (g) k ∈ G are uniquely determined by γ. If gt = geta , then t ∈ [0, 1] → xt = pgt is the unique geodesic connecting x and γx. Moreover mγ = |a| .

(3.1.10)

Finally, k ∈ K is the parallel transport along the above geodesic. Proof. If x ∈ X (γ), the unique geodesic connecting x and γx is of minimal length among the geodesics connecting y and γy, for y ∈ X. We claim that x ∈ X (γ) if and only if x is represented by g ∈ G such that there is a ∈ p, k ∈ K with γ = C (g) ea k −1 , Ad (k) a = a. (3.1.11) Indeed the geodesic connecting x and γx is represented in G by t ∈ [0, 1] → gt = geta , and its length is |a|. There is k ∈ K such that gea = γgk,

(3.1.12)

and (3.1.8), (3.1.12) show that k is precisely the parallel transport along x· . The condition Ad (k) a = a is exactly the one that guarantees that x minimizes dγ . As we saw before Theorem 3.1.2, in (3.1.9), C (g) ea and C (g) k are uniquely determined. The proof of our theorem is completed.

3.2 The norm of the canonical vector fields Take a ∈ g. Since the bilinear form B is G-invariant, by (2.17.4), we get T X 2 N 2 a − a = B (a, a) . (3.2.1) Set 2 1 1 N 2 1 a = aT X − B (a, a) . (3.2.2) 2 2 2 We denote by ∇ψa the gradient of ψa , and by ∇T X ∇ψa its Hessian, i.e., the covariant derivative of ∇ψa with respect to ∇T X . If e ∈ T X, we identify e with the corresponding element in g via the identification T X ⊕ N ' g. ψa =

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Proposition 3.2.1. The following identities hold: ∇ψa = aT X , aN , ∇T X ∇ψa = ad2 aT X − ad2 aN .

(3.2.3)

If e ∈ T X,

2 ∇Te X ∇ψa , e = |[a, e]| .

(3.2.4)

In particular the function ψa is convex. Proof. Clearly,

N ∇ψa = aN , ∇N . · a

(3.2.5) Using (2.17.10) and (3.2.5), and the fact that ad aT X acts as a symmetric operator on T X ⊕ N , we get the first identity in (3.2.3). By (2.17.10) and TX the first identity in (3.2.3), we get the second identity. Since ad a acts on T X ⊕ N as a symmetric operator, and ad aN as an antisymmetric operator, equation (3.2.4) follows from (3.2.3). Since (3.2.4) is nonnegative, the function ψa is convex. The proof of our proposition is completed. Remark 3.2.2. Using Proposition 3.2.1, the functions aT X , aN can be shown to be convex. Since along geodesics, aT X is a Jacobi field, and since X has nonpositive curvature, the convexity of aT X is also a consequence of [BaGSc85, Remark p.5]. Let aT X0 be the 1-form dual to aT X by the metric. Proposition 3.2.3. If U, V ∈ T X, then

daT X0 (U, V ) = 2 aN , U , V .

(3.2.6)

Proof. Since ∇T· X aT X is antisymmetric,

daT X0 (U, V ) = 2 ∇TU X aT X , V .

(3.2.7)

By (2.17.10) and (3.2.7), we get (3.2.6). In the sequel, we take a ∈ p. Set x = p1. The geodesic coordinate system centered at x allows us to identify p with X. More precisely, the identification is given by Y p ∈ p → exp (Y p ) x ∈ X. Along the geodesics centered at x, we trivialize T X, N by parallel transport with respect to the connections ∇T X , ∇N . Proposition 3.2.4. The following identities hold: aT X (Y p ) = cosh (ad (Y p )) a,

aN (Y p ) = − sinh (ad (Y p )) a.

so that T X 2 p a (Y ) = |cosh (ad (Y p )) a|2 ,

N 2 a = |sinh (ad (Y p )) a|2 .

(3.2.8)

(3.2.9)

In particular, T X 2 p a (Y ) ≥ |a|2 + |[a, Y p ]|2 .

(3.2.10)

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Chapter 3

Proof. Recall that aN vanishes at x. Since Y p is a parallel section of T X along the geodesic t ∈ R → exp (tY p ) x ∈ p, equation (3.2.8) follows from (2.17.10). Equation (3.2.9) follows from (3.2.8). By (3.2.9), we get T X 2 p a (Y ) = 1 |a|2 + 1 hcosh (2ad (Y p )) a, ai . 2 2

(3.2.11)

Since ad2 (Y p ) is a symmetric nonnegative endomorphism of p, we can diagonalize it on an orthonormal basis of p, with nonnegative eigenvalues. The inequality (3.2.10) is now a consequence of the fact that for t ∈ R, cosh (t) dominates its asymptotic expansion. Remark 3.2.5. In the trivialization used in Proposition 3.2.4, we get aT X + aN = exp (−ad (Y p )) a. (3.2.12) On the other hand, exp (ad (Y p )) aT X + aN ∈ g corresponds to aT X + aN in the identification T X ⊕ N ' g. By (3.2.12), we find that this is precisely a, which has indeed to be the case. Since ψa is convex, its critical set coincides with the set where it reaches its minimum value. Also this set is convex. Theorem 3.2.6. In the geodesic coordinate system centered at x = p1, X (ea ) = z (a) ∩ p.

(3.2.13)

If a ∈ p, the minimum value of ψa is 0, and the minimum set of ψa is just X (ea ). Equivalently, X (ea ) = x ∈ X, aN = 0 . (3.2.14) The section aN is nondegenerate along the manifold X (ea ), and the function ψa is a Morse-Bott function. sa Proof. Clearly T Xthe length of the path s ∈ [0, 1]a → e x that connects x and a e x is just ax . By Theorem 3.1.2, p1 ∈ X (e ), and so by equation (3.1.10) in Theorem 3.1.2, mea = |a|. Therefore, |a| ≤ dea ≤ aT X . (3.2.15) By (3.2.2), the minimum value of ψa is 0, and the minimum value of aT X is |a|. As we just saw, p1 ∈ X (ea ). Take f ∈ p, and assume that x = pef ∈ X (ea ). Since X (ea ) is convex, t ∈ [0, 1] → xt = petf ∈ X is a geodesic included in X (ea ). Set

ϕf (t) = dea (xt ) .

(3.2.16)

φf (t) = |a| .

(3.2.17)

Then

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Let ct (s) , 0 ≤ s ≤ 1 be the unique geodesic connecting xt and ea xt , and let Ef (t) be its energy. By (3.2.17), Ef (t) =

1 2 |a| . 2

(3.2.18)

By (3.2.18), we get Ef00 (0) = 0.

(3.2.19)

By Theorem 3.1.2, c0 (s) = pesa . Let Jf,s , 0 ≤ s ≤ 1 be the Jacobi field ∂ Jf,s = ∂t ct (s) |t=0 . In the trivialization given by parallel transport, J¨f − ad2 (a) Jf = 0, Jf,0 = f, Jf,1 = f.

(3.2.20)

Classically, Ef00 (0) =

Z 0

1

˙ 2 2 Jf + |[a, Jf ]| ds.

(3.2.21)

[a, f ] = 0,

(3.2.22)

By (3.2.19), (3.2.21), we get

so that f ∈ z (a) ∩ p. Conversely, if f ∈ z (a) ∩ p, then ef ∈ Z (ea ). By Theorem 3.1.2, pef ∈ X (ea ), which gives (3.2.13). By Proposition 3.2.4 and by (3.2.13), aN vanishes on X (ea ). Combining this with (3.2.15), we get (3.2.14). On X (ea ), aT X = a. By (2.17.10), we find that over X (ea ), if A ∈ T X, N ∇N A a = [a, A] .

(3.2.23)

Since ad (a) exchanges p and k, z (a) is the direct sum of its p and its k parts. Let z⊥ (a) ⊂ g be the orthogonal to z (a) in g with respect to B, which is also the orthogonal to z (a) with respect to the scalar product of g. Since ad (a) is symmetric, it acts as an invertible endomorphism of z⊥ (a). By (3.2.23), we conclude that aN is nondegenerate along its zero set. By (3.2.3), we get ∇T X ∇ψa |X(ea ) = ad2 (a) |p .

(3.2.24)

By the same argument as before, (3.2.24) is a nondegenerate quadratic form on z⊥ (a) ∩ p, i.e., ψa is a Morse-Bott function. The proof of our theorem is completed. Remark 3.2.7. There is a corresponding statement when replacing a ∈ p by b ∈ k, and exchanging the roles of T X and N . However, since X esb may change when s ∈ R∗ , some care has to be given to the precise formulation. Proposition 3.2.8. The following identity holds: Z (ea ) = Z (a) .

(3.2.25)

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Proof. Clearly Z (a) ⊂ Z (ea ). Conversely take g ∈ Z (ea ). By Theorem 3.1.2, pg ∈ X (ea ). We can write uniquely g in the form g = eb k 0 , b ∈ p, k 0 ∈ K.

(3.2.26)

For t ∈ R, set xt = etb p1.

(3.2.27) a

Then x· is a geodesic in X connecting p1 and pg. Since X (e ) is totally geodesic, for any t ∈ R, xt ∈ X (ea ). Using equation (3.2.13) in Theorem 3.2.6, we conclude that b ∈ z (a), and so eb ∈ Z (a). Using (3.2.26), we find that k 0 ∈ Z (ea ). Moreover, 0

C (k 0 ) ea = eAd(k )a ,

(3.2.28)

and so 0

ea = eAd(k )a .

(3.2.29)

0

Moreover, a ∈ p, Ad (k ) a ∈ p, and so by (3.2.29), we get Ad (k 0 ) a = a,

(3.2.30)

0

i.e., k ∈ Z (a). Since eb ∈ Z (a) , k 0 ∈ Z (a), we find that g ∈ Z (a). This completes the proof of our proposition.

3.3 The subset X (γ) as a symmetric space Let γ ∈ G be semisimple. We fix one g0 ∈ G such that (3.1.9) holds with g = g0 . Set pγ,g0 = ad (g0 ) p, kγ,g0 = ad (g0 ) k.

(3.3.1)

Then g = pγ,g0 ⊕ kγ,g0 is a Cartan decomposition of g. Because of the above, we may as well assume that g0 = 1, so that γ = ea k −1 , k ∈ K,

a ∈ p, Ad (k) a = a.

(3.3.2)

By (3.1.7) and (3.3.2), we get Z (γ) = Z (ea ) ∩ Z (k) .

(3.3.3)

Using Proposition 3.2.8, we can rewrite (3.3.3) in the form Z (γ) = Z (a) ∩ Z (k) .

(3.3.4)

The Lie algebra z (k) of Z (k) is given by z (k) = {f ∈ g, Ad (k) f = f } .

(3.3.5)

z (γ) = z (a) ∩ z (k) .

(3.3.6)

By (3.3.4), we obtain

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Put p (γ) = z (γ) ∩ p,

k (γ) = z (γ) ∩ k.

(3.3.7)

From (3.3.6), we find that z (γ) = p (γ) ⊕ k (γ) .

(3.3.8)

By (3.3.8), the restriction of B to z (γ) is nondegenerate. Set K (γ) = K ∩ Z (γ) .

(3.3.9)

By (3.3.6), a lies in the center of z (γ). Let za,⊥ (γ) be the orthogonal to a in z (γ), let pa,⊥ (γ) be the orthogonal to a in p (γ). By (3.3.8), we get za,⊥ (γ) = pa,⊥ (γ) ⊕ k (γ) .

(3.3.10)

a,⊥

Moreover, z (γ) is a Lie algebra. Let Z a,⊥,0 (γ) be the connected Lie subgroup of Z 0 (γ) that is associated with the Lie algebra za,⊥ (γ). Note that if a 6= 0, Z 0 (γ) ' Z a,⊥,0 (γ) × R,

(3.3.11)

so that eta maps into t |a|. Let z⊥ (γ) be the orthogonal space to z (γ) in g with respect to B. Then ⊥ z (γ) splits as z⊥ (γ) = p⊥ (γ) ⊕ k⊥ (γ) , ⊥

(3.3.12)

⊥

where p (γ) ⊂ p, k (γ) ⊂ k are the orthogonal spaces to p (γ) , k (γ) with respect to the scalar products induced by B. Moreover, g = z (γ) ⊕ z⊥ (γ) .

(3.3.13)

Theorem 3.3.1. The set X (γ) is preserved by θ. Moreover, X (γ) = X (ea ) ∩ X (k) .

(3.3.14)

The set X (γ) is a submanifold of X. In the geodesic coordinate system centered at p1, then X (γ) = p (γ) .

(3.3.15)

0

The action of Z (γ) on X (γ) is transitive. More precisely the map g ∈ Z 0 (γ) → pg ∈ X induces the identification of Z 0 (γ)-manifolds, X (γ) ' Z 0 (γ) /K ∩ Z 0 (γ) .

(3.3.16)

Also K∩Z 0 (γ) coincides with the connected component of the identity K 0 (γ) in K (γ). Similarly, the action of Z (γ) on X (γ) is transitive, and we have the identification of Z (γ)-manifolds, X (γ) ' Z (γ) /K (γ) .

(3.3.17)

The embedding K (γ) → Z (γ) induces the isomorphism of finite groups, K 0 (γ) \ K (γ) ' Z 0 (γ) \ Z (γ) .

(3.3.18)

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The map g ∈ Z a,⊥,0 (γ) → pg ∈ X surjects on a convex submanifold X (γ) of X (γ), and moreover, a,⊥

X a,⊥ (γ) = Z a,⊥,0 (γ) /K 0 (γ) .

(3.3.19)

If a 6= 0, we have the identification of Riemannian Z 0 (γ) manifolds, X (γ) ' X a,⊥ (γ) × R

(3.3.20)

so that the action of eta on X (γ) is just the translation by t |a| on R. In particular, γ acts on X (γ) by translation by |a|. Proof. By Theorem 3.1.2 and by (3.3.2), we know that p1 ∈ X (γ). More generally, the uniqueness of the decomposition of γ into the product of commuting hyperbolic and elliptic elements shows that g ∈ G represents x ∈ X (γ) if and only if there exist a0 ∈ p, k 0 ∈ K such that 0 C g −1 ea = ea , C g −1 k = k 0 , Ad (k 0 ) a0 = a0 . (3.3.21) By (3.3.21), X (γ) ⊂ X (ea ) ∩ X (k). If x ∈ X (ea ) ∩ X (k), x is represented by g ∈ G such that the first two conditions in (3.3.21) are verified. Since 0 k ∈ Z (ea ), then k 0 ∈ Z(ea ). By Proposition 3.2.8, k 0 ∈ Z(a0 ), so that the last condition in (3.3.21) is also verified, i.e., x ∈ X (γ). This completes the proof of (3.3.14). Also X (k) ⊂ X is the fixed point set of k. In the geodesic coordinate system centered at p1, X (k) is represented by p (k) ⊂ p. By Theorem 3.2.6, in the same coordinate system, X (ea ) is represented by z (a) ∩ p. Therefore X (γ) is represented by p (γ), which is just (3.3.15). Then X (γ) is preserved by θ. It follows that the map g ∈ Z (γ) → pg ∈ X (γ) is surjective. Since X (γ) is connected, p induces a surjection Z 0 (γ) → X (γ). Then Z 0 (γ) acts transitively on X (γ), and the stabilizer of p1 ∈ X (γ) in Z 0 (γ) is just K ∩ Z 0 (γ), so that (3.3.16) holds. Since Z 0 (γ) is connected and X (γ) is contractible, K ∩ Z 0 (γ) is also connected. Therefore K ∩ Z 0 (γ) is the connected component of the identity in K (γ). By the above, the action of Z (γ) on X (γ) is transitive, so that (3.3.17) holds. By (3.3.16), (3.3.17), we get (3.3.18). Since K is compact, the group in (3.3.18) is finite. Equation (3.3.20) follows from (3.3.11). The fact that the action of eta on X (γ) is just translation by t |a| also follows from (3.3.11). Since X (γ) ⊂ X (k), k acts trivially on X (γ), so that the actions of γ on X (γ) coincides with the action of ea . The proof of our theorem is completed. Remark 3.3.2. Identity (3.3.20) is proved in [BaGSc85, Lemma 6.5] in a more general context. The identity in [BaGSc85] can be rewritten in the form Z 0 (γ) ×K 0 (γ) p⊥ (γ) = Z a,⊥,0 (γ) ×K 0 (γ) p⊥ (γ) × R. (3.3.22) The action of γ on the right-hand side of (3.3.22) is given by (g, f, s) → g, Ad k −1 f, s + |a| .

(3.3.23)

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3.4 The normal coordinate system on X based at X (γ) a

On X (ea ), the vector field aT X restricts to the vector field aT X(e ) , and the restriction of aT X to X (γ) is equal to aT X(γ) . By (2.17.10) and by Theorem a 3.2.6, the vector field aT X(e ) is of constant norm |a|, its integral curves are geodesics, and it is a parallel section of T X (ea ). A similar result holds for aT X(γ) . Let NX(γ)/X be the orthogonal bundle to T X (γ) in T X. Under the identification X (γ) ' Z 0 (γ) /K 0 (γ), one verifies easily that NX(γ)/X = Z 0 (γ) ×K 0 (γ) p⊥ (γ) .

(3.4.1)

Let NX(γ)/X be the total space of NX(γ)/X . Using the geodesics normal to X (γ), we obtain an identification of a neighborhood of X (γ) in NX(γ)/X to a tubular neighborhood of X (γ) in X. As we shall see, this diffeomorphism identifies NX(γ)/X to X. Let pγ : X → X (γ) be the projection defined in [BaGSc85, p. 8], i.e., if x ∈ X, pγ x ∈ X (γ) is the unique element in X (γ) that minimizes the distance d (x, y) , y ∈ X (γ). Let |∇dγ | be the norm of ∇dγ with respect to the metric of T X. Theorem 3.4.1. The map ργ : (g, f ) ∈ Z 0 (γ) ×K 0 (γ) p⊥ (γ) → pgef = gpef ∈ X

(3.4.2) is a diffeomorphism of Z 0 (γ)-spaces. The map (g, f ) → ea g, Ad k −1 f corresponds to the action of γ on X, and the map Z 0 (γ) ×K 0 (γ) p⊥ (γ) → Z 0 (γ) /K 0 (γ) corresponds to pγ : X → X (γ). If (g, f ) ∈ Z 0 (γ) ×K 0 (γ) p⊥ (γ), then dγ (ργ (g, f )) = dγ (ργ (1, f )) .

(3.4.3)

There exists Cγ > 0 such that if f ∈ p⊥ (γ) , |f | ≥ 1, then dγ (ργ (1, f )) ≥ |a| + Cγ |f | .

(3.4.4)

There exist Cγ0 > 0, Cγ00 > 0 such that if f ∈ p⊥ (γ) , |f | ≥ 1, |∇dγ (ργ (1, f ))| ≥ Cγ0 ,

(3.4.5)

2 ∇dγ (ργ (1, f )) /2 ≥ Cγ00 |f | .

(3.4.6)

and for |f | ≤ 1,

In particular the function d2γ /2 is a Morse-Bott function, whose critical set is X (γ), and its Hessian on X (γ) is given by the symmetric positive endomorphism of p⊥ (γ), ∇T X ∇d2γ /2|X(γ) =

ad (a) 2ch (ad (a)) − Ad (k) + Ad k −1 . sinh (ad (a)) (3.4.7)

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Proof. If g, f are taken as in our theorem, t ∈ R → pgetf ∈ X is a geodesic normal to X (γ). Let Jt be the Jacobi field corresponding to the variation of such a geodesic. Since X (γ) is totally geodesic in X, we know that J0 ∈ D J0 ∈ NX(γ)/X , so that T X (γ) , Dt D d 2 |J| = 2 J0 , J0 = 0. (3.4.8) dt t=0 Dt Since X has nonpositive curvature, by [BaGSc85, Remark p.5], |Jt | is a convex function of t. We deduce from (3.4.8) that |Jt | ≥ |J0 |. If J0 6= 0, for any t > 0, Jt 6= 0. If J0 = 0, if for a given t > 0, Jt = 0, by convexity, Js = 0 D J0 = 0, and J· vanishes identically. We have thus for any s ∈ [0, t], so that Dt shown that the map ργ is nonsingular. The unique geodesic that connects pγ x to x is normal to X (γ), so that ργ is surjective. Therefore ργ is a covering map. Since X is contractible, ργ is a diffeomorphism. If g ∈ Z (γ), g preserves X (γ) and maps geodesics normal to X (γ) into geodesics of the same type. One verifies trivially that the obvious left actions of Z (γ) correspond via ργ . Clearly, if (g, f ) ∈ Z 0 (γ) ×K 0 (γ) p⊥ (γ), using (3.3.3), we get −1 γργ (g, f ) = pγgef = pea k −1 gef = pea geAd(k )f .

(3.4.9)

By (3.4.9), we find that the action of γ on X corresponds to the diffeomorphism (g, f ) → ea g, Ad k −1 f of Z 0 (γ) ×K 0 (γ) p⊥ (γ). Since dγ is Z (γ)-invariant, we get (3.4.3). We use the same notation as in the proof of Theorem 3.2.6. For t ∈ R, set ϕf (t) = dγ petf .

(3.4.10)

Since the function dγ is convex, the function t ∈ R → ϕf (t) ∈ R is convex. Assume first that γ is elliptic, so that a = 0, and γ = k −1 . In this case, by Toponogov’s theorem [BaGSc85, section 1.4], we get −1 d pef , peAd(k )f ≥ Ad k −1 − 1 f . (3.4.11) Since no eigenvalue of Ad k −1 is equal to 1 on p⊥ (γ), by (3.4.11), there is Cγ > 0 such that for any f ∈ p⊥ (γ), dγ pef ≥ Cγ |f | , (3.4.12) from which (3.4.4) follows. By (3.4.12), since the function ϕf is convex, for f ∈ p⊥ (γ) , |f | = 1, ϕ0f (0)+ ≥ Cγ .

(3.4.13)

Using (3.4.13) and the convexity of ϕf (t), for any t > 0, we get ϕ0f (t) ≥ Cγ .

(3.4.14)

By (3.4.12) and (3.4.14), we get (3.4.5) and (3.4.6). Moreover, by (3.4.13), the function d2γ is a Morse-Bott function. Also from the above, we get easily ∇T X ∇d2γ /2|X(γ) = 1 − Ad k −1 (1 − Ad (k)) , (3.4.15)

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which is just (3.4.7) when a = 0. Let us now assume that γ is nonelliptic, i.e., a 6= 0. Take f ∈ p⊥ (γ) , |f | = 1. The minimum of ϕf (t) is at t = 0, and ϕf (t) ≥ ϕf (0) = |a|. Since a 6= 0, ϕf is smooth. We claim that there is cγ > 0 such that for any f taken as before, ϕ00f (0) ≥ cγ .

(3.4.16)

Indeed let ct (s) , 0 ≤ s ≤ 1 be the unique geodesic in X connecting petf and γpetf . Let Ef (t) be the energy of ct , so that 1 (3.4.17) Ef (t) = ϕ2f (t) . 2 By (3.4.17), we get Ef00 (0) = |a| ϕ00f (0) . Let Jf,s , 0 ≤ s ≤ 1 be the Jacobi field Jf,s = ization given by parallel transport, J¨f − ad2 (a) Jf = 0, Jf,0 = f, Jf,1 = Ad k

−1

(3.4.18) ∂ ∂t ct

(s) |t=0 . In the trivial(3.4.19)

f.

As in (3.2.21), we get Ef00 (0) =

1

Z 0

˙ 2 2 Jf + |[a, Jf ]| ds.

(3.4.20)

If Ef00 (0) = 0, by (3.4.19), (3.4.20), we would get [a, f ] = 0, Ad k −1 f = f , which contradicts the fact that f ∈ p⊥ (γ) , f 6= 0. Therefore there is Cγ > 0 such that Ef00 (0) ≥ Cγ .

(3.4.21)

By (3.4.18) and (3.4.21), we get (3.4.16). Since ϕ0f (0) = 0, using (3.4.16) and Taylor’s formula, there exist c0γ > 0, γ ∈]0, 1/2] such that for f taken as before, ϕ0f (γ ) ≥ c0γ .

(3.4.22)

Since ϕf is convex and ϕf ≥ |a|, by (3.4.22), for t ∈ R, ϕf (t) ≥ |a| + c0γ (t − γ ) .

(3.4.23)

Since γ ∈]0, 1/2], for |f | ≥ 1, 1 |f | . (3.4.24) 2 By (3.4.23), (3.4.24), we deduce that if f ∈ p⊥ (γ) , |f | ≥ 1, then |f | − γ ≥

c0γ dγ pef ≥ |a| + |f | , (3.4.25) 2 which is just (3.4.4). If f ∈ p⊥ (γ) , |f | = 1, since ϕf is convex, by (3.4.22), for t ≥ γ , ϕ0f (t) ≥ c0γ ,

(3.4.26)

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which gives (3.4.5). Moreover, ϕ0f

Z (t) = 0

t

ϕ00f (s) ds.

(3.4.27)

Since ϕ00f (t) ≥ 0, and ϕ00f (0) > 0, there exists Cγ > 0 such that for 0 ≤ t ≤ 1, ϕ0f (t) ≥ Cγ t.

(3.4.28)

This gives (3.4.6). Also by (3.4.16), the function dγ is a Morse-Bott function with critical set X (γ), and so is the function d2γ /2. Let us now compute its Hessian on X (γ). By (3.4.19), (3.4.20), we get D E D E ∇∇d2γ /2|X(γ) (f, f ) = Jf,1 , J˙f,1 − Jf,0 , J˙f,0 . (3.4.29) Moreover, the unique solution of equation (3.4.19) is given by sinh (sad (a)) Ad k −1 − cosh (a) f. Jf,s = cosh (sad (a)) f + sinh (ad (a))

(3.4.30)

By (3.4.30), we get ad (a) Ad k −1 − cosh (ad (a)) f, sinh (ad (a)) ad (a) = ad (a) sinh (ad (a)) f + cosh (ad (a)) sinh (ad (a)) Ad k −1 − cosh (ad (a)) f.

J˙f,0 = J˙f,1

(3.4.31)

By (3.4.29), (3.4.31), we get (3.4.7). The proof of our theorem is completed. Remark 3.4.2. Take x, x0 ∈ X, x0 6= x, and let ux , vx0 be the unit tangent vectors along the geodesic connecting x to x0 . Take γ as in Theorem 3.4.1, and x ∈ X such that x0 = γx 6= x. Then ∇dγ (x) = (γ ∗ v − u) (x) .

(3.4.32)

The inequalities in (3.4.5) and (3.4.6) can be reformulated using (3.4.32). By [BaGSc85, p. 10], the distance function to X (γ), denoted d (·, X (γ)), is a convex function. Tautologically, d (ργ (1, f ) , X (γ)) = |f | .

(3.4.33)

The inequalities in (3.4.4)–(3.4.6) can be rewritten using (3.4.33). Let dx be the volume element on X, let dy be the volume element on X (γ), let df be the volume element on p⊥ (γ). Then dydf is a volume element on Z 0 (γ) ×K 0 (γ) p⊥ (γ) that is Z 0 (γ)-invariant. There is a smooth positive function r (f ) on p⊥ (γ) that is K 0 (γ)-invariant, such that we have the identity of volume elements on X, dx = r (f ) dydf,

(3.4.34)

r (0) = 1.

(3.4.35)

and moreover,

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Using the equation of Jacobi fields, one finds easily that there exist C > 0, C 0 > 0 such that for any f ∈ p⊥ (γ), r (f ) ≤ C exp (C 0 |f |) .

(3.4.36)

The group K 0 (γ) acts on the left on K and on p⊥ (γ), and so it also acts on the left on p⊥ (γ) × K. Over K 0 (γ) \ K, one obtains the vector bundle p⊥ (γ)K 0 (γ) × K. Similarly K acts on the right on K, and this action extends to an action on the right on p⊥ (γ) × K. Theorem 3.4.3. The map σγ : (g, f, k 0 ) ∈ Z 0 (γ) ×K 0 (γ) p⊥ (γ) × K → gef k 0 ∈ G

(3.4.37)

is a diffeomorphism of left Z 0 (γ)-spaces, and of right K-spaces. The map (g, f, k 0 ) → (g, f ) corresponds to the projection p : G → X = G/K. Moreover, under this diffeomorphism, we have the identity of right K-spaces, p⊥ (γ)K 0 (γ) × K = Z 0 (γ) \ G.

(3.4.38)

Proof. The first two statements in our theorem follow from Theorem 3.4.1. The remainder of our theorem is now obvious. Remark 3.4.4. Equation (3.4.38) just says that Z 0 (γ)\G fibres over K 0 (γ)\ K with fibre p⊥ (γ). Also note that in Theorem 3.4.3, we may as well replace Z 0 (γ) , K 0 (γ) by Z (γ) , K (γ). Let dk be the Haar measure on K that gives volume 1 to K. Let dg be the measure on G, dg = dxdk.

(3.4.39)

Then dg is a left-invariant Haar measure on G. Since G is unimodular, it is also a right-invariant Haar measure. By (3.4.34) and (3.4.39), we get dg = r (f ) dydf dk.

(3.4.40)

Let dk 00 be the Haar measure on K 0 (γ) that gives volume 1 to K 0 (γ), and let du0 be a K-invariant volume form on K 0 (γ) \ K, so that dk = dk 00 du0 .

(3.4.41)

dg = r (f ) dydk 00 df du0 .

(3.4.42)

dz 0 = dydk 00 .

(3.4.43)

By (3.4.40), (3.4.41), we get

Set 0

0

Then dz is a left and right Haar measure on Z (γ), so that (3.4.42) can be written in the form dg = r (f ) dz 0 df du0 .

(3.4.44)

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Also by (3.4.38), r (f ) df du0 can be viewed as a measure on Z 0 (γ) \ G such that dg = dz 0 r (f ) df du0 .

(3.4.45)

Let dv 0 be the measure on Z 0 (γ) \ G, dv 0 = r (f ) df du0 .

(3.4.46)

0

0

By (3.4.45), we see that dv is exactly the canonical measure on Z (γ) \ G that is canonically associated with dg and dz 0 , so that dg = dz 0 dv 0 . 0

(3.4.47) 0

Another interpretation for the identity dv = r (f ) df du can be given via equation (3.4.38), which simply expresses the obvious factorization of dv 0 . When replacing Z 0 (γ) , K 0 (γ) by Z (γ) , K (γ), one can define measures dk 0 , du, dz, dv on K (γ) , K (γ) \ K, Z (γ) , Z (γ) \ G such that the analogue of the above identities still holds.

3.5 The return map along the minimizing geodesics in X (γ) Let π : X → X be the total space of the tangent bundle T X to X, let X ∗ be the total space of the cotangent bundle T ∗ X. The manifold X ∗ is a symplectic manifold, whose symplectic form is denoted ω. The identification of the fibres T X and T ∗ X by the metric g T X identifies the manifolds X and X ∗. Let V be the vector field on X ∗ that generates the geodesic flow. The vector field V is the Hamiltonian vector field associated with the Hamiltonian 2 H (x, p) = 21 |p| . Let ϕt |t∈R be the corresponding group of diffeomorphisms of X . When identifying X and X ∗ , we may consider ϕtas a flow of symplectic diffeomorphisms of X . If x, Y T X ∈ X , if xt , YtT X = ϕt x, Y T X , then t ∈ R → xt ∈ X is the unique geodesic in X such that x0 = x, x˙ 0 = Y T X . Let γ ∈ G be a semisimple element such that (3.3.2) holds. The action of γ on X lifts to X and X ∗ . Since γ is an isometry, these actions correspond by the above identification. Also γ preserves the symplectic form of X or X ∗ . It follows from the above that γ −1 ϕ1 is a symplectic diffeomorphism of X. Recall that by (3.3.16), X (γ) ' Z 0 (γ) /K 0 (γ). Let ia : X → X be the embedding x ∈ X → x, aT X ∈ X . Set Fγ = z ∈ X , γ −1 ϕ1 (z) = z . (3.5.1) Proposition 3.5.1. The following identity of submanifolds of X holds: Fγ = ia X (γ) .

(3.5.2)

Proof. By the considerations we made in the proof of Theorem 3.1.2, it is clear that ia X (γ) ⊂ Fγ . Conversely if (x, y) ∈ Fγ , then x ∈ X is a critical point for the displacement function dγ . As we saw above, from the results

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of [BaGSc85], we find that x minimizes dγ , i.e., x ∈ X (γ). The geodesic xt = πϕt (x, y) is the minimizing geodesic connecting x and γx, so that x˙ = aT X . This completes the proof of our proposition. If x ∈ X (γ), then x, aT X ∈ Fγ , and the differential d γ −1 ϕ1 is an automorphism of T(x,aT X ) X . Recall that z (a) = ker ad (a) is the Lie algebra of Z (a). From now on, we will use the notation z (a) = z0 .

(3.5.3)

Put p0 = ker ad (a) ∩ p,

k0 = ker ad (a) ∩ k.

(3.5.4)

Then z0 = p0 ⊕ k0 . Let

⊥ ⊥ z⊥ 0 , p0 , k0

(3.5.5)

be the orthogonal spaces to z0 , p0 , k0 in g, p, k, so that ⊥ ⊥ z⊥ 0 = p0 ⊕ k0 .

(3.5.6)

Moreover, p⊥ 0 = ad (a) k,

k⊥ 0 = ad (a) p.

(3.5.7)

The above vector spaces are preserved by ad (a) , Ad (k) , Ad (γ). By (3.3.6), z (γ) is a Lie subalgebra of z0 . We will view p (γ) as a vector subspace of p0 ⊕ {0} ⊂ p0 ⊕ k0 = z0 . Using the Levi-Civita connection ∇T X on T X, we have the splitting, T X ' π ∗ (T X ⊕ T X) .

(3.5.8)

In (3.5.8), the first copy of T X is identified with its horizontal lift, and the second copy is the tangent bundle along the fibre. ⊥ Observe that ad (a) is an odd invertible endomorphism of p⊥ 0 ⊕ k0 . Let ρ ⊥ ⊥ ⊥ ⊥ be the isomorphism from p0 ⊕ k0 → p0 ⊕ p0 given by ρ (e, f ) = (e, −ad (a) f ) .

(3.5.9)

Our computations will not depend on the choice of x ∈ X (γ). It will be convenient to take x = p1. Theorem 3.5.2. The following identities hold at x, aT X ∈ Fγ : Ad (k) |p0 Ad (k) |p0 dγ −1 ϕ1 |p0 ⊕p0 = , (3.5.10) 0 Ad (k) |p0 −1 dγ −1 ϕ1 |p⊥ |z⊥ ρ−1 . ⊥ = ρAd γ 0 ⊕p0 0 The eigenspace of dγ −1 ϕ1 associated with the eigenvalue 1 is just T Fγ ' p (γ) ⊕ {0} ⊂ p0 ⊕ p0 .

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Proof. Let t ∈ [0, 1] → xt ∈ X be the geodesic connecting x and γx. Consider the Jacobi field Jt ∈ Txt X, so that J¨ + RxT X (J, x) ˙ x˙ = 0. (3.5.11) t

˙ J¨ are taken with respect to the Levi-Civita In (3.5.11), the differentials J, connection along x· . With respect to the splitting T X , the (3.5.8) of differential dϕ1 at (x0 , x˙ 0 ) is given by the linear map J0 , J˙0 → J1 , J˙1 . Recall that RT X was calculated at the beginning of section 3.1. In the parallel transport trivialization with respect to the Levi-Civita connection, x˙ is the constant a ∈ p, and as in (3.4.19), equation (3.5.11) can be written in the form J¨ − ad2 (a) J = 0. (3.5.12) In (3.5.12), Jt ∈ p. As we saw before, dϕ1 J0 , J˙0 = J1 , J˙1 .

(3.5.13)

By (3.3.2) and (3.5.13), we deduce that dγ −1 ϕ1 J0 , J˙0 = Ad (k) J1 , J˙1 .

(3.5.14)

By (3.5.12), (3.5.14), we get the in (3.5.10). Observe first identity that ρ in ⊥ ⊥ ⊥ (3.5.9) commutes with Ad (k). If J0 , J˙0 ∈ p0 ⊕p0 , then Jt , J˙t ∈ p⊥ 0 ⊕p0 . ⊥ Let H ∈ p⊥ 0 ⊕ k0 be given by

H = ρ−1 J, J˙ .

(3.5.15)

H˙ + [a, H] = 0,

(3.5.16)

H1 = Ad e−a H0 .

(3.5.17)

By (3.5.12), we obtain so that By (3.5.14), (3.5.15), (3.5.17), we get the second identity in (3.5.10). By (3.5.10), the kernel of dγ −1 ϕ1 − 1 in p0 ⊕ p0 is just the kernel of Ad (k) − 1 in the first copy of p0 ⊕ p0 , which, by (3.3.6), coincides with p (γ) ⊕ {0}. Equivalently, ker dγ −1 ϕ1 − 1 is just T Fγ . The proof of our theorem is completed. Remark 3.5.3. We know that dγ −1 ϕ1 is a symplectic transformation of p0 ⊕ p0 . The conditions in (3.3.2) on γ show directly that the right-hand side of (3.5.10) defines a symplectic transformation of p0 ⊕ p0 .

3.6 The return map on Xb Recall that Xb is the total space of T X ⊕ N . Let τ : Xb → X be the obvious projection. We lift the flow ϕt |t∈R to a flow of diffeomorphisms of Xb. The

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lifted flow acts on N by parallel transport with respect to the connection ∇N . If x, Y T X , Y N ∈ Xb, then ϕt x, Y T X , Y N = xt , Y Tt X , Y N t , x· being the geodesic in X with speed Y Tt X , and Y N · being the parallel transport of N b Y along x· . Similarly γ acts on X . As in (3.5.1), set n o Fbγ = z ∈ Xb, γ −1 ϕ1 (z) = z . (3.6.1) Then τ Fbγ = Fγ .

(3.6.2) −1 Recall that by (3.3.14), X (γ) ⊂ X k . In particular, k acts as a parallel isometry on the vector bundle N |X(γ) . If x = p1, the action of k −1 on N |X(γ) is induced by the action of Ad k −1 on k. Set N k −1 = Y N ∈ N |X(γ) , Ad k −1 Y N = Y N . (3.6.3) −1 Then N k is associated with the eigenspace of k corresponding to the −1 −1 eigenvalue 1 of Ad k , i.e., with the Lie algebra k k of the centralizer of k −1 in K. Clearly N k−1 is a vector bundle on X (γ). Let N k −1 be the total space of N k −1 . Let bia be the embedding x, Y N ∈ N k −1 → x, aT X , Y N ∈ Xb. Now we establish the obvious analogue of Proposition 3.5.1. −1

Proposition 3.6.1. The following identity of submanifolds of Xb holds: Fbγ = bia N k −1 . (3.6.4) Proof. Using Proposition 3.5.1 and (3.6.2), we get (3.6.4).

3.7 The connection form in the parallel transport trivialization Put x0 = p1. Let a ∈ p → x = pea be the geodesic coordinate system centered at x0 . This coordinate system induces a trivialization of the Kbundle p : G → X = G/K that is given by (a, k) ∈ p × K → ea k ∈ G. Equivalently, this is the trivialization of G using the parallel transport with respect to the canonical connection on the K-bundle p : G → X = G/K along geodesics centered at x0 . Let Γ be the k-valued connection form in the above trivialization. Let J· be a Jacobi field along the geodesic xs = esa x0 such that J0 = 0, J˙0 = U ∈ p. By proceeding as in (3.4.19), (3.4.30), we get Js =

sinh (sad (a)) U. ad (a)

(3.7.1)

Let | | denote the norm in Tx0 X, and k k the norm in Tx X. By (3.7.1), we get sinh (ad (a)) kU kTx X = U . (3.7.2) ad (a)

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Recall that Ω denotes the curvature of the canonical connection. One finds easily that Z 1 Γa (U ) = − Ω (J, x) ˙ ds. (3.7.3) 0

By (2.1.10), we can rewrite (3.7.3) in the form Z 1 [J, a] ds. Γa =

(3.7.4)

0

By (3.7.1), (3.7.4), we get cosh (ad (a)) − 1 U. (3.7.5) ad (a) As it should be, the expression after ad in the right-hand side of (3.7.5) is in k. Put Γa (U ) = −

V = J1 .

(3.7.6)

By (3.7.1), we get U=

ad (a) V. sinh (ad (a))

(3.7.7)

By (3.7.5), (3.7.7), we obtain cosh (ad (a)) − 1 V. (3.7.8) sinh (ad (a)) The critical fact on Γa (U ) is that as a linear function of V , it is uniformly bounded. The above trivialization of G induces a corresponding trivialization of T X ⊕ N with respect to the connection ∇T X⊕N along geodesics centered at x0 . Let ΓT X⊕N be the associated connection form on T X ⊕ N ' p ⊕ k. By (3.7.5), we get cosh (ad (a)) − 1 ΓTa X⊕N (U ) = −ad U . (3.7.9) ad (a) Remark 3.7.1. There is a more direct way to derive the above formulas. Indeed, consider the exponential map f ∈ g → exp (f ) ∈ G. Recall that the left-invariant forms ω g , ω p , ω k were defined in section 2.1. One verifies easily that if f ∈ g, 1 − exp (−ad (f )) (exp∗ ω g )f = . (3.7.10) ad (f ) From (3.7.10), if a ∈ p, we get sinh (ad (a)) cosh (ad (a)) − 1 (exp∗ ω p )a |p = |p , exp∗ ω k a |p = − |p , ad (a) ad (a) (3.7.11) cosh (ad (a)) − 1 sinh (ad (a)) (exp∗ ω p )a |k = − |k , exp∗ ω k a |k = |k . ad (a) ad (a) Then equations (3.7.1), (3.7.5) follow from the first line in (3.7.11). Γa (U ) = −

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3.8 Distances and pseudodistances on X and Xb If x, x0 ∈ X, let s ∈ [0, 1] → xs ∈ X be the geodesic connecting x and x0 . 0 Let τxx be the parallel transport from Tx0 X into Tx X with respect to the Levi-Civita connection along this geodesic. If (x, f ) , (x0 , f 0 ) ∈ X , set 0 (3.8.1) δ ((x, f ) , (x0 , f 0 )) = d (x, x0 ) + τxx f 0 − f . Note that δ is symmetric, and only vanishes on the diagonal of X . Also it is invariant under the left action of G on X . However, δ is not a distance. Moreover, δ ((x, f ) , (x0 , f 0 )) ≤ d (x, x0 ) + |f | + |f 0 | . (3.8.2) We will say that δ is a pseudodistance on X . Take x0 ∈ X. If (x, f ) , (x0 , f 0 ) ∈ X , set 0 dx0 ((x, f ) , (x0 , f 0 )) = d (x, x0 ) + τxx0 f 0 − τxx0 f . (3.8.3) Then dx0 is a distance on X . Moreover, δ ((x, f ) , (x0 , f 0 )) = dx ((x, f ) , (x0 , f 0 )) = dx0 ((x, f ) , (x0 , f 0 )) . (3.8.4) In the sequel, we may and we will assume that x0 = p1. We use the notation of section 3.7. As mentioned after equation (3.7.8), ΓT X is a bounded section of T ∗ X ⊗ End (T X), when T ∗ X and End (T X) are equipped with their natural metrics. Since the connection ∇T X preserves the metric, we deduce easily that x x0 x0 (3.8.5) τx0 τx τx0 − 1 ≤ Cd (x, x0 ) . If (x, f ) , (x0 , f 0 ) ∈ X , 0 0 0 0 x τx0 f − τxx0 f 0 ≤ f − τxx f 0 + τxx0 τxx τxx00 − 1 τxx0 f 0 . By (3.8.5), (3.8.6), we get 0 0 x τx0 f − τxx0 f 0 ≤ f − τxx f 0 + Cd (x, x0 ) |f 0 | .

(3.8.6)

(3.8.7)

Similarly, 0 0 x 0 τx f − f ≤ τxx0 f − τxx0 f 0 + Cd (x, x0 ) |f 0 | .

(3.8.8)

By (3.8.1), (3.8.3), and (3.8.7), we get dx0 ((x, f ) , (x0 , f 0 )) ≤ δ ((x, f ) , (x0 , f 0 )) + Cd (x, x0 ) |f 0 | . (3.8.9) Similarly, using (3.8.1), (3.8.3), and (3.8.8), we obtain δ ((x, f ) , (x0 , f 0 )) ≤ dx0 ((x, f ) , (x0 , f 0 )) + Cd (x, x0 ) |f 0 | . (3.8.10) 0 b The above definitions extend to X . Indeed, if x, x ∈ X, we still denote 0 by τxx the parallel transport from (T X ⊕ N )x0 into (T X ⊕ N )x with respect to the connection ∇T X⊕N along the geodesic connecting x and x0 . If (x, f ) , (x0 , f 0 ) ∈ Xb, we still define the pseudodistance δ ((x, f ) , (x0 , f 0 )) as in (3.8.1). Similarly, if x0 ∈ X, we define the distance dx0 on Xb as in (3.8.3). All the above inequalities established over X remain valid on Xb. In particular equation (3.7.8) still implies inequality (3.8.5).

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3.9 The pseudodistance and Toponogov’s theorem We use the notation of section 3.5. By Proposition 3.5.1, the fixed pointset Fγ of the map γ −1 ϕ1 in X is just It follows that if x, Y T X ∈ ia X (γ). −1 TX TX X,x ∈ / X (γ), then γ ϕ1 x, Y 6= x, Y . For a ∈ R, let δa be the dilation δa x, Y T X = x, aY T X . (3.9.1) Since for t 6= 0, ϕt = δ1/t ϕ1 δt ,

(3.9.2)

we deduce from the above that if x ∈ / X (γ), for any t ∈ R, γ −1 ϕt x, Y T X 6= x, Y T X . (3.9.3) TX Take x, Y ∈ X,x ∈ / X (γ). In the sequel, we will assume that TX Y = 1. (3.9.4) Set x0 , Y T X0 = γ x, Y T X . From the above, it follows that for any t ∈ R, ϕt/2 x, Y T X 6= ϕ−t/2 x0 , Y T X0 .

(3.9.5)

(3.9.6)

Let s ∈ [0, 1] → xs ∈ X be the geodesic connecting x to x0 . Set Y0T X = x˙ 0 ,

Y1T X = x˙ 1 .

(3.9.7)

Then x0 , Y1T X = ϕ1 x, Y0T X .

(3.9.8)

dγ (x) = Y0T X = Y1T X .

(3.9.9)

Moreover,

By equation (3.4.32), ∇dγ (x) =

1 γ ∗ Y1T X − Y0T X . dγ (x)

(3.9.10)

Assume that x = ργ (1, f ) , f ∈ p⊥ (γ). By (3.4.4), (3.4.5), and (3.9.10), for |f | ≥ 1, ∗ TX γ Y1 − Y0T X ≥ Cγ0 dγ (x) ≥ Cγ0 (|a| + Cγ |f |) . (3.9.11) Moreover, by (3.4.6) and (3.9.10), for |f | ≤ 1, ∗ TX γ Y1 − Y0T X ≥ Cγ00 |f | .

(3.9.12)

Theorem 3.9.1. Given β > 0, there exists Cγ,β > 0 such that if x ∈ X is such that d (x, X (γ)) ≥ β, if Y T X ∈ Tx X, Y T X = 1, for t ≥ 0, then δ ϕt x, Y T X , ϕ−t γ x, Y T X ≥ Cγ,β . (3.9.13)

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Proof. We may as well assume that x = ργ (1, f ) , f ∈ p⊥ (γ). Let θ, 0 ≤ θ ≤ π be the angle between Y0T X and γ ∗ Y1T X . Because of (3.9.9), (3.9.11) and (3.9.12), this angle cannot be 0. By (3.9.9), (3.9.11), (3.9.12), we find that given β > 0, and |f | ≥ β, there is θγ,β , 0 < θγ,β ≤ π such that θγ,β ≤ θ ≤ π. We use the notation in (3.9.5). Set x0t = πϕ−t x0 , Y T X0 . xt = πϕt x, Y T X , Then

x· , x0·

(3.9.14)

(3.9.15)

are geodesics in X. Moreover, d (x, xt ) = d (x0 , x0t ) = t.

(3.9.16)

dt = d (xt , x0t ) .

(3.9.17)

Set

Since d is a convex function, the function t ∈ R → dt ∈ R+ is convex. Let s ∈ [0, 1] → xs,t be the geodesic connecting xt and x0t . Of course, when t = 0, this is just the geodesic s ∈ [0, 1] → xs . By (3.4.32), 1 ∗ TX d˙0 = − γ Y1 + Y0T X , Y T X . (3.9.18) dγ (x) Assume first that

γ ∗ Y1T X + Y0T X , Y T X ≤ 0.

(3.9.19)

Since t → dt is convex, by (3.9.18) and (3.9.19), for any t ≥ 0, dt ≥ dγ (x) .

(3.9.20)

Also by (3.4.4), there is aγ,β > 0 such that for x ∈ X, d (x, X (γ)) ≥ β, dγ (x) ≥ aγ,β .

(3.9.21)

dt ≥ aγ,β ,

(3.9.22)

Using (3.9.20), (3.9.21), we get

which is compatible with (3.9.13). In the remainder of the proof, we will assume that

∗ TX γ Y1 + Y0T X , Y T X > 0,

(3.9.23)

so that d˙0 < 0.

(3.9.24)

dt ≥ dγ (x) − 2t.

(3.9.25)

By (3.9.16), we get

By (3.9.21), (3.9.25), we find that if t ≤ aγ,β /4, then dt ≥ aγ,β /2.

(3.9.26)

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By (3.9.26), we deduce that to establish (3.9.13), we may as well take t > aγ,β /4. Note that if u, v are unit vectors in p, their angle α (u, v) , 0 ≤ α (u, v) ≤ π is just the distance of u, v in the unit sphere S p . It follows that if u, v, w are unit vectors, α (u, w) ≤ α (u, v) + α (v, w) .

(3.9.27)

Put θ0 = α Y0T X , Y T X ,

θ1 = α Y1T X , γ∗ Y T X .

(3.9.28)

Using (3.9.14), (3.9.23), and (3.9.27), we find that θγ,β ≤ θ ≤ θ0 + θ1 < π.

(3.9.29)

Given t > 0, consider the quadrangle Γt in Figure 3.1 made of the geodesics s ∈ [0, 1] → xs connecting x and x0 , u ∈ [0, t] → x0u connecting x0 and x0t , s ∈ [0, 1] → xs,t connecting xt and x0t , and u ∈ [0, t] → xu connecting x to xt . The two lower angles of this quadrangle are just θ0 , θ1 . We denote by σ0,t , σ1,t the two upper angles. Using the form of Toponogov’s theorem given in [BaGSc85, p7, Exercice (i)], we get θ0 + θ1 + σ0,t + σ1,t ≤ 2π.

(3.9.30)

Set θ0,t = π − σ0,t ,

θ1,t = π − σ1,t .

(3.9.31)

Then (3.9.30) can be written in the form θ0,t + θ1,t ≥ θ0 + θ1 .

(3.9.32)

The same arguments as before show that θ0,t + θ1,t is an increasing function of t. However, this fact will not be used in the sequel. Set z t = x1/2,t .

(3.9.33) x0t .

Equivalently z t is the middle point between xt and Consider the geodesic triangle Tt in Figure 3.1 with vertices x, x0 , z t . Let 0 u ∈ [0, t] → xtu , x0t u be the geodesics connecting x, x to z t . Let α0,t , α1,t , γt 0 be the angles at x, x , z t . By Toponogov’s theorem, α0,t + α1,t + γt ≤ π.

(3.9.34)

Set ∂xtu ∂x0t u |u=t , gt = |u=t . ∂u ∂u Then ft , gt lie in Tzt X, and their angle is γt . Therefore, ft =

2

2

2

|ft + gt | = |ft | + |gt | + 2 |ft | |gt | cos (γt ) .

(3.9.35)

(3.9.36)

By (3.9.34), (3.9.36), we obtain 2

2

2

|ft + gt | ≥ |ft | + |gt | − 2 |ft | |gt | cos (α0,t + α1,t ) .

(3.9.37)

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zt

xt

xt

γt xtu

xut

α0,t

α1,t

α00,t x

α1,t

xs Figure 3.1

x

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Chapter 3

In the sequel, we will fix the value of a parameter c ∈]0, 1]. If dt > inf(caγ,β /4, 1), 2 then (3.9.13) is verified. Therefore, we may as well assume that dt ≤ inf(caγ,β /4, 1). 2 Since t > aγ,β /4, from (3.9.39), we get

(3.9.38)

(3.9.39)

dt ≤ 2ct.

(3.9.40)

Clearly, dt dt ≤ t |ft | ≤ t + , 2 2 By (3.9.40), (3.9.41), we get t−

1 − c ≤ |ft | ≤ 1 + c,

t−

dt dt ≤ t |gt | ≤ t + . 2 2

1 − c ≤ |gt | ≤ 1 + c.

By (3.9.37), (3.9.42), we get 2 2 2 |ft + gt | ≥ 2 (1 − c) − (1 + c) cos (α0,t + α1,t ) . z

(3.9.41)

(3.9.42)

(3.9.43)

z

Let τxtt , τx0t be the parallel transport from Tzt X into Txt X, Tx0t X along t the geodesic x·,t with respect to the Levi-Civita connection. Using (3.9.16), (3.9.40), and equation (13.1.10) in Proposition 13.1.1, we get dt zt 0 tx˙ t − τxzt tft ≤ (1 + Ct) dt , − τ x ˙ tg (3.9.44) t ≤ (1 + Ct) . 0 t t xt t 2 2 In (3.9.44), the norms are calculated with respect to the metric in Txt X or Tx0t X. x0

Recall that τxtt is the parallel transport from Tx0t X into Txt X along the geodesic x·,t with respect to the Levi-Civita connection. By (3.9.43) and (3.9.44), we get 1/2 dt x0 2 2 − (1 + Ct) . x˙ t + τxtt x˙ 0t ≥ 2 (1 − c) − 2 (1 + c) cos (α0,t + α1,t ) t (3.9.45) ∂ t 0 0 Let α0,t , α1,t denote the angles at x, x0 made by x˙ 0 , ∂u xu |u=0 and by 0 ∂ 0t x˙ 0 , ∂u xu |u=0 . By Toponogov’s theorem, these angles are smaller than the corresponding angles in the Euclidean triangles with similar lengths. Because of (3.9.39), there is C 0 > 0 such that 0 α0,t ≤

C 0 dt , 2 t

0 α1,t ≤

C 0 dt . 2 t

(3.9.46)

Also by (3.9.27), 0 α0,t ≥ θ0 − α0,t ,

0 α1,t ≥ θ1 − α1,t .

(3.9.47)

By (3.9.46), (3.9.47), we get α0,t + α1,t ≥ θ0 + θ1 − C 0

dt . t

(3.9.48)

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By (3.9.29), (3.9.40), and (3.9.48), we see that for c ≤ θγ,β /4C 0 , α0,t + α1,t ≥

θγ,β . 2

(3.9.49)

By (3.9.45) and (3.9.49), we conclude that 1/2 dt x0 2 2 − (1 + Ct) . x˙ t + τxtt x˙ 0t ≥ 2 (1 − c) − 2 (1 + c) cos (θγ,β /2) t (3.9.50) Clearly there is cγ,β ∈]0, 1] such that for c ∈]0, cγ,β ], 1/2 2 2 2 (1 − c) − 2 (1 + c) cos (θγ,β /2) ≥ sin (θγ,β /4) . (3.9.51) Recall that we assumed that t > aγ,β /4. Then there is ηγ,β ∈]0, 1] such that if dt ≤ ηγ,β , (1 + Ct)

1 dt ≤ sin (θγ,β /4) . t 2

By (3.9.50)–(3.9.52), we see that if dt ≤ ηγ,β , 1 x0 x˙ t + τxtt x˙ 0t ≥ sin (θγ,β /4) , 2

(3.9.52)

(3.9.53)

which is compatible with (3.9.13). If dt > ηγ,β this is also the case. The proof of our theorem is completed. Theorem 3.9.2. Given β > 0, M > 0, there exists Cγ,β,M > 0 such that if x ∈ X is such that d (x, X (γ)) ≥ β, if Y T X ∈ Tx X, for 0 ≤ t ≤ M , δ ϕt x, Y T X , ϕ−t γ x, Y T X ≥ Cγ,β,M . (3.9.54) Proof. Clearly, ϕt (x, 0) = (x, 0) .

(3.9.55)

Therefore, by (3.9.21), (3.9.54) holds for Y T X = 0. In the sequel, we may as well assume that Y T X 6= 0. By (3.9.2), Y TX TX ϕt x, Y = δ|Y T X | ϕ|Y T X |t x, T X . (3.9.56) |Y | dγ (x) Assume first that Y T X ≤ 4M . For 0 ≤ t ≤ M , Y TX Y TX d πϕ|Y T X |t x, T X , πϕ−|Y T X |t γ x, T X ≥ dγ (x) /2. (3.9.57) |Y | |Y | By (3.9.21), (3.9.57) is compatible with (3.9.54). dγ (x) So we may as well assume that Y T X > 4M . By the same arguments as before, there exists dγ,β,M > 0 such that Y T X ≥ dγ,β,M . Then (3.9.54) follows from (3.9.13) in Theorem 3.9.1. The proof of our theorem is completed.

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Theorem 3.9.3. Given β > 0, µ > 0, there exists Cγ,β,µ > 0 such that if f ∈ p⊥ (γ) , |f | ≤ β, x = ργ (1, f ) , Y T X − aT X ≥ µ, δ ϕ1/2 x, Y T X , ϕ−1/2 γ x, Y T X ≥ Cγ,β,µ . (3.9.58) Proof. We will use the notation of the proof of Theorem By a compact- 3.9.1. Y TX ness argument, we find that given M > 0, if |f | ≤ β, Y T X ≤ M , if x, TX TX stays away from Fγ = ia X (γ), then δ ϕ1/2 x, Y , ϕ−1/2 γ x, Y has a positive lower bound. For |f | ≤ β, dγ (x) remains uniformly bounded. We claim that there exists Mγ,β > 0, Cγ,β > 0 such that for t ≥ Mγ,β , Y T X = 1, δ ϕt x, Y T X , ϕ−t γ x, Y T X ≥ Cγ,β . (3.9.59) We still denote by z t the middle point of the geodesic segment connecting xt to x0t . Consider the geodesic triangle Tt with vertices x, x0 , z t . By Toponogov’s theorem, the angles of this triangle are smaller than the angles of the Euclidean triangle whose edges have the same lengths. In particular as t → +∞, the angle at z t tends uniformly to 0. By proceeding as in the proof of Theorem 3.9.1, we get (3.9.59). Now we establish (3.9.58). By the first part of our proof, we may assume that Y T X ≥ 2Mγ,β . Then we use equation (3.9.56) with t = 1/2, and also (3.9.59), and we get (3.9.58). The proof of our theorem is completed. Theorem 3.9.4. Given ν > 0, there exists Cν > 0 such that if f ∈ p⊥ (γ) , |f | ≤ 1, x = ργ (1, f ) , Y T X ∈ Tx X, Y T X ≤ ν, then δ ϕ1/2 x, Y T X , ϕ−1/2 γ x, Y T X ≥ Cν |f | + Y T X − aT X . (3.9.60) Proof. Here x, Y T X varies in a compact set. If x, Y T X stays away from Fγ = ia X (γ), (3.9.60) is trivial. For f = 0, Y T X = aT X , both sides of (3.9.60) vanish. Moreover, by Theorem 3.5.2, the return map γ −1 ϕ1 is nondegenerate along Fγ . More precisely T Fγ is the eigenspace of dγ −1 ϕ1 |Fγ that is associated with the eigenvalue 1. It follows that for |f | and Y T X − aT X small enough, (3.9.60) still holds. The proof of our theorem is completed. 0

If x, x0 ∈ X, we still denote by τxx the parallel transport from Nx0 into Nx along the geodesic connecting x to x0 with respect to the connection ∇N . Now we use the notation of section 3.6. If (x, Y ) ∈ / bia N k −1 , this means that either x ∈ / X (γ), or that if x ∈ X (γ), either Y T X 6= aT X or Y N ∈ / −1 N k . Take (x, Y ) ∈ Xb. As in (3.9.5), set (x0 , Y 0 ) = γ (x, Y ) .

(3.9.61)

As in (3.9.15), set bϕt (x, Y ) , xt = π N0 YN · ,Y · N

x0t = π bϕ−t (x0 , Y 0 ) .

(3.9.62)

Let be the parallel transports of Y N , Y N 0 along x· , x0· with respect N0 to ∇ . Equivalently, Y N are the projections on N of the vertical com· ,Y · ponents of ϕ· (x, Y ) , ϕ−· (x0 , Y 0 ).

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75 TX − aT X ≤ β, Assume that x = ργ (1, f ). Given β > 0, if |f | ≤ β, Y TX then x, Y varies in a compact subset of X . Moreover, x1/2 , x01/2 remain at finite distance from each other, this distance tending to 0 as β → 0. We trivialize N along the geodesics that are normal to X (γ) with respect to the connection ∇N . In particular Ad k −1 now acts on the vector bundle N. The displacement function and the return map

⊥ Theorem 3.9.5. T X There exists cγ > 0 such that f ∈ p (γ) , |f | ≤ 1, x = T X ργ (1, f ), if Y − a ≤ 1, then 0 x1/2 N 0 Ad k −1 − 1 Y N − cγ |f | + Y T X − aT X Y N . τx Y 1/2 − Y N 1/2 ≥ 1/2

(3.9.63) Proof. If we make f = 0, Y T X = aT X , x0 = γx, then 0 τxx Y N 0 − Y N = Ad k −1 − 1 Y N .

(3.9.64)

Since f, Y T X vary in a compact set, using finite increments, we get (3.9.63). The proof of our theorem is completed.

3.10 The flat bundle (T X ⊕ N ) (γ) ⊥

Let (T X ⊕ N ) (γ) , (T X ⊕ N ) (γ) be the vector subbundles of T X ⊕ N associated with z (γ) , z⊥ (γ) via the identification T X ⊕ N = g. Then ⊥

T X ⊕ N = (T X ⊕ N ) (γ) ⊕ (T X ⊕ N ) (γ) ,

(3.10.1)

and the splitting (3.10.1) is orthogonal with respect to B. Moreover, the vector bundles in the right-hand side of (3.10.1) are flat with respect to ∇T X⊕N,f . Since the action of γ on T X ⊕ N just corresponds to the action of ⊥ Ad (γ) on g, it follows that γ preserves (T X ⊕ N ) (γ) and (T X ⊕ N ) (γ). Since X (γ) is of the same type as X, over X (γ), there is a vector bundle N (γ) on X (γ) such that T X (γ) ⊕ N (γ) ' z (γ). One verifies easily that the restriction of (T X ⊕ N ) (γ) to X (γ) coincides with T X (γ) ⊕ N (γ). Also we have the dual splitting, ⊥

T ∗ X ⊕ N ∗ = (T ∗ X ⊕ N ∗ ) (γ) ⊕ (T ∗ X ⊕ N ∗ ) (γ) .

(3.10.2)

Note that by (3.1.4), if g ∈ G, then g maps X (γ) into X (C (g) γ) isometrically, and it maps (T X ⊕ N ) (γ) into (T X ⊕ N ) (C (g) γ). In particular Z (γ) acts on (T X ⊕ N ) (γ). The flat vector bundle (T X ⊕ N ) (γ) will play an important role in chapter 9 in our construction of a Getzler rescaling.

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Chapter Four Elliptic and hypoelliptic orbital integrals The purpose of this chapter is to construct semisimple orbital integrals associated with the heat kernel for the hypoelliptic Laplacian LX b . By making b → 0, we show that the corresponding supertrace coincides with the orbital integral associated with the standard elliptic heat kernel. In the next chapters, the evaluation of these elliptic orbital integrals will be obtained by making b → +∞. This chapter is organized as follows. In section 4.1, we introduce an algebra Q of invariant kernels q (x, x0 ) acting on the vector space of bounded continuous sections of F . In section 4.2, if γ ∈ G is semisimple, the orbital integrals of such kernels are constructed, and they are shown to vanish on commutators. In section 4.3, when replacing F by Λ· (T ∗ X ⊕ N ∗ ) ⊗ S · (T X ⊕ N ) ⊗ F , we construct an associated algebra Q of continuous kernels acting on C b Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . We obtain the corresponding orbital integrals, and we show that they vanish on supercommutators. X In section 4.4, if A is a constant section of End (F ) and if LX A = L + A, we construct the orbital integrals associated with the heat kernel for , which is known to lie in Q. exp −tLX A X In section 4.5, if LX Laplacian, the orbital A,b = Lb + A is the hypoelliptic X integrals associated with the heat kernel for exp −LA,b are obtained. The proof that such heat kernels lie in the algebra Q is deferred to chapter 14. In section 4.6, we prove the fundamental fact that the above elliptic and hypoelliptic orbital integrals coincide. In section 4.7, we give a slightly different approach to the orbital integrals associated with a semisimple element γ = ea k −1 , by incorporating the action of a in the considered elliptic or hypoelliptic Laplacians. Finally, in section 4.8, if Γ ⊂ G is a cocompact subgroup of G, and Z = Γ \ X, Zb = Γ \ Xb, the elliptic and hypoelliptic traces and supertraces are shown to be equal. In the whole chapter, we make the same assumptions as in chapters 2 and 3, and we use the corresponding notation. Also if V = V+ ⊕ V− is a finite dimensional Z2 -graded vector space, if τ = ±1 is the involution of V that defines the Z2 -grading, if A ∈ End (V ), we define its supertrace Trs [A] by Trs [A] = Tr [τ A] . (4.0.1) The same notation will be used for trace class operators when V is infinite dimensional.

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77

4.1 An algebra of invariant kernels on X Let C b (X, F ) be the Banach space of continuous bounded sections of F over b X. Let CK (G, E) be the vector space of continuous bounded functions s on G with values in E such that if k ∈ K, s (gk) = ρE k −1 s (g) . (4.1.1) Of course we have the canonical isomorphism b C b (X, F ) = CK (G, E) . (4.1.2) b Let Q be an operator acting on C (X, F ) with continuous kernel q (x, x0 ) with respect to the volume dx0 . If x, x0 ∈ X, then q (x, x0 ) ∈ Hom (Fx0 , Fx ). It will often be more convenient to view q as a continuous function q (g, g 0 ) defined on G × G with values in End (E), which is such that if k ∈ K, q (gk, g 0 ) = ρE k −1 q (g, g 0 ) , q (g, g 0 k) = q (g, g 0 ) ρE (k) . (4.1.3) In the sequel, we will assume that Q commutes with the left action of G on C b (X, F ). This is just to say that if x, x0 ∈ X, g ∈ G, q (gx, gx0 ) = gq (x, x0 ) g −1 , (4.1.4) where in the right-hand side of (4.1.4), g −1 maps Fgx0 onto Fx0 , and g maps Fx into Fgx . Equivalently, if we consider instead the kernel q (g, g 0 ), if g 00 ∈ G, then q (g 00 g, g 00 g 0 ) = q (g, g 0 ) . (4.1.5) Set q (g) = q (1, g) . (4.1.6) By (4.1.3), (4.1.5), we get q (g, g 0 ) = q g −1 g 0 , q k −1 g = ρE k −1 q (g) , (4.1.7) q (gk) = q (g) ρE (k) . From (4.1.7), we obtain q k −1 gk = ρE k −1 q (g) ρE (k) .

(4.1.8)

E

By (4.1.8), we find that Tr [q (g)] is invariant when replacing g by k −1 gk, with k ∈ K. In the sequel, since there is no risk of confusion, we will use the same notation q for the various versions of the corresponding kernel Q. Also |q (g)| will denote the norm of q (g) ∈ End (E). Definition 4.1.1. Let Q be the vector space of continuous kernels q (g) taken as above such that there exist C > 0, C 0 > 0 for which |q (g)| ≤ C exp −C 0 d2 (p1, pg) . (4.1.9) 0 Equation (4.1.9) is equivalent to the fact that x, x ∈ X, |q (x, x0 )| ≤ C exp −C 0 d2 (x, x0 ) . (4.1.10) In (4.1.10), |q (x, x0 )| denotes the norm of q (x, x0 ) ∈ Hom (Fx0 , Fx ) with respect to the Hermitian product on F .

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Chapter 4

Let dY p be the volume form on p with respect to its scalar product. If Y → x ∈ X is the geodesic coordinate system considered in section 3.2, the volume form dx can be written as p

dx = δ (Y p ) dY p ,

(4.1.11)

where δ is a positive smooth function such that δ (0) = 1. As in (3.4.36), from the equation for Jacobi fields, we deduce that there exist c > 0, C > 0 such that δ (Y p ) ≤ c exp (C |Y p |) .

(4.1.12)

2 |q (Y p )| ≤ C exp −C 0 |Y p | .

(4.1.13)

By (4.1.10), if q ∈ Q,

By (4.1.11), (4.1.13), we get Z |q (p1, x)| dx < +∞.

(4.1.14)

X

From (4.1.14), we deduce that there exists C > 0 such that for x ∈ X, Z |q (x, x0 )| dx0 ≤ C. (4.1.15) X

If q ∈ Q, s ∈ C b (X, F ), set Z Qs (x) =

q (x, x0 ) s (x0 ) dx0 .

(4.1.16)

X

By (4.1.10) and (4.1.15), Q is a continuous operator from C b (X, F ) into itself. Proposition 4.1.2. The vector space Q is an algebra with respect to the composition of operators acting on C b (X, F ). Proof. If q, q 0 ∈ Q and if Q, Q0 are the corresponding operators, the kernel of the operator QQ0 should be given by Z q ∗ q 0 (x, x0 ) = q (x, y) q 0 (y, x0 ) dy. (4.1.17) X

Equations (4.1.10), (4.1.15) and dominated convergence guarantee that q ∗ q 0 (x, x0 ) is continuous and uniformly bounded. If y ∈ X, either d (x, y) ≥ d (x, x0 ) /2 or d (x0 , y) ≥ d (x, x0 ) /2. It is now obvious that q ∗ q 0 verifies (4.1.10). The proof of our proposition is completed.

4.2 Orbital integrals Let γ ∈ G be semisimple. Because the properties of γ that will be used later are conjugation independent, we may as well assume that γ is as in (3.3.2), so that γ = ea k −1 , k ∈ K,

a ∈ p, Ad (k) a = a.

(4.2.1)

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Let q ∈ Q. If x ∈ X, q (x, γx) ∈ Hom (Fγx , Fx ). By still denoting by γ the obvious element of Hom (Fx , Fγx ), we find that γq (x, γx) ∈ End (Fγx ), so that TrF [γq (x, γx)] is well-defined. Let h (y) be a compactly supported bounded measurable function on X (γ). Theorem 4.2.1. The function TrF [γq (x, γx)] h (pγ x) is integrable on X, and moreover, Z TrF [γq (x, γx)] h (pγ x) dx X Z Z = TrE q e−f γef r (f ) df h (y) dy. (4.2.2) p⊥ (γ)

X(γ)

Proof. We temporarily assume the function TrF [γq (x, γx)] h (pγ x) to be integrable on X. Using (3.4.2) and (3.4.34), we get Z TrF [γq (x, γx)] h (pγ x) dx X Z = TrF γq pyef , γpyef h (y) r (f ) df dy. (4.2.3) Z 0 (γ)×K 0 (γ) p⊥ (γ)

By (4.1.5), Z

TrF γq pyef , γpyef h (y) r (f ) df dy

Z 0 (γ)×K 0 (γ) p⊥ (γ)

Z =

E

Tr q e−f γef r (f ) df

Z

p⊥ (γ)

h (y) dy. (4.2.4) X(γ)

By (4.2.3), (4.2.4), we get (4.2.2). Moreover, d ef p1, γef p1 = dγ (ργ (1, f )) .

(4.2.5)

By (3.4.4) and (4.2.5), we see that if q has compact support, the integral in the variable f in (4.2.4) is only over a bounded set of f , so that it is indeed well-defined. More generally, if q is such that (4.1.9) or (4.1.10) hold, using (3.4.4), we get q ef p1, γef p1 ≤ C exp −C 0 |f |2 . (4.2.6) By (3.4.36) and (4.2.6), the integral in the variable f in (4.2.4) is still finite. The proof of our theorem is completed. We will reformulate the above results in a different way. Indeed using (3.4.39), (4.1.5)–(4.1.8) and the fact that Vol (K) = 1, we get Z Z TrF [γq (x, γx)] h (pγ x) dx = TrE q g −1 γg h (pγ pg) dg. (4.2.7) X

G

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By (3.4.37) and (3.4.47), we get Z TrE q g −1 γg h (pγ pg) dg G Z Z = TrE q v −1 γv dv Z 0 (γ)\G

h pz 0 dz 0 .

(4.2.8)

Z 0 (γ)

Using (3.4.46) and the fact that Vol K 0 (γ) \ K = 1, we get Z Z TrE q v −1 γv dv = TrE q e−f γef r (f ) df.

(4.2.9)

p⊥ (γ)

Z 0 (γ)\G

By (3.4.43) and using the fact that Vol K 0 (γ) = 1, we have the identity Z Z 0 0 h pz dz = h (y) dy. (4.2.10) Z 0 (γ)

X(γ)

Therefore (4.2.7)–(4.2.10) is equivalent to (4.2.2). We will denote by [γ] the conjugacy class of γ in G. Definition 4.2.2. If γ ∈ G is semisimple, put Z Z Tr[γ] [Q] = TrE q v −1 γv dv =

TrE q e−f γef r (f ) df.

p⊥ (γ)

Z 0 (γ)\G

(4.2.11) By proceeding as in (4.2.3), (4.2.4), we get Z [γ] Tr [Q] = TrF γq γ −1 ef p1, ef p1 r (f ) df p⊥ (γ) Z = TrF γq ef p1, γef p1 r (f ) df.

(4.2.12)

p⊥ (γ)

Integrals like (4.2.11), (4.2.12) are called orbital integrals. If h ∈ G, the map g ∈ G → C (h) g ∈ G induces a map Z (γ) \ G → Z (C (h) γ)\G, which maps the volume dv of Z 0 (γ)\G into the corresponding volume on Z 0 (C (h) γ) \ G. Also dv is invariant under the right action of G on Z 0 (γ) \ G. Therefore, as the notation indicates, Tr[γ] [Q] only depends on the conjugacy class [γ] of γ in G. By Proposition 4.1.2, Q is an algebra. If q, q 0 ∈ Q, the function q ∗ q 0 (g) associated with QQ0 is given by Z Z 0 0 −1 q ∗ q (g) = q (h) q h g dh = q gh−1 q 0 (h) dh. (4.2.13) G

G 0

In the sequel, [Q, Q ] denotes the commutator of Q and Q0 . Theorem 4.2.3. For any semisimple element γ ∈ Γ, Tr[γ] [[Q, Q0 ]] = 0.

(4.2.14)

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Proof. If γ = 1, then Tr[γ] [Q] = TrE [q (1)] . In this case, (4.2.14) follows from (4.2.13) and (4.2.15). Let δγ be the current on G so that Z Z f δγ = f v −1 γv dv.

(4.2.15)

(4.2.16)

Z 0 (γ)\G

G

Since dv is invariant under the right action of G on Z 0 (γ) \ G, δγ is a current on G that is invariant by conjugation. By (4.2.11), Z Tr[γ] [Q] = (4.2.17) TrE [q] δγ = TrE q ∗ δγ −1 (1) . G

Let Rγ −1 be the operator Z Rγ −1 f (g) =

f gv −1 γ −1 v dv.

(4.2.18)

Z 0 (γ)\G

Then δγ −1 can be thought of as the kernel associated with the operator Rγ −1 . We can rewrite (4.2.17) in the form Tr[γ] [Q] = Tr[1] QRγ −1 . (4.2.19) Since δγ −1 is invariant under conjugation, then q ∗ δγ −1 = δγ −1 ∗ q,

(4.2.20)

QRγ −1 = Rγ −1 Q.

(4.2.21)

which just says that Then using (4.2.14) with γ = 1, (4.2.19) and (4.2.21), we get Tr[γ] [[Q, Q0 ]] = Tr[1] [Q, Q0 ] Rγ −1 = Tr[1] Q, Q0 Rγ −1 = 0,

(4.2.22)

which is precisely (4.2.14). The proof of our theorem is completed.

4.3 Infinite dimensional orbital integrals In the sequel, we will use the above formalism, when replacing the finite dimensional vector space E by the infinite dimensional vector space E = Λ· (p∗ ⊕ k∗ )⊗S · (p∗ ⊕ k∗ )⊗E. Equivalently, we will replace the vector bundle F on X by the infinite dimensional vector bundle F = Λ· (T ∗ X ⊕ N ∗ ) ⊗ S · (T ∗ X ⊕ N ∗ ) ⊗ F. Most of the time, S · (p∗ ⊕ k∗ ) and S · (T ∗ X ⊕ N ∗ ) will be replaced by their completion S · (p∗ ⊕ k∗ ) = L2 (p ⊕ k) and S · (T ∗ X ⊕ N ∗ ) = L2 (T X ⊕ N ). To avoid introducing extra notation, we still denote by E, F the completions of the above vector spaces. Of course the fact that E, F are infinite dimensional creates new analytic difficulties, but the algebraic formalism that we described above remains

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valid. A minor change with respect to what we did before is that traces will be replaced by supertraces. b Let C Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) denote the vector space of continuous bounded sections of π b∗ (Λ· (T ∗ X ⊕ N ∗ )) ⊗ F over Xb. b · ∗ Let CK (G × g, Λ (p ⊕ k∗ ) ⊗ E) denote the vector space of continuous bounded sections of Λ· (p∗ ⊕ k∗ ) ⊗ E on G × g such that if k ∈ K, · ∗ ∗ (4.3.1) s gk, Ad k −1 Y = ρΛ (p ⊕k )⊗E k −1 s (g, Y ) . Instead of (4.1.2), we now have b C b Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) = CK (G × g, Λ· (p∗ ⊕ k∗ ) ⊗ E) . (4.3.2) Recall that dY p is the volume element on p. Let dY k denote the volume element of k with respect to its scalar product. Let dY be the volume element of g = p ⊕ k, dY = dY p dY k .

(4.3.3)

The above volume elements define volume elements dY T X , dY N , dY on the fibres of T X, N, T X ⊕ N . Recall that K acts unitarily on the vector space E. Our kernel q (g) now acts as an endomorphism of E, and verifies (4.1.7). In what follows, the operator q (g) is given by continuous kernels q (g, Y, Y 0 ) , Y, Y 0 ∈ g, with q (g, Y, Y 0 ) ∈ End (Λ· (g∗ ) ⊗ E). Let q ((x, Y ) , (x0 , Y 0 )), (x, Y ) , (x0 , Y 0 ) ∈ Xb be the corresponding kernel on Xb. Definition 4.3.1. Let Q denote the vector space of continuous kernels q (g, Y, Y 0 ) taken as above such that there exist C > 0, C 0 > 0 for which 2 2 |q (g, Y, Y 0 )| ≤ C exp −C 0 d2 (p1, pg) + |Y | + |Y 0 | . (4.3.4) Equation (4.3.4) is equivalent to 2 2 |q ((x, Y ) , (x0 , Y 0 ))| ≤ C exp −C 0 d2 (x, x0 ) + |Y | + |Y 0 | .

(4.3.5)

Of course the norm in the right-hand side of (4.3.5) is calculated with respect to the Hermitian product on Λ· (T ∗ X ⊕ N ∗ ) ⊗ F . b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) , set If q ∈ Q, s ∈ C b Xb, π Z Qs (x, Y ) = q ((x, Y ) , (x0 , Y 0 )) s (x0 , Y 0 ) dx0 dY 0 . (4.3.6) b X

By proceeding as in section 4.1, by (4.3.5), (4.3.6), the operator Q acts on ∗ · ∗ ∗ b b (Λ (T X ⊕ N ) ⊗ F ) . C X,π b

Proposition 4.3.2. The vector spaceQ is an algebra with respect to the b ∗ · ∗ ∗ b composition of operators acting on C X , π b (Λ (T X ⊕ N ) ⊗ F ) .

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Proof. If q, q 0 ∈ Q, if Q, Q0 are the corresponding operators, the kernel of the operator QQ0 should be given by Z q ∗ q 0 ((x, Y ) , (x0 , Y 0 )) = q ((x, Y ) , (z, Z)) q 0 ((z, Z) , (x0 , Y 0 )) dzdZ. b X

(4.3.7) By proceeding as in the proof of Proposition 4.1.2, one finds easily that q ∗ q 0 ∈ Q, which completes the proof of our proposition. If q ∈ Q, by (4.3.4), if g ∈ G, the operator q (g) ∈ End (L2 (p ⊕ k) ⊗ E) is trace class. Moreover, the supertrace Trs E [q (g)] is given by Z · ∗ ∗ E Trs [q (g)] = Trs Λ (p ⊕k )⊗E [q (g, Y, Y )] dY. (4.3.8) g

As before, γ ∈ G is a semisimple element of G, such that (4.2.1) holds. Definition 4.3.3. We define Trs [γ] [Q] as in (4.2.11), i.e., Z · ∗ ∗ Trs [γ] [Q] = Trs Λ (p ⊕k )⊗E q v −1 γv, Y, Y dvdY (Z 0 (γ)\G)×g Z · ∗ ∗ = Trs Λ (p ⊕k )⊗E q e−f γef , Y, Y r (f ) df dY. (4.3.9) p⊥ (γ)×g

By Theorem 3.4.1 and by (4.3.5), the integral in (4.3.9) is well defined. As in (4.2.12), we can rewrite (4.3.9) in the form Trs [γ] [Q] Z Z = p⊥ (γ)

Z

·

Trs Λ (T

X⊕N ∗ )⊗F

γq γ −1 ef p1, Y , ef p1, Y dY

T X⊕N

Z

= p⊥ (γ)

∗

r (f ) df · ∗ ∗ Trs Λ (T X⊕N )⊗F γq

ef p1, Y , γ ef p1, Y dY

T X⊕N

r (f ) df.

(4.3.10)

Expressions such as (4.3.9), (4.3.10) will be called orbital supertraces. We now use the notation [ ] for the supercommutator. Let Q, Q0 ∈ Q be two operators taken as before. Theorem 4.3.4. For any semisimple element γ ∈ G, Trs [γ] [[Q, Q0 ]] = 0.

(4.3.11)

Proof. Recall that if V = V+ ⊕ V− is a finite dimensional vector space, the supertrace vanishes on supercommutators. The proof of our theorem is then an easy modification of the proof of Theorem 4.2.3.

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4.4 The orbital integrals for the elliptic heat kernel of X Let A be a self-adjoint element of End (E) that is K-invariant. Then A descends to a self-adjoint parallel section of End (F ). Recall that the operator LX acting on C ∞ (X, F ) was defined in (2.13.3). ∞ Definition 4.4.1. Let LX (X, F ), A be the operator acting on C X LX A = L + A.

(4.4.1)

∞,c

Let C (X, F ) be the vector space of smooth sections of F over X that have compact support. Since X is complete, LX A is an essentially self-adjoint operator on C ∞,c (X, F ). Let C ∞,b (X, F ) be the vector space of smooth sections of F , which are bounded together with their of arbitrary order. For t > 0, the derivatives acts on C ∞,b (X, F ). This operator commutes heat operator exp −tLX A 0 with the left action of G on C ∞,b (X, F ). Let pX t (x, x ) be the smooth kernel X for the operator exp −tLA with respect to the volume element dx0 on X. Proposition 4.4.2. For any t > 0, pX t ∈ Q. 0 Proof. The fact that pX t (x, x ) verifies (4.1.10) is well-known, and can be established using finite propagation speed for the wave equation [CP81, section 7.8], [T81, section 4.4].

Let γ ∈ G be a semisimple element. It follows from the results of section 4.2 and from Proposition 4.4.2 that for t > 0, the orbital integral is well-defined. Tr[γ] exp −tLX A 4.5 The orbital supertraces for the hypoelliptic heat kernel Recall that the operator LX was defined in equation (2.13.4), and acts on b C ∞ Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . Let LX A,b be the operator acting on the same vector space, X LX A,b = Lb + A. ∂ ∂t

(4.5.1)

LX A,b

By Theorem 2.13.2, the operator + is a hypoelliptic operator, and LX is formally self-adjoint with respect to η. A,b 2 By (2.13.5), with the exception of the quartic term 21 Y N , Y T X , the operator LX A,b has exactly the same structure as the hypoelliptic Laplacian constructed in [B05], whose analytic study was done in detail in BismutLebeau [BL08]. Moreover, the above quartic term is nonnegative, and it commutes with C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

∇Y T X

.

We will see in section 11.8 that for t > 0, the heat operator exp −tLX A,b is X well-defined. Also, there is an associated smooth kernel qb,t ((x, Y ) , (x0 , Y 0 )).

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Definition 4.5.1. Let P be the projection from Λ· (T ∗ X ⊕ N ∗ ) ⊗ F on Λ0 (T ∗ X ⊕ N ∗ ) ⊗ F = F . For t > 0, (x, Y ) , (x0 , Y 0 ) ∈ Xb, put 1 2 X 0 −(m+n)/2 0 2 |Y | + |Y q0,t ((x, Y ) , (x0 , Y 0 )) = PpX (x, x ) π exp − | P. t 2 (4.5.2) Theorem 4.5.2. Given > 0, M > 0, ≤ M , there exist C > 0, C 0 > 0 such that for 0 0, t > 0, qb,t ∈ Q. Moreover, as b → 0, X X qb,t ((x, Y ) , (x0 , Y 0 )) → q0,t ((x, Y ) , (x0 , Y 0 )) .

(4.5.4)

Proof. The proof of our theorem will be given in chapter 14.

4.6 A fundamental equality Let γ ∈ G be semisimple as in h(4.2.1). By the iresults of section 4.3 and by [γ] X Theorem 4.5.2, for t > 0, Trs exp −tLA,b is well-defined. Now we establish the main result of this chapter, which should be viewed as an analogue of [B08b, Theorem 4.1], where a corresponding result was established in the case where G is a compact Lie group. Theorem 4.6.1. For any b > 0, t > 0, the following identity holds: . (4.6.1) Trs [γ] exp −tLX = Tr[γ] exp −tLX A A,b Proof. A first step in the proof will be to show that ∂ Trs [γ] exp −tLX = 0. A,b ∂b Indeed using Theorem 4.3.4, we find easily that ∂ ∂ X [γ] X Trs [γ] exp −tLX = −tTr L exp −tL . s A,b A,b ∂b ∂b A,b

(4.6.2)

(4.6.3)

By (2.13.4) and (4.5.1), we get 1 ∂ X ∂ X LA,b = DX , D . b ∂b 2 ∂b b

(4.6.4)

X X Db , Lb = 0.

(4.6.5)

X Db , A = 0.

(4.6.6)

By Proposition 2.15.1,

Also, we have the trivial

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By (4.5.1), (4.6.5), and (4.6.6), we obtain X X Db , LA,b = 0.

(4.6.7)

By Theorem 4.3.4 and by (4.6.3), (4.6.4) and (4.6.7), we get t ∂ [γ] X ∂ X X Trs [γ] exp −tLX = − Tr D , D exp −tL = 0, s A,b b A,b ∂b 2 ∂b b (4.6.8) which is just (4.6.2). We claim that as b → 0, Trs [γ] exp −LX → Tr[γ] exp −LX . (4.6.9) A,b A Indeed using (3.4.4) and (4.5.3), we find that given t > 0, there exist C > 0, C 0 > 0 such that for 0 < b ≤ 1, f ∈ p⊥ (γ) , Y ∈ (T X ⊕ N )ef p1 , X qb,t ef p1, Y , γ ef p1, Y ≤ C exp −C 0 |f |2 + |Y |2 . (4.6.10) By (3.4.36), (4.2.12), (4.3.10), (4.5.4), and (4.6.10), using dominated convergence, we get (4.6.9). Our theorem now follows from (4.6.2) and (4.6.9).

4.7 Another approach to the orbital integrals We still take γ as in (4.2.1). We have the identity of operators acting on C ∞ (G), γ = e−∇a,r k −1 .

(4.7.1)

As explained in section 2.17, equation (4.7.1) descends to an identity of operators acting on C ∞ (X, F ), γ = e−La k −1 .

(4.7.2)

0 0 Take t > 0. Recall that pX t (x, x ) , x, x ∈ X is the smooth kernel associX X −1 0 ated with exp −tL . Then γp γ x, x is the smooth kernel associated t A . By (4.7.2), we get with γ exp −tLX A −1 γ exp −tLX exp −tLX (4.7.3) A =k A − La .

We will denote by pX (x, x0 ) , x, x0 ∈ X the smooth kernel associated with a,t X exp −tLA − La . By (4.7.3), we get 0 γpX γ −1 x, x0 = k −1 pX (4.7.4) t a,t (kx, x ) . Equations (4.7.3), (4.7.4) can be read off on G instead of on X. We already 0 X know that pX t (x, x ) lifts to a function pt (g) on G that verifies (4.1.6) X 0 0 and (4.1.7). Moreover, pa,t (x, x ) lifts to a smooth function pX a,t (g, g ) on G × G that verifies (4.1.3). By (3.3.4), exp −tLX A − La commutes with Z (γ). Therefore (4.1.5) holds only when g ∈ Z (γ). In particular the function

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0 pX a,t (kg, g) is well-defined on Z (γ) \ G. It follows from (4.7.4) that we have the equality of smooth functions on Z 0 (γ) \ G with values in End (E), pX g −1 γg = pX (4.7.5) t a,t (kg, g) .

By (4.2.11) and (4.7.5), we get Z Tr[γ] exp −tLX = TrE pX v −1 γv dv A t Z 0 (γ)\G Z = TrE pX a,t (kv, v) dv. (4.7.6) Z 0 (γ)\G

Of course a similar equation holds tautologically when replacing the operX X ator LX . If qa,b,t ((x, Y ) , (x0 , Y 0 )) denotes the smooth A by the operator LA,b kernel associated with exp −tLX A,b − La , then we have the obvious analogue of (4.7.4), i.e., X X γqb,t γ −1 (x, Y ) , (x, Y ) = k −1 qa,b,t (k (x, Y ) , (x, Y )) . (4.7.7) From (4.7.7), the orbital integral Trs [γ] exp −tLX , which is given by A,b X the integral (4.3.9), with q replaced by qb,t , can be reexpressed using the X kernel qa,b,t as in equation (4.7.6).

4.8 The locally symmetric space Z Let Γ ⊂ G be a cocompact subgroup of G. Then the elements of Γ are semisimple. Also Γ acts isometrically on X, and this action lifts to all the considered Euclidean or Hermitian vector bundles, and preserves the corresponding connections. Set Z = Γ\G/K.

(4.8.1)

Z = Γ\X.

(4.8.2)

Equivalently,

Then Z is a compact locally symmetric space. If Γ is torsion free, or equivalently if Γ does not contain elliptic elements, then Z is a smooth manifold. Otherwise, in Γ there are only a finite number of elliptic conjugacy classes in Γ, and Z is an orbifold. If Γ is torsion free, since X is contractible, Γ = π1 (Z) ,

(4.8.3)

and X is the universal cover of Z. A vector bundle like F descends to a vector bundle on Z, which we still denote by F . Except for the tangent bundle T X of X, which descends to the orbifold tangent bundle T Z of Z, we will use the same notation for the other vector bundles over X when descended to Z.

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Chapter 4

By definition, C (Z, F ) is the part of C b (X, F ) that is fixed under the left action of Γ, i.e., Γ

(4.8.4)

Γ

(4.8.5)

C (Z, F ) = C b (X, F ) . By (4.1.2) and (4.8.4), we obtain b C (Z, F ) = CK (G, E) .

We use the notation of section 4.1. Let Q ∈ Q. Since the operator Q commutes with the left action of G, Q descends to an operator QZ acting on C (Z, F ). Let q Z (z, z 0 ) , z, z 0 ∈ Z be the continuous kernel of QZ over Z. If z, z 0 are identified with their lifts in X, then X X q Z (z, z 0 ) = γq γ −1 z, z 0 = q (z, γz 0 ) γ. (4.8.6) γ∈Γ

γ∈Γ

If we use the representation (4.8.5) of C (Z, F ), the kernel q Z (z, z 0 ) can be identified with the function over G × G, X q Z (g, g 0 ) = q g −1 γg 0 . (4.8.7) γ∈Γ

Let C be the set of conjugacy classes in Γ. If [γ] ∈ C, set X q X,[γ] (g, g 0 ) = q g −1 γg 0 .

(4.8.8)

γ∈[γ]

Then q Z (z, z 0 ) =

X

q X,[γ] (g, g 0 ) .

(4.8.9)

The trace Tr QZ of QZ is given by Z Tr QZ = Tr q Z (z, z) dz.

(4.8.10)

[γ]∈C

Z

Set h i Z h i Tr QZ,[γ] = Tr q X,[γ] (z, z) dz.

(4.8.11)

Z

Then h i X Tr QZ = Tr QZ,[γ] .

(4.8.12)

[γ]∈C

On the other hand, one finds easily that h i Z Tr QZ,[γ] = Tr q g −1 γg dg.

(4.8.13)

Γ∩Z(γ)\G

We use the notation of subsection 3.4. By (4.8.13), we get Z h i Tr QZ,[γ] = Vol (Γ ∩ Z (γ) \ Z (γ)) Tr q g −1 γg dv. Z(γ)\G

(4.8.14)

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Using Theorems 3.4.1, 3.4.3, and 4.2.1, if Γ ∩ Z (γ) is torsion free, so that Γ ∩ Z (γ) acts freely on X (γ), we obtain Z i h Z,[γ] = Vol (Γ ∩ Z (γ) \ X (γ)) Tr Q Tr q e−f γef r (f ) df. p⊥ (γ)

(4.8.15) By (4.2.11), we can rewrite (4.8.15) in the form h i Tr QZ,[γ] = Vol (Γ ∩ Z (γ) \ X (γ)) Tr[γ] [Q] .

(4.8.16)

Since K (γ) acts freely on Γ ∩ Z (γ) \ Z (γ), we get Vol (Γ ∩ Z (γ) \ Z (γ)) = Vol (K (γ)) Vol (Γ ∩ Z (γ) \ X (γ)) . Also by (3.4.46) and its analogue for Z (γ) , K (γ), we obtain Z Vol (K (γ)) [γ] Tr q g −1 γg dv. Tr [Q] = Vol (K) Z(γ)\G

(4.8.17)

(4.8.18)

Since Vol (K) = 1, from (4.8.16)–(4.8.18), we recover (4.8.14). If Γ ∩ Z (γ) is not torsion free, let δ (γ) be the subgroup of the elements in K (γ) that act on the right like the identity on Γ ∩ Z (γ) \ Z (γ). Then δ (γ) is a subgroup of the finite group Γ ∩ K (γ). Instead of (4.8.17), we get Vol (Γ ∩ Z (γ) \ Z (γ)) =

Vol (K (γ)) Vol (Γ ∩ Z (γ) \ X (γ)) . |δ (γ)|

Instead of (4.8.16), we obtain h i Vol (Γ ∩ Z (γ) \ X (γ)) Tr QZ,[γ] = Tr[γ] [Q] . |δ (γ)|

(4.8.19)

(4.8.20)

If Γ is not torsion free, then K acts locally freely on Γ\G, and Z = Γ\G/K is an orbifold. Let E ⊂ C be the finite set of elliptic conjugacy classes in Γ. If e ∈ E, let [e] be the set of conjugacy classes in E that are conjugate to e in G. Let k ∈ K represent the classes in [e]. By (4.8.14), we get Z h i X X Z,[γ] Tr Q = Vol (Γ ∩ Z (γ) \ Z (γ)) Tr q g −1 kg dv. [γ]∈[e]

Z(k)\G

[γ]∈[e]

(4.8.21) Now we give a geometric interpretation of the right-hand side of (4.8.21). Let (Γ \ G)k ⊂ Γ \ G be the fixed point set of the right action of k. For each [γ] ∈ [e], take one γ ∈ [γ], and let gγ ∈ G be such that gγ kgγ−1 = γ. Then one verifies easily that [ (Γ \ G)k = (Z (γ) ∩ Γ \ Z (γ)) gγ , (4.8.22) [γ]∈[e]

and the components in (4.8.22) do not intersect. Moreover, K (k) acts on the right on (Γ \ G)k . By (4.8.22), we can rewrite (4.8.21) in the form Z h i X Tr QZ,[γ] = Vol ((Γ \ G)k ) Tr q g −1 kg dv. (4.8.23) [γ]∈[e]

Z(k)\G

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Chapter 4

Let ∆ (k) be the subgroup of K (k) of the elements that act like the identity on (Γ \ G)k . Then ∆ (k) is a subgroup of the finite group Γ ∩ K (k). Instead of (4.8.17), we get Vol (K (k)) Vol ((Γ \ G)k /K (k)) . (4.8.24) Vol ((Γ \ G)k ) = |∆ (k)| Still using (3.4.46), we obtain Z Vol (K (k)) Tr q g −1 kg dv. Tr[k] [Q] = (4.8.25) Vol (K) Z(k)\G By (4.8.23)–(4.8.25), we get i Vol ((Γ \ G) /K (k)) h X k Tr[k] [Q] . (4.8.26) Tr QZ,[γ] = |∆ (k)| [γ]∈[e]

Equation (4.8.26) is of special interest in connection with the Kawasaki formula [Ka79], an analogue for orbifolds of the Atiyah-Singer index theorem, that will be briefly considered in section 7.7. The vector bundle T X ⊕ N descends to T Z ⊕ N . The total space Xb of the vector bundle T X ⊕ N over X descends to the total space Zb of the vector bundle T Z ⊕ N over Z. We still denote by π b the projection Zb → Z. The same arguments as before apply to the kernels Q ∈ Q. Z LX The operator LX A descends to an operator LA , and the operator A,b de Z X scends to an operator LA,b . For t > 0, the operator exp −tLA is obviously trace class. Similarly, by proceeding as in [BL08, chapter 3], we find that the Z operator exp −tLA,b is also trace class. Since the base Z is compact, the results of [BL08] can be imported without any change. Theorem 4.8.1. For any t > 0, b > 0, Trs exp −tLZ = Tr exp −tLZ . A,b A

(4.8.27)

Proof. We give two proofs. A first method is to use Theorem 4.6.1, and (4.8.12), (4.8.16). Another method is first to establish the analogue of (4.6.2), ∂ Trs exp −tLZ = 0. (4.8.28) A,b ∂b This can be done by proceeding as in (4.6.4)–(4.6.8), while invoking at the last stage the fact that standard supertraces vanish on supercommutators. The next step is to show that as b → 0, Trs exp −tLZ → Tr exp −tLZ . (4.8.29) A,b A To prove (4.8.29), we can use Theorem 4.5.2. Since the base Z is compact, we can also use the results of [BL08, section 3.4], and obtain (4.8.29). This is because by (2.16.1), (2.16.2), the structure of the operator LX b is essentially the same as the structure of the operator considered in [BL08]. The extra 2 quartic term 21 Y N , Y T X can be dealt with by the methods of sections 11.7 and 11.8. Finally, by equation (2.16.4) in Theorem 2.16.1, we identify the limit in (4.8.29). The proof of our theorem is completed.

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Remark 4.8.2. Theorem 4.8.1 will not be used in the sequel, i.e., we will ignore the spectral side of the trace formula, to concentrate on the orbital integrals.

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Chapter Five Evaluation of supertraces for a model operator In this chapter, given a semisimple element γ ∈ G, we evaluate the supertrace of the heat kernel of a hypoelliptic operator acting over p×g. In section 9.10, this operator will appear in the asymptotics as b → +∞ of a rescaled version of the operator LX b . In particular we obtain a function Jγ Y0k , Y0k ∈ k (γ), which will play a fundamental role in our final formula for the orbital integrals. This chapter is organized as follows. In section 5.1, if Y0k ∈ k (γ), we introduce the hypoelliptic operator Pa,Y0k , its heat kernel, and a corresponding supertrace Jγ Y0k . In section 5.2, by a conjugation of Pa,Y0k , we obtain a simpler operator Qa,Y0k , where p × p and k have been decoupled. The operator Qa,Y0k splits naturally into a scalar part and a matrix part. In section 5.3, we evaluate the trace of the heat kernel of the scalar part of Qa,Y0k . In section 5.4, we compute the supertrace of the matrix part of the heat kernel. Finally, in section 5.5, we give an explicit formula for Jγ Y0k .

5.1 The operator Pa,Y0k and the function Jγ Y0k

Let γ be a semisimple element of G, which is written in the form given in (4.2.1). ∗

Definition 5.1.1. Let z (γ) denote another copy of z (γ), and let z (γ) ∗ denote the corresponding copy of the dual of z (γ). If α ∈ Λ· z (γ) , let α ∗ be the corresponding element in Λ· z (γ) . Also c (z (γ)) denotes the Clifford algebra of z (γ) , B|z(γ) . We will continue using the notation of chapter 2. In particular, E is taken b · z∗ (γ) ⊗E b . as in section 2.12. Our operators act on C ∞ p × g, Λ· (g∗ ) ⊗Λ Of course, p × g = p × (p ⊕ k) .

(5.1.1)

We denote by y the tautological section of the first copy of p in the right-hand side of (5.1.1), and by Y g = Y p + Y k the tautological section of g = p ⊕ k. Let dy be the volume form on p, let dY g be the volume form on g = p ⊕ k.

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Evaluation of supertraces for a model operator

Set p = dim p (γ) ,

q = dim k (γ) ,

r = dim z (γ) ,

(5.1.2)

so that r = p + q.

(5.1.3)

Let e1 , . . . , ep be an orthonormal basis of p (γ), let ep+1 , . . . , er be an orthonormal basis of k (γ). We denote with upper scripts the corresponding ∗ dual bases. Then e1 , . . . , er is a basis of z (γ) . Put r X α= (5.1.4) c (ei ) ei . i=1 p⊕k

Recall that ∆ is the standard Laplacian on p ⊕ k. Let ∆p , ∆k be the Laplacians on p, k ⊂ g = p ⊕ k. These operators act along p ⊕ k, i.e., on the second factor in the right-hand side of (5.1.1). Let ∇H denote differentiation in the variable y ∈ p, and let ∇V denote differentiation in the variable Y g ∈ g. Definition 5.1.2. If Y0k ∈ k (γ), set Pa,Y0k =

1 k k p 2 1 p⊕k Y , a + Y0 , Y − ∆ + α − ∇H Yp 2 2 − ∇V[a+Y k ,[a,y]] − b c (ad (a)) + c ad (a) + iθad Y0k . (5.1.5) 0

By H¨ ormander [H¨ o67], the operator

∂ ∂t

+ Pa,Y0k is hypoelliptic.

Proposition 5.1.3. The following identity holds: 1 k k p 2 H V Y , a + Y0 , Y , ∇Y p + ∇[a+Y k ,[a,y]] = 0. 0 2

(5.1.6)

Proof. The term ∇H Y p does not contribute to the commutator. As to the second term, its contribution will be

k k p Y , a + Y0 , Y , a, [y, a] , Y0k + [a, [a, y]] , Y0k . (5.1.7) k Using Jacobi’s identity and the fact that a, Y0 = 0, we get a, [y, a] , Y0k + [a, [a, y]] , Y0k = [y, a] , a, Y0k = 0. (5.1.8) By (5.1.7), (5.1.8), we get (5.1.6). Let Op be the set of scalar differential operators over p × g. Clearly, ∗ b b · z∗ (γ) . Pa,Y0k ∈ Op ⊗ End Λ· z⊥ (γ) ⊗c (z (γ)) ⊗Λ (5.1.9) Let RY0k ((y, Y g ) , (y 0 , Y g0 )) be the smooth kernel of exp −Pa,Y0k with respect to the volume dydY g on p × g. From (5.1.9), we deduce that ∗ b (z (γ)) ⊗Λ b · z∗ (γ) . RY0k ((y, Y g ) , (y 0 , Y g0 )) ∈ End Λ· z⊥ (γ) ⊗c (5.1.10)

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Chapter 5

∗ cs be the If A ∈ End Λ· z⊥ (γ) , let Trs [A] be its supertrace. Let Tr · ∗ b functional defined on c (z (γ)) ⊗Λ z (γ) that vanishes on monomials of r nonmaximal length, and gives the value (−1) to c (e1 ) e1 . . . c (er ) er . We cs that maps combine these two functionals, and we get a functional Tr ∗ b b · z∗ (γ) End Λ· z⊥ (γ) ⊗c (z (γ)) ⊗Λ

into R. ∗ Observe that Ad k −1 induces an automorphism of Λ· z⊥ (γ) . Set Jγ

Y0k

= (2π)

r/2

Z p⊥ (γ)×(p⊕k⊥ (γ))

" c Trs Ad k −1

g

RY0k (y, Y ) , Ad k

−1

g

(y, Y )

#

dydY g . (5.1.11)

The main purpose of the present chapter is to compute Jγ Y0k .

5.2 A conjugate operator Definition 5.2.1. Put

Qa,Y0k = exp [y, a] , Y0k , Y k Pa,Y0k exp − [y, a] , Y0k , Y k ,

[y, a] , Y0k , Y k + h[a, [a, y]] , Y p i Pa,Y0k (5.2.1) Qa,Y k = exp 0

k k p exp − [y, a] , Y0 , Y − h[a, [a, y]] , Y i . Proposition 5.2.2. The following identities hold: 2 2 2 1 −∆p⊕k + a, Y k + Y0k , Y p + [y, a] , Y0k Qa,Y0k = 2 − ∇H − ∇V[a,[a,y]] + α − b c (ad (a)) + c ad (a) + iθad Y0k , (5.2.2) Yp 2 2 2 1 2 −∆p⊕k + a, Y k + Y0k , Y p + [y, a] , Y0k + |[a, Y p ]| Qa,Y k = 0 2 1 2 + |[a, [a, y]]| − ∇H c (ad (a)) + c ad (a) + iθad Y0k . Yp +α −b 2 Proof. Let e1 , . . . , em be an orthonormal basis of p and let em+1 , . . . , em+n be an orthonormal basis of k. Then m+n

2 1 1 X − ∆k − ∇V[[y,a],Y k ] = − ∇Vei + [y, a] , Y0k , ei 0 2 2 i=m+1

+

2 1 [y, a] , Y0k . (5.2.3) 2

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By (5.2.3), we get

k 1 p⊕k V k H exp [y, a] , Y0 , Y − ∆ − ∇Y p − ∇[[y,a],Y k ] 0 2

k k exp − [y, a] , Y0 , Y 2 1 1 = − ∆p⊕k − ∇H [y, a] , Y0k + [Y p , a] , Y0k , Y k . (5.2.4) Yp + 2 2 Moreover, using the fact that Y0k , a = 0, the G-invariance of B and the Jacobi identity, we get

p [Y , a] , Y0k , Y k = Y0k , Y p , a, Y k . (5.2.5) By (5.1.5), (5.2.4), (5.2.5), we get the first identity in (5.2.2). Moreover, 2 1 1 X 1 2 ∇Vei + h[a, [a, y]] , ei i + |[a, [a, y]]| . − ∆p − ∇V[a,[a,y]] = − 2 2 2 1≤i≤m

(5.2.6) By (5.2.6), we get

1 p V exp (h[a, [a, y]] , Y i) − ∆ − ∇[a,[a,y]] exp (− h[a, [a, y]] , Y p i) 2 1 1 2 = − ∆p⊕k + |[a, [a, y]]| . (5.2.7) 2 2 Moreover, p H p 2 exp (h[a, [a, y]] , Y p i) ∇H Y p exp (− h[a, [a, y]] , Y i) = ∇Y p −|[a, Y ]| . (5.2.8) By the first identity in (5.2.2), by (5.2.7), and (5.2.8), we get the second identity in (5.2.2). The proof of our proposition is completed. p

Remark 5.2.3. It is remarkable that in equation (5.2.2) for Qa,Y0k and Qa,Y k , 0

(y, Y p ) and Y k have become independent variables. In the sequel, we will mostly use the operator Qa,Y0k , but the operator Qa,Y k could be used as well. 0

We denote by RY0 k ((y, Y ) , (y 0 , Y 0 )) the smooth kernel associated with 0 exp −Qa,Y0k . Since Ad (k) a = a, Ad (k) Y0k = Y0k , we get

Ad k −1 y, a , Y0k , Ad k −1 Y k = [y, a] , Y0k , Y k . (5.2.9) k It follows that in our evaluation of Jγ Y0 in (5.1.11), we may and we will replace RY0k by RY0 k . 0

5.3 An evaluation of certain infinite dimensional traces Let SY0k be the scalar part of Qa,Y0k , so that 2 2 2 1 −∆p⊕k + Y k , a + Y0k , Y p + [y, a] , Y0k SY0k = 2 V − ∇H Y p − ∇[a,[a,y]] . (5.3.1)

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Chapter 5

Let TY0k ((y, Y g ) , (y 0 , Y g0 )) be the smooth kernel of exp −SY0k with respect to the volume dy 0 dY g0 on p × g. It will also be convenient to consider the operator U acting on p × p, 1 V U = − ∆p − ∇H (5.3.2) Y p − ∇[[y,a],a] , 2 and the associated heat kernel V ((y, Y p ) , (y 0 , Y p0 )). Recall that dγ −1 ϕ1 is given by equation (3.5.10) in Theorem 3.5.2. Then dγ −1 ϕ1 preserves p (γ)⊕p (γ) and p⊥ (γ)⊕p⊥ (γ). By Theorem 3.5.2, p (γ)⊕ {0} ⊂ p0 ⊕ p0 is the eigenspace of dγ −1 ϕ1 associated with the eigenvalue 1. We denote by d⊥ γ −1 ϕ1 the restriction of dγ −1 ϕ1 to p⊥ (γ) ⊕ p⊥ (γ). Then 1 − d⊥ γ −1 ϕ1 is invertible. Let p⊥ 0 (γ) be the orthogonal to p (γ) in p0 . By (4.2.1), on p0 , Ad (γ) and Ad k −1 coincide. By equation (3.5.10) in Theorem 3.5.2, we get 2 det 1 − d⊥ γ −1 ϕ1 = det (1 − Ad (k)) |p⊥ det 1 − Ad γ −1 |z⊥ . 0 (γ) 0 (5.3.3) Moreover, z⊥ is invertible, it 0 is even dimensional, the linear map ad (a) |z⊥ 0 ⊥ ⊥ −1 exchanges p0 and k0 , and it intertwines the action of Ad k on p⊥ 0 and ⊥ k0 . Therefore (5.3.3) can be rewritten in the form 2 det 1 − d⊥ γ −1 ϕ1 = det 1 − Ad k −1 |p⊥ det (1 − Ad (γ)) |z⊥ . 0 (γ) 0 (5.3.4) ⊥ Let k⊥ 0 (γ) be the orthogonal to k (γ) in k0 . If z0 (γ) denotes the orthogonal to z (γ) in z0 , then ⊥ ⊥ z⊥ (5.3.5) 0 (γ) = p0 (γ) ⊕ k0 (γ) . b (x) is given by Recall that the function A x/2 b (x) = . (5.3.6) A sinh (x/2) If V is a finite dimensional Hermitian vector space and if B ∈ End (V ) is B/2 self-adjoint, note that sinh(B/2) is a self-adjoint positive endomorphism. Set B/2 b (B) = det 1/2 A . (5.3.7) sinh (B/2) In (5.3.7), the square root is taken to be the positive square root. Put F = ad2 (a) , F = ad2 (a) − ad2 Y0k . (5.3.8)

Theorem 5.3.1. The following identity holds: Z TY0k (y, Y g ) , Ad k −1 (y, Y g ) dydY g p⊥ (γ)×(p⊕k⊥ (γ))

1 b2 iad Y k |p(γ) (2π)−r/2 A = 0 det (1 − Ad (γ)) |z⊥ (γ) h √ i−1 √ det e F /2 − e− F /2 Ad k −1 |p⊥ (γ) . (5.3.9)

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Proof. As explained in Remark 5.2.3, the operator SY0k splits naturally as a sum of two commuting operators acting over p ⊕ p and over k. We will evaluate the trace of the heat kernels of the various components of SY0k , and take their products. First, we evaluate the contribution of p (γ)×z (γ), while using the splitting z (γ) = p (γ) ⊕ k (γ). Let ∆z(γ) be the Laplacian along z (γ) with respect to its natural metric. Let Y p(γ) be the canonical section of p (γ). Set i 2 1 1 h H,p(γ) p(γ)×z(γ) (5.3.10) SY k = Y0k , Y p(γ) − ∆z(γ) − ∇Y p(γ) . 0 2 2 p(γ)×z(γ)

p(γ)×z(γ)

Let TY k ((y, Y ) , (y 0 , Y 0 )) be the smooth kernel for exp −SY k 0 0 We claim that Z p(γ)×z(γ) TY k 0, Y p(γ) , 0, Y p(γ) dY p(γ) p(γ)

.

0

−r/2

= (2π)

b2 iad Y k |p(γ) . (5.3.11) A 0

−q/2

Indeed k (γ) obviously contributes by (2π) . If λ ∈ R+ , let Bλ be the operator on R2 , 1 ∂ 1 ∂2 + λ2 Y 2 − Y . (5.3.12) Bλ = − 2 ∂Y 2 2 ∂y Let exp (−Bλ ) ((y, Y ) , (y 0 , Y 0 )) be the smooth kernel associated with the operator exp (−Bλ ). We will establish the identity Z −1/2 b A (λ) , (5.3.13) exp (−Bλ ) ((0, Y ) , (0, Y )) dY = (2π) R

from which (5.3.11) follows. First, we will assume λ > 0. Then 2 1 ∂ 1 ∂2 1 ∂2 1 2 λ Y − − Bλ = − + . 2 ∂Y 2 2 λ2 ∂y 2λ2 ∂y 2

(5.3.14)

∂ By making the translation Y → Y + λ12 ∂y as in [B05, section 3.10], or by using a simple Fourier transform argument in the variable y as in [BL08, Proposition 4.11.2], we find easily that Bλ can be replaced by λ 1 ∂2 ∂2 2 Cλ = + Y − 2 2. (5.3.15) − 2 2 ∂Y 2λ ∂y Now λ ∂2 1 2 Tr exp − − + Y = , (5.3.16) 2 ∂Y 2 2 sinh (λ/2) from which (5.3.13) follows for λ > 0. By making λ → 0, we also obtain (5.3.13) when λ = 0. This completes the proof of (5.3.11). Now we study the contribution of p⊥ (γ) × p⊥ (γ) to the integral in (5.3.9). ⊥ ⊥ We denote by U p (γ)×p (γ) the obvious analogue of the operator U in (5.3.2) that is associated with this vector space. Namely, ⊥ ⊥ 1 ⊥ H,p⊥ (γ) V,p⊥ (γ) (5.3.17) U p (γ)×p (γ) = − ∆p (γ) − ∇ p⊥ (γ) − ∇[a,[a,y]] . Y 2

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Chapter 5 ⊥

⊥

Let V p (γ)×p(γ) ((y, Y ) , (y 0 , Y 0 )) be the smooth kernel associated with the ⊥ ⊥ operator exp −U p (γ)⊕p (γ) . p⊥ (γ)

∈ p⊥ (γ) be a standard Brownian motion with

Let t ∈ [0, 1] → wt

p⊥ (γ) w0

= 0. Consider the differential equation

⊥ ⊥ ⊥ y˙ = Y p (γ) , Y˙ p (γ) = [a, [a, y]] + w˙ p (γ) , p⊥ (γ) with given y0 , Y0 ∈ p⊥ (γ) × p⊥ (γ). By (5.3.18), we get ⊥

y¨ − ad2 (a) y = w˙ p

(γ)

(5.3.18)

.

(5.3.19) ⊥

⊥

Let h be a bounded measurable function defined on p (γ) × p (γ) with values in R. If E denotes the expectation with respect to the Brownian p⊥ (γ) motion w· , by Itˆ o’s formula, we get h i ⊥ p⊥ (γ)×p⊥ (γ) p⊥ (γ) exp −UY k h y0 , Y0p = E h y1 , Y1 . (5.3.20) 0

Set p⊥ (γ) Z· = y· , Y· .

(5.3.21)

Let Ma be the endomorphism of p × p, which is given in matrix form by 0 1 Ma = . (5.3.22) ad2 (a) 0 Of course Ma preserves p⊥ (γ) × p⊥ (γ) and commutes with Ad (k). p⊥ (γ)

⊥ Here w· ⊥ will be considered as taking its values in {0} × p (γ). Put p Z0 = y0 , Y0 . By (5.3.18), we get ⊥ Z˙ = Ma Z + w˙ p (γ) ,

Z|t=0 = Z0 .

Then we can rewrite equation (5.3.20) in the form p⊥ (γ)×p⊥ (γ) h (Z0 ) = E [h (Z1 )] . exp −UY k

(5.3.23)

(5.3.24)

0

⊥

⊥

In particular V p (γ)×p (γ) (Z0 , Z00 ) dZ00 is the probability law of Z10 . Let RZ0 ,Z00 be the probability law of the process Z· in (5.3.23) conditional on Z1 = Z00 . ⊥ d acting on smooth functions Let J p (γ) be the antisymmetric operator dt ⊥ on [0, 1] with values in p (γ), with the boundary condition y1 = Ad k −1 y0 . ⊥ The corresponding operator J p (γ),2 is symmetric and nonpositive. The cru⊥ cial fact is that the operator J p (γ),2 − ad2 (a) |p⊥ (γ) is symmetric and negative. Among the solutions of (5.3.19) there is the solution ⊥ −1 ⊥ y = J p (γ),2 − ad2 (a) |p⊥ (γ) w˙ p (γ) , (5.3.25) which is the unique solution of (5.3.18) such that y˙ 1 = Ad k −1 y˙ 0 . y 1 = Ad k −1 y 0 ,

(5.3.26)

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−2 ⊥ . Then y is a Gaussian process with covariance −J p (γ),2 + ad2 (a)

⊥

Note that if Y p

(γ)

⊥

= y, ˙ (5.3.26) is equivalent to p⊥ (γ) p⊥ (γ) y1 , Y 1 = Ad k −1 y 0 , Y 0 .

(5.3.27)

⊥

d (γ) Let J p (γ)×p be the antisymmetric operator dt acting on smooth func⊥ tions Z = y, Y p (γ) on [0, 1] with values in p⊥ (γ)×p⊥ (γ), and the bound p⊥ (γ)×p⊥ (γ) ary conditions Z1 = Ad k −1 Z − Ma is 0 . Then the operator J invertible. Moreover, Z = y, y˙ is given by ⊥ −1 ⊥ ⊥ Z = J p (γ)×p (γ) − Ma w˙ p (γ) . (5.3.28) Let Q be the probability law of Z on the set Ck [0, 1] , p⊥ (γ) ⊕ p⊥ (γ) of continuous functions Z t from [0, 1] into p⊥ (γ) ⊕ p⊥ (γ) such that Z 1 = Ad k −1 Z 0 . It is the probability law of a Gaussian process with covariance C given by ⊥ −1 −1 ⊥ ⊥ ⊥ C = J p (γ)×p (γ) − Ma −J p (γ)×p (γ) − Ma∗ . (5.3.29)

By stationarity, the probability law of Z t on p⊥ (γ) ⊕ p⊥ (γ) does not depend on t ∈ S 1 . Also it is a centered Gaussian. We claim that its covariance is nondegenerate. Indeed this is an easy consequence of the fact that the covariance C in (5.3.29) is invertible. A more direct proof is to show that the covariance of Z t is given by Z −1 1 sMa sM ∗ ∗ −1 Ad k −1 − eMa e e a ds Ad (k) − eMa , (5.3.30) 0

which is obviously invertible. Equation (5.3.30) follows easily from (5.3.28). We conclude that the probability law of Z 0 is a centered Gaussian of the form r (Z)dZ. p⊥ (γ)

If Z· = y· , Y·

⊥

p⊥ (γ)

is the solution of (5.3.23) with Z0 = y0 , Y0

∈

⊥

p (γ) × p (γ), we get Zt = etMa Z0 − Z0 + Z t .

(5.3.31)

By (3.5.12) and (3.5.14), dγ −1 ϕ1 = Ad (k) eMa = eMa Ad (k) . Since Z 1 = Ad k −1 Z 0 , by (5.3.31), (5.3.32), we get Z1 = eMa Z0 + Ad k −1 1 − d⊥ γ −1 ϕ1 Z 0 .

(5.3.32)

(5.3.33)

Put uZ0 (Z 0 ) = Ad (k) 1 − d⊥ γ −1 ϕ1

−1

Z 0 − eMa Z0 .

(5.3.34)

For Z ∈ p⊥ (γ) ⊕ p⊥ (γ), let QZ be the probability law of Z · conditional on Z 0 = Z. Let f be a bounded measurable function defined on

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Chapter 5

Ck [0, 1] , p⊥ (γ) × p⊥ (γ) with values in R, let f 0 be a bounded measurable function defined on p⊥ (γ) × p⊥ (γ) with values in R. By (5.3.33), we obtain Z f (Z) f 0 (Z1 ) dQ (Z)

Ck ([0,1],p⊥ (γ)⊕p⊥ (γ))

Z E

=

Qu

p⊥ (γ)⊕p⊥ (γ)

Z0 ( Z

0)

[f (Z · )] f 0 (Z 0 ) r (uZ0 (Z 0 ))

dZ 0 . |det (1 − d⊥ γ −1 ϕ1 )| (5.3.35) 0

By (5.3.35), the probability law of Z1 is just r (uZ0 (Z 0 )) |det(1−ddZ⊥ γ −1 ϕ1 )| . Now observe that by (5.3.32)–(5.3.34), uZ0 Ad k −1 Z0 = Z0 . (5.3.36) By (5.3.35), (5.3.36), we get ⊥ ⊥ r (Z0 ) V p (γ)×p (γ) Z0 , Ad k −1 Z0 = . (5.3.37) |det (1 − d⊥ γ −1 ϕ1 )| Moreover, if we make Z1 = Ad k −1 Z0 in (5.3.33), we get Z0 = Z 0 , and in (5.3.31), Z· = Z · . It follows from the above that we have the identity of positive measures on Ck [0, 1] , p⊥ (γ) ⊕ p⊥ (γ) , ⊥ ⊥ Q RZ0 ,Ad(k−1 )Z0 V p (γ)×p (γ) Z0 , Ad k −1 Z0 dZ0 = . |det (1 − d⊥ γ −1 ϕ1 )| (5.3.38) p⊥ (γ)×p⊥ (γ)

Let SY k 0

p⊥ (γ)×p⊥ (γ)

SY k 0

be the analogue of SY0k for p⊥ (γ) × p⊥ (γ), i.e., h i 2 2 ⊥ ⊥ 1 −∆V,p (γ) + Y0k , Y p (γ) + [y, a] , Y0k = 2 H,p⊥ (γ)

V,p⊥ (γ)

− ∇ p⊥ (γ) − ∇[a,[a,y]] . (5.3.39) Y ⊥ ⊥ ⊥ ⊥ p (γ)×p (γ) Let TY k y, Y p (γ) , y 0 , Y p (γ)0 be the heat kernel associated 0 ⊥ ⊥ p (γ)×p (γ) . Using (5.3.17), (5.3.39) and the Feynman-Kac with exp −SY k 0 formula, we get R p⊥ (γ)×p⊥ (γ) Z0 , Ad k −1 Z0 = E Z0 ,Ad(k−1 )Z0 TY k 0 Z 1 1 h k p⊥ (γ) i 2 k 2 ds exp − Y0 , Y + [y, a] , Y0 2 0 ⊥ ⊥ V p (γ)×p (γ) Z0 , Ad k −1 Z0 . (5.3.40) By combining (5.3.38) and (5.3.40), we obtain Z ⊥ ⊥ T p (γ)×p (γ) Z0 , Ad k −1 Z0 dZ0 p⊥ (γ)×p⊥ (γ) Z 1 = |det (1 − d⊥ γ −1 ϕ1 )| Ck (p⊥ (γ)×p⊥ (γ)) Z 2 1 1 h k p⊥ (γ) i 2 E Q exp − Y0 , Y + [y, a] , Y0k ds . (5.3.41) 2 0

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Since Q is the probability law associated with the Gaussian process y· with −2 ⊥ covariance −J p (γ),2 + ad2 (a) |p⊥ (γ) , we deduce from (5.3.41) the identity Z ⊥ ⊥ 1 T p (γ)×p (γ) Z0 , Ad k −1 Z0 dZ0 = |det (1 − d⊥ γ −1 ϕ1 )| p⊥ (γ)×p⊥ (γ) 2 ⊥ det1/2 −J p (γ),2 + ad2 (a) . (5.3.42) det1/2 −J p⊥ (γ),2 + ad2 (a) −J p⊥ (γ),2 + ad2 (a) − ad2 Y0k The notation in (5.3.42) should be self-explanatory. Indeed each of the determinants in the right-hand side of (5.3.42) is given by a diverging infinite product of eigenvalues. However, the eigenvalues, which can be explicitly calculated, are of the order 4k 2 π 2 , k ∈ Z. It is then a trivial matter that the ratio of the infinite products in (5.3.42) can be made to converge. Similar considerations will apply later. Using the notation in (5.3.8), we can rewrite (5.3.42) in the form Z ⊥ ⊥ T p (γ)×p (γ) Z0 , Ad k −1 Z0 dZ0 p⊥ (γ)×p⊥ (γ)

h i ⊥ det1/2 −J p (γ),2 + F |p⊥ (γ) 1 = . (5.3.43) |det (1 − d⊥ γ −1 ϕ1 )| det1/2 −J p⊥ (γ),2 + F |p⊥ (γ) We will give a more explicit formula for (5.3.43). By [BL08, eq. (7.8.19)], ⊥ if no eigenvalue of Ad (k) on p⊥ (γ) is equal to 1, the operator −J p (γ),2 is invertible, and we have the identity √ √ det e F /2 − e− F /2 Ad k −1 |p⊥ (γ) = det 1 − Ad k −1 |p⊥ (γ) ⊥ det 1/2 1 − J p (γ),−2 F |p⊥ (γ) . (5.3.44) By (5.3.44), if no eigenvalue of Ad (k) on p⊥ (γ) is equal to 1, h i √ √ ⊥ det1/2 −J p (γ),2 + F |p⊥ (γ) det e F /2 − e− F /2 Ad k −1 |p⊥ (γ) h i = √ √ . det1/2 −J p⊥ (γ),2 + F p⊥ (γ) det e F /2 − e− F /2 Ad (k −1 ) |p⊥ (γ) (5.3.45) ⊥ If Ad (k) is equal to the identity, the operator −J p (γ),2 is no longer in⊥ vertible, and the constants span its kernel. Still −J p (γ),2 is invertible on the vector space of the L2 functions that are orthogonal to the constants. We denote by det∗ the determinant of operators acting precisely on this vector space. By [BL08, equation (7.8.2)], we have the identity √ h i √ ⊥ det e F /2 − e− F /2 |p⊥ (γ) = det 1/2 [F ] |p⊥ (γ) det ∗1/2 1 − F J p (γ),−2 . (5.3.46)

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Chapter 5

By (5.3.46), we still get (5.3.45) when Ad (k) = 1. Therefore (5.3.45) holds in full generality. By (5.3.43) and (5.3.45), we get Z ⊥ ⊥ T p (γ)×p (γ) Z0 , Ad k −1 Z0 dZ0 p⊥ (γ)×p⊥ (γ)

√ √ F /2 − F /2 −1 |p⊥ (γ) det e − e Ad k 1 √ √ . (5.3.47) = |det (1 − d⊥ γ −1 ϕ1 )| det e F /2 − e− F /2 Ad (k −1 ) | ⊥ p (γ) Now we will evaluate the contribution of k⊥ (γ) to the trace in (5.3.9). The ⊥ relevant operator S k (γ) acting on k⊥ (γ) is given by h ⊥ i 2 ⊥ ⊥ 1 (5.3.48) S k (γ) = −∆k (γ) + Y k (γ) , a . 2 ⊥ ⊥ ⊥ ⊥ Let T k (γ) Y k (γ) , Y k (γ)0 be the smooth heat kernel for exp −S k (γ) . By proceeding as in (5.3.16), one finds easily that Z ⊥ ⊥ ⊥ ⊥ T k (γ) Y k (γ) , Ad k −1 Y k (γ) dY k (γ) k⊥ (γ)

1

= det

√ e F /2

−

√ e− F /2 Ad (k −1 ) |

. (5.3.49) k⊥ (γ)

By (5.3.4), (5.3.11), (5.3.47), (5.3.49), we obtain Z TY0k (y, Y g ) , Ad k −1 (y, Y g ) dydY g p⊥ (γ)×(p⊕k⊥ (γ))

−r/2

b2 iad Y0k |p(γ) A 1 2 det (1 − Ad (k −1 )) |p⊥ (1 − Ad (γ)) | ⊥ det z0 0 (γ) √ √ det e F /2 − e− F /2 Ad k −1 |p⊥ (γ) √ √ det e F /2 − e− F /2 Ad (k −1 ) |p⊥ (γ) = (2π)

1 det

√ e F /2

−

√ e− F /2 Ad (k −1 ) |

. (5.3.50) k⊥ (γ)

⊥ ⊥ Since ad (a) is an invertible map from z0 into itself that exchanges p0 and −1 and commutes with Ad k , we get √ √ det e F /2 − e− F /2 Ad k −1 |p⊥ (γ) det 1 − Ad k −1 |p⊥ 0 (γ) √ = . √ −1 )) | ⊥ det (1 − Ad (k F /2 − F /2 −1 k det e −e Ad (k ) |k⊥ (γ) 0 (γ) (5.3.51)

k⊥ 0

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Moreover, we have the trivial −1 det 1 − Ad k −1 |p⊥ det 1 − Ad k |k⊥ det (1 − Ad (γ)) |z⊥ (γ) 0 0 (γ) 0 = det (1 − Ad (γ)) |z⊥ (γ) . (5.3.52) By (5.3.50)–(5.3.52), we get (5.3.9). The proof of our theorem is completed.

5.4 Some formulas of linear algebra Proposition 5.4.1. The following identities hold: √ √ > 0, det e F /2 − e− F /2 Ad k −1 |p⊥ 0 (γ) √ √ det e F /2 − e− F /2 Ad k −1 |p⊥ > 0, 0 √ √ det e F /2 − e− F /2 Ad k −1 |z⊥ 0 h √ i2 √ = det e F /2 − e− F /2 Ad k −1 |p⊥ . 0

(5.4.1)

Moreover, Ad k −1 |z⊥ 0 h √ i2 √ dim p⊥ F /2 − F /2 −1 0 − e Ad k = (−1) det (Ad (k)) |p⊥ det e | , ⊥ p 0 0 (5.4.2) det 1 − exp −iθad Y0k Ad k −1 |z⊥ 0 (γ) √ √ = det e F /2 − e− F /2 Ad k −1 |z⊥ . 0 (γ)

det 1 − exp −ad (a) − iθad Y0k

In particular, dim p⊥

0 det (1 − Ad (γ)) |z⊥ = (−1) det (Ad (k)) |p⊥ 0 0 h √ i2 √ F /2 − F /2 −e Ad k −1 |p⊥ det e . (5.4.3) 0

Proof. Since F commutes with Ad (k) and is nonnegative on p⊥ 0 (γ), to establish the first identity in (5.4.1), it is enough to consider the case where F is a nonnegative constant, and Ad (k) is an isometry with no eigenvalue equal to 1. If Ad (k) is equal to −1, the first identity in (5.4.1) is obvious. If Ad (k) is a non trivial rotation in dimension 2, the first identity in (5.4.1) is also obvious, so that we get the first inequality in (5.4.1) in full generality. To establish the second inequality in (5.4.1), we may now assume that F is a positive constant. If Ad (k) = 1, the inequality is obvious, and the other cases for Ad (k) have already been covered. Since ad (a) is one to one from ⊥ p⊥ 0 into k0 , the third identity in (5.4.1) is obvious. Now we establish (5.4.2). First, note that since ad (a) commutes with ad Y0k and anticommutes with θ, it anticommutes with θad Y0k , so that 2 ad (a) + iθad Y0k = ad2 (a) − ad2 Y0k = F . (5.4.4)

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Chapter 5

k

Clearly, ad (a)+iθad Y0 commutes with F . Since ad (a) induces an isomor- ⊥ k phism from p⊥ 0 into k0 , it follows that the restriction of ad (a) + iθad Y0 ⊥ to√z0 is diagonalizable, and its eigenvalues coincide with the eigenvalues of ± F . The first equality in (5.4.2) follows. On z⊥ 0 (γ), ad (a) vanishes, so that the above argument cannot be used any more. However, by splitting k z⊥ 0 (γ) ⊗R C according to the eigenvalues of ad Y0 , we obtain easily the second identity in (5.4.2). ∗ If A ∈ End z⊥ 0 , let A be its adjoint with respect to the scalar product ⊥ a −1 of z0 . Since γ = e k , we find that Ad (γ) |∗−1 = Ad e−a k −1 |g . (5.4.5) g By the first identity in (5.4.2) and by (5.4.5), we get −1 det 1 − Ad (γ) |z⊥ 0 i2 h √ √ ⊥ dim p0 F /2 − F /2 −1 − e Ad k | . = (−1) det (Ad (k)) |p⊥ det e ⊥ p0 0 (5.4.6) Moreover, the right-hand side of (5.4.6) is unchanged when replacing a by −a and k by k −1 , which allows us to replace γ −1 by γ, so that we get (5.4.3). The proof of our proposition is completed. Remark 5.4.2. Since when Y0k = 0, F = F , the results of Proposition 5.4.1 also hold when making Y0k = 0 and when replacing F by F . By (5.4.2), we get det 1 − exp −ad (a) − iθad Y0k

Ad k −1

|z⊥ 0

−1

det 1 − exp (−ad (a)) Ad k |z⊥ 0 " √ √ = det e F /2 − e− F /2 Ad k −1 |p⊥ 0

det e

√ F /2

√ − F /2

−e

Ad k

−1

#2 |p⊥ 0

. (5.4.7)

Therefore the left-hand side of (5.4.7) has a natural positive square root, which is given by " det 1 − exp −ad (a) − iθad Y0k Ad k −1 |z⊥ 0 #1/2 det 1 − exp (−ad (a)) Ad k

−1

|z⊥ 0

√ √ = det e F /2 − e− F /2 Ad k −1 |p⊥ 0 √ √ F /2 − F /2 det e −e Ad k −1 |p⊥ . (5.4.8) 0

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Moreover, det 1 − exp −iθad Y0k

Ad k −1

z⊥ 0 (γ)

det 1 − Ad k −1

|z⊥ (5.4.9) 0 (γ)

has a natural square root, which depends analytically on Y0k . Indeed no ⊥ eigenvalue of Ad (k) acting on z⊥ 0 (γ) is equal to 1. Moreover, if z0 (γ) is k 1-dimensional, then Ad (k) |z⊥ = −1 and ad Y0 |z⊥ = 0, the square 0 (γ) 0 (γ) ⊥ root is 2. If z (γ) is of dimension 2, if Ad (k) is a rotation of angle φ and 0 0 θad Y0k |z⊥ acts by an infinitesimal rotation of angle φ , such a square 0 (γ) root is given by φ + iφ0 φ sin . (5.4.10) 4 sin 2 2 We will denote the above square root by h i1/2 det 1 − exp −iθad Y0k Ad k −1 z⊥ (γ) det 1 − Ad k −1 |z⊥ . (γ) 0 0 (5.4.11) ⊥ ⊥ Of course in (5.4.11), we may as well replace z⊥ 0 (γ) by p0 (γ) or k0 (γ). It follows that det 1 − exp −ad (a) − iθad Y0k Ad k −1 |z⊥ (γ) det 1 − exp (−ad (a)) Ad k −1 |z⊥ (γ) has a natural square root, which will be denoted by " det 1 − exp −ad (a) − iθad Y0k Ad k −1 |z⊥ (γ) #1/2 det 1 − exp (−ad (a)) Ad k

−1

|z⊥ (γ)

. (5.4.12)

Incidentally observe that the all above quantities are unchanged when replacing a by −a. This is easily seen by conjugating the above matrices by θ. cs was defined in section 5.1. Recall that Tr Theorem 5.4.3. The following identities hold: ⊥,∗ · Trs Λ (z0 ) Ad k −1 exp b c (ad (a)) − c ad (a) + iθad Y0k √ √ √ √ F /2 − F /2 −1 det e − e Ad k |p⊥ , = det e F /2 − e− F /2 Ad k −1 |p⊥ 0 0 (5.4.13) ) Ad k −1 exp −c iθad Y k Trs Λ ( 0 h i1/2 = det 1 − exp −iθad Y0k Ad k −1 z⊥ (γ) det 1 − Ad k −1 |z⊥ . 0 (γ) ·

z0⊥,∗ (γ)

0

Moreover, cs exp −α − c iθad Y0k |z(γ) Tr −1 b−1 iad Y k |p(γ) A b =A iad Y0k |k(γ) . (5.4.14) 0

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Chapter 5

In particular, cs Ad k −1 exp −α + b Tr c (ad (a)) − c ad (a) + iθad Y0k −1 b−1 iad Y0k |p(γ) A b =A iad Y0k |k(γ) " det 1 − exp −ad (a) − iθad Y0k Ad k −1 |z⊥ (γ) #1/2 det 1 − exp (−ad (a)) Ad k

−1

|z⊥ (γ)

. (5.4.15)

Proof. Observe that if R is the canonical Euclidean 1-dimensional vector space equipped with the canonical scalar product or with its negative, if e = ±1, among the monomials in the c (e) , b c (e∗ ), up to permutation, c (e) b c (e∗ ) · ∗ is the only monomial whose supertrace on Λ (R ) does not vanish, and moreover, Trs [c (e) b c (e∗ )] = −2.

(5.4.16)

∗

Similarly, the only monomial in c (e) , b c (e ) whose trace does not vanish is 1, and moreover, Tr [1] = 2.

(5.4.17)

Also observe that u = −1 acts like 1 on Λ0 (R∗ ) ' R, and like −1 on Λ1 (R∗ ) ' R, so that if C ∈ End (Λ· (R∗ )), ·

∗

·

∗

Trs Λ (R ) [uC] = TrΛ (R ) [C] .

(5.4.18) z⊥ 0

To establish the first equation in (5.4.13), we split ⊗R C according to −1 ⊥ the eigenvalues of Ad k −1 . Assume that Ad k = ±1, and that p⊥ 0 , k0 k are 1-dimensional, so that ad Y0 |z⊥ = 0. The matrix of ad (a) with respect 0 ⊥ ⊥ to the splitting z⊥ = p ⊕ k and a corresponding basis e, f consisting of 0 0 0 vectors of norm 1 is given by 0 λ ad (a) = . (5.4.19) λ 0 Then c (ad (a)) =

λ c (e) c (f ) , 2

λ b c (ad (a)) = − b c (e) b c (f ) . 2

(5.4.20)

Moreover, 2

(c (e) c (f )) = 1,

2

(b c (e) b c (f )) = 1.

(5.4.21)

By (5.4.20), (5.4.21), we get exp (−c (ad (a))) = cosh (λ/2) − sinh (λ/2) c (e) c (f ) , exp (b c (ad (a))) = cosh (λ/2) − sinh (λ/2) b c (e) b c (f ) . By (5.4.16), (5.4.17), (5.4.22), we obtain ⊥,∗ · Tr Λ (z0 ) [exp (b c (ad (a)) − c (ad (a)))] = 4 sinh2 (λ/2) , s

Λ· (z0⊥,∗ )

Tr

2

[exp (b c (ad (a)) − c (ad (a)))] = 4 cosh (λ/2) .

(5.4.22)

(5.4.23)

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By (5.4.18), both equations in (5.4.23) are compatible with the first identity in (5.4.13). ⊥ 0 0 Let us now suppose that p⊥ 0 , k0 are both of dimension 2. Let e, e and f, f ⊥ ⊥ be orthonormal bases of p0 , k0 . Also we assume that Ad (k) is a rotation of angle φ on both vector spaces, that ad (a)maps e, e0 into λf, λf 0 and f, f 0 into λe, λe0 , and that the matrix of ad Y0k with respect to e, e0 and f, f 0 is given by 0 −φ0 k ad Y0 = . (5.4.24) φ0 0 0 0 Incidentally observe that z⊥ 0 is canonically oriented by the basis e, e , f, f . By (5.4.20), we get

λ (c (e) c (f ) + c (e0 ) c (f 0 )) , 2 λ b c (ad (a)) = − (b c (e) b c (f ) + b c (e0 ) b c (f 0 )) . 2 c (ad (a)) =

(5.4.25)

Moreover, c θad Y0k

=−

φ0 (c (e) c (e0 ) + c (f ) c (f 0 )) . 2

(5.4.26)

By (5.4.22), (5.4.25), we get exp (b c (ad (a))) = (cosh (λ/2) − sinh (λ/2) b c (e) b c (f )) (cosh (λ/2) − sinh (λ/2) b c (e0 ) b c (f 0 )) . (5.4.27) Set φ φ (c (e) c (e0 ) − b c (e) b c (e0 )) − (c (f ) c (f 0 ) − b c (f ) b c (f 0 )) . (5.4.28) 2 2 Then Ad k −1 acts on Λ· z⊥∗ like exp (−B). 0 By (5.4.16), the supertrace in the first identity in (5.4.13) is obtained as 16 times the coefficient of B=

c (e) c (e0 ) c (f ) c (f 0 ) b c (e) b c (e0 ) b c (f ) b c (f 0 ) in the expansion of exp −B + b c (ad (a)) − c ad (a) + iθad Y0k

.

First, we make the c equal to 0. This way, we will get the coefficient of b c (e) b c (e0 ) b c (f ) b c (f 0 ). By (5.4.27), (5.4.28), this coefficient is given by 2 φ + iλ − cosh2 (λ/2) sin2 (φ/2) − sinh2 (λ/2) cos2 (φ/2) = − sin . 2 (5.4.29) Now we will obtain the coefficient of c (e) c (e0 ) c (f ) c (f 0 ). Put c (f ) = −ic (f ) , c (f 0 ) = −ic (f 0 ) .

(5.4.30)

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By (5.4.25), (5.4.26), iλ (c (e) c (f ) + c (e0 ) c (f 0 )) , (5.4.31) 2 iφ0 c iθad Y0k = − (c (e) c (e0 ) − c (f ) c (f 0 )) . 2 Moreover, if c (B) is the component of B that only contains c variables, we get from (5.4.28), φ c (B) = (c (e) c (e0 ) + c (f ) c (f 0 )) . (5.4.32) 2 Put   0 φ − iφ0 iλ 0 −φ + iφ0 0 0 iλ  . M = (5.4.33)  −iλ 0 0 φ + iφ0  0 0 −iλ −φ − iφ 0 as Let End z⊥ of z⊥ 0 be the set of antisymmetric endomorphisms 0 with respect as ⊥ ⊥ to the scalar product of z0 . Then M ∈ End z0 ⊗R C. Let c0 z⊥ be the Cliffordalgebra associated with the Euclidean vector 0 0 0 ⊥ 0 space z⊥ . The 0 algebra c z0 is generated by c (e) , c (e ) , c (f ) , c (f ). Let 0 ⊥ c (M ) ∈ c z0 ⊗R C be associated to the antisymmetric matrix M as in (1.1.9) with respect to the scalar product of z⊥ 0 . By (5.4.31), (5.4.32), one verifies easily that −c (B) − c ad (a) + iθad Y0k = c0 (M ) . (5.4.34) c (ad (a)) =

We have to evaluate the coefficient Trs c [exp (c0 (M ))] of c (e) c (e0 ) c (f )c (f 0 ) in exp (c0 (M )). To do this, we first consider the case of N ∈ Endas z⊥ 0 . We will evaluate the coefficient Trs c [exp (c0 (N ))] of exp (c0 (N )), and obtain the corresponding coefficient for exp (c0 (M )) by analytic continuation. There is an orthonormal oriented basis i, j, k, l of z⊥ 0 such that the matrix ν of N in this basis is given by   0 −α 0 0 α 0 0 0  . (5.4.35) ν= 0 0 0 −β  0 0 β 0 Then c0 (ν) =

1 (αc0 (i) c0 (j) + βc0 (k) c0 (l)) . 2

(5.4.36)

Also, exp (c0 (ν)) = (cos (α/2) + sin (α/2) c0 (i) c0 (j)) (cos (β/2) + sin (β/2) c0 (k) c0 (l)) . (5.4.37) Moreover, using (5.4.30) and the fact that i, j, k, l is an oriented basis, we get c0 (i) c0 (j) c0 (k) c0 (l) = −c (e) c (e0 ) c (f ) c (f 0 ) .

(5.4.38)

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109

By (5.4.36)–(5.4.38) we get Trs c [exp (c0 (N ))] = − sin (α/2) sin (β/2) .

(5.4.39)

We rewrite (5.4.39) in the form b−1 (iα) A b−1 (iβ) . 4Trs c [exp (c0 (N ))] = −αβ A

(5.4.40)

Now the Pfaffian Pf [N ] of N is given by Pf [N ] = αβ.

(5.4.41)

Then (5.4.40) can be written in the form b−1 (N ) . 4Trs c [exp (c0 (N ))] = −Pf [N ] A

(5.4.42)

By analytic continuation, we deduce from (5.4.42) that b−1 (M ) . 4Trs c [exp (c0 (M ))] = −Pf [M ] A

(5.4.43)

As we already saw p in the proof of Proposition 5.4.1, the eigenvalues of M are given by ±iφ ± λ2 + φ02 , so that p p b−1 (M ) = A b−1 iφ + λ2 + φ02 A b−1 −iφ + λ2 + φ02 . A (5.4.44) ⊥∗ We identify z⊥ by its scalar product. In particular Λ· z⊥∗ ' 0 to z0 0 2 Λ· z⊥ z⊥∗ ⊗R C be such that if U, V ∈ z⊥ 0 . Let ωM ∈ Λ 0 0 , then ωM (U, V ) = hU, M V i . A straightforward computation shows that 1 ωM ∧ ωM = φ2 + λ2 + φ02 e ∧ e0 ∧ f ∧ f 0 , 2 so that Pf [M ] = φ2 + λ2 + φ02 . By (5.4.43), (5.4.44), and (5.4.47), we obtain ! 2 p 2 + φ02 φ + i λ Trs c [exp (c0 (M ))] = − sin . 2

(5.4.45)

(5.4.46)

(5.4.47)

(5.4.48)

By the considerations that follow (5.4.28), and by (5.4.29), (5.4.48), we get ⊥,∗ · Trs Λ (z0 ) Ad k −1 exp b c (ad (a)) − c ad (a) + iθad Y0k ! 2 p 2 2 + φ02 φ + iλ λ φ + i sin = 16 sin . (5.4.49) 2 2 Again (5.4.49) fits with the first identity in (5.4.13). To establish the second identity in (5.4.13), we only need to consider the ⊥ contribution of p⊥ 0 (γ), the contribution of k0 (γ) being of the same type. Clearly, ⊥∗ · Trs Λ (p0 (γ)) Ad k −1 = det (1 − Ad (k)) |p⊥ 0 (γ) = det 1 − Ad k −1 |p⊥ . (5.4.50) 0 (γ)

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Chapter 5

Equation (5.4.50) is compatible with the second identity in (5.4.13) when p⊥ 0 (γ) is of dimension 1, and Ad (k) acts on it like −1. Now we assume that p⊥ 0 (γ) is of dimension 2, and that Ad (k) acts on ⊥ p0 (γ) by a rotation of angle φ, and that θad Y0k acts on p⊥ 0 (γ) by an infinitesimal rotation of angle φ0 . Then one verifies easily that in this case, φ + iφ0 Λ· (p⊥∗ (γ)) −1 k 0 . Trs Ad k exp −c iθad Y0 = 4 sin (φ/2) sin 2 (5.4.51) Comparing with (5.4.10), we also get the second equation in (5.4.13) in this case. We have completed the proof of this second equation. To establish (5.4.14), we may as well assume that q = 0, so that z (γ) is simply p (γ). If p (γ) is of dimension 1, then ad Y0k acts on p (γ) like 0, and our identity is obvious. If p (γ) is of dimension 2, our identity is an identity of Mathai-Quillen [MaQ86, eq. (2.13)]. In the general case, we obtain (5.4.14) by splitting p (γ) ⊗R C into the eigenspaces of ad Y0k . Using (5.4.8), (5.4.13) and (5.4.14), we get (5.4.15). The proof of our theorem is completed.

5.5 A formula for Jγ Y0k

Set det 1 − exp −iad Y0k Ad k −1 |k⊥ 1 (γ) 0 . A= k −1 ) | ⊥ det (1 − Ad (k −1 )) |z⊥ det 1 − exp −iad Y Ad (k (γ) p0 (γ) 0 0 (5.5.1) We claim that A has a natural square root. Indeed we have −1 det 1 − Ad k −1 |z⊥ = det 1 − Ad k |p⊥ (γ) 0 0 (γ) det 1 − Ad k −1 |k⊥ . (5.5.2) 0 (γ) As we already saw after (5.4.9) and (5.4.11), det 1 − exp −iad Y0k Ad k −1 |p⊥ det 1 − Ad k −1 |p⊥ 0 (γ) 0 (γ) ⊥ has a natural square root. The same is true when replacing p⊥ 0 (γ) by k0 (γ). Also det 1 − exp −iad Y0k Ad k −1 |k⊥ 0 (γ)

det (1 − Ad (k −1 )) |k⊥ (γ) 0 k det 1 − exp −iad Y0 Ad k −1 |k⊥ det 1 − Ad k −1 |k⊥ 0 (γ) 0 (γ) = . h i2 det (1 − Ad (k −1 )) |k⊥ 0 (γ) (5.5.3) Using again the results we mentioned, we see that (5.5.3) also has a natural square root.

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From the above, we conclude that A has a natural square root. It will be denoted " 1 1/2 A = det (1 − Ad (k −1 )) |z⊥ 0 (γ) #1/2 det 1 − exp −iad Y0k Ad k −1 |k⊥ 0 (γ) . (5.5.4) det 1 − exp −iad Y0k Ad (k −1 ) |p⊥ 0 (γ) Theorem 5.5.1. The following identity holds: b iad Y k |p(γ) A 0 Jγ = 1/2 b iad Y k A 0 k(γ) det (1 − Ad (γ)) |z⊥ 0 " #1/2 det 1 − exp −iad Y0k Ad k −1 |k⊥ 1 0 (γ) . det (1 − Ad (k −1 )) |z⊥ det 1 − exp −iad Y0k Ad (k −1 ) |p⊥ 0 (γ) 0 (γ) (5.5.5) Y0k

1

Proof. By equations (5.1.11), (5.3.9), (5.4.3), (5.4.8), and (5.4.13)–(5.4.15), we get b iad Y k |p(γ) A 0 k Jγ Y0 = b iad Y k |k(γ) A 0

1 1/2 det (1 − Ad (γ)) |z⊥ det (1 − Ad (k −1 )) |z⊥ 0 0 (γ) h

det 1 − exp −iθad Y0k

Ad k −1

|z⊥ det 1 − Ad k −1 0 (γ)

|z⊥ 0 (γ)

i1/2

1 √ √ . (5.5.6) F /2 − F /2 det e −e Ad (k −1 ) |p⊥ 0 (γ) In the right-hand side of (5.5.6), the determinants of the square roots over ⊥ ⊥ z⊥ 0 (γ) can be factored as a product of determinants over k0 (γ) and p0 (γ). ⊥ The contribution of k0 (γ) is given by " #1/2 det 1 − exp −iθad Y0k Ad k −1 |k⊥ (γ) 0 . (5.5.7) det (1 − Ad (k −1 )) |k⊥ 0 (γ) Since θ = 1 on k, we may as well remove θ in (5.5.7). Using the second identity in (5.4.2) applied to p⊥ 0 (γ), the contribution of ⊥ p0 (γ) to the product of determinants in (5.5.6) is given by " #1/2 det 1 − exp −iθad Y0k Ad k −1 |p⊥ 0 (γ) det (1 − Ad (k −1 )) |p⊥ 0 (γ) −1 det 1 − exp −iθad Y0k Ad k −1 |p⊥ . (5.5.8) (γ) 0

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Clearly, det 1 − exp −iθad Y0k Ad k −1 |p⊥ (γ) 0 " = det 1 − exp −iθad Y0k Ad k −1 |p⊥ 0 (γ) #1/2 det 1 − exp iθad

Y0k

Ad k

−1

|p⊥ 0 (γ)

. (5.5.9)

Using (5.5.9), we conclude that (5.5.8) is given by i−1/2 h −1 det 1 − exp iθad Y0k Ad k −1 |p⊥ det 1 − Ad k . ⊥ (γ) p0 (γ) 0 (5.5.10) Since θ = −1 on p, we can replace iθ by −i in (5.5.10). By (5.5.6)–(5.5.10), we get (5.5.5). The proof of our theorem is completed. Remark 5.5.2. Observe that Jγ Y0k is unchanged when replacing the bilinear form B by B/t, t > 0. Note that z (1) = g, p (1) = p, k (1) = k. By (5.5.5), if Y0k ∈ k, we get b (iad (Y0 ) |p ) A b−1 iad Y0k |k . J1 Y0k = A (5.5.11)

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Chapter Six A formula for semisimple orbital integrals This chapter is the central part of the book. First, we give an explicit formula for the orbital integrals associated with the heat kernel of LX A in terms of a Gaussian integral on k (γ). In chapter 9, this formula will be obtained by the explicit computation of the asymptotics as b → +∞ of the orbital integrals associated with LX A,b . From the formula for the heat kernel, we derive a corresponding formula for the semisimple orbital integrals associated with the wave operator of LX A. This chapter is organized as follows. In section 6.1, we give the formula for the orbital integral associated with the heat kernel of LX A. In section 6.2, we obtain the formula for the orbital integral associated q X LA , when µ ∈ S (R) is even, and its Fourier transform µ b decays with µ like a Gaussian at infinity. Finally, in section 6.3, we obtain a formula for the orbital integrals associated with the wave operator of LX A. In the whole chapter, γ ∈ G is supposed to be semisimple, and is written as in (4.2.1). Also we use the notation of chapters 2–5.

6.1 Orbital integrals for the heat kernel Recall that Z (γ) ⊂ G is the centralizer of γ, and that z (γ) = p (γ)⊕k (γ) is its Lie algebra. As before, we use the notation p = dim p (γ) , q = dim k (γ) , r = dim z (γ). For Y0k ∈ k (γ), Jγ Y0k is given by equation (5.5.5) in Theorem 5.5.1. By (5.5.5), there exist cγ > 0, Cγ > 0 such that if Y0k ∈ k (γ), Jγ Y0k ≤ cγ exp Cγ Y0k . (6.1.1) Theorem 6.1.1. For any t > 0, the following identity holds: 2 exp − |a| /2t Tr[γ] exp −tLX = A p/2 (2πt) Z Jγ Y0k TrE ρE k −1 exp −iρE Y0k − tA k(γ)

2 exp − Y0k /2t

dY0k (2πt)

q/2

. (6.1.2)

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Proof. The proof of our theorem will be given in chapter 9.

6.2 A formula for general orbital integrals Let ∆z(γ) be the standard Laplacian on z (γ) = p (γ) ⊕ k (γ) with respect to the scalar product induced by the scalar product of g. For t > 0, let exp t∆z(γ) /2 be the corresponding heat operator. We denote by exp t∆z(γ) /2 y, Y0k , y 0 , Y0k0 the associated Gaussian heat kernel. Here the heat kernel is calculated with respect to the given volume element on z (γ), which is fixed once andfor all. Then Jγ Y0k ρE k −1 exp −iρE Y0k δy=a is a distribution on z (γ) = p (γ) ⊕ k (γ) with values in End (E). The heat kernel exp t∆z(γ) /2 − tA can be applied to this distribution, and the resulting smooth function over z (γ) will be denoted by exp t∆z(γ) /2 − tA Jγ Y0k ρE k −1 exp −iρE Y0k δy=a . (6.2.1) When taking the trace of (6.2.1), we obtain a smooth function on z (γ) with values in C. This function can be evaluated at 0 ∈ z (γ). Theorem 6.2.1. For any t > 0, the following identity holds: " E [γ] X exp −t L + A = Tr exp t∆z(γ) /2 − tA Tr # E −1 k E k Jγ Y0 ρ k exp −iρ Y0 δy=a (0) . (6.2.2) Proof. Observe that exp t∆z(γ)/2 (0, 0) , y, Y0k =

1 (p+q)/2

(2πt)

2 2 exp − |y| /2t − Y0k /2t . (6.2.3)

By (6.1.2), (6.2.3), we get (6.2.2). Let S (R) be the Schwartz space of R, let S even (R) be the space of even functions in S (R). Take µ ∈ S even (R). Let µ b ∈ S even (R) denote its Fourier transform, i.e., Z µ b (y) = e−2iπyx µ (x) dx. (6.2.4) R

We will assume that there exists C > 0 such that for any k ∈ N, there exists ck > 0 such that (k) µ (y) ≤ ck exp −Cy 2 . (6.2.5) b √ √ LX + A is self-adjoint. Let µ LX + A (x, x0 ) , x, x0 ∈ X be Then µ √ the corresponding smooth kernel, which we will still denote µ LX + A .

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The algebra of operators Q was defined in Definition 4.1.1. Using (6.2.5) √ X and finite propagation speed for the wave operator cos t L + A [CP81, √ section 7.8], [T81, section 4.4], it can be easily shown that µ LX + A ∈ h √ i Q. Therefore Tr[γ] µ LX + A is well-defined. √ −∆z(γ) + A on Similarly from (6.2.5), one finds that the kernel of µ z (γ) has a Gaussian-like decay. Combining this with (6.1.1), we find that in (6.2.1), we can replace the function exp −tx2 /2 by our function µ, and still obtain a continuous function on z (γ). We give the following extension of Theorem 6.2.1. Theorem 6.2.2. The following identity holds: " q h p i E [γ] X z(γ) µ L + A = Tr µ −∆ /2 + A Jγ Y0k Tr # ρ

E

k

−1

E

exp −iρ

Y0k

δy=a (0) . (6.2.6)

Proof. For t > 0, if µ (x) = exp −tx2 /2 , equation (6.2.6) is just (6.2.2). More generally, by differentiating (6.2.2) with respect to t, we find that (6.2.6) still holds for µ (x) = x2k exp −tx2 /2 , t > 0, k ∈ N. Using basic results on the harmonic oscillator, we know that given t > 0, linear combinations of such functions are dense in S even (R). Set νb (y) = exp Cy 2 /2 µ b (y) . (6.2.7) Using (6.2.5), we find that νb ∈ S even (R). From the above, given M ∈ N, > 0, there is a finite linear combination νbM, of functions of the type x2k exp −x2 /2 , k ∈ N such that for p ∈ N, p ≤ M , (p) ν − νbM, ) (y) ≤ . (6.2.8) (b Set µ bM, (y) = exp −Cy 2 /2 νb,M (y) .

(6.2.9)

Let µM, ∈ S even (R) be the Fourier transform of µ bM, . By (6.2.7)–(6.2.9), for p ∈ N, p ≤ M , (p) µ−µ bM, ) (y) ≤ cM exp −Cy 2 /4 . (6.2.10) (b As we saw above, equation (6.2.6) is valid for µM, . By taking M ∈ N large enough and > 0 small enough, we find that the left- and right-hand sides of (6.2.6) for µ are approximated by the corresponding expressions for µM, , which turn out to be equal. Therefore (6.2.6) is valid for µ. The proof of our theorem is completed.

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Remark 6.2.3. The density result used at the beginning of the proof of Theorem 6.2.2 follows from basic properties of the harmonic oscillator. The harmonic oscillator, which had just disappeared from the geometric picture, resurrects itself in the proof, with a completely different interpretation. The deformation argument for the heat kernel given in Theorem 4.6.1 is not valid for the more general kernels considered in Theorem 6.2.2. Equivalently, the elliptic trace in the left-hand side of (6.2.6) cannot be replaced in general by a hypoelliptic supertrace, with LX replaced by LX b . This is especially true when µ b has compact support. 6.3 The orbital integrals for the wave operator i h √ Let Tr[γ] cos s LX + A be the even distribution on R such that for any µ ∈ S even (R) with µ b having compact support, h p i Z h i p [γ] X Tr µ L +A = µ b (s) Tr[γ] cos 2πs LX + A ds. (6.3.1) √ R √ The wave operator cos 2s LX + A defines a distribution on R × X × X. By finite propagation speed for the wave equation [CP81, section 7.8], [T81, section 4.4], its support is included in (s, x, x0 ) , |s| ≥ d (x, x0 ). Recall that we have identified T X and T ∗ X by the metric, and that s ∈ R → ϕs is the geodesic flow on X = X ∗ . Let τ be the variable dual to s. By [Hör85a, √ √ Theorem 23.1.4 and remark], the wave front set WF cos 2s LX + A √ √ of the distribution cos 2s LX + A is the conic set in R2 × T ∗ X × T ∗ X generated by (x0 , −Y 0 ) = ϕ±s (x, Y ) , |Y | = 1, τ = ±1. Conic here means that the dilations by λ > 0 are applied to the variables Y, Y 0 , τ . As we saw in section 3.4, the map f ∈ p⊥ (γ) → ργ (1, f ) ∈ X identifies p⊥ (γ) to a smooth submanifold P ⊥ (γ) of X. Let NP ⊥ (γ)/X be the orthogonal bundle to T P (γ) in T X. Set ∆γX = (x, γx) , x ∈ P ⊥ (γ) . (6.3.2) Then ∆γX is a smooth submanifold of X × X. The conormal bundle to R × ∆γX ⊂ R × X × X can be identified with the set ((s, τ ) , (x, Y ) , (x0 , Y 0 )) ∈ R2 × X × X such that τ = 0, x ∈ P ⊥ (γ) , x0 = γx, γ ∗ Y 0 + Y ∈ NP ⊥ (γ)/X . √ √ By [H¨ o83, Theorem 8.2.10], cos 2s LX + A ∆γX is a well-defined distribution on R × X × X, and its wave front set is the formal sum of the wave front sets of the two above distributions. Using (3.4.4) and the property √ √ of the support of cos 2s LX + A , which was given before, the pushh √ √ i forward of the distribution TrF γ cos 2s LX + A by the projection R × X × X → R is well-defined. It will be denoted Z h √ p i F Tr γ cos 2s LX + A . (6.3.3) ∆γ X

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This is an even distribution on R. Tautologically, we have the identity of even distributions on R, h p i Z h p i Tr[γ] cos s LX + A = TrF γ cos s LX + A .

(6.3.4)

∆γ X

h √ √ i Proposition 6.3.1. The singular support of Tr[γ] cos 2s LX + A is included in s = ± |a|, the ordinary support is includedh in {s |s| ≥ |a|}. √∈ R, i √ [γ] cos For a = 0, if p (γ) = 0, the singular support of Tr 2s LX + A is empty. Proof. The wave front set of the distribution √ √ (6.3.3) can be obtained from the wave front set of the product cos 2s LX + A ∆γX by [Hö83, Theorem 8.2.13]. Using this result, we find that s ∈ R lies in the singular support of this distribution if and only if there is x ∈ X, Y ∈ Tx X, |Y | = 1 such that if ϕs (x, Y ) = (x0 , Y 0 ) ,

(6.3.5)

then x ∈ P (γ) ,

x0 = γx,

γ ∗ Y 0 − Y ∈ NP ⊥ (γ)/X .

(6.3.6)

By (6.3.5), we get ϕ1 (x, sY ) = (x0 , sY 0 ) .

(6.3.7)

By (6.3.6), (6.3.7), x is a critical point of the restriction of d2γ to P ⊥ (γ). Since d2γ is a convex function, we find that x = p1, and that |s| = |a|. This i h √ √ shows that the singular support of Tr[γ] cos 2s LX + A is included in ± |a|. If a 6= 0, note that in (6.3.5), Y = ±a/ |a|. Since d (x, γx) = dγ (x) ≥ |a|, using h the √considerations i we made after √ [γ] X equation (6.3.1), the support of Tr cos 2s L + A is included in {s ∈ R, |s| ≥ |a|}. Assume now that a = 0, so that γ = k −1 . We already know that in (6.3.5), (6.3.6), s = 0, and so Y 0 = Y . By (6.3.6), (Ad (k) − 1) Y is fixed by Ad (k), so that Ad (k) Y = Y,

(6.3.8)

so that Y ∈ p (γ). If p (γ) = 0, we cannot have |Y | = 1. This completes the proof of our proposition. We define the even distribution on R, q E −1 E k E k z(γ) Tr cos s −∆ /2 + A Jγ Y0 ρ k exp −iρ Y0 δy=a (0) (6.3.9)

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Chapter 6

by the formula q −∆z(γ) /2 + A Jγ Y0k ρE k −1 exp −iρE Y0k δy=a (0) TrE µ " Z q E = µ b (s) Tr cos 2πs −∆z(γ) /2 + A Jγ Y0k ρE k −1 R

# exp −iρE Y0

k

δy=a (0) . (6.3.10)

We will denote by z = y, Y0k the generic element of z (γ) = p (γ) ⊕ k (γ). Also we equip z (γ) with its Euclidean norm. By finite propagation for the wave equation [CP81, section 7.8], √ speed p z(γ) 2s −∆ /2 + A is a distribution on R × z (γ) × [T81, section 4.4], cos 0 z (γ) whose support is included in (s, z, z 0 ) , |s| ≥ By [Hör85a, Theo |z −√z|. p rem 23.1.4 and remark], its wave front set WF cos 2s −∆z(γ) /2 + A

is equal to the conic set associated with (y 0 , −Y 0 ) = (y ± sY, Y ) , |Y | = 1, τ = ±1. Conic set means again that the dilations by λ > 0 are applied to the variables Y, Y 0 , τ . Set H γ = {0} × (a, k (γ)) ⊂ z (γ) × z (γ) .

(6.3.11)

The wave front set associated with R × H γ ⊂ × z (γ) × z (γ)is such that R p 0k(γ) Y = 0, τ = 0, so that the product cos s −∆z(γ) /2 + A H γ is well defined. The function Jγ Y0k ρE k −1 exp −iρE Y0k can be viewed as a smooth function on the second copy of z (γ) in z (γ) × z (γ). It lifts to a smooth function on z (γ) × z (γ). Therefore, Trs

E

q E −1 γ k E k z(γ) exp −iρ Y0 cos s −∆ /2 + A H Jγ Y0 ρ k

(6.3.12) is a well-defined distribution on R × z (γ) × z (γ). The pushforward of this distribution by the projection R × z (γ) × z (γ) → R will be denoted Z Hγ

q Trs E cos s −∆z(γ) /2 + A Jγ Y0k ρE k −1 exp −iρE Y0k .

(6.3.13) √ This is an even distribution supported in |s| ≥ 2 |a|, with singular support √ included in s = ± 2 |a|. Note that if a = 0 and if p (γ) = 0, the singular support of this distribution is empty.

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Tautologically, q E −1 E k E k z(γ) exp −iρ Y0 δy=a (0) Tr cos s −∆ /2 + A Jγ Y0 ρ k " q Z = Trs E cos s −∆z(γ) /2 + A Jγ Y0k Hγ

ρE k

−1

# k

exp −iρE Y0

. (6.3.14)

Theorem on R supported √ √ 6.3.2. We have the identity of even distributions on |s| ≥ 2 |a| with singular support included in ± 2 |a|, Z h p i TrF γ cos s LX + A ∆γ X

Z =

" E

Tr Hγ

q z(γ) cos s −∆ /2 + A Jγ Y0k ρ

E

k

−1

E

exp −iρ

Y0k

#

. (6.3.15)

Proof. By (6.2.6), (6.3.1), (6.3.4), (6.3.10), and (6.3.14), we get (6.3.15). Remark 6.3.3. Theorem 6.3.2 is a microlocal version of Theorem 6.2.2. Along the lines of Remark 6.2.3, note that although the hypoelliptic Laplacian does not have a wave operator, equation (6.3.15) has been obtained via the hypoelliptic Laplacian and its corresponding heat kernel.

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Chapter Seven An application to local index theory The purpose of this chapter is to verify the compatibility of our formula in Theorem 6.1.1 for the orbital integrals of heat kernels to the index formula of Atiyah-Singer [AS68a, AS68b], to the fixed point formulas of Atiyah-Bott [ABo67, ABo68], and to the index formula for orbifolds of Kawasaki [Ka79]. Recall that the McKean-Singer formula [McKS67] expresses the index of a Dirac operator over a compact manifold Z as the supertrace of a heat kernel. If Z is the quotient of X by a cocompact torsion free group, this supertrace can be evaluated explicitly by the formulas we gave in chapter 6. Here we will directly check these formulas to be compatible with the index formulas. This chapter is organized as follows. In section 7.1, we establish identities that relate the characteristic forms of T X to the characteristic forms of N . In section 7.2, we construct the Dirac operator DX acting on twisted spinors over X. In section 7.3, we state the McKean-Singer formula for the index of a Dirac operator. In section 7.4, we evaluate the orbital integrals associated with the index of a Dirac operator, when γ is the identity, and when γ is semisimple and nonelliptic. In section 7.5, we establish the results of section 7.4. In section 7.6, we establish a compatibility result when X is a complex manifold. In section 7.7, we extend our results to the case when γ is elliptic. In section 7.8, we consider the de Rham-Hodge operator dX + dX∗ . Finally, in section 7.9, we evaluate the integrand that appears in the definition of the Ray-Singer analytic torsion [RS71] for the de Rham complex. In particular, we recover the vanishing results of Moscovici-Stanton [MoSt91] for this integrand.

7.1 Characteristic forms on X We will work here on the manifold X, but the statements that follow can be easily obtained in more general situations. Recall that as we saw in equation (2.2.2), the vector bundle T X ⊕ N is equipped with a flat connection ∇T X⊕N,f , which preserves the bilinear form B. On the other hand, the connection ∇T X⊕N preserves the splitting, and also the bilinear form B. The connection ∇T X⊕N is a metric connection for

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An application to local index theory

the metric g T X⊕N = g T X ⊕ g N induced by B on T X ⊕ N . In this context, we can use the formalism of Bismut-Lott [BLo95, section 1(g)]. Namely, set −1 T X⊕N,f T X⊕N ω T X ⊕ N, g T X⊕N = g T X⊕N ∇ g . (7.1.1) T X⊕N Then ω T X ⊕ N, g is a 1-form with values in symmetric endomorphisms of T X ⊕ N . One readily verifies that ω T X ⊕ N, g T X⊕N = −2ad (·) . (7.1.2) By comparing with (2.2.2), we get 1 (7.1.3) ∇T X⊕N = ∇T X⊕N,f + ω T X ⊕ N, g T X⊕N . 2 In the formalism of [BLo95], ∇T X⊕N is the metric preserving connection canonically associated with the flat connection ∇T X⊕N,f and the metric g T X⊕N . If H, g H , ∇H is a complex Hermitian vector bundle equipped with a metric connection, we denote by RH = ∇H,2 the curvature of ∇H . The Chern character form ch H, ∇H is given by RH . (7.1.4) ch H, ∇H = Tr exp − 2iπ 0 0 If H 0 , g H , ∇H is a real Euclidean vector bundle equipped with an Eu 0 b H 0 , ∇H 0 is given by clidean connection with curvature RH , the form A !!#1/2 " H0 0 R 0 H b H ,∇ b − . (7.1.5) A = det A 2iπ b H 0 , ∇H 0 are real and closed, and the corThen the forms ch H, ∇H , A b (H 0 ). responding cohomology classes are denoted ch (H) , A Proposition 7.1.1. The following identities of closed forms hold on X: b T X, ∇T X A b N, ∇N = 1. ch T X, ∇T X + ch N, ∇N = dim g, A (7.1.6) Proof. We can use (7.1.3) and obtain (7.1.6) as a consequence of [BLo95, Proposition 1.3]. We will give here a direct proof, by following [BLo95]. The curvature of ∇T X and of ∇N is obtained by restricting ad (Ω) ∈ Λ2 (T ∗ X) ⊗ End (g) to p and to k. By (2.1.10), ad (Ω) = −ad2 (ω p ) .

(7.1.7) −degα/2

Let ϕ be the endomorphism of Λ· (T ∗ Z) given by α → (2iπ) α. We consider g = p ⊕ k as a trivially Z2 -graded vector space, and T X ⊕ N as the corresponding trivially Z2 -graded vector bundle. By (7.1.7), ch T X, ∇T X + ch N, ∇N = ϕTr exp ad2 (ω p ) . (7.1.8)

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Clearly, 1 [ad (ω p ) , ad (ω p )] . 2 Since traces vanish on supercommutators, we get ad2 (ω p ) =

(7.1.9)

∂ Tr exp sad2 (ω p ) = Tr ad2 (ω p ) exp sad2 (ω p ) ∂s 1 = Tr ad (ω p ) , ad (ω p ) exp sad2 (ω p ) = 0. (7.1.10) 2 Using (7.1.10), and making s = 0, we get the first identity in (7.1.6). Similarly, h i1/2 b T X, ∇T X = ϕ det A b ad2 ω p |p A , (7.1.11) h i 1/2 b ad2 ω p |k b N, ∇T N = ϕ det A . A By (7.1.11), we get h i b ad2 (ω p ) b T X, ∇T X A b N, ∇N = 1 ϕTr log A . (7.1.12) log A 2 By using the same deformation argument as before, from (7.1.12), we get the second identity in (7.1.6).

7.2 The vector bundle of spinors on X and the Dirac operator Here we will assume K to be connected and simply connected, and that p is even dimensional and oriented. Then the adjoint representation K → Aut (p) preserves the orientation of p, and lifts to a unitary representation p p p ρS : K → Auteven (S p ), where S p = S+ ⊕ S− is the Z2 -graded vector space TX p TX of ⊕ S− of p-spinors. Then S descends to the vector bundle S T X = S+ TX T X, g spinors. Here c (p) still denotes the Clifford algebra of (p, B|p ). To avoid confusion with earlier parts of the book, if e ∈ p, c (e) will denote the action of e ∈ c (p) on S p . More generally, the objects attached to c (p) that were considered in chapter 1 will be overlined when acting on S p . By (1.1.9), if f ∈ k, p

ρS (f ) = c (ad (f ) |p ) . ∞

X

X, S Let D be the Dirac operator acting on C is an orthonormal basis of T X, then m X TX DX = c (ei ) ∇Sei ⊗F .

TX

(7.2.1) ⊗ F . If e1 , . . . , em

(7.2.2)

i=1

The operator DX can be written in matrix form with respect to the obvious splitting of C ∞ S T X ⊗ F , so that X 0 D− DX = . (7.2.3) X D+ 0

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Let e1 , . . . , em+n be a basis of g taken as in (2.5.2). Let C g,X be the p Casimir operator acting on C ∞ X, S T X ⊗ F . Let C k,S ⊗E be the Casimir operator for k associated with the representation of K on S p ⊗ E. Of course this Casimir operator is still associated with the restriction of B to k, so that C k,S

p

⊗E

m+n X

=

2 c (ad (ei ) |p ) + ρE (ei ) .

(7.2.4)

i=m+1

Recall that for e ∈ p, ad (e) acts as a symmetric endomorphism of g with respect to its scalar product, which exchanges p and k. Then we can rewrite (7.2.4) in the form !2 m+n X 1 X E k,S p ⊗E hei , [ej , ek ]i c (ej ) c (ek ) − ρ (ei ) . (7.2.5) C = 4 i=m+1 1≤j,k≤m

As in the last identity in (2.12.17), we get C g,X = C g,H + C k,S

p

⊗E

.

(7.2.6)

k,E

Recall that C ∈ End (E) was defined in (2.5.8). Let ∆H,X be the Bochner Laplacian acting on C ∞ X, S T X ⊗ F . We have the Lichnerowicz formula RX 1 X DX,2 = −∆X,H + + c (ei ) c (ej ) RF (ei , ej ) . (7.2.7) 4 2 1≤i,j≤m

X

Let L

be the operator defined in (2.13.3), with E replaced by S p ⊗ E. Then

1 g,X 1 ∗ g g C + B (κ , κ ) . (7.2.8) 2 8 Now we prove an identity in the spirit of an identity in [W88, section 9.3, Lemma]. LX =

Theorem 7.2.1. The following identity holds: 1 DX,2 1 = LX − B ∗ κk , κk − C k,E . 2 8 2

(7.2.9)

Proof. Let Dg be the obvious analogue of the operator in (2.7.2), in which the Clifford variables c (ei ) have been replaced by the c (ei ), which are now defined for 1 ≤ i ≤ m + n. Put g DH =

m X

c (ei ) ei , i=1 m+n X

DVg = −

(7.2.10)

1 c (ei ) (ei + c (ad (ei ) |p )) + c κk , 2 i=m+1

so that as in (2.7.5), (2.7.6), g Dg = DH + DVg .

(7.2.11)

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By Theorem 2.7.2, 1 Dg,2 = C g + B ∗ (κg , κg ) , 4

g,2 Dg,2 = DH + DVg,2 .

(7.2.12)

g Clearly the operator DH descends to the operator DX . On the other hand g the operator DV descends to an operator DVg,X given by

DVg,X =

m+n X

1 c (ei ) ρE (ei ) + c κk . 2 i=m+1

(7.2.13)

By Theorem 2.7.2 applied to K, we get 1 DVg,X,2 = C k,E + B ∗ κk , κk . (7.2.14) 4 By (7.2.8), (7.2.12) and (7.2.14), we get (7.2.9). The proof of our theorem is completed. Put 1 k k,k 1 k,E Tr C − C . (7.2.15) 48 2 Then A has the properties of A given in section 4.4. By (2.6.7), (4.4.1), (7.2.9), and (7.2.15), we get A=−

DX,2 = LX A. 2

(7.2.16)

7.3 The McKean-Singer formula on Z This section serves only as a motivation for the next sections. We make the assumptions of section 4.8 and we use the corresponding notation. In particular Z is a compact orbifold. The bundle of T X-spinors S T X descends to the bundle of T Z-spinors S T Z . X The Dirac operator operator DZ , which acts D descends to the Dirac ∞ TZ X on C Z, S ⊗ F . Similarly the operator LA descends to an operator Z . The operator D+ is a Fredholm operator. When not contain LZ A Γ does Z Z elliptic elements, Z is smooth, and the index Ind D+ of D+ is given by the Atiyah-Singer index formula Z Z b (T X) ch (F ) . Ind D+ = A (7.3.1) Z

Z In the case where Z is an orbifold, Ind D+ is given by the Kawasaki formula of [Ka79]. This formula of Kawasaki is an elaboration of the fixed point formula of Atiyah-Bott [ABo67, ABo68]. Z Z The McKean-Singer formula [McKS67] for the index ind D+ asof D+ serts that for any t > 0, Z Trs exp −tDZ,2 = Ind D+ . (7.3.2)

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By (7.2.16), (7.3.2), for t > 0, Z Trs exp −tLZ = Ind D+ , (7.3.3) A Z and Ind D+ is given by (7.3.1). On the other hand, Theorem 6.1.1 gives us an explicit formula for the lefthand side of (7.3.3). In the next sections, we will verify that Theorem 6.1.1 is compatible with (7.3.3) and with the index formulas of Atiyah-Singer and Kawasaki.

7.4 Orbital integrals and the index theorem We make again the same assumptions as in sections 7.1 and 7.2. We may disregard the considerations we made on Γ and the manifold Z. Let η be the unit volume form on p or on T X that defines the orientation of p or of T X. If α ∈ Λ· (p∗ ) or if α ∈ Λ· (T ∗ X), for 0 ≤ p ≤ m, let α(p) be the component of α in Λp (p∗ ) or in Λp (T ∗ X). Let αmax ∈ R be defined by α(m) = αmax η.

(7.4.1)

Let γ be a semisimple element of G as in (4.2.1). Theorem 7.4.1. If γ ∈ G is nonelliptic, i.e., if a 6= 0, for Y0k ∈ k (γ), h p i p p Trs S %S k −1 exp −iρS Y0k = 0, (7.4.2) and for any t > 0, Trs [γ] exp −tDX,2 /2 = 0.

(7.4.3)

Moreover, for any t > 0, Trs [1] exp −tDX,2 /2 = Trs S

p

⊗E

h

Z

1 m/2

J1 Y0k

k (2πt) i 2 exp −iρS ⊗E Y0k − tA exp − Y0k /2t

p

dY0k n/2

(2πt) h imax b T X, ∇T X ch F, ∇F = A . (7.4.4)

Proof. Recall that a ∈ p is fixed by Ad (k), and also that ad Y0k vanishes on a. Since a 6= 0, 1 is an eigenvalue of Ad k −1 exp −iad Y0k . Also it is well-known that h p i p p Trs S %S k −1 exp −iρS Y0k is a square root of det 1 − Ad k −1 exp −iad Y0k p , which vanishes. This proves (7.4.2). By equation (6.1.2) in Theorem 6.1.1 and by (7.4.2), we get (7.4.3). The first part of (7.4.4) follows from Theorem 6.1.1. The proof of the second part is deferred to section 7.5.

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7.5 A proof of (7.4.4) First, we assume that K is connected, simply connected and simple. Let T ⊂ K be a maximal torus in K and let t ⊂ k be its Lie algebra. Let W be the Weyl group. Let R ⊂ t∗ be the root system, and let CR ⊂ t be the system of coroots, so that T ' t/CR. Our conventions are such that if ∗ α ∈ R, h ∈ CR, hα, hi ∈ Z. Let CR ⊂ t∗ be the dual lattice to CR, so that ∗ if h ∈ CR, λ ∈ CR , hλ, hi ∈ Z. Let R+ ⊂ R be a system of positive roots, and let C+ ∈ t be the positive ∗ Weyl chamber. Let P++ ⊂ CR be the system of dominant weights. Set 1 X ρ= α. (7.5.1) 2 α∈R+

As is well-known, ρ ∈ P++ . Let π : t → C be the polynomial function Y π (t) = h2iπα, ti .

(7.5.2)

α∈R+

Let σ : T → C be the denominator in Weyl’s character formula Y σ (t) = (exp (iπ hα, ti) − exp (−iπ hα, ti)) .

(7.5.3)

α∈R+

Proposition 7.5.1. The following identity holds: 1 1 2 (7.5.4) 4π 2 |ρ| = − Trk C k,k = − B ∗ κk , κk . 24 4 Proof. The first identity is Kostant’s strange formula [Ko76]. The second one is just equation (2.6.7). In the sequel, we will assume that ρE is an irreducible representation of K of highest weight λ ∈ P++ . Let χλ : K → C be the corresponding character, so that if g ∈ K, (7.5.5) χλ (g) = TrE ρE (g) . Proposition 7.5.2. The following identity holds: 2

A = 2π 2 |ρ + λ| .

(7.5.6)

Proof. Since ρE is the representation of K of highest weight λ, we get 2 2 C k,E = −4π 2 |ρ + λ| − |ρ| . (7.5.7) By (7.2.15), (7.5.4), and (7.5.7), we get (7.5.6). If A ∈ End (p) is antisymmetric, let Pf [A] be the Pfaffian of A. Then Pf [A] is a polynomial function of A, which is a square root of det [A]. The form ωA ∈ Λ2 (p∗ ) associated with A is given by U, V ∈ p → hU, AV i. Then Pf [A] is given by max

Pf [A] = [exp (ωA )]

.

(7.5.8)

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Theorem 7.5.3. For t > 0, 2 h exp −2π 2 t |ρ + λ| Z i p p J1 Y0k Trs S ⊗E exp −iρS ⊗E Y0k m/2 k (2πt) 2 dY k h imax 0 b T X, ∇T X ch F, ∇F exp − Y0k /2t . (7.5.9) = A n/2 (2πt) Proof. We may and we will assume that t = 1. Clearly, h i p p m Trs S ⊗E exp −iρS ⊗E Y0k = Trs S exp −ic ad Y0k . (7.5.10) TrE exp −iρE Y0k Also we have the fundamental identity −1 p b Trs S exp −ic ad Y0k = Pf ad Y0k |p A iad Y0k |p . (7.5.11) Set L=

1 (2π)

m/2

Z

h i p p J1 Y0k Trs S ⊗E exp −iρS ⊗E Y0k

k

2 exp − Y0k /2

dY0k n/2

(2π)

. (7.5.12)

By (5.5.11) and (7.5.10)–(7.5.12), we get Z −1 1 b L= Pf ad Y0k |p A iad Y0k |k TrE exp −iρE Y0k m/2 k (2π) 2 dY k 0 exp − Y0k /2 . (7.5.13) n/2 (2π) Let e1 , . . . , em be an oriented orthonormal basis of p. The 2-form ωad(Y k ) 0 on p associated with ad Y0k |p is given by 1 X k i ωad(Y k ) = ei , Y0 , ej e ∧ ej . (7.5.14) 0 2 1≤i,j≤m

Recall that the curvature Ω ∈ Λ2 (p∗ ) ⊗ k of the canonical connection on the K-principal bundle p : G → G/K is given by (2.1.10), so that 1 X Ω=− ei ∧ ej ⊗ [ei , ej ] . (7.5.15) 2 1≤i,j≤m

One can rewrite (7.5.14) in the form

ωad(Y k ) = − Y0k , Ω .

(7.5.16)

0

In (7.5.16), the scalar product is taken in k and only involves the k component of Ω.

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Using (7.5.8), (7.5.16), we get

max Pf ad Y0k |p = exp − Y0k , Ω .

(7.5.17)

In (7.5.17), the right-hand side is the coefficient of e1 ∧ . . . ∧ em in the expansion of the given form. By (7.5.13), (7.5.17), we obtain " 2 exp |Ω| /2 Z b−1 iad Y0k |k TrE exp −iρE Y0k A L= m/2 k (2π) # dY k max 2 0 . (7.5.18) exp − Y0k + Ω /2 n/2 (2π) 2

In (7.5.18), |Ω| ∈ Λeven (p∗ ) refers to the square of the norm of Ω where only the component of Ω in k is taken into account, while the forms in Λeven (p∗ ) are multiplied as usual. We claim that 2

|Ω| = 0.

(7.5.19)

Indeed 2

|Ω| = −

1 4

X

h[[ei , ej ] , ek ] , el i ei ∧ ej ∧ ek ∧ el .

(7.5.20)

1≤i,j,k,l≤m

Using the Jacobi identity, we get (7.5.19). Recall that ∆k is the standard Laplacian in k. From (7.5.18), (7.5.19), we get h imax −1 1 b L= exp ∆k /2 A iad Y0k |k χλ exp −iY0k . (−Ω) m/2 (2π) (7.5.21) Let ∆t be the Laplacian on t. It is well-known that when acting on Adinvariant functions on k, we have the identity of operators, 1 (7.5.22) ∆k = ∆t π. π The function b−1 iad Y0k |k χλ exp −iY0k A is Ad-invariant. By (7.5.21), (7.5.22), we get " −m/2 1 exp ∆t /2 L = (2π) π

b−1 (iad (t) |k ) χλ (exp (−it)) (−Ω) π (t) A

#max . (7.5.23)

In the right-hand side of (7.5.23), the function appearing in the right-hand side of (7.5.23) is viewed as a function of t ∈ t, which is W -invariant, and

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lifts to a central function on k. Of course we do not claim that −Ω takes its values in t, but that since the right-hand side defines a central function on k, it can be evaluated at −Ω. Equation (7.5.23) can also be obtained as a consequence of the Weyl integration formula, and of the localization formulas in equivariant cohomology of Duistermaat-Heckman [DH82, DH83], BerlineVergne [BeVe83] over the generic coadjoint orbits of K. Now if t ∈ t ⊗R C, we have the obvious identity b−1 (ad (t) |k ) = σ (t) . (7.5.24) A π If r+ = |R+ |, from (7.5.24), we get b−1 (iad (t) |k ) = ir+ σ (−it) . π (t) A

(7.5.25)

If w ∈ W , let w = ±1 be the determinant of w acting on t. The Lefschetz formula asserts that X σ (−it) χλ (exp (−it)) = w exp (h2π hρ + λ, wtii) . (7.5.26) w∈W

By (7.5.23)–(7.5.26), we finally obtain h imax −m/2 2 b−1 (−iad (Ω) |k ) χλ (exp (iΩ)) L = (2π) exp 2π 2 |ρ + λ| A . (7.5.27) b (x) is an even function, we can rewrite (7.5.27) in the form Since A max RN RF 2 2 −1 b L = exp 2π |ρ + λ| A − , (7.5.28) Tr exp − 2iπ 2iπ which is just h imax 2 b−1 N, ∇N ch F, ∇F A L = exp 2π 2 |ρ + λ| .

(7.5.29)

By the second identity in (7.1.6), by (7.5.12), and by (7.5.29), we get (7.5.9) for t = 1. The proof of our theorem is completed. Remark 7.5.4. From (7.5.6) and (7.5.9), we claim that one can obtain the last identity in (7.4.4) when K is connected and simply connected. Indeed we can write K in the form s Y K= Ki , (7.5.30) 1

the Ki being simply connected and simple. We may and we will assume E to be an irreducible representation of K. Then E can be written in the form E=

s O

Ei ,

(7.5.31)

i=1

the Ei being irreducible representations of the Ki . By using (7.5.9) for each i, 1 ≤ i ≤ s, we obtain (7.4.4).

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Chapter 7

7.6 The case of complex symmetric spaces For holomorphic locally symmetric spaces and bounded symmetric domains, we refer to Helgason [He78, Chapter VIII]. We will assume that p is equipped with a complex structure J p such that if a, b ∈ p, B (J p a, J p b) = B (a, b) .

(7.6.1)

Let p+ , p− ∈ p ⊗R C be the eigenspaces associated with the eigenvalues i, −i of J p , so that p ⊗R C = p+ ⊕ p− .

(7.6.2)

In particular p+ is now equipped with a Hermitian product. Also we assume that [p+ , p+ ] = 0, [p− , p− ] = 0.

(7.6.3)

Finally, we assume that the action of K on p preserves the complex structure J p , so that K acts on p+ by unitary automorphisms. Let ρdet(p+ ) : K → S 1 be the determinant of the representation. The manifold X = G/K is now a complex Kähler manifold. Indeed (7.6.2) guarantees that the almost complex structure on T X induced by J p is integrable. Since the connection ∇T X is torsion free and preserves the complex structure, X is K¨ ahler. Then p+ descends to the holomorphic tangent space T (1,0) X of X. As it should be, by (2.1.10) and (7.6.3), the curvature Ω is of complex type (1, 1). It follows that if ρE = K → Aut (E) is a unitary representation of K, the corresponding vector bundle F on X is a holomorphic Hermitian vector bundle. Also G acts on the left by holomorphic isometries of X. If Γ is a cocompact torsion-free subgroup of G, Z = Γ \ X is a compact Kähler manifold, if Γ is cocompact but not torsion free, Z is a Kähler orbifold. We do not assume any more K to be simply connected. Still K acts uni tarily on Λ· p∗− and preserves the grading. In the constructions of sections 7.2–7.3, we may as well replace S p by Λ· p∗− . The manifolds X and Z are not necessarily spin, but they are equipped with a Spinc structure. We can still define the operators DX as before. By [Hi74], we know that √ X X∗ . (7.6.4) DX = 2 ∂ + ∂ If Γ is a discrete group taken as before, a similar formula holds on Z. Let χ (Z, F ) be the Euler characteristic of F . Then χ (Z, F ) is the index of the Dirac operator DZ . If Z is smooth, the Riemann-Roch-Hirzebruch formula asserts that Z χ (Z, F ) = Td T (1,0) Z ch (F ) . (7.6.5) Z

We claim that a full analogue of the results of sections 7.2–7.5 can be 1/2 developed in this case. Let us briefly explain this point. Let ρdet(p+ ) → S 1

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det(p) be the square , which is well-defined up to sign. Let λ = root of ρ (1,0) det T Z be the line bundle on Z corresponding to the representation ρdet(p+ ) , and let λ1/2 be a locally defined square root. We still can define A as in (7.2.15), with E replaced by E ⊗ det1/2 (p+ ), since the Casimir operator is only obtained from the representation of the Lie algebra. The analogue of (7.4.2) is that if a 6= 0, Y0k ∈ k (γ), h i · ∗ · ∗ Tr Λ (p− ) Ad k −1 exp −iρΛ (p− ) Y k = 0, (7.6.6) s

0

the proof being the same as in Theorem 7.4.1. The analogue of (7.4.4) says that for any t > 0, Z 1 [1] X,2 J1 Y0k exp −tD /2 = Trs m/2 k (2πt) h i · ∗ Λ· (p∗ ⊗E Λ ) − Tr exp −iρ (p− )⊗E Y k − tA s

2 exp − Y0k /2t

0

dY0k n/2

(2πt)

h

= Td T (1,0) X, ∇T

(1,0)

X

ch F, ∇F

imax

.

(7.6.7) The proof essentially consists in extending the arguments of section 7.5. Note that K is no longer supposed to be simply connected. Let ZK be the center of K and let zK be its Lie algebra. Let ks be the semisimple part of k, so that we have the orthogonal splitting of Lie algebras, k = zK ⊕ ks . (7.6.8) The Q splitting (7.6.8) allows us to reduce the proof to the case where K = s T × i=1 Ki , with T a torus and the Ki being simple groups. The proof proceeds as in Remark 7.5.4.

7.7 The case of an elliptic element We still assume K to be simply connected. Let γ ∈ G be elliptic. We may and we will assume that γ ∈ K, so that γ = k −1 , k ∈ K. Then X (γ) ⊂ X is just the fixed point set of k. Note that γ acts naturally on T X|X(γ) . Moreover, T X (γ) is just the eigenspace of this action associated with the eigenvalue 1, and γ acts on NX(γ)/X as an isometry. The distinct angles ±θ1, . . . , ±θs , 0 < θi ≤ π are exactly the nonzero angles of the action of Ad k −1 on p. Let NX(γ)/X,θi , 1 ≤ i ≤ s be the part of NX(γ)/X on which Ad (γ) acts by a rotation of angle θi . The action of γ on T X|X(γ) is parallel. Therefore ∇T X induces metric connections on the above subbundles of T X|X(γ) . Let RT X(γ) , RNX(γ)/X,θi , 1 ≤ i ≤ s be the curvatures of these connections on T X (γ) , NX(γ)/X,θi , 1 ≤ i ≤ s. If θ ∈ R \ 2πZ, set 1 bθ (x) = . (7.7.1) A 2 sinh x+iθ 2

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Chapter 7

bθ (x) with the corresponding multiplicative genus. Put Given θ, we identify A bγ T X|X(γ) , ∇T X|Xγ = A b T X (γ) , ∇T X(γ) A r Y

bθi NX(γ)/X,θ , ∇NX(γ)/X,θi . (7.7.2) A i

i=1

bγ N |X(γ) , ∇N |X(γ) by a similar formula. We can also define the closed form A bγ|p (0) be the component of degree 0 of A bγ T X|X(γ) , ∇T X|Xγ , and Let A bγ|k (0) be the component of degree 0 of A b N |X(γ) , ∇N |X(γ) . These are let A constants on X (γ). Put bγ (0) = A bγ|p (0) A bγ|k (0) . A

(7.7.3)

Similarly, set ch

γ

F |X(γ)

F |X(γ) , ∇

" = Tr ρ

E

k

−1

RF |X(γ) exp − 2iπ

!# .

(7.7.4)

The closed forms in (7.7.2), (7.7.4) on X (γ) are exactly the ones that appear in the Lefschetz fixed point formula of Atiyah-Bott [ABo67, ABo68]. Note that there are questions of signs to be taken care of, because of the need to distinguish between θi and −θi . We refer to the above references for more detail. By proceeding as in the proof of Proposition 7.1.1, one can easily prove the analogue of (7.1.6), i.e., we get the identity of differential forms on X (γ), chγ T X|X(γ) , ∇T X|X(γ) + chγ N |X(γ) , ∇N |X(γ) = Trg Ad k −1 , (7.7.5)

bγ T X|X(γ) , ∇T X|X(γ) A bγ N |X(γ) , ∇N |X(γ) = A bγ (0) . A Now we prove an analogue of Theorem 7.4.1. Here the notation to forms on X (γ).

max

refers

Theorem 7.7.1. If γ = k −1 , k ∈ K, for any t > 0, Trs [γ] exp −tDX,2 /2 Z h p i p p 1 = Jγ Y0k Trs S ⊗E ρS ⊗E k −1 exp −iρS ⊗E Y0k − tA p/2 k(γ) (2πt) 2 dY k 0 exp − Y0k /2t q/2 (2πt) h imax bγ T X|X(γ) , ∇T X|X(γ) chγ F, ∇F = A . (7.7.6) Proof. The first part of the identity follows from Theorem 6.1.1. So we concentrate on the proof of the second part. We may and we will assume that

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t = 1. The proof proceeds very much as the proof of Theorems 7.4.1 and 7.5.3. First, we work under the assumptions of Theorem 7.5.3. We may and we will assume that k lies in a maximal torus T ⊂ K, whose Lie algebra is still denoted by t. Let L be the obvious analogue of the expression in (7.5.12). Instead of (7.5.11), we now have the identity h p i p Trs S ρS k −1 exp −ic ad Y0k −1 k −1 keiY0 |p⊥ (γ) k k b b = Pf ad Y0 |p(γ) A iad Y0 |p(γ) A (0) . (7.7.7) By (5.4.10), (5.4.11), (5.5.5), (7.5.12), and (7.7.7), we get dim p⊥ (γ)/2 Z (−1) Pf ad Y0k |p(γ) L= p/2 k(γ) (2π) k| ⊥ b p (γ) (0) b A−1 iad Y0k |k(γ) A " #1/2 det 1 − exp −iad Y0k Ad k −1 |k⊥ (γ) det (1 − Ad (k −1 )) |k⊥ (γ) 2 dY k 0 exp − Y0k /2 TrE ρE k −1 exp −iρE Y0k . (7.7.8) q/2 (2π) dim p⊥ (γ)/2

Note that the factor (−1) in the right-hand side of (7.7.8) comes from an identity for the expression in (5.4.10), which gives φ + iφ0 iφ iφ − φ0 φ sin = −4 sinh sinh . (7.7.9) 4 sin 2 2 2 2 Let Ωz(γ) be the analogue of Ω in (2.1.10), when replacing g by z (γ). By proceeding as in the proof of Theorem 7.5.3, from (5.4.10) and (7.7.8), we get the following analogue of (7.5.27), 2 exp 2π 2 |ρ + λ| ⊥ dim p (γ)/2 L = (−1) p/2 (2π) −1 h imax k k z(γ) −1 z(γ) b b A (0) A −iad Ω |k χλ k exp iΩ . (7.7.10) By the second identity in (7.7.5), we get −1 bk (0) A bk bk −iad Ωz(γ) |p . A −iad Ωz(γ) |k = A

(7.7.11)

Finally, we have the trivial dim p⊥ (γ)/2 bk bk−1 iad Ωz(γ) |p . (−1) A −iad Ωz(γ) |p = A

(7.7.12)

By (7.7.10)–(7.7.12), we obtain 2 h iimax exp 2π 2 |ρ + λ| h bγ iad Ωz(γ) |p χλ k −1 exp iΩz(γ) A . L= p/2 (2π) (7.7.13)

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Chapter 7

By (7.7.13), we get (7.7.6). To complete the proof of (7.7.6) in full generality, we proceed as in Remark 7.5.4. The proof of our theorem is completed. Remark 7.7.2. Using equation (4.8.26), we find that Theorem 7.7.1 is compatible with the Kawasaki formula [Ka79].

7.8 The de Rham-Hodge operator Let Ω· (X) , dX be the de Rham complex of smooth forms on X with compact support. Let dX∗ be the formal adjoint with respect to the obvious L2 product. Set DX = dX + dX∗ .

(7.8.1) Then DX,2 = dX , dX∗ is the Hodge Laplacian of X. The operator LX associated with E = Λ· (p∗ ) acts on Ω· (X). First, we establish a special case of Theorem 7.2.1. Proposition 7.8.1. The following identity holds: 1 1 DX,2 = LX − B ∗ κk , κk − Trp C k,p . 2 8 16

(7.8.2)

Proof. We will assume temporarily that K is simply connected, and that p is even dimensional. Then ∗

b p . Λ· (p∗ ) = S p ⊗S

(7.8.3)

It follows that DX is exactly a Dirac operator of the kind that was considered in section 7.2, with E = S p∗ . Moreover, C k,S

p∗

m+n X

=

2

b c (ad (ei ) |p ) .

(7.8.4)

i=m+1

By (1.1.11) and (7.8.4), we get C k,S

p∗

=

m+n 1 X 16 i=m+1

X

h[ei , ej ] , [ek , el ]i b c (ei ) b c (ej ) b c (ek ) b c (el ) . (7.8.5)

1≤i,j≤m 1≤k,l≤m

Using (2.6.8) and the Jacobi identity in (7.8.5), we finally obtain C k,S

p∗

=

1 p k,p Tr C . 8

(7.8.6)

From (7.2.9) and (7.8.6), we get (7.8.2). When K is not simply connected, the above arguments remain valid. When p is odd dimensional, the proof of (7.8.2) is essentially the same.

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Put β=−

1 k k,k 1 Tr C − Trp C k,p . 48 16

(7.8.7)

By (7.8.6), β is just A in (7.2.15) when E = S p∗ . By (7.2.16), (7.8.6), equation (7.8.2) can be written in the form DX,2 = LX β . 2

(7.8.8)

Let e T X, ∇T X be the Euler form of T X that is associated with the connection ∇T X . If p is even-dimensional, then TX R . (7.8.9) e T X, ∇T X = Pf 2π If p is odd dimensional, then e T X, ∇T X vanishes identically. Now we establish a special case of Theorems 7.4.1 and 7.7.1, when S p ⊗ E is replaced by Λ· (p∗ ). In view of (7.8.3), this means that when K is simply connected and p is even dimensional, E = S p∗ . Note that if Y0k ∈ k, then h i · ∗ · Trs Λ (p ) exp −iρΛ (p∗) Y0k = det 1 − exp iad Y0k |p . (7.8.10) In particular if p is odd dimensional, (7.8.10) vanishes identically. Let γ be a semisimple element of G as in (4.2.1). Theorem 7.8.2. If γ ∈ G is nonelliptic, for any t > 0, Trs [γ] exp −tDX,2 /2 = 0.

(7.8.11)

For any t > 0, the following identities hold: Trs [1] exp −tDX,2 /2 =

exp (−tβ) m/2

(2πt)

Z

J1 Y0k det 1 − exp iad Y0k |p

k

exp − Y0k /2t

dY0k (2π)

n/2

max = e T X, ∇T X . (7.8.12)

If γ ∈ G is elliptic, for any t > 0, imax h Trs [γ] exp −tDX,2 /2 = e T X (γ) , ∇T X(γ) .

(7.8.13)

Proof. By proceeding as in the proof of Theorem 7.4.1, we get (7.8.11). If p is odd dimensional, the last two expressions in (7.8.12) vanish identically. If p is even dimensional, then (7.8.12) is a consequence of (7.4.4), (7.8.8), and (7.8.10). Equation (7.8.13) follows from Theorem 7.7.1.

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Chapter 7

7.9 The integrand of de Rham torsion ·

∗

Recall that N Λ (p ) is the number operator of Λ· (p∗ ), which acts by multiplication by j on Λj (p∗ ). If g is an isometry of p, · ∗ Trs Λ (p ) [g] = det 1 − g −1 , (7.9.1) h i · ∗ · ∗ ∂ det 1 − g −1 eb (0) . Trs Λ (p ) N Λ (p ) g = ∂b If the eigenspace associated with the eigenvalue 1 is of dimension ≥ 1, the first quantity in (7.9.1) vanishes. If it is of dimension ≥ 2, the second expression in (7.9.1) also vanishes. Also if m is even and g preserves the orientation, then h · ∗ · ∗ m i g = 0. (7.9.2) Trs Λ (p ) N Λ (p ) − 2 Now we use the notation of section 7.8. In particular DX = dX + dX∗ . Let γ ∈ G be a semisimple element, which is written as in (4.2.1). By equation (6.1.2) in Theorem 6.1.1, we get h · ∗ i m Trs [γ] N Λ (T X) − exp −tDX,2 /2 2 Z 1 2 Jγ Y0k exp −tβ − |a| /2t = p/2 k(γ) (2πt) h i dY0k · ∗ · ∗ m Trs Λ (p ) N Λ (p ) − . (7.9.3) exp −iad Y0k Ad k −1 q/2 2 (2πt) Let T ⊂ K be a maximal torus, and let t ⊂ k be its Lie algebra. Set b = {e ∈ p, [e, t] = 0} .

(7.9.4)

h = b ⊕ t.

(7.9.5)

Put

By [Kn86, p. 129], h is a Cartan subalgebra of g. Also dim t is the complex rank of K, and dim h is the complex rank of G. If m is odd, then b is of odd dimension ≥ 1. Theorem 7.9.1. If m is even, or if m is odd and dim b ≥ 3, for any t > 0, h · ∗ i m Trs [γ] N Λ (T X) − exp −tDX,2 /2 = 0. (7.9.6) 2 Proof. This follows from the considerations which follow (7.9.1), from (7.9.2), and (7.9.3). Remark 7.9.2. For p, q ∈ N, let SO0 (p, q) be the connected component of the identity in the real group SO (p, q). By [He78, Table V p. 518] and [Kn86, Table C1 p. 713, and Table C2 p. 714], among the noncompact simple connected complex groups such that m is odd and dim b = 1, there is only

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SL2 (C), and among the noncompact simple real connected groups, there are only SL3 (R), SL4 (R), SL2 (H), and SO0 (p, q) with pq odd > 1. Also by [He78, pp. 519, 520], sl2 (C) = so (3, 1), sl4 (R) = so (3, 3), and sl2 (H) = so (5, 1). Therefore the above list can be reduced to SL3 (R) and SO0 (p, q) with pq odd > 1.1 Let Z be a compact locally symmetric space as in section 4.8. Let DZ be the analogue of DX . The Ray-Singer analytic torsion of Z is the derivative · ∗ Z,2 /2 . at 0 of the Mellin transform of −Trs N Λ (T Z) − m 2 exp −tD Now we recover a result of Moscovici-Stanton [MoSt91, Corollary 2.2]. Theorem 7.9.3. If m is even, or if m is odd and dim b ≥ 3, for any t > 0, h · ∗ i m exp −tDZ,2 /2 = 0. Trs N Λ (T Z) − (7.9.7) 2 Proof. This follows from equations (4.8.12), (4.8.26), from Theorem 7.9.1, and Remark 7.9.2.

1I

am indebted to Yves Benoist for providing the above information.

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Chapter Eight The case where [k (γ) , p0 ] = 0 The purpose of this chapter is to evaluate explicitly the Gaussian integral that appears in the right-hand side of our formula in (6.1.2) for the orbital integrals of the heat kernel, when γ is nonelliptic and [k (γ) , p0 ] = 0. Our computations can be easily extended to the more general kernels considered in chapter 6. It is remarkable that the index formulas of chapter 7 play here a key role. This chapter is organized as follows. In section 8.1, we consider the case where G = K. In section 8.2, we compute explicitly the Gaussian integral when γ is nonelliptic and [k (γ) , p0 ] = 0. Finally, in section 8.3, the case where G = SL2 (R) is worked out. We recover the evaluation in [McK72] of the trace of the scalar heat kernel in Selberg’s trace formula. We make the same assumptions as in chapters 6 and 7, and we use the corresponding notation.

8.1 The case where G = K In this section, we assume that G = K, so that g = k, p = 0. Otherwise we make the same assumptions as in chapter 6, and we use the corresponding notation. Here X = G/K is reduced to a point. If γ = 1, k (γ) = k,

k⊥ (γ) = 0.

(8.1.1)

By (5.5.11), if Y0k ∈ k, b−1 iad Y k . J1 Y0k = A 0

(8.1.2)

We still define A as in (7.2.15). Proposition 8.1.1. For any t > 0, the following identity holds: Z 2 dY k 0 J1 Y0k TrE exp −iρE Y0k − tA exp − Y0k /2t n/2 k (2πt) = dim E. (8.1.3) Proof. This is a special case of equation (7.4.4) in Theorem 7.4.1.

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The case where [k (γ) , p0 ] = 0

Now we make the same assumptions and we use the same notation as before, except that K is no longer supposed to be connected. Let K 0 be the −1 connected component of the identity. Let γ = k ∈ K. k k For Y0 ∈ k (γ), we define Jγ Y0 as in (5.5.5), with a = 0, p = 0. Namely, b−1 iad Y0k |k(γ) Jγ Y0k = A #1/2 " det 1 − exp −iad Y0k Ad k −1 |k⊥ (γ) . (8.1.4) det (1 − Ad (k −1 )) |k⊥ (γ) By (5.4.10), we can rewrite (8.1.4) in the form −1 bk A iad Y0k Jγ Y0k = . −1 bk A (0)

(8.1.5)

Let ρE : K → Aut (E) be a finite dimensional unitary representation of K. We still define A as in (7.2.15). Now we establish the following extension of Proposition 8.1.1. Theorem 8.1.2. For any t > 0, Z Jγ Y0k TrE ρE k −1 exp −iρE Y0k − tA k(γ)

2 exp − Y0k /2t

dY0k q/2

(2πt)

= TrE ρE k −1 . (8.1.6)

Proof. If k ∈ K 0 , then (8.1.6) is just a special case of (7.7.6) in Theorem 7.7.1. Now we will consider the general case, where k does not necessarily lie in K 0 . We will indeed use the formalism of chapter 2 in the case where G = K, so that X = G/K is reduced to a point. Clearly, LX = 0.

(8.1.7)

Then (8.1.6) is just a special case of (6.1.2) in Theorem 6.1.1. Indeed none of the arguments going into the proof ever uses the fact that K is connected.

8.2 The case a 6= 0, [k (γ) , p0 ] = 0 We take γ ∈ G semisimple as in (4.2.1). We assume that γ is nonelliptic, i.e., a 6= 0. We also assume that [k (γ) , p0 ] = 0.

(8.2.1)

Note that if G is of real rank 1, then p0 is the vector subspace generated by a, so that (8.2.1) holds. Set K0 = K ∩ Z (a) .

(8.2.2)

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Then K0 is a Lie subgroup of K with Lie algebra k0 . Moreover, k ∈ K0 . By (5.5.5), (8.2.1), we get 1 1 Jγ Y0k = 1/2 det (1 − Ad (k −1 )) |p⊥ 0 (γ) det (1 − Ad (γ)) |z⊥ 0 #1/2 " det 1 − exp −iad Y0k Ad k −1 |k⊥ (γ) −1 k 0 b . A iad Y0 |k(γ) det (1 − Ad (k −1 )) |k⊥ 0 (γ) (8.2.3) We equip k0 with the bilinear form Bk0 . This form is negative on k0 . Let C k0 be the corresponding Casimir. Since K0 acts on is a corresponding k0 , there operator C k0 ,k0 ∈ End (k0 ). In particular Trk0 C k0 ,k0 is well-defined. Similarly, we can also define the operator C k0 ,E . Let A[γ] be the analogue of A for the group K0 . By (7.2.15), we get 1 1 A[γ] = − Trk0 C k0 ,k0 − C k0 ,E . (8.2.4) 48 2 Theorem 8.2.1. For any t > 0, 2 exp − |a| /2t 1 = Tr[γ] exp −tLX 1/2 A det (1 − Ad (k −1 )) |p⊥ 0 (γ) det (1 − Ad (γ)) |z⊥ 0 1 (2πt)

p/2

TrE ρE k −1 exp −t A − A[γ] . (8.2.5)

Proof. Again, we use equation (6.1.2) in Theorem 6.1.1, and also equation (8.2.3). We combine these results with equation (8.1.6) in Theorem 8.1.2 applied to the Lie group K0 , and we get (8.2.5). 8.3 The case where G = SL2 (R) Assume that G = SL2 (R) , K = S 1 . We equip the Lie algebra g with the form B that is half of the Killing form. If α ∈ R, set α/2 0 a= . (8.3.1) 0 −α/2 Then 2 |a| = α2 . (8.3.2) If γ = ea , then 1/2 = 2 sinh (|α| /2) . (8.3.3) det (1 − Ad (γ)) |z⊥ 0 Let X = SL2 (R) /S 1 be the symmetric space associated with SL2 (R). Then X is the upper-half space in C. With the above conventions, its scalar curvature S X is equal to −2. By (2.6.11), B ∗ (κg , κg ) = −1. (8.3.4)

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The case where [k (γ) , p0 ] = 0

We take E to be the trivial representation. Let ∆X be the LaplaceBeltrami operator on X. By (2.13.2), (2.13.3), and (8.3.4), 1 1 LX = − ∆X − . 2 8 By (5.5.11), when Y0k ∈ k = R, we get b Y0k . J1 Y0k = A

(8.3.5)

(8.3.6)

By equation (6.1.2) in Theorem 6.1.1, by (8.3.5), and (8.3.6), we obtain Z 2 exp (−t/8) dY0k [1] X b Y0k √ Tr exp t∆ /2 = exp − Y0k /2t A . 2πt 2πt R (8.3.7) By (8.2.5) in Theorem 8.2.1, and by (8.3.2)–(8.3.5), if α 6= 0, we get 1 exp −α2 /2t − t/8 . (8.3.8) Tr[γ] exp t∆X /2 = √ 2 sinh (|α| /2) 2πt Equations (8.3.7) and (8.3.8) fit with the evaluation of Selberg’s trace formula for the heat kernel in [McK72, page 233].

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Chapter Nine A proof of the main identity The purpose of this chapter is to establish Theorem 6.1.1. The proof consists in making b → +∞ h in equation i (4.6.1), that is, we evaluate the limit as [γ] X exp −LA,b . b → +∞ of Trs This chapter is organized as follows. In section 9.1, we state various estimates on the hypoelliptic heat kernels, which are valid for b ≥ 1. The proofs of these estimates are deferred to chapter 15. They will be used for dominated convergence in the hypoelliptic orbital integrals as b → +∞. In section 9.2, we make a natural rescaling on the coordinates parametrizing Xb. In section 9.3, we introduce a conjugation on the Clifford variables, which is an analogue of the Getzler rescaling [Ge86] in local index theory. In section 9.4, we show that the norm of the term defining the conjugation can be adequately controlled. X In section 9.5, we introduce a conjugate LX A,b of LA,b and its associated heat kernel. In section 9.6, we obtain the limit as b → +∞ of the rescaled heat kernel. The proof is deferred to sections 9.8–9.11. In section 9.7, we establish Theorem 6.1.1. In section 9.8, we make the change of variables Y T X → aT X + Y T X in the X operator LX A,b , and we obtain a new operator Oa,A,b . In section 9.9, we choose a coordinate system on Xb that is based at x0 = p1 and we trivialize our vector bundles. We obtain this way an operator X Pa,A,b,Y k. 0

X In section 9.10, we show that as b → +∞, Pa,A,b,Y k converges in the proper 0

X sense to an operator Pa,A,∞,Y k acting over p × g. This operator is closely 0 related to the operator Pa,Y0k that was considered in chapter 5. Finally, in section 9.11, we state a result on convergence of heat kernels, which implies the convergence result of section 9.6. The proof is deferred to chapter 15.

X 9.1 Estimates on the heat kernel qb,t away from bia N k −1

X Recall that π b : Xb → X is the obvious projection, and qb,t ((x, Y ) , (x0 , Y 0 )) is the smooth kernel associated with exp −tLX A,b .

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A proof of the main identity

For b > 0, s (x, Y ) ∈ C ∞ Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) , set Fb s (x, Y ) = s (x, −bY ) .

(9.1.1)

−1 X LX A,b = Fb LA,b Fb .

(9.1.2)

Put By (2.13.7) and (4.5.1), we get b4 N T X 2 Y ,Y 2 · ∗ ∗ 1 1 N Λ (T X⊕N ) ∆T X⊕N 2 + + |Y | − (m + n) + − 2 b4 b2 b2 ∞ b )),f ∗,fb C (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F − c iθad Y N − iρE Y N − ∇Y T X

LX A,b =

Let

qX b,t

+ A. (9.1.3) ((x, Y ) , (x , Y )) be the kernel associated with exp −tLX A,b . When 0

0

t = 1, we will write q X instead of q X . b b,1 0 00 Take η > 0, C > 0, C > 0. Then there exists c > 0 such that for Y T X ∈ T X, b ≥ 1, 2 C 0 T X 2 C 0 Y T X + C 00 exp −2η Y T X b4 ≥ Y + c log2 (b) . (9.1.4) 2 TX TX > log (b) /η, and also for Y ≤ Indeed (9.1.4) is obviously true for Y log (b) /η. By (9.1.4), we find that if C 000 > 0 is another constant, there exists d > 0 such that 2 (9.1.5) C 0 Y T X + C 00 exp −2η Y T X b4 − C 000 log (b) ≥ −d. By (9.1.4), (9.1.5), we deduce that there exist c > 0, d > 0, e > 0 such that 2 C 0 T X 2 C 0 Y T X + C 00 exp −2η Y T X b4 − C 000 log (b) ≥ Y 2 C 0 T X 2 C 00 Y + e log2 (b) − d. (9.1.6) + exp −2η Y T X b4 − c ≥ 2 4 In the sequel, the notation of chapter 3 will be in force. Recall the map ργ was defined in Theorem 3.4.1. We use the same trivialization of N as the one before Theorem 3.9.5. Namely, recall that Ad k −1 acts on N |X(γ) . Along geodesics normal to X (γ), we trivialize the vector bundle N using the Euclidean connection ∇N , so that Ad k −1 acts on N fibrewise. 0 Theorem 9.1.1. Given > 0, M > 0, ≤ M , there exist C,M > 0, C,M > 0 0 b 0 such that for b ≥ 1, ≤ t ≤ M , (x, Y ) , (x , Y ) ∈ X , X q b,t ((x, Y ) , (x0 , Y 0 )) 2 2 0 ≤ C,M b4m+2n exp −C,M d2 (x, x0 ) + |Y | + |Y 0 | . (9.1.7)

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Chapter 9

0 Given β > 0, > 0, M > 0, ≤ M , there exist ηM > 0, C,M > 0, C,M > 00 b 0, Cγ,β,M > 0 such that for b ≥ 1, ≤ t ≤ M , (x, Y ) ∈ X , if d (x, X (γ)) ≥ β, X q b,t ((x, Y ) , γ (x, Y )) ≤ C,M b4m+2n 2 0 00 exp −C,M d2γ (x) + |Y | − Cγ,β,M exp −2ηM Y T X b4 . (9.1.8)

There exists η > 0 such that given β > 0, µ > 0, there exist C > 00 0, C 0 > 0, Cγ,β,µ > 0 such that for b ≥ 1, (x, Y ) ∈ Xb, if d (x, X (γ)) ≤ TX T X − a ≥ µ, β, Y X q b ((x, Y ) , γ (x, Y )) ≤ Cb4m+2n 2 00 exp −C 0 |Y | − Cγ,β,µ exp −2η Y T X b4 . (9.1.9) There exist c > 0, C > 0, Cγ0 > 0 such that for b ≥ 1, f ∈ p⊥ (γ) , |f | ≤ 1, x = ργ (1, f ) , Y ∈ (T X ⊕ N )x , Y T X − aT X ≤ 1, 2 X q b ((x, Y ) , γ (x, Y )) ≤ cb4m+2n exp −C Y N 2 2 − Cγ0 |f | + Y T X − aT X b4 − Cγ0 Ad k −1 − 1 Y N b2 ! 0 TX N 2 − Cγ a , Y b . (9.1.10) Proof. The proof of our theorem is deferred to section 15.7. Definition 9.1.2. A kernel K ((x, Y ) , (x0 , Y 0 )) acting on C b Xb, Λ· (T ∗ X ⊕ N ∗ ) ⊗ F is said to be rapidly decreasing if for any k ∈ N, k k 1 + |Y | + |Y 0 | |K ((x, Y ) , (x0 , Y 0 ))|

(9.1.11)

is uniformly bounded. If K also depends on b > 0, t > 0, if D ∈ R∗+ × R∗+ , we say that K is uniformly rapidly decreasing for (b, t) ∈ D if the previous bounds are uniform. Λ· (T ∗ X⊕N ∗ )⊗F,k

If k is a multi-index in 1, . . . , 2m + n, ∇x0 ,Y 0 denotes the co· ∗ ∗ variant derivative of order k with respect to the connection ∇Λ (T X⊕N )⊗F . Also | | denotes the obvious norm. Theorem 9.1.3. For b ≥ 1, ≤ t ≤ M , for any multi-index k, · ∗ Λ (T X⊕N ∗ )⊗F,k X q b,t ((x, Y ) , (x0 , Y 0 )) /b4m+2n+|2k| ∇(x0 ,Y 0 ) is uniformly rapidly decreasing on Xb × Xb. Proof. This result will be established in section 15.8.

(9.1.12)

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9.2 A rescaling on the coordinates (f, Y ) By equation (2.14.4) and by Remark 5.5.2, it is enough to prove (6.1.2) with t = 1. We hstartfrom identity (4.6.1) in Theorem 4.6.1, which says that for b > 0, i [γ] X Trs exp −LA,b is independent of b and equal to Tr[γ] exp −LX A . The h i proof of (6.1.2) will consist in getting the asymptotics of Trs exp −LX A,b as b → +∞. When f ∈ p⊥ (γ), we identify ef and ef p1. Using the same notation as in equation (4.3.10), we get Trs [γ] exp −LX A,b Z h i · ∗ ∗ f f e , Y , γ e , Y r (f ) dY df. = Trs Λ (T X⊕N )⊗F γq X b π b−1 p⊥ (γ)

(9.2.1) Take β ∈]0, 1]. By (3.4.4), (3.4.36), (9.1.6), and by equation (9.1.8) in Theorem 9.1.1, as b → +∞, Z h i Λ· (T ∗ X⊕N ∗ )⊗F X f f Tr e , Y , γ e , Y γq r (f ) dY df s −1 ⊥ b (f,Y )∈b π p (γ) |f |>β

→ 0. (9.2.2)

By (9.1.6), (9.1.9), given β > 0, µ > 0, we may as well obtain a similar result for the integral of h i · ∗ ∗ f f e , Y , γ e , Y Trs Λ (T X⊕N )⊗F γq X r (f ) b over the region considered in (9.1.9). As we saw in section 3.10, the vector bundle N (γ) on X (γ) is the analogue of N on X, and moreover, N (γ) ⊂ N |X(γ) . Let N ⊥ (γ) be the orthogonal to N (γ) in N |X(γ) . Clearly, N ⊥ (γ) = Z 0 (γ) ×K 0 (γ) k⊥ (γ) .

(9.2.3)

We trivialize the vector bundles T X, N by parallel transport along the geodesics orthogonal to X (γ) with respect to the connections ∇T X , ∇N , so that T X, N can be identified with p∗γ T X|X(γ) , p∗γ N |X(γ) . At x = p1, N (γ) = k (γ) ,

N ⊥ (γ) = k⊥ (γ) .

(9.2.4)

Therefore at ργ (1, f ), we may write Y N ∈ N in the form Y N = Y0k + Y N,⊥ ,

Y0k ∈ k (γ) , Y N,⊥ ∈ k⊥ (γ) .

(9.2.5)

Let dY0k , dY N,⊥ be the volume elements on k (γ) , k⊥ (γ), so that dY N = dY0k dY N,⊥ .

(9.2.6)

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Chapter 9

Ultimately, to evaluate the limit of (9.2.1) as b → +∞, given β > 0, we may as well consider the integral Z

·

|f |≤β |Y T X −aT X |≤β

∗

Trs Λ (T

X⊕N ∗ )⊗F

h

r (f ) dY T X dY N df = b−4m−2n+2r " γq X b

γ e

f /b2

TX

,a

+Y

TX

i ef , Y , γ ef , Y

γq X b Z

·

|f |≤βb2 |Y T X |≤βb2

Trs Λ (T

∗

X⊕N ∗ )⊗F

2 ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 , 2

/b

, Y0k

+Y

N,⊥

2

/b

!# r f /b2 dY T X dY0k dY N,⊥ df. (9.2.7)

By (9.1.10), for |f | ≤ βb2 , Y T X ≤ βb2 , we get

b−4m−2n q X b

2 ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 ,

! 2 2 Y N,⊥ 0 k f /b TX TX 2 k N,⊥ 2 γ e ,a +Y /b , Y0 + Y /b ≤ C exp −C Y0 + b2 2 2 − Cγ0 |f | + Y T X − Cγ0 Ad k −1 − 1 Y N,⊥ ! h i 0 TX 2 k N,⊥ − Cγ aef /b2 , b Y0 + Y . (9.2.8) Clearly, k Y N,⊥ 2 k 2 Y N,⊥ 2 Y0 + = Y0 + . b2 b4 By (3.2.8), in the given trivialization of T X, |f | ≤ βb2 , 2 aTefX/b2 = a + O |f | /b4 .

(9.2.9)

(9.2.10)

By (9.2.10), since a, Y0k = 0, !! h i k Y N,⊥ TX 2 k N,⊥ N,⊥ = a, Y + βO |f | Y0 + aef /b2 , b Y0 + Y . b2 (9.2.11) By (9.2.9), (9.2.11), for β > 0 small enough, the last term can be absorbed by the other terms appearing in the right-hand side of (9.2.8), so that for

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|f | ≤ βb2 , Y T X ≤ βb2 , 2 −4m−2n X ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 , b q b

γ e

f /b2

TX

,a

+Y

TX

2

/b

, Y0k

+Y

N,⊥

! /b 2

2 ≤ C exp −C 0 Y0k −

Cγ0

! T X 2 N,⊥ 0 −1 0 N,⊥ . −1 Y − Cγ a, Y − Cγ Ad k |f | + Y 2

(9.2.12) The marvelous fact in the right-hand side of (9.2.12) is that it is integrable. Still we have the extra diverging term b2r in the right-hand side of (9.2.7). This term will be dealt with in the next section.

9.3 A conjugation of the Clifford variables We use the notation of section 1.1 for (g, B), and also the notation of section 5.1. Let e1 , . . . , em+n be a basis of g, and let e∗1 , . . . , e∗m+n be the dual basis of g with respect to B. By (1.1.15), ·

N Λ (g

∗

)

−

m+n m+n 1 X = c (e∗i ) b c (ei ) . 2 2 i=1

(9.3.1)

Recall that r = dim z (γ) .

(9.3.2)

By (5.1.3), r = p + q.

(9.3.3) ∗

∗

∗

Let i = z (γ) → g be the obvious embedding, and let i : g → z (γ) be its adjoint. ∗ Recall that z (γ) is another copy of z (γ), and that z (γ) is the correspond∗ ing copy of the dual of z (γ). Also if u ∈ z (γ) , we denote by u the corre∗ sponding element in z (γ) . Let e1 , . . . , er be a basis of z (γ), let e1 , . . . , er be ∗ the corresponding dual basis of z (γ) . Set r r X X α= c (ei ) ei , α b= b c (ei ) ei . (9.3.4) 1

i=1

Then ∗

b · z (γ) α ∈ c (z (γ)) ⊗Λ

,

b · z (γ) α b∈b c (z (γ)) ⊗Λ

∗

.

(9.3.5)

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Chapter 9

Of course α, α b do not depend on the choice of the basis e1 , . . . , er . Clearly, c (z (γ)) ⊂ c (g) ,

b c (z (γ)) ⊂ b c (g) .

(9.3.6)

By (9.3.5), (9.3.6), we get b · z (γ)∗ , α ∈ c (g) ⊗Λ

b · z (γ)∗ . α b∈b c (g) ⊗Λ

(9.3.7)

∗

Recall that ϕ : g → g is defined by (1.1.2) with respect to (g, B). As explained in section 1.1, c (g) , b c (g) act on Λ· (g∗ ). If e ∈ z (γ), c (e) , b c (e) will be identified with their corresponding actions on Λ· (g∗ ). Also note that when using the identification T X ⊕ N = g, and when identifying T X ⊕ N to T ∗ X ⊕ N ∗ by its scalar product, then ϕ = −θ. By (1.1.3) and (9.3.4), we get r X (ϕei − iei ) ei , α=

α b=

(9.3.8) r X

(ϕei + iei ) ei .

(9.3.9)

i=1

i=1

Put Bγ =

r X

ϕei ∧ ei ,

Jγ =

r X

ei iei .

(9.3.10)

i=1

i=1

We can rewrite (9.3.9) in the form α = Bγ + J γ ,

α b = Bγ − J γ . (9.3.11) ∗ ∗ · · b b The group Z (γ) acts on c (z (γ)) ⊗Λ z (γ) and on b c (z (γ)) ⊗Λ z (γ) via its adjoint action on z (γ) and z (γ). Clearly, Ad (γ) α = α, Ad k −1 α = α,

Ad (γ) α b=α b, −1 Ad k α b=α b.

(9.3.12)

Since a ∈ z (γ), the Lie derivative operator La also acts on the above vector spaces. Since ad (a) vanishes on z (γ), we get La α = 0,

La α b = 0.

Definition 9.3.1. For b ≥ 0, e ∈ z (γ), set b cb (e) = exp −b2 α b b c (e) exp b2 α b .

(9.3.13)

(9.3.14)

Using the commutation relations in (1.1.4), we get b cb (e) = b c (e) + 2b2 i∗ ϕe ∧ .

(9.3.15)

[b cb (e) , b cb (f )] = 2B (e, f ) .

(9.3.16)

Moreover, if e, f ∈ g, Proposition 9.3.2. The following identities hold: · ∗ · ∗ exp −b2 α b N Λ (g ) exp b2 α b = N Λ (g ) + b2 α, exp −b2 α b Ad (γ) exp b2 α b = Ad (γ) , exp −b2 α b La exp b2 α b = La .

(9.3.17)

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Proof. In (9.3.1), we may as well assume that e1 , . . . , er are taken as in ⊥ (9.3.4), and that er+1 , . . . , em+n is a basis of z (γ) . By (9.3.1), (9.3.15), we get r X · ∗ · ∗ (9.3.18) exp −b2 α b N Λ (g ) exp b2 α b = N Λ (g ) + b2 c (e∗i ) i∗ ϕei , i=1

which is equivalent to the first identity in (9.3.17). The last two identities in (9.3.17) follow from (9.3.12), (9.3.13). The proof of our proposition is completed. Let e1 , . . . , em+n be a basis of g. Then one verifies easily that up to permutation c (e∗1 ) b c (e1 ) . . . c e∗m+n b c (em+n ) is the only monomial in the c (e∗i ) , b c (ei ) , 1 ≤ i ≤ m + n whose supertrace on Λ· (g∗ ) is nonzero, and moreover, · ∗ m+n Trs Λ (g ) c (e∗1 ) b c (e1 ) . . . c e∗m+n b c (em+n ) = (−2) . (9.3.19) A basis e1 , . . . , em+n will be said to be B-unimodular if the determinant of n the matrix of B on this basis is (−1) . If the basis e1 , . . . , em+n is unimodular, up to permutation, c (e1 ) b c (e1 ) . . . c (em+n ) b c (em+n ) is the only monomial whose supertrace is nonzero, and moreover, ·

∗

m+n

Trs Λ (g ) [c (e1 ) b c (e1 ) . . . c (em+n ) b c (em+n )] = (−2)

n

(−1) .

(9.3.20)

Put b · z (γ)∗ . G = End (Λ· (g∗ )) ⊗Λ

(9.3.21)

cs that was considered in section 5.1. Let Now we slightly redefine the map Tr e1 , . . . , em+n be a unimodular basis of g, which is such that e1 , . . . , er is a basis of z (γ). Let e1 , . . . , er be the basis of z (γ) that is dual to e1 , . . . , er . cs be the linear map from G into R that, up to permutation, vanishes Let Tr on all the monomials in the c (ei ) , b c (ei ) , 1 ≤ i ≤ m + n, ej , 1 ≤ j ≤ r except 1 r on c (e1 ) e . . . c (er ) e c (er+1 ) b c (er+1 ) . . . c (em+n ) b c (em+n ), and moreover, cs c (e1 ) e1 . . . c (er ) er c (er+1 ) b Tr c (er+1 ) . . . c (em+n ) b c (em+n ) r

m+n−r

= (−1) (−2)

n−q

(−1)

⊥

. (9.3.22)

e∗r+1 , . . . , e∗m+n

If we assume that er+1 , . . . , em+n is a basis of z (γ), and that the dual basis to er+1 , . . . , em+n with respect to B|z⊥ (γ) , then (9.3.22) can be replaced by cs c (e1 ) e1 . . . c (er ) er c e∗r+1 b c (er+1 ) . . . c e∗m+n b c (em+n ) Tr r

m+n−r

= (−1) (−2)

. (9.3.23)

Set H = G ⊗ End (E) .

(9.3.24)

cs to a linear map from H to C, so that if u ∈ G, v ∈ End (E), We extend Tr cs [uv] = Tr cs [u] TrE [v] . Tr

(9.3.25)

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Definition 9.3.3. For b ≥ 0, U ∈ End (Λ· (g∗ )), set Ub = exp −b2 α b U exp b2 α b .

(9.3.26)

Then Ub ∈ G. Proposition 9.3.4. For b > 0, the following identity holds: · ∗ cs [Ub ] . Trs Λ (g ) [U ] = b−2r Tr

(9.3.27)

Proof. Using (9.3.15), (9.3.19), (9.3.22), we get (9.3.27).

9.4 The norm of α As we saw in section 3.10, the isomorphism T X ⊕ N = g induces a corresponding isomorphism (T X ⊕ N ) (γ) = z (γ). Let (T X ⊕ N ) (γ) be the vector bundle corresponding to z (γ). This vector bundle is another copy of ∗ (T X ⊕ N ) (γ). We denote its dual by (T X ⊕ N ) (γ) . ∗ · b Then α is a section of c (T X ⊕ N ) ⊗Λ (T X ⊕ N ) (γ) , and α b is a section ∗ · b of b c (T X ⊕ N ) ⊗Λ (T X ⊕ N ) (γ) . Moreover, α and α b can also be consid b · (T X ⊕ N ) (γ)∗ . We will deered as sections of End (Λ· (T ∗ X ⊕ N ∗ )) ⊗Λ note by k k the norm on this last vector bundle, which is induced by the metric of T X ⊕ N . · ∗ ∗ b From the flat connection ∇Λ (T X⊕N ),f ∗,f and from the trivial connection ∗ b · z (γ)∗ , on Λ· z (γ) , we obtain a flat connection on Λ· (T ∗ X ⊕ N ∗ ) ⊗Λ ·

which is still denoted ∇Λ (T

∗

X⊕N ∗ ),f ∗,fb

.

Proposition 9.4.1. There is C > 0 such that kαk ≤ C,

kb αk ≤ C.

(9.4.1)

Moreover, Λ· (T ∗ X⊕N ∗ ),f ∗,fb

∇·

α b = 0.

(9.4.2)

Proof. Equation (9.4.1) follows from (9.3.8), (9.3.10), and (9.3.11). · ∗ ∗ Recall that the flat connection ∇Λ (T X⊕N ),f on Λ· (T ∗ X ⊕ N ∗ ) was defined in section 2.4, and that it is given by (2.4.1). One has the trivial ·

∇Λ (T

∗

X⊕N ∗ ),f

α b = 0.

(9.4.3)

Comparing (2.4.1) and (2.4.5), from (9.4.3), we get (9.4.2).

9.5 A conjugation of the hypoelliptic Laplacian We denote by Op Xb, π b∗ F the algebra of differential operators that act on C ∞ Xb, π b∗ F . By (9.1.3), b b∗ F ⊗ c (g) ⊗b b c (g) . LX (9.5.1) A,b ∈ Op X , π

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Definition 9.5.1. Set 2 2 b . (9.5.2) LX b LX A,b = exp −b α A,b exp b α ∞ b ⊗Λ b · z (γ)∗ . The operator LX Xb, π b∗ Λ· (T ∗ X ⊕ N ∗ ) ⊗F A,b acts on C Theorem 9.5.2. The following identity holds: X LX A,b = LA,b + α.

(9.5.3)

Proof. Equation (9.5.3) follows from (9.1.3), (9.3.17), and (9.4.2). Definition 9.5.3. For t > 0, let qX ((x, Y ) , (x0 , Y 0 )) denote the smooth b,t X kernel associated with exp −tLX A,b . Also we use the notation qb instead of qX b,1 . By (9.5.2), we get 0 0 2 qX b qX ((x, Y ) , (x0 , Y 0 )) exp b2 α b . (9.5.4) b ((x, Y ) , (x , Y )) = exp −b α b Proposition 9.5.4. For b > i0, the following identity holds: h X cs γqX ((x, Y ) , γ (x, Y )) . Trs γq b ((x, Y ) , γ (x, Y )) = b−2r Tr b

(9.5.5)

Proof. Equation (9.5.5) follows from the second equation in (9.3.17) and from equation (9.3.27) in Proposition 9.3.4. Remark 9.5.5. Note that contrary to the standard situation in local index theory [Ge86], we have used a global Getzler rescaling method, which does not necessitate the choice of local coordinates. This can be done because T X ⊕ N = g is a flat vector bundle. By (9.5.5), we get i h b−4m−2n+2r Trs γq X ((x, Y ) , γ (x, Y )) b cs γqX = b−4m−2n Tr b ((x, Y ) , γ (x, Y )) . (9.5.6) 0 0 In the sequel, the norm of qX evaluated with respect b ((x, Y ) , (x , Y )) will be ∗ · ∗ ∗ · to the norms of Λ (T X ⊕ N ) , Λ (T X ⊕ N ) (γ) , and F . Theorem 9.5.6. Given β > 0, there exist C > 0, Cγ0 > 0 such that for b ≥ 1, f ∈ p⊥ (γ) , |f | ≤ βb2 , and Y T X ≤ βb2 , 2 b−4m−2n qX ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 , b ! f /b2 TX TX 2 k N,⊥ 2 γ e ,a +Y /b , Y0 + Y /b 2 ≤ C exp −C 0 Y0k −

Cγ0

! T X 2 N,⊥ −1 N,⊥ 0 0 |f | + Y − Cγ Ad k −1 Y − Cγ a, Y . 2

(9.5.7)

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Proof. The proof of our theorem will be given in section 15.9.

9.6 The limit of the rescaled heat kernel By (9.2.7), (9.5.5), we get Z h Λ· (T ∗ X⊕N ∗ )⊗F Tr γq X s |f |≤β b |Y T X −aT X |≤β

r (f ) dY T X dY N df cs γqX ef /b2 , aT X + Y T X /b2 , Y k + Y N,⊥ /b2 , b−4m−2n Tr 0 b "

Z =


|f |≤βb2

|Y T X |≤βb2 2 γ ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2

!# r f /b2 dY T X dY0k dY N,⊥ df. (9.6.1)

The fundamental fact in (9.6.1) is that by (9.5.7), the integrand in the righthand side of (9.6.1) is such that " −4m−2n c f /b2 TX TX 2 k N,⊥ 2 b e , a + Y /b , Y + Y /b , Trs γqX b 0 !# 2 γ ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 2 ≤ C exp −C 0 Y0k −

Cγ0

! T X 2 N,⊥ 0 −1 0 N,⊥ − Cγ Ad k . − Cγ a, Y |f | + Y −1 Y 2

(9.6.2) ·

∗

Recall that γ acts on Λ (g ) like Ad (γ). Moreover, γ maps Fx into Fγx . On X (γ), we trivialize F |X(γ) , T X|X(γ) , N |X(γ) by parallel transport with respect to the connections ∇F , ∇T X , ∇N along geodesics starting at p1, so that T X|X(γ) = p, N |X(γ) = k. In particular, Λ· (T ∗ X ⊕ N ∗ ) |X(γ) ' Λ· (p∗ ⊕ k∗ ) .

(9.6.3)

Also along geodesics normal to X (γ), we trivialize the given vector bundles with respect to their canonical Euclidean and Hermitian connections. Ultimately, Λ· (T ∗ X ⊕ N ∗ ) ' Λ· (p∗ ⊕ k∗ ) .

(9.6.4)

Recall that the kernel RY0k on p × g was defined in section 5.1. Also Ad k −1 acts on Λ· (p∗ ⊕ k∗ ).

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Theorem 9.6.1. As b → +∞, b−4m−2n γqX b


γ e

f /b2

TX

,a

+Y

TX

2

/b

, Y0k

+Y

N,⊥

2

/b

!

2 2 → exp − |a| /2 − Y0k /2 Ad k −1 RY0k (f, Y ) , Ad k −1 (f, Y ) ρE k −1 exp −iρE Y0k − A . (9.6.5) Proof. Our theorem will be established in sections 9.8–9.11.

9.7 A proof of Theorem 6.1.1 Using (3.4.35), (9.6.1), (9.6.2), and (9.6.5), we find that as b → +∞, Z

·

|f |≤β

Trs Λ (T

∗

X⊕N ∗ )⊗F

h γq X b


|Y T X −aT X |≤β Z 2 r (f ) dY T X dY N df → exp − |a| /2

(y,Y g ,Y0k )

∈p⊥ (γ)×(p⊕k⊥ (γ))×k(γ)

TrE ρE

h i cs Ad k −1 RY k (y, Y g ) , Ad k −1 (y, Y g ) Tr 0 2 −1 E k k exp −iρ Y0 − A exp − Y0k /2 dydY g dY0k . (9.7.1)

By (5.1.11), (9.2.1), by (9.2.2) and the considerations that follow, and by (9.7.1), we get (6.1.2). This completes the proof of Theorem 6.1.1.

9.8 A translation on the variable Y T X Definition 9.8.1. Set Ta s x, Y T X , Y N = s x, aT X + Y T X , Y N .

(9.8.1)

X −1 Na,A,b = Ta LX A,b Ta .

(9.8.2)

Put

Recall that if e ∈ T X ⊕ N , ∇Ve denotes differentiation along e.

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Proposition 9.8.2. The following identity holds: X Na,A,b =

2 b4 N T X Y ,a + Y TX 2 2 1 1 ∆T X⊕N T X + a + Y − 2 (m + n) + − 2 b4 b ·

∗

N Λ (T X⊕N + b2

∗

)

+α−

− c iθad Y

N

b )),f ∗,fb C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ∇aT X +Y T X

E

− iρ

Y

N

!

+ ∇V[aN ,aT X +Y T X ] + A. (9.8.3)

Proof. By (2.17.10), (9.1.3), and (9.5.3), we get (9.8.3). Put X X Oa,A,b = Na,A,b + La .

(9.8.4)

Proposition 9.8.3. The following identity holds: X Oa,A,b =

2 b4 N T X Y ,a + Y TX 2 2 1 ∆T X⊕N T X − 1 (m + n) + − + a + Y 2 b4 b2 ·

+

∗

N Λ (T X⊕N b2

∗

)

+α−

b )),f ∗,fb C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ∇Y T X

! TX

− c ad a

N

−a

+b c (ad (a)) − c iθad Y

N

−ρ

E

iY

N

N

−a

+ ∇V[aN ,aT X +2Y T X +Y N ] + A. (9.8.5) Proof. By (2.4.5), (2.18.1), (2.18.2), and (9.8.3), we get (9.8.5). Equation (9.8.5) also follows from (2.18.4). Remark 9.8.4. Recall that the operator LX a,b was defined in (2.18.3). Set X LX a,A,b = La,b + A.

(9.8.6)

By (2.17.10), (2.18.1), and (2.18.2), we get Ta La Ta−1 = La .

(9.8.7)

From (9.3.17), (9.8.6), and (9.8.7), we obtain −1 X 2 Oa,A,b = Ta exp −b2 α b Fb LX b Ta−1 . a,A,b Fb exp b α

(9.8.8) LX a,b

Equation (9.8.5) can also be derived from equation (2.18.4) for and from (9.8.6)–(9.8.8). Also observe that by Theorem 2.18.2, LX a,A,b commutes with Z (a). ThereX fore Oa,A,b also commutes with Z (a).

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0 0 X Let oX a,b ((x, Y ) , (x , Y )) be the smooth kernel for exp −Oa,A,b .Then 0 0 a X −a x, aT X + Y , x0 , aT X + Y 0 . (9.8.9) oX a,b ((x, Y ) , (x , Y )) = e qb e

Since γ = ea k −1 , from (9.8.9), we obtain γqX x, aT X + Y , γ x, aT X + Y b a a −1 = k −1 oX (x, Y ) . (9.8.10) a,b e (x, Y ) , e k Definition 9.8.5. Set

N T X T X X

4 OX a ,a ,Y Oa,A,b exp b4 aN , aT X , Y T X . a,A,b = exp −b (9.8.11) Note that a, Y T X = aT X , Y T X + aN , Y T X is a section of T X ⊕ N . Proposition 9.8.6. The following identity holds: 2 b4 N T X Y ,a + Y TX 2 2 1 1 ∆T X⊕N T X + + a + Y − 2 (m + n) − 2 b4 b Λ· (T ∗ X⊕N ∗ ) 4 2 N b N T X 2 + +α+ a ,a + b4 a, Y T X 2 b 2

C ∞ (T X⊕N,π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),f ∗ ,fb + b4 aN , aT X , aN , 2Y T X − ∇Y T X

OX a,A,b =

TX

− c ad a

N

−a

+b c (ad (a)) − c iθad Y

N

−ρ

E

iY

N

N

−a

!

+ ∇V[aN ,2Y T X +Y N ] + A. (9.8.12) X Proof. We start from equation (9.8.5) for Oa,A,b . Let ∆T X be the Laplacian along the fibre T X. Let e1 , . . . , em be an orthonormal basis of T X. Clearly,

2 1 1 X − 4 ∆T X + ∇V[aN ,aT X ] = − 4 ∇Vei − b4 aN , aT X , ei 2b 2b 1≤i≤m

+

b4 N T X 2 a ,a . (9.8.13) 2

By (9.8.13), we get

N T X T X 1 T X⊕N V exp −b a ,a ,Y − 4∆ + ∇[aN ,aT X ] 2b

1 b4 N T X 2 a ,a . (9.8.14) exp b4 aN , aT X , Y T X = − 4 ∆T X⊕N + 2b 2 Also by equation (3.2.3) in Proposition 3.2.1, T X 2 a

N T X T X . (9.8.15) a ,a ,Y = −∇Y T X 2 4

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From (9.8.15), we deduce that

exp −b4 aN , aT X , Y T X ∇Y T X exp b4 aN , aT X , Y T X T X 2 a 4 TX . (9.8.16) = ∇Y T X − b ∇Y T X ∇Y T X 2 By (3.2.4) and (9.8.16), we get

exp −b4 aN , aT X , Y T X ∇Y T X exp b4 aN , aT X , Y T X 2 = ∇Y T X − b4 a, Y T X . (9.8.17) Also, exp −b4

N T X T X V

a ,a ,Y ∇[aN ,2Y T X ] exp b4 aN , aT X , Y T X

= ∇V[aN ,2Y T X ] + b4 aN , aT X , aN , 2Y T X . (9.8.18)

By (9.8.5), (9.8.11), (9.8.14), (9.8.17), and (9.8.18), we get (9.8.12). The proof of our proposition is completed.

9.9 A coordinate system and a trivialization of the vector bundles Put x0 = p1.

(9.9.1)

The map y ∈ p → pey ∈ X gives a coordinate system on X. In this coordinate system, z (γ) is identified with X (γ). Moreover, −1 k −1 pey = peAd(k )y .

(9.9.2)

The same procedure produces natural coordinates on Xb, the total space of T X ⊕ N . This coordinate system is given by ψb : (y, Y g ) ∈ p × g → (ey , Y g ) ∈ Xb. Equivalently, this is the coordinate system on Xb one obtains by parallel transport with respect to the connection ∇T X⊕N along the geodesics t ∈ R → pety , y ∈ p. Moreover, k −1 ψb (y, Y g ) = ψb Ad k −1 y, Ad k −1 Y g . (9.9.3) We trivialize F along the geodesic t → pety by parallel transport with respect to the connection ∇F , and we trivialize Λ· (T ∗ X ⊕ N ∗ ) along this ∗ geodesic by parallel transport with respect to ∇T X⊕N . Also Λ· z (γ) is b · (T X ⊕ N ) (γ)∗ ⊗ already naturally trivialized. On X, Λ· (T ∗ X ⊕ N ∗ ) ⊗Λ b · z (γ)∗ ⊗ E. F is identified with Λ· (g∗ ) ⊗Λ X b · z (γ)∗ ⊗ E . The operator Oa,A,b now acts on C ∞ p × g, Λ· (g∗ ) ⊗Λ Definition 9.9.1. If Y0k ∈ k⊥ (γ), set Hb,Y0k s (y, Y ) = s y/b2 , Y0k + Y /b2 .

(9.9.4)

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Put −1 X X Pa,A,b,Y , k = Hb,Y k Oa,A,b H b,Y k 0

X −1 (9.9.5) PX a,A,b,Y0k = Hb,Y0k O a,A,b Hb,Y0k . ∗ ∞ · ∗ b · The operators in (9.9.5) also act on C p × g, Λ (g ) ⊗Λ z (γ) ⊗ E . 0 0 X Let pX ((y, Y ) , (y , Y )) be the smooth kernel for exp −P k k a,b,Y a,A,b,Y 0

0

0

0

with respect to the volume dy 0 dY 0 . Recall that the function δ (Y p ) was defined in (4.1.11). Then 0 2 y /b k 0 2 y/b2 k 2 , e , Y + Y /b e , Y + Y /b b−4m−2n oX 0 0 a,b 0 2 X δ y /b = pa,b,Y k ((y, Y ) , (y 0 , Y 0 )) . (9.9.6) 0

By (9.8.9), (9.9.6), we get 2 2 0 0 pX a,b,Y0k b (y, Y ) , b (y , Y ) y0 T X −a y TX k k 0 = b−4m−2n ea qX e e , a + Y + Y , e , a + Y + Y δ (y 0 ) . b 0 0 (9.9.7) Since

LX A,b

a

commutes with e , we get

qb ((x, Y ) , (x0 , Y 0 )) = ea qb e−a (x, Y ) , e−a (x0 , Y 0 ) e−a . (9.9.8) In (9.9.8), e−a maps (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )x0 into (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )e−a x0 , and ea maps (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )e−a x into (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )x . By (9.8.10), (9.9.6), and (9.9.8), we obtain b−4m−2n γqX b

γ e


f /b2

TX

,a

+Y

TX

2

/b

, Y0k

+Y

N,⊥

2

/b

!

δ f /b2

−1 = ea k −1 pX (f, Y ) e−a . (9.9.9) a,b,Y k (f, Y ) , k 0

In the given coordinate system, k −1 (f, Y ) = Ad k −1 (f, Y ) . (9.9.10) · ∗ ∗ Moreover, in the trivialization of Λ (T X ⊕ N ) ⊗ F that was considered before, the action of k −1 on Λ· (T ∗ X ⊕ N ∗ ) ⊗ F is just given by Ad k −1 ⊗ ρE k −1 . Also the action of ea on (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F )x0 is the parallel transport along the geodesic t ∈ [0, 1] → eta x0 ∈ X (γ) with respect to the obvious connection. In the above trivialization of Λ· (T ∗ X ⊕ N ∗ ) ⊗ F , we can rewrite (9.9.9) in the form b−4m−2n γqX b

2 ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2 , 2

γ ef /b , aT X + Y T X /b2 , Y0k + Y N,⊥ /b2

! δ f /b2

−1 = Ad k −1 ⊗ ρE k −1 pX (f, Y ) . (9.9.11) a,b,Y k (f, Y ) , Ad k 0

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X 9.10 The asymptotics of the operator Pa,A,b,Y k as b → +∞ 0

Definition 9.10.1. Put 2 1 k k p 2 1 2 −∆p⊕k + |a| + Y0k Y , a + Y0 , Y + 0 2 2 V k E k +α−∇H −∇ −b c (ad (a))+c ad (a) + iθad Y +iρ Y +A. k Yp 0 0 [a+Y ,[a,y]] X Pa,A,∞,Y k =

0

(9.10.1) Recall that the operator Pa,Y0k was defined in (5.1.5). By comparing (5.1.5) and (9.10.1), we obtain 1 2 k 2 X |a| + Y0 + iρE Y0k + A. (9.10.2) Pa,A,∞,Y k = Pa,Y k + 0 0 2 Take β ∈]0, 1]. In the asymptotic expansion that follows, we will assume that |y| /b2 ≤ β. Theorem 9.10.2. As b → +∞, 2 4 X X 2 Pa,A,b,Y +O |Y | /b2 +O Y0k |y| /b4 k = Pa,A,∞,Y k +O (1 + |y|) /b 0 0 2 2 2 4 + O Y k |y| /b8 + O Y k |Y p | /b4 2 2 + O Y k , a + Y0k , Y p Y0k |y| /b2 + Y k |y| /b4 + Y k |Y p | /b2 4 2 + O |y| /b8 + |Y | /b4 2 2 + O |a| + Y0k |y| /b4 + |Y | /b2 + O |y| |Y | /b4 ∇V + O |Y | Y0k |y| /b2 ∇V + O |y| |Y | /b2 ∇V 3 + O |y| 1 + Y0k /b4 ∇V . (9.10.3) X Proof. We use equation (9.8.5) for Oa,A,b . Note that by (2.17.10), since aN vanishes on X (γ), in the given trivialization of T X, 2 3 X 2 6 aTy/b |y| /b4 , aN . (9.10.4) 2 = a + O y/b2 = [a, y] /b + O |y| /b

Note that (9.10.4) also follows from (3.2.8). equation Since a, Y0k = 0, from (9.10.4), we find that as b → +∞, i 2 2 b4 h k 1 X p 2 Y0 + Y k /b2 , aTy/b = Y k , a + Y0k , Y p 2 + Y /b 2 2 2 2 2 2 4 4 + O Y0k |y| /b4 + O Y k |y| /b8 + O Y k |Y p | /b4 2 2 + O Y k , a + Y0k , Y p Y0k |y| /b2 + Y k |y| /b4 + Y k |Y p | /b2 . (9.10.5)

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Also, 2 2 TX 2 4 2 ay/b2 + Y0k + Y /b2 = |a| + Y0k + O |y| /b8 + |Y | /b4 2 + O |a| + Y0k |y| /b4 + |Y | /b2 . (9.10.6) Moreover, as b → +∞, b )),f ∗,fb C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F −1 H Hb,Y0k ∇Y T X Hb,Y k = ∇Y p 0 2 + O |y| |Y | /b4 ∇V + O |Y | Y0k |y| /b2 ∇V + O |Y | /b2 . (9.10.7) In (9.10.7), we have used the fact that the connection form for ∇T X⊕N in the given trivialization vanishes at x0 , and so it gives a contribution of the order O |y| /b2 . For more details on this computation, we refer to equations (15.10.3), (15.10.4). Moreover, −1 V h i . (9.10.8) Hb,Y0k ∇V[aN ,aT X +2Y T X +Y N ] Hb,Y k = ∇ N p k 2 k 2 TX 0

a

,2Y y/b2

+Y +b Y0 +b a

y/b2

By (9.10.4), (9.10.8), we obtain V −1 V 2 Hb,Y0k ∇V[aN ,aT X +2Y T X +Y N ] Hb,Y k = −∇ [a+Y0k ,[a,y]] + O |y| |Y | /b ∇ 0 3 + O |y| 1 + Y0k /b4 ∇V . (9.10.9) By (9.8.5), and (9.10.5)–(9.10.9), we get (9.10.3). The proof of our theorem is completed. Remark 9.10.3. A similar asymptotics can be obtained for P X a,A,b,Y k . The 0

limit operator P X a,A,∞,Y k is closely related with the operator Qa,Y k in equa0

0

X tion (5.2.1). Also the conjugation in (9.8.11) that transforms Oa,A,b into X Oa,A,b is reflected in equation (5.2.1), where Qa,Y k is obtained from Qa,Y0k 0 by a related conjugation. 2 Also the reason for the term O |y| |Y | /b4 ∇V to appear in (9.10.3) is that in any trivialization of T X, the scalar operator ∇Y T X is quadratic in Y = Y T X + Y N in the vertical direction. This term is not as bad as it seems; since the connection ∇T X preserves the norm Y T X , the corresponding vector field is norm preserving. A similar question is discussed at length in [BL08, sections 15.4 and 17.8] in a different context. More precisely, we refer to [BL08, eqs. (15.4.38) and(17.8.6)] and to the references to these equations.

9.11 A proof of Theorem 9.6.1 By (9.10.2), 2 2 X exp −Pa,A,∞,Y = exp − |a| /2 − Y0k /2 k 0 exp −Pa,Y0k exp −iρE Y0k − A . (9.11.1)

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Chapter 9

X Let pX be the smooth kernel associated with exp −Pa,A,∞,Y k . By a,∞,Y0k 0 (9.11.1), we get k 2 2 /2 RY k exp −iρE Y0k − A . (9.11.2) pX a,∞,Y k = exp − |a| /2 − Y0 0 0

Then Theorem 9.6.1 is an obvious consequence of (9.9.11), (9.11.2), and of the following result. Theorem 9.11.1. As b → +∞, 0 0 X 0 0 pX a,b,Y k ((y, Y ) , (y , Y )) → pa,∞,Y k ((y, Y ) , (y , Y )) . 0

0

Proof. The proof of our theorem will be given in section 15.10.

(9.11.3)

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Chapter Ten The action functional and the harmonic oscillator The purpose of this chapter is to solve explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. The action is the one introduced in [B05, eq. (0.10)]. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b → 0 and when b → +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action. The above variational problem has already been considered by Lebeau in [L05, section 3.6] as a warm up to the more general problem on Riemannian manifolds, in order to study in detail the small time asymptotics of the hypoelliptic heat kernel. The results of the present chapter will be used in chapter 12 in combination with the Malliavin calculus to obtain a uniform control of the regularity of the scalar hypoelliptic heat kernel for bounded b > 0, and also for b → +∞. The relevant tangent variational problems take place in the vector spaces p and g = p ⊕ k. Also, when G is an Euclidean vector space, in which case the main results of the book are trivial, we give a direct computational verification of the intermediate steps. This chapter is organized as follows. In section 10.1, we describe the variational problem over a Riemannian manifold. In section 10.2, we obtain the relevant Hamiltonian version of the problem, by an application of Pontryagin’s maximum principle. In section 10.3, we consider the case of an Euclidean vector space E. We compute explicitly the solution to the variational problem as a function of b, and we study its behavior as b → 0, as well as the behavior of the minimum value of the action. In section 10.4, we state Mehler’s formula for the heat kernel of the harmonic oscillator on E, which we relate to one of the variational problems considered before. In section 10.5, we make a similar computation for the hypoelliptic heat kernel on E ⊕ E. In section 10.6, when G = E, K = {0} , X = E, we give a direct computational verification of the results contained in the book. In this case, our main result, which is Theorem 6.1.1, gives just the explicit formula for the standard heat kernel on E. The intermediate estimates used in the proof of Theorem 6.1.1 can be verified by hand. Even though the isometry group

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Chapter 10

of E is not reductive, we also apply the approach used in the book to the orbital integrals associated with an isometry of E. In section 10.7, we make simple computations involving the heat kernel of the harmonic oscillator. The obtained estimates will play an important role in the sequel. Finally, in section 10.8, we construct the diffusion process associated with the harmonic oscillator, and we establish estimates on the Lp norm of certain random variables. The results of sections 10.5 and 10.6 are not used in the rest of the book.

10.1 A variational problem We will now describe a variational problem that comes naturally from the theory of the hypoelliptic Laplacian. Let X be a smooth complete Riemannian manifold, and let g T X be a smooth Riemannian metric on T X. Let ∇T X denote the Levi-Civita connection on T X, and let RT X be its curvature. D denote covariant Let s ∈ [0, 1] → xs ∈ X be a smooth path. Let Ds differentiation with respect to the Levi-Civita connection. In particular, we use the notation D x ¨= x. ˙ (10.1.1) Ds For b > 0, t > 0, set Z Z 2 1 t 2 1 t 2 4 Hb,t (x) = |x| ˙ + b |¨ x| ds, Kb,t (x) = x˙ + b2 x ¨ ds. 2 0 2 0 (10.1.2) Note that b2 2 2 |x˙ t | − |x˙ 0 | . (10.1.3) Kb,t (x) = Hb,t (x) + 2 The action Hb,t appears naturally in the functional analytic description of the hypoelliptic Laplacian in [B05]. Set ys = xb2 s . (10.1.4) Then 1 1 Hb,t (x) = 2 H1,t/b2 (y) , Kb,t (x) = 2 K1,t/b2 (y) . (10.1.5) b b In what follows, we will fix (x0 , x˙ 0 ) = (x, Y ) and (xt , x˙ t ) = (x0 , Y 0 ), and we will find the extrema of Hb,t and of Kb,t , which by (10.1.3) are equivalent problems. This problem was considered in detail by Lebeau [L05, section 3]. When b = +∞, the corresponding variational problem is associated with the functional Z 1 t 2 H∞ (x) = |¨ x| ds. (10.1.6) 2 0 The existence of minima for the above actions can be easily established. In the sequel, our paths will be supposed to verify the above constraints.

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Definition 10.1.1. Given a smooth path x, let L be the differential operator acting along smooth sections J of T X along this path, LJ =

D2 J + RT X (J, x) ˙ x. ˙ Ds2

(10.1.7)

Now we state a result in [L05, Theorem 3.2]. Theorem 10.1.2. A necessary and sufficient condition for a path x to be an extremum of Hb,t is that 1 − b4 L x ¨ = 0. (10.1.8) Proof. Let u ∈ R → xu,· be a smooth family of paths verifying the given boundary conditions with x0,· = x· the given extremal path. Set J=

D J˙ = J. Ds

∂ xu,· |u=0 , ∂u

(10.1.9)

Then J0 = J˙0 = 0,

Jt = J˙t = 0.

An elementary computation shows that Z t

d Hb,t (xu ) |u=0 = − J, 1 − b4 L x ¨ ds. du 0

(10.1.10)

(10.1.11)

0 By (10.1.11), the vanishing of Hb,t (x) is equivalent to (10.1.8). The proof of our theorem is completed.

Remark 10.1.3. For b = 0, we get the equation of geodesics x ¨ = 0,

(10.1.12)

L¨ x = 0.

(10.1.13)

and for b = +∞, we get We will now reformulate the above as a control problem. Let L2 denote the vector space of square-integrable functions on [0, t] with values in E. If Y· is a smooth section of T X over a path x· , we will use the notation D Y˙ = Ds Y . For u ∈ L2 , consider the differential equation Y˙ = u, (xt , Yt ) = (x0 , Y 0 ) .

x˙ = Y, (x0 , Y0 ) = (x, Y ) ,

(10.1.14)

Set L ((x, Y ) , u) =

b4 2 1 2 |Y | + |u| . 2 2

(10.1.15)

L ((x, Y ) , u) ds.

(10.1.16)

Then Z Hb,t (x) =

t

0

The problem of finding the extremals of Hb,t becomes a control theoretic problem.

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A similar treatment can be applied to the functional Kb,t . For v ∈ L2 , the controlled differential equation is now given by 1 Y˙ = 2 (−Y + v) , b (xt , Yt ) = (x0 , Y 0 ) .

x˙ = Y, (x0 , Y0 ) = (x, Y ) ,

(10.1.17)

Set 2

M ((x, Y ) , v) =

|v| . 2

(10.1.18)

Then Z Kb,t (x) =

t

M ((x, Y ) , v) ds.

(10.1.19)

0

Let X be the total space of T X. The connection ∇T X induces the splitting T X ' T X ⊕ T X, the first copy of T X being the horizontal part of T X , and the second copy of T X being the tangent bundle along the fibre. Let X∗ be the total space of the cotangent bundle of X . Then X∗ can be identified with the total space of T X ⊕ (T ∗ X ⊕ T ∗ X). The first copy of T ∗ X consists of the pull back of 1-forms on the basis X, and the second copy of the 1-forms along the fibres T X of X that vanish horizontally. The generic element of X∗ will be denoted ((x, Y ) , (p, q)). Let π : X∗ → X be the obvious projection. Let T X∗ be the total space of the tangent bundle of X∗ . Then the fibres of T X∗ can be identified with (T X ⊕ T X) ⊕ (T ∗ X ⊕ T ∗ X). Here T X ⊕ T X is the horizontal part of T X∗ , and is identified with T X , and T ∗ X ⊕ T ∗ X is the tangent bundle to the fibres.

10.2 The Pontryagin maximum principle Now we will apply the Pontryagin maximum principle to the above control problems, which we will consider in succession. Put Hb ((x, Y ) , (p, q)) = sup {hp, Y i + hq, ui − L ((x, Y ) , u)} .

(10.2.1)

u

Then Hb ((x, Y ) , (p, q)) = hp, Y i −

1 1 2 2 |Y | + 4 |q| . 2 2b

(10.2.2)

In the sequel, we will identify T X and T ∗ X by the given metric g T X . Let θ be the canonical 1-form on X∗ , and let ω = dθ be the canonical symplectic form. With the above identifications, θ is given by θ = (p, q) .

(10.2.3)

θ = hp, dxi + hq, DY i .

(10.2.4)

Equivalently,

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Then an easy computation shows that

ω = hDp, dxi + hDq, DY i + q, RT X Y . Hb

(10.2.5)

∗

Let Y be the Hamiltonian vector field on X associated with the Hamiltonian Hb , so that dHb + iY Hb ω = 0.

(10.2.6)

Using (10.2.2), (10.2.5), and (10.2.6), we find that the vector field Y Hb is given by Y Hb = Y, q/b4 , RT X (q, Y ) Y, −p + Y . (10.2.7) Let z = ((x, Y ) , (p, q)) be the solution of the differential equation z˙ = Y Hb .

(10.2.8)

The Pontryagin principle asserts that (x, Y ) = πz is an extremum of Hb,t as long as it verifies the given boundary conditions, and any extremum can be obtained this way. Moreover, Hb is conserved along the above trajectory. The Hamilton equation (10.2.8) on z = (x, Y, p, q) can be written in the form q x˙ = Y, Y˙ = 4 , (10.2.9) b p˙ = RT X (q, Y ) Y, q˙ = −p + Y. By (10.2.9), we get q q x ¨ − 4 = 0, q¨ + RT X (q, Y ) Y − 4 = 0. (10.2.10) b b It is then obvious that (10.1.8) and (10.2.10) are equivalent. Moreover, Hb is preserved by the Hamiltonian flow. Using (10.2.2) and (10.2.9), we find that the conserved quantity is given by 1 2 b4 2 ... −b4 h x , xi ˙ + |x| ˙ + |¨ x| . (10.2.11) 2 2 Also by comparing (10.1.14) and (10.2.9), we get q u = 4. (10.2.12) b Now we consider the variational problem associated with the functional Kb,t . We define the associated Hamiltonian Kb as in (10.2.1), i.e., 1 Kb ((x, Y ) , (p, q)) = sup hp, Y i + 2 hq, −Y + vi − M ((x, Y ) , v) . b v (10.2.13) We get D E q 1 2 Kb ((x, Y ) , (p, q)) = p − 2 , Y + 4 |q| . (10.2.14) b 2b The Hamiltonian vector field Y Kb associated with Kb is given by Y q q Y Kb = Y, − 2 + 4 , RT X (q, Y ) Y, −p + 2 . (10.2.15) b b b

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Instead of (10.2.9), we now have q Y Y˙ = − 2 + 4 , b b q p˙ = RT X (q, Y ) Y, q˙ = −p + 2 . b By comparing (10.1.17) and (10.2.16), we obtain q v = 2. b Put S (x, Y, p, q) = x, Y, p, q − b2 Y . x˙ = Y,

(10.2.16)

(10.2.17)

(10.2.18)

By (10.2.5), we find that S is a symplectic transformation. The transformation S is associated with the generating function σ (x, Y, p, q) given by σ (x, Y, p, q) = −

b2 2 |Y | , 2

(10.2.19)

so that S ∗ θ − θ = dσ.

(10.2.20)

Kb = Hb ◦ S.

(10.2.21)

Moreover,

As should be the case, equation (10.2.16) is obtained from (10.2.9) by replacing q by q − b2 Y . The above results were obtained by Lebeau in [L05, Theorem 3.3].

10.3 The variational problem on an Euclidean vector space With the notation in (10.1.14), (10.1.17), Z Z 2 2 1 t 1 t 2 4 ˙ |Y | + b Y ds, Kb,t (x) = Hb,t (x) = Y + b2 Y˙ ds. 2 0 2 0 (10.3.1) Let E be an Euclidean vector space of dimension m. Here we will take X = E. In a first step, we will assume that x0 and xt are free to vary, while Y0 = Y, Y1 = Y 0 . This means that we can entirely disregard x. Let Hb∗ (Y, q) , Kb∗ (Y, q) be the Hamiltonians associated with the above variational problems. Then Dq E 1 1 1 2 2 2 Hb∗ (Y, q) = − |Y | + 4 |q| , Kb∗ (Y, q) = − 2 , Y + 4 |q| . 2 2b b 2b (10.3.2) The Hamiltonians in (10.3.2) are obtained from the ones in (10.2.2), (10.2.14) by making p = 0.

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Making p = 0 in (10.2.9), (10.2.16), the solution to the variational problem associated with the actions in (10.3.1) is such that b4 Y¨ − Y = 0. (10.3.3) By (10.3.3), we obtain 2

2

Ys = Aes/b + Be−s/b .

(10.3.4)

Adjusting A, B for the conditions Y0 = Y, Yt = Y 0 leads to the formulas, 2

2

Y − e−t/b Y 0 Y − et/b Y 0 , B = . (10.3.5) A= 2 1 − e2t/b 1 − e−2t/b2 ∗ ∗ Let Hb,t (Y, Y 0 ) , Kb,t (Y, Y 0 ) be the values of Hb,t , Kb,t over the path Y· given by (10.3.4), (10.3.5). By (10.1.3), b2 0 2 2 ∗ ∗ Kb,t (Y, Y 0 ) = Hb,t (Y, Y 0 ) + |Y | − |Y | . (10.3.6) 2 Proposition 10.3.1. The following identities hold: ! 2 2 b2 |Y 0 − Y | ∗ 0 2 0 2 Hb,t (Y, Y ) = tanh t/2b |Y | + |Y | + , 2 sinh (t/b2 ) ∗ Kb,t (Y, Y 0 ) =

2 b2 2 tanh t/2b2 |Y | + |Y 0 | 2

(10.3.7)

2

|Y 0 − Y | 2 2 + |Y 0 | − |Y | + sinh (t/b2 ) 2 b2 −t/2b2 t/2b2 0 = Y − e Y e . 2 sinh (t/b2 ) Also we have the Hamilton-Jacobi equations, ∂ ∗ ∂ 0 ∗ ∗ 0 0 H (Y, Y ) + Hb Y , H (Y, Y ) = 0, ∂t b,t ∂Y 0 b,t ∂ ∗ ∂ 0 ∗ ∗ 0 0 K (Y, Y ) + Kb Y , K (Y, Y ) = 0. ∂t b,t ∂Y 0 b,t

!

(10.3.8)

Proof. Equation (10.3.7) is a trivial consequence of (10.3.4)–(10.3.6). Equation (10.3.8) follows from general arguments in the calculus of variations. It can also be directly verified. By (10.3.7), we get ∗ Hb,t (Y, Y 0 ) =

1 ∗ H1,t/b2 b2 Y, b2 Y 0 , 2 b

∗ Kb,t (Y, Y 0 ) =

1 ∗ K1,t/b2 b2 Y, b2 Y 0 . 2 b (10.3.9)

Equation (10.3.9) also follows from (10.1.4), (10.1.5). By (10.1.17), (10.3.4), 2

v = b2 Y˙ + Y = 2Aes/b .

(10.3.10)

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Moreover, ∗ Kb,t (Y, Y 0 ) =

1 2

Z

t

2

|v| ds.

(10.3.11)

0

Observe that by (10.3.4), (10.3.5), when b → 0, on compact subsets of ]0, t[, Y· converges uniformly to 0, while remaining equal to Y at s = 0 and to Y 0 at s = t. Moreover, as b → 0, ∗ Hb,t (Y, Y 0 ) → 0,

∗ Kb,t (Y, Y 0 ) → 0.

(10.3.12)

From (10.3.12), we find that as b → 0, Y· → 0 in L2 .

(10.3.13)

Also as b → 0, 1 2 2 |Y | + |Y 0 | , 2 2 ∗ Kb,t (Y /b, Y 0 /b) → K ∗0 (Y, Y 0 ) = |Y 0 | . ∗ Hb,t (Y /b, Y 0 /b) → H ∗0 (Y, Y 0 ) =

(10.3.14)

By (10.3.4), (10.3.5), when replacing Y, Y 0 by Y /b, Y 0 /b, as b → 0, Y· still converges to 0 uniformly on compact subsets of ]0, t[. Note that Z 2 1 s −u/b2 e du = b 1 − e−s/b , (10.3.15) b 0 so that 1 b

Z

s

2

e−u/b du ≤ b.

(10.3.16)

0

We still replace Y, Y 0 by Y /b, Y 0 /b. For 1 ≤ p < +∞, Lp denotes the obvious vector space of functions defined on [0, t] with values in E. By (10.3.4), (10.3.5), and (10.3.16), as b → 0, Y· → 0 in Lp , 1 ≤ p < 2,

Y· → 0 weakly in L2 .

(10.3.17)

By (10.3.4), (10.3.5), in general, Y· does not converge strongly to 0 in L2 . More precisely, as b → 0, Z Z 1 t b4 t ˙ 2 1 2 1 2 2 2 0 2 |Y | ds → |Y | + |Y | , |Y | + |Y 0 | , Y ds → 2 0 4 2 0 4 (10.3.18) which fits with (10.3.1), (10.3.14). Moreover, given 0 < ≤ M < +∞, as long as Y, Y 0 remain uniformly bounded in E, and ≤ t ≤ M , the convergence of Y· to 0 in Lp , 1 ≤ p < 2 is uniform. Also, Y· remains uniformly bounded in L2 , and bY· remains uniformly bounded. We fix again Y, Y 0 . When b → +∞, we have the uniform convergence on [0, t], s s (10.3.19) Ys → 1 − Y + Y 0. t t

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Moreover, as b → +∞, ∗ ∗ Hb,t Kb,t 1 1 2 2 0 0 (Y, Y (Y, Y 0 ) → ) → |Y − Y | , |Y 0 − Y | . 4 4 b 2t b 2t Also as b → +∞, t t 2 2 ∗ ∗ Kb,t (Y, Y ) → |Y | . Hb,t (Y, Y ) → |Y | , 2 2 Now we consider the original problem on E ⊕ E. Set

Hb ((x, Y ) , (p, q)) =

1 2 |p| + Hb∗ (Y, q) . 2

(10.3.20)

(10.3.21)

(10.3.22)

By (10.3.2), (10.3.22), we get 1 2 1 1 2 2 |p| − |Y | + 4 |q| . (10.3.23) 2 2 2b A remarkable feature of Hb is that the variables (x, p) and (Y, q) are uncoupled, which is not the case for Hb in (10.2.2). Recall that since X = E, then X ∗ = E ⊕ E ⊕ E ∗ ⊕ E ∗ . The vector space ∗ X is equipped with the obvious symplectic form ω in (10.2.5), where D should just be d, and RT X = 0. Let T be the map Hb ((x, Y ) , (p, q)) =

T (x, Y, p, q) = (x + q, Y + p, p, q) .

(10.3.24)

τ (x, Y, p, q) = hp, qi .

(10.3.25)

Set

Then τ is a generating function for T , i.e., T ∗ θ − θ = dτ.

(10.3.26)

Moreover, one verifies easily that Hb = Hb ◦ T.

(10.3.27)

Let us now replace in the above (Y, q) by (iY, −iq), so that the transformation T becomes the complex symplectic transformation T 0 given by T 0 (x, Y, p, q) = (x − iq, Y − ip, p, q) .

(10.3.28)

If Hb0 , H0b are obtained from Hb , Hb by replacing Y by iY , by (10.3.27), we get H0b = Hb0 ◦ T 0 .

(10.3.29)

Incidentally, note that (10.3.29) is correct in spite of the fact that we did not change q into −iq. Equation (10.3.27) gives the Hamiltonian counterpart to the conjugation arguments developed in [B05, section 3.10] and in [B09a, section 1]. In these references, it is shown that when X = S 1 , the scalar hypoelliptic Laplacian on S 1 is conjugate to the sum of the Laplacian on S 1 and of a scaled version of the harmonic oscillator along the fibre of X , via an unbounded self-adjoint conjugation. This conjugation is a simple form of Egorov’s theorem [Hör85b,

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Theorem 25.3.5], [T81, chapter 8, p. 147] with a conjugating operator formally associated with a Fourier integral operator with real phase. Equations (10.5.7)–(10.5.8) will provide an explicit illustration to these somewhat mysterious considerations. Let us now return to the full variational problem in (10.3.1), with x0 = x, Y0 = Y, xt = x0 , Yt = Y 0 . By (10.2.9) or (10.2.16), p is constant, but no longer 0. Instead of (10.3.3), we get b4 Y¨ − Y + p = 0.

(10.3.30)

Of course, (10.3.30) is equivalent to b4 Y (3) − Y 0 = 0,

(10.3.31)

which by (10.1.14) or (10.1.17) can be written in the form b4 x(4) − x ¨ = 0.

(10.3.32)

Note that as it should be, (10.3.32) is just (10.1.8). By (10.3.30), 2

2

Ys = p + Aes/b + Be−s/b .

(10.3.33)

By (10.3.5), we get 2

2

A=

Y − p − et/b (Y 0 − p) , 1 − e2t/b2

B=

Y − p − e−t/b (Y 0 − p) . 1 − e−2t/b2

(10.3.34)

Taking into account the fact that x˙ = Y, x0 = x, xt = x0 , by (10.3.33) and (10.3.34), we get t/b2 − 2 tanh t/2b2 p = (x0 − x) /b2 − tanh t/2b2 (Y + Y 0 ) . (10.3.35) Equation (10.3.35) determines p. Instead of (10.3.10), we get 2

v = p + 2Aes/b .

(10.3.36)

Recall that Hb ((x, Y ) , (p, q)) , Kb ((x, Y ) , (p, q)) were defined in (10.2.2) and (10.2.14). Let Hb,t ((x, Y ) , (x0 , Y 0 )) , Kb,t ((x, Y ) , (x0 , Y 0 )) be the value of Hb,t , Kb,t on the trajectory (x· , Y· ). By (10.1.3), b2 0 2 2 Kb,t ((x, Y ) , (x0 , Y 0 )) = Hb,t ((x, Y ) , (x0 , Y 0 )) + |Y | − |Y | . 2 (10.3.37) The analogue of equation (10.3.11) still holds, i.e., Z 1 t 2 Kb,t ((x, Y ) , (x0 , Y 0 )) = |v| ds. (10.3.38) 2 0 We denote by dx0 ,Y 0 Hb,t ((x, Y ) , (x0 , Y 0 )) the gradient of Hb,t in the variables (x0 , Y 0 ). A similar notation will be used for Kb,t .

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Proposition 10.3.2. The following identities hold: ! 2 2 |Y 0 − Y | b2 2 0 2 0 0 |Y | + |Y | + tanh t/2b Hb,t ((x, Y ) , (x , Y )) = 2 sinh (t/b2 ) 0 1 x − x − b2 tanh t/2b2 (Y + Y 0 ) 2 , (10.3.39) + 2 (t − 2b2 tanh (t/2b2 )) 2 b2 −t/2b2 t/2b2 0 Y − e Y Kb,t ((x, Y ) , (x0 , Y 0 )) = e 2 sinh (t/b2 ) 0 1 x − x − b2 tanh t/2b2 (Y + Y 0 ) 2 . + 2 2 2 (t − 2b tanh (t/2b )) Moreover, we have the Hamilton-Jacobi equations, ∂ Hb,t ((x, Y ) , (x0 , Y 0 )) + Hb ((x0 , Y 0 ) , dx0 ,Y 0 Hb,t ((x, Y ) , (x0 , Y 0 ))) = 0, ∂t (10.3.40) ∂ Kb,t ((x, Y ) , (x0 , Y 0 )) + Kb ((x0 , Y 0 ) , dx0 ,Y 0 Kb,t ((x, Y ) , (x0 , Y 0 ))) = 0. ∂t Proof. By (10.3.7), (10.3.33), and (10.3.37), we get b2 2 2 Hb,t ((x, Y ) , (x0 , Y 0 )) = tanh t/2b2 |Y − p| + |Y 0 − p| (10.3.41) 2 ! 2 |Y 0 − Y | 1 2 + + hp, x0 − xi − |p| t, sinh (t/b2 ) 2 b2 2 2 tanh t/2b2 |Y − p| + |Y 0 − p| (10.3.42) 2 ! 2 1 2 |Y 0 − Y | 2 2 + |Y 0 | − |Y | + hp, x0 − xi − |p| t. + sinh (t/b2 ) 2

Kb,t ((x, Y ) , (x0 , Y 0 )) =

By (10.3.35), (10.3.41), we get (10.3.39). As to (10.3.40), it follows from general arguments in the calculus of variations or from a direct computation. By (10.3.39), we get 1 H1,t/b2 x, b2 Y , x0 , b2 Y 0 , 2 b 1 0 0 Kb,t ((x, Y ) , (x , Y )) = 2 K1,t/b2 x, b2 Y , x0 , b2 Y 0 . b Equation (10.3.43) also follows from (10.1.4), (10.1.5). By (10.3.35), as b → 0, Hb,t ((x, Y ) , (x0 , Y 0 )) =

(10.3.43)

x0 − x . (10.3.44) t By (10.3.33), (10.3.34), (10.3.44), Y· remains uniformly bounded on [0, t], the uniform bound only depends on x0 − x, Y, Y 0 . Moreover, over compact p→

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subsets of ]0, t[, Y· converges uniformly to x −x t . It follows that as b → 0, we have the uniform convergence on [0, t], s s xs → 1 − x + x0 . (10.3.45) t t As b → 0, 1 0 2 |x − x| , (10.3.46) Hb,t ((x, Y ) , (x0 , Y 0 )) → H0,t ((x, Y ) , (x0 , Y 0 )) = 2t 1 0 2 Kb,t ((x, Y ) , (x0 , Y 0 )) → K0,t ((x, Y ) , (x0 , Y 0 )) = |x − x| . 2t By (10.3.36), one finds easily that as b → 0, x0 − x v→ in L2 . (10.3.47) t Now we replace Y, Y 0 by Y /b, Y 0 /b. Inspection of (10.3.35) shows that (10.3.44) still holds. Also by (10.3.33), (10.3.35), Y· converges uniformly to x0 −x on compact subsets of ]0, t[. However, in general, Y· does not remain t uniformly bounded. By (10.3.33), (10.3.34), and by the analogue of (10.3.44), we conclude easily that as b → 0, x0 − x Y· → in Lp , 1 ≤ p < 2. (10.3.48) t Moreover, given 0 < ≤ M < +∞, as long as x0 − x, Y, Y 0 remain uniformly bounded, and ≤ t ≤ M , as b → 0, the convergence in (10.3.48) is uniform, and bY· remains uniformly bounded on [0, t]. By (10.3.48), we conclude that the uniform convergence in (10.3.45) still holds, and that it is uniform as long as ≤ t ≤ M , and x0 − x, Y, Y 0 remain bounded. Under the same conditions, Y· remains uniformly bounded in L2 . Moreover, as b → 0, 1 0 2 Hb,t ((x, Y /b) , (x0 , Y 0 /b)) → H 0,t ((x, Y ) , (x0 , Y 0 )) = |x − x| (10.3.49) 2t 1 2 2 + |Y | + |Y 0 | , 2 1 0 2 2 |x − x| + |Y 0 | . Kb,t ((x, Y /b) , (x0 , Y 0 /b)) → K 0,t ((x, Y ) , (x0 , Y 0 )) = 2t Inspection of (10.3.34)–(10.3.36) shows that as b → 0, x0 − x x0 − x v→ in Lp , 1 ≤ p < 2, v→ weakly in L2 . (10.3.50) t t By (10.3.38) and (10.3.49), in general, the convergence in (10.3.50) is not a strong convergence in L2 . For ≤ t ≤ M , under uniform bounds on x0 − x, Y, Y 0 , for 0 < b ≤ 1, v remains bounded in L2 . Also as b → 0, Z 1 t 1 0 1 2 2 2 2 |Y | ds → |x − x| + |Y | + |Y 0 | , (10.3.51) 2 0 2t 4 Z b4 t ˙ 2 1 2 2 |Y | + |Y 0 | , Y ds → 2 0 4 which fits with (10.3.49).

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Proposition 10.3.3. Given > 0, M > 0, ≤ M , there exists C,M > 0 such that for ≤ t ≤ M, 0 < b ≤ M , and x, x0 , Y, Y 0 ∈ E, 2 2 2 Hb,t ((x, Y /b) , (x0 , Y 0 /b)) ≥ C,M |x0 − x| + |Y | + |Y 0 | . (10.3.52) Proof. This is an easy consequence of (10.3.39). When b → +∞, a necessary and sufficient condition for p in (10.3.35) to converge is that x0 − x =

t (Y + Y 0 ) , 2

(10.3.53)

in which case, p=

1 x0 − x = (Y + Y 0 ) . t 2

(10.3.54)

If (10.3.53) is verified, by (10.3.19), (10.3.33), and (10.3.54), as b → +∞, we have the uniform convergence over [0, t], s s Ys → 1 − (10.3.55) Y + Y 0. t t By (10.3.55), as b → +∞, we have the uniform convergence over [0, t], xs → x + sY +

s2 (Y 0 − Y ) . 2t

(10.3.56)

Under (10.3.53), if Y = Y 0 , as b → +∞, Hb,t ((x, Y ) , (x0 , Y )) →

1 0 2 |x − x| , 2t

Kb,t ((x, Y ) , (x0 , Y )) →

1 0 2 |x − x| . 2t (10.3.57)

We no longer assume (10.3.53) to hold. By (10.3.39), as b → +∞, 2 1 6 0 t Hb,t 2 0 0 0 0 ((x, Y ) , (x , Y )) → |Y − Y | + 3 x − x − (Y + Y ) , b4 2t t 2 (10.3.58) 2 Kb,t 1 6 0 t 2 0 0 0 0 ((x, Y ) , (x , Y )) → |Y − Y | + x − x − (Y + Y ) . 4 3 b 2t t 2 10.4 Mehler’s formula Let E still be an Euclidean vector space of dimension m, and let Y be the generic element of E. Let ∆E be the Laplacian on E. Let OE be the harmonic oscillator 1 2 OE = −∆E + |Y | − m . (10.4.1) 2

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0 E . Given t > 0, let hE t (Y, Y ) be the smooth kernel associated with exp −tO By Mehler’s formula [GlJ87], we get hE t

m/2 et (Y, Y ) = 2π sinh (t) 1 tanh (t/2) 2 2 0 2 0 |Y | + |Y | − exp − |Y − Y | . (10.4.2) 2 2 sinh (t)

0

Let P E be the operator 2 2 P E = exp |Y | /2 OE exp − |Y | /2 .

(10.4.3)

Then PE =

1 −∆E + 2∇Y . 2

The corresponding heat kernel ktE (Y, Y 0 ) is given by 1 2 E 0 0 2 0 kt (Y, Y ) = exp |Y | − |Y | hE t (Y, Y ) . 2

(10.4.4)

(10.4.5)

Equivalently, et 2π sinh (t)

m/2

2 1 t/2 0 −t/2 exp − Y . (Y, Y ) = e Y − e 2 sinh (t) (10.4.6) Equation (10.4.6) indicates that given Y , ktE (Y, Y 0 ) dY 0 is the probability law of a Gaussian variable, centered at e−t Y , with variance given by 1 − e−2t /2. If f : E → R is smooth, for a > 0, set ktE

0

Ka f (Y ) = f (aY ) .

(10.4.7)

Set ObE = Kb

OE −1 K , b2 b

PbE = Kb

P E −1 K . b2 b

(10.4.8)

Then ObE

1 = 2

∆E m 2 − 4 + |Y | − 2 b b

,

PbE = −

1 E 1 ∆ + 2 ∇Y . 2b4 b

Also by (10.4.3), we get 2 2 PbE = exp b2 |Y | /2 ObE exp −b2 |Y | /2 .

(10.4.9)

(10.4.10)

0 E Let hE (Y, Y 0 ) be the smooth kernels associated with the opb,t (Y, Y ) , k b,t E erators exp −tOb , exp −tPbE . By (10.4.8), we get 0 m E 0 hE b,t (Y, Y ) = b ht/b2 (bY, bY ) ,

E E 0 kb,t (Y, Y 0 ) = bm kt/b 2 (bY, bY ) . (10.4.11)

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Proposition 10.4.1. The following identities hold: #m/2 " 2 b2 et/b ∗ E 0 exp −Hb,t (Y, Y 0 ) , hb,t (Y, Y ) = 2 2π sinh (t/b ) " #m/2 2 t/b2 b e E ∗ kb,t (Y, Y 0 ) = exp −Kb,t (Y, Y 0 ) . 2π sinh (t/b2 )

(10.4.12)

Proof. This follows from (10.3.7), (10.4.2), (10.4.6), and (10.4.11). Remark 10.4.2. Equation (10.4.11) can also be viewed as a consequence of (10.3.9) and (10.4.12).

10.5 The hypoelliptic heat kernel on an Euclidean vector space Now we replace E by E ⊕ E, whose generic element is denoted (x, Y ). The operators ObE , PbE still act on the variable Y . We denote by ∇H differentiation in x. Also ∆E,H , ∆E,V denote the Laplacians on the first and second copies of E. E Let QE b , Rb be the operators on E ⊕ E, E H QE b = Ob − ∇ Y ,

RbE = PbE − ∇H Y .

(10.5.1)

By (10.4.10), we get 2 2 2 RbE = exp b2 |Y | /2 QE exp −b |Y | /2 . b

(10.5.2)

0 0 E Let hE )) , kb,t ((x, Y ) , (x0 , Y 0 )) be the smooth kernels asb,t ((x, Y ) , (x , Y E sociated with exp −tQb , exp −tRbE . By (10.5.2), 2 b 2 E 0 0 0 2 0 0 kb,t ((x, Y ) , (x , Y )) = exp |Y | − |Y | hE b,t ((x, Y ) , (x , Y )) . 2 (10.5.3) Also one verifies easily that 0 0 E 0 0 hE b,t ((x, Y ) , (x , Y )) = h1,t/b2 ((x/b, bY ) , (x /b, bY )) , E kb,t

0

0

((x, Y ) , (x , Y )) =

E k1,t/b 2

0

(10.5.4)

0

((x/b, bY ) , (x /b, bY )) .

Clearly, QE b

1 = 2

∆E,V 1 m H 2 − 4 + Y − ∇· − 2 − ∆E,H . b b 2

(10.5.5)

Let e1 , . . . , em be an orthonormal basis of E, and let xi , Y i , 1 ≤ i ≤ m be the coordinates of x, Y with respect to this basis. Set N

E

=

m X i=1

∂2 . ∂xi ∂Y i

(10.5.6)

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E

The operator exp N acts on functions f (x, Y ) that are polynomial in the variables x, Y . Using (10.5.5), a formal calculation shows that 1 E,H E E E exp N E . (10.5.7) Qb = exp −N Ob − ∆ 2 Equation (10.5.7) can be viewed as the quantization of equation (10.3.29). Here we use the correspondence p → i∇H . Equivalently, (10.5.7) is an exact form of Egorov’s theorem, in which the conjugating operator is unbounded. Sense can be made of the conjugation, once we use the fact that the eigenfunctions of the harmonic oscillator are analytic in the variable Y . This issue is discussed in detail in [B05, section 3.10], [B08c, section 2.9], and [B09a, section 1.2]. By (10.5.7), we get 1 E,H E E ∆ exp N E . (10.5.8) exp −tQE = exp −N exp −t O − b b 2 Proposition 10.5.1. The following identities hold: " #m/2 2 t/b2 b e 0 0 hE b,t ((x, Y ) , (x , Y )) = 4π 2 sinh (t/b2 ) (t − 2b2 tanh (t/2b2 )) exp (−Hb,t ((x, Y ) , (x0 , Y 0 ))) , #m/2 " 2 t/b2 b e E kb,t ((x, Y ) , (x0 , Y 0 )) = 4π 2 sinh (t/b2 ) (t − 2b2 tanh (t/2b2 )) exp (−Kb,t ((x, Y ) , (x0 , Y 0 ))) . (10.5.9) Proof. By (10.5.8), we deduce that 0 0 E H 0 H hE b,t ((x, Y ) , (x , Y )) = hb,t Y − ∇· , Y − ∇· 1 2 0 exp − |x − x| /2t . (10.5.10) m/2 (2πt) By (10.4.12) and (10.5.10), we obtain "

hE b,t

2

b2 et/b ((x, Y ) , (x , Y )) = 2π sinh (t/b2 ) 0

#m/2

0

∗ 0 H exp −Hb,t Y − ∇H + t∆E,H /2 (x, x0 ) . (10.5.11) · , Y − ∇· Moreover, using (10.3.7), we get ∗ 0 H ∗ Hb,t Y − ∇H − t∆E,H /2 = Hb,t (Y, Y 0 ) · , Y − ∇· b2 − t/b2 − 2 tanh t/2b2 ∆E,H − b2 tanh t/2b2 ∇H Y +Y 0 . (10.5.12) 2

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By (10.5.11), (10.5.12), we obtain "

hE b,t

2

b2 et/b ((x, Y ) , (x , Y )) = 4π 2 sinh (t/b2 ) (t − 2b2 tanh (t/2b2 )) 0

#m/2

0

∗ exp −Hb,t (Y, Y 0 ) −

1 2 (t − 2b2 tanh (t/2b2 ))

! 0 x − x − b2 tanh t/2b2 (Y + Y 0 ) 2 . (10.5.13) Using (10.3.7), (10.3.39), and (10.5.13), we get the first identity in (10.5.9). The second identity then follows from (10.3.37), (10.5.3), and from the first identity. ∗ Remark 10.5.2. The fact that Hb,t (Y, Y 0 ) and Hb,t ((x, Y ) , (x0 , Y 0 )) appear E 0 0 in equations (10.4.12) for hb,t (Y, Y 0 ) and (10.5.9) for hE b,t ((x, Y ) , (x , Y )) can also be viewed as a consequence of the Hamilton-Jacobi equations in (10.3.8), (10.3.40). Since Z E kb,t ((x, Y ) , (x0 , Y 0 )) dx0 dY 0 = 1, (10.5.14) E⊕E

the normalizing factor in the right-hand side of (10.5.9) reflects the explicit form of Kb,t ((x, Y ) , (x0 , Y 0 )) in (10.3.39). Also (10.5.4) can be viewed as a consequence of (10.3.43) and (10.5.9). By combining (10.3.52) and (10.5.9), we find that given > 0, M > 0, ≤ M , there exist c,M > 0, C,M > 0 such that for ≤ t ≤ M, 0 < b ≤ M , 0 0 b−m hE b,t ((x, Y /b) , (x , Y /b)) 2 2 2 ≤ c,M exp −C,M |x0 − x| + |Y | + |Y 0 | . (10.5.15)

Equation (10.5.15) is just a version of the estimate in (4.5.3). Moreover, by E (10.3.58), (10.5.9), as b → +∞, the kernels hE b,t , kb,t concentrate along the trajectories of the geodesic flow of E in E ⊕ E.

10.6 Orbital integrals on an Euclidean vector space The results contained in this section will not be needed later. We use again the formalism of chapters 2, 4, and 6, in the case where G = E. Multiplication in G will be denoted additively. Then K = {0}. The Lie algebra g of G is given by g = E,

(10.6.1)

so that p = E,

k = 0.

(10.6.2)

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Also the scalar product of E is defined to be the nondegenerate bilinear symmetric form B on g. The symmetric space X is just E, and G acts on X by translations. Moreover, T X = E, N = {0}, and the Euclidean and flat connections on T X coincide. Clearly, X = Xb, and moreover, X = E ⊕ E.

(10.6.3)

In (10.6.3), the first copy of E is identified with X, and the second copy with T X. Since K = {0}, the quotient procedures of section 2.12 become irrelevant. We use the notation of section 10.5. In particular, (x, Y ) still denotes the generic section of E ⊕E. In the sequel, the operators that will be constructed act on C ∞ (E ⊕ E, Λ· (E ∗ )). Let e1 , . . . , em be an orthonormal basis of E. By (2.7.2), we get m m X X bg = c (ei ) ∇H , D b c (ei ) ∇H (10.6.4) Dg = ei ei . i=1

i=1

By (2.7.7), Dg,2 = ∆E,H .

(10.6.5)

b g and D b g,X coincide. Since K = {0}, we may as well take F = R. Then D By (2.9.4), (2.9.5), or by (2.12.19), we get √ b g,X + 2 dE + dE∗ . b g,X + 1 DE + E E = D (10.6.6) DX = D b b b By (2.11.4) or by (2.12.19), we get N Λ· (E ∗ ) 1 1 1 1 X,2 2 Db = ∆E,H + 2 −∆E,V + |Y | − m + + ∇H . (10.6.7) 2 2 2b b2 b Y By (2.13.3), 1 LX = − ∆E,H . (10.6.8) 2 By (2.13.5), N Λ· (E ∗ ) 1 1 2 X E,V Lb = 2 −∆ + ∇H . (10.6.9) + |Y | − m + 2b b2 b Y 0 For t > 0, let pt (x, x ) be the smooth heat kernel on E associated with exp t∆E /2 . Then 1 1 0 2 pt (x, x0 ) = exp − |x − x| . (10.6.10) m/2 2t (2πt) X For b > 0, t > 0, let qb,t ((x, Y ) , (x0 , Y 0 )) be the smooth kernel associated with exp −tLX b .

Proposition 10.6.1. For b > 0, t > 0, the following identity holds: X qb,t ((x, Y ) , (x0 , Y 0 ))

0 0 Λ· (E ∗ ) 2 = b−m hE /b . (10.6.11) b,t ((x, −Y /b) , (x , −Y /b)) exp −tN

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Proof. This follows from (10.4.9), (10.5.1), and (10.6.9). Let a ∈ p = E. Then γ = ea acts by translation by a on the first copy of E ⊕ E. Since G = E is commutative, we will use the notation γ instead of [γ]. Proposition 10.6.2. For b > 0, t > 0, the following identities hold: (10.6.12) Trγ exp t∆E,H /2 = pt (0, a) , Z 2 m Trs γ exp −tLX = 1 − e−t/b hE b b,t ((0, Y ) , (a, Y )) dY. E

Proof. The first identity in (10.6.12) is trivial, and the second is a consequence of equation (10.6.11) in Proposition 10.6.1. The crucial identity (4.6.1) in Theorem 4.6.1 asserts that Trs γ exp −tLX = Trγ exp t∆E,H /2 . b

(10.6.13)

In the special case we are considering, equation (10.6.13) can also be obtained as a consequence of (10.5.8). Let us give a computational proof of (10.6.13). By equation (10.3.39) in Proposition 10.3.2 for Hb,t , we get easily 2 b2 t tanh t/2b2 a 2 |a| Hb,t ((0, Y ) , (a, Y )) = + − . (10.6.14) Y t − 2b2 tanh (t/2b2 ) t 2t By (10.6.14), we obtain Z exp (−Hb,t ((0, Y ) , (a, Y ))) dY E

=

t − 2b2 tanh t/2b2 π b2 t tanh (t/2b2 )

!m/2

2

|a| exp − 2t

! . (10.6.15)

By equation (10.5.9) in Proposition 10.5.1 and by (10.6.15), we get Z hE b,t ((0, Y ) , (a, Y )) dY E

2

=

et/b 4πt tanh (t/2b2 ) sinh (t/b2 )

!m/2

2

|a| exp − 2t

! . (10.6.16)

Equivalently, ! Z 2 −m |a| −m/2 E −t/b2 hb,t ((0, Y ) , (a, Y )) dY = 1 − e (2πt) exp − . 2t E (10.6.17) Equation (10.6.13) follows from (10.6.12), (10.6.17). Finally, let us reinterpret equation (10.6.13), by getting rid of γ. The enveloping algebra U (g) can be identified with the algebra of differential

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X operators with constant coefficients on E. The operator Lb can Xbe viewed ∞ · ∗ as acting on C (E, Λ (E ) ⊗ U (g)). For t > 0, Trs exp −tLb is the supertrace of a trace class operator, with values in a completion of U (g). Equation (10.6.13) then says that for any b > 0, t > 0, Trs exp −tLX = exp t∆E,H /2 . (10.6.18) b

Equation (10.6.18) can also be obtained as a consequence of (10.5.8). Equation (10.6.18) should be thought of as an analogue of the McKeanSinger formula [McKS67] for the index of a Dirac operator. In this case, for a given t > 0, the index is the operator exp t∆E,H /2 . The fact that for t > 0, the right-hand side of (10.6.18) does not depend on b > 0 is one key feature of a McKean-Singer formula. From the point of view of the present book, as b → 0, Trs exp −tLX to the right-hand side, which is an b converges converges to the explicit formula operator. As b → +∞, Trs exp −tLX b for the heat kernel of E. In index theory, this is called a local index theorem [ABoP73, Gi73, Gi84]. Let O (E) be the orthogonal group of E, let I (E) be the group of isometries of E. Then I (E) is the semidirect product I (E) = E o O (E) .

(10.6.19)

Clearly, we have the identification of symmetric spaces, E = I (E) /O (E) .

(10.6.20)

The group I (E) is not reductive, so that, strictly speaking, the methods and results contained in this book do not apply. However, if γ ∈ I (E), for t > 0, the orbital integral Tr[γ] exp t∆E /2 is well-defined, and its computation is elementary. The objects introduced in equations (10.6.4)–(10.6.9) are still adequate to study the above orbital integral by the methods used in this book. In particular we still take k = 0, although this is not the Lie algebra of O (E), and equation (4.6.1) in Theorem 4.6.1 still holds, i.e., for t > 0, (10.6.21) Trs [γ] exp −tLX = Tr[γ] exp t∆E,H /2 . b We will now verify (10.6.21) directly. With respect to what we did before, it is enough to take k ∈ O (E) with no eigenvalue equal to 1, and to assume that γ = k −1 . Proposition 10.6.3. For b > 0, t > 0, the following identities hold: 1 , det (1 − k −1 ) −t/b2 −1 Trs [γ] exp −tLX = det 1 − e k b Z E kb,t (x, Y ) , k −1 (x, Y ) dxdY.

Tr[γ] exp t∆E,H /2 =

E×E

(10.6.22)

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Proof. Note that with X = E, then X (γ) is reduced to 0. By (4.2.12), we get Z Tr[γ] exp t∆E,H /2 = pt x, k −1 x dx. (10.6.23) E

By (10.6.10), (10.6.23), we obtain the first identity in (10.6.22). Moreover, Z [γ] X X Trs exp −tLb = Trs k −1 qb,t (x, Y ) , k −1 (x, Y ) dxdY. E×E

(10.6.24) Also by (10.5.3), and by equation (10.6.11) in Proposition 10.6.1, we get X qb,t (x, Y ) , k −1 (x, Y ) · ∗ E = b−m kb,t (x, −Y /b) , k −1 (x, −Y /b) exp −tN Λ (E ) /b2 . (10.6.25) The second identity in (10.6.22) follows from (10.6.24), (10.6.25). Remark 10.6.4. With the proviso indicated before, the methods used in the proof of Theorem 6.1.1 still apply. In particular, the first identity in (10.6.22) can be viewed as a consequence of equation (6.1.2) in Theorem 6.1.1. Now we will verify (10.6.21) directly using Proposition 10.6.3. By equation (10.3.39) in Proposition 10.3.2, we get 2 2 b2 et/b −t/b2 1 − e k Y 2 sinh (t/b2 ) 1 − k −1 x + b2 tanh t/2b2 1 + k −1 Y 2 .

Kb,t (x, Y ) , k −1 (x, Y ) = +

1 2 (t − 2b2 tanh (t/2b2 ))

(10.6.26) By (10.6.26), we get Z exp −Kb,t (x, Y ) , k −1 (x, Y ) dxdY E×E

1 −1 det (1 − k ) 1 − e−t/b2 k h im/2 2 4π 2 b−2 e−t/b sinh t/b2 t − 2b2 tanh t/2b2 . (10.6.27) =

By (10.5.9), (10.6.27), we obtain Z E kb,t (x, Y ) , k −1 (x, Y ) dxdY =

1 . −1 det (1 − k ) 1 − e−t/b2 k E×E (10.6.28) 2 2 Since k is an isometry, we can replace 1 − e−t/b k by 1 − e−t/b k −1 in the right-hand side of (10.6.28). By the second identity in (10.6.22) and by (10.6.28), we get 1 Trs [γ] exp −tLX = . (10.6.29) b det (1 − k −1 )

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Equation (10.6.29) fits with (10.6.21) and (10.6.22). The behavior of the integral in the right-hand side of the second identity in (10.6.22) says much about the hypoelliptic orbital integrals on symmetric spaces. One can directly observe the concentration as b → +∞ of this integral near X (γ) = {0}.

10.7 Some computations involving Mehler’s formula By (10.4.6), if q < et / sinh (t), we get Z q 1 2 |Y 0 | dY 0 = ktE (Y, Y 0 ) exp m/2 −t 2 E (1 − qe sinh (t)) q 2 e−2t |Y | / 1 − qe−t sinh (t) . (10.7.1) exp 2 In particular, from (10.7.1), we deduce that if q < 2, Z q q −2t 1 2 2 ktE (Y, Y 0 ) exp |Y 0 | dY 0 ≤ exp e |Y | . m/2 2 2−q E (1 − q/2) (10.7.2) By (10.7.1), we find that if q ≤ et / cosh (t), Z q 1 2 2 ktE (Y, Y 0 ) exp exp |Y | /2 . (10.7.3) |Y 0 | dY 0 ≤ m/2 2 E (1 − q/2) Take > 0. By (10.7.3), for t ≥ , q = e / cosh (), Z q 1 2 2 ktE (Y, Y 0 ) exp exp |Y | /2 . |Y 0 | dY 0 ≤ m/2 2 E (1 − q/2) From (10.7.4), we deduce that for p ∈ N, t ≥ , Z 2 p 2 exp − |Y | /2 ktE (Y, Y 0 ) exp |Y 0 | /2 |Y 0 | dY 0 ≤ C,p .

(10.7.4)

(10.7.5)

E

By taking q = 1 + u in (10.7.1), with u < coth (t), we get Z 1+u 0 2 2 E 0 exp − |Y | /2 kt (Y, Y ) exp |Y | dY 0 2 E m/2 et 1 u coth (t) − 1 2 = exp |Y | . (10.7.6) cosh (t) − u sinh (t) 2 coth (t) − u By (10.7.6), if 0 ≤ u < 1, t ≥ , Z 1 2 2 exp − |Y | /2 ktE (Y, Y 0 ) exp (1 + u) |Y 0 | dY 0 2 E m/2 2 1 − u coth () 2 ≤ exp − tanh () |Y | /2 . (10.7.7) 1−u 1 − u tanh ()

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Note that (10.7.7) does not follow from (10.7.2) as can be seen by making q = 1, or u = 0. From (10.7.7), we deduce that for t ≥ , p ∈ N, Z 2 p 2 exp − |Y | /2 ktE (Y, Y 0 ) exp |Y 0 | /2 |Y 0 | dY 0 E 1 2 ≤ C,p exp − tanh () |Y | . (10.7.8) 4 By (10.7.1), Z 2 2 exp − |Y | /2 ktE (Y, Y 0 ) exp |Y 0 | /2 dY 0 E

=

et cosh (t)

m/2

tanh (t) 2 exp − |Y | ≤ 2m/2 . (10.7.9) 2

By (10.4.5), (10.7.6), we find that if u < coth (t), m/2 Z et 0 0 2 0 hE (Y, Y ) exp u |Y | /2 dY = t cosh (t) − u sinh (t) E 1 u coth (t) − 1 2 |Y | . (10.7.10) exp 2 coth (t) − u By (10.7.10), we deduce that for u < coth (t), Z 0 0 2 hE (Y, Y ) exp u |Y | /2 dY 0 t E

≤

2 1 − u tanh (t)

By (10.7.10), we get Z 0 0 hE (Y, Y ) dY = t E

m/2

et cosh (t)

exp

m/2

1 u coth (t) − 1 2 |Y | . (10.7.11) 2 coth (t) − u

tanh (t) 2 exp − |Y | , 2

(10.7.12)

and the right-hand side of (10.7.12) is uniformly bounded for t > 0. By (10.4.5), (10.7.8), given > 0, p ∈ N, there is C,p > 0 such that for t ≥ , Z tanh () 2 0 0 p 0 hE (Y, Y ) |Y | dY ≤ C exp − |Y | . (10.7.13) ,p t 4 E 10.8 The probabilistic interpretation of the harmonic oscillator Let P be the probability measure on C (R+ , E) of the Brownian motion s ∈ R+ → ws ∈ E, with w0 = 0. For 1 ≤ p < +∞, k kp denotes the standard norm on the vector space Lp associated with the measure P on C (R+ , E). For Y ∈ E, consider the stochastic differential equation Y˙ = −Y + w, ˙ Y0 = Y. (10.8.1)

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This equation can be written in the form Z t Yt = Y − Ys ds + wt .

(10.8.2)

0

Note that (10.8.2) can be integrated for any w· ∈ C (R+ , E). The explicit solution of (10.8.2) is given by Z t −t −t es δws . (10.8.3) Yt = e Y + e 0

In the right-hand side of (10.8.3), the last term is an Itô integral. In particular, from (10.8.3), we deduce that the probability law of Yt is a Gaussian centered at e−t Y with variance 1 − e−2t /2, i.e., the probability law of Yt is given by ktE (Y, Y 0 ) dY 0 . Note that since es is a smooth function, Z t Z t es δws = et wt − es ws ds, (10.8.4) 0

0

so that the machinery of stochastic integration is not needed to make sense of the integral in (10.8.4). Let E be the expectation operator with respect to P . It is a standard consequence of the Itˆ o calculus that if f ∈ C ∞,c (E, R), for t ≥ 0, exp −tP E f (Y ) = E [f (Yt )] . (10.8.5) By (10.8.5), we recover the above result on the probability law of Yt . Let Y ∗ be the solution of Y˙ ∗ = w, ˙ Y0∗ = Y, (10.8.6) so that Yt∗ = Y + wt .

(10.8.7)

∞,c

Let f ∈ C (E, R). Then using the Itô calculus and the Feynman-Kac formula, we get Z mt 1 t ∗ 2 − |Ys | ds f (Yt∗ ) . (10.8.8) exp −tOE f (Y ) = E exp 2 2 0 By (10.4.3), (10.8.5), we also obtain h i 2 2 exp −tOE f (Y ) = exp − |Y | /2 E exp |Yt | /2 f (Yt ) .

(10.8.9)

One can verify directly that the right-hand side of (10.8.9) is finite. Indeed 2 by (10.7.1), given Y , for any t ≥ 0, exp |Yt | /2 ∈ L2 , and its L2 norm remains uniformly bounded for bounded t. Proposition 10.8.1. Given p ∈ [1, +∞[ there is Cp > 0 such that for t ≥ 0, kYt kp ≤ e−t |Y | + Cp .

(10.8.10)

Given M ≥ 0, p ∈ [1, +∞[, there is CM,p > 0 such that for 0 ≤ t ≤ M, b > 0,

Z 1/2

t 4

2 Ys/b2 ds (10.8.11)

≤ CM,p 1 + b |Y | .

0

p

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The action functional and the harmonic oscillator t

e For t ≥ 0, and 1 ≤ q ≤ cosh(t) ,

1

2 2 exp |Y | /2q .

exp |Yt | /2 ≤ m/2q q (1 − q/2)

(10.8.12)

Proof. Equation (10.8.10) is an easy consequence of the above results on the probability law of Yt . Let Y·0 be the solution of (10.8.1) with Y0 = 0, so that in (10.8.3), Yt = e−t Y + Yt0 .

(10.8.13)

Then Z

t

1/2 ! Z t 0 4 . b |Y | + Ys/b2 ds

1/2 Ys/b2 4 ds ≤C

2

(10.8.14)

0

0

Moreover, Z t 1/2 Z t 1 0 4 0 4 ≤ ds 1 + ds . Ys/b2 Ys/b2 2 0 0 By H¨ older’s inequality, for 0 ≤ t ≤ M , 1/p Z t Z t 0 4 0 4p . Ys/b2 ds ≤ CM,p Ys/b2 ds 0

(10.8.16)

0

By (10.8.16), we get

Z t

Z t 1/p

0 4 0 4p

≤ C ds E ds . Ys/b2 Ys/b2 M,p

0

(10.8.15)

(10.8.17)

0

p

h i 4p By (10.8.10) with Y = 0, for s ≥ 0, E Ys0 is uniformly bounded, so that from (10.8.17), for 0 ≤ t ≤ M ,

Z t

0 4

(10.8.18) Ys/b2 ds

≤ CM,p . 0

p

By (10.8.14)–(10.8.18), we get (10.8.11). Since the probability law of Yt is equal to ktE (Y, Y 0 ) dY 0 , we get h i Z 2 2 E exp q |Yt | /2 = ktE (Y, Y 0 ) exp q |Y 0 | /2 dY 0 . (10.8.19) E

By (10.7.3) and (10.8.19), we get (10.8.12). The proof of our proposition is completed. For b > 0, instead of equation (10.8.1), we consider the stochastic differential equation Yb w˙ Y˙ b = − 2 + , b b The solution of (10.8.20) is given by 2

Y0b = Y.

2

Ytb = e−t/b Y + e−t/b

Z 0

t

2

es/b

w˙ s ds. b

(10.8.20)

(10.8.21)

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Instead of (10.8.5), we have the more general, t E f (Y ) = E f Ytb . exp − 2 P b

(10.8.22)

There is still another method to construct the probability law of the process Y·b . Set w·b = bw·/b2 .

(10.8.23)

Then it is well-known that w·b is still a Brownian motion. Put Yb,t = Yt/b2 .

(10.8.24)

Then Yb w˙ b Y˙ b = − 2 + , b b

Yb,0 = Y.

(10.8.25)

By (10.8.1) and (10.8.25), the processes Y·b and Yb,· have the same probability law. Proposition 10.8.2. Given p ∈ [1, +∞[ there is Cp > 0 such that for t ≥ 0,

b

Yt ≤ e−t/b2 |Y | + Cp . (10.8.26) p Given M ≥ 0, p ∈ [1, +∞[, there is CM,p > 0 such that for 0 ≤ t ≤ M, b > 0,

Z 1/2

t

2 b 4 Ys ds (10.8.27)

≤ CM,p 1 + b |Y | .

0

p

Proof. This is an obvious consequence of Proposition 10.8.1 and of the fact that we may as well replace Y·b by Y·/b2 in the above inequalities. A direct proof can also be given along the lines of the proof of Proposition 10.8.1.

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Chapter Eleven The analysis of the hypoelliptic Laplacian The purpose of this chapter is to construct a functional analytic machinery that is adapted to the analysis of the hypoelliptic Laplacian LX A,b . Our constructions are inspired by our previous work with Lebeau [BL08, chapter 15]. There are two differences with [BL08]. The first difference is that the symmetric space X is noncompact, while the base manifold in [BL08] was assumed to be compact. A second difference is that in LX A,b , the quartic N T X 2 1 appears. Strictly speaking, it is not directly accessible term 2 Y , Y to the methods of [BL08]. The analysis of the hypoelliptic Laplacian essentially consists in the construction of Sobolev spaces on which the operators LX A,b act as unbounded operators, and in the proof of regularizing properties of their resolvents and of their heat operators. The heat operators are shown to be given by smooth kernels. In [BL08, chapter 15], when the base X is compact, one idea is to construct a Littlewood-Paley decomposition of functions along the fibres of the vector bundle T X, and to adapt the method developed by Kohn [Koh73] in his treatment of hypoelliptic second order operators. While still using the results of [BL08, chapter 15], a new ingredient in the present chapter is the construction of a Littlewood-Paley decomposition of functions also on the base X. To make the analysis easier, we first work with a scalar hypoelliptic operator AX b over X , to which part of the analysis of [BL08] can be adapted. This operator does not contain a quartic term. The results on AX b then easily b, which also does not contain extend to a scalar operator AX acting over X b b the quartic term. A scalar operator AX b on X containing the quartic term is introduced. Finally, we extend the analysis to the operator LX A,b . This chapter is organized as follows. In section 11.1, we define the scalar X operator AX b over X , and a conjugate operator Bb . In section 11.2, in the case where the base manifold X is compact, we describe a Littlewood-Paley decomposition of functions along the fibres T X, and the chain of Sobolev spaces constructed in [BL08, chapter 15]. In section 11.3, we construct another Littlewood-Paley decomposition of functions on X. In section 11.4, we combine the Littlewood-Paley decompositions of sections 11.2 and 11.3, and we obtain a Littlewood-Paley decomposition of functions on X , and a corresponding chain of Sobolev spaces. Also, along

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the lines of [BL08], we establish key estimates on the resolvent of AX b with respect to the Sobolev spaces. In section 11.5, using the results of sections 11.2–11.4, we construct the X heat operators associated with AX b , Bb and the corresponding smooth heat X X kernels rb,t , sb,t . In section 11.6, we obtain corresponding results for the scalar operators X b AX b , Bb over X . b In section 11.7, we extend these results to a scalar operator AX b over X N T X 2 1 . The main difficulty is containing the extra quartic term 2 Y , Y that the methods of [BL08] have to be modified to accommodate this term. Finally, in section 11.8, we prove corresponding results for the hypoelliptic Laplacian LX A,b and its heat kernel. As explained in the introduction, except in the present chapter, most of the analysis in the book is based on the probabilistic construction of the heat kernels. In chapter 12, we establish the nontrivial fact that the heat kernels that are obtained via probability theory coincide with the heat kernels that are obtained here. In the whole chapter, we fix the parameter b > 0. X 11.1 The scalar operators AX b , Bb on X

In this section, we will consider scalar differential operators acting on X . Recall that ∆T X denotes the Laplacian acting along the fibres T X of X . Also ∇Y T X denotes the first order operator associated with the vector field V , which is the generator of the geodesic flow on X . The notation is compatible with the notation of chapter 2. Let AX b be the scalar differential operator on X , 1 2 1 AX −∆T X + Y T X − m − ∇Y T X . (11.1.1) b = 2 2b b Set 2 T X 2 BbX = exp Y T X /2 AX exp − Y /2 . (11.1.2) b Then BbX =

1 1 −∆T X + 2∇VY T X − ∇Y T X . 2 2b b

(11.1.3)

There is an associated operator BbG on G×p. Indeed if Y p ∈ p, we still denote by ∇Y p the associated left-invariant vector field on G. Also ∇VY p denotes the obvious radial vector field along p. Set 1 1 BbG = 2 −∆p + 2∇VY p − ∇Y p . (11.1.4) 2b b Let p still denote the obvious projection G × p → X . If F ∈ C ∞ (X, R), then BbG p∗ F = p∗ BbX F.

(11.1.5)

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11.2 The Littlewood-Paley decomposition along the fibres T X Recall that b > 0 is fixed once and for all. In the sequel, positive constants may well depend on b, but the dependence will not be written explicitly. For x ∈ X, r > 0, let B (x, r) be the open ball of center x and radius r, and let B (x, r) be the corresponding closed ball. We use the notation x0 = p1. Let Z be the m-dimensional sphere obtained from B (x0 , 3) by collapsing the boundary to a point. The compact group K acts on Z. Let g T Z be a K-invariant Riemannian metric on T Z, which restricts to the metric g T X on B (0, 2). Let Z be the total space of the vector bundle T Z over Z, and let π still denote the projection Z → Z. Let ∇T Z be the Levi-Civita connection on T Z. Let T H Z ⊂ T Z be the horizontal subbundle associated with ∇T Z . Clearly T H Z ' T Z, so that T Z = T Z ⊕ T Z. Let g T Z denote the obvious direct sum metric on T Z. Let H be the Hilbert space of real square-integrable functions on Z, and let | |H be the corresponding L2 norm. Now we follow [BL08, section 15.2]. Take r0 ∈]1, 2[. Let φ (r) be a smooth function defined on R+ with values in [0, 1], which is decreasing and is such that φ (r) = 1 if r ≤ r10 , and φ (r) = 0 if r ≥ 1. Set χ (r) = φ (r/2) − φ (r) . The support of χ is included in

[ r10 , 2].

(11.2.1)

For j ∈ N, put

χj (r) = χ(2−j r).

(11.2.2)

Clearly, φ (r) +

∞ X

χj (r) = 1.

(11.2.3)

j=0

By (11.2.3), +∞ X

χ2j ≤ 1.

(11.2.4)

j=0

If Y T Z ∈ T Z, put 2 1/2 < Y T Z >= 1 + Y T Z .

(11.2.5)

For u ∈ C ∞ (Z, R) , j ∈ N, set δj (u) = χj < Y T Z > u.

(11.2.6)

By (11.2.3), (11.2.6), we obtain the Littlewood-Paley decomposition, u=

∞ X

δj (u).

(11.2.7)

j=0

Set n o 2 B = Y T Z ∈ T Z, Y T Z ≤ 3 .

(11.2.8)

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The support of δ0 (u) is included in the ball B. For j ≥ 1, the support of the δj (u) is included in the annulus Cj given by n o Cj = Y T Z , < Y T Z >∈ [2j /r0 , 2j+1 ] . (11.2.9) Note that Cj ∩ Cj+2 = ∅.

(11.2.10)

Using (11.2.4), (11.2.7) and (11.2.10), we get +∞ X

2

2

|δj (u)|H ≤ |u|H ≤ 3

j=0

+∞ X

2

|δj (u)|H .

(11.2.11)

j=0

Definition 11.2.1. For u ∈ C ∞ (Z, R), set Uj (x, Y T Z ) = δj (u)(x, 2j Y T Z ).

(11.2.12)

Let R be the annulus, n 2 1 o 1 R = Y T Z , Y T Z ∈ [ 2 − , 4] . r0 4

(11.2.13)

For any j ∈ N, Uj ∈ C ∞,c (Z, R). Moreover, the support of U0 is included in the ball B, and for j ≥ 1, the support of the Uj is included in R. We recover u from U by the formula ∞ X

Uj (x, 2−j Y T Z ).

(11.2.14)

n o 2 B0 = Y T Z ∈ T Z, Y T Z ≤ 5 .

(11.2.15)

u(x, Y

TZ

)=

j=0

Put

Let Y be the total space of the projectivization P (T Z ⊕ R) of the vector bundle T Z over Z. Then K acts on Y . Let g T Y be a K-invariant metric on Y , which coincides with g T Z on B0 . Let ∆Y be the Laplace-Beltrami operator on Y . As in [BL08, eq. (15.3.1)], set S = −∆Y + 1.

(11.2.16)

As in [BL08, eq. (15.4.11)], for τ > 0, set Λτ = S + τ −4

1/2

.

(11.2.17)

Now we recall a definition given in [BL08, eqs. (15.3.3) and (15.4.12)]. Definition 11.2.2. Let H be the Hilbert space of real square-integrable functions on Y , and let | |H be the corresponding L2 norm. For s ∈ R, U ∈ C ∞ (Y, R), put |U |τ,s = τ −m/2 |Λsτ U |H .

(11.2.18)

The completion of C ∞ (Y, R) for the norm | |τ,s is the Sobolev space Hs on Y.

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Note that in [BL08, eq. (14.4.12)], the factor τ −m/2 has been omitted, while it is present in [BL08, eq. (15.3.3)] when τ = 2−j . All the computations in [BL08] are correct even without the factor τ −m/2 anyway. Take u ∈ C ∞,c (Z, R). For j ∈ N, the support of Uj is included in B0 , so that Uj can be viewed as an element of C ∞ (Y, R). Now we follow [BL08, Definition 15.3.1 and eqs. (15.5.16), (15.5.20)]. Definition 11.2.3. If u ∈ C ∞,c (Z, R), if the Uj are defined as in (11.2.12), put ∞ X 2 2 kuks = (11.2.19) |Uj |2−j ,s , j=0 2

|||u|||s =

+∞ X

2 2 23j/2 |Uj |2−j ,s + 2−5j/4 ∇V Uj 2−j ,s−1/8 .

j=0

We denote by H , W s the completions of C ∞,c (Z, R) with respect to the norms k ks , |||u|||s . Note that W s is denoted W s−1/4 in [BL08]. 1/2 Observe that τ −1 ≤ Λτ , so that s

τ −3/4 Λsτ ≤ Λs+3/8 . τ

(11.2.20)

By (11.2.19), (11.2.20), we get kuks ≤ |||u|||s ≤ C kuks+7/8 .

(11.2.21)

From (11.2.21), we find that Hs+7/8 ⊂ W s ⊂ Hs .

(11.2.22) 0

0

Also, by (11.2.11), H = H and there exist C > 0, C > 0 such that if u ∈ C ∞,c (Z, R), C |u|H ≤ kuk0 ≤ C 0 |u|H .

(11.2.23)

As explained in [BL08, Remark 15.3.2], the Hs form a chain of Sobolev 0 spaces, and if s0 > s, the embedding of Hs in Hs is compact. Put H∞ = ∩s∈R Hs .

(11.2.24)

Let S (Z) be the vector space of smooth functions on Z that are rapidly decreasing in the fibre direction T Z together with their derivatives of arbitrary order. Equivalently S (Z) consists of the u ∈ C ∞ (Z, R), such that for any multi-index k, and k 0 ∈ N, k0 T Z,k ∇· 1 + Y T Z u (11.2.25) is uniformly bounded. By [BL08, eq. (15.3.7)], H∞ = S (Z) .

(11.2.26)

In [BL08, section 15.2], global Sobolev spaces H s , s ∈ R are also defined s on Z. By [BL08, Remark 15.3.2] for any s ∈ R, Hs ⊂ H s . Let Hloc be the local Sobolev space of index s on Z. By the above, we get s Hs ⊂ Hloc ,

(11.2.27)

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and the embedding in (11.2.27) is continuous. Over Z, we can define a scalar operator AZ b by a formula strictly similar to (11.1.1). Over π −1 B (x0 , 2), it restricts to the operator AX b . For λ0 > 0, set Pb = AZ b + λ0 .

(11.2.28)

AZ b s

(11.2.29)

For s ∈ R, let Ds be the domain of

s

in H , i.e.,

s Ds = {u ∈ H , AZ b u ∈ H }.

We now have a result stated in [BL08, Theorem 15.5.1 and eq. (15.5.22)]. Theorem 11.2.4. Take λ0 > 0 large enough. Then if s ∈ R, if u is a tempered distribution on Z such that Pb u ∈ Hs , then u ∈ W s+1/4 , and there exists Cs > 0 such that for u taken as before, |||u|||s+1/4 ≤ Cs kPb uks .

(11.2.30)

∞,c

Moreover, C (Z, R) is dense in Ds , i.e., for any u ∈ Ds , there exists a sequence uk ∈ C ∞,c (Z, R) , k ∈ N such that as k → +∞, ku − uk ks + kPb (u − uk )ks → 0.

(11.2.31)

11.3 The Littlewood-Paley decomposition on X For x ∈ X, r > 0, the open balls B (x, r) ⊂ X are isometric to the ball B (x0 , r). Let xn ∈ X, n ∈ N be a sequence in X such that the open balls B (xn , 1/2) form an open covering of X. If xn , xm are such that d (xm , xn ) ≤ 1/2, then B (xm , 1/2) ⊂ B (xn , 1). Therefore we may and we will assume that for m 6= n, d (xm , xn ) ≥ 1/2, and the balls B (xn , 1) form an open covering of X. In particular, {xn , n ∈ N} is a discrete subset of X. As the notation suggests, we assume that x0 is precisely the element of X defined in section 11.2, i.e., x0 = p1. After renumbering, we may and we will assume that d (x0 , xn ) increases with n. We claim that given n ∈ N, there is a uniformly bounded number of m ∈ N such that B (xn , 2) ∩ B (xm , 2) is nonempty. Indeed, each such B (xm , 2) is included in B (xn , 4). Moreover, the corresponding B (xm , 1/4) are mutually disjoint and have the same volume. Since the volume of B (xn , 4) does not depend on n ∈ N, the number of such m is uniformly bounded. Let ϕ (r) : R+ → [0, 1] be a smooth function such that ϕ (r) = 1 for r ≤ 1, and φ (r) = 0 for r ≥ 2. For n ∈ N, put ϕn (x) = ϕ (d (xn , x)) .

(11.3.1)

If ϕn (x) > 0, then xn ∈ B (x, 2). Since the B (xn , 1/4) do not intersect and have the same volume, given x ∈ X, the number of n ∈ N such that ϕn (x) > 0 is uniformly bounded. Therefore there is N > 1 such that 1≤

+∞ X n=0

ϕn ≤ N.

(11.3.2)

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In the sequel, covariant derivatives will be taken with respect to the connection ∇T X . By (11.3.1), the ϕn are uniformly bounded together with their P+∞covariant derivatives of arbitrary order. It follows from the above that n=0 ϕn and its covariant derivatives of arbitrary order are also uniformly bounded. For n ∈ N, set ϕn . (11.3.3) ψn = P+∞ n=0 ϕn Then +∞ X

ψn = 1.

(11.3.4)

n=0

More precisely the ψn , n ∈ N form a partition of unity of X that is subordinated to the open covering B (xn , 2) , n ∈ N. By the considerations we made after (11.3.2), the ψn and their covariant derivatives of arbitrary order are uniformly bounded. By (11.3.4), we get +∞ X

ψn2 ≤ 1.

(11.3.5)

n=0

If f ∈ C ∞ (X, R), for n ∈ N, set n (f ) = ψn f.

(11.3.6)

By (11.3.4), f=

+∞ X

n (f ) .

(11.3.7)

n=0

Let H be the Hilbert space of real square-integrable functions on X, and let | |H denote the corresponding norm. By the above results and in particular by (11.3.5), (11.3.7), if f ∈ C ∞,c (X, R), then +∞ X

2

2

|n (f )|H ≤ |f |H ≤ C

1

+∞ X

2

|n (f )|H .

(11.3.8)

n=0

11.4 The Littlewood Paley decomposition on X If u ∈ C ∞ (X , R), we define n (u) ∈ C ∞ (X , R) by the formula n (u) = (π ∗ ψn ) u.

(11.4.1)

The obvious analogue of (11.3.7) holds, i.e., u=

+∞ X n=0

n (u) .

(11.4.2)

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Let H0 be the Hilbert space of real square integrable functions on X , and let | |H0 denote the corresponding norm. If u ∈ C ∞,c (X , R), as in (11.3.8), we get +∞ X n=0

2 |n (u)|H0

≤

2 |u|H0

≤C

+∞ X n=0

2

|n (u)|H0 .

(11.4.3)

Moreover, by (11.2.23), (11.4.3), if u ∈ C ∞,c (X , R), we have the analogue of (11.3.8), X X 2 2 2 (11.4.4) |ψn Uj |2−j ,0 . |ψn Uj |2−j ,0 ≤ |u|H0 ≤ C 0 j,n∈N

j,n∈N

If u ∈ C

∞,c

(X , R), n (u) can be viewed as an element of C ∞,c (Z, R).

Definition 11.4.1. For u ∈ C ∞,c (X , R), set 02

kuks =

+∞ X

02

2

kn (u)ks ,

|||u|||s =

+∞ X

2

|||n (u)|||s .

(11.4.5)

n=0

n=0

By (11.2.21), (11.4.5), we get 0

0

0

kuks ≤ |||u|||s ≤ C kuks+7/8 .

(11.4.6) 0

0

Let H0s , W 0s be the completions of C ∞,c (X , R) for the norms k ks , |||u|||s . By (11.4.6), we get H0s+7/8 ⊂ W 0s ⊂ H0s . 0s

(11.4.7)

0s

The vector spaces H , W still form families of Sobolev spaces. However, 0 0 for s0 > s, the embeddings of H0s , W 0s into H0s , W 0s are no longer compact. For u ∈ C ∞,c (X , R), equation (11.4.4) can be rewritten in the form 02

2

02

kuk0 ≤ |u|H0 ≤ C 0 kuk0 .

(11.4.8)

By (11.4.8), we deduce that H00 = H0 . For s ∈ R, let (11.2.27), we get

0s Hloc

(11.4.9)

be the local Sobolev space of index s on X . Using 0s H0s ⊂ Hloc .

(11.4.10)

Also the embedding in (11.4.10) is continuous. Set H0∞ = ∩s∈R H0s .

(11.4.11)

H0∞ ⊂ C ∞ (X , R) .

(11.4.12)

By (11.4.10), Let S (X ) be the vector space of the u ∈ C ∞ (X , R) that are rapidly decreasing along the fibres T X together with their derivatives of arbitrary

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order. The definition just consists in replacing ∇T Z by ∇T X in (11.2.25). Note that here, the rapid decay is uniform over X. By (11.2.26), we get the following refinement of (11.4.12), H0∞ ⊂ S (X ) .

(11.4.13)

Pb = AX b + λ0 .

(11.4.14)

0s Ds0 = u ∈ H0s , AX . b u∈H

(11.4.15)

For λ0 > 0, set For s ∈ R, put We establish an extension of [BL08, Theorem 15.5.1 and eq. (15.5.22)], which was stated here as Theorem 11.2.4. Theorem 11.4.2. For λ0 > 0 large enough, for any s ∈ R, there exists Cs > 0 such that if u ∈ Ds0 , then u ∈ W 0s+1/4 , and moreover, 0

0

|||u|||s+1/4 ≤ Cs kPb uks . ∞,c

Ds0 ,

Also, C (X , R) is dense in i.e., for s ∈ R, u ∈ up ∈ C ∞,c (X , R) , p ∈ N such that as p → +∞, 0

(11.4.16) Ds0 ,

0

ku − up ks + kPb (u − up )ks → 0.

there is a sequence (11.4.17)

Proof. For simplicity, we first assume that u ∈ C ∞,c (X , R). This will guarantee that all the expressions we deal with are finite. By equation (11.2.30) in Theorem 11.2.4, for λ0 large enough, for any n ∈ N, |||n (u)|||s+1/4 ≤ Cs kPb n (u)ks .

(11.4.18)

Pb n (u) = n (Pb u) + [Pb , ψn ] u.

(11.4.19)

Moreover,

By (11.1.1), 1 [Pb , ψn ] = − ∇Y T X ψn . b By (11.4.19), (11.4.20), we get kPb n (u)ks ≤ kn (Pb u)ks +

(11.4.20)

1 k(∇Y T X ψn ) uks . b

(11.4.21)

By (11.2.19), we have the identity 2

k(∇Y T X ψn ) uks =

+∞ X

2

22j |∇Y T X ψn Uj |2−j ,s .

(11.4.22)

j=0

Also given m ∈ N, n ∈ N, if ψm dψn 6= 0, then d (xm , xn ) ≤ 4. Given m ∈ N, there is a uniformly bounded family of n ∈ N such that ψm dψn 6= 0. Combining this fact with (11.3.4) and (11.4.22), we get X X 2 2 k(∇Y T X ψn ) uks ≤ Cs 22j |ψn Uj |2−j ,s . (11.4.23) n∈N

j,n∈N

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By (11.4.5), (11.4.18)–(11.4.23), we obtain   X 02 02 2 |||u|||s+1/4 ≤ Cs kPb uks + 22j |ψn Uj |2−j ,s  .

(11.4.24)

j,n∈N

By (11.2.19) and (11.4.5), we get X 2 2 02 23j/2 |ψn Uj |2−j ,s+1/4 + 2−5j/4 ψn ∇V Uj 2−j ,s+1/8 . |||u|||s+1/4 = j,n∈N

(11.4.25) From (11.2.17), (11.2.18), we obtain 2

2

|ψn Uj |2−j ,s+1/4 ≥ 2j |ψn Uj |2−j ,s .

(11.4.26)

By (11.4.24)–(11.4.26), we obtain 1 5j/2 1 X 3j/2 2 2 2 |ψn Uj |2−j ,s + 2 |ψn Uj |2−j ,s+1/4 2 2 j,n∈N j,n∈N   ! X 2 02 2 + 2−5j/4 ψn ∇V Uj 2−j ,s+1/8 ≤ Cs kPb uks + 22j |ψn Uj |2−j ,s  . X

j,n∈N

(11.4.27) From (11.4.25), (11.4.27), we conclude that there is js ∈ N such that   X   02 02 2 |||u|||s+1/4 ≤ Cs kPb uks + |ψn Uj |2−j ,s  . (11.4.28) 0≤j≤js n∈N

By (11.2.19), (11.4.5), we get X X 2 2 02 |ψn Uj |2−j ,s ≤ |ψn Uj |2−j ,s = kuks . 0≤j≤js n∈N

(11.4.29)

j,n∈N

By (11.4.28), (11.4.29), we find that 02 02 02 |||u|||s+1/4 ≤ Cs kPb uks + kuks .

(11.4.30)

For α > 0, A > 0, s ≥ 0, α2s ≤ A−1/2 α2s+1/2 + A2s .

(11.4.31)

By (11.2.18), (11.2.19), (11.4.5), and (11.4.6), we get 02

02

02

02

02

kuks ≤ A−1/2 kuks+1/4 + A2s kuk0 ≤ A−1/2 |||u|||s+1/4 + A2s kuk0 . (11.4.32) By (11.4.30), and by taking A large enough in (11.4.32), we get 02 02 02 (11.4.33) |||u|||s+1/4 ≤ Cs kPb uks + kuk0 .

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By [BL08, eqs. (15.4.7)–(15.4.9)], for λ0 > 0 large enough, 2

2

|u|H0 ≤ C |Pb u|H0 .

(11.4.34)

By (11.4.8), we can rewrite (11.4.34) in the form 02

02

kuk0 ≤ C kPb uk0 .

(11.4.35)

By (11.4.33), (11.4.35), we obtain 02 02 02 |||u|||s+1/4 ≤ Cs kPb uks + kPb uk0 .

(11.4.36)

For s ≥ 0, 02

02

kPb uk0 ≤ kPb uks .

(11.4.37)

Using (11.4.36), (11.4.37), for s ≥ 0, we get (11.4.16) when u ∈ C ∞,c (X , R). We will now establish (11.4.16) for s ≥ 0, when only assuming that u ∈ Ds0 . For the moment, we take s ∈ R. We still have equation (11.4.24). However, it is no longer clear that any of the two sides in this equation is finite. Of course (11.4.24) remains valid for t ≤ s, i.e.,   X 02 02 2 |||u|||t+1/4 ≤ Ct kPb ukt + 22j |ψn Uj |2−j ,t  . (11.4.38) j,n∈N

Again both sides of (11.4.38) may well be infinite. Since Λτ ≥ τ −2 , τ −1 ≤ τ −3/4 Λ1/8 τ .

(11.4.39)

From (11.2.19), (11.4.5), and(11.4.39), we deduce that X 2 02 22j |ψn Uj |2−j ,t ≤ |||u|||t+1/8 .

(11.4.40)

j,n∈N

Still there is no guarantee that the right-hand side of (11.4.40) is finite. By (11.4.7), since u ∈ H0s , then u ∈ W 0s−7/8 . Therefore when making t = s − 1 in (11.4.40), both sides of the equation are finite. Therefore for t = s − 1, both sides of (11.4.38) are finite, so that u ∈ W 0s−3/4 . By iterating (11.4.38), we find that u ∈ W 0s+1/4 , and that (11.4.38) holds with s = t, which is equation (11.4.24), where both sides of the equation are known to be finite. We can then proceed as in (11.4.25)–(11.4.29), and obtain (11.4.30), while knowing that both sides of the equation are finite. For s ≥ 0, we can proceed as in (11.4.31)–(11.4.37), and we get (11.4.16) for s ≥ 0. Let AX∗ be the formal L2 adjoint of AX b b with respect to the scalar product 0 ∗ on H . This operator has the same structure as AX b . Let Pb be the L2 adjoint of Pb . For λ0 > 0 large enough, for s ≥ 0, we get the analogue of (11.4.16), 0

0

|||u|||s+1/4 ≤ Cs kPb∗ uks .

(11.4.41)

By duality, from (11.4.41), for s ≥ 0, we obtain 0

kuk−s ≤ Cs kPb uk−s−1/4 .

(11.4.42)

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By (11.4.42), (11.4.16) also holds for s ≤ −1/4. By interpolation, we get (11.4.16) for any s ∈ R. Take s ∈ R, and assume that u ∈ Ds0 , so that u ∈ W 0s+1/4 . For p ∈ N, set X ψn δj (u) . (11.4.43) up = 0≤j,n≤p

We claim that as p → +∞, up converges to u in W 0s+1/4 , and Pb up converges to Pb u in H0s . Note that X u − up = ψn δj (u) . (11.4.44) j≥p+1 or n≥p+1

Recall that d (x0 , xn ) is an increasing sequence that tends to +∞ as n → +∞. From the properties of the sequence xn given in section 11.3, one deduces easily that as p → +∞, u − up converges to 0 in W 0s+1/4 . Clearly,   X X Pb up = ψn δj (Pb u) + Pb , ψn χj < Y T X >  u. (11.4.45) 0≤j,n≤p

0≤j,n≤p

By the same arguments as before, the first term in the right-hand side of (11.4.45) lies in H0s and converges to Pb u in H0s as p → +∞. Also,     X X   Pb , ψn χj < Y T X >  ψn χ j < Y T X >  = −  . Pb , 0≤j,n≤p

j≥p+1 or n≥p+1

(11.4.46) Moreover, we have the obvious Pb , ψn χj < Y T X > = [Pb , ψn ] χj < Y T X > + ψn Pb , χj < Y T X > . (11.4.47) The term [Pb , ψn ] has been computed in (11.4.20). Using the fact that u ∈ W 0s+1/4 , equation (11.4.20), and also the easycomputation in [BL08, eq. (15.5.8)] of the commutator Pb , χj < Y T X > , we find that as p → +∞, Pb up → Pb u in H0s . By the above, we find that to establish the last part of our theorem, we may as well assume that u ∈ Ds0 has compact support. Let H s,c be the standard Sobolev space with compact support on X . The condition on u ∈ Ds0 just says that u ∈ H s,c , Pb u ∈ H s,c . Let Ψp , p ∈ N be a family of smooth regularizing pseudodifferential operators of order 0 on X that converges to the identity as p → +∞. We may and we will assume that the support of the Ψp , p ∈ N is arbitrarily close to the diagonal. Such a family of operators can easily be constructed using a regularization of the wave kernel on X . Set up = Ψp u. (11.4.48) Then the up are smooth functions with uniformly bounded compact support, and as p → +∞, up converges to u in HXs,c so that up converges to u in H0s . Moreover, Pb up = Ψp Pb u + [Pb , Ψp ] u. (11.4.49)

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Since Pb u ∈ H0s has compact support, the same argument as before shows that as p → +∞, Ψp Pb u converges to Pb u in H0s . Also by (11.1.1), we can write [Pb , Ψp ] in the form [Pb , Ψp ] = Ap ∇V + Bp ,

(11.4.50)

where Ap , Bp are pseudodifferential operators of order 0 with support arbitrarily close to the diagonal, which converge to 0 in the space of pseudodifferential operators of order 0. Clearly Bp u ∈ H s,c , and Bp u converges to 0 in H s,c , so that Bp u → 0 in H0s . Since u ∈ W 0s+1/4 , and since u has compact support, by (11.2.19) and (11.4.5), ∇V u ∈ H0s+1/8 , so that ∇V u ∈ H s,c . Therefore Ap ∇V u converges to 0 in H s,c , so that Ap ∇V u converges to 0 in H0s . From the above, we conclude that Pb up converges to Pb u in H0s . The proof of our theorem is completed. Remark 11.4.3. It is important to observe that while the proof of [BL08, Theorems 15.4.2 and 15.5.1] uses the fact that the Sobolev spaces Hs are compactly embedded into each other, because here X is noncompact, such an argument cannot be used any more. As a substitute, we used instead (11.4.24)–(11.4.35). We will use the notation D0 = D00 . We establish an analogue of [BL08, Theorem 15.6.1]. Theorem 11.4.4. There exist λ0 > 0, C0 > 0 such that for any u ∈ D0 , λ ∈ C, Re(λ) ≤ −λ0 ,

0 1/6 0 0

|λ| kuk0 + kuk1/4 ≤ C0 AX (11.4.51) b −λ u 0. 1/6

There exists C1 > 0 such that for σ, τ ∈ R, σ ≤ C1 |τ | , if λ = −λ0 +σ +iτ , then

0 1/6 0

(1 + |λ|) kuk0 ≤ C1 AX (11.4.52) b −λ u 0. Proof. We proceed exactly as in the proof of [BL08, Theorem 15.6.1], while using the concepts and techniques introduced in the proof of Theorem 11.4.2. As explained in [BL08], equation (11.4.52) is a consequence of (11.4.51). So we concentrate on the proof of (11.4.51). As in [BL08, eqs. (15.6.7) and (15.6.15)], set 1/2 2 Λλ,τ = S + τ −4 + τ 2 |λ| . (11.4.53) If U ∈ C ∞ (Y, R), we define |U |λ,τ,s as in (11.2.18), by replacing Λτ by Λλ,τ . Similarly we define norms k kλ,s , ||| |||λ,s as in (11.2.19) on C ∞,c (Z, R). The completions Hs , W s are the same as in Definition 11.2.3. Finally, we 0 0 define the norms k kλ,s , ||| |||λ,s on C ∞,c (X , R) as in Definition 11.4.1. The 0s 0s completions H , W do not depend on λ. Set Pb,λ = AX b − λ.

(11.4.54)

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To prove (11.4.51), we take λ0 > 0 large enough as in Theorems 11.2.4 and 11.4.2. We define Pb as in (11.4.14). If α ∈ R+ , β ∈ R, let λ ∈ C be given by λ = −λ0 − α + iβ,

(11.4.55)

Pb,λ = Pb + α − iβ.

(11.4.56)

so that

The same argument as in [BL08, eqs. (15.6.9)–(15.6.11), proof of Theorem 15.6.1] shows that (11.4.51) is a consequence of the estimate, 0

0

|||u|||λ,s+1/4 ≤ Cs kPb,λ ukλ,s ,

(11.4.57)

for s = 0. As the notation indicates, the constant Cs > 0 should not depend on λ. As explained in [BL08, p. 243], an analogue of Theorem 11.2.4 holds with the new norms indexed by λ, s, the crucial point being that the positive constants that appear only depend on s and not on λ. The proof of (11.4.57) then proceeds as the proof of equation (11.4.16) in Theorem 11.4.2, which completes the proof of our theorem. In the sequel, the Hilbert space H0 is equipped with its canonical scalar product h iH0 . The formal adjoint of the operator AX b is taken with respect to this scalar product. If U is a bounded operator acting on H0 , we denote by kU k its norm with respect to the norm of H0 . We establish now an analogue of [BL08, Theorem 15.7.1]. 0 Theorem 11.4.5. The adjoint of the operator AX b acting on H with domain 0 X∗ X ∞,c D is the formal adjoint Ab of Ab acting on C (X , R), with domain 0 D0∗ = {u ∈ H0 , AX∗ b u ∈ H }. There exist c0 > 0, λ0 > 0, C > 0 such that if U ⊂ C is given by

U = {λ = −λ0 + σ + iτ, σ, τ ∈ R, σ ≤ c0 |τ |1/6 }, −1 if λ ∈ U, the operator AX exists, and moreover b −λ

X

(Ab − λ)−1 ≤

C 1/6

(1 + |λ|)

.

There exists C > 0 such that if λ ∈ R, λ ≤ −λ0 , then

−1

X

−1

Ab − λ

≤ C (1 + |λ|) .

(11.4.58)

(11.4.59)

(11.4.60)

−1 −1 If λ ∈ U, AX maps H0s into H0s+1/4 . In particular AX b −λ b −λ maps H0∞ into itself. If s ∈ R, there exists Cs > 0 such that for λ ∈ U, u ∈ H0s ,

−1

X

0 4|s|+1 0 u ≤ Cs (1 + |λ|) kuks . (11.4.61)

Ab − λ s+1/4

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201 ∞,c Proof. By Theorem 11.4.2, AX is the closure of AX (X , R) , b ,D b ,C which implies the first part of our theorem. Using Theorems 11.4.2 and 11.4.4, the proof of (11.4.59) proceeds exactly as the proof of [BL08, Theorem 15.7.1]. To establish (11.4.60), we proceed as in [BL08]. We use the notation in the proof of Theorem 11.4.4, and we take The analysis of the hypoelliptic Laplacian

0

λ = −λ0 − α, α ≥ 0.

(11.4.62)

By (11.1.1), we get 2 1 1 (11.4.63) Pb,λ = 2 −∆T X + Y T X − m + λ0 + α − ∇T X . 2b b By (11.4.63), for λ0 > 0 large enough, there is C > 0 such that if u ∈ C ∞,c (X , R), 2

(C + α) |u|H0 ≤ hPb,λ u, uiH0 .

(11.4.64)

(1 + α) |u|H0 ≤ C |Pb,λ u|H0 .

(11.4.65)

From (11.4.64), we get

−1 Also if λ is as in (11.4.62), λ ∈ U, so that Pb,λ exists. By (11.4.65), we get (11.4.60). By (11.4.6), (11.4.16), if u ∈ C ∞,c (X , R), if λ ∈ C, 0 0 0

+ |λ + λ | kuk (11.4.66) kuks+1/4 ≤ Cs AX − λ u 0 b s . s

By (11.4.32), for A > 0, s ≥ 0, we get 0

0

0

kuks ≤ A−1/4 kuks+1/4 + As kuk0 .

(11.4.67)

4

By taking A = 24 Cs4 |λ + λ0 | in (11.4.67), we deduce from (11.4.66) that for s ≥ 0, 0 4s+1 0 0

+ |λ + λ | kuk (11.4.68) kuks+1/4 ≤ Cs AX − λ u 0 b 0 . s −1 By (11.4.59), (11.4.68), we find that if λ ∈ U, for s ≥ 0, AX maps b −λ H0s into H0s+1/4 , and also that (11.4.61) holds when u ∈ C ∞,c (X , R). By density this equation extends to u ∈ H0s . The case of a general s ∈ R can be obtained using duality and an interpolation argument. The proof of our theorem is completed.

X 11.5 The heat kernels for AX b , Bb

Let γ be the contour in C in Figure 11.1. This contour separates the domains δ± , which contain ±∞. Let c0 > 0, λ0 > 0 be the constants that were defined in Theorem 11.4.5. These constants depend on b > 0. Then n o 1/6 γ = λ = −λ0 + σ + iτ, σ, τ ∈ R, σ = c0 |τ | . (11.5.1)

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g

d-

-l0

0

Figure 11.1

d+

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0 The domain of AX b is dense in H . Moreover, by equation (11.4.60) in Theorem 11.4.5 and by the theorem of Hille-Yosida [Y68, section IX-7, p. 246], there is a unique well-defined continuous semigroup exp −tAX , t ≥ 0 b acting on H0 . We have the obvious analogue of [BL08, Proposition 3.3.1]. Proposition 11.5.1. For t > 0, the heat operator exp −tAX is given by b the contour integral, Z −1 1 e−tλ λ − AX dλ. (11.5.2) exp −tAX = b b 2iπ γ

Proof. Using Theorem 11.4.5, the proof of our proposition is the same as the proof of [BL08, Proposition 3.3.1]. Using (11.4.59), (11.5.2), and integration by parts, for N ∈ N, Z −(N +1) (−1)N N ! X e−tλ λ − AX dλ. (11.5.3) exp −tAb = b 2iπtN γ −1 By Theorem 11.4.5, if λ ∈ δ− , AX maps H0s into H0s+1/4 with b −λ −N 4|s|+1 a norm dominated by Cs (1 + |λ|) . Therefore AX maps H0s b −λ (4|s|+1)N into H0s+N/4 with a norm dominated by Cs,N (1+ |λ|) . X By (11.5.3), for t > 0, the operator exp −tAb maps H0s into H0∞ . More 0 precisely given s, s0 ∈ R, this operator is continuous from H0s into H0s with a bounded norm. Combining these results with (11.4.13), standard arguments show that exp −tAX has a smooth kernel on X that is rapidly decreasing b along the fibres T X × T X of X × X together with its derivatives of any order. Let F ∈ C ∞,c (X , R). For t ≥ 0, x, Y T X ∈ X , set TX Φ t, x, Y T X = exp −tAX . (11.5.4) b F x, Y Proposition 11.5.2. For any t ≥ 0, the function Φ (t, ·) lies in H0∞ . As t → 0, Φ (t, ·) converges to F in H0∞ . The function Φ is smooth on R+ × X , and is such that ∂ Φ + AX (11.5.5) b Φ = 0. ∂t Finally, for bounded t ≥ 0, Φ (t, ·) is uniformly rapidly decreasing along the fibres T X of X together with its covariant derivatives of any order in all variables. Proof. For t ≥ 0, exp −tAX maps H0∞ into H0∞ , and so for t ≥ 0, b 0∞ Φ (t, ·) ∈ H . By (11.5.3), Φ is smooth on ]0, +∞[×X . Since exp −tAX b is a continuous semigroup acting on H0 , as t → 0, Φ (t, ·) converges to F in H0 , and t ∈ R+ → Φt ∈ H0 is a smooth map. Take λ ∈ U. For any k ∈ N, −k X k exp −tAX = AX exp −tAX Ab − λ F. (11.5.6) b b −λ b

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Chapter 11

k X k Since AX F ∈ C ∞,c (X , R), t ∈ R+ → exp −tAX Ab − λ F is a b −λ b smooth map with values in H0 . By equation (11.4.61) in Theorem 11.4.5 and by (11.5.6), for any k ∈ N, t ∈ R+ → exp −tAX b F is a smooth map with values in H0k/4 . By (11.4.10), we conclude that Φ is smooth on R+ × X , and also that Φ (t, ·) converges to F in H0∞ . Using (11.4.13), we conclude that for bounded t ≥ 0, Φ is uniformly rapidly decreasing together with its derivatives of arbitrary order along the fibres T X. Since Φ is smooth, equation (11.5.5) follows from (11.5.4). The proof of our proposition is completed. By (11.1.2), BbX is conjugated to AX b . Therefore all the results we obtained for AX can be tautologically transferred to BbX . Of course the functional b analytic machinery has to be adequately modified. If F ∈ C ∞,c (X , R), set Ψ t, x, Y T X = exp −tBbX F x, Y T X . (11.5.7) 2 Let Φ be the function in (11.5.4), with F replaced by exp − Y T X /2 F . By (11.1.2), we get 2 Ψ t, x, Y T X = exp Y T X /2 Φ t, x, Y T X . (11.5.8) By Proposition 11.5.2, the function Ψ is smooth on R+ × X , and is such that ∂ Ψ + BbX Ψ = 0. ∂t

(11.5.9)

Definition 11.5.3. For b > 0, t > 0, we denote by X rb,t x, Y T X , x0 , Y T X0 , sX x, Y T X , x0 , Y T X0 b,t X the smooth kernels associated with exp −tAX b , exp −tBb . By (11.1.2), sX b,t

2 x, Y T X , x0 , Y T X0 = exp Y T X /2 2 X rb,t x, Y T X , x0 , Y T X0 exp − Y T X0 /2 . (11.5.10)

X We know that given b > 0, t > 0, rb,t TX T X0 decreasing in the variables Y ,Y .

x, Y T X , x0 , Y T X0 is rapidly

Remark 11.5.4. Using the estimate on ∆X d2 (x0 , x) /2 in Proposition 13.1.1, and combining this estimate with corresponding commutator estimates as X in [BL91, section 11], given b > 0, t > 0, rb,t x, Y T X , x0 , Y T X0 can be shown to be rapidly decreasing together with its derivatives in the variables x0 , Y T X0 , the decay in the variable x0 being measured via d (x, x0 ). However, this result is superseded by the Gaussian estimate in Theorem 13.2.4, which is uniform for bounded b > 0.

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11.6 The scalar hypoelliptic operators on Xb Here ∆T X⊕N still denotes the standard Laplacian acting along the fibres of T X ⊕ N of Xb, and Y = Y T X + Y N denotes the tautological section of T X ⊕ N over Xb. b Let AX b be the differential operator acting on X , 1 1 2 T X⊕N −∆ + |Y | − m − n − ∇Y T X . (11.6.1) AX b = 2b2 b Set 2 2 X BX (11.6.2) b = exp |Y | /2 Ab exp − |Y | /2 . Then BX b =

1 1 −∆T X⊕N + 2∇VY − ∇Y T X . 2 2b b

(11.6.3)

X Note that when acting on smooth functions on X , the operators AX b , Bb X X restrict to the operators Ab , Bb . We claim that the results that were obtained before for the operators X X X AX b , Bb can be easily transferred to the operators Ab , Bb . First, the Euclidean vector bundle N with connection on B (x0 , 2) extends to an Euclidean vector bundle with connection over Z, which we still denote by N . The action of K on Z extends to the vector bundles T Z, N . Let Zb be the total space of the vector bundle T Z ⊕ N over Z. The Euclidean connection ∇T Z⊕N b Let g T Zb be the obvious direct induces a horizontal subbundle T H Zb of T Z. b b Then g T Z is K-invariant. sum metric on T Z. We fix r0 ∈]1, 2[. Set n o 2 B 0 = Y ∈ T X ⊕ N, |Y | ≤ 3 ,

1 1 2 R0 = {Y ∈ T Z ⊕ N, |Y | ∈ [ 2 − , 4], r0 4 n o 2 0 B0 = Y ∈ T Z ⊕ N, |Y | ≤ 5 .

(11.6.4)

Let Yb be the total space of P (T Z ⊕ N ⊕ R) over Z. Let g T Y be a Kb b invariant metric on T Yb that coincides with g T Z on B00 . Let ∆Y be the b be the Hilbert space of squareLaplace-Beltrami operator on Yb . Let H integrable real functions on Yb , and let | |Hb be the corresponding norm. We still define S as in (11.2.16), and Λτ as in (11.2.17), and the norms k ks , ||| |||s as in Definition 11.2.3. The same techniques as in sections 11.2–11.5 can be used to establish properties of the resolvent of AX b and of its heat kernel. In particular for t > 0, the heat operators for AX b are well-defined, and are given by smooth kernels, which are rapidly decreasing on Xb × Xb together with their covariant derivatives of arbitrary order. By (12.2.10), the heat operators for BX b are also given by smooth kernels. b

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Chapter 11

0 Y 0 )) , sX ) , (x0 , Y 0 )) Definition 11.6.1. For t > 0, let rX b,t ((x, Y ) , (x , b,t ((x, Y X X be the smooth kernels associated with exp −tAb , exp −tBb . By (11.6.2), we get 2 0 0 sX b,t ((x, Y ) , (x , Y )) = exp |Y | /2 0 0 0 2 rX b,t ((x, Y ) , (x , Y )) exp − |Y | /2 . (11.6.5)

Remark 11.5.4 is still valid for the kernel rX b,t . 11.7 The scalar hypoelliptic operator on Xb with a quartic term Again b > 0 is fixed. Set ∇ TX 1 N T X 2 1 2 T X⊕N Y , Y + −∆ + |Y | − m − n − Y . AX = b 2 2b2 b (11.7.1) b. By (11.6.1) and is a scalar differential operator acting on X Then AX b (11.7.1), we get 1 N T X 2 AX Y ,Y + AX (11.7.2) b = b . 2 Set 2

2

X BX b = exp |Y | /2 Ab exp − |Y | /2 .

(11.7.3)

Then ∇ TX 1 N T X 2 1 Y ,Y + 2 −∆T X⊕N + 2∇VY − Y . (11.7.4) 2 2b b In the analysis of the operator AX b , the arguments of sections 11.2–11.5 have to be adequately modified, because of the presence of the quartic term in (11.7.2). b There is an associated operator AZ b acting over Z. Difficulties already appear in the proof of [BL08, Theorem 15.5.1 and eq. (15.5.22)], which is here Theorem 11.2.4. For λ0 > 0, instead of (11.2.28), set BX b =

Qb = AZ b + λ0 .

(11.7.5)

Let us review some of the arguments given in [BL08, chapter 15]. Recall that for a > 0, Ka was defined in (2.14.2). We use the notation in [BL08, eq. (15.4.1)]. For τ > 0, set −1 Z AZ b,τ = Kτ Ab Kτ .

(11.7.6)

By (11.7.1), (11.7.6), we get AZ b,τ =

τ −4 N T Z 2 1 2 Y ,Y + 2 −τ 2 ∆T Z⊕N + τ −2 |Y | − m − n 2 2b ∇ TZ − τ −1 Y . (11.7.7) b

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Put Qb,τ = AZ b,τ + λ0 .

(11.7.8)

We now state an analogue of [BL08, Theorem 15.4.2]. Theorem 11.7.1. If λ0 > 0 is large enough, for any s∈ R, there exists b R whose support is Cs > 0 such that for any j ∈ N, for any U ∈ C ∞,c Z, included in the ball B 0 for j = 0 and in the annulus R0 for j ≥ 1, then 2 2 2 24j |U |2−j ,s + ∇V U 2−j ,s + 23j/2 |U |2−j ,s+1/4 2 2 ≤ Cs Qb,2−j U −j . (11.7.9) + 2−5j/4 ∇V U −j 2

2

,s+1/8

,s

b be the Hilbert space Proof. In the sequel, we will make τ = 2−j , j ∈ N. Let H b let | | b be the corresponding L2 of real square integrable functions on Z, H Z norm, and let h iHb be the scalar product. Let AZ b,τ,+ , Ab,τ,− be the symmetric and antisymmetric parts of AZ b . Then b,τ with respect to h iH −4 τ N T X 2 1 2 2 T Z⊕N −2 AZ = Y , Y + −τ ∆ + τ |Y | − m − n , b,τ,+ 2 2b2 (11.7.10) −1 ∇Y T Z . AZ b,τ,− = −τ b Let Qb,τ,+ , Qb,τ,− be the symmetric and antisymmetric parts of Qb,τ . By proceeding as in [BL08, eq. (15.4.7)], for λ0 > 0 large enough, for any U ∈ ∞,c b C Z, R with support in the annulus R0 , we get V 2

∇ U b + τ −4 |U |2b + τ −6 Y N , Y T X U 2b ≤ C Qb,τ,+ U, τ −2 U b . H H H H (11.7.11) By proceeding as in [BL08, eqs. (15.4.8), (15.4.9)], from (11.7.11), we get V 2 ∇ U b + τ −4 |U |2b + τ −6 Y N , Y T X U 2b ≤ C |Qb,τ U |2b . (11.7.12) H H H H If τ = 1, and if the support of U is included in B 0 , (11.7.12) still holds for λ0 > 0 large enough. The only difference with [BL08] is the presence of the third term in the left-hand side of (11.7.12). Let ρ (r) : R+ → [0, 1] be a smooth function, which is equal to 1 when r2 ≤ 5, and to 0 for r2 ≥ 6. Put θ0 (Y ) = ρ (|Y |) .

(11.7.13)

Set R = θ02 Qb,τ θ02 . With respect to [BL08, eq. (15.4.18)], replacing θ0 by that will appear in (11.7.17). If the support of U is included in B00 , then Qb,τ U = RU.

(11.7.14) θ02

is done for reasons

(11.7.15)

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Let R+ , R− be the symmetric and antisymmetric components of R. Clearly, R+ = θ02 Qb,τ,+ θ02 ,

R− = θ02 Qb,τ,− θ02 .

(11.7.16)

We define the algebra of pseudodifferential operators E · on Yb exactly as in [BL08, p. 231]. Let us briefly indicate that the weight function associated 1/2 2 with this algebra is just τ −4 + |ζ| , ζ ∈ T ∗ Yb . Take E ∈ E d , d ∈ R. Instead of [BL08, eq. (15.4.23)], and using the fact that τ −2 ∈ E 1 , we now have the commutation relation, [R, E] ∈ τ 2 E d ∇V + τ −1 E d + τ −4 θ0 Y N , Y T X E d−1 . (11.7.17) The first two terms in the right-hand side of (11.7.17) were already obtained in [BL08, eq. (15.4.23)]. Since τ −2 ∈ E 1 , from (11.7.17), we get [R, E] ∈ τ 2 E d ∇V + τ −1 E d + τ −2 θ0 Y N , Y T X E d . (11.7.18) N TX Note that as it should be, we could as well have the factor θ0 Y , Y to the right of E d . From (11.7.18), we deduce the weaker [R, E] ∈ τ 2 E d ∇V + τ −2 E d .

(11.7.19)

Equation (11.7.19) is exactly similar to [BL08, eq. (15.4.22)], which itself is weaker than [BL08, eq. (15.4.23)]. We claim that [BL08, Lemma 15.4.3] still holds. Indeed equation (11.7.12) is stronger than [BL08, eq. (15.4.25)], and the commutation relation (11.7.19) is the other main ingredient of the proof of this lemma. By using the same argument as before, [BL08, Lemma 15.4.4 and eq. (15.4.51)] still hold. Let θ1 (Y ) be a smooth function of |Y | with values in [0, 1], which has compact support and is constructed in the same way as the function θ0 . We assume that θ1 is equal to 1 near the annulus R0 and to 0 near Y = 0. For j ≥ 1, set θj = θ1 . Also θ0 has been defined before. For j ∈ N, we use the notation θ = θj . Instead of [BL08, eq. (15.4.56)], because of (11.7.18), for s ∈ R, we get [R, Λs ] ∈ τ 2 E s ∇V + τ −1 E s + τ −2 θ0 Y N , Y T X E s . (11.7.20) Using [BL08, (eq. (15.4.52)–(15.4.55)], and equation (11.7.20) instead of [BL08, eq. (15.4.56)], instead of [BL08, eq. (15.4.57)] we obtain 2

|U |τ,s+1/4 ≤ τ 3/2

2 2 2 |RU |τ,s + Cs τ 4 ∇V U τ,s + Cs τ −2 |U |τ,s + Cs τ

−4

N T X 2 Y ,Y U

τ,s

! . (11.7.21)

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If we apply [BL08, eq. (15.4.51)] to θΛs+1/8 U , by using the same commutation arguments as before, [BL08, eq. (15.4.58)] should be replaced by V 2 ∇ U

τ,s+1/8

+ Cs τ

−4

2 2 2 ≤ δ C |RU |τ,s + Cs τ 4 ∇V U τ,s + Cs τ −2 |U |τ,s

N T X 2 Y ,Y U

τ,s

! +

τ −4 2 2 C |U |τ,s+1/4 + Cs τ 4 |U |τ,s δ 2

+ Cs |U |τ,s+1/8 . (11.7.22) Using (11.7.21), (11.7.22) instead of [BL08, eqs. (15.4.57), (15.4.58)] and proceeding as in [BL08, eq. (15.4.59)], for A > 0, we get 2 2 2 2 τ −3/2 |U |τ,s+1/4 + Aτ 5/4 ∇V U τ,s+1/8 ≤ C |RU |τ,s + Cs τ 4 ∇V U τ,s 2 2 2 + Cs τ −2 |U |τ,s + Cs τ −4 Y N , Y T X U τ,s + CA2 τ −3/2 |U |τ,s+1/4 2 2 + Cs A2 τ 5/2 |U |τ,s + Aτ 5/4 |U |τ,s+1/8 . (11.7.23) By taking A > 0 small enough so that CA2 ≤ 1/2, from (11.7.23), we deduce an analogue of [BL08, eq. (15.4.60)], 2 2 2 2 τ −3/2 |U |τ,s+1/4 + τ 5/4 ∇V U τ,s+1/8 ≤ C |RU |τ,s + Cs τ 4 ∇V U τ,s 2 2 2 + Cs τ −2 |U |τ,s + Cs τ −4 Y N , Y T X U τ,s + Cs τ 5/4 |U |τ,s+1/8 . (11.7.24) Now we will use equation (11.7.12) applied to θΛs U , and we use again (11.7.20). Instead of [BL08, eq. (15.4.61)], we get V 2 ∇ U + τ −4 |U |2 + τ −6 Y N , Y T X U 2 τ,s τ,s τ,s 2 2 2 4 V 2 −2 ≤ C |RU |τ,s + Cs τ ∇ U τ,s + Cs τ |U |τ,s + Cs τ −4 Y N , Y T X U τ,s . (11.7.25) By adding (11.7.24) and (11.7.25), instead of [BL08, eq. (15.4.62)], we obtain V 2 5/4 V 2 ∇ U + τ −4 |U |2 + τ −3/2 |U |2 + τ ∇ U τ,s τ,s+1/4 τ,s τ,s+1/8 N T X 2 2 2 4 V 2 −6 +τ Y ,Y U τ,s ≤ C |RU |τ,s + Cs τ ∇ U τ,s + Cs τ −2 |U |τ,s 2 2 + Cs τ 5/4 |U |τ,s+1/8 + Cs τ −4 Y N , Y T X U τ,s . (11.7.26) Moreover, given > 0, τ −4 ≤ τ −6 +

1 . 2

(11.7.27)

By (11.7.26), (11.7.27), we get V 2 5/4 V 2 ∇ U + τ −4 |U |2 + τ −3/2 |U |2 ∇ U τ,s+1/8 τ,s τ,s+1/4 + τ τ,s 2 2 2 + τ −6 Y N , Y T X U τ,s ≤ C |RU |τ,s + Cs τ 4 ∇V U τ,s 2

2

+ Cs τ −2 |U |τ,s + Cs τ 5/4 |U |τ,s+1/8 . (11.7.28)

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The term containing τ −6 can be deleted in the left-hand side of (11.7.28). Then for τ = 2−j , for j large enough, (11.7.9) follows from (11.7.28). For the missing finite family of j, we proceed as in [BL08, proof of Theorem 15.4.2], or instead we use the same argument as in (11.4.32). The proof of our theorem is completed. We can now deduce the obvious analogue for the operator Qb of [BL08, Theorem 15.5.1 and eq. (15.5.22)], which was stated here as Theorem 11.2.4. Indeed the proof in [BL08] only uses Theorem 11.7.1, and also commutations 2 in which the extra quartic term 21 Y N , Y T X disappears. Similarly the results of [BL08, section 15.6] remain valid for the operator AX b . One can then easily extend the arguments of sections 11.4–11.6 to the operator AX b . X Definition 11.7.2. For t > 0, let rb,t ((x, Y ) , (x0 , Y 0 )) , sX Y ) , (x0 , Y 0 )) b,t ((x, X X be the smooth kernels associated with exp −tAb , exp −tBb . By (11.7.3),

2 0 0 sX b,t ((x, Y ) , (x , Y )) = exp |Y | /2 2 X rb,t ((x, Y ) , (x0 , Y 0 )) exp − |Y 0 | /2 . (11.7.29) X Again rb,t ((x, Y ) , (x0 , Y 0 )) is rapidly decreasing in Y, Y 0 . Also Remark 11.5.4 can still be applied to this heat kernel.

11.8 The heat kernel associated with the operator LX A,b We fix b > 0. We consider the hypoelliptic operator LX A,bgiven by (2.13.5), ∞ ∗ · ∗ ∗ b (4.5.1), which acts on C X,π b (Λ (T X ⊕ N ) ⊗ F ) . Recall that by C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

that Theorem 2.13.2, except for the term 1b ∇Y T X is skew-adjoint, the other components of LX are formally self-adjoint with A,b respect to the Hermitian product h i on C ∞,c Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . For simplicity, even though the operator LX A,b is no longer a scalar operator, we will use the same notation as in sections 11.2–11.7 for the function spaces defined there and for the corresponding norms. As in (11.7.5), for λ0 > 0, set Qb = LX A,b + λ0 .

(11.8.1)

b H,Y We still denote by π b the projection Yb → be the Bochner Z. Let ∆ Laplacian acting on smooth sections of C ∞ Yb , π b (Λ· (T ∗ Z ⊕ N ∗ ) ⊗ F ) . Instead of (11.2.16), we define the operator S by

S = −∆Y + 1. b

(11.8.2)

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We claim that the obvious analogue of [BL08, Theorem 15.4.2], which was stated as Theorem 11.7.1, still holds. Indeed set (Y ) = b c ad Y T X − c ad Y T X + iθad Y N − iρE Y N . (11.8.3) Let | (Y )| be the norm of (Y ) associated with the Hermitian product on Λ· (T ∗ X ⊕ N ∗ ) ⊗ F . Then 2 (Y ) C ≤ |Y | ≤ |Y | + C 0 . (11.8.4) b b 4b2 Using (11.8.4), for λ0 > 0 large enough, the obvious analogue of (11.7.11) still holds. The remainder of the proof of the analogue of Theorem 11.7.1 for LX A,b continues as in [BL08] and in the proof of Theorem 11.7.1. b The analysis of the operator LX as for the opA,b on X continues exactly X erator Ab . In particular for t > 0, the heat operator exp −tLX A,b is welldefined. X Definition 11.8.1. For t > 0, let qb,t ((x, Y ) , (x0 , Y 0 )) denote the smooth kernel associated with the operator exp −tLX A,b .

Again this kernel is rapidly decreasing in the variables Y, Y 0 . Also the conclusions of Remark 11.5.4 still hold.

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Chapter Twelve Rough estimates on the scalar heat kernel The purpose of this chapter is to establish rough estimates on the heat kernel X rb,t for the scalar hypoelliptic operator AX b on X defined in chapter 11. By rough estimates, we mean just uniform bounds on the heat kernel, and not a Gaussian-like decay like in (4.5.3). Such refined estimates will be obtained in chapter 13 for bounded b, and in chapter 15 for b large. We will also obtain X corresponding bounds for the heat kernels associated with operators AX b , Ab over Xb. The case of X should be thought of as a warm-up for the case of Xb, but the arguments used in both cases are essentially the same. Note that the X X operators BbX , BX b , which are conjugate to Ab , Ab , also play an important role in the present chapter. Moreover, we give a probabilistic construction of the heat kernels. As explained in the introduction, most of the analytic results in the book will be obtained by probabilistic methods. We also explain the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X. In this context, recall that the hypoelliptic Laplacian does not have a wave equation. However, we show that after integration in the fibre variable Y T X ∈ T X, the projected heat kernel verifies a wave-like equation. This behavior becomes stronger and stronger as b → +∞. This idea is used to obtain the proper estimates on the hypoelliptic heat kernel in the present chapter as well as in chapter 13. The rough bounds we mentioned before cannot be directly derived from our previous work with Lebeau [BL08], where only the case of a compact Riemannian manifold X is considered. Still the methods of [BL08] could lead to such rough estimates when b > 0 remains bounded and stays away from 0. In [BL08, chapter 17], the control of the behavior of the heat operator as b → 0 is obtained by functional analytic arguments, from which uniform pointwise estimates cannot be easily extracted. Recall that Malliavin [M78, M97] invented the Malliavin calculus to study the heat kernel associated with the second order hypoelliptic operators considered by H¨ ormander [Hö67]. Malliavin used the fact that the heat kernel can be obtained as the image of a classical flat Brownian measure P via the solution of a stochastic differential equation. Integration by parts on Wiener space ultimately explains the regularity of the heat kernel. Other methods of integration by parts on Wiener space have been given by Stroock [St81a, St81b] and ourselves [B81a, B84]. One key feature of Malliavin’s approach is that it connects the calculus of variations with the analysis of the heat kernel for hypoelliptic operators. In particular, the estimates on the

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heat kernel rely on estimates of a certain Malliavin covariance matrix. To obtain our rough estimates, we use the methods of the Malliavin calculus, part of which were already applied in [BL08, chapter 14] to the hypoelliptic Laplacian. It turns out that in the context of our hypoelliptic operators, estimating the Malliavin covariance matrix can be very easily done via the computations we made in chapter 10 of the explicit solutions to certain elementary variational problems. The parameter b > 0 entering explicitly in the computations, it is easy to obtain this way whatever uniformity is needed for the rough estimates. X We also prove that as b → 0, the heat kernel rb,t converges to the standard heat kernel of X, by a method that is different from what is done in [BL08]. In [BL08], the convergence of the heat operators was obtained by functional analytic arguments, which did not produce pointwise estimates. Here the convergence will be obtained by dynamical arguments, involving a detailed analysis of the convergence as b → 0 of the relevant hypoelliptic diffusion on X with parameter b > 0 to the standard Brownian motion of X. The fact that the symmetric space X has constant curvature does not play a fundamental role in the arguments, except through the uniform control of the geometry of X. This chapter is organized as follows. In section 12.1, we briefly recall the application of the Malliavin calculus given in [B84] to the Brownian motion on X and to the standard elliptic heat kernel, the key point being a simple integration by parts formula on Wiener space. In section 12.2, we give a probabilistic description of the heat kernel for BbX , by constructing an associated Markov diffusion x· , Y·T X ∈ X . In section 12.3, we relate the heat equation for BbX to the wave equation. In section 12.4, along the lines of [BL08, chapter 14], we obtain an integration by parts formula for the diffusion x· , Y·T X . In section 12.5, we use the results of chapter 10 to obtain a more explicit integration by parts formula. In section 12.6, we obtain a uniform control of the integration by parts formula as b → 0. In section 12.7, when b > 0 is bounded, we obtain the rough uniform X estimates for rb,t . X In section 12.8, we study the limit of rb,t as b → 0. X In section 12.9, we obtain rough estimates on a rescaled version of rb,t as b → +∞. In section 12.10, we extend the above results to the heat kernels associated b with the scalar operator AX b on X . Finally, in section 12.11, we obtain corresponding results for the heat kerb nels associated with the operator AX b over X .

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12.1 The Malliavin calculus for the Brownian motion on X Here we assume again X = G/K to be the symmetric space that was considered in the previous chapters of the book. Let ∆X be the Laplace-Beltrami operator on X. Then −∆X is just the action of the Casimir operator on C ∞ (X, R). We denote by C ∞,c (X, R) the vector space of smooth real functions on X that have compact support. Since X is complete, the restriction of ∆X to 0 C ∞,c (X, R) is essentially self-adjoint. For t > 0, let pt (x, x ) be the smooth X heat kernel associated with exp t∆ /2 . If e ∈ p, we identify e with the corresponding left invariant vector field on G. Let e1 , . . . , em be an orthonormal basis of p. Let ∆G,H be the differential operator on G, m X ∇2ei . (12.1.1) ∆G,H = i=1

Then ∆G,H is a self-adjoint nonelliptic operator on G. If f ∈ C ∞ (X, R), it follows from the previous considerations that ∆G,H p∗ f = p∗ ∆X f.

(12.1.2)

Let C (R+ , p) be the vector space of continuous functions from R+ into p, and let w· be its generic element. For t ∈ [0, +∞[, let Ft be the σ-algebra generated by ws , s ≤ t. Let Ft |t≥0 denote the corresponding filtration. Let F∞ be the σ-algebra generated by the Ft , t ≥ 0. Let P be the Brownian measure on C (R+ , p), with w0 = 0, and let E be the corresponding expectation operator. In the sequel, we use the probability space (C (R+ , p) , F∞ , P ). Consider the stochastic differential equation on X, x˙ = w, ˙

x0 = p1.

(12.1.3)

τt0 w˙

along the trajectory In (12.1.3), w˙ is identified with its parallel transport x· . Recall that w· is P a.s. nowhere differentiable. The theory of stochastic differential equations takes care of the difficulties in making sense of the above. If we identify T G to G × g via the left-invariant vector fields, we may as well solve the stochastic differential equation on G, g˙ = w, ˙

g0 = 1,

(12.1.4)

and set ∞,c

x· = pg· .

(12.1.5)

Let f ∈ C (X, R). The crucial link between the heat equation semigroup exp t∆X /2 and Brownian motion is that exp t∆X /2 f (x0 ) = E [f (xt )] . (12.1.6) Similarly, if u ∈ C ∞,c (G, R), then exp t∆G,H /2 u (1) = E [u (gt )] .

(12.1.7)

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Let ht be a bounded process with values in p that is predictable with respect to the filtration Ft |t≥0 , let Bt be a bounded predictable process taking values in k. Now we describe the integration by parts formula of [B84, Theorem 2.2]. Let δw be the Itô differential of w· , let dw = wds ˙ be its Stratonovitch differential. For ` ∈ R, set Z t eàd(Bs ) (δws + `hs ds) . (12.1.8) wt` = 0

Then using the Girsanov formula [KSh91, Theorem 3.5.1 and Corollary 3.5.13] as in [B84, equation (2.18)], we know that for any t ∈ R+ , the probability law w·` on Ft is equivalent to the probability law of w· . More precisely, put Z t Z 1 2 t 2 ` |hs | ds . (12.1.9) Zt = exp −` hhs , δws i − ` 2 0 0 Then Zt` is an integrable martingale with respect to the filtration Ft |t≥0 . Let P ` be the probability measure on C (R+ , p), such that for any t ∈ R+ , dP ` |F = Zt` . (12.1.10) dP t Using Girsanov’s theorem and also the rotational invariance of Brownian motion, we find that under P ` , the probability law of w·` is equal to P . In (12.1.3), we replace w· by w·` , and we denote by x`· the solution of the corresponding stochastic differential equation. Let f : X → R be a smooth bounded function with bounded first derivative. By the above, we get E f x`t Zt` = E [f (xt )] . (12.1.11) The integration by parts formula of [B84] is obtained by differentiating (12.1.11) at ` = 0. First, we calculate the differential of x`· with respect to ` at ` = 0. It is a nontrivial fact that differentiation is legitimate. It is made possible via the theory of stochastic flows. Ultimately, and forgetting about technicalities, the rules of classical differential calculus do apply, as long as one uses Stratonovitch differentials. Recall that the curvature Ω of the canonical connection on the principal bundle p : G → G/K is given by (2.1.10). Also the Ricci tensor RX of X is given by (2.6.8). In particular RX is parallel. Consider the differential equation on the processes (ϑs , $s ) ∈ g = p ⊕ k along the path xs , 1 X d$ = [θ, dw] , (12.1.12) dϑ = − R ϑ + h ds + [$ + B, δw] , 2 ϑ0 = 0,

$0 = 0.

In (12.1.12), ϑ· ∈ p is identified to the corresponding section of T X over the path x· that is obtained by parallel transport with respect to ∇T X . Also in

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(12.1.12), dw still denotes the Stratonovitch differential of w, and δw its Itô differential. The integration by parts formula of [B84, Theorem 2.2] asserts in its simplest form that if f : X → R is a smooth function, then Z t E [hf 0 (xt ), ϑt i] = E f (xt ) hhs , δwi . (12.1.13) 0

Consider the differential equation, 1 ϑ˙ = − RX ϑ + h, 2 ϑ0 = 0,

d$ = [θ, dw] ,

(12.1.14)

$0 = 0.

Observe that (12.1.14) is a special case of (12.1.12), by simply taking B = −$. Then the key observation in [B84] is that (12.1.13) still holds. From the above, we find that if ϑ· is a C 1 predictable path with values in p with uniformly bounded first derivative such that ϑ0 = 0, then Z t 1 (12.1.15) E [hf 0 (xt ) , ϑt i] = E f (xt ) ϑ˙ + RX ϑ, δw . 2 0 Observe that " Z 2 # Z t t 2 |hs | ds . hδw = E E

(12.1.16)

0

0

More generally, by Doob’s inequality [Do53, Chapter VII, Theorem 3.4], [ReY99, Theorem II (1.7)] and by the Burkholder-Davis-Gundy inequality [BuGu70], [ReY99, Theorem IV (4.1)], for 1 < p < +∞,

Z Z s

Z t 1/2

t

2

' |h| ds sup hh, δwi ' hh, δwi

. (12.1.17)

0≤s≤t

0 0 0 p p p

By (12.1.16), if h· is a nonrandom function, " Z 2 # Z t t 2 E hδw = |hs | ds. 0

(12.1.18)

0

Now we fix e ∈ p, and we impose the condition ϑt = e in (12.1.14). It is natural to choose h so as to minimize the L2 norm of h Z t 1/2 2 |h|L2 = |hs | ds . (12.1.19) 0

X

Since R is a constant matrix, the first equation in (12.1.14) is deterministic and the question of finding the optimal h is a simple control theoretic problem. We get sinh sRX /2 RX 2 ϑt = e, |h| = e, e . (12.1.20) L2 sinh (tRX /2) 1 − e−tRX

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Of course (12.1.20) includes the case where RX vanishes, and (12.1.20) becomes 2

|e| s 2 |h|L2 = . (12.1.21) ϑt = e, t t Also note that if ϑs = se/t, then ϑ· is a solution of the first equation in (12.1.14), with 1 1 X h= 1 + sR e. (12.1.22) t 2 If h is given by (12.1.22), then 1 t X t2 X,2 2 |h|L2 = 1+ R + R e, e . t 2 12

(12.1.23)

The advantage of this choice of h is that it is valid even if RX is non constant. 12.2 The probabilistic construction of exp −tBbX over X X Recall that the scalar operators AX b , Bb over X were defined in (11.1.1)– X (11.1.3), and the corresponding smooth heat kernels rb,t , sX b,t were introduced in Definition 11.5.3. Again, we consider the probability space (C (R+ , p) , F∞ , P ). Fix Y T X = p Y ∈ p. As explained in [B05, section 3.9] and in [BL08, section 14.2], the dynamics of the path x· , Y·T X ∈ X associated with the operator BbX is given by

Y TX , b x0 = p1, x˙ =

Y Y˙ T X = −

TX

b2

Y0T X

w˙ , b = Y p. +

(12.2.1)

In (12.2.1), Y˙ T X denotes the covariant derivative of Y T X with respect to ∇T X . The first line of equation (12.2.1) can be rewritten in the form b2 x ¨ + x˙ = w, ˙

Y T X = bx. ˙

(12.2.2)

Note that (12.2.1) is still a stochastic differential equation on X . However, Y T X is a continuous process, so that x· is C 1 . On a general smooth Riemannian manifold, there is no difficulty in defining parallel transport along such paths. Finally, observe that as b → 0, the first equation in (12.2.2) degenerates formally to equation (12.1.3). We said before that equations (12.2.1), (12.2.2) are stochastic differential equations. However, the probabilistic machinery is not needed to solve this equation, even on an arbitrary Riemannian manifold. More precisely, we claim that these equations can be solved for an arbitrary continuous path w· . Indeed let x· be a C 1 path in X such that x0 = p1. Let us fix a local coordinate system near x0 on X, disregarding the fact that here, such a

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coordinate system can be chosen globally. Let ΓT X be the connection form for ∇T X in this coordinate system. The parallel transport τt0 from Tx0 X to Txt X is obtained by solving the differential equation, τ˙t0 + ΓTxtX (x) ˙ τt0 = 0, τ00 = 1. (12.2.3) Put −1 . (12.2.4) τ0t = τt0 Set ZtT X = τ0t YtT X , (12.2.5) TX so that Z· takes its values in Tx0 X. In local coordinates, equation (12.2.1) can be rewritten in the form TX TX w˙ ZT X 0 TX 0Z 0Z , τ˙t + Γxt τt τt0 = 0, Z˙ T X = − 2 + , x˙ = τt b b b b (12.2.6) x0 = x, Z0T X = Y p , τ00 = 1. By rewriting the last equation in (12.2.6) in the form Z t TX wt Zs TX p ds + , (12.2.7) Zt = Y − 2 b b 0 it is easy to see that at least locally, equation (12.2.6) has a unique solution for any continuous path w· . This solution depends continuously on w· for the topology of uniform convergence of continuous functions over compact sets in R+ . Such an argument can be globalized easily. Let us give the probabilistic counterpart for the operator BbG in (11.1.4). Let w· be a Brownian motion in p. Then we consider the differential equation on (g, Y p ) ∈ G × p, Yp Yp w˙ g˙ = , Y˙ p = − 2 + , (12.2.8) b b b p p g0 = 1, Y0 = Y , so that xt = pgt , xt , YtT X = p gt , Ytp . (12.2.9) By proceeding as in (12.2.7), we find that (12.2.8) makes sense for an arbitrary continuous function w· , and also that the solution of (12.2.8) depends continuously on w· for the topology of uniform convergence over compact sets in R+ . Recall that in section 11.5, for t ≥ 0, we have constructed the heat operX ators exp −tAX , exp −tB b b . Now we establish the obvious extension of (12.1.6), (12.1.7). Theorem 12.2.1. Let F : X → R be a smooth function with compact support. Then for t ≥ 0, exp −tBbX F (x0 , Y p ) = E F xt , YtT X . (12.2.10) Similarly, if u : G × p → R is a smooth function with compact support, then exp −tBbG u (1, Y p ) = E u gt , Ytp . (12.2.11)

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Proof. Let Ψ : R+ × X → R be given by (11.5.7). By Proposition 11.5.2 and by (11.5.8), this is a smooth function. Take t > 0. By (11.5.9), we know that s ∈ [0, t] → Ψ t − s, xs , YsT X is a local martingale. We will prove that it is a martingale. By (11.5.8), we get 2 Ψ t − s, xs , YsT X = exp YsT X /2 Φ t − s, xs , YsT X . (12.2.12) By Proposition 11.5.2, for 0 ≤ s ≤ t, the function Φ s, x, Y T X is smooth and uniformly bounded. Recall that the kernel ktp on p is given by (10.4.6). 0 p By the results of section 10.8 and by (12.2.1), for s > 0, ks/b2 Y p , Y p dY p0 TX s TX p is hthe probability ilaw of Zs = τ0 Ys . By (10.7.1), given Y ∈ p, t ≥ 0, 2 is uniformly bounded for 0 ≤ s ≤ t. Using (12.2.12) and E exp YsT X the above, we find that the local martingale in (12.2.12) is a martingale. Since Ψ (0, ·) = F , we get (12.2.10). Equation (12.2.11) is just a trivial lift of (12.2.10).

Remark 12.2.2. Since G acts transitively on X, equation (12.2.10) is still valid for any x, Y T X ∈ X . An identity similar to (12.2.10) for the heat operator associated with a conjugate of AX b will also be given in equation (12.9.9). The proof of Theorem 12.2.1 is the model of the proofs given in the book, which identify the analytically defined heat kernels of chapter 11 with their probabilistic counterpart. 12.3 The operator BbX and the wave equation As before, ∆T X denotes the Laplacian acting along the fibres T X of X . Let e1 , . . . , em be an orthonormal basis of T X. Let C be the differential operator on X , m X C= ∇ei ∇Vei . (12.3.1) i=1

If f ∈ C

∞

(X, R),

∇T· X ∇· f

denotes the Hessian of f .

Proposition 12.3.1. The following identity holds: 2 1 1 b2 BbX,2 − BbX = 2 −∆T X + 2∇VY T X − 2 −∆T X + 2∇VY T X 4b 2b 1 C − ∇Y T X −∆T X + 2∇VY T X + + ∇TY X (12.3.2) T X ∇Y T X . b b In particular, if f (x) is a smooth function on X, then b2 BbX,2 − BbX f = ∇TY X (12.3.3) T X ∇Y T X f. Equation (12.3.3) can also be written in the form −BbX (f + b∇Y T X f ) = ∇TY X T X ∇Y T X f.

(12.3.4)

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Proof. Equation (12.3.2) follows from a trivial computation, which is left to the reader. By (12.3.2), we get (12.3.3). Also, −b2 BbX f = b∇Y T X f,

(12.3.5)

so that (12.3.4) follows from (12.3.3). Remark 12.3.2. From the probabilistic point of view, equation (12.3.3) connects the hypoelliptic operator BbX with −∆X /2. In fact it is because of equation (12.3.3) that as we shall see in section 12.8, as b → 0, the operators X X AX b and Bb deform to the elliptic operator −∆ /2. An identity very similar to (12.3.2) can be established for the operator BbG . Finally, an identity similar to (12.3.4) is established in Proposition 14.2.1 for an operator MX b conjugate to the operator LX b . For t > 0, set Sb,t = exp −tBbX .

(12.3.6)

If f (x) is a smooth bounded function on X, by (12.3.3), we get ∂ ∂2 Sb,t f = Sb,t ∇TY X (12.3.7) b2 2 + T X ∇Y T X f. ∂t ∂t Recall that sX x, Y T X , x0 , Y T X0 is the smooth kernel associated b,t X with Sb,t . By Theorem 12.2.1, sX b,t is nonnegative. We claim that sb,t is everywhere positive. Indeed by a theorem of Stroock and Varadhan for 0 T[StV72], X0 x, Y T X ∈ X , the support of sb,t x, Y T X , x0 , Y T X0 dx dY is equal to X . By taking adjoints, the role of the couples x, Y T X and x0 , Y T X0 can be reversed. Also, Z X TX 0 T X0 TX sb,t x, Y , x ,Y = sX b,t/2 (x, Y ) , z, Z X sX z, Z T X , x0 , Y T X0 dzdZ T X . (12.3.8) b,t/2 If the couples x, Y T X , x0 , Y T X0 are such that vanishes, by the (12.3.8) TX TX above, as a function of z, Z T X , sX x, Y , z, Z vanishes idenb,t/2 tically. However, this is impossible in view of the fact that Z sX x, Y T X , z, Z T X dzdZ T X = 1. (12.3.9) b,t/2 X

Put Z

x, Y T X , x0 , Y T X0 dY T X0 , Tx0 X Z 0 1 TX Mb,t x, Y ,x = sX x, Y T X , x0 , Y T X0 b,t T X 0 σb,t ((x, Y ) , x ) Tx0 X (12.3.10) Y T X0 ⊗ Y T X0 dY T X0 .

σb,t

x, Y T X , x0 =

sX b,t

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TX

Note that Mb,t x, Y morphisms of Tx0 X. In the sequel, we fix x, Y T X ∈ X . We can associate to Mb,t the second order elliptic operator acting on C ∞ (X, R),

Mb,t x, Y T X g (x0 ) = ∇T· X ∇· , Mb,t x, Y T X , x0 g (x0 ) . (12.3.11) In (12.3.11), the operator ∇T· X ∇· acts on the variable x0 . By (12.3.7), (12.3.11), we get ∂ ∂2 − Mb,t x, Y T X σb,t x, Y T X , · = 0. b2 2 + ∂t ∂t

(12.3.12)

Equation (12.3.12) is a hyperbolic equation. Of course, it does not determine σb,t , since Mb,t depends on sb,t . As we shall see in (12.8.46), as b → 0, T X0 2 TX 0 T X0 −m/2 0 Y sX x, Y , x , Y → π p (x, x ) exp − . (12.3.13) t b,t In (12.3.13), the convergence of the functions and their derivatives of arbitrary order is uniform over compact sets. Let 1 be the identity of T X. We identify 1 to the scalar product of T X. By (12.3.13), as b → 0, 1 σb,t x, Y T X , x0 → pt (x, x0 ) , Mb,t x, Y T X , x0 → . (12.3.14) 2 It is interesting to compare (12.3.12) with the heat equation, ∆X ∂ − pt (x, ·) = 0. (12.3.15) ∂t 2 By (12.3.11), (12.3.13), and (12.3.14), as b → 0, equation (12.3.12) converges in the proper sense to equation (12.3.15). While equation (12.3.15) is parabolic, equation (12.3.12) is hyperbolic. This should give a sense that when only the variable x is considered, there is a wave equation quality to the heat operator for the hypoelliptic Laplacian. In that respect, it is tempting to also consider the genuinely hyperbolic operator ∂2 ∂ ∆X + − . (12.3.16) 2 ∂t ∂t 2 This wave operator propagates at the speed √12b . When b → 0, it converges Hbp = b2

∂ in the proper sense to the parabolic heat operator ∂t − ∆X /2. However, it certainly does not interpolate between this parabolic operator and the geodesic flow, and so it cannot be used as a substitute to the hypoelliptic Laplacian. The computations that follow are a path integral version of equation (12.3.4) in Proposition 12.3.1. Let f : X → R be a smooth function. By (12.2.1), we get Z t ∇Y T X f (xs ) f (xt ) = f (x0 ) + ds. (12.3.17) b 0

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Still using (12.2.1) and integrating by parts in (12.3.17), we obtain Z t ∇Y T X f (xs ) ds = −b ∇YtT X f (xt ) − ∇Y p f (x0 ) b 0 Z t Z t TX + ∇YsT X ∇YsT X f (xs ) ds + ∇δws f (xs ) . (12.3.18) 0

0

In (12.3.18), the last term is an Itô integral with respect to the Brownian motion w· . By (12.3.17), (12.3.18), we get f (xt ) = f (x0 ) − b ∇YtT X f (xt ) − ∇Y p f (x0 ) Z t Z t ∇δws f (xs ) . (12.3.19) ∇TYsX + T X ∇Y T X f (xs ) ds + s 0

0

Note that if f has uniformly bounded first and second derivatives, when taking the expectation in (12.3.19), and using (12.2.10), we recover (12.3.4). Recall that if e ∈ g, e is identified with the corresponding left-invariant vector field on G. If u : G → R is a smooth function, if g· , Y·p are taken as in (12.2.8), instead of (12.3.19), we have the identity u (gt ) = u (1) − b ∇Ytp u (gt ) − ∇Y p u (1) Z t Z t p p + ∇Ys ∇Ys u (gs ) ds + ∇δws u (gs ) . (12.3.20) 0

0

Of course, if u = p∗ f , (12.3.20) is equivalent to (12.3.19). 12.4 The Malliavin calculus for the operator BbX In this section, we follow Bismut-Lebeau [BL08, chapter 14]. In that reference, the Malliavin calculus is developed when X is instead a compact Riemannian manifold. We will obtain a formula of integration by parts for the process (x· , Y·T X ) in (12.2.1). ` As in section 12.1, in equation (12.2.1), we replace w· by w· given by T X,` (12.1.8). Let x`· , Y· be the corresponding solution of (12.2.1). We will compute the differential of this solution with respect to ` at ` = 0. In the D denotes covariant differentiation with respect to the Levi-Civita sequel, Ds connection along x· . Set ∂ ` x . (12.4.1) JsT X = ∂` s D T X D2 T X We use the notation J˙T X , J¨T X instead of Ds J , Ds2 J . By the first equation in (12.2.1), 1 D T X,` Y . (12.4.2) J˙sT X = b D` s

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By (12.2.1), (12.2.2), we obtain δw b2 J¨T X + J˙T X + b2 RT X J T X , x˙ x˙ = h + B + $, , ds $ ˙ = J T X , x˙ , J0T X = 0, J˙0T X = 0, $0 = 0.

(12.4.3)

Here J·T X ∈ p is identified with its parallel transport along x· with respect to ∇T X . Note that by the arguments given in section 12.2, differentiation with respect to ` does not necessitate sophisticated probabilistic arguments. Now we state a result established in [BL08, section 14.2]. This result gives an analogue of equation (12.1.13), i.e., it is a formula of integration by parts. Theorem 12.4.1. Let F : X → R be a smooth function with compact support. Then Z hD Ei t E dF xt , YtT X , JtT X , bJ˙tT X = E F xt , YtT X hh, δwi . 0

(12.4.4) Proof. The proof of (12.4.4) still relies on Girsanov’s transformation, and can be given by following the same arguments as in section 12.1. The same argument as in (12.1.11) shows that h i E F x`t , YtT X,` Zt` = E F xt , YtT X . (12.4.5) Using (12.4.1), (12.4.2) and (12.4.5), we get (12.4.4). Now we proceed as in [BL08, section 14.2]. Consider the differential equation b2 J¨T X + J˙T X + b2 RT X J T X , x˙ x˙ = h, $ ˙ = J T X , x˙ , (12.4.6) TX TX ˙ J0 = 0, J0 = 0, $0 = 0. Then (12.4.6) is a special case of (12.4.3), by taking B = −$.

(12.4.7)

In particular Theorem 12.4.1 applies to the solution of (12.4.6). Remark 12.4.2. The presence of the expression b2 RT X J T X , x˙ x, ˙ which is quadratic in x, ˙ is still another reflection of the considerations we made in Remark 9.10.3. The facts outlined in this remark are also obvious in (12.2.6).

12.5 The tangent variational problem and integration by parts Consider equation (12.4.6). It has been obtained by differentiating equation (12.2.2) in the parameter `. Except for the term containing the curvature RT X , when replacing formally h by w, ˙ we recover equation (12.2.2). Take e, f ∈ p. Let J·T X be any smooth deterministic function defined on [0, t] with values in p, such that J0T X = 0, J˙0T X = 0, JtT X = e, J˙tT X = f /b. (12.5.1)

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Then equation (12.4.6) determines a predictable h to which (12.4.4) can be applied. This principle was used in [BL08, section 14.3], with a very simple choice of J·T X . However, the choice made in [BL08] is not adequate for our purpose, because with this choice, the norm |h|L2 diverges as b → 0. So instead, when E = p, we consider the variational problem of section 10.1 for the functional Kb,t in (10.1.2). Also we use the notation in (10.1.17)– (10.1.19). We fix e, f ∈ p. Given a control v· ∈ L2 , consider the differential equation b2 J¨T X + J˙T X = v, J0T X = 0, J˙0T X = 0, (12.5.2) J˙tT X = f /b.

JtT X = e,

The control problem consists in finding v with minimal norm |v|L2 . This question was solved in section 10.3, in equations (10.3.30)–(10.3.39), in which we make x = 0, Y = 0, x0 = e, Y 0 = f /b. Moreover, by (10.3.38), 1 2 (12.5.3) |v| = Kb,t ((0, 0) , (e, f /b)) . 2 L2 By the considerations that follow (10.3.47), as b → 0, we have the uniform convergence on [0, t], s JsT X → e. (12.5.4) t Moreover, by (10.3.49), (10.3.50), and (12.5.3), as b → 0, 1 2 e 1 2 2 |v|L2 → |e| + |f | , v → weakly in L2 . (12.5.5) 2 2t t Take > 0, M > 0 such that 0 < ≤ M . Using again the considerations after (10.3.47), we know that as long as ≤ t ≤ M , and e, f remain uniformly bounded, the above convergence results are uniform with respect to t, e, f . Also bJ˙T X remains uniformly bounded. Let τt0 be the parallel transport operator from Tx0 X into Txt X with respect to ∇T X . We fix unit vectors e, f ∈ p, and for a given b > 0, we denote by v the optimal control that has been determined above. By Theorem 12.4.1, and by (12.4.6), (12.5.2), we get

E dF xt , YtT X , τt0 (e, f ) Z t

= E F xt , YtT X v + b2 RT X J T X , x˙ x, ˙ δw . (12.5.6) 0

Equation (12.5.6) is still not enough for our purpose. If a ∈ g, recall that aT X denotes the corresponding Killing vector field on X. We fix c, d ∈ g. Let e1 , . . . , em be an orthonormal basis of p. Given b > 0, we denote by vi , v i the controls determined above, with e = ei , f = 0, and e = 0, f = ei , and by JiT X , J Ti X the associated solutions of (12.5.2). Finally, we define $i , $i as in (12.4.6), i.e., $ ˙ i = JiT X , x˙ , $i,0 = 0, (12.5.7) TX $ ˙ i = J i , x˙ , $i,0 = 0.

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Theorem 12.5.1. Let F : X → R be a smooth function with compact support. The following identity holds:

E dF xt , YtT X , cTxtX , dTxtX " Z X TX 0 t

= E F xt , YtT X cxt , τt ei vi + b2 RT X JiT X , x˙ x, ˙ δw 0

1≤i≤m

+ dTxtX , τt0 ei −

X

Z 0

t

!#

˙ δw v i + b2 RT X J Ti X , x˙ x,

TX 0

E F xt , YtT X cxt , τt $i,t ei + dTxtX , τt0 $i,t ei . (12.5.8)

1≤i≤m

Proof. Clearly, we have the identities X

X

cTxtX = cTxtX , τt0 ei τt0 ei , dTxtX = dTxtX , τt0 ej τt0 ej . (12.5.9) 1≤i≤m

1≤j≤m

Set Ui,t = F xt , YtT X U i,t

TX 0 cxt , τt ei ,

= F xt , YtT X dTxtX , τt0 ei .

(12.5.10)

` Let h be a given bounded predictable process. Let Ui,t , U ì,t be the random variables Ui,t , U i,t in (12.5.10) in which w· has been replaced by w·` given by (12.1.8). Then the obvious analogue of (12.1.11), (12.4.5) is that h i ` ` E Ui,t Zt = E [Ui,t ] , E U ì,t Zt` = E U i,t . (12.5.11) In the first equation, we replace h by vi +b2 RT X JiT X , x˙ x, ˙ and in the second ˙ We take the differential of equation, we replace h by v i + b2 RT X J Ti X , x˙ x. the identities in (12.5.11) at ` = 0, and we sum them in i, 1 ≤ i ≤ m. Also we use (12.5.9) and the fact that since cT X is a Killing vector field, D E ∇Tτ 0Xei cT X , τt0 ei = 0, (12.5.12) t

and we get (12.1.15). The proof of our theorem is completed. The principle of the Malliavin calculus is to iterate a formula like (12.5.8), in order to obtain on the left-hand side of (12.5.8) the expectation of a TX TX differential operator of arbitrary order in the vector fields c ,TdX applied TX in terms of expectations where only F xt , Yt appears. to F at xt , Yt Equation (12.5.8) is the model of such an equation in the case where the order of the operator is 1. Let us explain how to do this when the order is 2. We use (12.5.8) with F x, Y T X replaced by uc,d x, Y T X = dF x, Y T X , cTx X , dTx X . On the right-hand side of (12.5.8), the function uc,d x, Y T X appears. Then the same method as before can be used on this right-handside so as to obtain another expression where only the function F x, Y T X appears. Of course

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extra terms have now to be differentiated. However, they are algebraically very simple. Moreover, since F has compact support, except for the terms containing x˙ and δw, they are uniformly bounded on the domain where F xt , YtT X is nonzero. 12.6 A uniform control of the integration by parts formula as b → 0 Let K be the compact support of F . Over K, the norms of cT X , dT X are uniformly bounded. We will write (12.5.8) in the form h i

E dF xt , YtT X , cTxtX , dTxtX = E F xt , YtT X Mtc,d . (12.6.1) The random variable Mtc,d is explicitly determined in (12.5.8). Note that Mtc,d is linear in cTxtX , dTxtX . We denote by Mt the corresponding linear form on Txt X × Txt X. Theorem 12.6.1. Given > 0, M > 0, ≤ M, p ∈ [1, +∞[, there is C,M,p > 0 such that for 0 < b ≤ M, ≤ t ≤ M , 2 kMt kp ≤ C,M,p 1 + |Y p | . (12.6.2) Proof. As we saw in section 12.5, for 0 < b ≤ M, ≤ t ≤ M , the vi , v i are deterministic, and their L2 norm is uniformly bounded. By (12.1.17), for 1 < Rt Rt p < +∞, we get a uniform control of the Lp norm of 0 hvi , δwi , 0 hv i , δwi. In the sequel, KiT X denotes one of the JiT X , J Ti X . By (12.1.17), for 1 < p < +∞,

Z

Z t

1/2

t

2 TX 2

b2 RT X KiT X , x˙ x˙ ds b R KiT X , x˙ x, ˙ δw

.

'

0 0 p p

(12.6.3) As we saw in section 12.5, for 0 < b ≤ M, ≤ t ≤ M , the Ki remain uniformly bounded. Therefore, Z t Z t 2 TX 2 4 TX b R Ki , x˙ x˙ ds ≤ C,M |bx| ˙ ds. (12.6.4) 0

0

By (12.2.1), we get Z

t

Z

4

|bx| ˙ ds = 0

t

T X 4 Y ds.

(12.6.5)

0

By equation (10.8.27) in Proposition 10.8.2, by (12.6.3)–(12.6.5), we find that for 1 < p < +∞, for 0 < b ≤ M, ≤ t ≤ M ,

Z t

2 TX TX

≤ C,M,p 1 + |Y p |2 . b R K , x ˙ x, ˙ δw (12.6.6) i

0

p

We denote by σi either $i or $i . By (12.5.7), Z t σi,t = − RT X KiT X , x˙ ds. 0

(12.6.7)

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By (12.2.2), we can rewrite (12.6.7) in the form Z t σi,t = − RT X KiT X , w˙ − b2 x ¨ ds.

(12.6.8)

Recall that KiT X is deterministic and smooth, so that Z t Z t RT X KiT X , w˙ ds = RT X KiT X , δw .

(12.6.9)

0

0

0

Using (12.1.17) and the fact that for 0 < b ≤ M, ≤ t ≤ M , KiT X is uniformly bounded, the Lp norm of (12.6.9) remains uniformly bounded. Moreover, if KiT X = JiT X , using (12.2.2) and the fact that RT X is parallel, we get Z t Z t RT X KiT X , x b2 ¨ ds = bRT X ei , YtT X − RT X bK˙ iT X , Y T X ds. 0

0

(12.6.10) If KiT X = J Ti X , the first term in the right-hand side of (12.6.10) does not appear. By equation (10.8.26) in Proposition 10.8.2, under the given conditions on b, t, we get

TX

bR ei , YtT X p ≤ Cp,M (1 + |Y p |) . (12.6.11) As we saw after (12.5.5), for 0 < b ≤ M, ≤ t ≤ M , bK˙ iT X remains uniformly bounded on [0, t]. Therefore Z t Z t TX TX TX TX ˙ Ys ds. R (12.6.12) bKi , Y ds ≤ C,M 0

0

The same arguments as before show that

Z t

p TX TX TX ˙

R bKi , Y ds

≤ C,M,p (1 + |Y |) .

0

(12.6.13)

p

Using (12.6.7)–(12.6.13), for 0 < b ≤ M, ≤ t ≤ M , kσi,t kp ≤ C,M,p (1 + |Y p |) .

(12.6.14)

By (12.5.8), (12.6.1), and by (12.6.3)–(12.6.14), we get (12.6.2). The proof of our theorem is completed. Remark 12.6.2. Similar arguments can be used when iterating the integration by parts formula (12.5.8) by the procedure outlined in section 12.5. Bounds similar to the bound in Theorem 12.6.1 can be easily obtained. If k is the order of the differential operator acting on F , in the right-hand side 2 2k of (12.6.2), 1 + |Y p | is replaced by 1 + |Y p | . Since Mt has uniformly controlled Lp norm for p ∈ [1, +∞[, random variables like Mt can be multiplied and their product still has uniformly controlled Lp norm.

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X for bounded b 12.7 Uniform rough estimates on rb,t TX 0 0 Recall that the probability law of YtT X is given by kt/b 2 (Y, Y ) dY . As we already proof of Theorem 12.2.1, given b > 0, t ≥ 0, by (10.7.1), saw in the 2 exp YtT X /2 ∈ L2 .

Let F : X → R be a smooth function with compact support. Theorem 12.7.1. The following identity holds: h 2

i E exp YtT X /2 dF xt , YtT X , cTxtX , dTxtX " X 2 TX 0 = E exp YtT X /2 F xt , YtT X cxt , τt ei 1≤i≤m

Z

t

0

+ dTxtX , τt0 ei

−

vi + b2 RT X JiT X , x˙ x, ˙ δw

!#

t

Z

0

˙ δw v i + b2 RT X J Ti X , x˙ x,

h 2

i E exp YtT X /2 F xt , YtT X YtT X , dTxtX

X 1≤i≤m

"

−

X

2 E exp YtT X /2 F xt , YtT X

1≤i≤m

cTxtX , τt0 $i,t ei

#

TX 0 + dxt , τt $i,t ei . (12.7.1)

2 Proof. We replace F x, Y T X by F x, Y T X exp Y T X /2 in (12.5.8) and we get (12.7.1). 2 Remark 12.7.2. Since exp YtT X /2 ∈ L2 , in (12.7.1), we may replace this random variable by 2 2 p exp YtT X /2 1 + YtT X , p ∈ N, and obtain a similar formula. The same is true when replacing dTxtX by any vertical polynomial vector field like ∇VY T X . By equation (10.8.12) in Proposition 10.8.1, for 0 < b ≤ M, ≤ t ≤ M , since t/b2 stays away from exists q = q,M , 1 < q,M < 2 such that 0, there T X 2 for t ≥ , 0 < b ≤ M , exp Yt /2 ∈ Lq , and moreover,

2

2 (12.7.2)

exp YtT X /2 ≤ Cq exp |Y p | /2q . q 2 Let Ntc,d be the expression appearing after exp YtT X /2 F xt , YtT X in the right-hand side of (12.7.1). Again Ntc,d can be viewed as a linear form

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Nt on Txt X ×Txt X. The same arguments as in the proof of equation (12.6.2) in Theorem 12.6.1 show that for 0 0, c,M ∈]0, 1[ such that under the given conditions on b, t,

2

2 (12.7.4)

exp YtT X /2 Nt ≤ C,M exp c,M |Y p | /2 . 1

By (12.7.4), we get 2

2 exp − |Y p | /2 exp YtT X /2 Nt 1 2 ≤ C,M exp − (1 − c,M ) |Y p | /2 . (12.7.5) By (11.5.10) and (12.2.10), if F ∈ C ∞,c (X , R), we get Z X rb,t (x0 , Y p ) , x0 , Y T X0 F x0 , Y T X0 dx0 dY T X0 X h 2 i 2 = exp − |Y p | /2 E exp YtT X /2 F xt , YtT X . (12.7.6) By (12.7.1) and (12.7.6), we obtain Z

X rb,t (x0 , Y p ) , x0 , Y T X0 dF x0 , Y T X0 , cTx0X , dTx0X dx0 dY T X0 X i h 2 2 = exp − |Y p | /2 E exp Y T X /2 F xt , YtT X Ntc,d . (12.7.7)

By proceeding as before, in (12.7.7), dF x0 , Y T X0 , cTx0X , dTx0X can be replaced by the action on F of an arbitrary differential operator in the vector fields cT X , dT X with coefficients that are polynomials in Y T X0 , and we still obtain an analogue of (12.7.7). Moreover, the obvious analogue of (12.7.5) holds. Now we give an analogue of Definition 9.1.2. Definition 12.7.3. A function f : X × X → Ris said to be rapidly decreas k k ing if for any k ∈ N, 1 + Y T X + Y T X0 f x, Y T X , x0 , Y T X0 is uniformly bounded. If f also depends on b > 0, t > 0, if D ∈ R∗+ × R∗+ , we say that f is uniformly rapidly decreasing for (b, t) ∈ D if the previous bounds are uniform. X Theorem 12.7.4. For 0 < b ≤ M, ≤ t ≤ M , rb,t x, Y T X , x0 , Y T X0 and its covariant derivatives of arbitrary order in x0 , Y T X0 with respect to ∇T X are uniformly rapidly decreasing on X × X . Proof. To establish our theorem, we may as well assume that x, Y T X = (x0 , Y p ). Let u (r) : R+ → [0, 1] be a smooth function that is equal to 1

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Chapter 12

for r ≤ 1 and to 0 for r ≥ 2. Take x0 ∈ X. We take geodesic coordinates centered at x0 ∈ X, and we trivialize T X by parallel transport with respect to ∇T X along such geodesics, so that X ∼ p × p. Given α, β ∈ p, set gα,β (y, Y p0 ) = u (|y|) exp (i hα, yi + i hβ, Y p0 i) .

(12.7.8)

Recall that the smooth positive function δ : p → R+ was defined in (4.1.11). Set Z X Φb,t,Y p ,x0 (α, β) = rb,t ((x0 , Y p ) , (y, Y p0 )) gα,β (y, Y p0 ) δ (y) dydY p0 . p×p

(12.7.9) We use equations (12.7.5) and (12.7.7) with F replaced by gα,β , and their higher order analogues. We find that for any k ∈ N, there exist C,M,k > 0, c,M,k > 0 such that for 0 < b ≤ M, ≤ t ≤ M , 2 k k 1 + |α| + |β| |Φb,t,Y p ,x0 (α, β)| ≤ C,M,k exp −c,M,k |Y p | . (12.7.10) Using elementary properties of Fourier transform, we see that for ≤t≤ X M, 0 < b ≤ M , rb,t (x0 , Y p ) , x0 , Y T X0 and its covariant derivatives of any order in the variable x0 , Y T X0 ∈ X are not only uniformly bounded, but they also exhibit a Gaussian decay in the variable Y p . Using (10.8.26) 0 and proceeding as before, the same arguments show that for any k ∈ N, 0 2k X p 0 T X0 T X0 these results also apply to rb,t (x0 , Y ) , x , Y 1+ Y . The proof of our theorem is completed. Remark 12.7.5. In the above argument, we used the fact that X is a symmetric space, because the coordinate system centered at x0 can be chosen uniformly on the full X. Also the above proof gives a uniform Gaussian deX cay of rb,t in the variables Y T X , Y T X0 . We will give a related proof of this fact in Theorem 13.2.4. Also note that in equation (12.5.2), we chose the control v so as to minimize the norm |v|L2 . However, we can instead assume that v vanishes on [0, t/2] and pick the control v such that it minimizes |v|L2 among such controls. This problem is again a problem that was already considered in section 10.3, with the time interval [t/2, t] replacing the time interval [0, t]. The considerations of the previous sections still hold in this case. Of course, using the fact that the kernels sb,· , rb,· form semigroups, the interpretation of such manipulations is quite obvious.

12.8 The limit as b → 0 Theorem 12.8.1. For t > 0, as b → 0, we have the pointwise convergence, X rb,t x, Y T X , x0 , Y T X0 1 1 T X 2 T X0 2 Y + Y . (12.8.1) → m/2 pt (x, x0 ) exp − 2 π

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Proof. In (12.8.1), we may as well take x = x0 , Y T X = Y p ∈ p. To establish (12.8.1), we will show that there is a corresponding narrow convergence of measures, which in view of Theorem 12.7.4, immediately leads to (12.8.1). To establish the narrow convergence, we will use a method inspired by StroockVaradhan [StV72]. In the whole proof, we will assume that 0 < b ≤ 1. Let f : X → R be a continuous function with derivatives of order ≤ 2 that are uniformly bounded and continuous . Put Mf,t = f (xt ) − f (x0 ) − b ∇YtT X f (xt ) − ∇Y p f (x0 ) Z t (12.8.2) − ∇TYsX T X ∇Y T X f (xs ) ds. s 0

By (12.3.19), Mf,t is a martingale given by Z t Mf,t = ∇δw f (xs ) .

(12.8.3)

0

By (12.1.17), (12.8.3), for 0 ≤ s ≤ t, 1 < p < +∞, 1/2

kMf,t − Mf,s kp ≤ Cf,p |t − s|

.

Moreover, if as is a real continuous function, for 1 < p < +∞, Z t p Z t p p−1 ≤ a du |au | du (t − s) . u s

(12.8.5)

s

Using (10.8.26) and (12.8.5), we get

Z t

T X 2

TX (t − s) . Y

T X f (xu ) du ≤ Cf,p ∇ ∇ 1 + T X Y Yu

u s

(12.8.4)

(12.8.6)

p

For s ≤ t, let τts denote parallel transport from Txs X into Txt X along x· with respect to the Levi-Civita connection, and let τst be its inverse. Then ∇YtT X f (xt ) − ∇YsT X f (xs )

= τst YtT X − YsT X , τst df (xt ) + YsT X , τst df (xt ) − df (xs ) . (12.8.7) Conditionally on Fs , the probability law of τst YtT X is a Gaussian with mean TX −2(t−s)/b2 e Ys , and variance 1 − e /2, so that using (10.8.26), we get

t TX

τs Yt − YsT X ≤ Cp 1 − e−(t−s)/b2 1 + Y T X p 2 1/2 + Cp 1 − e−2(t−s)/b . (12.8.8) −(t−s)/b2

Moreover, √ 2 b 1 − e−(t−s)/b ≤ t − s. √ √ Indeed, (12.8.9) holds for b ≤ t − s, and for b > t − s, t−s √ 2 ≤ t − s. b 1 − e−(t−s)/b ≤ b

(12.8.9)

(12.8.10)

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Chapter 12

Finally, by the first inequality in (12.8.10), we get p 2 1/2 b 1 − e−2(t−s)/b ≤ 2 (t − s).

(12.8.11)

By (12.8.8)–(12.8.11), we obtain

√

b τst YtT X − YsT X p ≤ Cp 1 + Y T X t − s.

(12.8.12)

Also for s ≤ t, τst df

Z (xt ) − df (xs ) = s

t

τsu ∇TYuX T X /b df (xu ) du.

By (10.8.26), (12.8.5), and (12.8.13), we obtain

TX t

b Ys , τs df (xt ) − df (xs ) ≤ Cf,p 1 + Y T X 2 (t − s) . p

(12.8.13)

(12.8.14)

By (12.8.2), (12.8.4), (12.8.6), (12.8.7), (12.8.12), and (12.8.14), for 1 < p < +∞, |t − s| ≤ 1, 2 √ kf (xt ) − f (xs )kp ≤ Cf,p 1 + Y T X t − s. (12.8.15) Let k be a left and right K-invariant real smooth function on G with values in R that vanishes on K. Then k descends to a smooth function from X into R, which is invariant under the left action of K on X. If g, g 0 ∈ X, pg = x, pg 0 = x0 , put k (x, x0 ) = k g −1 g 0 . (12.8.16) Then k (x, x0 ) is a smooth function on X ×X, which vanishes on the diagonal. We may and we will assume that when d (x, x0 ) ≥ 1, k is equal to 1. By using (12.8.15) with f (y) = k (x0 , y), for 0 ≤ t ≤ 1, 2 √ kk (x0 , xt )kp ≤ Ck,p 1 + Y T X t. (12.8.17) More generally, using (10.8.26), (12.8.17), and the Markov property of the process x· , Y·T X , we find that for 0 < s ≤ t, t − s ≤ 1, 2 √ kk (xs , xt )kp ≤ Ck,p 1 + Y T X t − s. (12.8.18) For t ≥ 0, let Gt be the σ-algebra on C (R+ , X) generated by xs , s ≤ t, and let Gt |t≥0 be the corresponding filtration. Given T ≥ 0, we use a similar notation over C ([0, T ], X). For T ≥ 0, let Qb,T be the probability law of x· on C ([0, T ], X). Using (12.8.18) and [StV79, page 61, 2.4.2], we deduce from (12.8.18) that for 0 0, let bn be a decreasing sequence in R+ that converges to 0, and is such that Qbn ,T converges to Q0,T for the topology of the narrow convergence. We will show that Q0,T is precisely the probability law of the Brownian motion starting at x0 on C ([0, T ], X). Let f : X → R be a smooth function with compact support. Take s0 , t0 , t such that 0 ≤ s0 < t0 ≤ t ≤ T . Since Mf,· is an F· -martingale, it is also

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a G· -martingale. Let H (x· ) : C ([0, T ], X) → R be a bounded continuous function that is Gs0 -measurable. Then E Qb,T [H (x· ) Mf,t ] = E Qb,T [H (x· ) Mf,t0 ] .

(12.8.19)

By (10.8.26), as n → +∞, h i E Qbn ,T H (x· ) bn ∇YtT X f (xt ) − ∇YtT X f (xt0 ) → 0.

(12.8.20)

0

Take b > 0 small enough so that t0 −b ≥ s0 . To make the notation simpler, we will omit the parallel transport operators along x· . In other words, along x· , T X is trivialized by parallel transport with respect to ∇T X . Then Z t Z t T X ∇TYsX ∇ f (x ) ds = ∇TYsX TX T X ∇Y T X (f (xs ) − f (xs−b )) ds s Ys s t0

t0

Z

t−b

+ t0 −b

∇TY X (12.8.21) T X ∇Y T X f (xs ) ds. s+b s+b

By (10.8.26), (12.8.5), and (12.8.15), for 0 < b ≤ 1, we get

Z t

√ TX

≤ Cf,p 1 + Y T X 4

b. ∇ T X ∇Y T X (f (xs ) − f (xs−b )) ds Y

s s t0

p

(12.8.22) As we saw after (12.8.7), conditionally on Fs , the probability law of Ys+b is a Gaussian centered at e−1/b YsT X and with covariance equal to 1 − e−2/b /2. Since t0 − b ≥ s0 , by the above, we get " # Z t−b

E Qb,T H (x· )

t0 −b

∇TY X T X ∇Y T X f (xs ) ds s+b s+b

" = e−2/b E H (x· )

Z

#

t−b

t0 −b

∇TYsX T X ∇Y T X f (xs ) ds s

" Z −2/b + 1−e E H (x· )

t−b

t0 −b

# 1 X ∆ f (xs ) ds . (12.8.23) 2

We make b = bn in (12.8.23). By (10.8.26), as n → +∞, the first term in the right-hand side of (12.8.23) tends to 0. Since Qbn ,T converges to Q0,T in the sense of narrow convergence, as n → +∞, " # Z t−b Z t 1 1 X Qbn ,T X Q0,T E H (x· ) ∆ f (xs ) ds → E H (x· ) ∆ f (xs ) ds . t0 −b 2 t0 2 (12.8.24) By (12.8.21)–(12.8.24), as n → +∞, Z t Qbn ,T TX E H (x· ) ∇YsT X ∇YsT X f (xs ) ds t0

→E

Q0,T

Z H (x· )

t

t0

1 X ∆ f (xs ) ds . (12.8.25) 2

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Chapter 12

Put t

Z M0,f,t = f (xt ) − f (x0 ) − 0

1 X ∆ f (xs ) ds. 2

(12.8.26)

Using (12.8.2), (12.8.19), (12.8.20), and (12.8.25) we get E Q0,T [H (x· ) M0,t,f ] = E Q0,T [H (x· ) M0,t0 ,f ] .

(12.8.27)

Since (12.8.27) is valid for any bounded continuous function H that is Gs0 -measurable with 0 ≤ s0 < t0 , (12.8.27) still holds for s0 = t0 . Therefore M0,f,· is a G· -martingale. Using a result of Stroock and Varadhan [StV79, chapter 7], Q0,T is just the Brownian measure on C ([0, T ] , X), with initial value x0 . Since Q0,T is uniquely determined, it follows that as b → 0, Qb,T converges to Q0,T . For T = +∞, we will write Qb instead of Qb,+∞ . It follows from the above that for any t ≥ 0, as b → 0, E Qb [f (xt )] → E Q0 [f (xt )] .

(12.8.28)

Let F : X → R be a smooth function with compact support. Given t > 0, and b > 0 such that b ≤ t, t E F xt , YtT X = E F xt−b , τt−b YtT X t + E F xt , YtT X − F xt−b , τt−b YtT X . (12.8.29) t Using the above arguments on the probability law of τt−b YtT X conditional on Ft−b , we get t E F xt−b , τt−b YtT X "Z 1/2 TX =E F xt−b , e−1/b Yt−b + 1 − e−2/b ZT X Txt−b X

# TX T X 2 dZ . (12.8.30) exp − Z π m/2 Moreover, 1/2 TX −2/b TX TX F xt−b , e−1/b Yt−b + 1−e Z − F xt−b , Z TX ≤ CF e−1/b Yt−b + e−2/b Z T X . (12.8.31) By (10.8.26), (12.8.30), and (12.8.31), for 0 < b ≤ 1, t E F xt−b , τt−b YtT X "Z # TX Qb TX T X 2 dZ −E F xt−b , Z exp − Z π m/2 Txt−b X = O e−1/b (1 + |Y p |) . (12.8.32)

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Also since F has compact support, we deduce from (12.8.15) that if in the second term in the left-hand side of (12.8.32), we replace xt−b by xt , this √ p 2 b 1 + |Y | . Using (12.8.28), (12.8.32), introduces an error that is O and the above argument, we find that as b → 0, t E F xt−b , τt−b YtT X # "Z 2 dZ T X TX T X Q0 F xt , Z exp − Z →E . (12.8.33) π m/2 Txt X Now we will estimate the second term in the right-hand side of (12.8.29). We identify F to the smooth function Fe = p∗ F , which maps G × p into R. The function Fe is such that if k ∈ K, Fe gk, Ad k −1 Y = Fe (g, Y ) . (12.8.34) Using the notation in (12.2.8), we get t F xt , YtT X − F xt−b , τt−b YtT X = Fe gt , Ytp − Fe gt−b , Ytp . (12.8.35) Recall that θ denotes the Cartan involution. We equip g = p ⊕ k with the scalar product −B (·, θ·). Also we equip T G with the associated left-invariant metric. We denote by dG the corresponding distance on G. Set (12.8.36) dG = inf dG , 1 . Since F has compact support, we get e p p F gt , Yt − Fe gt−b , Yt ≤ CF dG (gt−b , gt ) .

(12.8.37)

Let u : G → R be a smooth function that is constant outside of the ball of center 0 and radius 1. Instead of (12.8.2), we now set Z t Mu,t = u (gt ) − u (1) − b ∇Ytp u (gt ) − ∇Y p u (1) − ∇Ysp ∇Ysp u (gs ) ds. 0

(12.8.38)

By (12.3.20), Z Mu,t =

t

∇δws u (gs ) ds.

(12.8.39)

0

By proceeding as in (12.8.4)–(12.8.15), for 0 < b ≤ 1, 1 < p < +∞, s ≤ t, |t − s| ≤ 1, we get the analogue of (12.8.15), 2 √ t − s. (12.8.40) ku (gt ) − u (gs )kp ≤ Cu,p 1 + |Y p | Using (10.8.26) and the Markov property of the process g· , Y·p , under the same conditions as above, we get √

u gs−1 gt − u (1) ≤ Cu,p 1 + |Y p |2 t − s. (12.8.41) p By (12.8.41), we deduce that 2 √ kdG (gs , gt )kp ≤ Cp 1 + |Y p | t − s.

(12.8.42)

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By (12.8.35)–(12.8.42), we get √ t E F xt , YtT X − F xt−b , τt−b YtT X ≤ CF b.

(12.8.43)

By (12.8.29), (12.8.33), (12.8.43), as b → 0, "Z # TX TX Q0 TX T X 2 dZ F xt , Z E F xt , Yt →E exp − Z . π m/2 Txt X (12.8.44) By (12.1.6), (12.2.10), we can rewrite (12.8.44) in the form Z 0 T X0 p 0 T X0 F x0 , Y T X0 sX dx dY b,t (x0 , Y ) , x , Y X Z 2 dY T X0 F x0 , Y T X0 pt (x0 , x0 ) exp − Y T X0 dx0 m/2 . (12.8.45) → π X By (11.5.10) and by Theorem 12.7.4, for 0 < b ≤ 1, given x, Y T X , p 0 T X0 as functions of x0 , Y T X0 , the functions sX are unib,t (x0 , Y ) , x , Y formly bounded on compact subsets of X as well as their derivatives of arbitrary order in these variables. Using (12.8.45), as b → 0, we get the pointwise convergence 2 sX x0 , Y T X , x0 , Y T X0 → π −m/2 pt (x0 , x0 ) exp − Y T X0 . b,t (12.8.46) By (11.5.10) and (12.8.46), we get (12.8.1). The proof of our theorem is completed. Remark 12.8.2. The proof of Theorem 12.8.1 shows that given x, Y T X ∈ X , the convergence in (12.8.1) is uniform when x0 , Y T X0 varies in compact subsets of X , and also that the derivatives of arbitrary order in x0 , Y T X0 also converge uniformly on the compact subsets. The proof gives a probabilistic counterpart to arguments of Bismut-Lebeau [BL08, chapter 17]. Establishing the narrow convergence of Qb,T to Q0,T is difficult because, contrary to equations (12.2.1) and (12.2.8), which can be solved pointwise for any w· ∈ C (R+ , p), equations (12.1.3) and (12.1.4) are genuinely stochastic differential equations. Still by proceeding as in [B81b, Théorème 1.2], we can replace the narrow convergence of probability measures by uniform convergence in probability over the probability space (C (R+ , p) , F∞ , P ). This way, one can avoid the apparent duplication of arguments on the martingales Mf,· and Mu,· . The method used in the proof of Theorem 12.8.1 also shows that as b → 0, the probability law of g· on C ([0, T ] , G) converges narrowly to the probability law of the solution of equation (12.1.4). The proof uses the bound in (12.8.42), and the same martingale argument as in the proof of Theorem 12.8.1. Let P(gt ,Y T X ) be the probability law of gt , YtT X , let Pg0t be the t

probability law of gt in (12.1.4). For t > 0, the same argument as in the proof

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of Theorem 12.8.1 shows that as b → 0, we have the narrow convergence of probability measures on G × p, 2 dY T X . (12.8.47) P(gt ,Y T X ) → Pg0t exp − Y T X t π m/2 12.9 The rough estimates as b → +∞ For a > 0, set Ka s x, Y T X = s x, aY T X .

(12.9.1)

X −1 AX b = Kb Ab Kb .

(12.9.2)

Put By (11.1.1), we get AX b

1 V T X 2 m − 2 − ∇Y T X . − 4∆ + Y (12.9.3) b b x, Y T X , x0 , Y T X0 be the smooth kernel associated Then X x0 , Y T X0 = bm rb,t x, bY T X , x0 , bY T X0 . (12.9.4)

1 = 2

For t > 0, let rX b,t with exp −tAX b . rX x, Y T X , b,t

Take Y p ∈ p. Instead of (12.2.1), we consider the stochastic differential equation, w˙ x˙ = Y T X , Y˙ T X = 2 , (12.9.5) b x0 = p1, Y0T X = Y p . The first line of (12.9.5) can be written in the form w˙ x ¨ = 2, Y T X = x. ˙ (12.9.6) b By (12.9.5), we obtain 1 τ0t YtT X = Y p + 2 wt . (12.9.7) b By (12.9.7), the probability law of τ0t YtT X is a Gaussian centered at Y p , and with variance bt4 . Then we get an analogue of (10.8.10), i.e., for 1 0, t ≥ 0, √

TX

Yt ≤ |Y p | + Cp t . (12.9.8) p b2 Let F : X → R be a smooth function with compact support. By using Proposition 11.5.2 and the Itô calculus as in the proof of Theorem 12.2.1, and the Feynman-Kac formula, for t > 0, we get Z mt 1 t T X 2 X p TX − Y exp −tAb F (x0 , Y ) = E exp ds F xt , Yt . 2b2 2 0 (12.9.9)

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Observe that when proving (12.9.9), the considerations of integrability in the proof of Theorem 12.2.1 are no longer necessary. Let k be a multi-index, where the indices run in 1, . . . , 2m. Let ∇T(xX,k 0 ,Y T X0 ) be the corresponding covariant derivative of order k with respect to ∇T X in 0 T X0 the variables x , Y . Theorem 12.9.1. Given , M with 0 < ≤ M , there exists C,M > 0 such that for b ≥ 1, ≤ t ≤ M , x, Y T X , x0 , Y T X0 ≤ C,M b4m . (12.9.10) rX b,t For b ≥ 1, ≤ t ≤ M , for any multi-index k, 4m+2|k| T X,k TX 0 T X0 x, Y , x , Y /b ∇(x0 ,Y T X0 ) rX b,t

(12.9.11)

is uniformly rapidly decreasing on X × X . Proof. If h, B are taken as in section 12.1, instead of (12.4.3), we consider the system δw 2 TX TX TX ¨ b J +R J , x˙ x˙ = h + B + $, , (12.9.12) ds $ ˙ = J T X , x˙ , J0T X = 0, J˙0T X = 0, $0 = 0. By proceeding as in the proof of Theorem 12.4.1, instead of (12.4.4), we get D Z T X T X E 1 t T X 2 TX ˙ ds dF xt , Yt , Jt , Jt Y E exp − 2 0 " Z 1 t T X 2 = E exp − Y ds F xt , YtT X 2 0 Z t D # Z t E T X ˙T X Y ,J ds + hh, δwi . (12.9.13) 0

0

Consider the system b2 J¨T X + RT X J T X , x˙ x˙ = h, J0T X

=

0, J˙0T X

$ ˙ = J T X , x˙ ,

(12.9.14)

$0 = 0.

= 0,

Then (12.9.14) is a special case of (12.9.12), so that (12.9.13) still holds. As an aside, let us note that the integration by parts formulas in (12.4.4), (12.4.6) and in (12.9.13), (12.9.14) can be shown to be equivalent. Now we proceed as in [BL08, section 14.3]. Set ψ s = s2 (−1 + s) .

(12.9.15)

ψ0 = ψ00 = 0,

ψ1 = 1, ψ10 = 0,

(12.9.16)

0 ψ0

0 0, ψ 1

ψs = s2 (3 − 2s) , Then

ψ0 =

= 0,

ψ1 =

= 1.

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For e, f ∈ p, set JsT X = ψs/t e + tψ s/t f,

(12.9.17)

so that J0T X = 0, J˙0T X = 0,

JtT X = e, J˙tT X = f.

(12.9.18)

Incidentally, note that since JsT X is a polynomial of degree 3 in the variable s, J·T X is the solution of a variational problem associated with the functional 2 R t H∞,t (J) = 21 0 J¨T X ds. By (12.9.13), (12.9.14), and (12.9.18), we get Z

1 t T X 2 Y ds dF xt , YtT X , τt0 (e, f ) E exp − 2 0 " Z 1 t T X 2 = E exp − Y ds F xt , YtT X 2 0 # Z t D Z tD E E T X ˙T X T X T X T X 2 Y ,J J¨ + R J , x˙ x, ˙ δw . (12.9.19) ds + b 0

0

Formulas similar to (12.5.8) can also be obtained. In particular, inspection of (12.1.17), (12.9.5), and (12.9.8) shows that for 1 < p < +∞, b ≥ 1, ≤ TX t ≤ M , the Lp norm of the relevant term Mt to the right of F xt , Y in t 2

the right-hand side of (12.9.19) is dominated by C,M,p b2 1 + |Y p | . In the integration by parts formula where k derivatives of F appear in the 2k left-hand side, the corresponding bound is given by C,M,p,k b2k 1 + |Y p | . Now we use the same notation as in the proof of Theorem 12.7.4. In particular we fix x0 ∈ X, and we choose the same coordinate system on X . Also we define gα,β (y, Y p0 ) as in (12.7.8). Set Z p p0 p0 p0 Φb,t,Y p ,x0 (α, β) = rX b,t ((x0 , Y ) , (y, Y )) gα,β (y, Y ) δ (y) dydY . p×p

(12.9.20) By the above, we find that given k ∈ N, there exists C,M > 0 such that for b ≥ 1, ≤ t ≤ M , 4k k k 1 + |α| + |β| /b2k Φb,t,Y p ,x0 (α, β) ≤ C,M,k 1 + |Y p | . (12.9.21) Some care has to be given to the derivation of (12.9.21). Indeed, in the coordinates (y, Y p0 ), e is a combination with smooth bounded coefficients of ∂ V , where ∇V denotes differentiation in the variable Y p0 , ∂yi − ∇ T X ∂ p0 Γy

∂y i

Y

∂ and ΓT X denotes the connection form for ∇T X . Equivalently, ∂y i is a smooth linear combination of (e, f ), the coefficients growing at most linearly in Y p0 . This explains the power 4k in the right-hand side of (12.9.21) instead of the more natural 2k. This issue did not appear explicitly in (12.7.10), where polynomials in Y p were absorbed by a Gaussian factor.

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Chapter 12

The critical fact is that the constant C,M,k > 0 in the right-hand side of (12.7.10) does not depend on Y p , x0 . By taking (12.7.10) with k = 2m + 1 and using an inverse Fourier transform, from (12.7.10), we get p 4(2m+1) p 0 T X0 4m 1 + |Y | . (12.9.22) rX (x , Y ) , x , Y ≤ C b 0 ,M b,t Of course (12.9.22) remains valid when replacing (x0 , Y p ) by any x, Y T X ∈ X , so that T X 4(2m+1) TX 0 T X0 4m Y . (12.9.23) 1 + x, Y , x , Y ≤ C b rX ,M b,t 0 Using (12.9.8) and proceeding as before, in (12.9.23), for k ∈ N, we replace 2k0 X TX 0 T X0 X TX 0 T X0 rb,t x, Y 1 + Y T X0 , , x ,Y by rb,t x, Y , x ,Y and still obtain an estimate similar to (12.9.22), i.e., 2k0 rX x, Y T X , x0 , Y T X0 1 + Y T X0 b,t 4(2m+1)+2k0 ≤ C,M,k0 b4m 1 + Y T X . (12.9.24)

Clearly, rX b,t

x, Y

TX

0

, x ,Y

T X0

Z =

rX b,t/2

rX b,t/2

x, Y T X , z, Z T X X z, Z T X , x0 , Y T X0 dzdZ T X . (12.9.25)

By (12.9.22), (12.9.25), we get rX b,t

x, Y

TX

0

, x ,Y

T X0

0 4m

≤Cb

Z

rX b,t/2

x, Y T X , z, Z T X b X T X 2(2m+1) 1+ Z dzdZ T X . (12.9.26)

By (10.4.9), (10.4.11), and (12.9.3), we get the crucial identity, Z T X 2(2m+1) p TX Z (x , Y ) , z, Z 1 + dzdZ T X rX 0 b,t/2 X p 2(2m+1) ! Z Z p p p dZ p . (12.9.27) = ht/2b2 (bY , Z ) 1 + b p By (10.7.10), for u < b2 coth t/b2 , we get Z 2 hpt/b2 (bY p , Z p ) exp u |Z p | /2b2 dZ p p

=

!m/2 2 et/b cosh (t/b2 ) − u sinh (t/b2 ) /b2 ! 1 u coth t/b2 /b2 − 1 2 p 2 exp b |Y | . (12.9.28) 2 coth (t/b2 ) − u/b2

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In (12.9.28), we take u = 12 b2 tanh t/b2 , so that u coth t/b2 /b2 − 1 2 tanh t/b2 b2 b =− ≤ −b2 tanh t/b2 , 2 1 2 2 2 coth (t/b ) − u/b 2 1 − 2 tanh (t/b ) (12.9.29) e cosh t/b2 − u sinh t/b2 /b2 ≥

t/b2

4

.

With such a choice of u, we get Z 2 2 hpt/b2 (bY p , Z p ) exp u |Z p | /2b2 dZ p ≤ C exp −u |Y p | .

(12.9.30)

p

For b ≥ 1, ≤ t ≤ M , u has a positive lower bound. From (12.9.26)–(12.9.28), 0 00 and (12.9.30), we conclude that there exist C,M > 0, C,M > 0 such that in the above range of parameters, p 0 T X0 0 4m 00 p 2 (x , Y ) , x , Y ≤ C b exp −C |Y | . (12.9.31) rX 0 ,M ,M b,t In (12.9.31), we may replace (x0 , Y p ) by any x, Y T X ∈ X , and so we get (12.9.10). By (12.9.24), for any k 0 ∈ N, the above arguments can be applied to T X0 2k0 TX 0 T X0 Y rX . x, Y , x , Y 1 + b,t By the analogue of (12.9.31), rX x, Y T X , x0 , Y T X /b4m is uniformly b,t rapidly decreasing on X × X , i.e., we obtain the second part of our theorem for k = 0. Let k be a multi-index. By equation (12.9.21) with k replaced by 2m + 1 + |k|, for b ≥ 1, ≤ t ≤ M , we get T X,k x, Y T X , x0 , Y T X0 ∇(x0 ,Y T X0 ) rX b,t 4(2m+1+|k|) ≤ C,M,k b4m+2|k| 1 + Y T X . (12.9.32) By proceeding as in (12.9.25)–(12.9.31), and using (12.9.32), we get an T X,k X X analogue of (12.9.31), in which rb,t is replaced by ∇(x0 ,Y T X0 ) rb,t . As be T X0 2k0 X fore, in the above, rX . So we have b,t can be replaced by r b,t 1 + Y established the second part of our theorem, which completes the proof. Remark 12.9.2. Needless to say, joint derivatives in all variables can be introduced in (12.9.11), by using group invariance, or by exchanging the roles of x, Y T X and x0 , Y T X0 . b 12.10 The heat kernel rX b,t on X X b Recall that the scalar operators AX b , Bb over X were defined in (11.6.1)– X (11.6.3), and their heat kernels rX , s in Definition 11.6.1. b,t b,t

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242

Chapter 12

In the sequel, C (R+ , p) will be replaced by C (R+ , p ⊕ k). However, we will still use the same notation as before for this new space. For example w· = w·T X + w·N is the generic path, and P denotes the Brownian measure on (C (R+ , p ⊕ k) , F∞ ). Take Y ∈ g. Equation (12.2.1) should be replaced by Y TX , b x0 = p1,

x˙ =

w˙ Y Y˙ = − 2 + , b b Y0 = Y.

(12.10.1)

Let F : Xb → R be a smooth function with compact support. Using the Itô calculus, instead of (12.2.10) in Theorem 12.2.1, we have the identity exp −tBX (12.10.2) b F (x0 , Y ) = E [F (xt , Yt )] . The considerations of section 12.3 remain valid for the operator BX b . Let h· = hp + hk be a bounded process with values in g = p ⊕ k, which is predictable with respect to the filtration Ft |t≥0 . We still take B· as in sections 12.1 and 12.4. We define w·` as in (12.1.8). Of course w·` takes now its values in g. Let J· = J T X + J N be the solution of the differential equation, YN δw b2 J¨ + J˙ + b2 RT X⊕N J T X , x˙ x˙ + = h + B + $, , (12.10.3) b ds $ ˙ = J T X , x˙ , J0 = 0, J˙0 = 0, $0 = 0. Then the analogue of (12.4.1), (12.4.2) is just ∂ ` 1 DYs x , J˙s = . (12.10.4) ∂` s b D` Also J T X , $ still verify equation (12.4.3), with h replaced by hp , and w· replaced by w·p . Note that J N appears in (12.10.3) only through its first two differentials. If F : Xb → R is a smooth function with compact support, then the obvious analogue of (12.4.4) holds, i.e., Z t hD Ei TX ˙ E dF (xt , Yt ) , Jt , bJt = E F (xt , Yt ) hh, δwi . (12.10.5) JsT X =

0

Instead of (12.4.6), we consider now the differential equation, YN = h, $ ˙ = J T X , x˙ , (12.10.6) b2 J¨ + J˙ + b2 RT X⊕N J T X , x˙ x˙ + b ˙ J0 = 0, J0 = 0, $0 = 0. Then (12.10.6) is a special case of (12.10.3), so that (12.10.5) still holds. Now L2 denotes the Hilbert space of square integrable function defined on [0, t] with values in g. Instead of (12.5.2), for e ∈ p, f ∈ g, we consider the differential equation, b2 J¨ + J˙ = v, J0 = 0, J˙0 = 0, (12.10.7) JtT X = e,

J˙t = f /b.

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The control problem still consists in finding v with minimal norm |v|L2 . Note that in (12.10.7), JtN is allowed to vary freely. This variational problem can be decoupled into a problem on v p associated with (e, f p ), and a problem on v k associated with f k . These two problems were considered in section 10.3. By (10.3.11) and (10.3.38), we get 1 2 ∗ |v|L2 = Kb,t ((0, 0) , (e, f p /b)) + Kb,t 0, f k /b . (12.10.8) 2 Instead of (12.5.6), we get

E dF (xt , Yt ) , τt0 (e, f ) Z t YN v + b2 RT X⊕N J T X , x˙ x˙ + , δw . = E F xt , YtT X b 0 (12.10.9) We extend Definition 12.7.3 to functions f : Xb × Xb → R. Using the results of chapter 10, and in particular the remarks after (10.3.17) together with the above considerations, it is easy to proceed as in sections 12.5–12.7, and to obtain the following analogue of Theorem 12.7.4. 0 0 Theorem 12.10.1. For 0 < b ≤ M, ≤ t ≤ M , rX b,t ((x, Y ) , (x , Y )) and its 0 0 covariant derivatives of arbitrary order in (x , Y ) with respect to ∇T X⊕N are uniformly rapidly decreasing on Xb × Xb.

We now state the analogue of Theorem 12.8.1. Theorem 12.10.2. For t > 0, as b → 0, we have the pointwise convergence, 1 1 2 0 0 2 0 0 p (x, x ) exp − |Y | + |Y | . rX ((x, Y ) , (x , Y )) → t b,t 2 π (m+n)/2 (12.10.10) Proof. The proof follows the same lines as the proof of Theorem 12.8.1. Indeed the first part of the proof concerning the probability law of x· is identical. Then we handle the extra component Y · exactly as in that proof. For a > 0, recall that Ka was defined in (2.14.2). Put X −1 AX b = Kb Ab Kb .

By (11.6.1), (12.10.11), we get 1 1 T X⊕N m+n 2 X Ab = − 4∆ + |Y | − − ∇Y T X . 2 b b2

(12.10.11)

(12.10.12)

For t > 0, let rX ((x, Y ) , (x0 , Y 0 )) be the smooth kernel associated with the b,t operator exp −tAX b . Instead of (12.9.4), we get 0 0 m+n X rX rb,t ((x, bY ) , (x0 , bY 0 )) . b,t ((x, Y ) , (x , Y )) = b

(12.10.13)

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Chapter 12

Let w· be a Brownian motion valued in g = p ⊕ k. Instead of equation (12.9.5), we now consider the differential equation w˙ Y˙ = 2 , b Y0 = Y.

x˙ = Y T X , x0 = p1,

(12.10.14)

Instead of (12.9.9), if F : Xb → R is a smooth function with compact support, for t > 0, we get Z (m + n) t 1 t 2 X − |Y | ds F (xt , Yt ) . exp −tAb F (x0 , Y ) = E exp 2b2 2 0 (12.10.15) be the covariant derivaIf k is a multi-index in 1, . . . , 2m+n, let ∇T(xX⊕N,k 0 ,Y 0 ) T X⊕N tive of order k with respect to the connection ∇ . We have the following analogue of Theorem 12.9.1. Theorem 12.10.3. Given > 0, M > 0 with ≤ M , there exists C,M > 0 such that for b ≥ 1, ≤ t ≤ M , 0 0 4m+2n rX . b,t ((x, Y ) , (x , Y )) ≤ C,M b

For b ≥ 1, ≤ t ≤ M , for any multi-index k, T X⊕N,k X ∇(x0 ,Y 0 ) rb,t ((x, Y ) , (x0 , Y 0 )) /b4m+2n+2|k|

(12.10.16)

(12.10.17)

is uniformly rapidly decreasing on Xb × Xb. Proof. The proof proceeds exactly as the proof of Theorem 12.9.1. Remark 12.10.4. The considerations of Remark 12.9.2 also apply to the kernel rX b,t . X 12.11 The heat kernel rb,t on Xb X Recall that the scalar differential operators AX b , Bb were defined in (11.7.1), X (11.7.3), and (11.7.4), and their smooth heat kernels rb,t , sX b,t in Definition 11.7.2. Let (x· , Y· ) be as in (12.10.1). Let F : Xb → R be a smooth function with compact support. An application of the Itô calculus and of the Feynman-Kac formula similar to what we did in the proof of equation (12.2.10) in Theorem 12.2.1, in (12.9.9), and in (12.10.15) shows that for t > 0, Z 1 t N T X 2 exp −tBX F (x , Y ) = E exp − Y , Y ds F (x , Y ) . 0 t t b 2 0 (12.11.1) By comparing (12.10.2) and (12.11.1), we get X sX b,t ≤ sb,t .

(12.11.2)

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By (11.6.5), (11.7.29), equation (12.11.2) is equivalent to X rb,t ≤ rX b,t .

(12.11.3)

The analogue of Theorem 12.10.1 holds. Indeed, as we shall see in the proof of Theorem 12.11.2, the exponential term in (12.11.1) does not create any new difficulty. The analogue of equation (12.10.10) says that as b → 0, t k,p 1 X Tr C rb,t ((x, Y ) , (x0 , Y 0 )) → (m+n)/2 exp 8 π 1 2 2 pt (x, x0 ) exp − |Y | + |Y 0 | . (12.11.4) 2 For a hint on the proof of (12.11.4), we refer to equation (2.16.25), to Proposition 14.10.1, and also to Theorem 14.11.2, where a more difficult result is proved for the operator LX A,b . Definition 12.11.1. Set X −1 AX b = Kb Ab Kb .

(12.11.5)

By (11.7.2), (12.10.11), and (12.11.5), we get AX b =

b4 N T X 2 + AX Y ,Y b . 2

(12.11.6)

By (12.10.12), (12.11.6), we obtain m+n b4 N T X 2 1 1 T X⊕N 2 AX Y , Y + ∆ + |Y | − = − − ∇Y T X . b 2 2 b4 b2 (12.11.7) Let (x· , Y· ) be as in (12.10.14). If F : Xb → R is a smooth function with compact support, instead of (12.10.15), we get " Z (m + n) t b4 t N T X 2 X exp −tAb F (x0 , Y ) = E exp − Y ,Y ds 2b2 2 0 ! # Z 1 t 2 − |Y | ds F (xt , Yt ) . (12.11.8) 2 0 For t > 0, let rX ((x, Y ) , (x0 , Y 0 )) be the smooth kernel associated with the b,t operator exp −tAX b . By (12.10.15), (12.11.8), we get X rX b,t ≤ rb,t ,

(12.11.9)

which also follows from (12.11.3). By (12.11.9), we deduce that any upper X bound for rX b,t is a fortiori valid for rb,t . Now we establish an analogue of Theorem 12.10.3 for the kernel rX b,t .

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Chapter 12

Theorem 12.11.2. Given > 0, M > 0 with ≤ M , there exists C,M > 0 such that for b ≥ 1, ≤ t ≤ M , 0 0 4m+2n . rX b,t ((x, Y ) , (x , Y )) ≤ C,M b

(12.11.10)

For b ≥ 1, ≤ t ≤ M , for any multi-index k, T X⊕N,k X ∇(x0 ,Y 0 ) rb,t ((x, Y ) , (x0 , Y 0 )) /b4m+2n+2|k|

(12.11.11)

is uniformly rapidly decreasing on Xb × Xb. Proof. By equation (12.10.16) in Theorem 12.10.3 and by (12.11.9), we get (12.11.10). However, this argument cannot be used to get (12.11.11). To establish (12.11.11), we will use again the Malliavin calculus as in the proof of Theorems 12.9.1 and 12.10.3. We take (x· , Y· ) as in (12.10.14), and we use (12.11.8). We take ψs , ψ s as in (12.9.15). If e ∈ p, f ∈ g = p ⊕ k are unit vectors, as in (12.9.17), put Js = ψs/t e + tψ s/t f.

(12.11.12)

Now we apply to (12.11.8) the method used in the proof of Theorem 12.9.1. We get the following analogue of (12.9.19), " 4Z t Z N T X 2 b 1 t 2 E exp − Y ,Y ds − |Y | ds 2 0 2 0 #

0 dF (xt , Yt ) , τt (e, f ) "

b4 = E exp − 2

Z 0

t

N T X 2 Y ,Y ds − 1 2

Z

t

|Y | ds 2

0

Z tD Z tD E N T X h N T X i h N T X iE b4 Y ,Y , J˙ , Y + Y , J˙ ds + Y, J˙ ds 0

0

! # Z tD E + b2 J¨ + RT X⊕N J T X , x˙ x˙ + Y N , δw F (xt , Yt ) . (12.11.13) 0

In the right-hand side of (12.11.13), the main new difficulty with respect to the proof of Theorems 12.9.1 and 12.10.3 is the presence of the term starting with b4 in the right-hand side of (12.11.13). However, note that if Ks is a smooth process with values in Tx X, Z t Z Z t 2

N T X b4 t N T X 2 2 b ≤ Y ,Y ds + 1 Y , Y , K ds |K| ds. 2 0 2 0 0 (12.11.14) From (12.11.14), for b ≥ 1, we get 4Z t Z t N T X 2 4

N T X b exp − Y ,Y ds b Y ,Y , K ds 2 0 0 Z 1 t 2 ≤ b2 C + |K| ds . (12.11.15) 2 0

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Note that the fateful factor b2 appears in the right-hand side of (12.11.15). We will use (12.11.15) with i h i h (12.11.16) K = J˙N , Y T X + Y N , J˙T X . In this case, Z

t

2

Z

|K| ds ≤ C 0

t

2

|Y | ds.

(12.11.17)

0

Proceeding as in the proof of Theorem 12.9.1 while using (12.11.13)– (12.11.17), the right-hand side of (12.11.13) can be controlled in the same way as the right-hand side of (12.9.19). We claim that the above argument can be iterated. Let us just explain the case where instead a second order operator acts on F . The term with the factor b4 is no longer a problem because this factor is acceptable when |k| = 2. The only new difficulty comes from the products of terms like the one with b4 , which introduce a factor b8 . By squaring (12.11.14), we get Z t 2

N T X 2 b Y ,Y , K ds 0 Z t 2 Z t 2 ! N T X 2 2 4 ≤C b Y ,Y ds + . (12.11.18) |K| ds 0

0

From (12.11.18), for b ≥ 1, we obtain Z t 4Z t N T X 2 4

N T X 2 b Y ,Y , K ds Y ,Y ds b exp − 2 0 0 Z t 2 ! 2 4 ≤b C 1+ |K| ds . (12.11.19) 0 4

Again the right weight b appears in the right-hand side of (12.11.19). The above argument can be easily iterated, and leads to (12.11.11) for arbitrary k. The proof of our theorem is completed.

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Chapter Thirteen Refined estimates on the scalar heat kernel for bounded b In this chapter, for bounded b > 0, we obtain uniform bounds for the kernels X b rb,t , rX b,t , with the proper decay at infinity on X or X . In chapter 14, these X bounds will be used to obtain corresponding bounds for the kernel qb,t , in order to prove Theorem 4.5.2. The arguments developed in section 12.3, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates. This chapter is organized as follows. In section 13.1, we establish estimates on the Hessian of the distance function on X. X In section 13.2, we obtain the bounds on the heat kernel rb,t . In section 13.3, we establish the bounds on the heat kernel rX b,t . In the whole chapter, we will use the notation and the results of chapter 12.

13.1 The Hessian of the distance function Clearly, there is C > 0 such that if a ∈ g, |ad (a)| ≤ C |a| . 2

(13.1.1)

0

Recall that the function d (x, x ) is smooth on X × X. We fix x ∈ X. If x0 ∈ X, x0 6= x, let u ∈ Tx0 X be the unit vector that is the derivative in the time parameter of the geodesic connecting x to x0 , with speed 1. Then ∇d (x, ·) = u.

(13.1.2)

From (13.1.2), we get d2 (x, ·) = d (x, ·) u. (13.1.3) 2 Since u is of norm 1, and since u is parallel along geodesics centered at x, from (13.1.2), (13.1.3) we get ∇

X 2 2 ∇T∇d 2 (x,·)/2 ∇d (x, ·) /2 = ∇d (x, ·) /2.

(13.1.4)

Also since the function d is convex, ∇T X ∇d (x, ·) ≥ 0 on X \ {x}.

(13.1.5)

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Refined estimates on the scalar heat kernel for bounded b

Taking the trace in (13.1.5), we get ∆X d (x, ·) ≥ 0 on X \ {x}.

(13.1.6)

We may and we will assume that x = p1. We use the geodesic coordinate system on X centered at x, which is given by a ∈ p → x0 = ea x ∈ X. We identify T X with p by parallel transport along the geodesics s ∈ R → esa x ∈ X with respect to the Levi-Civita connection. Note that d (x, x0 ) = |a| .

(13.1.7)

Proposition 13.1.1. The following identity holds: d2 (x, ·) = ad (a) coth (ad (a)) , (13.1.8) ∇T X ∇ 2 so that d2 (x, ·) ≤ 1 + |ad (a)| . (13.1.9) 1 ≤ ∇T X ∇ 2 Moreover, d2 (x, ·) 1 ≤ ∇T X ∇ ≤ 1 + Cd (x, ·) . (13.1.10) 2 In particular, d2 (x, ·) m ≤ ∆X ≤ m (1 + Cd (x, ·)) . (13.1.11) 2 Proof. Let Js be a Jacobi field along the geodesic s ∈ [0, 1] → esa x ∈ X that vanishes at s = 0. By proceeding as in (3.4.19), (3.4.30), we get sinh (sad (a)) J1 , (13.1.12) Js = sinh (ad (a)) so that J˙1 = ad (a) coth (ad (a)) J1 . (13.1.13) By (13.1.13), we get (13.1.8). Moreover, for x ∈ R, 1 ≤ x coth (x) ≤ 1 + |x| .

(13.1.14)

By (13.1.8), (13.1.14), we get (13.1.9). Also equation (13.1.10) follows from (13.1.1), (13.1.7), and (13.1.9). By taking the trace in (13.1.10), we get (13.1.11). The proof of our proposition is completed. Now we will establish corresponding results for the function d (x, ·). Proposition 13.1.2. The following identity holds on X \ {x}, ∇T X ∇d (x, ·) = −

∇· d (x, ·) ∇· d (x, ·) ∇T X ∇d2 (x, ·) /2 + . d (x, ·) d (x, ·)

(13.1.15)

Moreover, 1 − ∇· d (x, ·) ∇· d (x, ·) ≤ ∇T X ∇d (x, ·) d (x, ·) 1 ≤ (1 − ∇· d (x, ·) ∇· d (x, ·) + |ad (a)|) . (13.1.16) d (x, ·)

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There is C > 0 such that on X \ {x}, 1 − ∇· d (x, ·) ∇· d (x, ·) ≤ ∇T X ∇d (x, ·) d (x, ·) 1 (1 − ∇· d (x, ·) ∇· d (x, ·) + Cd (x, ·)) . (13.1.17) ≤ d (x, ·) In particular, on X \ {x}, m−1 m−1 ≤ ∆X d (x, ·) ≤ + Cm. d (x, ·) d (x, ·)

(13.1.18)

Finally, on X \ B (x, 1), the covariant derivatives of order ≥ 1 of d (x, ·) are uniformly bounded. Proof. Equation (13.1.15) is trivial. Equations (13.1.16)–(13.1.18) can be derived from (13.1.1), (13.1.2), from Proposition 13.1.1 and from (13.1.15). By (13.1.2) and (13.1.17), the covariant derivatives of order 1 and 2 of d (x, ·) are uniformly bounded on X \ B (x, 1). Let us explain how to handle the higher covariant derivatives. We will first rederive the bound on the second derivatives. Recall that x = p1. In the sequel, the scalar product is taken in Tx X. Also we still use the coordinate system a ∈ p → ea x ∈ X. By (13.1.7), we get a . (13.1.19) ∇d (x, ·) = |a| We trivialize T X by parallel transport with respect to the connection ∇T X along geodesics centered at x by parallel transport with respect to the connection ∇T X . Let ΓT X denote the associated connection form. By (13.1.19), we obtain ! 1 a a ∇TU X ∇d (x, ·) = U − 2 ha, U i + ΓTa X (U ) . (13.1.20) |a| |a| |a| By (3.7.5), (13.1.20), we obtain ∇TU X ∇d (x, ·) = − ha, U i

a 3

|a|

+

1 cosh (ad (a)) U. |a|

(13.1.21)

Using (3.7.7), we can rewrite (13.1.21) in the form ∇TU X ∇d (x, ·) = − ha, V i

a 3

|a|

+

ad (a) coth (ad (a)) V. |a|

(13.1.22)

In view of (13.1.8) and (13.1.19), (13.1.22) is just a form of (13.1.15). By (13.1.1), (13.1.7), (13.1.14), and (13.1.15) or (13.1.22), we recover the fact that the tensor ∇T X ∇d (x, ·) is uniformly bounded outside of B (x, 1). Using (3.7.7), (3.7.8), and (13.1.22), the same is true for the higher order covariant derivatives of d (x, ·). This completes the proof of our proposition.

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13.2 Bounds on the scalar heat kernel on X for bounded b X Recall that the kernel rb,t nition 11.5.3.

x, Y T X , x0 , Y T X0 on X was defined in Defi-

Theorem 13.2.1. Given > 0, M > 0, ≤ M , there exist C > 0, C 0 > 0, C 00 > 0 such that for 0 0, M > 0, ≤ M , there exist C > 0, C 0 > 0 such that for 0 < b ≤ M, ≤ t ≤ M , x, Y T X , x0 , Y T X0 ∈ X , X rb,t x, Y T X , x0 , Y T X0 2 2 ≤ C exp −C 0 b2 d2 (x, x0 ) + Y T X + Y T X0 . (13.2.2) Proof. Clearly, X rb,t

x, Y

TX

0

, x ,Y

T X0

Z

X rb,t/2

x, Y T X , z, Z T X X X rb,t/2 z, Z T X , x0 , Y T X0 dzdZ T X . (13.2.3)

=

X By Theorem 12.7.4, under the given conditions on b, t, the kernel rb,t/2 is 0 uniformly bounded by a constant C > 0. By (13.2.3), we get Z X TX 0 T X0 0 X rb,t x, Y , x ,Y ≤C rb,t/2 x, Y T X , z, Z T X dzdZ T X . X

(13.2.4) Recall that x0 = p1. We may and we will assume that x, Y T X = (x0 , Y p ), with Y p ∈ p. Now we proceed as in (12.9.27). By comparing (10.4.1) and (11.1.1), we get Z Z X p TX TX rb,t/2 (x0 , Y ) , z, Z dzdZ = hpt/2b2 (Y p , Z p ) dZ p . (13.2.5) X

p

By (10.7.12), (13.2.4), and (13.2.5), we obtain X rb,t (x0 , Y p ) , x0 , Y T X0 !m/2 t/2b2 p2 1 e 2 0 exp − tanh t/2b |Y | . (13.2.6) ≤C cosh (t/2b2 ) 2 By (13.2.6), there exist C 0 > 0, c0 > 0 such that for 0 < b ≤ M, ≤ t ≤ M , 2 X rb,t (x0 , Y p ) , x0 , Y T X0 ≤ C 0 exp −c0 |Y p | . (13.2.7) More generally, by (13.2.7), we get 2 X rb,t x, Y T X , x0 , Y T X0 ≤ C 0 exp −c0 Y T X .

(13.2.8)

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of AX By (11.1.1), the formal adjoint AX∗ b b with respect to the standard L2 scalar product on X is obtained from AX by replacing − 1b ∇Y T X by 1b ∇Y T X . b X∗ X The operator Ab is of the same type as Ab . By proceeding as in the proof of (13.2.8), by exchanging the roles of Y T X and Y T X0 , we also get 2 X (13.2.9) rb,t x, Y T X , x0 , Y T X0 ≤ C 0 exp −c0 Y T X0 . In the integral in (13.2.3), either d (x, z) ≥ d (x, x0 ) /2 or d (x0 , z) ≥ d (x, x0 ) /2. Therefore we can refine (13.2.4) into x, Y T X , x0 , Y T X0 Z X TX , z, Z T X dzdZ T X ≤ C0 (z,Z T X )∈X rb,t/2 x, Y d(x,z)≥d(x,x0 )/2 Z X TX , x0 , Y T X0 dzdZ T X . (13.2.10) + C0 (z,Z T X )∈X rb,t/2 z, Z

X rb,t

d(x0 ,z )≥d(x,x0 )/2

By symmetry, it is enough to estimate the first integral in the right-hand side of (13.2.10). In estimating this integral, we may as well assume that x, Y T X = (x0 , Y p ). Instead of (12.9.5), we consider the stochastic differential equation on X , Y TX , b x0 = p1,

w˙ Y˙ T X = , b Y0T X = Y p ,

x˙ =

(13.2.11)

so that the first equation in (12.9.6) still holds. Let F : X → R be a smooth function with compact support. By proceeding as in (12.9.9), we get Z t T X 2 mt 1 TX p ds F x , Y exp −tAX F (x , Y ) = E exp Y . − t 0 t b 2b2 2b2 0 (13.2.12) By (13.2.12), we obtain Z (z,Z T X )∈X

X rb,t/2 (x0 , Y p ) , z, Z T X

dzdZ T X

d(x0 ,z)≥d(x0 ,x0 )/2

" = E exp

mt 1 − 2 4b2 2b

t/2

Z

! T X 2 Y ds 1

d(x0 ,xt/2 )≥d(x0 ,x0 )/2

0

# . (13.2.13)

Let x· be a path as in (13.2.11). Clearly 1 2b2

Z 0

t/2

2 T X 2 Y ds ≥ d x0 , xt/2 . t

(13.2.14)

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Take α ∈]0, 1[. By (13.2.13), (13.2.14), we get Z X p TX r (x , Y ) , z, Z dzdZ T X TX 0 b,t/2 z,Z ∈X ( ) d(x0 ,z)≥d(x0 ,x0 )/2

≤ exp − 1 − α2 d2 (x0 , x0 ) /4t !# " Z α2 t/2 T X 2 mt Y ds . (13.2.15) − 2 E exp 4b2 2b 0 Let Oαp be the operator Oαp =

1 2 −∆p + α2 |Y p | − m , 2

(13.2.16)

and for t > 0, let hpα,t (Y p , Z p ) be the associated heat kernel. Set 1 Ytp = Y p + wt . (13.2.17) b Using again the Feynman-Kac formula, we get !# " Z Z α2 t/2 p 2 mt p p p p − 2 |Y | ds . (13.2.18) hα,t/2b2 (Y , Z ) dZ = E exp 4b2 2b 0 p Recall that the harmonic oscillator Op was defined in (10.4.1). For a > 0, we define Ka as in (10.4.7). Clearly, −1 p √ O K α = αOp − K√ α α

m (1 − α) . 2

(13.2.19)

By (13.2.19), we get hpα,t (Y p , Z p ) = exp (mt (1 − α) /2) αm/2 hpαt

√ p √ p αY , αZ .

(13.2.20)

By (10.7.12) and (13.2.20), we obtain !m/2 t/2b2 e hpα,t/2b2 (Y p , Z p ) dZ p = cosh (αt/2b2 ) p 1 2 exp − α tanh αt/2b2 |Y p | /2 . (13.2.21) 2

Z

By (13.2.15), (13.2.18), and (13.2.21), we get Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2

2

≤ exp − 1 − α

2

2

0

d (x0 , x ) /4t

et/2b cosh (αt/2b2 )

!m/2

1 2 exp − α tanh αt/2b2 |Y p | /2 . (13.2.22) 2

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By taking α = 1/2, we deduce from (13.2.22) that there exist C 0 > 0, c0 > 0 such that under the given conditions on the parameters b, t, Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2

≤ C 0 exp −c0 d2 (x0 , x0 ) + C 00 /b2 . (13.2.23) 1/2 For 0 0, c0 > 0 such that Z X (x0 , Y p ) , z, Z T X dzdZ T X rb,t/2 (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2

≤ C 0 exp −c0 b2 d2 (x0 , x0 ) . (13.2.24) As explained after equation (13.2.10), the bounds in (13.2.23) and (13.2.24) X give corresponding bounds for rb,t x, Y T X , x0 , Y T X0 . By (13.2.8), (13.2.9), (13.2.23) and (13.2.24), we get (13.2.1) and (13.2.2). The proof of our theorem is completed. Now we establish a second critical estimate. 0 Theorem 13.2.2. Given > 0, M > 0, ≤ M , there exist C > 0, C > 0 TX 0 T X0 such that for 0 < b ≤ M, ≤ t ≤ M , x, Y , x ,Y ∈ X, X rb,t

x, Y T X , x0 , Y T X0 ≤ C exp −C 0 d2 (x, x0 ) + exp −C 00 d (x, x0 ) /b2 . (13.2.25)

Proof. We will still use arguments taken from the proof of Theorem 13.2.1, but we will complement them by a new crucial step. By (13.2.2), we may and we will assume that d (x, x0 ) ≥ 4, and also that b > 0 is small. We still start from equation (13.2.10). As before, we will only estimate the first term in the right-hand side of this equation. Recall that sX b,t is the smooth kernel associated with exp −tBbX . By (11.5.10), we get Z

X TX , z, Z T X dzdZ T X (z,Z T X )∈X rb,t/2 x, Y d(x,z)≥d(x,x0 )/2 Z 2 X TX = exp − Y T X /2 , z, Z T X (z,Z T X )∈X sb,t/2 x, Y d(x,z)≥d(x,x0 )/2 2 exp Z T X /2 dzdZ T X . (13.2.26)

Again, we may and we will assume that x, Y T X = (x0 , Y p ). Instead of (13.2.11), we consider the stochastic differential equation in (12.2.1). By

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(12.2.10) and (13.2.26), we get Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2

T X 2 2 = exp − |Y p | /2 E 1d(x0 ,xt/2 )≥d(x0 ,x0 )/2 exp Yt/2 /2 . (13.2.27) Let k : R+ → R+ be a smooth increasing function such that k (u) = u2 for u ≤ 1/2, = u for u ≥ 1.

(13.2.28)

For z ∈ X, set f (z) = k (d (x0 , z)) .

(13.2.29)

0

0

Then f is smooth. Since d (x0 , x ) ≥ 4, if d (x0 , z) ≥ d (x0 , x ) /2, then f (z) = d (x0 , z). Since f and its first derivative vanish at x0 , if d x0 , xt/2 ≥ d (x0 , x0 ) /2, by (12.3.19), we get Z t/2 d x0 , xt/2 = −b∇Y T X f xt/2 + ∇δws f (xs ) t/2

0

Z

t/2

∇TYsX (13.2.30) T X ∇Y T X f (xs ) ds. s

+ 0

By (13.2.30), if d x0 , xt/2 ≥ d (x0 , x0 ) /2, at least one of the terms in the right-hand side of (13.2.30) is larger than d (x0 , x0 ) /6. We will consider the contribution of each term to the expectation in the right-hand side of (13.2.27). This will be in increasing order of difficulty. By (13.1.2), ∇· f is uniformly bounded, so that (13.2.31) |∇Z T X f (z)| ≤ c Z T X . Therefore if −b∇Z T X f (z) ≥ d (x0 , x0 ) /6, then T X c0 Z ≥ d (x0 , x0 ) . b By an obvious analogue of (13.2.26), (13.2.27), we get Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X (z,Z T X )∈X

(13.2.32)

0

|Z T X |≥ cb d(x0 ,x0 ) 2 = exp − |Y p | /2 E 1 Y T X ≥ c0 d(x t/2

b

0

T X 2 exp /2 . Yt/2 ,x0 )

(13.2.33)

By proceeding as in (12.9.27) and in (13.2.5), we obtain Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X (z,Z T X )∈X 0

|Z T X |≥ cb d(x0 ,x0 )

Z = p

hpt/2b2 (Y p , Z p ) 1|Z p |≥ c0 d(x0 ,x0 ) dZ p . (13.2.34) b

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By Chebyshev’s inequality, for u ≥ 0, Z u hpt/2b2 (Y p , Z p ) 1|Z p |≥ c0 d(x0 ,x0 ) dZ p ≤ exp − c02 d2 (x0 , x0 ) /b2 b 2 p Z u 2 |Z p | dZ p . (13.2.35) hpt/2b2 (Y p , Z p ) exp 2 p In (13.2.35), we take 1 tanh /2M 2 . (13.2.36) 2 Note that u< 1 < coth t/2b2 . Also for t ≥ , b ≤ M , coth t/2b2 ≤ coth /2M 2 . Since 0 < u < 1/2, we get u coth /2M 2 − 1 u coth t/2b2 − 1 −u ≤ = ≤ −u. (13.2.37) coth (t/2b2 ) − u coth (/2M 2 ) − u 1 − 2u2 u=

By (10.7.11), (13.2.35)–(13.2.37), we obtain Z p

hpt/2b2

m/2 2 (Y , Z ) 1|Z p |≥ c0 d(x0 ,x0 ) dZ ≤ b 1 − tanh (/2M 2 ) /2 1 2 p 2 02 2 0 2 exp − tanh /2M |Y | + c d (x0 , x ) /b . (13.2.38) 4 p

p

p

By (13.2.31)–(13.2.38), the contribution of the first term in the right-hand side of (13.2.30) to estimating the right-hand side of (13.2.27) is compatible with (13.2.25). By H¨ older’s inequality, for θ ∈]1, +∞[, T X 2 R E 1 t/2 ∇ f (xs )≥d(x0 ,x0 )/6 exp Yt/2 /2 0

δws

≤

1/θ T X 2 E exp θ Yt/2 /2

"Z P

t/2

#(θ−1)/θ ∇δws f (xs ) ≥ d (x0 , x0 ) /6

. (13.2.39)

0

Since ∇· f is uniformly bounded, by a well-known inequality on Itô integrals [StV79, eq. (2.1) in Theorem 2.4.1], there exist C > 0, C 0 > 0 such that for ≤ t ≤ M, "Z # t/2 0 P ∇δws f (xs ) ≥ d (x0 , x ) /6 ≤ C exp −C 0 d2 (x0 , x0 ) . (13.2.40) 0

In the sequel, we take θ = 3/2. By (10.8.12), given > 0, for b > 0 small enough and t ≥ , we get T X 2 2 E exp θ Yt/2 ≤ C exp |Y p | /2 . (13.2.41) /2

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By (13.2.39)–(13.2.41), we get 2 exp − |Y p | /2 E 1R t/2 ∇ 0

T X 2 /2 exp Y 0 t/2 δws f (xs )≥d(x0 ,x )/6 2 ≤ C exp −C 0 d2 (x0 , x0 ) − C 00 |Y p | . (13.2.42)

By (13.2.42), the contribution of the second term in the right-hand side of (13.2.30) to the right-hand side of (13.2.27) is also compatible with (13.2.25). We now come to the most difficult step in the proof, i.e., to the control of the contribution of the third term in the right-hand side of (13.2.30). We still take θ = 3/2. As in (13.2.39), we get T X 2 R E 1 t/2 ∇T X ∇ T X f (xs )ds≥d(x0 ,x0 )/6 exp Yt/2 /2 0

YsT X

Ys

≤ "Z P 0

t/2

1/θ T X 2 E exp θ Yt/2 /2 #(θ−1)/θ

0 ∇TYsX T X ∇Y T X f (xs ) ds ≥ d (x0 , x ) /6 s

. (13.2.43)

By (13.1.17), there is C > 0 such that ∇T· X ∇· f ≤

C . 2

(13.2.44)

From (13.2.44), we get Z t/2 Z C t/2 T X 2 Ys ds. (13.2.45) ∇TYsX T X ∇Y T X f (xs ) ds ≤ s 2 0 0 By (13.2.45), using Chebyshev’s inequality, we find that for any α > 0, # "Z t/2 0 ≤ exp −αd (x0 , x0 ) /6b2 P ∇TYsX T X ∇Y T X f (xs ) ds ≥ d (x0 , x ) /6 s 0

" E exp

αC 2b2

Z 0

t/2

!# T X 2 Ys ds . (13.2.46)

In the sequel, we take β, 0 < β < 1 and we choose α given by α=

β2 . C

(13.2.47)

Set Pβp =

1 β2 p 2 (−∆p + 2∇Y p ) − |Y | . 2 2

(13.2.48) p For t > 0, let kβ,t (Y p , Z p ) be the smooth kernel associated with exp −tPβp . Let Y·p be taken as in (12.2.8). By the Feynman-Kac formula, we get " !# Z Z β 2 t/2 p 2 p p p p E exp |Y | ds = kβ,t/2b (13.2.49) 2 (Y , Z ) dZ . 2b2 0 p

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Now we use the notation in the proof of Theorem 13.2.1. Let ρ ∈]0, 1[ be given by ρ2 = 1 − β 2 . (13.2.50) By (10.4.3), (10.4.4), (13.2.16), and (13.2.48), we get 2 2 exp − |Y p | /2 Pβp exp |Y p | /2 = Oρp . (13.2.51) By (13.2.49) and (13.2.51), we get " !# Z β 2 t/2 p 2 p 2 |Y E exp | ds = exp |Y | /2 s 2b2 0 Z 2 hpρ,t/2b2 (Y p , Z p ) exp − |Z p | /2 dZ p . (13.2.52) p

By (13.2.20) and (13.2.52), we obtain !# " Z β 2 t/2 p 2 2 p 2 |Y | ds = exp mt (1 − ρ) /4b exp |Y | /2 E exp s 2b2 0 Z √ 2 hpρt/2b2 ( ρY p , Z p ) exp − |Z p | /2ρ dZ p . (13.2.53) p

Now we use equation (10.7.10), with t replaced by ρt/2b2 , and u = −1/ρ, together with (13.2.53), and we get !# " Z β 2 t/2 p 2 |Ys | ds = exp mt (1 − ρ) /4b2 E exp 2 2b 0 " ! #m/2 2 eρt/2b 1 β 2 tanh ρt/2b2 p 2 |Y | . exp 2 2 ρ + tanh (ρt/2b2 ) cosh (ρt/2b2 ) + sinh(ρt/2b ) ρ

(13.2.54) Incidentally, observe that if we replace β by γb with γ ≥ 0, and if we make b → 0 in (13.2.54), the limit is exp mtγ 2 /8 , which indicates that as b → 0, 2 R t/2 the probability law of 0 YsT X ds converges to the Dirac mass at mt/4. This fact is very important from a probabilistic point of view, and plays a key role in the proof of our main result. We refer to Proposition 14.10.1 for more details on this point. We now go back to the case where β is fixed with 0 < β < 1. By (13.2.41), (13.2.43), (13.2.46), (13.2.47), (13.2.54), for b > 0 small enough and ≤ t ≤ M , we obtain T X 2 2 exp − |Y p | /2 E 1R t/2 ∇T X ∇ T X f (xs )ds≥d(x0 ,x0 )/6 exp Yt/2 /2 0 Ys YsT X " β2 2 ≤ C 0 exp − |Y p | /2 exp − d (x0 , x0 ) /6b2 C !#(θ−1)/θ 1 β 2 tanh ρt/2b2 2 p 2 exp mt (1 − ρ) /4b exp |Y | . (13.2.55) 2 ρ + tanh (ρt/2b2 )

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Equivalently, 2 T X exp − |Y | /2 E 1R t/2 ∇T X ∇ T X f (xs )ds≥d(x0 ,x0 )/6 exp Yt/2 /2 0 Y Y TX " β2 0 0 2 ≤ C exp − d (x0 , x ) /6b C !#(θ−1)/θ 1 ρ 1 + ρ tanh ρt/2b2 2 p 2 exp mt (1 − ρ) /4b exp − |Y | . 2 ρ + tanh (ρt/2b2 )

p 2

(13.2.56) Now observe that since ρ ≤ 1, 1 + ρ tanh ρt/2b2 ≥ 1. ρ + tanh (ρt/2b2 ) By (13.2.56), (13.2.57), we get 2 exp − |Y p | /2 E 1R t/2 ∇T X 0

Y TX

∇Y

(13.2.57)

T X 2 /2 exp Y 0 t/2 T X f (xs )ds≥d(x0 ,x )/6

2 #(θ−1)/θ β ρ p2 0 2 2 ≤ C exp − d (x0 , x ) /6b − mt (1 − ρ) /4b − |Y | . C 2 "

0

(13.2.58) For d (x0 , x0 ) large enough, β2 β2 d (x0 , x0 ) /6 − mt (1 − ρ) /4 ≥ d (x0 , x0 ) /12. C C

(13.2.59)

By (13.2.2), (13.2.58), and (13.2.59), the contribution of the third term in (13.2.30) to the right-hand side of (13.2.27) is also compatible with the estimate (13.2.25). The proof of our theorem is completed. Remark 13.2.3. A complete explanation for the remark that follows equation (13.2.54) will be given in Proposition 14.10.1. Also note that if we only assume 0 < t ≤ M, 0 < b ≤ M , using (10.7.11), the bound in (13.2.38) can be replaced by the fact that given 0 < u < 1, Z hpt/2b2 (Y p , Z p ) 1|Z p |≥ c0 d(x0 ,x0 ) dZ p b

p

≤

2 1−u

m/2

exp

1 c02 2 p 2 0 u |Y | − 2 d (x0 , x ) . (13.2.60) 2 b

Similarly, when 0 < t ≤ M , there is still a bound like (13.2.42), with a negative constant C 00 in the right-hand side. We will now obtain the estimate we were looking for.

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Chapter 13

0 Theorem 13.2.4. Given > 0, M > 0, ≤ M , there exist C > 0, C > 0 TX 0 T X0 such that for 0 < b ≤ M, ≤ t ≤ M , x, Y , x ,Y ∈ X, X rb,t x, Y T X , x0 , Y T X0 2 2 . (13.2.61) ≤ C exp −C 0 d2 (x, x0 ) + Y T X + Y T X0

Proof. When d (x, x0 ) remains uniformly bounded, our estimate follows from equation (13.2.2) in Theorem 13.2.1. Also by taking d (x, x0 ) large enough, equation (13.2.61) follows from (13.2.1), (13.2.2), and (13.2.25).

13.3 Bounds on the scalar heat kernel on Xb for bounded b 0 0 b Recall that the kernel rX b,t ((x, Y ) , (x , Y )) on X was defined in Definition 11.6.1. Now we state the obvious extension of Theorem 13.2.4 to the kernel rX b,t .

Theorem 13.3.1. Given > 0, M > 0, ≤ M , there exist C > 0, C 0 > 0 such that for 0 < b ≤ M, ≤ t ≤ M , (x, Y ) , (x0 , Y 0 ) ∈ X , 2 2 0 0 0 rX d2 (x, x0 ) + |Y | + |Y 0 | . (13.3.1) b,t ((x, Y ) , (x , Y )) ≤ C exp −C Proof. The proof of our theorem will be obtained by following the same steps that led to the proof of Theorem 13.2.4, i.e., by establishing the obvious extensions of Theorems 13.2.1 and 13.2.2. Of course, we will also use the rough estimates for the kernel rX b,t that were established in Theorem 12.10.1. First, we review the proof of Theorem 13.2.1. The obvious analogue of equation (13.2.5) holds, i.e., if Y ∈ g, Z Z rX ((x , Y ) , (z, Z)) dzdZ = hp⊕k (13.3.2) 0 b,t/2 t/2b2 (Y, Z) dZ. b X

g

By proceeding as in (13.2.6), we get the obvious analogue of (13.2.8), 2 0 0 0 0 rX ((x , Y ) , (x , Y )) ≤ C exp −c |Y | . (13.3.3) 0 b,t p k , The obvious analogue of (13.2.10) holds for rX b,t . Moreover, if Y = Y , Y Z rX b b,t/2 ((x0 , Y ) , (z, Z)) dzdZ (z,Z)∈X d(x0 ,z)≥d(x0 ,x0 )/2 Z X rb,t/2 (x0 , Y p ) , z, Z T X dzdZ T X = (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2 Z hkt/2b2 Y k , Z k dZ k . (13.3.4) k

Using (10.7.12) , (13.2.23), and (13.2.24), we get the analogue of these bounds for the left-hand side of (13.3.4). This completes the proof of the analogue of Theorem 13.2.1.

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Now we review the proof of the analogue of Theorem 13.2.2. We combine (10.7.12), (13.2.26) and the subsequent estimates in the proof of Theorem 13.2.2 together with (13.3.4), so that the analogue of Theorem 13.2.2 also holds for rX b,t . Then we can use the arguments of the proof of Theorem 13.2.4 and obtain (13.3.1). The proof of our theorem is completed.

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Chapter Fourteen X The heat kernel qb,t for bounded b

The purpose of this chapter is to establish the estimates of Theorem 4.5.2 X for the hypoelliptic heat kernel qb,t ((x, Y ) , (x0 , Y 0 )) on Xb. More precisely, we show that for bounded b > 0, the heat kernel verifies uniform Gaussian type estimates. Also we study the limit as b → 0 of this heat kernel. The method consists in using the techniques of chapters 12 and 13 for the X scalar heat kernels over X and Xb, and to control the kernel qb,t by using X a Feynman-Kac formula. Still, because the operator LA,b contains matrix terms that themselves diverge as b → 0, we have to be careful. In particular, the contribution of the representation ρE will be obtained by using results of chapter 12 applied to the symmetric space attached to the complexification KC of K. This chapter is organized as follows. In section 14.1, we give a probabilistic construction of the elliptic heat operators exp −tLX A on X. In section 14.2, we extend the wave equation considerations of section 12.3 to the operator LX b . In section 14.3, by a trivial change of variables on Xb, we obtain a new operator LX0 A,b . Insection 14.4, we give a probabilistic construction of the heat operators · ∗ ∗ exp −tLX0 A,b . A matrix valued process U· acting on Λ (T X ⊕ N ) ⊗ F appears, which can be naturally expressed in the form U· = V· ⊗ W· . The X estimation of the heat kernel qb,t will be obtained by estimating |Vt | and |Wt |. In section14.5, we estimate |Vt |. In section 14.6, we estimate |Wt | via the introduction of the symmetric space attached to KC . In section 14.7, when E is the trivial representation, we prove uniform X estimates on qb,t for bounded b > 0. In section 14.8, we establish such estimates in the general case, i.e., we obtain the first part of Theorem 4.5.2. In section 14.9, we obtain uniform rough estimates on higher order derivatives of our hypoelliptic heat kernels. Such estimates will be needed in the proof of the existence of their limit as b → 0. In section 14.10, we study the behavior of the process V· as b → 0. We prove in particular that it converges in probability to an explicit deterministic process. Finally, in section 14.11, we obtain the limit as b → 0 of the heat kernel

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263

X qb,t , i.e., we establish the second part of Theorem 4.5.2.

14.1 A probabilistic construction of exp −tLX A

We use the same notation as in section 12.1. Let s ∈ R+ → ws ∈ p be a Brownian motion with values in p, with w0 = 0. Let E denote the corresponding expectation operator. Again, we consider equation (12.1.3), i.e., x˙ = w, ˙ x0 = p1, (14.1.1) and also equation (12.1.4), g˙ = w, ˙ g0 = 1, (14.1.2) so that as in (12.1.5), x· = pg· . (14.1.3) 0 Here, τt denotes parallel transport on F with respect to the connection ∇F from Fx0 into Fxt , and τ0t is its inverse. Note that since g· is the horizontal lift of x· , τt0 is immediately obtained from g· . We will establish the obvious extension of equation (12.1.6). Proposition 14.1.1. Let u ∈ C ∞,c (X, F ). For any t ≥ 0, t k,F t ∗ g g X − tA E τ0t u (xt ) . exp −tLA u (x0 ) = exp − B (κ , κ ) − C 8 2 (14.1.4) Proof. By (2.13.1)–(2.13.3), and (4.5.1), we get 1 1 H,X 1 ∗ g g + B (κ , κ ) + C k,F + A. (14.1.5) LX A =− ∆ 2 8 2 Recall that the last three terms in (14.1.5) are parallel with respect to ∇F . Equation (14.1.4) now follows from (14.1.5) and from Itô’s formula. 14.2 The operator LX b and the wave equation Set 2 2 X MX A,b = exp |Y | /2 LA,b exp − |Y | /2 .

(14.2.1)

For A = 0, we use instead the notation MX b . By (2.13.5), (4.5.1), we get ·

MX A,b

∗

∗

2 N Λ (T X⊕N ) 1 1 = Y N , Y T X + 2 −∆T X⊕N + 2∇VY + 2 2b b2 1 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) + ∇Y T X +b c ad Y T X b ! TX N E N − c ad Y + iθad Y − iρ Y + A. (14.2.2)

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Set C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F ))

R = ∇Y T X

+b c ad Y T X − c ad Y T X + iθad Y N − iρE Y N . (14.2.3)

By (14.2.2), (14.2.3), we get ·

∗

N Λ (T X⊕N 1 1 N T X 2 Y ,Y + 2 −∆T X⊕N + 2∇VY + 2 2b b2

∗

)

R . b (14.2.4) Note that R maps smooth sections of Λ· (T ∗ X ⊕ N ∗ ) ⊗ F over X to smooth sections of π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) over Xb. Now we establish an extension of equation (12.3.4) in Proposition 12.3.1. MX b =

+

Proposition 14.2.1. If s ∈ C ∞ (X, Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ), then −1 1 N T X 2 Λ· (T ∗ X⊕N ∗ ) Y , Y 1 − b 1 + N R π b∗ s MX − b 2 · ∗ ∗ −1 · ∗ ∗ N Λ (T X⊕N ) ∗ = π b s − R 1 + N Λ (T X⊕N ) Rb π ∗ s. (14.2.5) 2 b Proof. The eigenspace of the fibrewise operator 21 −∆T X⊕N + 2∇VY associated with the eigenvalue 0 is generated by the constants, and the linear functions of Y generate the eigenspace associated with the eigenvalue 1. Equation (14.2.5) then follows from (14.2.4) and from the fact that Rb π∗ s depends linearly on Y .

14.3 Changing Y into −Y Definition 14.3.1. Let I be the map s (x, Y ) → s (x, −Y ). Set X −1 LX0 . A,b = ILA,b I

(14.3.1)

By (2.13.5), (4.5.1), and (14.3.1), we get 1 1 N T X 2 2 T X⊕N Y , Y + −∆ + |Y | − m − n LX0 = A,b 2 2b2 · ∗ ∗ N Λ (T X⊕N ) 1 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) + ∇Y T X +b c ad Y T X − 2 b b ! − c ad Y T X + iθad Y N − iρE Y N + A. (14.3.2) X0 For t > 0, let qb,t ((x, Y ) , (x0 , Y 0 )) be the smooth kernel associated with the operator exp −tLX0 A,b . Then X0 X qb,t ((x, Y ) , (x0 , Y 0 )) = qb,t ((x, −Y ) , (x0 , −Y 0 )) .

(14.3.3)

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14.4 A probabilistic construction of exp −tLX0 A,b

Let s ∈ R+ → ws = wsp + wsk ∈ g = p ⊕ k be a Brownian motion, with w0 = 0. Instead of (13.2.11), we consider the stochastic differential equation Y TX , b x0 = p1,

x˙ =

w˙ Y˙ = , b Y0 = Y.

(14.4.1)

Let τt0 denote parallel transport from x0 to xt with respect to the connection · ∗ ∗ ∇Λ (T X⊕N )⊗F , and let τ0t be its inverse. Along the path x· , we trivialize Λ· (T ∗ X ⊕ N ∗ ) , F by parallel transport · ∗ ∗ with respect to the connections ∇Λ (T X⊕N ) , ∇F . Consider the differential equation along the path x· , " · ∗ ∗ N Λ (T X⊕N ) 1 dU =U − + b c ad Y T X 2 ds b b # i E 1 TX N N − c ad Y + iθad Y − ρ Y , (14.4.2) b b U0 = 1. Theorem 14.4.1. Let s ∈ C ∞,c Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . For t > 0, the following identity holds: "

exp

−tLX0 A,b

Z (m + n) t 1 t N T X 2 − Y ,Y ds 2b2 2 0 ! # Z t 1 2 t |Y | ds − tA Ut τ0 s (xt , Yt ) . (14.4.3) − 2 2b 0

s (x0 , Y ) = E exp

Proof. Using the results of section 11.8, equation (14.4.3) can be proved by the same arguments as equations (12.2.10), (12.9.9), (12.11.1), (12.11.8), and (13.2.12). The only significant difference is that in (14.4.2), (14.4.3), we have used a matrix version of the Feynman-Kac formula. Incidentally, since A is parallel and commutes with ρE Y N , its contribution has been factored out in (14.4.3). Definition 14.4.2. Set ·

∗

N Λ (T X⊕N Mb = − b2

∗

)

1 1 + b c ad Y T X − c ad Y T X + iθad Y N . b b (14.4.4) As we saw in the proof of Theorem 2.13.2, Mb is self-adjoint. We use the same trivialization of Λ· (T ∗ X ⊕ N ∗ ) , F along the path x· as

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Chapter 14

in (14.4.2). Let V· , W· be the solutions of the differential equations, dV = V Mb , V0 = 1, (14.4.5) ds dW i = W − ρE Y N , W0 = 1. ds b In (14.4.5), V· ∈ Aut Λ· (T ∗ X ⊕ N ∗ )x0 , and W· ∈ Aut (Ex0 ). Then we have the obvious identity U· = V· ⊗ W· . (14.4.6) 14.5 Estimating V· As before, we identify Λ· (T ∗ X ⊕ N ∗ ) with Λ· (T ∗ X ⊕ N ∗ ) ⊗R C. We still denote by h i the Hermitian product on Λ· (T ∗ X ⊕ N ∗ ). Proposition 14.5.1. There is C > 0 such that if b > 0, η > 0, f ∈ Λ· (T ∗ X ⊕ N ∗ ), η 1 C2 2 2 + 2 |Y | |f | . (14.5.1) hMb f, f i ≤ 2 η b There exists C 0 > 0 such that for b > 0, f ∈ Λ· (T ∗ X ⊕ N ∗ ), E ∗ C0 1 D · ∗ 2 2 hMb f, f i ≤ |Y | |f | − 2 N Λ (T X⊕N ) f, f . (14.5.2) 2 2b Proof. If f ∈ Λ· (T ∗ X ⊕ N ∗ ), then C 2 (14.5.3) hMb f, f i ≤ |Y | |f | . b By (14.5.3), we get (14.5.1). Let e1 , . . . , em be an orthonormal basis of T X, let em+1 , . . . , em+n be an orthonormal basis of N . By (2.16.11), we get

b c ad Y T X − c ad Y T X f, f X

T X

= −2 Y , ei , ej Re iei iej f, f . (14.5.4) 1≤i≤m m+1≤j≤m+n

Using (14.5.4), we conclude that

p b c ad Y T X − c ad Y T X f, f ≤ C Y T X N Λ· (T ∗ X⊕N ∗ ) f |f | . (14.5.5) By (2.16.11), (2.16.12), we get

p c iθad Y N f, f ≤ C Y N N Λ· (T ∗ X⊕N ∗ ) f |f | .

(14.5.6)

Also for any η > 0, p E 1 1 η D Λ· (T ∗ X⊕N ∗ ) 1 2 2 · (T ∗ X⊕N ∗ ) Λ |Y | N f |f | ≤ N f, f + |Y | |f | . b 2 b2 η (14.5.7) By (14.4.4), (14.5.5), (14.5.6) and by taking η small enough in (14.5.7), we get (14.5.2). The proof of our proposition is completed.

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We take V· as in (14.4.5). Theorem 14.5.2. There exists C > 0 such that for η > 0, b > 0, t > 0, Z t 1 C2 η 2 |Ys | ds . (14.5.8) |Vt | ≤ exp t+ 2 2 η 2b 0 Moreover, there exists C 0 > 0 such that for any b > 0, t > 0, 0Z t C 2 |Vt | ≤ exp |Ys | ds . 2 0

(14.5.9)

Proof. Let V·∗ be the adjoint of V· . Then V·∗ is the solution of the differential equation dV ∗ = Mb V ∗ , V0∗ = 1. (14.5.10) ds If f ∈ Λ· (T ∗ X ⊕ N ∗ )x0 , by (14.5.10), we get d 2 |V ∗ f | = 2 hMb V ∗ f, V ∗ f i . (14.5.11) ds By (14.5.1), (14.5.2), and (14.5.11), using Gronwall’s lemma, we get (14.5.8) and (14.5.9) for |Vt∗ |, which is equivalent to the corresponding estimates for |Vt |.

14.6 Estimating W· We take W· as in (14.4.5). A first crude estimate for Wt is the obvious Z t N C Y ds . (14.6.1) |Wt | ≤ exp b 0 From (14.6.1), we find that for any η > 0, Z t 1 C2 η 2 |Wt | ≤ exp t+ 2 |Y | ds , 2 η 2b 0

(14.6.2)

which is exactly of the same type as (14.5.8). Still, the estimate (14.6.2) is useless for small b > 0. Let KC be the complexification of K. Let kC be the Lie algebra of KC . Then kC = ik ⊕ k is the splitting of kC corresponding to the splitting g = p ⊕ k. The symmetric bilinear form Re B|kC has the same properties as the symmetric bilinear form B on g. Let 1KC denote the unit element in KC . Let XK = KC /K be the symmetric space associated with the pair (KC , K). More generally, we will use the same notation for the objects attached to KC as for the objects attached to G. Temporarily, we denote by 1G the unit element in G. Using the unique decomposition of elements in KC as in (3.2.26), we find that there exists C > 0 such that if h ∈ KC , E E −1 ρ (h) ≤ eCd(p1,ph) , ρ h ≤ eCd(p1,ph) . (14.6.3)

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Chapter 14

Consider the differential equation on XK , w˙ k iY N , Y˙ N = , (14.6.4) y˙ = − b b y0 = p1KC , Y0N = Y N . As in (12.2.8), we may as well consider the differential equation on h ∈ KC , w˙ k −iY N , Y˙ N = , (14.6.5) h˙ = b b h0 = 1KC , Y0N = Y N , so that in (14.6.4), y· = ph· . By (14.4.5) and (14.6.5), we get Wt = ρE (ht ) . (14.6.6) By (14.6.3), (14.6.6), we get −1 Wt ≤ eCd(y0 ,yt ) . (14.6.7) |Wt | ≤ eCd(y0 ,yt ) , 14.7 A proof of (4.5.3) when E is trivial Here, we assume ρE to be trivial, so that U· = V· . Also we make A = 0, so that we eliminate the subscript A. By (14.4.3), exp −tLX0 s (x0 , Y ) b Z t 1 (m + n) t 2 − |Y | ds |V | |s (x , Y )| . (14.7.1) ≤ E exp t t t 2b2 2b2 0 By (14.5.9) and (14.7.1), we obtain " Z t 2 X0 exp −tLb s (x0 , Y ) ≤ E exp (m + n) t − 1 |Y | ds 2b2 2b2 0 ! # Z C0 t 2 + |Y | ds |s (xt , Yt )| . (14.7.2) 2 0 b Recall that the operator scalar AX b on X was defined in (11.6.1). Set C0 2 X AX0 |Y | . (14.7.3) b = Ab − 2 As we shall see later, for b > 0 small enough and t > 0, the heat opera0 0 tor exp −tAX0 and its smooth heat kernel rX0 b b,t ((x, Y ) , (x , Y )) are welldefined. By as in (12.9.9), (13.2.12), and (14.4.3), we find that if proceeding F ∈ C ∞,c Xb, R , if (x· , Y· ) is taken as in (14.4.1), then " Z t (m + n) t 1 2 X0 exp −tAb F (x0 , Y ) = E exp − 2 |Y | ds 2b2 2b 0 ! # Z C0 t 2 |Y | ds F (xt , Yt ) . (14.7.4) + 2 0

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Then (14.7.2) can be written in the form exp −tLX0 s (x0 , Y ) ≤ exp −tAX0 |s| (x0 , Y ) . b b

(14.7.5)

From (14.7.5), we get X0 0 0 qb,t ((x, Y ) , (x0 , Y 0 )) ≤ rX0 b,t ((x, Y ) , (x , Y )) .

(14.7.6)

For b > 0 small enough, let αb ∈]0, 1[ be such that αb2 = 1 − C 0 b2 .

(14.7.7)

1 1 2 T X⊕N 2 −∆ + α |Y | − m − n − ∇Y T X . b 2b2 b

(14.7.8)

Then AX0 b =

For a > 0, we define Ka as in (2.14.2). Then −1 X0 √ K√ αb Ab K αb =

αb 2 T X⊕N −∆ + |Y | − m − n 2b2 1 − αb 1 (m + n) . (14.7.9) − √ ∇Y T X − b αb 2b2

Set βb =

b 3/2 αb

,

τb =

1 . αb2

(14.7.10)

Then we can rewrite (14.7.9) in the form −1 X0 √ X K√ αb Ab K αb = τb Aβb −

1 − αb (m + n) . 2b2

(14.7.11)

By (14.7.11), it is now clear that for b > 0 small enough, for t > 0, the operator exp −tAX0 is well-defined, and moreover, b rX0 b,t

0

0

((x, Y ) , (x , Y )) = exp

1 − αb (m+n)/2 (m + n) t αb 2b2 √ 0 √ 0 rX βb ,τb t ((x, αb Y ) , (x , αb Y )) . (14.7.12)

By combining equation (13.3.1) in Theorem 13.3.1 with (14.3.3), (14.7.6), (14.7.7), and (14.7.12), for b > 0 small enough, and ≤ t ≤ M , we get (4.5.3). Take now µ > 0, M > 0, µ ≤ M , and assume that µ ≤ b ≤ M . Instead of (14.5.9), we will now use (14.5.8), while taking η = 14 . The same arguments as before still lead to the proof of (4.5.3). The proof of this estimate is then completed. Remark 14.7.1. Using (14.5.8) with b > 0 small would not lead to the proof of (4.5.3).

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14.8 A proof of the estimate (4.5.3) in the general case Note that if µ, M are taken as in section 14.7, if µ ≤ b ≤ M , using the estimates (14.5.8) and (14.6.2), the proof of equation (4.5.3) in Theorem 4.5.2 proceeds exactly as in section 14.7. Therefore, in the sequel, we may as well assume b > 0 to be small. By (14.4.3), (14.4.6), (14.5.9), and (14.6.7), we obtain " Z t 1 (m + n) t 2 exp −tLX0 − |Y | ds s (x , Y ) ≤ E exp 0 A,b 2b2 2b2 0 ! # Z C0 t 2 |Y | ds + Cd (y0 , yt ) + |A| t |s (xt , Yt )| . (14.8.1) + 2 0 Theorem 14.8.1. There exists b0 ∈]0, 1] such that given > 0, M > 0, ≤ 0 00 M , there exist C,M > 0, C,M > 0, C,M > 0 such that for b ∈]0, b0 ], t ∈ [, M ], p ≥ 0, " Z t (m + n) t 1 2 E exp − |Y | ds 2b2 2b2 0 !# Z C0 t 2 + |Y | ds + pd (y0 , yt ) 2 0 2 0 00 ≤ C,M exp −C,M |Y | + C,M p2 . (14.8.2) XK Proof. We use the notation of section 14.7. Let rb,t be the analogue of the X smooth kernel rb,t on the symmetric space XK . By (14.6.4) and (14.7.12), we get " Z t 1 (m + n) t 2 − 2 |Y | ds E exp 2b2 2b 0 !# Z C0 t 2 + |Y | ds + pd (y0 , yt ) 2 0 Z √ 1 − αb = exp (m + n) t hpτb t/β 2 αb Y T X , Y T X0 dY T X0 2 b 2b p Z 0 rβXbK,τb t y0 , −iY N , y, −iY N exp (pd (y0 , y)) dydY N 0 . (14.8.3) XK

By (10.7.12), we obtain Z √ hpτb t/β 2 αb Y T X , Y T X0 dY T X0 b

p

2

=

eτb t/βb cosh (τb t/βb2 )

!m/2

! T X 2 tanh τb t/βb2 . (14.8.4) exp − αb Y 2

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Also, using (4.1.11), (4.1.12), and equation (13.2.61) in Theorem 13.2.4, we find that for b > 0 small enough, and ≤ t ≤ M , Z 0 rβXbK,τb t y0 , −iY N , y, −iY N exp (pd (y0 , y)) dydY N 0 XK 2 ≤ C exp −C 0 Y N + C 00 p2 . (14.8.5) By (14.7.7) and (14.8.3)–(14.8.5), we get (14.8.2). The proof of our theorem is completed. b Recall that the operator BX b on X was defined in (11.6.2), (11.6.3). Put C0 2 X |Y | . (14.8.6) BX0 b = Bb − 2 By (11.6.2), (14.7.3), and (14.8.6), we get 2 2 X0 BX0 = exp |Y | /2 A exp − |Y | /2 . (14.8.7) b b To construct the semigroup exp −tBX0 , we consider again equation b (12.10.1) and the suitable modification of (14.6.4), i.e., iY N Y w˙ Y TX , y˙ = − , Y˙ = − 2 + , (14.8.8) x˙ = b b b b x0 = p1G , y0 = p1KC , Y0 = Y. The first equations in (12.2.8) and (14.6.5) still hold, so that iY k Yp , h˙ = − , (14.8.9) g˙ = b b g0 = 1G , h0 = 1KC . ∞,c If F ∈ C Xb, R , for t > 0, 0Z t C 2 X0 |Y | ds F (xt , Yt ) . (14.8.10) exp −tBb F (x0 , Y ) = E exp 2 0 Theorem 14.8.2. There exists b0 ∈]0, 1] such that given > 0, M > 0, ≤ 00 0 > 0 such that for b ∈]0, b0 ], t ∈ > 0, C,M M , there exist C,M > 0, C,M [, M ] , p ≥ 0, " !# Z C0 t 2 2 2 exp − |Y | /2 E exp |Yt | /2 + |Y | ds + pd (y0 , yt ) 2 0 2 0 00 ≤ C,M exp −C,M |Y | + C,M p2 . (14.8.11) There exists b0 ∈]0, 1] such that given > 0, M > 0, ≤ M , there ex0 00 ist C,M > 0, C,M > 0, C,M > 0 such that for b ∈]0, b0 ], t ∈ [, M ] , s ∈ [t/2, t] , p ≥ 0, " !# 2 2 exp − |Y | /2 E exp |Yt | /2 + pd (y0 , ys ) 2 0 00 ≤ C,M exp −C,M |Y | + C,M p2 . (14.8.12)

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Proof. We claim that the right-hand side of (14.8.2) is just the left-hand side of (14.8.11). If p = 0, this is just a consequence of (14.7.4), (14.8.7), and (14.8.10). When p is nonzero, either one includes the process y· as part of the Markov process (x· , y· , Y· ) and uses a similar argument. Alternatively, a proper use of the Girsanov transformation leads to the equality we just mentioned. Then (14.8.11) is equivalent to (14.8.2). Recall that the kernel k·p⊕k was defined in section 10.4. For s ≥ 0, let τ0s still denote the parallel transport from (T X ⊕ N )xs into (T X ⊕ N )x0 with respect to the connection ∇T X⊕N . Using the Markov property of the process (x· , Y· ), for 0 ≤ s ≤ t, we get " !# 2

E exp |Yt | /2 + pd (y0 , ys ) Z 2 p⊕k pd(y0 ,ys ) s =E e k(t−s)/b2 (τ0 Ys , Y ) exp |Y | /2 dY . (14.8.13) g

By (10.7.9), (14.8.13), we obtain " !# 2

E exp |Yt | /2 + pd (y0 , ys )

h i 2 ≤ 2(m+n)/2 E exp |Ys | /2 epd(y0 ,ys ) .

(14.8.14) By (14.8.11) with C 0 = 0, and by (14.8.14), we find that for s ≥ t/2, (14.8.12) holds. The proof of our theorem is completed. Remark 14.8.3. The condition s ∈ [t/2, t] in (14.8.12) is imposed by the fact that the proof relies on equation (14.8.2) in Theorem 14.8.1. The proof of this equation itself relies on equation (14.8.5), where the pointwise estimates of chapter 13 on the kernel rβXbK,τb t are used, which require t > 0 to stay away from 0. However, the estimate (14.8.5) being in integrated form, it can be obtained by the methods of chapter 13, and more specifically by using equation (13.2.30), without having to use any pointwise estimate on the kernel rβXbK,τb t . Ultimately, in (14.8.12), we can as well take s ∈ [0, t]. Let k (u) be a smooth function as in (13.2.28). For y, y 0 ∈ XK , set f (y, y 0 ) = k (d (y, y 0 )) .

(14.8.15)

|f (y, y 0 ) − d (y, y 0 )| ≤ 1.

(14.8.16)

Clearly,

0

00

By (14.8.16), we deduce that if y, y , y ∈ XK , then f (y, y 00 ) ≤ f (y, y 0 ) + f (y 0 , y 00 ) + 3.

(14.8.17)

By (14.8.16), in equation (14.8.11), we may as well replace d (y0 , y) by the function f (y0 , y). We fix p > 0. The constants that follow will depend on p > 0. The value of p will be determined later.

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273 Take F ∈ C ∞,c Xb, R . We claim that for b ∈]0, b0 ], ≤ t ≤ M , the methods of chapter 12 can be applied to the expression ! # " Z C0 t 2 2 2 |Y | ds + pf (y0 , yt ) F (xt , Yt ) . exp − |Y | /2 E exp |Yt | /2 + 2 0 (14.8.18) Indeed, an analogue of equation (12.7.1) can be easily obtained, because, by Proposition 13.1.2, the covariant derivatives of order ≥ 1 in the variable y of f (y0 , y) are uniformly bounded on XK . The fact that estimates like (14.8.11) hold for any value of p > 0 results in that the arguments in the proofs of Theorems 12.7.4 and 12.10.1 can be used without any change. Ultimately, 0 0 we find that there is a nonnegative smooth kernel rX b,t ((x, Y ) , (x , Y )), such that " ! # Z C0 t 2 2 2 exp − |Y | /2 E exp |Yt | /2 + |Y | ds + pf (y0 , yt ) F (xt , Yt ) 2 0 Z 0 0 0 0 0 0 rX = b,t ((x0 , Y ) , (x , Y )) F (x , Y ) dx dY . (14.8.19)

b X

b Also rX b,t and its covariant derivatives are uniformly bounded on X . X Contrary to what the notation suggests, the kernels rb,t do not form a semigroup. Still, using (14.8.17) and the Markov property of the process (x· , y· , Y· ), instead of the equality in (13.2.3), we have the inequality, Z 0 0 rX ((x, Y ) , (x , Y )) ≤ C rX b,t b,t/2 ((x, Y ) , (z, Z)) b X

0 0 rX b,t/2 ((z, Z) , (x , Y )) dzdZ.

(14.8.20)

Using (14.8.20) and proceeding as in the proofs of Theorems 13.2.1, 13.2.2, 13.2.4, and 13.3.1, the analogue of (13.3.1) holds, i.e., there exist C > 0, C 0 > 0 such that for 0 0 is the constant that appears in (14.8.1), we take p = C. In the right-hand side of (14.8.1), by (14.8.16), we can as well replace d (y0 , yt ) by f (y0 , yt ) and still get an inequality with an extra constant in the righthand side. Using (14.8.1), by proceeding as in the beginning of the proof of Theorem 14.8.2, and using (14.8.19), instead of (14.7.5), we get Z 0 0 0 0 0 0 exp −tLX0 rX A,b s (x0 , Y ) ≤ C b,t ((x0 , Y ) , (x , Y )) |s (x , Y )| dx dY . b X

By (14.8.22), instead of (14.7.6), we obtain X0 0 0 qb,t ((x, Y ) , (x0 , Y 0 )) ≤ rX b,t ((x, Y ) , (x , Y )) .

(14.8.22) (14.8.23)

By (14.3.3), (14.8.21), and (14.8.23), we get (4.5.3). This completes the proof of equation (4.5.3) in the general case.

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X0 for bounded b 14.9 Rough estimates on the derivatives of qb,t X0 Equation (4.5.3) gives a uniform estimate on qb,t ((x, Y ) , (x0 , Y 0 )). Now we briefly explain how to obtain uniform bounds on the covariant derivatives of arbitrary order. To make the argument simpler, we will always assume that b0 ∈]0, 1] is taken as in Theorem 14.8.2, and we will assume that 0 < b ≤ b0 . The case of b bounded and bounded away from 0 can be dealt with by the same methods. Set 2 2 X0 MX0 (14.9.1) A,b = exp |Y | /2 LA,b exp − |Y | /2 . X Note that MX0 A,b can be obtained from MA,b in (14.2.2) by conjugation by I. We consider again the stochastic We de differential equation (12.10.1). b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) , instead of fine U as in (14.4.2). If s ∈ C ∞,c Xb, π (14.4.3), we get exp −tMX0 A,b s (x0 , Y ) Z 1 t N T X 2 t Y ,Y ds − tA Ut τ0 s (xt , Yt ) . (14.9.2) = E exp − 2 0

The idea is now to apply the techniques of the Malliavin calculus to (14.9.2). We will proceed as in chapter 12, and more precisely in sections 12.4–12.7 and in section 12.10. Also, instead of obtaining bounds of the type (12.7.3), we will show directly an inequality that is slightly stronger than (12.7.5). Namely, we will show that for > 0, M > 0, ≤ M, p ≥ 1, if b ∈]0, b0 ], t ∈ [, M ], if Nt is one of the random variables produced by integration by parts, then

2 p 2 2 0 |Y | . exp − |Y | /2 exp |Yt | /2 |Nt | ≤ C,M,p exp −C,M,p 1

(14.9.3) Needless to say, as shown in (12.7.3)–(12.7.5), such bounds are verified by the random variables that appear in chapter 12. The need for these stronger estimates appears because by (14.6.7), |Wt | is big compared to the terms that appeared in chapter 12. In relation with the above, we will systematically use controls v that vanish identically on the interval [0, t/2], while still picking the optimal control on the interval [t/2, t]. This possibility was already explored in Remarks 12.7.5 and 14.8.3. As explained in the proof of Theorem 12.11.2, the term Z 1 t N T X 2 exp − Y ,Y ds 2 0 does not create any difficulty with respect to what was done in sections 12.4– 12.7, no more than τ0t . Here we will concentrate on the term Ut . Of course we still write Ut as in (14.4.6).

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First, we explain how to obtain the derivative of V· τ01 with respect to the parameter ` at ` = 0, when replacing w· by w·` as in (12.1.8). Here, as in section 12.10, w·` takes its values in g = p ⊕ k. Set ∂ V· τ · |`=0 . (14.9.4) V· = ∂` 0 We use the trivialization of Λ· (T ∗ X ⊕ N ∗ ) by parallel transport with respect to the connection ∇T X⊕N . Also we use the notation in (12.10.3). By (14.4.5), we get D d V = VMb + V Mb + RT X⊕N J T X , x˙ , V0 = 0. (14.9.5) ds D` From (14.9.5), we deduce that Z t D Mb + RT X⊕N J T X , x˙ Vs−1 dsVt . Vt = Vs D` 0 By (14.4.4), we get D Mb = b c ad J˙T X − c ad J˙T X + iθad J˙N . D` By (14.9.7), we get D ≤ C J˙ . M b D` For s ≤ t, by (14.5.9) applied to the interval [s, t], we get 0Z t −1 2 Vs Vt ≤ exp C |Yu | du . 2 s

(14.9.6)

(14.9.7)

(14.9.8)

(14.9.9)

We take unit vectors e ∈ p, f ∈ k, and we take J· as in (12.10.7), so that J0 = 0, JtT X = e, J˙t = f /b. As explained before, we choose the control v that minimizes |v|L2 among the controls that vanish on [0, t/2]. In particular J· also vanishes on [0, t/2]. As we saw in (10.3.17), (10.3.48) and in the considerations that follow these two equations, when b → 0, ≤ t ≤ M , J˙ remains uniformly bounded in L2 . By (14.9.8), (14.9.9), we conclude that for 0 < b ≤ 1, Z t 0Z t D C 2 −1 |Y | ds . (14.9.10) V M V dsV ≤ C exp b s t s s D` 2 0 0 We will now estimate the contribution of the second term in (14.9.6) by the same technique as in (14.9.7)–(14.9.10). Let v· be the solution of the differential equation d v = vMb + V RT X⊕N J T X , x˙ , v0 = 0. (14.9.11) ds By (2.1.10) and (12.10.1), we get Y TX RT X⊕N J T X , x˙ = −ad J T X , . (14.9.12) b

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Chapter 14

Let Mb ∈ End (Λ· (T ∗ X ⊕ N ∗ ) ⊕ Λ· (T ∗ X ⊕ N ∗ )) be given by i# " h TX Mb −ad J T X , Y b . Mb = 0 Mb

(14.9.13)

Consider the differential equation d V = VMb , ds Then V is given by V V= 0

V0 = 1. v . V

(14.9.14)

(14.9.15)

To estimate v, V, we will instead estimate V. By the considerations that follow (10.3.47), for 0 < b ≤ 1, J·T X remains pointwise uniformly bounded. This is also a consequence of the fact that J˙T hX is uniformly bounded in L2 . Also when acting on Λ· (T ∗ X ⊕ N ∗ ), i TX Y TX preserves the total degree and vanishes on Λ0 (T ∗ X ⊕ N ∗ ). ad J , b It follows that if f, f 0 ∈ Λ· (T ∗ X ⊕ N ∗ ), TX 0 ad J T X , Y f, f b T X p p Y N Λ· (T ∗ X⊕N ∗ ) f N Λ· (T ∗ X⊕N ∗ ) f 0 . (14.9.16) ≤ C b By (14.5.2), (14.9.13), and (14.9.16), we conclude that C0 2 2 |Y | |(f, f 0 )| . (14.9.17) 2 By proceeding as in the proof of (14.5.9) in Theorem 14.5.2, we deduce from (14.9.17) that for 0 < b ≤ 1, ≤ t ≤ M , 0Z t C 2 |Vt | ≤ exp |Y | ds . (14.9.18) 2 0 Re hMb (f, f 0 ) , (f, f 0 )i ≤

Therefore we have a corresponding estimate for vt . By combining (14.9.10) with this estimate, we get a corresponding estimate for Vt . Now we will apply the same ideas to W· . Set

∂ W· τ0· |`=0 . (14.9.19) ∂` Using the second equation in (14.4.5) and proceeding as in (14.9.5), we get d W = WρE −iY N /b + W ρE −iJ˙N + RN J T X , x˙ , W0 = 0. ds (14.9.20) W· =

Using (2.1.10) and (14.9.20), we get Z t Wt = Ws ρE −iJ˙N − J T X , x˙ Ws−1 dsWt . 0

(14.9.21)

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Recall that J· vanishes on [0, t/2]. Using (14.6.7) and the fact that J˙ remains uniformly bounded in L2 , we get Z t E N −1 ˙ Ws ρ −iJ Ws dsWt 0 ! Z t 0 ≤ C exp (Cd (y0 , yt )) 1 + exp (4Cd (y0 , ys )) ds . (14.9.22) t/2

By (14.8.12), (14.9.22), and using Hölder’s inequality, we find that for b ∈]0, b0 ], ≤ t ≤ M, p > 1, " p # Z t 2 2 E N −1 Ws ρ −iJ˙ Ws dsWt exp − |Y | /2 E exp |Yt | /2 0

2 0 ≤ C,M,p exp −C,M |Y | . (14.9.23) Recall that T X has been trivialized by parallel transport with respect to the connection ∇T X along x· . By (14.8.8), Z t Z t Ws ρE J T X , x˙ Ws−1 dsWt = Ws ρE J T X , −b2 x ¨ + w˙ p Ws−1 dsWt . 0

0

(14.9.24)

Moreover, Z t Z t Ws ρE J T X , w˙ p Ws−1 ds = Ws ρE J T X , δwp Ws−1 , 0

(14.9.25)

0

i.e., (14.9.25) is an Itˆ o integral. By (14.6.7), " # Z t TX −1 p 2 2 E p exp − |Y | /2 E exp |Yt | /2 Ws ρ J , δw Ws Wt 0

h i 2 2 ≤ exp − |Y | /2 E exp |Yt | /2 e2Cpd(y0 ,yt ) #! Z t TX −1 2p E p + E exp |Yt | /2 Ws ρ J , δw Ws . (14.9.26) "

2

0

By (14.8.11), we can dominate the first term in the right-hand side of (14.9.26) by an expression similar to the right-hand side of (14.9.23). Take θ = 3/2. By H¨ older’s inequality, we get " Z 2p # 2 t 2 exp − |Y | /2 E exp Yt /2 Ws ρE J T X , δwp Ws−1 0

h ii1/θ 2 2 ≤ exp − |Y | /2 E exp θ |Yt | /2 " " Z ##(θ−1)/θ t TX −1 2pθ/(θ−1) 2 E p exp − |Y | /2 E Ws ρ J , δw Ws . h

0

(14.9.27)

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By (10.8.12), given > 0, for b > 0 small enough, and for t ≥ , the first term in the right-hand side of (14.9.27) is uniformly bounded. Moreover, using (12.1.17), (14.6.7), and the fact that JsT X vanishes for s ≤ t/2, for p > 1, we obtain " Z 2pθ/(θ−1) # t Ws ρE J T X , δwT X Ws−1 E 0

" Z ≤ Cp E 

#pθ/(θ−1) 

t

e4Cd(y0 ,ys ) ds

t/2

Moreover, for ≤ t ≤ M , "Z #pθ/(θ−1) Z t 4Cd(y0 ,ys ) e ds ≤ CM,p t/2

 . (14.9.28)

t

e4Cpd(y0 ,ys )θ/(θ−1) ds.

(14.9.29)

t/2

By (14.8.11), (14.9.28), and (14.9.29), we obtain " Z 2pθ/(θ−1) # t 2 Ws ρE J T X , δwT X Ws−1 exp − |Y | /2 E 0

"Z ≤ CM,p exp − |Y | /2 E

2

#

t

e

4Cpd(y0 ,ys )θ/(θ−1)

ds

t/2

2 0 ≤ C,M,p exp −C,M,p |Y | . (14.9.30) By (14.9.26)–(14.9.30), the left-hand side of (14.9.26) can be also dominated by an expression similar to the right-hand side of (14.9.23). Moreover, since JtT X = e, using (14.4.5) and (14.8.8), we get Z t Ws ρE J T X , −b2 x ¨ Ws−1 dsWt = bWt ρE e, −YtT X 0 Z t i h + Ws ρE −iY N , J T X , Y T X + bJ˙T X , Y T X Ws−1 dsWt . t/2

(14.9.31) By (14.6.7), for 0 ≤ b ≤ 1, bWt ρE e, −YtT X ≤ c e2Cd(y0 ,yt ) + |Yt |2 .

(14.9.32)

Take θ = 3/2. For p ≥ 1, using Hölder’s inequality, we obtain i h 2 2p E exp |Yt | /2 |Yt | h h i(θ−1)/θ ii1/θ h 2 2pθ/(θ−1) ≤ E exp θ |Yt | /2 . (14.9.33) E |Yt | By (10.8.12), (10.8.26), and (14.9.33), given > 0, for b > 0 small enough and t ≥ , h i 2 2 2p 2 0 exp − |Y | /2 E exp |Yt | /2 |Yt | ≤ C,p exp −C,p |Y | . (14.9.34)

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By (14.6.7), (14.8.12), (14.9.32), and (14.9.34), for b > 0 small enough, t ∈ [, M ], we get h p i 2 2 exp − |Y | /2 E exp |Yt | /2 bWt ρE e, −YtT X 2 0 ≤ C,M,p exp −C,M,p |Y | . (14.9.35) Using (14.6.7) and the fact that J T X remains uniformly bounded, we get Z t T X T X −1 E N Ws ρ −iY , J , Y Ws dsWt ≤ ce2Cd(y0 ,yt ) t/2 Z t Z t 8 +c e8Cd(y0 ,ys ) ds + c |Ys | ds. (14.9.36) t/2

t/2

The first two terms in the right-hand side of (14.9.36) can be dealt with as (14.9.22), (14.9.23). We will now explain how to deal with the last term in the right-hand side of (14.9.36). We take again θ = 3/2. By Hölder’s inequality, for ≤ t ≤ M , " # Z t h h ii1/θ 2 8p 2 E exp |Yt | /2 |Ys | ds ≤ CM E exp θ |Yt | /2 t/2

" "Z E

##(θ−1)/θ

t

|Ys |

8pθ/(θ−1)

ds

.

(14.9.37)

t/2

By (10.8.12), (10.8.26), and (14.9.37), given > 0, for b > 0 small enough and ≤ t ≤ M, p > 1, we obtain " # Z t 2 2 8p exp − |Y | /2 E exp |Yt | /2 |Ys | ds t/2

2 0 ≤ C,M,p exp −C,M,p |Y | .

(14.9.38)

By (14.9.36) and the considerations which follow this equation, and by (14.9.38), we get " 2 2 exp − |Y | /2 E exp |Yt | /2 Z p # t Ws ρE −iY N , J T X , Y T X Ws−1 dsWt t/2 2 0 ≤ C,M,p exp −C,M,p |Y | .

(14.9.39)

Moreover, since J˙T X remains uniformly bounded in L2 , the last term in the right-hand side of (14.9.31) can be treated by the same methods as in (14.9.22), (14.9.36).

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By (14.9.24)–(14.9.39), given > 0, for b > 0 small enough, ≤ t ≤ M , and p > 1, " p # Z t T X −1 2 2 E exp − |Y | /2 E exp |Yt | /2 Ws ρ J , x˙ Ws dsWt 0

2 0 ≤ C,M,p exp −C,M |Y | . (14.9.40) Ultimately, we have shown that the methods of chapter 12 can be applied to the semigroup exp −tLX0 A,b . We obtain this way a uniform control on the X0 first covariant derivatives of qb,t ((x, Y ) , (x0 , Y 0 )) in the variables (x0 , Y 0 ). The principle of iteration of the above to higher derivatives in the variables (x0 , Y 0 ) should be clear. We leave the details to the reader. By exchanging the roles of (x, Y ) and (x0 , Y 0 ), similar results can be proved also for the covariant derivatives in the variables (x, Y ). They will not be needed here. Remark 14.9.1. We assumed the control v to vanish on [0, t/2] in order to use the estimate (14.8.12) in Theorem 14.8.2, which requires s ∈ [t/2, t]. However, as explained in Remark 14.8.3, this estimate remains valid also for s ∈ [0, t]. Ultimately, this restriction on v is unnecessary.

14.10 The behavior of V· as b → 0 Let 1 ∈ End (g) be the identity. We identify 1 to the scalar product on g = 2 p⊕k. Also, if Y ∈ g, we denote by Y ⊗Y the quadratic form u ∈ g → hu, Y i . Let (V, kk) be a Banach space. For b > 0, let Ub be a random variable with values in V . Recall that Ub is said to converge to 0 in probability as b → 0 if for any > 0, P [kUb k ≥ ] → 0. (14.10.1) We still assume equations (14.8.8) and (14.8.9) to be verified. Proposition 14.10.1. Given α > 1/2, M > 0, as b → 0, the process −α |log (b)| Y· converges uniformly on [0, M ] to 0 in probability. Rt For any M > 0, as b → 0, the process 0 Ys ⊗ Ys ds converges uniformly R t p k 2 Y , Y ds converges and the process on [0, M ] to 2t 1 in probability, 0 k,p 1 t in probability. uniformly on [0, M ] to − 4 Tr C Proof. As in (10.8.3), by (14.8.8), we get Z t 2 2 2 1 es/b δws . (14.10.2) Yt = e−t/b Y + e−t/b b 0 By (14.10.2), to establish the first part of our proposition, we may as well assume that Y0 = 0. By a time change argument, there is a Brownian motion B· with values in g = p ⊕ k such that B0 = 0, and moreover, Z t 2 b es/b δws = √ Be2t/b2 −1 . (14.10.3) 2 0

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By (14.10.2), (14.10.3), we get 2

e−t/b √ Be2t/b2 −1 . (14.10.4) 2 By (14.10.4), we conclude that the uniform convergence result takes place as long as we assume that 0 ≤ t ≤ b2 . Now we assume that t ≥ b2 . It is well-known that Bs0 = sB1/s is still a Brownian motion starting at 0. We can write (14.10.4) in the form √ (14.10.5) Yt = 2 sinh t/b2 B 0 2t/b2 −1 . −1) (e −1 2 Since t ≥ b2 , e2t/b − 1 remains uniformly bounded. Moreover, by the law of iterated logarithm [ReY99, Theorem II (1.9)], we get Bt0 = 1. (14.10.6) lim sup p t→0 2t log |log t| Yt =

Also observe that 2 log e2t/b − 1 ≤ 2t/b2 ,

(14.10.7)

so that for b > 0 small enough and b2 ≤ t ≤ M , 2 log e2t/b − 1 ≤ 2M/b2 .

(14.10.8)

By (14.10.8), for b > 0 small enough and b2 ≤ t ≤ M , 2 log log e2t/b − 1 ≤ log 2M/b2 .

(14.10.9)

By (14.10.5)–(14.10.9), we obtain the first part of our proposition. For the solution of the third equation in (14.8.8) with b = 1, we use the notation Z· = Y· . Set Yb,t = Zt/b2 .

(14.10.10)

As we saw after (10.8.25), for b > 0, the probability law of the process Yb,· coincides with the probability law of Y· for the given b. Then Z t Z t/b2 Yb,s ⊗ Yb,s ds = b2 Zs ⊗ Zs ds. (14.10.11) 0

0

The ergodic theorem can be applied to the Markov process Z· , whose 2 invariant measure is given by exp − |Y | dY /π (m+n)/2 . As t → +∞, we get the almost sure convergence, Z 1 t 1 Zs ⊗ Zs ds → . (14.10.12) t 0 2 In the sequel, we assume that (14.10.12) is verified. Given > 0, there exists A > 0 such that for t ≥ A , Z t 1 1 Z ⊗ Z ds − ≤ . (14.10.13) s s t 2 M 0

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Chapter 14

By (14.10.11), (14.10.13), we deduce that if b2 A ≤ t ≤ M , Z t 1 Y ⊗ Y ds − t ≤ . b,s b,s 2 0

(14.10.14)

Moreover, by (14.10.11), there exists b > 0 such that for 0 0, there exists b > 0 such that for 0 < b ≤ b , (14.10.14) holds for any t ∈ [0, M ]. Therefore, Rt almost surely, as b → 0, 0 Yb,s ⊗ Yb,s ds converges uniformly on [0, M ] to t 12 . Since Y· and Yb,· have the same probability law, this implies the convergence in probability that is mentioned in our proposition. By (14.10.10), we get Z t/b2 Z t h i 2 p k 2 p 2 k Zs , Zs ds. (14.10.15) Yb,s , Yb,s ds = b 0

0

By the ergodic theorem, we know that as t → +∞, we have the almost sure convergence, Z 1 t p k 2 1 Zs , Zs ds → − Tr C k,p . (14.10.16) t 0 4 Using (14.10.16) and proceeding as before, we get the last part of our proposition. Recall that R (Y ) was defined in (2.16.10). By (14.4.4), we get ·

Mb = −

∗

N Λ (T X⊕N b2

∗

)

+

R (Y ) . b

(14.10.17)

Using (14.4.5), we obtain · ∗ ∗ Vt = exp −tN Λ (T X⊕N ) /b2 Z t · ∗ ∗ R (Ys ) exp − (t − s) N Λ (T X⊕N ) /b2 ds. (14.10.18) + Vs b 0 As in section 2.16, P denotes the projection operator from Λ· (T ∗ X ⊕ N ∗ ) on Λ0 (T ∗ X ⊕ N ∗ ) = R, and P⊥ = 1 − P. Proposition 14.10.2. For any p > 2, M > 0, there exist bp ∈]0, 1], Cp,M > 0, C 0 > 0 such that for b ∈]0, bp ],

sup Vt − exp −tN Λ· (T ∗ X⊕N ∗ ) /b2 P⊥

0≤t≤M

p

(p−2)/p

≤ Cp,M b

(1 + |Y |) exp

C 0 b2 2 |Y | . (14.10.19) 2

For any > 0, M > 0, ≤ M , V· P⊥ converges uniformly to 0 on [, M ] in probability.

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283

Proof. Clearly, Z t R (Ys ) Λ· (T ∗ X⊕N ∗ ) 2 ⊥ V exp − (t − s) N dsP /b s b 0 Z t R (Ys ) ⊥ Λ· (T ∗ X⊕N ∗ ) 2 ≤ /b P ds. (14.10.20) Vs b exp − (t − s) N 0 Let p > 2 and let q be such that 0 ≤ t ≤ M , we get

1 p

1 q

+

= 1. By Hölder’s inequality, for

Z t R (Ys ) ⊥ Λ· (T ∗ X⊕N ∗ ) 2 V s exp − (t − s) N /b P ds b 0 "Z #1/p M

≤ b(p−2)/p

p

|Vs R (Ys )| ds 0

(

·

∗

1 − exp −qM N Λ (T X⊕N qN Λ· (T ∗ X⊕N ∗ )

∗

)/ 2

b

)1/q

⊥

P

. (14.10.21)

Moreover, ( "Z E

#)1/p

M

p

|Vs R (Ys )| ds

( "Z ≤ E

0

M

#)1/2p 2p

|Vs |

ds

0

( "Z E

M

#)1/2p 2p

|R (Ys )|

ds

. (14.10.22)

0

By equation (10.8.26) in Proposition 10.8.2, we get ( "Z E

M

#)1/2p 2p

|R (Ys )|

ds

≤ Cp,M (1 + |Y |) .

(14.10.23)

0

Let C 0 be the positive constant that appears in (14.5.9). Then

( "Z ! #)1/2p Z

M C0 M

2 2p |Y | ds . (14.10.24) E |Vs | ds ≤ Cp,M exp

2 0 0 2p

By proceeding as in (14.7.7)–(14.7.12) and using the notation in these equations, and by (14.8.7), for b2 < 1/C 0 , we get 0Z t C 1 − αb 2 E exp |Y | ds = exp (m + n) t 2 0 2b2 !Z ! 2 2 √ |Y | |Y 0 | p⊕k 0 exp hαb t/b2 ( αb Y, Y ) exp − dY 0 . (14.10.25) 2 2α b g

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Chapter 14

By (10.7.10), we obtain Z g

hp⊕k αb t/b2

2 √ |Y 0 | ( αb Y, Y 0 ) exp − 2αb

=

! dY 0

!(m+n)/2 2 eαb t/b cosh (αb t/b2 ) + sinh (αb t/b2 ) /αb ! αb coth αb t/b2 + αb 2 exp − |Y | . (14.10.26) 2 coth (αb t/b2 ) αb + 1

By (14.10.25), (14.10.26), we get !(m+n)/2 0Z t 2 et/b C 2 |Y | ds E exp = 2 0 cosh (αb t/b2 ) + sinh (αb t/b2 ) /αb C 0 b2 1 2 |Y | exp 2 coth (αb t/b2 ) αb + 1 0 2 Cb 2 ≤ exp (1 − αb ) t (m + n) /2b2 exp |Y | . (14.10.27) 2 By (14.10.24) and (14.10.27), for b ∈]0, 1] small enough, ( "Z #)1/2p 0 2 M Cb 2 2p |Y | . E |Vs | ds ≤ Cp,M exp 2 0

(14.10.28)

By (14.10.18), (14.10.20)–(14.10.23), and (14.10.28), we get (14.10.19). Clearly, · ∗ ∗ (14.10.29) exp −tN Λ (T X⊕N ) /b2 P⊥ ≤ exp −t/b2 . By (14.10.19) and (14.10.29), we get the second part of our proposition. Now we establish a path integral version of equation (14.2.5) in Proposition 14.2.1. Proposition 14.10.3. The following identity holds: −1 Λ· (T ∗ X⊕N ∗ ) Vt 1 + b 1 + N R (Yt ) −1 · ∗ ∗ = 1 + b 1 + N Λ (T X⊕N ) R (Y0 ) Z

t

Vs

+ 0

! · ∗ ∗ −1 N Λ (T X⊕N ) Λ· (T ∗ X⊕N ∗ ) − + R (Ys ) 1 + N R (Ys ) ds b2 Z t −1 · ∗ ∗ + Vs 1 + N Λ (T X⊕N ) R (δw) . (14.10.30) 0

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Proof. By (14.10.17), the first equation in (14.4.5) can be written in the form ! · ∗ ∗ R (Y ) N Λ (T X⊕N ) dV + =V − . (14.10.31) ds b2 b Using (14.8.8) and (14.10.31), we get (14.10.30). By (14.10.30), if f ∈ Λ0 (T ∗ X ⊕ N ∗ ) = R, we get −1 Λ· (T ∗ X⊕N ∗ ) Vt 1 + b 1 + N R (Yt ) f −1 Λ· (T ∗ X⊕N ∗ ) = 1+b 1+N R (Y0 ) f t

Z + 0

−1 · ∗ ∗ R (Ys ) f ds Vs R (Ys ) 1 + N Λ (T X⊕N ) Z t −1 · ∗ ∗ Vs 1 + N Λ (T X⊕N ) R (δw) f. +

(14.10.32)

0

In the sequel, we simply take f = 1. Proposition 14.10.4. Given M > 0, as b → 0, the following processes converge uniformly over [0, M ] in probability, Z t −1 · ∗ ∗ Vs P⊥ R (Ys ) 1 + N Λ (T X⊕N ) R (Ys ) f ds → 0, (14.10.33) 0 Z t −1 · ∗ ∗ Vs 1 + N Λ (T X⊕N ) R (δw) f → 0. 0

Proof. Clearly, for 0 ≤ t ≤ M , Z t −1 ⊥ Λ· (T ∗ X⊕N ∗ ) V P R (Y ) 1 + N R (Y ) f ds s s s 0 Z M Vs P⊥ |Ys |2 ds. (14.10.34) ≤C 0

Take p > 2, and let q be such that "Z E

M

1 p

+

1 q

= 1. By Hölder’s inequality, we get

# 2 ⊥ Vs P |Ys | ds

0

( "Z ≤ E 0

M

#)1/p ( "Z p ⊥ Vs P ds E

#)1/q

M

|Ys |

2q

ds

By equation (10.8.26) in Proposition 10.8.2, we get ( "Z #)1/q M 2q 2 E |Ys | ds ≤ CM,q 1 + |Y | . 0

. (14.10.35)

0

(14.10.36)

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Chapter 14

By equation (14.10.19) in Proposition 14.10.2, and by (14.10.34)–(14.10.36), ⊥ to establish the first part of our theorem, we may as well replace ⊥ Vs P Λ· (T ∗ X⊕N ∗ ) 2 in the right-hand side of (14.10.35) by exp −sN /b P , which concludes the proof of the first convergence in (14.10.33). By (2.16.14), PR (Y ) P = 0, and so Z t −1 · ∗ ∗ Vs 1 + N Λ (T X⊕N ) R (δw) f 0 Z t −1 · ∗ ∗ = Vs P⊥ 1 + N Λ (T X⊕N ) R (δw) f. (14.10.37) 0

Take p > 1. By (12.1.17),

Z t −1

⊥ Λ· (T ∗ X⊕N ∗ )

sup R (δw) f V P 1 + N s

0≤t≤M 0 p

" #1/2

Z M

. (14.10.38) Vs P⊥ 2 ds ≤ Cp

0 p

Moreover, for p > 2,

" #1/2 ( "Z #)1/p

Z M

M

2

V P⊥ ds

≤ Cp,M E V P⊥ p ds .

0

0

(14.10.39)

p

By using again equation (14.10.19) in Proposition 14.10.2, we get the second part of our proposition. Recall that for Y ∈ T X ⊕ N , S (Y ) was defined in Definition 2.16.2, and was evaluated in Proposition 2.16.3. Note S (Y ) is a nonnegative scalar multiple of P, and that this scalar depends quadratically on Y . Definition 14.10.5. Put Z V t = exp

t

S (Ys ) ds .

(14.10.40)

0

Theorem 14.10.6. For > 0, M > 0, ≤ M , as b → 0, V· − V · P converges uniformly to 0 on [, M ] in probability. Proof. By Proposition 14.10.2, to establish our theorem, we may as well replace V· by V· P. Set Z t vt = Vt P − P − Vs PS (Ys ) ds. (14.10.41) 0

By (14.10.41), we get Z Vt P = V t P + vt +

0

t

vs S (Ys ) V −1 s dsV t .

(14.10.42)

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By (14.10.42), since S (Ys ) is nonnegative, we get |Vt P − V t P| ≤ |vt | + |V t − 1| sup |vs | .

(14.10.43)

0≤s≤M

By (14.5.9) and (14.10.27), given p > 1, for b > 0 small enough,

0 2

sup |Vt | ≤ CM,p exp C b |Y |2 . (14.10.44)

0≤t≤M 2 p

By Proposition 14.10.1 and by (14.10.44), as b → 0, −1 · ∗ ∗ bVt 1 + N Λ (T X⊕N ) R (Yt ) converges to 0 uniformly on [0, M ] in probability. Using (2.16.15), (14.10.32), and Proposition 14.10.4, as b → 0, v· converges uniformly to 0 on [0, M ] in probability. Moreover, it is obvious that V· also satisfies (14.10.44). By (14.10.43), we deduce that as b → 0, V· P − V · P converges to 0 uniformly on [0, M ] in probability. This completes the proof of our theorem. Recall that δ ∈ R was defined in Definition 2.16.4. Theorem 14.10.7. For > 0, M > 0, ≤ M , as b → 0, Vt converges uniformly to exp (δt) P on [, M ] in probability. Proof. By Theorem 14.10.6, to establish our theorem, it is enough to prove that as b → 0, V t converges to exp (δt) uniformly on [0, M ] in probability. By Rt (14.10.40), we have to show that 0 S (Ys ) ds converges to δt uniformly on [0, M ] in probability. This is a consequence of Proposition 2.16.5, of (2.16.23), and of Proposition 14.10.1. X0 as b → 0 14.11 The limit of qb,t

Recall that given b > 0, W· is the solution of the second equation in (14.4.5). As before if f ∈ kC , we identify f with the corresponding left-invariant vector field on KC . Note that h· in (14.6.5) is the obvious analogue for KC of g· for G in (12.2.8). Definition 14.11.1. Let w = wp , wk be a Brownian motion with values in g = p ⊕ k. Let g0,· , h0,· ∈ KC be the solutions of the stochastic differential equations, dg0,· = dwp , h˙ 0,· = −idwk , (14.11.1) g0,0 = 1G , h0,0 = 1KC . Put x0,· = pg0,· , y0,· = ph0,· , W0,· = ρE (h0,· ) . Then W0,· is the solution of the stochastic differential equation dW0,· = W0,· ρE −idwk , W0,0 = 1.

(14.11.2) (14.11.3)

X Recall that the kernel q0,t ((x, Y ) , (x0 , Y 0 )) was defined in (4.5.2).

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Chapter 14

Theorem 14.11.2. Let s ∈ C ∞,c Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . Then for t > 0, (x, Y ) ∈ Xb, as b → 0, Z X qb,t ((x, Y ) , (x0 , Y 0 )) s (x0 , Y 0 ) dx0 dY 0 b X Z X → q0,t ((x, Y ) , (x0 , Y 0 )) s (x0 , Y 0 ) dx0 dY 0 . (14.11.4) b X

Proof. To establish our theorem, we may and we will assume that x = x0 . Take (x· , Y· ) as in (12.10.1). By (14.1.4), (14.3.1), (14.4.6), (14.9.1), and (14.9.2), we need to show that for t > 0, as b → 0, Z 1 t N T X 2 t Y ,Y ds − tA Vt ⊗ Wt τ0 s (xt , Yt ) E exp − 2 0 t k,F t ∗ g g −(m+n)/2 − tA →π exp − B (κ , κ ) − C 8 2 " Z # 2 t E τ0 Ps (x0,t , Y ) exp − |Y | dY . (14.11.5) (T X⊕N )x

t

1. The case where ρE is trivial. First, we will assume that ρE is trivial, so that we can disregard Wt and C k,F in (14.11.5). By Theorem 14.10.7, we know that for t > 0, as b → 0, Vt converges in probability to exp (δt) P. By (14.10.44), to establish (14.11.5), we may as well replace Vt by exp (δt) P. Similarly, by the last part of 2 R t Proposition 14.10.1, we may as well replace 12 0 Y N , Y T X ds by − 18 Tr C k,p t. Using to prove (14.11.5), we only need to show (2.16.26), that if F ∈ C ∞,c Xb, R , as b → 0, E [F (xt , Yt )] →π

−(m+n)/2

"Z

E

F (x0,t , Y ) exp − |Y | (T X⊕N )x

2

# dY ,

0,t

(14.11.6) which is a consequence of (11.6.5), (12.1.6), (12.10.2), and of Theorems 12.10.1 and 12.10.2. Needless to say, we need much less than the convergence of kernels in Theorem 12.10.2, but just the intermediate narrow convergence result given in the proof of Theorem 12.8.1, and its analogue over Xb. 2. The case of a general ρE By (14.8.11), given > 0, M > 0, for t ∈ [, M ] , p > 0, 2 0 00 E [exp (pd (y0 , yt ))] ≤ C,M exp −C,M |Y | + C,M p2 .

(14.11.7)

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Using (14.6.7), (14.10.44), the considerations in the first part of the proof, and (14.11.7), to establish (14.11.5), we only need to show that ∞,c ∗ b if s ∈ C X,π b F , as b → 0, t E Wt τ0t s (xt , Yt ) → π −(m+n)/2 exp − C k,F 2 " Z # 2 t E τ0 s (x0,t , Y ) exp − |Y | dY .

(14.11.8)

(T X⊕N )x

0,t

Let H : KC → End [Ex0 ] be a smooth bounded function. By equation (12.8.47) in Remark 12.8.2 applied to G and KC , for t > 0, as b → 0, we get E H (ht ) τ0t s (xt , Yt ) → E [H (h0,t )] # " Z 2 s (x0,t , Y ) exp − |Y | dY . (14.11.9) π −(m+n)/2 E τ0t (T X⊕N )x

0,t

By (14.6.6), (14.6.7), and (14.11.7), equation (14.11.9) can be used with the unbounded H = ρE , so that H (ht ) = Wt . When H = ρE , by (14.11.2), H (h0,t ) = W0,t .

(14.11.10)

By transforming equation (14.11.3) for W0,· into an Itô stochastic differential equation, we get 1 dW0,· = − W0,· C k,E dt + W0,· ρE −iδwk , W0,0 = 1, (14.11.11) 2 so that t k,E E [W0,t ] = exp − C . (14.11.12) 2 From (14.11.9)–(14.11.12), we get (14.11.8). The proof of our theorem is completed.

Given (x, Y ) ∈ Xb, equation (4.5.4) in Theorem 4.5.2 is an obvious conX0 sequence of the uniform estimates on qb,t ((x, Y ) , (x0 , Y 0 )) and its covariant derivatives of arbitrary order in the variable (x0 , Y 0 ) ∈ Xb that were established in section 14.9, and of Theorem 14.11.2.

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Chapter Fifteen X The heat kernel qb,t for b large

The purpose of this chapter is to establish the estimates of Theorem 9.1.1 on the hypoelliptic heat kernel q X . More precisely, we show that as b → +∞, b,t q X exhibits the proper decay away from Fbγ = bia N k −1 ⊂ Xb. b,t

To avoid being overburdened with technicalities at the very beginning, first, we prove similar estimates on scalar hypoelliptic heat kernels over X . Such estimates are then extended to scalar hypoelliptic heat kernels over Xb. using the FeynmanUltimately, we extend the estimates to the kernel q X b,t N T X 2 1 in the right-hand side of equation Kac formula. The term 2 Y , Y (2.13.5) for LX plays a crucial role in proving the estimates. b This chapter is organized as follows. In section 15.1, we establish uniform Gaussian estimates on the scalar heat kernel rX b,t over X . In section 15.2, we establish important estimates on the deviation of the solution of the differential equation (12.9.5) from the geodesic flow as b → +∞. Such estimates are easy. Unavoidably the hyperbolic nature of the geodesic flow appears in the estimates. The main purpose of the chapter is to convert such pointwise estimates on the trajectories to estimates on the hypoelliptic heat kernels. In section 15.3, we show that as b → ∞, rX x, Y T X , γ x, Y T X deb,1 cays in the proper way away from ia X (γ). The estimates are based on the properties of the pseudodistance on X established in section 3.9, and also on the results of section 15.2. Theargument goes roughly as follows. Away from Fγ = ia X (γ), ϕ1/2 x, Y T X and ϕ−1/2 γ x, Y T X are far away. On the other hand, by the results of section 15.2, the hypoelliptic heat kernel propagates more and more along the geodesic flow. The point is to ultimately deduce from these two facts the proper decay of the hypoelliptic heat kernel. In section 15.4, we obtain uniform Gaussian estimates for rX b,1 near ia X (γ). In section 15.5, we obtain corresponding results for the scalar heat kernel b away from bia N k −1 . Because rX is dominated by rX , some of rX on X b,1 b,1 b,1 these estimates are proved instead for rX b,1 . −1 b In section 15.6, we obtain uniform estimates for rX . These b,1 near ia N k estimates are non-Gaussian. In section 15.7, we prove Theorem 9.1.1 by using the estimates on the X scalar heat kernel rX b,t to obtain corresponding estimates for q b,t . In section 15.8, we establish Theorem 9.1.3, i.e., we obtain uniform estimates for the covariant derivatives of q X . b,t

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X for b large The heat kernel qb,t

291

In section 15.9, we prove Theorem 9.5.6. Finally, in section 15.10, we establish Theorem 9.11.1. In the present chapter, we use the notation of the previous chapters of the book. In particular the semisimple element γ ∈ G is taken as in (3.3.2).

15.1 Uniform estimates on the kernel rX b,t over X We use the notation of section 12.9. For t > 0, rX x, Y T X , x0 , Y T X0 b,t is the smooth kernel associated with exp −tAX b . Set X rX b = r b,1 .

(15.1.1)

0 > Theorem 15.1.1. Given > 0, M > 0, ≤ M , there exist C,M > 0, C,M TX 0 T X0 0 such that for b ≥ 1, ≤ t ≤ M , x, Y , x ,Y ∈ X,

rX b,t

x, Y T X , x0 , Y T X0 2 2 0 . (15.1.2) ≤ C,M b4m exp −C,M d2 (x, x0 ) + Y T X + Y T X0

Proof. We follow the proof of Theorem 13.2.1. Using the uniform bounds on the kernel rX b,t that were given in equation (12.9.10) in Theorem 12.9.1, and using also the analogue of (13.2.3), we get the analogue of (13.2.4), rX x, Y T X , x0 , Y T X0 b,t Z 4m x, Y T X , z, Z T X dzdZ T X . (15.1.3) ≤ C,M b rX b,t/2 X

Recall that x0 = p1. We may and we will assume that x, Y T X = (x0 , Y p ), with Y p ∈ p. By proceeding as in (13.2.5), we get Z Z p TX TX rX (x , Y ) , z, Z dzdZ = hpt/2b2 (bY p , Z p ) dZ p . (15.1.4) 0 b,t/2 X

p

By (10.7.12), we obtain Z p

hpt/2b2

p

p

p

(bY , Z ) dZ =

!m/2 2 et/2b cosh (t/2b2 ) p2 1 2 2 exp − b tanh t/2b |Y | . (15.1.5) 2

By (15.1.3)–(15.1.5), we get T X 2 0 Y . rX x, Y T X , x0 , Y T X0 ≤ C,M b4m exp −C,M b,t

(15.1.6)

Also we get a corresponding estimate by exchanging the roles of x, Y T X and x0 , Y T X0 .

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Chapter 15

By proceeding as in (13.2.10) and using again the uniform bounds on rX b,t of Theorem 12.9.1, we get rX x, Y T X , x0 , Y T X0 b,t Z X 4m TX ≤ C,M b , z, Z T X dzdZ T X (z,Z T X )∈X rb,t/2 x, Y d(x,z)≥d(x,x0 )/2

+ C,M b4m

Z

X

(z,Z T X )∈X rb,t/2 d(x0 ,z )≥d(x,x0 )/2

z, Z T X , x0 , Y T X0 dzdZ T X . (15.1.7)

We only need to estimate the first term in the right-hand side of (15.1.7). By (13.2.22), we get Z p TX dzdZ T X rX b,t/2 (x0 , Y ) , z, Z (z,Z T X )∈X d(x0 ,z)≥d(x0 ,x0 )/2

≤ exp − 1 − α2 d2 (x0 , x0 ) /4t !m/2 2 p2 et/2b 1 2 2 exp − αb tanh αt/2b |Y | /2 . (15.1.8) cosh (αt/2b2 ) 2 By (15.1.7), and by taking α = √12 in (15.1.8), we find that the estimate we obtain is compatible with (15.1.2). By (15.1.6) and (15.1.8), we get (15.1.2). The proof of our theorem is completed. Remark 15.1.2. The estimate (15.1.2) is not so useful, because it diverges as b → +∞. However, given the structure of the operator AX b , this divergence is unavoidable. In the sequel, we will study the kernel rX b,t in more detail. 15.2 The deviation from the geodesic flow for large b We introduce the notation 1 . (15.2.1) b2 By (12.9.3), the operator AX b can be written in the form T X 2 1 2 V AX = −h ∆ + Y − hm − ∇Y T X . (15.2.2) b 2 The operator AX ∞ is well-defined and given by 1 T X 2 AX Y − ∇Y T X . (15.2.3) ∞ = 2 We consider equation (12.9.5), in which the starting point x, Y T X ∈ X is now arbitrary. This equation takes the form x˙ = Y T X , Y˙ T X = hw, ˙ (15.2.4) h=

x0 = x,

Y0T X = Y T X .

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293 TX

When h = 0, we will write x· , Y ·

instead of x· , Y·T X , so that TX = 0, Y˙

x˙ = Y T X , x0 = x,

Y

TX 0

=Y

TX

(15.2.5)

.

Recall that ϕt |t∈R denotes the geodesic flow on X . Then in (15.2.5), xt , Y Tt X = ϕt x, Y T X . (15.2.6) As we saw in section 12.2 after (12.2.7), equation (15.2.4) can be solved for any continuous path w· , and the solution depends continuously on the path w· . Recall that the pseudodistance δ on X was defined in (3.8.1). Theorem 15.2.1. Given M > 0, there exists ηM > 0 such that for h ∈ [0, 1] , t ∈ [0, M ] , x, Y T X ∈ X , Z t |ws | ds + h |wt | δ xt , Y Tt X , xt , YtT X ≤ exp ηM Y T X + h 0 Z t h |ws | ds + |wt | . (15.2.7) 0 TX

Proof. We may and we will assume that x, Y = (p1, Y p ). We will proceed as in (12.2.8), (12.2.9). Let w· be a Brownian motion with values in p. Instead of equation (12.2.8), and in view of (15.2.4), we consider the differential equation on G × p, g˙ = Y p , Y˙ p = hw, ˙ (15.2.8) Y0p = Y p .

g0 = 1,

We can rewrite the first equation in (15.2.8) in the form g˙ t = Y p + hwt ,

g0 = 1.

(15.2.9)

As in (12.2.9), we get xt , YtT X = gt , Ytp .

xt = pgt ,

(15.2.10)

Also the solution of (15.2.8) depends smoothly on h. Put ∂ g g ωs = ω gs . ∂h

(15.2.11)

By (2.1.8), we get ω g0 = 0.

ω˙ g = − [Y p , ω g ] + w, g

(15.2.12) p

k

Of course, we can split ω into its p and k components ω , ω , so that ωg = ωp + ωk .

(15.2.13)

Using (15.2.12) and Gronwall’s lemma, there exists η > 0 such that Z t Z t g ω t ≤ exp η |Ysp | ds |ws | ds. (15.2.14) 0

0

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Chapter 15

By (15.2.14), for 0 ≤ h ≤ 1, we get Z t Z t g ω t ≤ exp η t |Y p | + h |ws | ds. |ws | ds

(15.2.15)

By (15.2.15), for 0 ≤ h ≤ 1, Z t Z t d (xt , xt ) ≤ exp η t |Y p | + h |ws | ds h |ws | ds.

(15.2.16)

0

0

0

0

D denote the covariant derivative with respect to the Levi-Civita Let Dh connection. Then D p k p Y = ω t , Yt + wt . (15.2.17) Dh t TX Let ∇xt be the flat connection on T X that is associated with the trivialization of T X along radial geodesics centered at xt by parallel transport with respect to ∇T X . Set (15.2.18) ΓT X = ∇T X − ∇TxtX .

Using (3.7.8), we get TX Γ ≤ C.

(15.2.19) In (15.2.19), the norm of ΓTx X ∈ T ∗ X ⊗ End (T X) is calculated with respect to the natural metric of this vector bundle. D0 Let Dh denote covariant differentiation with respect to ∇TxtX . Clearly, D p ∂xt D0 p TX Y = Y − Γxt Ytp . (15.2.20) Dh t Dh t ∂h By (15.2.14)–(15.2.20), we get 0 Z t Z t D p p |ws | ds |ws | ds Dh Yt ≤ exp η t |Y | + h 0 0 (|Y p | + h |wt |) + |wt | . (15.2.21) By (15.2.16), (15.2.21), we get (15.2.7). The proof of our theorem is completed.

15.3 The scalar heat kernel on X away from Fγ = ia X (γ) Recall that the displacement function dγ was defined in (3.1.3). Theorem 15.3.1. Given β > 0, > 0, M > 0, ≤ M , there exist ηM > 0 00 0, C,M > 0, C,M > 0, Cγ,β,M > 0 such that for b ≥ 1, ≤ t ≤ M , TX x, Y ∈ X , if d (x, X (γ)) ≥ β, rX x, Y T X , γ x, Y T X b,t 2 0 ≤ C,M b4m exp −C,M d2γ (x) + Y T X −

00 Cγ,β,M

! T X 4 b . (15.3.1) exp −2ηM Y

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There exists η > 0 such that given β > 0, µ > 0, there exist C > 00 0, C 0 > 0, Cγ,β,µ > 0 such that for b ≥ 1, x, Y T X ∈ X , if d (x, X (γ)) ≤ TX T X β, Y − a ≥ µ, x, Y T X , γ x, Y T X rX b 2 00 ≤ Cb4m exp −C 0 Y T X − Cγ,β,µ exp −2η Y T X b4 . (15.3.2) 0 Given β > 0, > 0, M > 0, ≤ M, ν > 0, there exist C,M > 0, C,M > 00 TX 0 T X0 0, Cβ,M,ν > 0 such that for b ≥ 1, ≤ t ≤ M , if x, Y , x ,Y ∈ X, and Y T X ≤ ν, Y T X0 ≤ ν, if δ xt , Y Tt X , x0 , Y T X0 ≥ β, then x, Y T X , x0 , Y T X0 ≤ C,M b4m rX b,t 2 2 00 0 b4 . (15.3.3) exp −C,M d2 (x, x0 ) + Y T X + Y T X0 − Cβ,M,ν

Proof. As in (3.9.5), set x0 , Y T X0 = γ x, Y T X .

(15.3.4)

T X0 T X . Y = Y

(15.3.5)

Then Under the assumptions of the first part of our theorem, by equation (3.9.54) in Theorem 3.9.2, for 0 ≤ t ≤ M , (15.3.6) δ ϕt/2 x, Y T X , ϕ−t/2 x0 , Y T X0 ≥ Cγ,β,M . Set ϕt x, Y T X = xt , Y Tt X ,

ϕ−t x0 , Y T X0 = x0t , Y Tt X0 .

(15.3.7)

Then T X T X0 T X T X0 = Y . Y t = Y t = Y Equation (15.3.6) can be rewritten in the form X X0 δ xt/2 , Y Tt/2 , x0t/2 , Y Tt/2 ≥ Cγ,β,M .

(15.3.8)

(15.3.9)

Recall that if x0 ∈ X, the distance dx0 ((x, f ) , (x0 , f 0 )) on X was defined in (3.8.3). As in (3.9.33) let z t denote the middle point on the geodesic segment connecting xt and x0t . Then (15.3.9) can be written in the form X X dzt/2 xt/2 , Y Tt/2 , x0t/2 , Y Tt/2 ≥ Cγ,β,M . (15.3.10) X By (15.3.10), if z, Z T X ∈ X , either dzt/2 xt/2 , Y Tt/2 , z, Z T X ≥ X0 Cγ,β,M /2 or dzt/2 x0t/2 , Y Tt/2 , z, Z T X ≥ Cγ,β,M /2. Also by (3.8.9), dzt/2

X xt/2 , Y Tt/2 , z, Z T X X ≤ δ xt/2 , Y Tt/2 , z, Z T X + Cd xt/2 , z Y T X . (15.3.11)

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Chapter 15

By (12.9.25), we get rX x, Y T X , x0 , Y T X0 b,t Z x, Y T X , z, Z T X ≤ rX b,t/2 (z,Z T X )∈X X TX dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z rX z, Z T X , x0 , Y T X0 dzdZ T X b,t/2 Z rX x, Y T X , z, Z T X + b,t/2 (z,Z T X )∈X X0 TX dzt/2 ((x0t/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z X rb,t/2 z, Z T X , x0 , Y T X0 dzdZ T X . (15.3.12) By equation (15.1.2) in Theorem 15.1.1, for ≤ t ≤ M, b ≥ 1, Z x, Y T X , z, Z T X rX b,t/2 (z,Z T X )∈X TX X dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z X rb,t/2 z, Z T X , x0 , Y T X0 dzdZ T X Z 4m rX x, Y T X , z, Z T X ≤ C,M b b,t/2 (z,Z T X )∈X X TX dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z dzdZ T X . (15.3.13) We may and we will assume that x = x0 , with x0 = p1. By (12.9.9), Z (z,Z T X )∈X X TX dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z X rb,t/2 x, Y T X , z, Z T X dzdZ T X " ! Z mt 1 t/2 T X 2 = E exp − Y ds 4b2 2 0 # 1d x ,Y T X ,x ,Y T X ≥C z t/2 γ,β,M /2 t/2 t/2 t/2 t/2

. (15.3.14)

X TX By (15.3.11), if dzt/2 xt/2 , Y Tt/2 , xt/2 , Yt/2 ≥ Cγ,β,M /2, then X TX Cγ,β,M /2 ≤ δ xt/2 , Y Tt/2 , xt/2 , Yt/2 + Cd xt/2 , xt/2 Y T X . (15.3.15) By (15.2.7), (15.3.15), for b ≥ 1, 0 ≤ t ≤ M , there exist ηM > 0, Dγ,β,M > 0 such that for b ≥ 1, t ∈ [0, M ], under the above conditions, !!! Z t/2 TX 1 Dγ,β,M ≤ exp ηM Y + 2 |ws | ds + wt/2 b 0 ! Z t/2 1 |ws | ds + wt/2 . (15.3.16) b2 0

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By (15.3.14), (15.3.16), we get Z (z,Z T X )∈X

X TX dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z

rX b,t/2 "

1 b2

x, Y T X , z, Z T X dzdZ T X !! Z t/2 ηM |ws | ds + wt/2 ≤ CM P exp b2 0 # ! Z t/2 T X . (15.3.17) |ws | ds + wt/2 ≥ Dγ,β,M exp −ηM Y 0

For 0 ≤ t ≤ M , Z

t/2

|ws | ds + wt/2 ≤ CM

sup |ws | .

(15.3.18)

0≤s≤t/2

0

Note that given d > 0, there exists cd,M > 0 such that if x ≥ 0, 0 ≤ y ≤ d are such that exp (ηM x) x > y,

(15.3.19)

x > cd,M y.

(15.3.20)

then

By (15.3.18)–(15.3.20), for 0 ≤ t ≤ M , we obtain " !! ! Z t/2 Z t/2 ηM 1 P exp |ws | ds + wt/2 |ws | ds + wt/2 b2 b2 0 0 # T X ≥ Dγ,β,M exp −ηM Y " ≤P

sup

|ws | ≥

0≤s≤M/2

0 Dγ,β,M

# T X 2 b . (15.3.21) exp −ηM Y

By [ReY99, Proposition II (1.8)], there exist c > 0, c0 > 0 such that for t ≥ 0, ` ≥ 0, P sup |ws | ≥ ` ≤ c exp −c0 `2 /2t . (15.3.22) 0≤s≤t

00 By (15.3.22), there exists Dγ,β,M > 0 such that

" P

# sup 0≤s≤M/2

|ws | ≥

0 Dγ,β,M

exp −ηM Y T X b2

00 ≤ c exp −Cγ,β,M exp −2ηM Y T X b4 . (15.3.23)

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By (15.3.13), (15.3.17), (15.3.21), and (15.3.23), we get Z x, Y T X , z, Z T X rX b,t/2 (z,Z T X )∈X X TX dzt/2 ((xt/2 ,Y T ))≥Cγ,β,M /2 t/2 ),(z,Z X rb,t/2 z, Z T X , x0 , Y T X0 dzdZ T X 00 ≤ C,M b4m exp −Cγ,β,M exp −2ηM Y T X b4 . (15.3.24) Using (15.3.5) and interchanging the roles of x, Y T X and x0 , Y T X0 , we get a similar estimate for the second term in the right-hand side of (15.3.12). By combining equation (15.1.2) in Theorem 15.1.1 with the above estimates, we get (15.3.1). Now we establish (15.3.2), which is an estimate for t = 1. Instead of equation (3.9.54) in Theorem 3.9.2, we use equation (3.9.58) in Theorem 3.9.3, and we proceed exactly as before. Finally, we will establish (15.3.3). We no longer assume that x0 , Y T X0 = γ x, Y T X , but otherwise the previous notation will still be in force. For ρ > 0, set ( X , z, Z T X < ρ, Aρ = z, Z T X ∈ X , δ xt/2 , Y Tt/2 ) X0 , z, Z T X < ρ . (15.3.25) δ x0t/2 , Y Tt/2 We will show that under the assumptions of the third part of our theorem, for ρ > 0 small enough, Aρ is empty. Take z, Z T X ∈ Aρ . By (3.8.4), (15.3.25), we get X X0 , x0t/2 , Y Tt/2 ≤ 2ρ. (15.3.26) dz xt/2 , Y Tt/2 Moreover, X ϕt/2 xt/2 , Y Tt/2 = xt , Y Tt X ,

X0 ϕt/2 x0t/2 , Y Tt/2 = x0 , Y T X0 . (15.3.27)

Also by (15.3.27), we get T X T X , Y t/2 = Y

T X0 T X0 . Y t/2 = Y

(15.3.28)

Under of the third part of our theorem, we know that T X the assumptions Y ≤ ν, Y T X0 ≤ ν. By (3.8.10), (15.3.27), and (15.3.28), there exists ρβ,M,ν > 0 such that if ρ = ρβ,M,ν , if for a given t ∈ [0, M ], (15.3.26) holds, then β (15.3.29) δ xt , Y Tt X , x0 , Y T X0 ≤ . 2 By (15.3.29), under the assumptions of the third part of our theorem, Aρβ,M,ν is empty.

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By (12.9.25), as in (15.3.12), we get rX x, Y T X , x0 , Y T X0 b,t Z rX ≤ b,t/2 (z,Z T X )∈X

299

x, Y T X , z, Z T X

X TX δ ((xt/2 ,Y T ))≥ρβ,M,ν t/2 ),(z,Z

z, Z T X , x0 , Y T X0 dzdZ T X Z rX + x, Y T X , z, Z T X b,t/2 (z,Z T X )∈X X0 TX δ ((x0t/2 ,Y T ))≥ρβ,M,ν t/2 ),(z,Z X rb,t/2 z, Z T X , x0 , Y T X0 dzdZ T X . (15.3.30) rX b,t/2

Using (15.3.30) and proceeding as in (15.3.12)–(15.3.24), we get (15.3.3). The proof of our theorem is completed. Remark 15.3.2. A consequence of equation (15.3.3) in Theorem 15.3.1 and of its proof is that as should be the case, as b → +∞, the heat flow for AX b propagates more and more along the geodesic flow. 15.4 Gaussian estimates for rX b near ia X (γ) In the sequel, for x ∈ X, we will write aT X instead of aTx X . Theorem 15.4.1. Given ν > 0, there exist C > 0, C 0 > 0, Cν0 > 0 such that for b ≥ 1, f ∈ p⊥ (γ) , |f | ≤ 1, x = ργ (1, f ) , Y T X ∈ Tx X, Y T X ≤ ν, rX x, Y T X , γ x, Y T X ≤ Cb4m b 2 2 2 exp −C 0 Y T X − Cν0 |f | + Y T X − aT X b4 . (15.4.1) Proof. Let Cν be the positive constant appearing in Theorem 3.9.4. By equations (15.3.1) and (15.3.2) in Theorem 15.3.1, as long as |f | or Y T X − aT X stay from 0, (15.4.1) holds. So we may as well assume that |f | + T XawayT X Y − a ≤ c, with c such that 0 < cCν < 1. We use the notation in the first part of the proof of Theorem 15.3.1. By equation (3.9.60) in Theorem 3.9.4, instead of (15.3.6), we have δ ϕ1/2 x, Y T X , ϕ−1/2 x0 , Y T X0 ≥ Cν |f | + Y T X − aT X . (15.4.2)

The proof of our theorem continues exactly as the proof of the first part of Theorem 15.3.1. 15.5 The scalar heat kernel on Xb away from Fbγ = bia N k −1

We use the notation of sections 12.10 and 12.11. In particular, for t > 0, 0 0 b associated with exp −tAX , rX ((x, Y ) , (x , Y )) is the smooth kernel on X b b,t X X 0 0 and rb,t ((x, Y ) , (x , Y )) is the smooth kernel associated with exp −tAb .

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Chapter 15

Set X rX b = rb,1 ,

X rX b = rb,1 .

(15.5.1)

By (12.11.9), (15.1.1), and (15.5.1), we get X rX b ≤ rb .

(15.5.2)

0 Theorem 15.5.1. Given > 0, M > 0, ≤ M , there exist C,M > 0, C,M > 0 0 b 0 such that for b ≥ 1, ≤ t ≤ M , (x, Y ) , (x , Y ) ∈ X , 0 0 rX b,t ((x, Y ) , (x , Y ))

2 2 0 d2 (x, x0 ) + |Y | + |Y 0 | . (15.5.3) ≤ C,M b4m+2n exp −C,M Proof. We use the uniform bounds on the kernel rX b,t that were established in Theorem 12.10.3, and we proceed as in the proof of Theorem 15.1.1. As in (15.2.1), we still take h = 1/b2 . We consider equation (12.10.14), in which the starting point (x, Y ) ∈ Xb is arbitrary. Let w· = w·T X ⊕ w·N be a Brownian motion valued in Tx X ⊕ Nx . Equation (12.10.14) takes the form x˙ = Y T X , Y˙ = hw, ˙ (15.5.4) x0 = x,

Y0 = Y.

We denote by (x· , Y · ) the solution of (15.5.4) for h = 0. Recall that the pseudodistance δ on Xb was defined in section 3.8. As usual, TX N we write Y = Y , Y . We have the following extension of Theorem 15.2.1. Theorem 15.5.2. Given M > 0, there exists ηM > 0 such that for h ∈ [0, 1] , t ∈ [0, M ] , (x, Y ) ∈ Xb, Z t δ ((xt , Y t ) , (xt , Yt )) ≤ exp ηM Y T X + h |ws | ds + h |wt | 0 Z t h |ws | ds + |wt | 1 + Y N . (15.5.5) 0

Proof. With respect to Theorem 15.2.1, we only have to deal with the extra component Y N of Y . We use the notation on the proof of Theorem 15.2.1. We may and we will assume that x0 = p1. Instead of (15.2.8), we now have g˙ = Y p , Y˙ = hw, ˙ (15.5.6) g0 = 1,

Y0 = Y.

In (15.5.6), we split Y· in the form

Y· = Y·p + Y·k .

(15.5.7)

D Let dh also denote covariant differentiation with respect to the connection N ∇ . As in (15.2.17), we get D k k k Y = ω t , Yt + wtk . (15.5.8) Dh t

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Let ∇N xt be the flat connection on N that is associated with the trivialization of N along radial geodesics centered at xt by parallel transport with respect to the connection ∇N . Set ΓN = ∇N − ∇N xt .

(15.5.9)

A formula for ΓN is given in (3.7.9). As in (15.2.19), by (3.7.8), we obtain N Γ ≤ C. (15.5.10) As in (15.2.20), we get D0 k D k Y = Y − ΓN xt Dh Dh

∂xt ∂h

Ytk .

(15.5.11)

By equation (15.2.7) in Theorem 15.2.1, by (15.2.15), and (15.5.8)–(15.5.11), we get (15.5.5). The proof of our theorem is completed. We use the same conventions as before Theorems 3.9.5 and 9.1.1. In particular, Ad k −1 acts on the fibres on N . Also we still write aT X instead of aTx X . Theorem 15.5.3. Given β > 0, > 0, M > 0, ≤ M , there exist ηM > 0 00 0, C,M > 0, C,M > 0, Cγ,β,M > 0 such that for b ≥ 1, ≤ t ≤ M , (x, Y ) ∈ b X , if d (x, X (γ)) ≥ β, 2 4m+2n 0 rX exp −C,M d2γ (x) + |Y | b,t ((x, Y ) , γ (x, Y )) ≤ C,M b −

00 Cγ,β,M

! T X 4 exp −2ηM Y b . (15.5.12)

There exists η > 0 such that given β > 0, µ > 0, there exist C > 00 0, C 0 > 0, Cγ,β,µ > 0 such that for b ≥ 1, (x, Y ) ∈ Xb, if d (x, X (γ)) ≤ TX T X − a ≥ µ, β, Y rX b ((x, Y ) , γ (x, Y )) 2 00 ≤ Cb4m+2n exp −C 0 |Y | − Cγ,β,µ exp −2η Y T X b4 . (15.5.13) 00 Given ν > 0, ρ > 0, there exist C > 0, C 0 > 0, Cγ,ν,ρ > 0 such that for β ∈ TX ⊥ ) , Y − aT X ≤ ]0, 1] small enough, b ≥ 1,f ∈ p (γ) , |f | ≤ β, x = ργ (1, TfX N −1 N N β, Y ≥ ρ, if Ad k −1 Y ≥ ν Y , or if a , Y N ≥ ν Y N , then 4m+2n rX b ((x, Y ) , γ (x, Y )) ≤ Cb 2 2 −1 4 00 exp −C 0 Y N − Cγ,ν,ρ 1 + Y N b . (15.5.14)

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0 Given β > 0, > 0, M > 0, ≤ M, ν > 0, there exist C,M > 0, C,M > 00 0 0 b 0, Cβ,M,ν > 0 such that for b ≥ 1, ≤ t ≤ M , if (x, Y ) , (x , Y ) ∈ X , and |Y | ≤ ν, |Y 0 | ≤ ν, if δ ((xt , Y t ) , (x0 , Y 0 )) ≥ β, then 0 0 rX b,t ((x, Y ) , (x , Y ))

2 2 00 0 d2 (x, x0 ) + |Y | + |Y 0 | − Cβ,M,ν b4 . ≤ C,M b4m+2n exp −C,M (15.5.15) Proof. As in (3.9.61), set (x0 , Y 0 ) = γ (x, Y ) .

(15.5.16)

To obtain (15.5.12), we proceed exactly as in the proof of Theorem 15.3.1. Instead of (15.3.12), we get 0 0 rX b,t ((x, Y ) , (x , Y )) Z rX ≤ b b,t/2 ((x, Y ) , (z, Z)) (z,Z)∈X TX TX dzt/2 ((xt/2 ,Y t/2 ),(z,Z ))≥Cγ,β,M /2 0 0 rX b,t/2 ((z, Z) , (x , Y )) dzdZ

Z

X

+

rb,t/2 ((x, Y ) , (z, Z)) (z,Z T X )∈Xb ),(z,Z T X ))≥Cγ,β,M /2

X0 x0t/2 ,Y T t/2

dzt/2 ((

0 0 rX b,t/2 ((z, Z) , (x , Y )) dzdZ.

(15.5.17)

By equation (15.5.3) in Theorem 15.5.1, instead of (15.3.13), we get Z rX b b,t/2 ((x, Y ) , (z, Z)) (z,Z)∈X X TX dzt/2 ((xt/2 ,Y T , z,Z ≥C /2 )) γ,β,M t/2 ) ( 0 0 rX b,t/2 ((z, Z) , (x , Y )) dzdZ Z

≤ C,M b4m+2n

b (z,Z)∈X X dzt/2 ((xt/2 ,Y T t/2 ),(z,Z))≥Cγ,β,M /2 TX rX . (15.5.18) b,t/2 ((x, Y ) , (z, Z)) dzdZ

Now we use equation (12.10.15) instead of equation (12.9.9), and by proceeding as in the proof of Theorem 15.3.1, we get an estimate for (15.5.18) similar to (15.3.24), with b4m replaced by b4m+2n . The second term in the right-hand side of (15.5.18) can also be estimated by the same method. Combining equation (15.5.3) with these estimates, we get (15.5.12). By proceeding as in the proof of Theorem 15.3.1, we also obtain (15.5.13). Note that in the proof of (15.5.12) and (15.5.13), only Theorem 15.2.1 has been used, and not the more general Theorem 15.5.2. Now we concentrate on the proof of (15.5.14). Under the first set of asX sumptions, we will prove the estimate (15.5.14) in which rX b is replaced by rb .

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Using (15.5.2), this will also give (15.5.14) for rX b . We still use the notation in (15.5.16). Set ϕt (x, Y ) = (xt , Y t ) , ϕt (x0 , Y 0 ) = x0t , Y 0t . (15.5.19) N0 0 In particular Y N denote the components · ,Y · ·. N in N of Y · , Y−1 Take ν > 0, ρ > 0, and assume that Y ≥ ρ, Ad k − 1 Y N ≥ N ν Y . By equation (3.9.63) in Theorem 3.9.5, there exists βγ,ν > 0 such that for |f | ≤ βγ,ν , Y T X − aT X ≤ βγ,ν , we get 0 ν N ν x1/2 N 0 Y ≥ ρ. τx Y 1/2 − Y N (15.5.20) 1/2 ≥ 1/2 2 2

Let z 1/2 denote the middle point on the geodesic segment connecting x1/2 and x01/2 . We can rewrite (15.5.20) in the form 0 x1/2 N 0 ν N ν τz Y 1/2 − τzx1/2 Y N Y ≥ ρ. 1/2 ≥ 1/2 1/2 2 2

(15.5.21)

By (15.5.21), we get Z b (z,Z)∈X

X 0 0 rX b,1/2 ((x, Y ) , (z, Z)) rb,1/2 ((z, Z) , (x , Y )) dzdZ Z b ≤ rX (z,Z)∈X b,1/2 ((x, Y ) , (z, Z)) z x1/2 N N τ z1/2 Z −τz1/2 Y 1/2 ≥νρ/4

0 0 rX b,1/2 ((z, Z) , (x , Y )) dzdZ

Z +

b (z,Z)∈X rX b,1/2 x0 z 1/2 0 Z N −τz1/2 Y N τz ≥νρ/4 1/2 1/2

((x, Y ) , (z, Z))

0 0 rX b,1/2 ((z, Z) , (x , Y )) dzdZ.

(15.5.22)

As explained in section 3.8, equations (3.8.5)–(3.8.10) are also valid when f, f 0 ∈ N , so that x1/2 z z N Y N . (15.5.23) −YN τz1/2 Z N − τz1/2 Y N 1/2 ≤ τx1/2 Z 1/2 + Cd x1/2 , z By (15.5.5), (15.5.23), and keeping in mind that Y T X remains uniformly bounded, we obtain ! |ws | ds + w1/2 0 ! Z 1/2 |ws | ds + w1/2 1 + Y N . (15.5.24)

x1/2 x1/2 N τz1/2 Y1/2 − τz1/2 Y N 1/2 ≤ C exp 1 b2

0

η b2

Z

1/2

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By (12.10.15), (15.5.24), and proceeding as in (15.3.17)–(15.3.23), we get Z b rX (z,Z)∈X b,1/2 ((x, Y ) , (z, Z)) z x1/2 N N τ z1/2 Z −τz1/2 Y 1/2 ≥νρ/4

0 0 rX b,1/2 ((z, Z) , (x , Y )) dzdZ N 2 −1 4 4m+2n 00 ≤ Cb exp −Cγ,ν,ρ 1 + Y b . (15.5.25)

The second integral in the right-hand side of (15.5.22) can be dominated in the same way. Combining these bounds with Theorem 15.5.1, we obtain equation (15.5.14) for rX b , from which, as explained before, (15.5.14) follows for rX b . Now we will establish (15.5.14) under our second set of assumptions. We still use the notation in (15.5.16). By the analogue of equation (12.9.25) for the kernel rX b,t and by equation (12.11.10) in Theorem 12.11.2, we get Z 0 0 4m+2n ((x, Y ) , (x , Y )) ≤ Cb rX rX (15.5.26) b b,1/2 ((x, Y ) , (z, Z)) dzdZ. b X

ν > 0, for β > 0 small enough, if aT X , Y N ≥ ν Y N and Given Y T X − aT X ≤ β, then N T X ν N Y ,Y ≥ Y . (15.5.27) 2 In the sequel, we may assume that (15.5.27) holds, and we will estimate the right-hand side of (15.5.26). We may and we will assume that x = x0 . By (12.11.8), we obtain Z rX b,1/2 ((x, Y ) , (z, Z)) dzdZ X " !# Z Z (m + n) 1 1/2 b4 1/2 N T X 2 2 = E exp − |Y | ds − Y ,Y ds . 4b2 2 0 2 0 (15.5.28) Since x = x0 , we can take Y g ∈ g with Y g = Y p + Y k , Y p ∈ p, Y k ∈ k, so that Y g , Y p , Y k represent Y, Y T X , Y N . By (15.5.27), we get k p ν k Y , Y ≥ Y . (15.5.29) 2 Let w·g = w·p + w·k be a Brownian motion in g = p ⊕ k with w0 = 0. Let E still denote the corresponding expectation. Put wsg . b2

(15.5.30)

Y·g = Y·p + Y·k .

(15.5.31)

Ysg = Y g + The process Y·g splits as

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Also equation (15.5.28) can be written in the form Z rX b,1/2 ((x, Y ) , (z, Z)) dzdZ X " !# Z Z (m + n) 1 1/2 g 2 b4 1/2 k p 2 = E exp − |Ys | ds − Ys , Ys ds . 4b2 2 0 2 0 (15.5.32) By (15.5.30), since |Y p | remains uniformly bounded, we get 0 0 k p k p C 0 k p Y |ws | − C wsk − C |wsp | wsk . (15.5.33) Ys , Ys ≥ Y , Y − 2 2 4 b b b By (15.5.29), (15.5.33), we obtain 0 0 0 k p Ys , Ys ≥ ν − C |wsp | Y k − C wsk − C |wsp | wsk . (15.5.34) 2 2 4 2 b b b By (15.3.22), there exists C 00 > 0 such that " # b2 ν p P sup |ws | ≥ ≤ c exp −C 00 b4 . 0 4C 0≤s≤1/2 Moreover, if

sup 0≤s≤1/2

|wsp |
0, Cν,ρ > 0 such that k p νρ P inf ≤ c exp −Cν,ρ b4 . (15.5.37) Ys , Ys ≤ 8 0≤s≤1/2

By (15.5.28) and (15.5.37), we conclude that given ν > 0, ρ > 0, there exist c > 0, Cν,ρ > 0 such that for β ∈]0, 1] small enough, under the conditions of the second part of the statement leading to (15.5.14), Z 4 rX . (15.5.38) b,1/2 ((x, Y ) , (z, Z)) dzdZ ≤ c exp −Cν,ρ b X

By equation (15.5.3) in Theorem 15.5.1, by (15.5.26), and by (15.5.38), we get a stronger statement than (15.5.14) under the second set of assumptions, 2 −1 since in this case, the factor 1 + Y N is unnecessary. As to the proof of equation (15.5.15), using Theorem 15.5.2 instead of Theorem 15.2.1, it is the same as the proof of equation (15.3.3) in Theorem 15.3.1. The proof of our theorem is completed. Remark 15.5.4. The conclusions of Remark 15.3.2 remain valid for the opb erator AX b , if we replace the geodesic flow in X by the geodesic flow in X . By (12.11.9), this is still true for the operator AX . b

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Note that for b ≥ 1, 2 −1 4 2 2 00 1 + Y N b ≥ cγ,ν,ρ Y N + b2 . C 0 Y N + Cγ,ν,ρ

(15.5.39)

2 2 Indeed, (15.5.39) is trivially true for Y N ≥ b2 and for Y N < b2 . By (15.5.39), we can then give a weaker version of (15.5.14). The estimates (15.5.14) and (15.5.15) will not be used in the sequel. 15.6 Estimates on the scalar heat kernel on Xb near bia N k −1

−1 b Now, we will give an upper bound for rX . b ((x, Y ) , γ (x, Y )) near ia N k Recall that we write aT X instead of aTx X . Theorem 15.6.1. There exist c > 0, C > 0, Cγ0 > 0 such that for b ≥ 1, f ∈ p⊥ (γ) , |f | ≤ 1, x = ργ (1, f ) , Y ∈ (T X ⊕ N )x , Y T X − aT X ≤ 1, 2 4m+2n exp −C Y N rX b ((x, Y ) , γ (x, Y )) ≤ cb 2 2 − Cγ0 |f | + Y T X − aT X b4 − Cγ0 Ad k −1 − 1 Y N b2 ! 0 TX N 2 b . (15.6.1) − Cγ a , Y Proof. We can handle f and Y T X by the same methods as in the proof of Theorem 15.4.1. So we concentrate on how to deal with Y N . We will use arguments that were already used in the proof of Theorem 15.5.3. Let C > 0 be a fixed constant. We claim that if Ad k −1 − 1 Y N ≤ C |f | + Y T X − aT X Y N , (15.6.2) then the third term in the right-hand side of (15.6.1) can be easily obtained. Indeed, by (15.6.2), we get 2 2 2 b2 Ad k −1 − 1 Y N ≤ C 0 b4 |f | + Y T X − aT X + Y N , (15.6.3) so that the first two exponential terms in the right-hand side of (15.6.1) already incorporate the third term. Therefore in the sequel, we may as well assume that for a given C > 0, Ad k −1 − 1 Y N > C |f | + Y T X − aT X Y N . (15.6.4) Let cγ > 0 be the constant that appears in Theorem 3.9.5. By taking C = 2cγ in (15.6.4), by equation (3.9.63) in Theorem 3.9.5, there exists C 0 > 0 such that for |f | ≤ 1, Y T X − aT X ≤ 1, we get 0 x1/2 N 0 0 τx Y 1/2 − Y N Ad k −1 − 1 Y N . (15.6.5) 1/2 ≥ C 1/2

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Using (15.6.5), we can proceed as in the proofs of Theorems 15.3.1 and 15.5.3, and we get the estimate, 0 0 4m+2n rX b ((x, Y ) , (x , Y )) ≤ Cb N 2 −1 4 N 2 0 −1 1+ Y b . (15.6.6) exp −C Ad k −1 Y

2 −1 Note that as in (15.5.25), the term 1 + Y N is forced upon us by equation (15.5.5). If 2 (15.6.7) 1 + Y N > Ad k −1 − 1 Y N b2 , 2 then the presence of the term Y N in the right-hand side of (15.6.1) can be made to incorporate the term Ad k −1 − 1 Y N b2 . If 2 (15.6.8) 1 + Y N ≤ Ad k −1 − 1 Y N b2 , the term (15.6.6) is enough to produce the required exponential in the righthand side of of (15.6.1). We will now obtain the remaining term in (15.6.1), which will be more difficult. Take C > 0. We claim that when T X N a , Y ≤ C Y T X − aT X , Y N , (15.6.9) then the remaining estimate in (15.6.1) follows easily. Indeed from (15.6.9), we get 2 2 (15.6.10) b2 aT X , Y N ≤ C 0 b4 Y T X − aT X + Y N . Therefore, the first two terms in the right-hand side of (15.6.1) can be made to absorb the last term. In the sequel, we may as well assume that C > 0 is a fixed constant and that T X N a , Y > C Y T X − aT X , Y N . (15.6.11) Recall that x = ργ (1, f ). We use the notation of equation (15.5.4). Then wsT X , b2 + YsN .

YsT X = Y T X + Ys = YsT X

YsN = Y N +

wsN , b2

(15.6.12)

Equation (15.5.28) still holds. Observe that for u, v ∈ Rn , α ≥ 1, 2

2

2

|u + v| ≥ (1 − 1/α) |u| − (α − 1) |v| .

(15.6.13)

In particular, for 1 ≤ α ≤ 2, 2

|u + v| ≥ (α − 1)

1 2 2 |u| − |v| . 2

(15.6.14)

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Set

TX

w· =

sup 0≤s≤1/2

TX ws ,

N

w· =

sup 0≤s≤1/2

N ws .

(15.6.15)

Let η ∈]0, 1] be a constant whose precise value will be chosen later. Set η α=1+ (15.6.16) 2 2. T X 1 + kw· k + kw·N k By (15.6.12), (15.6.14), we get 1 T X N 2 a ,Y 2 TX TX ! N N 2 w w w w s s − aT X , Y N + Y T X , s2 + ,Y N + , s2 . b b2 b2 b

T X N 2 Ys , Ys ≥ (α − 1) − Y T X

(15.6.17) By (15.6.17), we obtain Z 1/2 T X N 2 Ys , Ys ds 0 T X α − 1 1 T X N 2 0 TX N 2 ≥ a ,Y −C Y − a ,Y 2 2 −

C0 (α − 1) 2

1 N 2

w·T X 2 + 1 Y T X 2 w·N 2 Y 4 4 b b !

1 T X 2 N 2

+ 8 w· w· . (15.6.18) b

By choosing the constant C > 0 adequately in (15.6.11), we deduce from (15.6.19) that we may as well assume that Z 1/2 T X N 2 4 Ys , Ys ds ≥ 1 (α − 1) aT X , Y N 2 b4 b 8 0 −

2

2 2

2 C0 (α − 1) Y N w·T X + Y T X w·N 2 !

1 T X 2 N 2

w· . (15.6.19) + 4 w· b

We will use the estimate (15.6.19) to obtain an upper bound for the righthand side of (15.5.28). We will control the four terms in the right-hand side of (15.6.19). The first term in the second line of (15.6.19) is the most annoying, and this is where the right choice of α in (15.6.16) is crucial. Indeed, let C > 0 be the positive constant that appears in (15.6.1). Then 2

2 2 2 C 0 C0 (α − 1) Y N w·T X ≥ C − η Y N . (15.6.20) C Y N − 2 2

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By taking η ∈]0, 1] small enough, we deduce from (15.6.20) that

2 2 2 2 C 0 C C Y N − (α − 1) Y N w·T X ≥ Y N . 2 2

(15.6.21)

Combining equation (15.5.28) with (15.6.21) indicates that the first term in 2 the second line of (15.6.19) can be absorbed by the quadratic term Y N already appearing in the right-hand side of (15.6.1). Of course, this fixes the constant η once and for all. Recall that Y T X remains uniformly bounded, so that 2

2 (α − 1) Y T X w·N ≤ C 00 . (15.6.22) Therefore the second term in the second line of (15.6.19) is irrelevant in our estimate. Ultimately, in the right-hand side of (15.6.19), what remains to control is

1 T X N 2 4 C 0 1 T X 2 N 2

b − 4 w· a ,Y w· . (15.6.23) (α − 1) 2 4 b Using [ReY99, Proposition II (1.8)] as in (15.3.22), for y ≥ 0, we get

√ P w·T X w·N ≥ y ≤ P w·T X ≥ y

√ + P w·N ≥ y ≤ c exp (−c0 y) . (15.6.24) By (15.6.24), given d > 0, we get

P w·T X w·N ≥ d aT X , Y N b4 ≤ c exp −c0 d aT X , Y N b4 . (15.6.25) By (15.6.25), we find that to find an upper bound for (15.5.28) compatible with (15.6.1), we may as well replace the expression in (15.6.23) by 2 1 (α − 1) aT X , Y N b4 . 16

(15.6.26)

By still using (15.3.22), we obtain h

1/2 i P w·T X + w·N ≥ aT X , Y N b ≤ c exp −c0 aT X , Y N b2 . (15.6.27) By (15.6.16), (15.6.26), we have to find a proper lower bound for (15.6.26)

1/2 on the set w·T X + w·N < aT X , Y N b. Using (15.6.16), on this set, 2 (α − 1) aT X , Y N b4 ≥ η aT X , Y N b2 − 1 . (15.6.28) Equation (15.6.28) still produces the last exponential term in the right-hand side of (15.6.1). The proof of our theorem is completed. Remark 15.6.2. It is remarkable that in the uniform estimate (15.6.1), two important terms are non-Gaussian.

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15.7 A proof of Theorem 9.1.1 Theorem 9.1.1 is just the analogue of Theorems 15.5.1, 15.5.3, and 15.6.1 for . To prove this theorem, we will combine the methods of the heat kernel q X b,t section 14.4 with the above results. b Recall that the operator LX A,b on X is given by (9.1.3), and that fort > 0, X 0 0 q b,t ((x, Y ) , (x , Y )) is the smooth kernel associated with exp −tLX A,b . Also instead of q X . when t = 1, we write q X b b,1 A first step in the proof of Theorem 9.1.1 is to obtain an analogue of . equation (14.4.3) for exp −tLX A,b Take (x· , Y· ) as in (12.10.14). Instead of equation (14.4.2), we consider the differential equation " · ∗ ∗ dU N Λ (T X⊕N ) +b c ad Y T X =U − ds b2 # TX N E N − c ad Y + iθad Y − iρ Y , (15.7.1) U0 = 1. Now we give an analogue of Theorem 14.4.1. Theorem 15.7.1. Let s ∈ C ∞,c Xb, π b∗ (Λ· (T ∗ X ⊕ N ∗ ) ⊗ F ) . For t > 0, the following identity holds: "

exp

−tLX A,b

Z (m + n) t b4 t N T X 2 − ds Y ,Y 2b2 2 0 ! # Z 1 t 2 − |Y | ds − tA Ut τ0t s (xt , Yt ) . (15.7.2) 2 0

s (x0 , Y ) = E exp

Proof. The proof of our theorem is the same as the proof of Theorem 14.4.1.

By proceeding as in the proof of Proposition 14.5.1 and of Theorem 14.5.2, we find that for b ≥ 1, Z t |Ut | ≤ exp ct + c0 |Y | ds . (15.7.3) 0

By (15.7.3), for 0 ≤ t ≤ M , |Ut | ≤ CM exp

Z t 1 2 |Y | ds . 4 0

(15.7.4)

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By (15.7.2), (15.7.4), we obtain exp −tLX A,b s (x0 , Y ) " Z (m + n) t b4 t N T X 2 ≤ CM E exp − Y ,Y ds 2b2 2 0 ! # Z 1 t 2 |Y | ds |s (xt , Yt )| . (15.7.5) − 4 0 b Let AX0 b be the scalar operator on X , b4 N T X 2 1 1 T X⊕N 1 m+n 2 Y + ∆ − ∇Y T X . AX0 = , Y − + |Y | − b 2 2 b4 2 b2 (15.7.6) For t > 0, let rX0 ((x, Y ) , (x0 , Y 0 )) be the smooth kernel associated with b,t the operator exp −tAX0 . Using the Itô calculus as in (12.11.8), if F ∈ b C ∞,c Xb, R , we get exp

−tAX0 b

"

Z (m + n) t b4 t N T X 2 F (x0 , Y ) = E exp − Y ,Y ds 2b2 2 0 ! # Z 1 t 2 − |Y | ds F (xt , Yt ) . (15.7.7) 4 0

By (15.7.5), (15.7.7), we obtain exp −tLX s (x0 , Y ) ≤ CM exp −tAX0 |s| (x0 , Y ) . A,b b By (15.7.8), we find that X 0 0 q b,t ((x0 , Y ) , (x0 , Y 0 )) ≤ CM rX0 b,t ((x0 , Y ) , (x , Y )) .

(15.7.8)

(15.7.9)

Of course, in (15.7.9), (x0 , Y ) can be replaced by an arbitrary (x, Y ) ∈ Xb. X0 Comparison of equations (12.11.7) and (15.7.6) for AX b and Ab shows that 2 1 1 only the coefficient of |Y | differ, this coefficient being 2 for AX b and 4 for AX0 b . However, a quick inspection of the arguments in the present chapter X0 shows that the arguments we gave to estimate rX b,t remain valid for rb,t . By X0 Theorems 15.5.1, 15.5.3, and 15.6.1 applied to rb,t , and by (15.7.9), we get Theorem 9.1.1.

15.8 A proof of Theorem 9.1.3 Now we establish Theorem 9.1.3. This result is just the analogue of Theorems 12.9.1 and 12.10.3 for the kernel q X . The proof consists of using the same b,t methods in the proofs of these results, combined with the methods of the

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Malliavin calculus as in section 14.9, while still using the estimates (15.7.3) and (15.7.4). Because here b ≥ 1, no sophisticated arguments like the ones used in section 14.9 are needed, the problem in that section being that as b → 0, some of the matrix terms became singular.

15.9 A proof of Theorem 9.5.6 The proof of Theorem 9.5.6 is essentially the same as the proof of the cor, which were established responding inequalities (9.1.10) and (9.2.12) for q X b in section 15.7. Let us now give more details. We use the notation of chapter 9. First, note that by (9.5.3), with respect X to LX A,b , LA,b contains the extra term α. By equation (9.4.1) in Proposition 9.4.1, the norm of this term is uniformly bounded, so that it can easily be handled by the Feynman-Kac arguments of sections 14.4 and 15.7. · ∗ ∗ b Recall that the flat connection ∇Λ (T X⊕N ),f ∗,f on Λ· (T ∗ X ⊕ N ∗ ) was defined in Definition 2.4.1. By Theorem 2.13.2, equation (2.4.5) gives the b )),f ∗,fb C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F splitting of the operator ∇Y T X into its skewadjoint and self-adjoint components with respect to the Hermitian product · ∗ ∗ b h i in (2.13.6). However, the restriction of ∇Λ (T X⊕N ),f ∗,f to the extra ∗ Λ· (T X ⊕ N ) (γ) does not preserve its metric. Let ∇(T X⊕N )(γ),f be the obvious flat connection on (T X ⊕ N ) (γ) that one · ∗ obtains via the isomorphism (T X ⊕ N ) (γ) = z (γ), and let ∇Λ (T X⊕N )(γ) ,f ∗ be the corresponding flat connection on Λ· (T X ⊕ N ) (γ) . Let Q be the orthogonal projection from T X ⊕ N on (T X ⊕ N ) (γ) with respect to the scalar product of T X ⊕ N . Let ∇(T X⊕N )(γ) be the projection of ∇T X⊕N on (T X ⊕ N ) (γ) with respect to this scalar product, and let · ∗ ∗ ∇Λ ((T X⊕N )(γ) ) be the induced connection on Λ· (T X ⊕ N ) (γ) . If e ∈ T X, set adQ (e) = Qad (e) Q. (15.9.1) Then adQ (e) ∈ End ((T X ⊕ N ) (γ)) is self-adjoint. By projecting the two sides of (2.2.2) on (T X ⊕ N ) (γ), we get ∇(T X⊕N )(γ),f = ∇(T X⊕N )(γ) + adQ (·) . (15.9.2) ∗ · f Let adQ (e) be the obvious action of −adQ (e) on Λ (T X ⊕ N ) (γ) .Then adQ (e) is self-adjoint. By (15.9.2), we get ·

∇Λ (T X⊕N )(γ) Let ∇

∗

,f

·

∗

= ∇Λ (T X⊕N )(γ) + adQ (·) .

(15.9.3)

b · ((T X⊕N )(γ)∗ ),f ∗ ,fb Λ· (T ∗ X⊕N ∗ )⊗Λ

be the connection on ∗ · b Λ (T X ⊕ N ) ⊗Λ (T X ⊕ N ) (γ) · X ∗ ∗ b · ∗ that is induced by ∇Λ (T ⊕N ),f ,f and ∇Λ ((T X⊕N )(γ) ),f . By equations (2.4.5), (15.9.3), we get ·

·

∇Λ (T

∗

∗

∗

b · ((T X⊕N )(γ)∗ ),f ∗ ,fb X⊕N ∗ )⊗Λ

·

∗

∗

·

∗

= ∇Λ (T X⊕N )⊗Λ ((T X⊕N )(γ) ) − c (ad (·)) + b c (ad (·)) + adQ (·) . (15.9.4) b

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Using (9.5.3) and (15.9.4), we find that LX A,b is of exactly the same form X as LA,b . The same arguments as in the proof of (9.2.12) lead us to (9.5.7). The proof of Theorem 9.5.6 is completed.

15.10 A proof of Theorem 9.11.1 We will only briefly sketch the proof of Theorem 9.11.1, which can be split into two main steps: 1. If Y0k ∈ k (γ) , (y, Y ) ∈ p ⊕ g, for b ≥ 1, pX ((y, Y ) , (y 0 , Y 0 )) and its a,b,Y0k derivatives of arbitrary order in (y 0 , Y 0 ) ∈ p × g are uniformly bounded on compact subsets of p × g. Here the bounds may well depend on Y0k . b · z (γ)∗ ⊗ E , then 2. If s (y 0 , Y 0 ) ∈ C ∞,c p × g, Λ· (g∗ ) ⊗Λ X X exp −Pa,A,b,Y s (y, Y ) → exp −Pa,A,∞,Y s (y, Y ) . (15.10.1) k k 0

0

From (1) and (2), we get (9.11.3), i.e., we have established Theorem 9.11.1. Let us explain how to get step (1). Take β > 0. By Theorem 9.1.3, we know that for b ≥ 1, ≤ t ≤ M , if |y| ≤ βb2 , |y 0 | ≤ βb2 , 0 2 −a y/b2 2 y /b 0 2 e e b−4m−2n q X , Y /b , e , Y /b b,t together with its derivatives of any order in the variables (y 0 , Y 0 ) are uniformly bounded. We fix Y0k ∈ k (γ) , N > 0. We claim that for b ≥ 1, ≤ t ≤ M, |y| ≤ N, |y 0 | ≤ N , 0 2 −a y/b2 TX k 2 y /b TX k 0 2 e e , a + Y + Y /b , e , a + Y + Y /b b−4m−2n q X 0 0 b,t (15.10.2) and its derivatives of any order in the variables (y 0 , Y 0 ) remain uniformly bounded by constants depending on Y0k . Indeed aT X + Y0k is now viewed as a section of T X ⊕ N , which is uniformly bounded together with its covariant derivatives of arbitrary order in the region |y 0 | ≤ β, the bounds still depending on the choice of Y0k . Since in the second term in (15.10.2), this section is evaluated at y 0 /b2 , covariant derivatives introduce the right power of b−2 . X Recall that the kernel qX b,t was defined in Definition 9.5.3, and that qb = X qb,1 . We claim that the considerations we made before also apply to the kernel qX b,t . The argument is exactly the same as in the proof of Theorem 9.5.6 that was given in section 15.9. It uses the estimate (9.4.1) in Proposition 9.4.1, and also equation (9.5.3) for LX A,b . By combining (9.9.7) and the above arguments, step (1) is completed. X We move to step (2). We start from equation (9.8.5) for Oa,A,b , from which X we get a corresponding equation for Pa,A,b,Y k in (9.9.5). 0 As indicated in section 9.9, the vector bundles T X, N have been trivialized by an Euclidean trivialization, and identified with p, k. On the other hand,

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the coordinate system y ∈ p → ey p1 ∈ X provides another trivialization of T X, which does not preserve the metric. Using the coordinate system y, the canonical identification of the fibre T X with the tangent bundle to X is induced by an isomorphism ry : p → p, which coincides with the identity at y = 0. Let ΓT X⊕N be the connection form for ∇T X⊕N in the Euclidean trivialization of T X ⊕ N . Then ΓT X⊕N is a 1-form with values in antisymmetric sections of End (p ⊕ k) that preserve p, k. In the coordinates (y, Y ), the scalar operator ∇Y T X can be written in the form V ∇Y T X = ∇H rY p − ∇ΓT X⊕N (rY p )Y .

(15.10.3)

When making the change of variables indicated in (9.9.4), (9.9.5), the operator in (15.10.3) is changed into the operator V ∇H ry/b2 Y p − ∇ΓT X⊕N (r

y/b2 Y

y/b2

p

)(Y0k +Y /b2 ) .

(15.10.4)

X,s X Let Pa,A,b,Y k be the scalar part of Pa,A,b,Y k . By (9.8.5), (9.9.4), (9.9.5), 0 0 and (9.10.8), we get i 2 1 h k X,s X p Pa,A,b,Y Y0 + Y k /b2 , b2 aTy/b 2 + Y k = 0 2 2 1 1 X k 2 −∆p⊕k + aTy/b (m + n) − ∇H + 2 + Y0 + Y /b − ry/b2 Y p 2 b2

+ ∇VΓT X⊕N (r y/b2

y/b2 Y

p

V h aN

)(Y0k +Y /b2 ) + ∇

y/b2

,2Y p +Y k +b2 Y0k +b2 aT X2

i.

(15.10.5)

y/b

X,s To the operator Pa,A,b,Y k , we associate the stochastic differential equation 0 in the variables (y, Y ),

y˙ b = ryb /b2 Ybp , X⊕N ryb /b2 Ybp Y0k + Yb /b2 Y˙ b = −ΓTy/b 2 h i p k 2 k 2 TX − aN ˙ yb /b2 , 2Yb + Yb + b Y0 + b ayb /b2 + w, yb,0 = y,

(15.10.6)

Yb,0 = Y.

Equation (15.10.6) is just the coordinate version of a stochastic differential equation over Xb, for which existence is already granted by the results of section 12.2. An application of the Feynman-Kac formula shows that if F : p × g → R is smooth with compact support, (m + n) t X,s exp −tPa,A,b,Y F (y, Y ) = exp k 0 2b2 " Z t h i 2 1 k p E exp − Y0 + Ybk /b2 , b2 aTybX/b2 + Yb ds 2 0 ! # Z t 2 1 TX k 2 − F (yb,t , Yb,t ) . (15.10.7) a 2 + Y0 + Yb /b ds 2 0 yb /b

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315

By proceeding as in (12.2.7), we rewrite (15.10.6) in the form Z t yb,t = y + ryb /b2 Ybp ds, 0 Z t Yb,t = Y − ΓTybX⊕N ryb /b2 Ybp Y0k + Yb /b2 ds (15.10.8) /b2 0 Z th Z th i i p k N 2 k 2 TX − aN , 2Y + Y , b ds + wt . ds − a Y + b a 2 2 2 b 0 yb /b yb /b yb /b b 0

0

By (9.10.4), (9.10.9), we find easily that as b → +∞, (yb,· , Yb,· ) converge uniformly over compact sets to (y∞,· , Y∞,· ), which is a solution of Z t Z t y∞,t = y + Y∞ ds, Y∞,t = Y + a + Y0k , [a, y∞ ] ds + wt . (15.10.9) 0

0

By (9.10.5), (9.10.6), and by the above convergence result, as b → ∞, "

Z i 2 1 t h k p X E exp − Y0 + Ybk /b2 , b2 aTy/b ds 2 + Yb 2 0 ! # Z 2 1 t T X t 2 k 2 k 2 − |a| + Y0 ay/b2 + Y0 + Yb /b ds F (yt , Yt ) → exp − 2 0 2 Z t k k p 2 t E exp − Y∞ , a + Y0 , Y∞ ds F (y∞,t , Y∞,t ) . (15.10.10) 2 0 X,s X Let Pa,A,∞,Y k be the scalar part of the operator Pa,A,∞,Y k in (9.10.1). 0 0 By (15.10.7) (15.10.10), and using again the Feynman-Kac formula, we can rewrite (15.10.10) in the form X,s X,s exp −tPa,A,b,Y F (y, Y ) → exp −tPa,A,∞,Y F (y, Y ) . (15.10.11) k k 0

0

For t = 1, (15.10.11) is just the scalar version of (15.10.1). Let us now establish equation (15.10.1) itself. Let Ua,b,· be the solution of the differential equation " · ∗ ∗ dUa,b N Λ (p ⊕k ) = Ua,b − − α − c ad Ybp /b2 + b c ad Ybp /b2 2 ds b − c ad aTybX/b2 − aN +b c (ad (a)) − c iθad Y0k + Y k /b2 yb /b2 # E k k 2 N − ρ iY0 + iYb /b − ayb /b2 , (15.10.12) Ua,b,0 = 1. By (2.4.5), (9.8.5), and (9.9.5), and proceeding as in the proofs of Theorems 14.4.1 and 15.7.1, if s (x, Y ) is taken as in (15.10.1), instead of (15.10.7), we

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316

Chapter 15

get exp

X −tPa,A,b,Y k 0

s (y, Y ) = exp

" E exp −

1 2

(m + n) t 2b2

Z t h i 2 k p Y0 + Ybk /b2 , b2 aTybX/b2 + Yb 0

! # Z 2 1 t T X k 2 − Ua,b,t τ0t s (yb,t , Yb,t ) . (15.10.13) a 2 + Y0 + Yb /b − tA 2 0 yb /b By (15.10.12), as b → +∞, Ua,b,· converges uniformly over compact sets to Ua,∞,· which is given by dUa,∞ = Ua,∞ −α + b c (ad (a)) − c ad (a) + iθad Y0k − iρE Y0k , ds (15.10.14) Ua,∞,0 = 1. By (9.10.1), (15.10.12)–(15.10.14), and proceeding as in (15.10.11), we find that as b → +∞, X X exp −tPa,A,b,Y s (y, Y ) → exp −tPa,A,∞,Y s (y, Y ) . (15.10.15) k k 0

For t = 1, we get (15.10.1). The proof of Theorem 9.11.1 is completed.

0

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Subject Index B-unimodular, 149 σ-algebra, 214, 232 Atiyah-Singer index formula, 124 Bargmann kernel, 17 Bianchi identity, 41 Bochner Laplacian, 37, 39, 123 Brownian motion, 98 Burkholder-Davis-Gundy inequality, 216 Cartan involution, 23, 235 Cartan subalgebra, 136 Casimir, 27 Chern character form, 121 Clifford algebra, 12 complex rank, 136 conormal bundle, 116 convergence in probability, 280 convex function, 49 convex set, 49 Dirac operator, 122 displacement function, 49, 294 dominant weights, 126 Doob’s inequality, 216 Egorov’s theorem, 169, 176 elliptic element, 49 enveloping algebra, 27, 179 ergodic theorem, 281 Euler form, 135 expectation, 98, 214, 263 Feynman-Kac formula, 100, 184, 237, 244, 253, 257, 265, 314, 315 filtration, 214, 215, 232 geodesic flow, 62 Girsanov, 272 Girsanov formula, 215 Haar measure, 61 Hamiltonian, 62 Hamiltonian vector field , 62 harmonic oscillator, 18, 173 Heisenberg algebra, 15 Hessian, 50, 219 hyperbolic element, 49

hypoelliptic Laplacian, 40 Itˆ o calculus, 184, 237, 242, 244 Itˆ o differential, 215, 216 Itˆ o integral, 184 Itˆ o’s formula, 98, 263 Kawasaki formula, 90, 124, 134 Kostant’s strange formula, 126 Laplace-Beltrami operator, 141, 214 law of iterated logarithm, 281 Littlewood-Paley decomposition, 189 locally symmetric space, 87 maximal torus, 126, 136 McKean-Singer formula, 5, 124, 180 Mehler’s formula, 174 metaplectic representation, 18 orbital integrals, 80 orbital supertraces, 83 parallel transport element, 50 Pfaffian, 126 Pontryagin maximum principle, 164 pseudodistance, 67 rapidly decreasing, 144, 191, 194, 229 Ray-Singer analytic torsion, 137 real rank, 139 Ricci tensor, 29, 215 scalar curvature, 29 semisimple element, 49 spinors, 122 Stratonovitch differential, 215, 216 supercommutator, 11 supertrace, 76 symmetric space, 24 Toponogov, 58 uniformly rapidly decreasing, 144, 229 wave front set, 116 Weyl group, 126 Weyl’s character formula, 126

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Index of Notation 1KC , 267 1, 221, 280 A, 124, 139 α, 41, 93, 147 αmax , 125 α b, 147 α−1 , 42 b (B), 96 A AX b , 188 AX b , 237 ad (a), 23 Ad (g) , 23 adQ (e), 312 adQ (e), 312 Ag , 30 A[γ] , 140 bg , 30 A bγ (0), 132 A bγ|k (0), 132 A bγ N |X(γ) , ∇N |X(γ) , 132 A bγ|p (0), 132 A b Aγ T X|X(γ) , ∇T X|Xγ , 132 b H 0 , ∇H 0 , 121 A b (H 0 ), 121 A aN , 46 aT X0 , 51 aT X , 45, 224 A (V ), 14 b (x), 96 A AX0 b , 268 AX b , 206 AX b , 245 AX0 b , 311 AX b , 205 B, 12, 17, 23 B ∗ , 12 B, 189 B0 , 205 β, 41, 135 b, 136 B00 , 205

B0 , 190 BbG , 188 BbX , 188 b (e), 16 b b (e), 16 Bγ , 148 BbG , 188 Bλ , 97 B (V ), 16 b b (V ), 15 b (V ), 15 BX b , 206 B (x, r), 189 B (x, r), 189 C, 88 C, 219 b c (A), 13, 14 c (A), 13, 14 b c (α), 13 b c (a), 13 c (α), 13 c (a), 13 b cb (e), 148 C ∞,c (X, R), 214 c (e), 122 C (g), 23 C g , 27 C g,H , 28 C g,X , 123 chγ F |X(γ) , ∇F |X(γ) , 132 ch H, ∇H , 121 ch (H), 121 χj (r), 189 χλ , 126 χ (r), 189 Cj , 190 C k0 , 140 b (G, E), 77 CK Ck [0, 1] , p⊥ (γ) ⊕ p⊥ (γ) , 99 k,V C , 28 b c (V ), 12 c (V ), 12 C ∞,b (X, F ), 84 C ∞,c (X, F ), 84

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326 C b (X, F ), 77 c (z (γ)), 92 D0 , 199 D, 19 D0 , 19 δ, 44, 287 d, 18 ∗ d , 18 d, 49 δa , 17, 68 x ¨, 162 ∆E , 173 ∆E,H , 175 ∆E,V , 175 df , 60 Dg , 30 b g , 30 D δ (γ), 89 g∗ d , 32 g d , 32 dγ , 49, 294 dg, 61 d⊥ γ −1 ϕ1 , 96 g DH , 123 g b DH , 30 g DV , 123 b g , 30 D V ∆γX , 116 b g,X , 37 D b g,X , 37 D H b g,X , 37 D V ∆H,X , 37, 39, 123 b ∆H,Y , 210 δj (u), 189 ∆ (k), 90 ∆k , 93 Dk , 33 k∗ d , 32 k d , 32 dk, 61 dk0 , 62 dk00 , 61 ∆p , 93 Dp , 32 p∗ d , 32 p d , 32 ∆p⊕k , 34, 93 δ± , 201 Ds0 , 195 Ds , 192 ∆t , 128 ∇T X⊕N,f ∗ , 25 ∆T X , 155, 188, 219, 250 ∆T X⊕N , 38

INDEX OF NOTATION ∇T X⊕N,f , 25 du, 62 du0 , 61 ∆V , 17 dV , 19 dv, 62 dv 0 , 62 dvXb , 38 δw, 216 dw, 216 DX , 122 ∆X , 214 dx, 60 DX b , 38 DX b,t , 41 δ ((x, f ) , (x0 , f 0 )), 67 dx0 ((x, f ) , (x0 , f 0 )), 67, 295 δ (Y p ), 78, 157, 230 dY , 82 dY g , 92 dy, 60, 92 Y k , 82 dY N , 82 dY p , 78 dY T X , 82 dz, 62 dz 0 , 61 ∆z(γ) , 97 E, 35, 214, 263 E, 19, 81 E 0 , 19 E · , 208 E, 98 E k , 33 n (f ), 193 n (u), 193 E p , 32 η, 38, 125 e T X, ∇T X , 135 (Y ), 211 F , 35, 96 F , 81 F∞ , 214 F , 96 Fb , 143 Φb,t,Y p ,x0 (α, β), 230 Φb,t,Y p ,x0 (α, β), 239 ϕf (t), 58 Fγ , 62 bγ , 65 F ϕ, 12, 148 ϕn , 192 φ (r), 189 Ft |t≥0 , 214 Ft , 214

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INDEX OF NOTATION ϕt , 62, 293 Φ t, x, Y T X , 203 G, 23 Γ, 65 G, 149 γ, 41 g, 23 [γ], 80 γ, 201 g F , 35 g N , 24 Gt , 232 g T X , 24 ΓT X⊕N , 66, 314 H, 42 H, 189 H, 193 H, 36, 149 H0 , 194 H, 190 b 207 H, h0,· , 287 H ∗0 (Y, Y 0 ), 168 Hb,t ((x, Y ) , (x0 , Y 0 )), 170 Hb,t (x), 162 ∗ (Y, Y 0 ), 167 Hb,t Hb ((x, Y ) , (p, q)), 164 Hb ((x, Y ) , (p, q)), 169 Hb∗ (Y, q), 166 0 hE b,t (Y, Y ), 174 hE ((x, Y ) , (x0 , Y 0 )), 175 b,t 0 ), 174 hE (Y, Y t H γ , 118 H∞ (x), 162 H∞ , 191 H 0,t ((x, Y ) , (x0 , Y 0 )), 172 hpα,t (Y p , Y p0 ), 253 H0s , 194 Hs , 191 Hs , 190 0s , 194 Hloc I, 264 i, 147 bia , 65 ia , 62 bia N k−1 , 65, 306 J γ , 148 Jγ Y0k , 94 J p , 130 ⊥ ⊥ J p (γ)×p (γ) , 99 ⊥ p (γ) J , 98 J T X , 222

K, 23 k, 23 K 0 , 139 K ∗0 (Y, Y 0 ), 168 Ka , 41, 174, 206, 237, 243, 253, 269 Kb ((x, Y ) , (p, q)), 165 Kb,t ((x, Y ) , (x0 , Y 0 )), 170 Kb,t (x), 162 ∗ (Y, Y 0 ), 167 Kb,t ∗ Kb (Y, q), 166 E (Y, Y 0 ), 174 kb,t E ((x, Y ) , (x0 , Y 0 )), 175 kb,t ktE (Y, Y 0 ), 174 K (γ), 55 κg , 28 k (γ), 55, 113 kγ,g0 , 54 k⊥ (γ), 55 κk , 29 k k−1 , 65 K0 , 139 k0 , 63 K 0 (γ), 55 k⊥ 0 (γ), 96 k⊥ 0 , 63 K 0,t ((x, Y ) , (x0 , Y 0 )), 172 p kβ,t (Y p , Z p ), 257 L, 163 L2 , 163 La , 45 Λλ,τ , 199 Lp , 168, 183 Λτ , 190 LV N , 47 [a ,Y ] V L[e,Y ] , 35, 46 LX , 39 LX A , 124 LX A,b , 84 LX0 A,b , 264 LX A,b , 151, 312 LX A,b , 143, 310, 312 LX a,b , 47, 154 LX b , 40 L ((x, Y ) , u), 163 m, 23 Ma , 98 Mb , 265 Mb,t x, Y T X , x0 , 220 c,d Mt , 226 µF (a), 45 Mf,t , 231 mγ , 49

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INDEX OF NOTATION

pa,⊥ (γ), 55 mt , 226 µ, 114 µ b, 114 MX A,b , 263 MX b , 220, 263 n, 23 ∇a,r , 45 ∞ π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) , 37 ∇C (T X⊕N,b c,d Nt , 228 N E , 175 ∇e,` , 37 ∇F , 35 N (γ), 75 | |H0 , 194 | |H , 190 | |H , 189 ∇H , 93 N k−1 , 65 N k−1 , 65 k kp , 183 · ∗ ∗ ˆ ∇Λ (T X⊕N ),f ∗,f , 26 b · ((T X⊕N )(γ)∗ ),f ∗ ,fb Λ· (T ∗ X⊕N ∗ )⊗Λ ∇ , 312 ·

∗

∇Λ (T X⊕N )(γ) ,f , 312 ∗ ∗ ∗ ∇Λ(T X⊕N ),f , 26 Λ(T ∗ X⊕N ∗ ),f ∇ , 25 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )),f ∗,fb ∇Y T X , 39 · ∗ N Λ (V ) , 14 ∇N , 24 NP ⊥ (γ)/X , 116 · ∗ N S (V ) , 17 ∇T X , 24 ∇T X,k , 238 (x0 ,Y T X0 ) T X⊕N ∇ , 24 ∇(T X⊕N )(γ),f , 312 ∇(T X⊕N )(γ) , 312 ∇V , 93 ∇V e , 46, 153 ∇V a , 35 ∇W , 24 X Na,A,b , 153 N ⊥ (γ), 145 NX(γ)/X , 57 NX(γ)/X , 57 ∇Y T X , 188 C ∞ (T X⊕N,b π ∗ (Λ· (T ∗ X⊕N ∗ )⊗F )) ∇Y T X , 38 Ω, 24 0 0 oX a,b ((x, Y ) , (x , Y )), 155 E O , 173 ObE , 174

ω g , 24, 66 ω k , 24, 66 ω, 62 ωA , 126 ω, 164 Op, 93 ω p , 24, 66 p Oα ,253 Op Xb, π b∗ F , 150 ω T X ⊕ N, g T X⊕N , 121 X Oa,A,b , 154 OX a,A,b , 155 Ωz(γ) , 133 P , 42, 214 P ⊥ , 42 P++ , 126 P, 43, 85, 282 P⊥ , 43, 282 p, 23 p, 24, 93, 113, 188 p+ , 130 p− , 130 Pa,Y k , 93, 158 0 Pb , 192, 195 Pb,λ , 199 P E , 174 PbE , 174 Pf [A], 126 P ⊥ (γ), 116 p (γ), 55, 113 pγ , 57 pγ,g0 , 54 p⊥ (γ), 55 π, 62, 126, 164, 189 π b, 36, 142 p0 , 63 p⊥ 0 , 63 p⊥ 0 (γ), 96 Pβp , 257 ψ, 156 ψa , 50 ψn , 193 Ψ t, x, Y T X , 204 0 pX t (x, x ), 84 0 ), 86 pX (x, x a,t 0 pt (x, x ), 214 PX k , 158 a,A,∞,Y0

PX

, 157

A,b,Y0k , 157 PX a,A,b,Y0k pX k ((y, Y ) , (y 0 , Y 0 )), a,b,Y0 P X,s , 314 a,A,b,Y0k

157

Traceglobfin

June 3, 2011

329

INDEX OF NOTATION Q, 77, 99, 312 Q, 77 Q, 82 q, 93, 113 X ((x, Y ) , (x0 , Y 0 )), 85, 287 q0,t Qa,Y k , 94 0 Qa,Y k , 94 0 Qb , 206, 210 E Qb , 175 q (g, g 0 ), 77 q ∗ q 0 (x, x0 ), 78 X qa,b,t ((x, Y ) , (x0 , Y 0 )), 87 X qb , 151, 313 qX , 143, 310 b qX b,t , 313 ((x, Y ) , (x0 , Y 0 )), 143, 310 qX b,t X ((x, Y ) , (x0 , Y 0 )), 84, 142, 178, 211 qb,t X qb,t ((x, Y ) , (x0 , Y 0 )), 151

q X,[γ] (g, g 0 ), 88 X0 ((x, Y ) , (x0 , Y 0 )), 264 qb,t q (x, x0 ), 77 QZ , 99 R, 190, 264 R0 , 205 r, 93, 113, 147 0 0 rX0 b,t ((x, Y ) , (x , Y )), 268 X T X rb,t x, Y , x0 , Y T X0 , 204 E ρ , 35 RbE , 175 RF , 35 r (f ), 60 ργ , 57, 143 ρ, 63, 126 ρE aN , 46 T X R , 45 RX , 29, 215 rX b , 300 rX b , 300 rX b , 291 0 0 rX b,t ((x, Y ) , (x , Y )), 206 X 0 0 rb,t ((x, Y ) , (x , Y )), 210 0 0 rX b,t ((x, Y ) , (x , Y )), 243, 299 X 0 0 rb,t ((x, Y ) , (x , Y )), 245, 299 0 0 rX0 b,t ((x, Y ) , (x , Y )), 311 0 0 rX b,t ((x, Y ) , (x , Y )), 273 X T X , x0 , Y T X0 , 237, 291 rb,t x, Y R (Y ), 43, 282 R0 k ((y, Y ) , (y 0 , Y 0 )), 95 Y0

RY k ((y, Y g ) , (y 0 , Y g0 )), 93, 152 0

r (Z) dZ, 99 RZ0 ,Z 0 , 98 0

S, 190 σb,t x, Y T X , x0 , 220 X T X sb,t x, Y , x0 , Y T X0 , 204, 220 σγ , 61 σ, 126 p p S p = S+ ⊕ S− , 122 Sp (V ), 16 S (R), 114 S even (R), 114 S T Z , 124 S · (V ∗ ), 15 S V , 15 S X , 29 0 0 sX b,t ((x, Y ) , (x , Y )), 206 X 0 0 sb,t ((x, Y ) , (x , Y )), 210 S (x, Y, p, q), 166 S (Y ), 43, 286 SY k , 95 0

S (Z), 191 T , 17, 126, 169 T 0 , 169 t, 126 t, 136 τ , 64 τt0 , 263, 265 Ta , 153 τ , 26 τb, 26 θ, 23, 164, 235 Tr[γ] exp −tLX A , 84 Tr[γ] [Q], 80 Tr QZ,[γ] , 88 Trs [A], 76 cs , 94, 149 Tr Trs [γ] [Q], 83 Tt , 70 τ0t , 263, 265 (T X ⊕ N ) (γ), 75 (T X ⊕ N )⊥ (γ), 75 0 τxx , 67 TY k ((y, Y g ) , (y 0 , Y g0 )), 96 0

p(γ)×z(γ) Y0k

T

((y, Y ) , (y 0 , Y 0 )), 97

U , 96 U· , 265 U , 200 Ub , 150 U (g), 27, 179 Uj , 190 kuk0s , 194 kuks , 191 |||u|||s , 191 |||u|||0s , 194 V , 62, 188

Traceglobfin

June 3, 2011

330 v· , 275 V ⊗ E, 35 V ((y, Y p ) , (y 0 , Y p0 )), 96 W , 126 W0,· , 287 √ p 2s LX + A , 116 WF cos W 0s , 194 W s , 191 X, 24 X , 62, 116, 164 X ∗ , 62, 116 X∗ , 164 Xb, 36 x· , 293 x0 , 189, 192 X (γ) ⊂ X, 49 x0t , 69, 74, 295, 303 xt , 69, 74, 295, 303 Y , 190 Y b , 185 Yb,t , 186 Y Hb , 165 Y Kb , 165 Y k , 32 Y0k , 93 Y0k , 145 Y N,⊥ , 145

INDEX OF NOTATION Y N , 36 0 YN · , 74, 303 YN · , 74, 303 Y p , 32 Y T X , 36 X0 , 295 YT t X , 293 YT · X , 295 YT t < Y T Z >, 189 Y ⊗ Y , 280 Z, 87, 189 Z, 189 b 90 Z, Z (a), 49 z (a), 49 z⊥ (a), 53 za,⊥ (γ), 55 Z a,⊥,0 (γ), 55 Z (γ), 49, 113 z (γ), 49, 113 z (γ), 92, 147 z⊥ (γ), 55 ZK , 131 zK , 131 z (k), 54 z0 , 63 Z 0 (γ), 49 z⊥ 0 (γ), 96 z⊥ 0 , 63

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