Posn̳(R) and Eisenstein series

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1868 Jay Jorgenson...

Author: Jay Jorgenson | Serge Lang

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1868

Jay Jorgenson · Serge Lang

Posn(R) and Eisenstein Series

ABC

Authors Jay Jorgenson City College of New York 138th and Convent Avenue New York, NY 10031 USA e-mail: [email protected] Serge Lang Department of Mathematics Yale University 10 Hillhouse Avenue PO Box 208283 New Haven, CT 06520-8283 USA

Library of Congress Control Number: 2005925188 Mathematics Subject Classification (2000): 43A85, 14K25, 32A50 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-25787-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-25787-5 Springer Berlin Heidelberg New York DOI 10.1007/b136063 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005

Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: TEX output by the author Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

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Preface

We are engaged in developing a systematic theory of theta and zeta functions, to be applied simultaneously to geometric and number theoretic situations in a more extensive setting than has been done up to now. To carry out our program, we had to learn some classical material in several areas, and it wasn’t clear to us what would simultaneously provide enough generality to show the effectiveness of some new methods (involving the heat kernel, among other things), while at the same time keeping knowledge of some background (e.g. Lie theory) to a minimum. Thus we experimented with the quadratic model of G/K in the simplest case G = GLn (R). Ultimately, we gave up on the quadratic model, and reverted to the G/K framework used systematically by the Lie industry. However, the quadratic model still serves occasionally to verify some things explicitly and concretely for instance in elementary differential geometry. The quadratic forms people see the situation on K\G, with right G-action. We retabulated all the formulas with left G-action. Just this may be useful for readers since the shift from right to left is ongoing, but not yet universal. Some other people have found our notes useful. For instance, we include some reduction theory and Siegel’s formula (after Hlawka’s work). We carry out with some variations material in Maass [Maa 71], dealing with GLn (R), but also include more material than Maass. We have done some things hinted at in Terras [Ter 88]. Her inclusion of proofs is very sporadic, and she leaves too many “exercises” for the reader. Our exposition is selfcontained and can be used as a naive introduction to Fourier analysis and special functions on spaces of type G/K, making it easier to get into more sophisticated treatments.

VI

Preface

Acknowledgements Jorgenson thanks PSC-CUNY and the NSF for grant support. Lang thanks Tony Petrello for his support of the Yale Mathematics Department. Both of us thank him for support of our joint work. Lang also thanks the Max Planck Institut for productive yearly visits. We thank Mel DelVecchio for her patience in setting the manuscript in TEX, in a victory of person over machine. February, 2005

J. Jorgenson S. Lang

Contents

1

GLn (R) Action on Posn (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Iwasawa-Jacobi Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Inductive Construction of the Grenier Fundamental Domain . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 The Inductive Coordinates on SPosn . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Grenier Fundamental Domain and the Minimal Compactification of Γn \SPosn . . . . . . . . . . . . . . . 17 5 Siegel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2

Measures, Integration and Quadratic Model . . . . . . . . . . . . . . . 1 Siegel Sets and Finiteness of Measure Mod SLn (Z) . . . . . . . . . . . . 2 Decompositions of Haar Measure on Posn (R) . . . . . . . . . . . . . . . . . 3 Decompositions of Haar Measure on SPosn . . . . . . . . . . . . . . . . . . . 4 Siegel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 25 36 39

3

Special Functions on Posn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Characters of Posn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Bengtson Bessel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Mellin and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 50 55 58 69

4

Invariant Differential Operators on Posn (R) . . . . . . . . . . . . . . . 1 Invariant Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Invariant Differential Operators on Posn the Maass-Selberg Generators . . . . . . . . . . . . . . . . . . . . . . . 3 The Lie Algebra Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Transpose of an Invariant Differential Operator . . . . . . . . . . . 5 Invariant Differential Operators on A and the Normal Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 78 84 87 90

VIII

Contents

5

Poisson Duality and Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . 95 1 Poisson Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2 The Matrix Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3 The Epstein Zeta Function: Riemann’s Expression . . . . . . . . . . . . . 99 4 Epstein Zeta Function: A Change of Variables . . . . . . . . . . . . . . . . . 104 5 Epstein Zeta Function: Bessel-Fourier Series . . . . . . . . . . . . . . . . . . 105

6

Eisenstein Series First Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 1 Adjointness Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2 Fourier Expansion Determined by Partial Iwasawa Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3 Fourier Coefficients from Partial Iwasawa Coordinates . . . . . . . . . . 114 4 A Fourier Expansion on SPosn (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 ∂ | . . . . . . . . . . . . . . . . . . . . . 118 5 The Regularizing Operator QY = |Y || ∂Y

7

Geometric and Analytic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 121 1 The Metric and Iwasawa Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 121 2 Convergence Estimates for Eisenstein Series . . . . . . . . . . . . . . . . . . . 125 3 A Variation and Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8

Eisenstein Series Second Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 1 Integral Matrices and Their Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2 The ζQ Fudge Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3 Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4 Adjointness and the ΓU \Γ-trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5 Changing to the (s1 , . . . , sn )-variables . . . . . . . . . . . . . . . . . . . . . . . . 152 6 Functional Equation: Invariance under Cyclic Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7 Invariance under All Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

1 GLn (R) Action on Posn (R)

Let G = GLn (R) or SLn (R) and Γn = GLn (Z). Let Posn (R) be the space of positive symmetric real n × n matrices. Recall that symmetric real n × n matrices Z have an ordering, defined by Z = 0 if and only if hZx, xi = 0 for all x ∈ Rn . We write Z1 = Z2 if and only if Z1 − Z2 = 0. If Z = 0 and Z is non-singular, then Z > 0, and in fact Z = λI if λ is the smallest, necessarily positive, eigenvalue. The group G acts on Posn (R) by associating with each g ∈ G the automorphism (for the C ∞ or real analytic structure) of Posn given by [g]Z = gZ t gs . We are interested in Γn \Posn (R), and we are especially interested in its topological structure, coordinate representations, and compactifications which then allow effective computations of volumes, spectral analysis, differential geometric invariants such as curvature, and heat kernels, and whatever else comes up. The present chapter deals with finding inductively a nice fundamental domain and establishing coordinates which are immediately applied to describe Grenier’s compactification, following Satake. Quite generally, let X be a locally compact topological space, and let Γ be a discrete group acting on X. Let Γ0 be the kernel of the representation Γ → Aut(X). A strict fundamental domain F for Γ is a Borel measurable subset of X such that X is the disjoint union of the translates γF for γ ∈ Γ/Γ0 . In most practices, X is also a C ∞ manifold of finite dimension. We define a fundamental domain F to be a measurable subset of X such that X is the union of the translates γF , and if γx ∈ F for some γ ∈ Γ, and γ does not act trivially on X, then x and γx are on the boundary of F . In practice, this boundary will be reasonable, and in particular, in the cases we look at, this boundary will consist of a finite union of hypersurfaces. By resolution of singularities, the boundary can then be parametrized by C ∞ maps defined on cubes of Euclidean space of dimension 5 dim X − 1. Thus the boundary has n-dimensional measure 0. Jay Jorgenson: Posn (R) and Eisenstein Series, Lect. Notes Math. 1868, 1–22 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com °

2


In this chapter, we have essentially reproduced aspects of Grenier’s papers [Gre 88] and [Gre 93]. He carried out on GLn (R) and SLn (R) Satake’s compactification of the Siegel upper half space [Sat 56], [Sat 58], see also Satake’s general results [Sat 60]. It was useful to have Grenier’s special case worked out in the literature, especially Grenier’s direct inductive method. Note that to a large extent, this chapter proves results compiled in Borel [Bor 69], with a variation of language and proofs. These are used systematically in treatments of Eisenstein series, partly later in this book, and previously for instance in Harish-Chandra [Har 68].

1 Iwasawa-Jacobi Decomposition Let: G = Gn = GLn (R) Posn = Posn (R) = space of symmetric positive real matrices K = O(n) = Unin (R) = group of real unitary n × n matrices U = group of real unipotent upper triangular matrices, i.e. of the form   1 xij 0 1    u(X) = u =  .  so u(X) = I + X, . ..  ..  0 0 ... 1 with

X = (xij ), 1 < i < j 5 n . A = group of diagonal matrices with positive components,   a1  a2 0      .. a=  ai > 0 all i . .   0  an

Theorem 1.1. The product mapping

U × A × K → U AK = G is a differential isomorphism. Actually, the map U × A → Posn (R) given by (u, a) 7→ uat u is a differential isomorphism.

1 Iwasawa-Jacobi Decomposition

3

Proof. Let {e1 , . . . , en } be the standard unit vectors of Rn , and let x ∈ Ln (R). Let vi = xei . We orthogonalize {v1 , . . . , vn } by the standard Gram-Schmidt process, so we let w1 = v1 , w2 = v2 − c21 w1 w1 , w3 = v3 − c32 w2 − c31 w2 ⊥ w1 and w2 and so on. Then e0i = wi /kwi k is a unit vector, and the matrix a having kwi k−1 for its diagonal elements is in A. Let k = aux so x = u−1 a−1 k. Then k is unitary, which proves that G = U AK. To show uniqueness, suppose that u1 at u1 = u2 bt u2 with u1 , u2 ∈ U and a, b ∈ A, then putting u = u−1 2 u1 we find ua = bt u . Since u and t u are triangular in opposite direction, they must be diagonal, and finally a = b. That the decomposition is differentially a product is proved by computing the Jacobian of the product map, done in Chap. 2. The group K is the subset of elements of G fixed under the involution g 7→ t g −1 . We write the transpose on the left to balance the inverse on the right. We have a surjective mapping G → Posn given by g 7→ g t g . This mapping gives a bijection of the coset space ϕ : G/K → Posn , and this bijection is a real analytic isomorphism. Furthermore, the group G acts on Posn by a homomorphism g 7→ [g] ∈ Aut(Posn ), where [g] is given by the formula [g]p = gpt g . This action is on the left, contrary to right wing action by some people. On the other hand, there is an action of G on the coset space G/K by translation τ : G → Aut(G/K) such that τ (g)g1 K = gg1 K . Under the bijection ϕ, a translation τ (g) corresponds precisely to the action [g]. Next we tabulate some results on partial (inductive) Iwasawa decompositions. These results are purely algebraic, and do not depend on real matrices

4


or positivity. They depend only on routine matrix computations, and it will prove useful to have gotten them out of the way systematically. Let G = GLn denote the general linear group, wherever it has its components. Vectors are column vectors. An element g ∈ GLn can be written

g=

µ

A t c



¶   b =  d 

A

 b1 ..  .     b n−1

c1 · · · cn−1

d

where b and c are (n − 1)-vectors, so t c is a row vector of dimension n − 1, A is an (n − 1) × (n − 1) matrix, and d is a scalar. We write d = dn (g) for this lower right corner of g. For induction purposes, we do not deal with fully diagonal matrices but with an inductive decomposition ! ! Ã Ã W 0 W 0(n−1) with W ∈ GLn−1 and v scalar 6= 0 . = t (n−1) 0 v 0 v We have the left action of GL on Matn given on a matrix M by [g]M = gM t g, so g 7→ [g] is a representation. For an (n − 1)-vector x, we denote ! Ã In−1 x . u(x) = 0 1 We write t x = (x1 , . . . , xn−1 ). Then x 7→ u(x) is an injective homomorphism. In particular, u(x)−1 = u(−x) . The usual matrix multiplication works to yield Ã ! A Ax + b gu(x) = t t (1) c cx + d An expression Z = [u(x)]

Ã

W 0

0 v

!

=

Ã

W + [x]v vt x

xv v

!

1 Iwasawa-Jacobi Decomposition

5

for a matrix Z will be called a first order Iwasawa decomposition of Z. We note that with such a decomposition, we have (2)

dn (Z) = v .

Straightforward matrix multiplication yields the expression: ! Ã W 0 (3) [g]Z = [g][u(x)] 0 v  AWc + (Ax + b)v(t xc + d)  . = t t t t t t cW A + ( cx + d)v (Ax + b) [ c]W + [ cx + d]v 

[A]W + [Ax + b]v

In particular, (4)

dn ([g]Z) = [t c]W + [t cx + d]v = [t c]W + (t cx + d)2 v .

Indeed, t cx is a scalar, so is t cx + d, so [t cx + d]v = (t x + d)2 v. Note that directly from matrix multiplication, one has also (5)

dn ([g]Z) = [t c, d]Z .

For later purposes, we record the action of a semidiagonalized matrix: ! #Ã #" # " " W 0 A 0 In−1 x A 0 (6) Z= 0 v 0 1 0 1 0 1 =

"

In−1 0

#Ã [A]W Ax 0 1

0 v

!

.

One way to see this is to multiply both sides of (6) by [u(−Ax)] = [u(Ax)]−1 , and to verify directly the identity !Ã !Ã Ã In−1 A 0 In−1 −Ax (7) 0 1 0 1 0 We have a trivial action " In−1 (8) 0

0 −1

#Ã

W 0

0 v

!

=

! x 1

Ã

W 0

=

Ã

0 v

A 0

!

0

.

1

!

.

6


"

0

In−1

#

acts as the identity on a semidiagonalized matrix In other words, 0 −1 ! Ã W 0 . On the other hand, on u(x) we can effect a change of sign by the 0 v transformation " #Ã ! Ã ! In−1 0 In−1 x In−1 −x (9) = . 0 −1 0 1 0 1 We then derive the identity " In−1 (10) 0 =

0 −1 "

#"

In−1 0

#Ã x W 1 0

#Ã −x W 1 0

In−1 0

Indeed, in the left side of (10) we insert #" # " In−1 0 In−1 0 0

just before (10).

µ

W 0

−1

0

−1

=

0 v

"

0 v

!

!

.

In−1 0

# 0

1

¶ 0 , and use (8), (9) to obtain the right side, and thus prove v

2 Inductive Construction of the Grenier Fundamental Domain This section is taken from [Gre 88]. Throughout the section, we let: G = GLn (R) Γ = GLn (Z) Posn = Posn (R) = space of symmetric positive n × n real matrices. We write Z > 0 for positivity. We use the action of G on Posn given by g 7→ [g], where [g]Z = gZ t g . Thus g 7→ [g] is now viewed as a representation of G in Aut(Posn ). We note that the kernel of this representation in Γn is ±In , in other words, if g ∈ Γn and [g] = id then g = ±In . We use the notation of Sect. 1. An element Z ∈ Posn has a first order Iwasawa decomposition

2 Inductive Construction of the Grenier Fundamental Domain

Z=

"

In−1 0

#Ã W x 1

0

0 v

7

!

with W ∈ Posn−1 and v ∈ R+ . Since we shall deal with the discrete group Γn , the following fact from algebra is useful to remember. Let R be a principal ideal ring. A vector in t Rn is primitive, i.e. has relatively prime components, if and only if this vector can be completed as the first (or any) row of a matrix in GLn (R). This fact is immediately proved by induction. In dealing with dn ([g]Z) and g ∈ Γn , we note that this lower right component depends only on the integral row vector (t c, d) ∈ t Zn . Here we continue to use the notation of Sect. 1, that is Ã ! A b g= t . c d Note that we have an obvious lower bound for v. If λ is the smallest eigenvalue of Z (necessarily > 0), then using the n-th unit vector en and the inequality [en ]Z = λken k2 we find v=λ.

(1)

For n = 2, we define the set Fn to consist of those Z ∈ Posn such that: Fun 1. dn (Z) 5 dn ([g]Z) for all g ∈ Γn , or in terms of coordinates, v 5 [t c]W + (t cx + d)2 v for all (t c, d) primitive in Zn . Fun 2. W ∈ Fn−1 Fun 3. 0 5 x1 5 12 and |xj | 5

1 2

for j = 2, . . . , n − 1.

Minkowski had defined a fundamental domain Minn by the following conditions on matrices Z = (zij ) ∈ Posn : Min 1. For all a ∈ Zn with (ai , . . . , an ) = 1 we have [t a]Z = zii . Min 2. zi,i+1 = 0 for i = 1, . . . , n − 1. Minkowski’s method is essentially that followed by Siegel in numerous works, for instance [Sie 40], [Sie 55/56]. Grenier’s induction (following Satake) is simpler in several respects, and we shall not use Minkowski’s in general. Grenier followed a recursive idea of Hermite. However, we shall now see that for n = 2, F2 is the same as Minkowski’s Min2 . The case n = 2. We tabulate the conditions in this case. The positivity conditions imply at once that v, w > 0, so Fun 2 doesn’t amount to anything more. We have with x ∈ R:

8


Z=

Ã

!Ã w 1 x 0

1

0

0 v

!Ã

1

0

x 1

!

=

"

#Ã w 1 x 0

0

1

0 v

!

.

The remaining Fun conditions read: Fun 1. v 5 c2 w + (cx + d)2 v for all primitive vectors (c, d). Fun 3. 0 5 x 5 12 . Proposition 2.1. For v, w > 0 the above conditions are equivalent to the conditions 1 v 5 w + x2 v and 0 5 x 5 for all x ∈ R . 2 Under these conditions, we have w=

3 v. 4

Proof. The inequality v 5 w + x2 v comes by taking c = 1, d = 0. Then w = v(1 − x2 ), and since 0 5 x 5 12 , the inequality w = 3v/4 follows. Then it also follows that for all primitive pairs (c, d) of integers, we have Fun 1 (immediate verification), thus proving the proposition. Write Z in terms of its coordinates: Ã ! Ã w + x2 v z11 z12 Z= = z12 z22 vx

xv v

!

.

Proposition 2.2. The inequalities Fun 1 and Fun 3 are equivalent to: 0 5 2z12 5 z22 5 z11 . Thus F2 is the same as the Minkowski fundamental domain Min2 . If z12 = 0, then the inequalities are equivalent to 0 < z22 5 z11 . Proof. The equivalence is immediate in light of the explicit determination of zij in terms of v, w and x, coming from the equality of matrices above. After tabulating the case n = 2, we return to the general case. We shall prove by induction: Theorem 2.3. The set Fn is a fundamental domain for GLn (Z) on Posn (R). Proof. The case n = 2 follows from the special tabulation in Proposition 2.2. So let n = 3 and assume Fn−1 is a fundamental domain. Let Z ∈ Posn and let g ∈ GLn (Z) have the matrix expression of Sect. 1, so A, b, c, d are integral matrices. We begin by showing:


9

Given a positive number r and Z ∈ Posn , there is only a finite number of primitive (t c, d) ∈ t Zn (bottom row of some g ∈ GLn (Z)) such that dn ([g]Z) 5 r . Proof. Since W ∈ Posn−1 we have W = λIn−1 for some λ > 0, and hence [t c]W = λt cc for all c ∈ Zn−1 . Hence there is only a finite number of c ∈ Zn−1 such that [t c]W 5 r. Then from the inequality (t cx + d)2 v 5 r, we conclude that there is only a finite number of d ∈ Z satisfying this inequality, as was to be shown. We next prove that every element of Posn may be translated by some element of Γ into a point of Fn . Without loss of generality, in light of the above finiteness, we may assume that dn (Z) = v is minimal for all elements in the Γ-orbit [Γ]Z. By induction, there exists a matrix A ∈ GLn−1 (Z) = Γn−1 such that [A]W ∈ Fn−1 . We let g=

Ã

A 0 0 1

!

∈ Γn .

Then by (6) of Sect. 1, [g]Z =

"

In−1 0

#Ã [A]W Ax 0 1

0 v

!

.

Thus we have at least satisfied the condition Fun 2. Without loss of generality, we my now assume that W ∈ Fn−1 , since the dn does not change under the action of a semidiagonalized element g as above, with dn (g) = 1. Now by acting with g = u(b) with b ∈ Zn−1 and using the homomorphic property u(x + b) = u(b)u(x), we may assume without loss of generality that |xij | 5 12 for all j. Finally, using (10) of Sect. 2, we may change the sign of x if necessary so that 0 5 x1 , thus concluding the proof that some element in the orbit [Γ]Z satisfies the three Fun conditions. There remains to prove that if Z and [g]Z ∈ Fn with g ∈ Γn then Z and [g]Z are on the boundary, or [g] = id on X, that is [g] = ±In . We again prove this by induction, it being true for n = 2 by Proposition 2.2, so we assume the result for Fn−1 , and we suppose [g]Z, Z are both in Fn . Then from Fun 1, dn (Z) = dn ([g]Z), that is, v = [t c]W + (t cx + d)2 v . If c 6= 0,then Z and [g]Z are on the boundary, because the boundary is defined among other conditions by this hypersurface equality coming from Fun 1.

10


If c = 0, then this equality reads v = vd2 so d = ±1 and ! Ã A b . g= 0 ±1 Since det(g) = ±1 because g ∈ GLn (Z), it follows that det A = ±1, or, in other words, A ∈ GLn−1 (Z). We have #Ã ! " In−1 ±Ax + b [A]W 0 . [g]Z = 0 1 0 v Then [A]W ∈ Fn−1 so by induction: either W, [A]W ∈ boundary of Fn−1 or A = ±In−1 . If W, [A]W ∈ boundary of Fn−1 , then Z and [g]Z ∈ boundary Fn . On the other hand, if A = ±In−1 , then Ã ! ±In−1 b g= , 0 ±1 and therefore [g]Z =

"

In−1

±x ± b

0

1

#Ã

W

0

0

v

!

.

That Z, [g]Z ∈ Fn implies that 1 1 and 0 5 ±x1 ± b1 5 ; 2 2 1 |xj |, | ± xj ± bj | 5 . 2 0 5 x1 5

Since b ∈ Zn−1 , we find: either xj = ± 21 , bj = ±1 for j = 2, . . . , n − 1 and x1 = 12 , b1 = 1; or xj 6= ± 12 and bj = 0 for all j. It follows that either A and [g]Z are on the boundary of Fn determined by the x-coordinate, or b = 0 in which case ! Ã In−1 0 . g=± 0 ±1 If g =

Ã

!

In−1

0

0

−1

, then by (10) of Sect. 2, we have


[g]Z =

"

In−1 0

#Ã W −x

0

0

1

v

!

11

,

and both 0 5 x1 5 12 and 0 5 −x1 5 12 , so x1 = 0 and Z, [g]Z are in the boundary. This concludes the proof of the theorem. Theorem 2.4. The fundamental domain Fn can be defined by a finite number of inequalities, which can be determined inductively explicitly. Its boundary consists of a finite number of hypersurfaces. Proof. Proposition 2.2 gives the result for n = 2, so let n = 3, and assume the result for Fn−1 . Conditions Fun 2 and Fun 3 clearly consist only of a finite number of inequalities, with equalities defining the boundary. So we are concerned whether the conditions Fun 1 involve only a finite number of conditions, that is v 5 [t c]W + (t cx + d)2 v for all primitive (t c, d) . Since W ∈ Fn−1 , we may write ! " #Ã In−2 x0 W0 0 1 with v 0 > 0, W 0 ∈ Fn−2 , |x0 | 5 , W = 2 0 1 0 v0 where |x0 | is the sup norm of x0 . By induction, there is only a finite number of inequalities v 0 5 [t c0 ]W 0 + (t c0 x0 + d0 )2 v 0 . By straight matrix multiplication, [t c]W = [t c(n−2) ]W 0 + (t c(n−2) x0 + cn−1 )2 v 0 where t c(n−2) = (c1 , . . . , cn−2 ). Thus there is only a finite number of vectors t c = (t c(n−2) , cn−1 ) because we may take c(n−2) among the choices for c0 . Then with the bounds on the coordinates xj , there is only a finite number of d ∈ Z which will satisfy the inequalities Fun 1. This concludes the proof of the general finiteness statements. In addition, as Grenier remarks, the finite number of inequalities can be determined explicitly. For this and other purposes, one uses: Lemma 2.5. Let Z ∈ Fn , Z=

"

In−1 0

#Ã x W 1 0

0 v

!

as before. let zi = zii be the i-th diagonal element of Z, and wi = wii the i-th diagonal element of W . Then for i = 1, . . . , n − 1, v 5 zi 5

4 wi . 3

12


Proof. From Fun 1, with all (t c, d) = 1, we consider the values d = 0 and c = ei (the i-unit vector). Then [t c]W = wi ,

t

cx = xi ,

so Fun 1 yields v 5 wi + x2i v = zi . Then also v 5 wi /(1 − x2i ) 5

4 wi 3

whence

4 1 zi 5 wi + v 5 wi , 4 3

thus concluding the proof. To get the explicit finite number of inequalities for the Grenier fundamental domain, one simply follows through the inductive procedure using Lemma 2.5, cf. [Gre 88], pp. 301-302. In the sequel we use systematically the above notation: zi = zii = i-th diagonal component of the matrix Z . We conclude this section with further inequalities which are usually stated and proved in the context of so-called “reduction theory”, for elements of Posn in the Minkowski fundamental domain. These inequalities, as well as their applications, hold for the Grenier fundamental domain, cf. [Gre 88], Theorem 2, which we reproduce. Theorem 2.6. For Z ∈ Posn we have |Z| 5 z1 . . . zn , and for Z ∈ Fn , µ ¶n(n−1)/2 4 |Z| 5 z1 . . . zn 5 |Z| . 3 So |Z| =

µ ¶n(n−1)/2 3 znn . 4

Proof. We prove the first (universal) inequality by induction. As before, we use the first order Iwasawa decomposition of Z with the matrix W . The theorem is trivial for n = 1. Assume it for n − 1. Then zi = wi + x2i v (i = 1, . . . , n − 1), so by induction, |W | 5 w1 . . . wn−1 5 z1 . . . zn−1 . Hence |Z| = |W |v = |W |zn 5 z1 . . . zn , which is the desired universal inequality. Next suppose Z ∈ Fn . Again, we use induction for the right inequality. For n = 2, from Proposition 2.2 we get 2z12 5 z2 5 z1 . Therefore 2 z12 5

1 z1 z2 4


13

whence

3 4 z1 z2 or z1 z2 5 |Z|, 4 3 which takes care of n = 2. Assume the inequality for n − 1. In the first order Iwasawa decomposition, we have W ∈ Fn−1 . Then z1 . . . zn z1 . . . zn−1 z1 . . . zn = = |Z| |W |v |W | µ ¶n−1 4 w1 . . . wn−1 5 by Lemma 2.5 3 |W | µ ¶n−1+(n−1)(n−2)/2 4 5 by induction and W ∈ Fn−1 3 µ ¶n(n−1)/2 4 5 , 3 |Z| =

thus proving the desired inequality z1 . . . zn 5 ( 43 )n(n−1)/2 |Z|. The final inequality then follows at once from Lemma 2.5, that is zi = v = zn for all i. This concludes the proof. We give an application following Maass [Maa 71], Sect. 9, formula (8), which is used in proving the convergence of certain Eisenstein series. In order to simplify the notation, we write µ ¶n(n−1)/2 4 . c1 = c1 (n) = 3 Theorem 2.7. let Z ∈ Fn . Let Zdia be the diagonal matrix whose diagonal components are the same as those of Z. Then as operators on Rn , 1 nn−1 c1

Zdia 5 Z 5 nZdia . −1

Proof. Let r1 , . . . , rn be the eigenvalues of Z[Zdia2 ]. Then −1

−1 r1 + . . . + rn = tr(Z[Zdia2 ]) = tr(ZZdia )=n

and −1 r1 . . . rn = |Z| · |Zdia | = c−1 1

by Theorem 2.7. Hence for all i = 1, . . . , n, ri < n and ri =

1 . nn−1 c1

Therefore

1 −1 −1 In 5 Zdia2 ZZdia2 5 nIn . nn−1 c1 If A > 0 and C is invertible symmetric, then CAC > 0, so if A 5 B then 1 2 to conclude the proof. CAC 5 CBC, and we use C = Zdia

14


3 The Inductive Coordinates on SPosn In some important and especially classical contexts, one consider the special linear group SLn (R) and the special symmetric space: SPosn (R) = space of positive matrices with determinant 1. For the rest of this chapter, we consider only this case, and we follow Grenier [Gre 93]. Observe that we keep the same discrete group Γn = GLn (Z), because the operation of GLn (Z) leaves SPosn (R) stable. We now take Z ∈ SPosn (R), with the same first order Iwasawa decomposition as before, so that with some v ∈ R+ , W ∈ Posn (R), we have #µ " ¶ In−1 x W 0 , Z= 0 v 0 1 but to take account of the additional property of determinant 1, we put dn (Z) = v = a−1 n , so that (1)

Z=Z

(n)

=

"

W = a1/(n−1) Z (n−1) , n

In−1 0

#Ã 1/(n−1) (n−1) x an Z 1 0

x = x(n−1) 0 a−1 n

!

,

with Z (n−1) ∈ SPosn−1 (R), in particular, det Z (n−1) = 1. This particular choice of coordinates is useful for some inductive purposes. In case n = 2, this decomposition corresponds to expressing z = x + iy in the upper half plane h2 , and the part going to infinity corresponds to y → ∞, y = an , Z (n−1) = 1. In (1), if we factor out a−1 n we can write the decomposition in the form Ã ! n/(n−1) (n−1) an Z 0 (n) −1 Z=Z = [u(x)]an (1.a) , 0 1 Ã ! In−1 x with u(x) = . 0 1 We may then iterate the above construction, applied to Z (n−1) which has determinant 1. We let n 7→ n − 1. Let x(n−2) be the x-coordinate in the first order Iwasawa decomposition of Z (n−1) . Identify   In−2 x(n−2) 0   u(x(n−2) ) =  0 1 0 . 0

0

1

(n−1) Then using (1), and letting a−1 ), we get n−1 = dn−1 (Z

3 The Inductive Coordinates on SPosn



1/(n−1) 1/(n−2) (n−2) an−1 Z

an

 Z = [C] 

(2)

where

1/(n−1) −1 an−1

an

a−1 n

15

  

C = u(x(n−1) )u(x(n−2) ) . (n−1) )u(x(n−2) ), and Factoring out a−1 n , putting u2 = u(x 2 yn−1 = an/(n−1) a−1 n n−1 ,

we get Z=

(2a)





n−1

n−2 2 (n−2)  yn−1 an−1 Z [u2 ]a−1  n

2 yn−1

1

  .

We may continue inductively. Let u3 = u(x(n−1) )u(x(n−2) )u(x(n−3) ) for instance. Then (3) Z = [u3 ]  1/(n−1) 1/(n−2) 1/(n−3) an an−1 an−2 Z (n−3)  1/(n−1) 1/(n−2) −1  an an−1 an−2  1/(n−1) −1  an an−1

 a−1 n

   

(n−i) where a−1 ). We factor out a−1 n . This gives rise to a factor n−i = dn−i (Z n/(n−1)

an

(3a)

on each diagonal component. Then we rewrite (3) in the form: 

n−2 n−3 2 2 yn−1 yn−2 an−2 Z (n−3)

  Z = [u3 ]a−1 n 

2 2 yn−1 yn−2



2 yn−1

1

  . 

Thus we define inductively the standard coordinates. Letting n 7→ n − 1 7→ n − 2 and so forth: 2 yn−1 = an/(n−1) a−1 n n−1 (n−1)/(n−2) −1 an−2 (n−2)/(n−3) −1 an−2 an−3

2 yn−2 = an−1 2 yn−3 =

and so forth. Then we obtain inductively the partial Iwasawa decomposition with ui = u(x(n−1) )u(x(n−2) ) . . . u(x(n−i) ).

16




n−i+1

    −1  [ui ]an     

(4)



n−i 2 2 yn−1 . . . yn−i+1 an−i+1 z (n−i)

2 2 yn−1 . . . yn−i+1

..

. 2 2 yn−2 yn−1 2 yn−1

1

      .    

We carry this out to the end, and put X = upper triangular nilpotent matrix formed with the column vectors (x(1) , . . . , x(n−1) ). Thus     0 xij 0 x(1) . . . x(n−1) 0   0 0 0     X =. =. . .   . . .. .. ..  ..    .. .. 0

0

...

0 0

0

...

0

and

u(X) = In + X . We obtain the full Iwasawa decomposition:  2 2 yn−1 yn−2 . . . y12  ..  . −1  2 2 (5) Z = [u(X)]an  yn−2 yn−1  

 2 yn−1

with a full diagonal matrix formed with the standard coordinates. From the property that det(Z) = 1, we conclude ann =

(6)

n−1 Y

2(n−j)

yn−j

1

     

.

j=1

One may also use another convenient normalization by letting (7)

Zn−1 = ann/(n−1) Z (n−1)

so that

and therefore (8)

Z = [u(x(n−1) )]a−1 n

µ

W = a−1 n Zn−1 ,

Zn−1 0

¶ 0 . 1

Of course, Zn−1 does not have determinant 1, contrary to Z (n−1) . From the 2 and (7) we obtain definition of yn−1 (9)

−2 Zn−1 = an−1 Z (n−1) . yn−1

This formula then remains valid inductively replacing n − 1 by n − i for i = 1, . . . , n − 1. Note that an−1 = an−1 (Z (n−1) ), similar to an = an (Z (n) ) = an (Z) .

4 The Grenier Fundamental Domain and the Minimal Compactification

17

Remark. What we call the standard coordinates are actually standard in the literature, dating back to Minkowski, Jacobi, Siegel, etc. Actually, if one puts qi = yi2 then Siegel calls (q1 , . . . , qn−1 ) the normal coordinates, see, for instance, the references [Sie 45], [Sie 59]. Formulas for dn . Because the fundamental domain is partly defined by a minimality condition on the lowest right corner of matrices, we record formulas for dn in the present context, where dn denotes the lower right corner of an n × n matrix. For any Z ∈ Posn and any row vector (t c, d) ∈ t Rn , just by matrix multiplication we have (10)

an [t c, d]Z = [t c]Zn−1 + (t cx + d)2 .

Note that an = dn (Z)−1 . Formula (10) is set up so that it holds inductively, say for the first step, and any row vector (t c0 , d0 ) of dimension n − 1, (11)

an−1 [t c0 , d0 ]Z (n−1) = [t c0 ]Zn−2 + (t c0 x0 + d0 )2 .

In light of (9), the formula can be rewritten (12)

2 [t c0 , d0 ]Zn−1 = yn−1 ([t c0 ]Zn−2 + (t c0 x0 + d0 )2 ) .

Formulas (11) and (12) are set up so that they are valid replacing n − 1 by n − i, with c0 of dimension n − i − 1, d0 equal to a scalar, x0 of dimension n − i − 1. The formulas are set up for immediate application in the next section where we consider Z in a fundamental domain. The first condition defining such a domain will specify that the expression on the right of (11), or in parentheses on the right of (12), is = 1.

4 The Grenier Fundamental Domain and the Minimal Compactification of Γn\SPosn We define the fundamental domain SFn of Γn acting on SPosn to be the set of all matrices Z ∈ SPosn (R) satisfying for all primitive (t c, d) ∈ t Zn and notation as in Sect. 3, (1), (7): SFun 1. an [t c, d]Z = 1, or equivalently n/(n−1) (n−1) [t c]an Z + (t cx + d)2 = 1, or equivalently t t [ c]Zn−1 + ( cx + d)2 = 1.

18


SFun 2. Z (n−1) ∈ SFn−1 . SFun 3. 0 5 x1 5 1/2, and |xj | 5 1/2 for j = 2, . . . , n − 1. Special Case n = 2. In this case, we set an = y. Then SFun 1 amounts to x2 + y 2 = 1, which is the usual condition defining the lower part of the fundamental domain. Condition SFun 3 when n = 2 states that 0 5 x 5 1/2 . Thus F2 corresponds to the elements x + iy in h2 such that x2 + y 2 = 1

and

0 5 x 5 1/2 .

Thus we get half the usual fundamental domain, because we took the discrete group to be GLn (Z) rather than SL2 (Z). That the above conditions define a fundamental domain follows at once from the case for GLn . In the rest of the section, we give further inequalities which will be used for the compactification subsequently. The first inequality generalizes the inequality x2 + y 2 = 1 from n = 2. Lemma 4.1. Let Z ∈ SFn . For i = 1, . . . , n − 1 we have 2 x2n−i + yn−i =1

and

2 yn−i =

3 . 4

Hence we get Hermite’s inequality an (Z) = (3/4)(n−1)/2 . Proof. We choose d = 0 and c = en−1 (the standard unit vector with all components 0 except for the (n − 1)-component which is 1). Then SFun 1 yields 2 x2n−1 + yn−1 = 1, 2 = 34 . The coordinates yn−i are designed and since |xn−1 | 5 12 , we get yn−1 in such a way that this argument can be applied step by step, thus proving the first statement of the lemma. The Hermite inequality then follows from Sect. 3, (6).

Lemma 4.2. Let Z ∈ SFn . For all c(n−i) ∈ Zn−i (i = 1, . . . , n − 1) for which c(n−i) 6= 0 we have 2 . [t c(n−i) ]Zn−i = yn−i 2 Proof. This comes from the fact that in Sect. 3, (12) we get yn−1 times a number = 1 according to SFun 1.

4 The Grenier Fundamental Domain and the Minimal Compactification

19

Lemma 4.3. Let z ∈ SFn . For c(n−1) ∈ Zn−1 having cn−j 6= 0 if j 5 k + 1, we have 2 2 [t c(n−1) ]Z1 = yn−1 . . . yn−k . Proof. This comes inductively from Sect. 3, (12), where on the right we obtain 2 2 . . . yn−k times a number = 1, plus a number = 0. the product yn−1 Grenier carried out an idea of Satake for the Siegel modular group to the case of GLn , and we continue to follow Grenier. There are actually several compactifications, and we begin with the simplest inductive one. It is not clear to what extent this simplest suffices and for which purposes. Since the present discussion deals with SLn , we shall write Fn instead of SFn for simplicity. In case both GLn and SLn are considered simultaneously, then of course a distinction has to be preserved. We shall first define a compactification of Fn . Quite simply, we let Fn∗ = Fn ∪ Fn−1 ∪ . . . ∪ F1 . We shall put a topology of this union and show that Fn∗ then provides a compactification of Fn . The topology is defined inductively. For n = 1, F1 = {∞} is a single point. For n = 2, F2 is the usual fundamental domain, as we have seen, and its compactification is F2 ∪ {∞} = F2 ∪ F1 . Let n = 2. ∗ Let P ∈ Fn−k with 1 5 k 5 n − 1, so P ∈ Fn−1 . Let U be a neighborhood ∗ of P in Fn−1 . Let M > 0. Let: V (U, M ) = set of Z ∈ SPosn (R) such that an (Z) > M, Z (n−1) ∈ U, 1 0 5 x1 , and |xj | 5 for all j = 1, . . . , n − 1 . 2 Lemma 4.4. For M sufficiently large, V (U, M ) is contained in Fn . Proof. There are three conditions to be met. We start with SFun 1. We need to show that for all primitive (t c, d) we have an (Z)n/(n−1) [t c]Z (n−1) + (t cx + d)2 = 1 . ∗ We are given Z (n−1) ∈ U ⊂ Fn−1 . If c = 0, then d = ±1, so the above inequality is clear. Assume c 6= 0. Then we shall prove the stronger inequality ∗ with the term (t cx + d)2 deleted. From Z (n−1) ∈ Fn−1 , we get

t

[ c]Z

(n−1)

= an−1 (Z

(n−1)

µ ¶(n−1)/2 3 )= 4

20


by Hermite’s inequality (Lemma 4.1), and hence n/(n−1) t

an (Z)

[ c]Z

(n−1)

=M

n/(n−1)

µ ¶(n−2) 3 /2 . 4

Hence as soon as M is sufficiently large, we have the desired inequality which proves SFun 1. As to SFun 2, the inductive condition on the x-coordinate is met by definition of V (U, M ), so we have to verify SFun 1 in lower dimensions, or in other words an−i [t c(n−i) ]Z (n−i) = 1 for i = 1, . . . , n − 1, where c(n−i) is primitive in Zn−i . This follows from Sect. 3, (9) and Lemma 4.1. Finally, SFun 3 holds by the definition of V (U, M ). This proves the lemma. We define the topology on Fn∗ inductively. A neighborhood of a point ∗ is defined to be V (U, M ) ∪ U as above, i.e. it is the union of the P ∈ Fn−1 ∗ . If P ∈ Fn , then a fundamental system of part in Fn and the part in Fn−1 neighborhoods is given from the standard topology on Fn . Theorem 4.5. The space Fn∗ is compact. Proof. It suffices to show that any sequence {Z(ν)} in Fn has a subsequence with a limit in Fn∗ . By induction, without loss of generality we may assume that {Z(ν)(n−1) } converges to something in Fn−1 ∪ . . . ∪ F1 . From Lemma 4.1 (Hermite inequality), {an (ν)} is bounded away from 0, actually bounded from below by ( 34 )(n−1)/2 , but the precise value is irrelevant here. If {an (ν)} has a subsequence bounded away from ∞, then it has a subsequence which converges to a real number, and then the subsequence {Z(ν)} converges in a natural way. If on the other hand an (ν) → ∞, then Z(ν) converges to the above something by definition of the topology on Fn∗ . This concludes the proof. Remark. Satake has told us that actually the compactification of Γ\SPosn (R) can be described as follows. Since SPosn (R) is contained in Symn (R), one simply takes the closure of F¯n of the fundamental domain of Fn in the projective space PSymn (R). Then the compactification is Γ\ΓF¯n , which is the union of the Fk for k = 1, . . . , n.

5 Siegel Sets We follow [Gre 93]. Let Dn be the group of diagonal matrices with ±1 as diagonal elements. We let [ [γ]Fn . Fn± = γ∈Dn

This is a domain defined inductively by the conditions:

5 Siegel Sets

21

SFun 1± . Same as SFun 1. ± . SFun 2± . Z (n−1) ∈ Fn−1 SFun 3± . |xi | 5 1/2 for i = 1, . . . , n − 1. Thus Fn± has symmetry about 0 for all the x-coordinates. (n) (n) For T > 0 we define the Siegel set SieT,1/2 = SieT (since we don’t deal with another bound on the x-coordinates) by: (n)

SieT = set of Z ∈ SPosn such that |xij | 5 1/2 and yi2 = T for all i = 1, . . . , n − 1. Remark. For n = 2, a Siegel set is just a rectangle to infinity inside the usual half vertical strip −1/2 5 x 5 1/2, y > 0.

T

–1/2

1/2

1/2

Note that the largest value of T such that the Siegel set contains the fundamental domain is 3/4. The shaded portion just reaches the two corners. We then have the following rather precise theorem of Grenier. (n)

Theorem 5.1. Sie1

(n)

⊂ Fn± ⊂ Sie3/4 .

Proof. The inclusion on the right is a special case of Lemma 4.1. Now for the inclusion on the left, note that condition SFun 2± follows at once by induction, and SFun 3± is met by definition, so the main thing is to prove SFun 1± , for which we give Grenier’s proof. The statement being true for n = 2, we give the inductive step, so we index things by n, with Siegel sets (n) being denoted by SieT in SPosn , for instance. So suppose (n−1)

Sie1

± ⊂ Fn−1 .

(n)

Given Z ∈ Sie1 , and writing t c = (t c0 , d0 ), we have 2 2 [t c0 ]Zn−2 + yn−1 (t c0 , d0 )2 + (t cx + d)2 . [t c]Zn−1 + (t cx + d)2 = yn−1 (n−1)

But Z (n−1) ∈ Sie1

± ⊂ Fn−1 , so Lemma 4.2 implies that for

c0 ∈ Zn−2 , c0 6= 0,

22


we have 2 , [t c0 ]Zn−2 = yn−2

and hence we get the inequality 2 2 2 yn−1 [t c0 ]Zn−2 = yn−1 yn−2 = 1,

which proves SFun 1± in the case c0 6= 0, If c0 = 0, then it is easy to show that either 2 = 1, [t c]Zn−1 + (t cx + d)2 = yn−1

or [t c]Zn−1 + (t cx + d)2 = d2 = 1, which proves SFun 1± , and concludes the proof of the theorem.

2 Measures, Integration and Quadratic Model

We shall give various formulas related to measures on GLn and its subgroups. We also compute the volume of a fundamental domain, a computation which was originally carried out by Minkowski. Essentially we follow Siegel’s proof [Sie 45]. We note historically that people used to integrate over fundamental domains, until Weil pointed out the existence of a Haar (invariant) measure on homogeneous spaces with respect to unimodular subgroups in his book [We 40], and observed that Siegel’s arguments could be cast in the formalism of this measure [We 46]. Siegel’s historical comments [Sie 45] are interesting. He first refers to a result obtained by Hlawka the year before [Hla 44], proving a statement by Minkowski which had been left unproved for 50 years. However, as Siegel says, Hlawka’s proof “does not make clear the relation to the fundamental domain of the unimodular group which was in Minkowski’s mind. This relation will become obvious in the theorem” which Siegel proves in his paper, and which we reproduce here. The Siegel formula is logically independent of most of the computations that precede it. For the overall organization, and ease of reference, we have treated each aspect of the Haar measures systematically before passing on to the next, but we recommend that readers read the section on Siegel’s formula early, without wading through the other computations. The present chapter can be viewed as a chapter of examples, both for this volume and subsequent ones. The discrete subgroups GLn (Z) and SLn (Z) will not reappear for quite some time, and in particular, they will not reappear in the present volume which is concerned principally with analysis on the universal covering space G/K with G = SLn (R) and K = Unin (R) (the real unitary group). Still, we thought it worthwhile to give appropriate examples jumping ahead to illustrate various concepts and applications. The next chapter will continue in the same spirit, with a different kind of application. Readers in a hurry to get to the extension of Fourier analysis can omit both chapters, with the exception of Sect. 1 in the present chapter. Even Sect. 1 will be redone in a different spirit when the occasion arises. Jay Jorgenson: Posn (R) and Eisenstein Series, Lect. Notes Math. 1868, 23–47 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com °

24

2 Measures and Integration

1 Siegel Sets and Finiteness of Measure Mod SLn(Z) We assume that the reader is acquainted with the basic computations of Haar measure in an Iwasawa decomposition, as in [JoL 01], Chap. 1, Sect. 2. In this section, we give an application of the basic Haar measure formulas. We recall that in Iwasawa coordinates G = U AK, for Haar measures dx, du, da, dk, Z Z Z Z f (x)dx = C f (uak)δ(a)−1 dudadk . G

U

A

K

For t, c > 0 we define the following subsets of U and A in SLn (R): Uc = subset of u ∈ U with |uij | 5 c At = subset of a ∈ A with ai = tai+1 for i = 1, . . . , n − 1. We then define the Siegel set Siet,c = Uc At K in SLn (R) , or we may work just on the subgroup U A, in which case we would specify that Siet,c = Ut Ac . On SLn (R), Siet,c thus consists of all elements uak with |uij | 5 c and ai = tai+1 for i = 1, . . . , n − 1. Since we are on SLn (R), we take the quotients qi = ai /ai+1 (i = 1, . . . , n − 1) for coordinates, which are called normal coordinates by Siegel [Sie 45] and [Sie 59]. Then the coordinates (q1 , . . . , qn−1 ) given an isomorphism (q1 , . . . , qn−1 ) : A → R+(n−1) . Theorem 1.1. A Siegel set in SLn (R) has finite Haar measure. Proof. [Sie 59] Since K is compact and Uc has bounded (euclidean) measure, it follows that Z Z Z Z δ(a)−1 d∗ a . dg = C Uc

At

K

At

Hence it suffices to prove that this integral over At is finite. Using the coordinates q1 , . . . , qn−1 , the fact that Haar measure on each factor of R+(n−1) is dqi /qi , and the fact that n−1 Y

qimi

with mi = 1 ,

d a=

Z∞

Z∞ Y

δ(a) =

i=1

we find that Z

At

−1 ∗

δ(a)

t

···

t

which is finite, thus proving the theorem.

qi−mi

Y dqi qi

,

2 Decompositions of Haar Measure on Posn (R)

25

In [Sie 59] Siegel used the above result to show that SLn (Z)\SLn (R) has finite measure. He also used the normal coordinates to construct a compactification of SLn (Z)\SLn (R). By Theorem 5.1 of Chap. 1, we know that a fundamental domain for SLn (Z) is contained in a Siegel set, and hence we have given one proof of Theorem 1.2. The quotient space SLn (Z)\SLn (R) has finite measure.

2 Decompositions of Haar Measure on Posn(R) Next we shall deal with formulas for integration on the space Posn = Posn (R). It is a homogeneous space, so has a unique Haar measure with respect to the action of G = GLn (R), up to a constant factor. We follow some notation systematically as follows. If Y = (yij ) is a system of coordinates from euclidean space, we write Y dµeuc (Y ) = dyij for the corresponding euclidean (Lebesgue) measure. Here we shall reserve the letter Y for a variable in Posn , so the indices range over 15i5j5n, and the product expression for dY is thus taken over this range of indices. Deviations from Lebesgue measure will be denoted by dµ(Y ), with µ to be specified. If ϕ is a local C ∞ isomorphism, we let J(ϕ) be the Jacobian factor of the induced map on the measure, so it is the absolute value of the determinant of the Jacobian matrix, when expressed in terms of local coordinates. Often the determinant will be positive. If g is a square matrix, we let |g| denote its determinant, and kgk is then the absolute value of the determinant. The exposition of the computation of various Jacobians and measure follows Maass [Maa 71], who based himself on Minkowski and Siegel, in particular [Sie 59]. Proposition 2.1. A GLn (R)-bi-invariant measure on Posn is given by dµ(Y ) = |Y |−(n+1)/2 dµeuc (Y ) . For g ∈ GLn (R), the Jacobian determinant J(g) of the transformation [g] is J(g) = kgkn+1 . The invariant measure satisfies dµn (Y −1 ) = dµn (Y ), i.e. it is also invariant under Y 7→ Y −1 .

26


Proof. We prove the second assertion first. Note that g 7→ J(g) is multiplicative, that is J(g1 g2 ) = J(g1 )J(g2 ), and it is continuous, so it suffices to prove the formula for a dense set of matrices g. We pick the set of matrices of the form gDg −1 , with D = diag(d1 , . . . , dn ) being diagonal. Then [D]Y is the matrix (di yij dj ). Hence Y J(gDg −1 ) = J(D) = |di dj | = kDkn+1 = kgDg −1 kn+1 , i5j

which proves the formula for J(g). Then dµn ([g]Y ) = k[g]Y k−(n+1)/2 J([g])

Y

dyij

i5j

= kgk−(n+1) kY k−(n+1)/2 kgkn+1

Y

dyij

i5j

= dµn (Y ) , thus concluding the proof of left invariance. Right invariance follows because J(g) = J(t g). Finally, the invariance under Y 7→ Y −1 can be seen as follows. If we let S(Y ) = Y −1 , then for a tangent vector H ∈ Symn , S 0 (Y )H = −Y −1 HY −1 , so det S 0 (Y ) = J(Y −1 ) = |Y |−(n+1) . Then dµn (Y −1 ) = |Y |(n+1)/2 |Y |−(n+1) dµeuc (Y ) = dµn (Y ) , thus concluding the proof of the proposition. Full Triangular Coordinates + Let Tri+ n (R) = Trin be the group of upper triangular matrices with positive diagonal coefficients. Then in the notation of Sect. 1, we have the direct decomposition AU = Tri+ n .

We also have the C ∞ isomorphism t Tri+ n → Posn given by T 7→ T T .

In Sect. 1 we recalled that a Haar measure on Tri+ n is given by dµ Tri (T ) = δ(T )−1 β(T )−1 dµeuc (T ) ,


where dµeuc (T ) =

Y

27

dtij

i5j

is the ordinary euclidean measure. Note that we are following systematic notation where we use a symbol µ to indicate deviation from euclidean measure. For the triangular group Tri+ , the variables i and j range over 1 5 i 5 j 5 n. We shall usually abbreviate tii = ti . First we decompose the Iwasawa coordinates stepwise, going down one step at a time. We write an element Y ∈ Posn in inductive coordinates µ ¶ t y z Y = with y ∈ R+ , z ∈ Rn−1 , Yn−1 ∈ Posn−1 . z Yn−1 Thus Y = Y (y, z, Yn−1 ) . We have the first decomposition of an element T ∈ Tri+ n: ¶ µ t x t1 = T (t1 , x, Tn−1 ) , T = 0 Tn−1 so (t1 , x, Tn−1 ) are coordinates for T , and we have the mapping + + t ϕ+ 1,n−1 : Trin → Posn given by ϕ1,n−1 (T ) = T T .

Direct matrix multiplication gives (1) whence (2)



Y = ϕ+ (T ) = ϕ+ (t1 , x, Tn−1 ) =   2t1 ∂(y, z, Yn−1 ) ∂(Y ) = = 0 ∂(T ) ∂(t1 , x, Tn−1 ) 0

t21 +t xx

t t

x Tn−1

Tn−1 x

Yn−1

∗

Tn−1 0

0 ∗

∂(Yn−1 ∂(Tn−1 ) .

 

 

Thus we obtain (3)

¯ ¯ ¯ ∂(Yn−1 ) ¯ ¯ ¯ J(ϕ ) = 2t1 |Tn−1 | ¯ ∂(Tn−1 ) ¯ = 2n (t1 · · · tn )(t2 · · · tn ) · · · tn n Y = 2n tii . +

i=1

Thus not only do we get the inductive expression (3), but we can state the full transformation formula:

28


+ t Proposition 2.2. Let ϕ+ : Tri+ n → Posn be the map ϕ (T ) = T T . Let ti = tii be the diagonal elements of T . Then

J(ϕ+ ) = 2n

n Y

tii = 2n β(T ) ;

i=1

or in terms of integration, Z Z f (T t T )J(ϕ+ )(T )dµeuc (T ) . f (Y )dµeuc (Y ) = Posn

Tri+ n

Then for the Haar measures of Propositions 1.4 and 2.1, we have dµn (Y ) = 2n dµ Tri (T ) . Written in full, this means Z

f (Y )dµn (Y ) =

Posn

Z

...

Z

f (T t T )

n Y

ti−n i

i=1

n Y 2dti Y

i=1

ti

dtij ,

i<j

where the integrals over t1 , . . . , tn are from 0 to ∞, and those over tij (with i < j) are from −∞ to ∞. There is a corresponding map ϕ− given by − t ϕ− : Tri+ n → Posn given by ϕ (T ) = T T .

Similarly, we have the map ϕ− 1,n−1 (T )

t

t

= T T when T =

µ

0

t1 x

Tn−1

¶

.

Direct multiplication gives (1− )



Y = ϕ− (T ) = ϕ− (t1 , x, Tn−1 ) = 

whence 

(2− )

   ∂(Y )   = ∂(T )    

2t1 0 .. .

t1 t x

t21 t

t1 x

Tn−1 Tn−1

∗ t1 ..

0

. 0 · · · t1

0

0···0

∂(Yn−1 ) ∂(Tn−1 )

          

 


29

so we obtain J(ϕ− ) = 2tn1 J(ϕ− 1,n−1 )

(3− )

= 2n

n Y

tn−i+1 i

i=1 n

= 2 δ(T )β(T ) . Thus we can state the analogous transformation formula: Proposition 2.3. Let ϕ− : Tri+ → Posn be the map ϕ− (T ) = t T T . Then J(ϕ− ) = 2n

n Y

tn−i+1 = 2n α(T )β(T ) ; i

i=1

or in terms of integration, Z Z f (t T T )J(ϕ− )(T )dµeuc (T ) f (Y )dµeuc (Y ) = Posn

Tri+ n

= 2n

Z

f (t T T )δ(T )β(T )dµeuc (T ) .

Tri+ n

The triangularization has a variation depending on how we write it. Let us write a positive diagonal matrix   1/2   a11 . . . 0 0 a11 . . .  . ..  ..  ..  so A1/2 =  .. .. . .  A =  ...  .   . .   0 . . . ann 1/2 0 . . . ann A full Iwasawa decomposition for Y can be written in the form 1

Y = [u(X)]A = T t T with T = u(X)A 2 . 1

1

2 Then tii = aii2 and 2dtii /tii = daii /aii . Furthermore, tij = xij ajj for i < j. Then 1 2 dxij + a term with daij . dtij = ajj

Plugging in the formula for dµn (Y ) in Proposition 2.2, we find: Proposition 2.4. Let Y = [u(X)]A. Then dµn (Y ) =

n Y

i=1

Similarly, on the other side:

i−(n+1)/2

aii

n Y daii Y

i=1

aii

i<j

dxij .

30


Proposition 2.5. Let Y = A[u(X)]. Then dµn (Y ) =

n Y

−i+(n+1)/2

aii

i=1

n Y daii Y

i=1

aii

dxij .

i<j

Remark. We meet here a density related to the Iwasawa density of Sect. 1. Let us define n Y −i+(n+1)/2 aii . δPos (a) = i=1

Then

1 2 δPos = δIw =ρ.

Block Decomposition of Triangular Coordinates Next we give similar formulas for the partial decomposition in blocks of arbitrary size. Let 0 < p < n and p + q = n. For x ∈ Rp×q let µ ¶ Ip X u(X) = . 0 Iq We can write an element Y ∈ Posn in the form µ ¶ W 0 (4) Y = [u(X)] = ϕ+ p,q (W, X, V ) 0 V with W ∈ Posp , V ∈ Posq , and X ∈ Rp×q . Then Y is decomposed into rectangular blocks illustrated on the following figure: 

p×p

 Y1   Y =   t  Y2

p×q

Y2

Y3

       

This decomposition gives partial coordinates Y = Y (W, X, V ), with the map p×q ϕ+ × Posq → Posn p,q : Posp × R

as above. A direct check of dimensions shows that they add up properly, that is n(n + 1) p(p + 1) (n − p)(n − p + 1) + + p(n − p) = . 2 2 2 Direct multiplication in (4) yields the explicit expression


(5)



 ϕ+ p,q (W, X, V ) =

W + [X]V

XV

V tX

V

31



 .

From (5), one sees that ϕ+ p,q is bijective, because first, V uniquely determines the lower right square Y3 . Then X is uniquely determined to give Y2 = XV , and finally W is uniquely determined to give Y1 . Note that aside from this formal matrix multiplication, one has Y > 0 if and only if W > 0 and V > 0, and X is arbitrary. Proposition 2.6. The Jacobian is given by p J(ϕ+ p,q ) = |V | .

For Y = ϕ+ p,q (W, X, V ) we have the change of variable formula dµn (Y ) = |W |−q/2 |V |p/2 dµeuc (X)dµp (W )dµq (V ) . Proof. We compute the Jacobian matrix,  Ir 0  ∂(Y )  =  ... ∂(W, X, V )  0 0

and find ∗...∗ V ...0 .. . . .. . .. 0...V 0...0

∗ ∗ .. .



     ∗ Is

with V occurring p times as blocks on the diagonal, r = p(p + 1)/2 and s = q(q +1)/2. Taking the determinant yields the stated value. For the change of variable formula, we just plug in using the definitions dµp (W ) = |W |−(p+1)/2 dµeuc (W ) , and similarly with n and q, combined with the value for the Jacobian. The formula comes out as stated. One may carry out the similar analysis with lower triangular matrices. Thus we let Tri− n be the space of lower triangular matrices, with the map − t ϕ− : Tri+ n → Posn defined by ϕ (T ) = T T .

Then we have the partial map (6)

Y = ϕ− p,q (W, X, V )    W 0 W = = [t u(X)]  t 0 V XW

WX W [X] + V



.

32


Proposition 2.7. The Jacobian is given by −q J(ϕ− . p,q ) = |W |

For Y = ϕ− p,q (W, X, V ) the change of variable formula is dµn (Y ) = |W |q/2 |V |−p/2 dµeuc (X)dµp (W )dµq (V ) . The proofs are exactly the same as for the other case carried out previously, and will therefore be omitted. Polar Coordinates There is another decomposition besides the Iwasawa decomposition, giving other types of information about the invariant measure, which we shall present below, namely polar coordinates. These have been considered in the general context of semisimple Lie groups and symmetric spaces (cf. Harish-Chandra [Har 58a,b] and Helgason’s book [Hel 84]). We learned from Terras [Ter 88] that statisticians dealt with certain special cases and computations on GLn (R), notably Muirhead, and we found her book helpful in writing up the rest of this section. See notably 4.1 Exercise 24, and 4.2 Proposition 2. A standard theorem from linear algebra states that given a positive definite symmetric operator on a finite dimensional real vector space with a positive definite scalar product, there exists an orthonormal basis with respect to which the operator is diagonalized. Since the eigenvalues are necessarily positive, this means in our set up that every element Y ∈ Posn can be expressed in the form Y = [k]a with k ∈ K and a ∈ A , where as before A is the group of diagonal matrices with positive diagonal elements. For those matrices with distinct eigenvalues (the regular elements) this decomposition is unique up to a permutation of the diagonal elements, and elements of k which are diagonal and preserve orthonormality, in other words, diagonal elements consisting of ±1. Hence the map p : K × A → Posn given by (k, a) 7→ kat k = kak−1 = Y is a covering of degree 2n n! over the regular elements. This map is called the polar coordinate representation of Posn , and (k, a) are called the polar coordinates of a point. As mentioned above, these coordinates are unique up to the above mentioned 2n n! changes over the regular elements. We want to give an expression for the Haar measure on Posn in terms of the polar coordinates, and for this we need to compute the Jacobian J(p). For k ∈ K, we have t kk = I, t k = k −1 , dk = ((dk)ij ), so (6)

(dt k)k + t kdk = 0

and so dt k = −k −1 dkk −1 .


33

Then we let ω(k) = k −1 dk so t ω = −ω .

Thus ω is a skew symmetric matrix of 1-forms,

((k −1 dk)ij ) = (ωij (k)) , with component being 1-forms ωij . Observe that each such form is necessarily left K-invariant, that is for any fixed k1 ∈ K, we have ω(k1 k) = ω(k) directly by substitution. Taking the wedge product ^ ωij i<j

in some definite order yields a volume form on K, which is left invariant, and therefore right invariant. The absolute value of this form then represents a Haar measure on K, which we denote by νn , or dνn (k) if we use it inside an integral sign. We call νn the polar Haar measure, so ¯ ¯ ¯^ ¯ ωij ¯¯ . dνn = ¯¯ i<j

Of course we can compare νn with the universally normalized Haar measure on compact groups,written µn , such that Z dµn = 1 = µn (K) . K

Then dνn (k) = νn (K)dµn (k) where νn (K) is the total νn -measure of K, which we have to compute. For the moment, we go on with the differential of the polar coordinate map. Formula 2.8. dp(k, a) = k(da + ωa − aω)k −1 = [k](da + ωa − aω) and (ωa − aω)ij = (aj − ai )ωij .

In the above formula, da is the Euclidean diag(da1 , . . . , dan ). Proof. Matrix multiplication being bilinear, we have dp(kat k) = (dk)at k + kdat k + kadt k . But also t kk = I implies (dt k)k + t kdk = 0 as we have seen, so dt k = −t k(dk)t k .

Substituting this value for dt k = dk −1 into the previous formula and using the skew symmetry t ω = −ω yields the formula.

34


We note that the computation of Formula 2.6 can also be written dY = [k](da + ωa − aω) = matrix of 1-forms (dyij ) . Taking the wedge product will allow us to determine various relations between previous measures. Proposition 2.9. These measures satisfy the infinitesimal relation Y Y dµeuc (Y ) = |ai − aj | dai dνn (k) . i<j

Letting Y

γ(a) =

i<j

|ai − aj |

n Y

−(n−1)/2

ai

i=1

this yields the comparison of Haar measures locally where p is bijective: Y dai /ai . dµn (Y ) = γ(a)d∗ adνn (k) where d∗ a = For a function f with compact support on Posn , we thus have globally Z Z Z f ([k]a)γ(a)d∗ adνn (k) . f (Y )dµn (Y ) = (2n n!)−1 Posn

A

K

Proof. We take the wedge product of the forms in Formula 2.8, that is ^ Y ^ ± dai ∧ (aj − ai ) ωij . i<j

i<j

Taking absolute values yields the first formula. The second formula concerning the Haar measures follows from the definition of the Haar measure in Proposition 2.1. This concludes the proof. We shall now compute the constant νn (K), following Muirhead and the computation reproduced in Terras [Ter 88]. First we need a remark on determinants. Let V = (vij ) be an n × n matric of vectors in a vector space, and C = (cij ) be a scalar matrix. Let W = V C = (wij ) . Then ±

^ i,j

wij =

^

vij (det C)n .

i,j

This is immediate from the usual rule, when one applies a matrix to a row (v1 , . . . , vn ) of vectors, in which case one gets a factor of det C. Here we perform this operation n times, whence the power (det C)n .


35

Lemma 2.10. Let K × Tri+ → G = GLn (R) be the product map (k, T ) 7→ kT . Put X = kT , and dX = (dxij ). Then n Y

dµeuc (X) =

tn−i ii dµeuc (T )dνn (k) .

i=1

Q Proof. By definition, dµeuc (X) = dxij taken over all i, j = 1, . . . , n. We have dX = (dk)T + kdT so k −1 dX = (dT T −1 + ω)T . Hence taking the wedge product of all the components, up to sign, we get n ^

dxij =

i,j=1

^

((dT T −1 )ij + ωij ) det(T )n .

i,j

But dT T −1 is an upper triangular matrix (with components 0 unless i 5 j), and we have seen that ω is skew symmetric. Hence separating the wedge products over i 5 j and i > j, we find up to sign ^ ^ ^ dxij = ((dT T −1 )ij + ωij ) ∧ ωij (det T )n . i,j

i>j

i5j

The product of ωij over i > j is a volume form (maximal degree form) on K. Hence wedging it with any one of these 1-forms yields 0. Hence we find ¯¯ ¯ ¯ ¯V ¯¯ V ¯ −1 ¯ ¯ ¯ ωij ¯¯(det T )n dµeuc (X) = ¯ ((dT T )ij ¯¯ i>j

i5j

=

n Q

i=1

n−i tii

as was to be shown.

Q

i<j

¯ ¯ ¯V ¯ ¯ dtij ¯ ωij ¯¯ i<j

n √ n(n+1)/2 Q Theorem 2.11. We have νn (K) = 2n π Γ(j/2)−1 . j=1

Proof. Let f (X) = exp(−tr(t XX)) for X ∈ Matn (R), so f (X) splits Y exp(−x2ij ) . f (X) = i,j

36


Then π

n2 /2

=

Z

dµeuc (X) =

f (X)

Matn

[by Lemma 2.10]

Z

f (X)dµeuc (X)

GLn

=

Z Z Tri+

f (kT )

K

= νn (K)

Z

Y

tn−i ii dµeuc (T )dνn (k)

exp(−t T T )

Tri+

Y

tn−i ii

Y

dtii

Y

dtij .

i<j

Now the integral over Tri+ splits into a product of several single integrals over i = 1, . . . , n. The integrals over the variables tii are from 0 to ∞, µ ¶ Z ∞ n−i+1 1 −t2 n−i e t dt = Γ 2 2 0 after we let uQ= t2 , du = 2tdt. The product of these integrals is therefore precisely 2−n Γ(i/2). The other integrals over the tij (i < j) are individually of the form Z ∞ √ 2 e−t dt = π , −∞

√ n(n−1) . and there are n(n − 1) of them, thus giving rise to the power π Multiplying these two types of products and solving for νn (K) gives the stated value. Corollary 2.12. For f ∈ Cc (Posn ) we have Z Z Z n −1 f (Y )dµn (Y ) = (2 n!) νn (K) f ([k]a)γ(a)d∗ adk , Posn

A

K

where dk is the Haar measure on K with total measure 1, and (2n n!)−1 νn (K) =

√ n(n+1)/2 1 π Q . Γ(i/2) n!

3 Decompositions of Haar Measure on SPosn This section complements the preceding one by giving a measure decomposition via SLn , and also working out some inductive block decompositions on SLn itself.

3 Decompositions of Haar Measure on SPosn

37

In the first place, every Y ∈ Posn can be written uniquely in the form Y = r1/n Z with r > 0 and Z ∈ SPosn . Thus we have a product decomposition Posn = R+ × SPosn−1 in terms of the coordinates (r, Z). There exists a unique SLn (R)-invariant (1) measure on SPosn , denoted by µn , such that for the above product decomposition, we have dr (1) dµn (Y ) = dµn (Z) . r Warning: This decomposition does not come from the Jacobian of the coordinate mapping! Immediately from the above decomposition, we obtain: Proposition 3.1. For a function f on SPosn , Z Z f (|Y |−1/n Y )|Y |dµn (Y ) =

f (Z)dµ(1) n (Z) .

SPosn

|Y |51

On the other hand, say for a continuous function g on R+ , Z

g(|Y |)dµn (Y ) = Voln

Γn \Posn 0 0 all i . .. T =  ... ii . .  0

...

tnn

For an n-tuple s = (s1 , . . . , sn ) of complex numbers, we define χs (T ) =

n Y

s

tjjj

j=1

and we define the basic character ρs on Posn to be ρs (Y ) = χ2s (T ) =

n Y

2s

tjj j if Y = T t T = [T ]I .

j=1

Since a character on Posn (R) amounts to a character on the n-fold product of the multiplication group R+ , it follows that all characters are parametrized by such n-tuples of complex numbers. However, for much of the formalism,

1 Characters of Posn

51

it is less cumbersome to leave out the complex variables. Note that for any α ∈ C, dα ρs = ρs+α where s + α = (s1 + α, . . . , sn + α) . Observe that the transpose maps Tri+ to Tri− and preserves the diagonal elements. We could have carried out the same construction with Tri− instead of Tri+ . To relate the two constructions, it is useful to introduce the reversing matrix 1

0 ω=

1 1

0

consisting of 1 on the antidiagonal, and 0 elsewhere. Then ω 2 = I, and for T ∈ Tri+ we have   tnn 0   tn−1,n−1   [ω]T = T [ω] =   ∈ Tri− . ..   . ∗ t11 Furthermore, the diagonal elements are permuted by the action of [ω]. The operation [ω] is multiplicative, and is simply conjugation by ω, defined on all of G. We note that [ω] : Tri+ → Tri−

is a group isomorphism. It allows us to deal with the map Y 7→ Y −1 . If we write Y = TY t TY , then [ω]Y −1 = ([ω] t TY−1 )[ω]TY−1 , so we obtain the formula T[ω]Y −1 = [ω] t TY−1 .

(2)

Define ρ∗ (Y ) by the formula (3)

ρ∗ (Y ) = ρ([ω]Y −1 ) = ψ([ω] t TY−1 ) so ρ(Y −1 ) = ρ∗ ([ω]Y ) .

If ψ = χ2s as above, and T ∈ Tri+ , then Y −2s −2sn 1 (4) tii n−i+1 = t−2s . ψ([ω] t T −1 ) = nn · · · t11 So we define

ψ ∗ (T ) = ψ([ω] t T −1 ) . Proposition 1.1. The map ρ 7→ ρ∗ is an involution on the group of characters. The character associated with ρ∗ is ψ ∗ . For α ∈ C, (dα )∗ = d−α .

52

3 Special Functions on Posn

Proof. Immediate from (3) and (4) above, and the definition of ρ∗ . The above involution gives one mechanism to go back and forth between the upper triangular action and the lower triangular action on the left. The non-commutativity forces the development of some sort of formalism to deal with it. The advantage of dealing with [ω] is that it allows us to define an involution on the group of characters. However, it stems from the fact that Posn is actually both a left G-space and a right G-space. The map S : Y 7→ Y −1 leaves Tri+ and Tri− stable, and interchanges left characters (for Tri+ ) to right characters. For a left character ρ, we may denote ρ0 the right character such that ρ0 (Y ) = ρ(Y −1 ) so ρ00 = ρ . On the other hand, [ω] interchanges Tri+ and Tri− , as well as it interchanges left and right characters. Indeed, for T ∈ Tri+ and abbreviating [ω]X = X ω , we have ρ ◦ [ω](Y [T ]) = ρ(t T ω Y ω T ω ) = ψ(t T ω )ρ([ω]Y ) ,

so ψρ◦[ω] (T ) = ψ(t T ω ) is the character on Tri+ associated with ρ ◦ [ω]. Taking the composite ρ∗ = ρ0 ◦ [ω] = (ρ ◦ [ω])0 then yields the involution on left characters. Next, let us define an involution on the space of n-complex variables, that is, s∗ = (−sn , . . . , −s1 ) .

Thus s∗1 = −sn−i+1 for i = 1, . . . , n. We want a character hs = ρs# with a suitable vector s# such that h∗s = hs∗ . Trivial fooling around with the index and solving a linear equation shows: Proposition 1.2. Let (4)

hs (Y ) =

n Y

(tn−i+1 )2si +i−(n+1)/2 .

i=1

Then h∗s = hs∗ . Furthermore, we have the identity (5)

hs = ρs#

where s# i = sn−i+1 +

n−i n−1 − . 2 4

1 Characters of Posn

53

For the record, we note that we can also write hs (Y ) =

(6)

n Y

t2sn−i+1 −i+(n+1)/2 ,

i=1

as well as hs (Y ) = |Y |−(n+1)/4

(7)

n Y

(tn−i+1 )2si +i .

i=1

Subdeterminants Let Y ∈ Posn . For each j with j = 1, . . . , n let Subj Y be the i × j upper left square submatrix of Y , and let Subj Y be the lower right square submatrix of Y , as shown on the figure. j Subj Y

j

Subj Y

j

j

Authors dealing with the right action of Tri+ then consider Subj Y and corresponding characters. With our left action, we consider Subj Y . The reason is given in the next proposition. Proposition 1.3. The map Y 7→ Subj Y is a homomorphism for the left action of Tri+ , and the map Y 7→ Subj Y is a homomorphism for the right action of Tri+ (left action of Tri− ). Proof. Let T be a square matrix decomposed into blocks t    A 0 A B  .  so t T =  T = t B tD 0 D Then t

t

TT = 

t

AA BA

so the proposition is clear.

t

AB ∗





 and T t T = 

∗

B tD

t

t

D B

D D



 ,

54


Proposition 1.4. Let j be a positive integer, 1 5 j 5 n. Let σ be a character on Posj . Then Y 7→ σ(Subj Y ) is a character on Posn . Proof. Proposition 1.3. Characters are of course meant in the sense we have fixed, for the left action of Tri+ . If one considers right action then Subj Y has to be replaced by Subj Y to make the assertion valid. Proposition 1.5. Index [ω] as [ωn ] for its action on n × n matrices. Then Subj [ωn ]Y = [ωj ]Subj Y . Proof. Check it out directly for n = 2, then n = 3, and then do what you want with induction and matrix multiplication. The Selberg Power Function For application to a later chapter, we shall consider a special character which plays a role in the theory of Eisenstein series. Readers may omit the following considerations until they are applied, because for the general theory, the particular choice of variables we are going to make is at beast irrelevant, and at worst it obscures the general latent structure. Two sets of variables are going to play a role, so we consider new complex variables z = (z1 , . . . , zn ). We define the Selberg power function qz(n) (Y ) =

n Y

i=1

|Subj Y |zj and qz(n−1) (Y ) =

n−1 Y i=1

|Subj Y |zj .

Thus the power function on the right depends only on n − 1 variables z1 , . . . , zn−1 . Note that for either function q−z = qz−1 . Both power functions are left characters. For right characters (as in other authors), one defines the power function pz by taking the product with Subj instead of Subj . Proposition 1.6. Write Y = T t T with T ∈ Tri+ . Let ti (i = 1, . . . , n) be the diagonal elements of T , so ti = ti (Y ). Then (5)

|Subj Y | =

n Y

i=n−j+1

t2i ;

2 The Gamma Function (n) q−z (Y

(6)

)= =

j n Y Y

55

(t2n−k+1 )−zj

j=1 k=1 n Y −2(zi +...+zn ) tn+1−i i=1

.

Proof. Immediate from the definitions. A change of variables relates the Selberg power function in terms of the function hs defined previously. Let zj = sj+1 − sj +

1 . 2

Proposition 1.7. On Posn , we have (n−1)

q−z

= hs , with s = (s1 , . . . , sn ) .

Proof. Immediate from the definitions. (n)

Remark. For q−z we are at the edge where the introduction of the additional variable sn+1 raises the need for a boundary condition. The relationship on Posn still reads (n) q−z = hs with s = (s1 , . . . , sn ) with the provision that in the relation zn = sn+1 − sn + 12 the special value sn+1 = −(n + 1)/4. With this provision, the variables (z1 , . . . , zn ) are still related by an invertible affine map with the variables (s1 , . . . , sn ).

2 The Gamma Function Classical integral transforms on the positive multiplicative group Pos1 and some of their properties extend to Posn , as we shall see. First we define the Mellin transform of a function f to be Z Mf (ρ) = f (Y )ρ(Y )dµn (Y ) . Posn

Note that the transform is a function of characters (left or right). If we express ρ as ρs , then one may write Mf (s), in which case we view Mf as a function of the n complex variables s1 , . . . , sn . As a first example, we define the gamma function Γn to be Z e−tr(Y ) ρ(Y )dµn (Y ) . Γn (ρ) = Posn

When n = 1 and ρ = ρs , then the above value coincides with the usual gamma function of a single variable s. Actually, the integral can be expressed in terms of the usual gamma function as follows.

56


Proposition 2.1. For ρ = ρs , the above integral is absolutely convergent for Re(si ) > (n − i)/2, and has the value ¶ µ √ n(n−1)/2 Y n−1 Γn (ρs ) = π Γ si − . 2 i In the normalization of Proposition 1.2, µ ¶ √ n(n−1)/2 Y n−1 Γ si − Γn (hs ) = π . 4 i Proof. We use the change of variables formula from Y to T in Chap. 2, Proposition 2.2, which yields Z e−tr(Y ) ρs (Y )dµn (Y ) Posn

=

Z

exp(−tr(T t T ))

Tri+

Y

i n t2s tii 2

Y

ti−n−1 ii

Y

dtij .

i5j

The matrix multiplication shows that the diagonal elements of T t T are   2 t11 + . . . + t21n   t222 + . . . + t22n   T tT =   . ..   2 tnn It follows that the integral splits into a product of single integrals as follows: For indices i < j, we have the product ∞ Y Z

exp(−t2ij )dtij =

√

π

n(n−1)/2

.

i<j −∞

For indices i = j, so over the variables t11 , . . . , tnn we have the product of i-th terms, i = 1, . . . , n, using a single variable s for simplicity, y = t2 , and dy/y = 2dt/t, Z∞

−t2 2(s+(i−n)/2)

e

t

0

dt 2 = t

Z∞

e−y y (s+(i−n)/2)

0

µ

i−n = Γ s+ 2 This concludes the proof.

¶

.

dy y

2 The Gamma Function

57

Remark. Mellin transforms and gamma functions as above and more generally hypergeometric functions occur notably in Gindikin [Gin 64]. In the early days, as in [Sie 35] and [Her 55], only the determinant character was used to define a gamma function. Selberg saw further with his power function [Sel 56]. We shall now extend systematically the standard formalism of the gamma integral and subsequently the K-Bessel integral to the matricial case. Proofs which used only the invariance of the measure dy/y on the positive multiplicative group, and an interchange of integration, go over systematically. We start with the standard result that for a > 0, we have Z∞

(5)

e−ay y s

dy = Γ(s)a−s . y

0

We may also cast the generalization in the formal context of a convolution, that is for any function h, under conditions of absolute convergence, we define its gamma transform to be its convolution with the kernel −1 (Z, Y ) 7→ e−tr(Z Y ) , for which we use the notation Z −1 e−tr(Y Z ) h(Y )dµn (Y ) . (Γn #h)(Z) = Posn

The integral is absolutely convergent in a half plane of characters ρ. Proposition 2.2. In the domain of absolute convergence, a left character ρ is an eigenfunction of the above integral operator, namely (Γn #ρ)(Z) = Γn (ρ)ρ(Z) . Or, in direct generalization of (5), for A ∈ Posn , Z e−tr(AY ) ρ(Y )dµn (Y ) = Γn (ρ)ρ(A−1 ) . Posn

Proof. Write Z = T t T with T ∈ Tri+ . Then starting with the measure invariance, we find: Z −1 e−tr([T ]Y Z ) ρ([T ]Y )dµn (Y ) (Γn #ρ)(Z) = Posn

=

Z

e−tr(T Y

t

T t T −1 T −1 )

ψ(T )ρ(Y )dµn (Y )

Posn

= ψ(T )

Z

e−tr(T Y T

Posn

= ρ(Z)Γn (ρ),

−1

)

ρ(Y )dµn (Y )

58


which proves the first formula. The second follows by putting Z = A−1 and using the fact that tr(AY ) = tr(Y A). Remark. The above result joins two properties. First, the gamma function was viewed as a function on the space of characters, whence the notation Γ(ρ). Second, we view the exponential function involving two variables (Z, Y ) as a “kernel function” giving rise to an integral operator, which we may apply to a space of functions which make the integral absolutely convergent, certainly including functions with “polynomial growth” in some sense, and certainly including a half plane of characters on the symmetric space Posn . A function h may be an eigenfunction for this integral transform, i.e. the gamma transform, and in this case, the corresponding eigenvalue may be denoted by λΓ (h). Then Proposition 2.2 may be formulated by saying that for a character ρ, we have λΓ (ρ) = Γn (ρ) . We recall Chap. 2, Proposition 2.1, that Y 7→ Y −1 and the action of GLn (R), both left and right, preserve the measure dµn (Y ) on Posn . In particular, [ω] is measure preserving. Proposition 2.3. For a left character ρ, A ∈ Posn , we have Z −1 e−tr(AY ) ρ(Y )dµn (Y ) = Γn (ρ∗ )ρ(A) . Posn

Proof. We make the two measure preserving transformations Y 7→ Y −1 and Y 7→ [ω]Y . The desired formula then follows from Proposition 2.2 and the definitions. Corollary 2.4. For a right character ρ, we have Z −1 (Γn #ρ)(Z) = e−tr(Y Z ) ρ(Y )dµn (Y ) = Γn (ρ ◦ [ω])ρ(Z) . Proof. We note that ρ = ρ00 , and ρ0 is a left character. We then make the measure-preserving change of variables Y 7→ Y −1 and apply Propositions 2.2 and 2.3 to conclude the proof.

3 The Bengtson Bessel Function Following an idea of Bochner [Boc 52], Herz [Her 55] defined a Bessel function using the determinant character. Bengtson [Be 83] extended this Bessel function to all characters, and extended the proofs of various classical formulas from Pos1 to Posn . We shall give the main results of his paper in this section

3 The Bengtson Bessel Function

59

and the next. The basis principle is that, once the original definition is given with matrices, then almost all the formulas for the ordinary K-Bessel function are valid, with essentially the same proofs as in the one-dimensional case, using the invariant measure dµn (Y ) on Posn instead of the invariant measure dy/y on the positive multiplicative group, which is Pos1 . We have found it convenient to adopt the convention that for n = 1, s ∈ C, a, b > 0, Ks (a, b) =

Z∞

e−(ay+by

−1

) s dy

y

y

.

0

Although we shall deal a lot with characters, we find it worth while to give general definition for the Bessel transform of a function h on Posn . Under conditions of absolute convergence, we define the K-Bessel transform of h for A, B ∈ Posn by the integral Z −1 Kh (A, B) = e−tr(AY +BY ) h(Y )dµn (Y ) . Posn

Thus we may also write Kh (A2 , B 2 ) =

Z

e−tr([A]Y +[B]Y

−1

)

h(Y ) dµn (Y ) .

Posn

Directly from the fact that for g ∈ GLn (R), the map [g] preserves the measure, we get the transformation formula Kh◦[g] (A, B) = Kh (A[g −1 ], [g]B) . We just let Y 7→ [g −1 ]Y in the defining integral. We now specialize to the case when h = ρ is a character. We then call Kρ a Bessel function. By a character, we mean a left character unless otherwise specified. Let ρ be a left or right character on Posn . If one parameterizes ρ by n complex variables s = (s1 , . . . , sn ), then the function Kρ (A, B) is entire in s, or as we shall also say, Kρ is entire in ρ. We shall prove that this is the case by estimating the higher dimensional Bessel function in terms of the classical Bessel function in one variable. For r ∈ R and x > 0, this classical function is defined by the integral Kr (x) =

Z∞

e−x(t+1/t) tr

dt . t

0

It is very easy to show that in a fixed bounded range for r, one has uniformly Kr (x) = O(e−2x ) for x → ∞ .

60


Cf. [La73/87], Chap. XX, Sect. 3, K7. Of course, as x → 0, the integral blows up. The next theorem will show not only that the higher dimensional BengtsonBessel function is entire in the n complex variables s1 , . . . , sn , but it also gives uniform estimates for the absolute convergence of the integral, in terms of the eigenvalues of A and B as these approach 0 (which is bad) or ∞ (which is good). Theorem 3.1. Let λ > 0 be such that A and B = λI. Put σj = Re(sj ) . Then the integral representing Kρs (A, B) is absolutely convergent and satisfies |Kρs (A, B)| 5

r

π λ

n(n−1)/2 Y n

Kσj −(n−j)/2 (λ) .

j=1

Proof. We write down the integral representing the Bessel function just with the real part, since the imaginary part does not contribute to the absolute value estimate. For X ∈ Rn , we have A[X] = λt XX and hence for X ∈ Rn×n we have tr(A[X]) = λ tr([X]I) . In the Bessel integral, we change the variable as in Chap. 2, Sect. 2, putting Y = T t T with T ∈ Tri+ , so tr(AY ) =

X i5j

λt2ij =

n X

λt2jj +

j=1

X

λt2ij .

i<j

We have similar inequalities with BY −1 . Let (tij ) = T −1 . Put tj = tjj . By Proposition 2.3+ of Chap. 2, and as in Proposition 1.4, we find the estimate |Kρs (A, B)| 5

Z∞ n Y

j=1

0

−2 2 2dtj 2σ e−λ(tj +tj ) tj j tjj−n tj

∞ Y Z

2

ij 2

e−λ(tij +(t

) )

dtij .

i<j −∞

In the product over i < j, we omit the term with (tij )2 which only makes the p n(n−1)/2 estimate worse, and then the integral gives precisely the factor π/λ . For the product with the diagonal variables tj = tjj , we change the variable putting u = t2 , du/u = 2dt/t. Then one gets just the Bessel integral in one variable, giving the other factor in the desired estimate. This concludes the proof. Next we deal systematically with an extensive formalism satisfied by the Bessel function. Instead of taking both variables A, B positive, we allow some degeneracy in one of them, say B, and with Z semipositive, we define


Kρ (A, Z) =

Z

e−tr(AY +ZY

−1

)

61

ρ(Y )dµn (Y ) .

Posn

Then the integral is absolutely convergent only in the half plane of convergence of the gamma integral, but we can substitute Z = 0. Through K 4 we Assume that ρ is a Left Character We then find (K1)

Kρ (A, O) = ρ(A−1 )Γn (ρ), so Kρ (I, O) = Γn (ρ) .

The formula follows from Proposition 2.2 For T ∈ Tri+ , A, B ∈ Posn , we have (K2)

Kρ (A[T ], [T −1 ]B) = ψρ (T )−1 Kρ (A, B) .

Proof. The integral representing the left side is Z t −1 t −1 −1 e−tr( T AT Y +T B T Y ) ρ(Y )dµn (Y ) . Posn

We use the commutativity of the trace to see that this integral is the same as Z −1 e−tr(A[T ]Y +B([T ]Y ) ) ρ(Y )dµn (Y ) . Posn

We make the translation Y 7→ [T −1 ]Y and use the definition of a character to conclude the proof. As a direct consequence, we can reduce the value of the Bessel function to the case when one of the arguments is I. Indeed, we choose T ∈ Tri+ such that A−1 = T t T , so A[T ] = I. Then for this T , (K3)

Kρ (A, B) = ψρ (T )Kρ (I, [T −1 ]B) .

On Pos1 , the Bessel function in two variables is reduced to a function of one variable Ks (c), defined by the formula Ks (c) =

Z∞

e−c(t+1/t) ts

dt . t

0

Indeed, a change of variables immediately shows that µ ¶s µ ¶2s b b Ks (ab) = Ks (b2 , a2 ) . Ks (a2 , b2 ) = a a

62


The non-commutativity on Posn apparently prevents a similar reduction when n > 1. One still has K 3, and an additional symmetry: (K4)

Kρ (A, B) = Kρ0 (B, A) = Kρ∗ ◦[ω] (B, A) = Kρ∗ ([ω]B, [ω]A) .

This is immediate by making the change of variables Y 7→ Y −1 in the integral defining Kρ , and using formula (3) of Sect. 1, namely ρ0 (Y ) = ρ(Y −1 ) = ρ∗ ([ω]Y ) . Of course, we have now indexed K by ρ0 = ρ∗ ◦ [ω], which is a right character, but not a left character. Inductive Formulas We conclude this section with inductive formulas for the Bessel function starting with the degenerate case as in Bengtson. We fix the notation for the rest of the section Proposition 3.1. Let 0 < p < n and p + q = n. Let P ∈ Posn , Q, D ∈ Posq . For X ∈ Rp×q , we let   Ip X  . u(X) =  0 Iq

A variable Y ∈ Posn has a unique expression in partial Iwasawa coordinates ¶ µ W 0 Y = Iw+ (W, X, V ) = [u(X)] 0 V with W ∈ Posp , V ∈ Posq , X ∈ Rp×q ; or also Y = Iw− (W, X, V ) = [t u(X)]

µ

W 0

0 V

¶

.

We specify each time which expression is used. A left, resp. a right, character ρ can be expressed uniquely as a product µ ¶ W 0 ρ = ρ1 (W )ρ2 (V ) , 0 V where ρ1 , ρ2 are left resp. right characters on Posp and Posq . We shall be computing Bessel integrals Z e−trM (Y ) ρ(Y )dµn (Y ) Kρ (A, B) = Posn

where


63

M (Y ) = AY + BY −1 and A, B are either in partial Iwasawa coordinates or degenerate, for instance µ ¶ 0 0 + . A = Iw (P, C, Q) and B = 0 D In the proof, we then use the coordinates Y = Iw− (W, X, V ) with Iw− . Many combinations can occur, as in the next four propositions, with the alternate Iwasawa decomposition or upper left hand corner blocks instead of lower right. The four propositions allow to determine similar answers for Bessel integrals formed by permuting variables, e.g. using t u(C) instead of u(C) in Proposition 3.2 below. For example, in that proposition, we let µ ¶ µ ¶ P 0 O O M (Y ) = [u(C)] Y + Y −1 = AY + BY −1 . 0 Q O D Let M 0 (Y ) =

µ

0 0 0 D

¶

Y + [u(C)]

µ

P 0

0 Q

¶

Y −1 = BY + AY −1 .

Then for a left character ρ, formula K 4 tells us Kρ (B, A) = Kρ0 (A, B) , and ρ0 is a right character to which we can apply Proposition 3.2. We fix a notation, useful when taking the trace of products of matrices. if M, N are square matrices of the same size, we define an equivalence M ∼N to mean that M can be written as a product, and N is obtained from M by a succession of interchanges of factors of type Z1 Z2 7→ Z2 Z1 . Thus if M ∼ N then tr(M ) = tr(N ). Proposition 3.2. Let C ∈ Rp×q . Let µ ¶ µ P 0 0 (1) M (Y ) = [u(C)] Y + 0 Q 0

0 D

¶

Y −1 = AY + BY −1 .

Let ρ be a right character on Posn . Then Z e−trM (Y ) ρ(Y )dµn (Y ) Kρ (A, B) = =

Posn √ pq

π |Q|−p/2 ρ1 (P −1 )Γp (ρ1 )Kρ2 d−p/2 (Q, D) . q

64


Proof. The pattern of the present proof will be repeated several times afterwards. We are computing Kρ (A, B), where A = Iw+ (P, C, Q). For certain matrix multiplications to come out, we use Y = Iw− (W, X, V ), i.e. we use the alternate partial Iwasawa decomposition for the variable Y . We apply Corollary 2.7 of Chap. 2, giving us the change of variables formula. We use the function f (Y ) = e−trM (Y ) ρ(Y ) = e−trM (Y ) ρ1 (W )ρ2 (V ).

(2)

Matrix multiplication shows that  (P + [C + X]Q)W (3) M (Y ) ∼  ∗ ∗ ∗∗

∗∗ QV





+

0 ∗

∗ DV

−1

 

so that (4) trM (Y ) = tr(P W + W [C + X]Q + QV + DV −1 ) = trM (W, X, V ) . Then by Corollary 2.7 of Chap. 2, the integral to be evaluated is Z Z f (Y )dµn (Y ) = e−trM (Y ) ρ(Y )dµn (Y ) = (5) Posn

ZZZ

Posn

e−trM (W,X,V ) ρ1 (W )ρ2 (V )|W |q/2 |V |−p/2 dµeuc (X)dµp (W )dµq (V )

The translation by C in the dµeuc (X)-integral does not change the integral, so we can omit C from the integrand. Multiplication by Q on the right and W on the left introduce linear changes of coordinates in this dµeuc (X)-integral, 1 1 in fact the change is by Q 2 and W 2 since the variable X is “squared”. The situation is the same as in Chap. 3, Sect. 2. Formally, we are dealing with the change of variables 1

1

Z = W 2 XQ 2 ,

dZ = dX|W |q/2 |Q|p/2 .

Making this change of variables shows that the dµeuc (X)-integral has the standard value Z √ pq t e−tr( XX) dµeuc (X) = π , Rp×q

divided by the Jacobian factor, i,.e, multiplied by the inverse factor |Q|−p/2 |W |−q/2 . Then (5) has the value √ pq (6) π |Q|−p/2 ZZ −1 e−tr(P W ) e−tr(QV +DV ) ρ1 (W )ρ2 (V )|V |−p/2 dµp (W )dµq (V ) .


65

The variables are separated in the double integral. From Proposition 2.2, the W -integral yields ρ1 (P −1 )Γp (ρ1 ) ; (7) while directly from the definition of the K-Bessel function, the V -integral yields (8) Kρ2 d−p/2 (Q, D) . q √ Putting these last two factors together with the power of π and the factor |Q|−p/2 proves the proposition. We tabulate the variation with [t u(C)] instead of [u(C)]. Proposition 3.3. Let (9)

M (Y ) = [t u(C)]

µ

P 0

0 Q

¶

Y +

µ

0 0

0 D

¶

Y −1 .

Let ρ be a left character on Posn . Then Z √ pq e−trM (Y ) ρ(Y )dµn (Y ) = π |P |q/2 Γp (ρ1 d−q/2 )ρ1 (P −1 )Kρ2 (Q, D) . Posn

Proof. The point of the method of Proposition 3.2 was to combine the expressions with (C, P, Q) and (X, W, V ). To do so in the present case, we have to use the alternative coordinates Y = Iw+ (W, X, V ). Then matrix multiplication now gives     0 ∗∗∗ PW ∗ ∗ ∗ ∗ ∗∗ +  , M (Y ) ∼  (10) ∗ DV −1 ∗∗ ([t (C + X)]P + Q)V whence

(11)

trM (Y ) = tr(P W + V [t (C + X)]P + QV + DV −1 ) = trM (W, X, V ) .

We can then argue as in Proposition 3.2, using Corollary 2.5 of Chap. 2, so the desired integral is equal to ZZZ e−trM (W,X,V ) ρ1 (W )ρ2 (V )|W |−q/2 |V |p/2 dµeuc (X)dµp (W )dµq (V ) . (12) We perform the W -integral, which yields Γ(ρ1 d−q/2 )ρ1 (P −1 ) by Proposition 2.2. Then we perform the X-integral, get rid of the C-translation, and change 1 1 variables, so let√Z = V 2 t XP 2 and dZ = |V |p/2 |P |−q/2 dX. Then the Xpq q/2 integral yields π |P | . The factor|V |p/2 disappears, and the V -integral then yields Kρ2 (Q, D). Putting all factors together gives the stated answer.

66


Proposition 3.4. Let A ∈ Posp , C ∈ Rp×q . Let µ µ ¶ A 0 P t M (Y ) = (13) Y + [ u(C)] 0 0 0

0 Q

¶

Y −1 .

Let ρ be a right character on Posn . Then Z √ pq e−trM (Y ) ρ(Y )dµn (Y ) = π |P |−q/2 Kρ1 dq/2 (A, P )ρ2 (Q)Γq (ρ∗2 ) . p

Posn

Proof. We let Y = [t u(X)]

µ

W 0

0 V

¶

= Iw− (W, X, V ) .

Matrix multiplication gives µ ¶ µ A 0 AW (14) Y = 0 0 ∗

∗ 0

¶

and (15) [t u(C)]

µ

P 0

0 Q

¶



Y −1 ∼ 

P W −1

∗ ∗ ∗∗

∗∗∗

([t u(C − X)]P + Q)V −1



 .

Then (16)

trM (Y ) = tr(AW + P W −1 + V −1 [t (C − X)]P + QV −1 ) .

Since dµn (Y ) = |W |q/2 |V |−p/2 dµeuc (X)dµp (W )dµq (V ) , we find that Z e−trM (Y ) ρ(Y )dµn (Y ) = (17) Posn

ZZZ

e−trM (W,X,V ) ρ1 (W )ρ2 (V )|W |q/2 |V |−p/2 dµeuc (X)dµp (W )dµq (V ).

The W -integral splits to give the factor (18)

Kρ1 dq/2 (A, P ) . p

We perform the dµeuc (X)-integral next, whereby we can eliminate translation by C, replace −X by X, but we have the stretching factor coming V −1/2 t XP 1/2 and from V −1 , P , so we need a change of variables Z = √ pq −p/2 q/2 t dZ = |V | |P | d X. The integral yields the factor π as usual. This


67

leaves the dµq (V )-integral, times the inverse |P |−q/2 , while there is a cancelation of |V |−p/2 , so the V -integral is Z −1 e−tr(QV ) ρ2 (V )dµq (V ) . (19) Applying Proposition 2.3 concludes the proof. Proposition 3.5. Notation as in the previous two propositions, let µ ¶ µ ¶ P 0 A 0 M (Y ) = [u(C)] Y + Y −1 . 0 Q 0 0 Let ρ be a right character. Then Z e−trM (Y ) ρ(Y )dµn (Y ) Posn

=

Z

Kρ1 dq/2 (P + [C + X]Q, A)Kρ2 d−p/2 (Q, A[X])dµeuc (X). p

q

Rp×q

Proof. The situation can use some of the computations of Proposition 3.2, but in evaluating trM (Y ) the term with D has to be replaced. A new computation shows that     A(W −1 + [X]V −1 ) ∗ A 0  .  Y −1 =   (20) ∗ ∗ ∗ ∗ ∗∗ 0 0 0 Hence (21)

trM (Y ) = tr(P W + W [C + X]Q + QV + AW −1 + A[X]V −1 ) .

Then we first perform the W -integral and V -integral, and the stated answer drops out. Remark. As we observed at the beginning of our discussion, the last two propositions can be used to derive variations. For instance, let µ µ ¶ ¶ A 0 P 0 0 −1 t M (Y ) = BY + with B = [ u(C)] Y . 0 0 0 Q Let ρ be a left character. Then by K 4, ¶¶ µ µ ¶ ¶ µµ A 0 A 0 = Kρ0 ,B Kρ B, 0 0 0 0 and we can apply Proposition 3.4 to find a value for the right side.

68


The inductive formulas of Propositions 3.2 and 3.3 have corresponding formulas in the non-degenerate case, as in [Ter 85], Proposition 2, which we now give. They depend on the Iw+ and Iw− Iwasawa coordinates. Proposition 3.6. Let P ∈ Posq , C ∈ Rp×q and n = p + q, giving rise to a partial Iwasawa decomposition, so we put µ ¶ P 0 A = [u(C)] = Iw+ (P, C, Q) . 0 Q Let ρ be a right character on Posn . Then Kρ (A, In ) Z =

Kρ1 dq/2 (P + [X + C]Q, Ip )Kρ2 d−p/2 (Q, Iq + t XX)dµeuc (X) . p

q

Rp×q

Proof. We let again Y = Iw− (W, X, V ), and Z Kρ (A, In ) = f (Y )dµn (Y ) with (16)

f (Y ) = e−trM (Y ) ρ1 (W )ρ2 (V ) .

The expression for M (Y ) = M (W, X, V ) = AY + Y −1 is the same as in (3), except that the matrix with DV −1 is replaced by  −1  W ∗ ∗ ∗∗  . Y −1 ∼  (17) t −1 ∗ ∗ ∗ ( XX + I)V Therefore we obtain (18)

trM (Y ) = tr(P W + [X + C]QW + QV + W −1 + (I + t XX)V −1 ) .

Then dµn (Y ) = |W |q/2 |V |−p/2 dµp (W )dµq (V )dµeuc (X), so Z f (Y )dµn (Y ) = Kρ (A, In ) = Posn

Z

Kρ1 dq/2 (P + [X + C]Q, Ip )Kρ2 d−p/2 (Q, Iq +t XX)dµeuc (X) . p

Rp×q

This proves the proposition. Finally, we come to the last variation.

q

4 Mellin and Fourier Transforms

69

Proposition 3.7. Put A = [t u(C)]

µ

P 0

0 Q

¶

= Iw− (P, C, Q) .

Let ρ be a left character on Posn . Then Kρ (A, In ) Z = Kρ1 d−q/2 (P, I + X t X)Kρ2 dp/2 (P [C + X] + Q, I)dµeuc (X). Rp×q

Proof. For A we have the same computation as in Proposition 3.3, but we now need the new computation   −1 W ∗ ∗ ∗∗  . Y −1 =  −1 −1 ∗∗ W [X] + V We let M (Y ) = AY + Y −1 . Then (22)trM (Y ) = tr(P W + V [t (C + X)]P + QV + W −1 + W −1 [X] + V −1 ) . The stated answer comes out.

4 Mellin and Fourier Transforms The next Bengtson result extends the classical Fourier transform formula Γ(s)

Z∞

−∞

(x2

eixr dx 1 √ = √ Ks− 12 (a2 , r2 /4) + a2 )s 2π 2

for a > 0, r = 0, Re(s) > 1/2. Theorem 4.1. Let Rp×q have the scalar product hX, Ri = tr( t XR). Let √ −pq Q dxij . Let A ∈ Posq and let ρ be a left character on Posq . dν(X) = 2π Then in a half plane of ρ, Z Γq (ρ) ρ((t XX + A2 ) e−ihX,Ri dν(X) Rp×q

√ = ( 2)−pq Kρd−p/2 (A2 , t RR/4) . q

Proof. We recall that the function e−<X,X>/2 is self-dual with respect to ν. Write down the defining integral for the gamma function, and interchange the order of integration to get the left side equal to

70

(1)


Z

Z

e−tr(Y ) ρ(Y )ρ((t XX + A2 )−1 )dµn (Y )e−ihX,Ri dν(X) .

Rp×q Posq

We write t XX + A2 = t TX TX with TX ∈ Tri+ . Make the left translation by [TX ] on Y . Then the above expression is equal to Z Z (2) e−tr([TX ]Y ) ψ(TX )ρ(Y )ψ(TX )−1 dµq (Y )e−ihX,Ri dν(X) . Rp×q Posq

Then ψ(TX ) cancels ψ(TX )−1 . Now interchange the integrals again. We have tr([TX ]Y ) = tr(TX Y t TX ) = tr(t TX TX Y ) = hXY 1/2 , XY 1/2 i + hAY 1/2 , AY 1/2 i, so (2) becomes Z Z 1/2 1/2 1/2 1/2 e−hXY ,XY i e−ihX,Ri dν(X)e−hAY ,AY i ρ(Y )dµq (Y ) . (3) Posq Rp×q

We make the change of variables √ √ Z/ 2 = XY 1/2 , so dZ = ( 2)pq |Y |p/2 dX , −hZ,Zi/2 evaluated and the inner √ integral becomes the Fourier transform of e −1/2 at RY / 2. Thus (3) becomes Z √ pq 2 t −1/4 ) (4) ρ(Y )|Y |−p/2 dµ(Y )/ 2 e−tr(A Y + RRY Posn

1 = √ Kρd−p/2 (A2 , t RR/4) . ( 2)pq This proves the theorem. The next theorem gives the Mellin transform of the Bessel function, and extends the one variable formula Z∞

Kz (1, y)y s

dy = Γ(s)Γ(s + z) . y

0

Theorem 4.2. let σ, ρ be two left characters on Posn . Then Z Kσ (1, y)ρ(Y )dµ(Y ) = Γn (ρ)Γn (σρ) . Posn


71

Proof. We start with the right side: ZZ Γn (ρ)Γn (σρ) = e−tr(Y ) ρ(Y )e−tr(Z) σ(Z)ρ(Z)dµ(Y )dµ(Z) . Write Z = TZ t TZ = [TZ ]I. Make the translation Y 7→ [TZ−1 ]Y in the dµ(Y )-integral. Then the double integral is equal to ZZ −1 e−tr(Z+Y Z ) ψρ (TZ−1 )ρ(Y )ψρ (TZ )σ(Z)dµ(Y )dµ(Z) . Then there is a cancelation of ψρ (TZ ). After an interchange of integrals, the desired formula comes out. In our discussion of Proposition 1.2, we had left one component unspecified. This component has its own importance, as in Bengtson, and we shall now deal with it. We consider partial Iwasawa decompositions on Posn . We let p, q be positive integers with p + q = n. We identify Matp,q (R) = Rp×q . As we have seen in Chap. 3, Sect. 2, we have a natural positive definite scalar product hX, Ri = tr(t XR) on this space. For X ∈ Rp×q we let   Ip X  so u(X) ∈ Tri+ . u(X) =  0 Iq Let σ be a left character on Posp . Let Y ∈ Posn . We define the upper Bengtson function βσ,Y on Rp×q by the formula βσ,Y (X) = σ ◦ [ω](Subp [u(X)]Y ) . We normalize the Fourier transform with the measure Y √ dxij , dν(X) = ( 2π)−pq so that ∧ βσ,Y (R) =

Z

βσ,Y (X)e−i<X,R> dν(X) .

Rp×q

The first thing we remark about this Fourier transform, also called the Bengtson function, is its eigenfunction property for the action of Rp×q on Posn . Proposition 4.3. With the above notation, for Z ∈ Rp×q , we have ∧ ∧ βσ,[u(Z)]Y (R) = ei βσ,Y (R) .

72


Proof. This is immediate, because Z ∧ (R) = βσ,[u(Z)]Y σ ◦ [ω](Subp [u(X)][u(Z)]Y )e−i<X,R> dν(X) . Rp×q

We make the translation X 7→ X − Z, and the formula falls out. We investigate the Bengtson function in connection with the inductive decomposition of an element of Posn . An element Y ∈ Posn has a unique partial Iwasawa decomposition   W 0  Y = [u(X)]  0 V with W ∈ Posn and V ∈ Posq . Matrix multiplication yields     W 0 W + [X]V XV =  . [u(X)]  (5) 0 V V tX V Note that V = Subq Y and W + [X]V = Subp Y . Furthermore, the expression on the right immediately gives both the existence and uniqueness of the partial Iwasawa decomposition. Indeed, V is determined first, then X is determined by the upper right and lower left components, and finally W is determined to solve for the upper left component. For the record we give the alternate version with   W 0  , Y = [t u(X)]  0 V       W 0 W WX W 0  =  [u(X)] = [t u(X)]  (6)  t XW W [X] + V 0 V 0 V   −1   W 0 W −1 −W −1 X [u(X)]   =   . (7) t −1 −1 −1 0 V − XW W [X] + V If Y has the diagonal decomposition Y = diag(W, V ) as above, then we may write βσ,Y = βσ,W,V . By Theorem 4.1, the Fourier transform then has the expression Z ∧ βσ,W,V (R) = σ ∗ ((W + [X]V )−1 )e−i<X,R> dν(X) . Rp×q


73

Theorem 4.4. let n = p + q as above. Let W ∈ Posp and V ∈ Posq . Let σ be a left character on Posp and β the upper Bengtson function. Then √ ∧ Γp (σ ∗ )βσ,W,V (R) = |V |−p/2 ( 2)−pq Kσ∗ d−q/2 (W, [R]V −1 /4) . p

Proof. By definitions, letting W = A2 , V = B 2 , we have βσ,W,V (X) = σ([ω]Subp [u(X)]Y ) = σ ∗ ((A2 + [X]B 2 )−1 ). Then (8)

∧ βσ,W,V

(R) =

Z

σ ∗ ((A2 + XB t (XB))−1 )e−ihX,Ri dν(X) .

Rp×q

We make the change of variables X 7→ XB −1 . Then hXB −1 , Ri = hX, RB −1 i and dν(XB −1 ) = dν(X)|B|−p . 1

Applying Theorems 4.1 with R replaced by RV − 2 (and taking transposes) concludes the proof. ∧ a Bessel Remark. Bengtson (see [Be 83]) calls the Fourier transform βσ,W,V function, and denotes it by kp,q .

For the convenience of the reader we tabulate the alternate formula with upper and lower triangular matrices reversed. Let τ be a left character on Posq . We define the lower Bengtson function βτ,Y (X) = τ ◦ [ω](Subq Y [u(X)]) . Theorem 4.5. Let βτ,Y be the lower Bengtson function. Then √ ∧ (R) = |W |−q/2 ( 2)−pq Kτ ∗ d−p/2 (V, W −1 [R]/4) . Γq (τ ∗ )βτ,W,V q

The proof is of course the same, mutatis mutandis. Remark. In the definitions of the Bengtson functions, note that the reversing operator [ω] might be denoted by [ωp ] resp. [ωq ], to denote the size of ω, which is p resp. q in the respective theorems. In practice, the context determines the size. Next we give formulas which reduce the computation of the Fourier transform to the case when W = Ip , or V = Iq , or Y = Ip+q = In . Proposition 4.6. Let the notation be as in Theorem 4.4, so σ is a left character on Posp and βσ the upper Bengtson function. Then ∧ ∧ (R) = |V |−p/2 βσ,W,I (RV −1/2 ) . βσ,W,V q

74


Proof. This is actually the content of (8), together with the change of variables X 7→ XB −1 , before we apply Theorem 4.1. Proposition 4.7. Let W = t T T with T ∈ Tri+ p , W ∈ Posp . Let σ be a left character on Posp and βσ the upper Bengtson function. Then ∧ ∧ (R) = |W |q/2 σ ∗ (W −1 )βσ,I (T R) . βσ,W,I q p ,Iq

Proof. By definition ∧ βσ,W,I (R) q

Z

=

σ ∗ ((W + X t X)−1 )e−i<X,R> dν(X) .

Rp×q

We make the change of variables X 7→ t T X. We note that ht T X, Ri = hX, T Ri and dν(t T X) = |T |q dν(X) = |W |q/2 dν(X) . Then ∧ (R) βσ,T,I q

q/2

= |W |

Z

σ([t T ω ](I + X t X)ω )ei<X,T R> dν(X)

Rp×q

Z

= |W |q/2 σ([ω]t T )

σ([ω](I + X t X))e−i<X,T R> dν(X)

Rp×q q/2 ∗

= |W |

σ (W

−1

)

Z

σ ∗ ((I + X t X)−1 )e−i<X,T R> dν(X) ,

Rp×q ∧ (T R). which yields the theorem by definition of βσ,I p ,Iq

Having given the two reduction formulas above in separate cases, we can combine them into one statement for the record. Theorem 4.8. Let the situation be as in Theorems 4.6 and 4.7, so σ is a left character on Posp , W ∈ Posp , W = t T T with T ∈ Tri+ p , and V ∈ Posq . Then 1

∧ ∧ (T RV − 2 ) . (R) = |V |−p/2 |W |q/2 σ ∗ (W −1 )βσ,I βσ,W,V p ,Iq

Observe that the change of position of V resp. W in the two preceding propositions was carried out independently, and each change is somewhat lighter than the combination, so we have given all the steps to lighten the computation.

4 Invariant Differential Operators on Posn (R)

1 Invariant Polynomials Let V be a finite dimensional vector space over the reals. We let: Pol(V ) = algebra of polynomial functions on V ; S(V ) = Pol(V ∨ ) = symmetric algebra of V , where V ∨ is the dual space. In non-invariant terms, if {λ1 , . . . , λN } is a basis of V ∨ , the monomials m1 N {λ1 . . . λm N } form a basis of Pol(V ). We apply this construction to two vector spaces as follows. First, let a = vector space of n × n diagonal matrices. W = group of permutations of the diagonal elements of a diagonal matrix. Because of the way W generalizes in the theory of Lie algebras, we call W the Weyl group. Let Eii be the diagonal matrix with 1 in the i-th component and 0 elsewhere. Then every element v ∈ a can be expressed as a linear combination v=

n X

hi Eii with coordinate functions hi .

i=1

Let: Pol(a)W = subalgebra of Pol(a) consisting of elements invariant under W. Thus Pol(a)W consists of the symmetric polynomials in the algebraically independent elements (variables) h1 , . . . hn . Next we consider V = Sym. Jay Jorgenson: Posn (R) and Eisenstein Series, Lect. Notes Math. 1868, 75–94 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com °

76


Let Eij be the matrix with ij-component equal to 1, and all other components equal to 0. Then Sym has a basis (actually orthogonal) consisting of the elements vii = Eii and vij =

1 (Eij + Eji ) for i < j . 2

Then the algebra Pol(Sym) can be viewed as the algebra of polynomials P (X) = P (. . . , xij , . . .)i5j where X is the coordinate matrix of a vector X v= xij vij = vX . i5j

The coordinate functions (xij ) form the dual basis of (vij ), and hi = xii . Let K be the usual compact group of real unitary matrices. We let: Pol(Sym)K = subalgebra consisting of the elements invariant under the conjugation action by K. Theorem 1.1. The restriction Pol(Sym) → Pol(a) induces an algebra isomorphism ≈ Pol(Sym)K −→ Pol(a)W . In other words, every W -invariant polynomial on a can be uniquely extended to a K-invariant polynomial on Sym. Proof. Every element of Sym can be diagonalized with respect to some orthonormal basis. This means that Sym = [K]a , so that every element of Sym is of the form kv t k = kvk−1 for some v ∈ a and k ∈ K. Thus the restriction map is injective. We have to prove that it is surjective. For this we recall that a symmetric polynomial in variables h1 , . . . , hn can be expressed uniquely as a polynomial in the elementary symmetric functions s1 , . . . , sn . Furthermore, these symmetric functions are the coefficients of the characteristic polynomial of elements v ∈ a: det(tI + v) = tn + s1 tn−1 + . . . + sn . But then a polynomial Q(s1 , . . . , sn ) can be viewed as an element of the symmetric algebra Pol(Sym), by taking v ∈ Sym, extending the polynomial in Pol(a), and obviously K-invariant, thus proving the theorem. Remark. The above result was proved by Chevalley for semisimple Lie algebras. Cf. Wallach [Wal 88], Theorem 3.1.2 and Helgason [Hel 84], Chap. 2,

1 Invariant Polynomials

77

Corollary 5.12 for a proof as a consequence of a much more analytic theorem. A more direct proof was given by Harish-Chandra essentially along the same lines as the proof we gave above for Theorem 1.1, but with technical complications. Cf. [Hel 62], Chapter X, Theorem 6.16 which gives a complete exposition of this proof, not kept in [Hel 84], but only mentioned in Exercise D1, p. 340, following Harish-Chandra. Actually, the algebra of symmetric polynomials on a has two natural sets of generators. The elementary symmetric polynomials as above, and the Newton polynomials Sr (h) = tr(hr ) = hr1 + . . . + hrn with r = 1, . . . , n . Thus the algebra Pol(Sym)K is generated by the algebraically independent elements tr(X), . . . , tr(X n ) . These elements restrict to tr(h), . . . , tr(hn ) on a. Let P = P (X) be an arbitrary K-invariant polynomial. Then there exists a unique polynomial PNew in n variables such that P (X) = PNew (tr(X), . . . , tr(X n )) . we call PNew the Newton polynomial of P . For the reader’s convenience, we recall specific properties of the duality for polynomial functions. We do so in a general context. Let V be a finite dimensional vector space over a field of characteristic 0. Let d be a positive integer, and let Pold (V ) be the vector space of homogeneous polynomials of degree d on V . The usual “variables” are the coordinate functions with respect to a basis, and such polynomials are therefore polynomial functions on V . If V ∨ is the dual space, then V = V ∨∨ , and V, V ∨ play a symmetric role with respect to each other. We denote elements of V by v and elements of V ∨ by λ. The vector spaces Pold (V ) and Pold (V ∨ ) are dual to each other, under the pairing whose value on monomials is given by X hλ1 . . . λd , v1 . . . vd i = hλ1 , vσ(1) i . . . hλd , vσ(d) i . σ

The sum is here taken over all permutations σ of {1, . . . , d}. Given a nondegenerate bilinear map between V and another vector space V ∨ , the same formula defines a duality on their algebras of polynomial functions. In practice, one is usually given some non-degenerate symmetric bilinear form on V itself, identifying V with its dual space. Note that the sum defining the scalar product on monomials is the same as the sum defining determinants, except that the alternating signs are replaced by all plus signs, thus making the sum symmetric rather than skew symmetric in the two sets of variables (λ1 , . . . , λd ) and (v1 , . . . , vd ). If {v1 , . . . , vn } is a basis for V and {λ1 , . . . , λn }

78


is the dual basis, then the value of the above pairing on their monomials is 1 or 0. Thus the distinct monomials of given degree d form dual bases for Pold (V ) and Pold (V ∨ ). Let K be a group acting on V . Then K also acts functorially on the dual space V ∨ . For a functional λ ∈ V ∨ , we have by definition ([k]λ)(v) = λ([k −1 ]v) . Proposition 1.2. The pairing described above between Pol(V ) and Pol(V ∨ ) is K-invariant, in the sense that for P ∈ Pol(V ) and Q ∈ Pol(V ∨ ), we have h[k]P, [k]Qi = hP, Qi . This is an immediate consequence of the definitions. Let a be a subspace of V . The exact sequence 0 → a → V has the dual exact sequence V ∨ → a∨ → 0 . The restriction map Pol(V ) → Pol(a) → 0 corresponds to the dual sequence 0 → Pol(a∨ ) → Pol(V ∨ ) . Let W be the subgroup of K leaving a stable, modulo the subgroup leaving a elementwise fixed. We have Pol(a∨ )W = S(a)W . Immediately from the definitions, we get: Proposition 1.3. The restriction Pol(V )K → Pol(a)W is an isomorphism if and only if the dual sequence Pol(a∨ )W → Pol(V ∨ )K is an isomorphism.

2 Invariant Differential Operators on Posn the Maass-Selberg Generators This section will describe the Maass-Selberg generators for invariant differential operators. We show how the invariant polynomials of the preceding section are related to differential operators. There are two natural charts for

2 Invariant Differential Operators

79

Posn : First as an open subset of Sym; and second as the image of the exponential map giving a differential isomorphism with Sym. Each one of these charts gives rise to a description of the invariant differential operators, serving different purposes. The algebra of invariant differential operators is isomorphic to a polynomial algebra, and each one of these charts gives natural algebraically independent generators for this algebra. The first set of generators is due to Maass-Selberg [Maa 55], [Maa 56], [Sel 56], see also [Maa 71], which we follow more or less. We let DO(M ) denote the algebra of C ∞ differential operators on a manifold M . Let G be a Lie group acting on M . As mentioned in the introduction, we let DO(M )G be the subalgebra of G-invariant differential operators, and 0 similarly DO(M )G for any subgroup G0 . When the subgroup is G itself, we often omit the reference to G, and speak simply of invariant differential operators. In the present chapter, we take M = Posn , and G = GLn (R). We let Y = (yij ) be the symmetric matrix of variables on Posn with Yij = yji for all i, j = 1, . . . , n. We let dY = (dyij ). We also let   1 ∂/∂y11 2 ∂/∂ij   .. ∂   . (1) =  ∂Y   1 ∂/∂ynn 2 ∂/∂yij 

  = 

1 2 ∂ij

∂11 ..

.

1 2 ∂ij

∂nn



   . 

The notation with partial derivatives ∂ij on the right is useful when we do not want to specify the variables. Note that the matrix of partial derivatives has a factor 1/2 in its components off the diagonal. We let tr be the trace. For any function f on Posn ,we have (2)

tr(dY

X ∂ )f = df = (∂ij f )(Y )dyij . ∂Y i5j

This follows at once from the multiplication of matrices dY and ∂/∂Y . When summing over all indices i, j, the factors 1/2 add up to 1, as desired to get the df . This justifies the notation ∂/∂Y . Next we consider a change of variables under the action of the group G = GLn (R). Let g ∈ G. Let (3)

Z = gY t g

so

dZ = gdY t g .

Then f (Y ) = f (g −1 Z t g −1 ) = f1 (Z) = f ◦ [g −1 ](Z), and

80


µ

∂ df1 (Z) = tr dZ ∂z

¶

¶ ∂ f1 (Z) = tr gdY g f ∂Z ¶ µ ∂ g f. = tr dY · t g ∂Z µ

t

Hence (4)

∂ −1 ∂ ∂ ∂ = tg g and = t g −1 g . ∂Y ∂Z ∂Z ∂Y Example. For any positive integer r, ¶r ¶r µ µ ∂ −1 ∂ ∂ ∂ Z = gY g (5) and Z =g Y g −1 . ∂Z ∂Y ∂Z ∂Y Consequently

(6)

µ

∂ tr Z ∂Z

¶r

= tr

µµ

∂ Z ∂Z

¶r ¶

= tr

µµ

∂ Y ∂Y

¶r ¶

,

from which we see that tr((Y ∂/∂Y )r ) is invariant for all positive integers r. Thus we have exhibited a sequence of invariant differential operators. For a positive integer r, we define the Maass-Selberg operators ¶r ¶ µµ ∂ δr = tr . Y ∂Y We return to these differential operators in Theorem 2.3. Here we continue with general properties. The left translation operation [Lg ]D is characterized by the property (7)

([Lg ]D)(Lg f ) = Lg (Df )

or

([Lg ]D)(Lg f )(Y ) = (Df )([g −1 ]Y ) .

Thus the invariance of D, namely that [Lg ]D = D for all g, means that for all f , (D(Lg f ))(Y ) = (Df )([g −1 ]Y ) = (Df )(g −1 Y t g −1 ) (8) or equivalently D(Lg f ) = Lg (Df )

or also

D(f ◦ Lg ) = (Df ) ◦ Lg .

A differential operator can be written uniquely in the form X Y D = P (Y, ∂/∂Y ) = ϕ(m) (Y ) (∂/∂yij )m(i,j) (m)

i5j

with functions ϕ(m) and integral exponents m(i, j) = 0. Let X = (xij ) be a variable in Sym, in terms of its coordinate functions. Let Fu be the ring of C ∞ functions on Posn . If we wish to specify the dependence of an element of Fu[X] on Y via its coefficients, we write an element of Fu[X] in the form


P (Y, X) =

X

ϕ(m) (Y )

(m)

Y

m(i,j)

xij

81

.

i5j

Thus P (Y, X) is C ∞ in the coordinates of Y , and polynomial in those of X. ∂ ∂ The map P (Y, X) 7→ P (Y, ∂Y ) also written D(Y, ∂Y ), obtained by substituting ϕij for Xij , establishes an Fu-linear (not ring) isomorphism between Fu[X] and the Fu-module of differential operators. As in freshman calculus, it is useful to have a formalism of differentiation both with the variable explicitly, and without the variable. Observe that the invariance formula (8) can be written with any letter, say Z, that is (D(Lg f ))(Z) = (Df )(g −1 Z t g −1 ) . Now we put Z = gY t g. Then the right side becomes (Df )(Y ). The left side is ¶ ¶ µ µ ∂ −1 t −1 t t −1 ∂ −1 (f (Y )) . g (D(Lg f ))(Z) = P Z, (f (g Z g )) = P gY g, g ∂Z ∂Y Therefore the invariance formula can be expressed as µ ¶ µ ¶ ∂ −1 ∂ P gY t g, t g −1 g (f (Y )) = P Y, (f (Y )) , ∂Y ∂Y from which we can omit the expression f (Y ) at the end. We obtain Proposition 2.1. Given P (Y, X) ∈ F u[X], the operator P (Y, ∂/∂Y ) is invariant if and only if for all g ∈ G, P (gY t g, t g −1 Xg −1 ) = P (Y, X) or also

µ ¶ µ ¶ ∂ t t −1 ∂ −1 D gY g, g g = D Y, . ∂Y ∂Y

We are now finished with the general remarks on invariant differential operators, and we relate them with operators with constant coefficients at the origin. The origin is the unit matrix, which we denote by I. We let X Y m(i,j) ϕ(m) (I) Xij = PD,I (X) . P (I, X) = (m)

i5j

Then PD,I (X) is a polynomial (ordinary), and PD,I (∂/∂Y ) is a polynomial differential operator with constant coefficients, called the polynomial expression of D(Y, ∂/∂Y ) at the origin. It gives the value of the differential operator at the origin, in the sense that for any function f on Posn , ¯ (Df )(I) = PD,I (∂/∂Y )f (Y ) ¯Y =I . (9)

Furthermore, this polynomial PD,I uniquely determines the invariant differential operator, i.e. the association

82


P (Y, ∂/∂Y ) 7→ P (I, ∂/∂Y ) = PD,I (∂/∂Y ) is an injective linear map of the real vector space of invariant differential operators into the space of polynomials with constant coefficients. Indeed, given a point Y , we select g ∈ G such that Y = [g −1 ]I. Then for any function f on Posn , (10) (Df )([g −1 ]Y ) = (D(Lg f ))(I) . Thus the family of values of derivatives (Df )(I) (for all f ∈ Fu) uniquely determines D. Hence so does the polynomial PD . Furthermore, the degree of the differential operator D is equal to the degree of PD,I (as an ordinary polynomial). This is true because of the invariance of the degree under local isomorphisms, and also more explicitly by Proposition 2.2. Theorem 2.2. The association D 7→ PD,I is a linear isomorphism DO(Posn )G → Pol(Sym)K . Proof. We have already seen above that PD,I is K-invariant, and that the association D 7→ PD,I is injective on DO(Posn )G . There remains only to prove the surjectivity. Given P (X) ∈ Pol(Sym)K and Z ∈ Posn , we may write Z = [g]I with some g ∈ G. Define Df by the formula µ µ ¶ ¶ ¯ ¯ ∂ ∂ (Df )(Z) = P f ([g]Y ) ¯Y =I = P (f ◦ [g])(Y ) ¯Y =I . ∂Y ∂Y This value is independent of the choice of g, because any other choice is of the form gk with some k ∈ K, and by the K-invariance of P , we get ¶ µ ¶ µ ∂ ∂ (f ◦ [g])([k]Y ) = P (f ◦ [g])(Y ) , P ∂Y ∂Y

so (Df )(Z) is well defined, and D is defined in such a way that its G-invariance is then obvious. Local charts show that it is a differential operator, thereby concluding the proof. Theorem 2.3. The Maass-Selberg invariant operators δ1 , . . . δn are algebraically independent, and the (commutative) ring C[δ1 , . . . δn ] is the full ring of invariant differential operators. Proof. In Sect. 1, we recalled the Newton polynomials, and we now define ¶ µ ¶n ¶ µ µ ∂ ∂ D1 (Y, ∂/∂Y ) = PD,New tr Y . , . . . , tr Y ∂Y ∂Y Then D1 is an invariant differential operator in the algebra generated by δ1 , . . . , δn , and


µ

D Y,

∂ ∂Y

¶

µ

− D1 Y,

∂ ∂Y

83

¶

is an invariant differential operator. Furthermore, PD − PD1 has degree less than the degree of PD , in other words, D − D1 is a differential operator of lower degree than D, because the terms of highest degree in two differential operators commute modulo operators of lower degree. Thus we can continue by induction to conclude the proof that δ1 , . . . , δn generate the algebra of invariant differential operators. Finally, we prove the algebraic independence. Let P (x1 , . . . , xn ) be a nonzero polynomial. We have to prove that P (δ1 , . . . , δn ) 6= 0. We shall prove this non-vanishing by applying the operator µ µ ¶ µµ ¶n ¶¶ ∂ ∂ P (δ1 , . . . , δn ) = P tr Y , . . . , tr Y ∂Y ∂Y to the function etr(Y ) , and showing that we don’t get 0, by a degree argument. Define the weight w of P (x1 , . . . , xn ) to be the degree of the polynomial P (x1 , x22 , . . . , xnn ). Then w is also the degree of the polynomial P (tr(Y ), . . . , tr(Y n )) in the variables (Y ) = (yij ). To see this, let Pw be the sum of all the monomial terms of weight w occurring in P (x1 , . . . , xn ). We suppose P 6= 0 so Pw 6= 0. Then Pw (tr(Y ), . . . , tr(Y n )) is homogeneous of degree w in (yij ), and 6= 0 since tr(Y ), . . . , tr(Y n ) are algebraically independent. All other monomials occurring in P have lower weight, and hence lower degree in yij ), thus proving the assertion about w being the degree in (Y ). Suppose that µ µ ¶ µ ¶n ¶ ∂ ∂ P (δ1 , . . . δn ) = 0 = P tr Y . , . . . , tr Y ∂Y ∂Y Observe that ∂ij etr(Y ) = etr(Y ) if i = j and 0 otherwise. If M is a monomial of power products of the δij , then M etr(Y ) = etr(Y ) if M contains only power products of ∂11 , . . . , ∂nn M etr(Y ) = 0 otherwise Lemma 2.4. Let Q(x1 , . . . , xn ) be a polynomial 6= 0. Then µ µ ¶ µµ ¶n ¶ ∂ ∂ Q tr Y , . . . , tr Y etr(Y ) ∂Y ∂Y = etr(Y ) {Q(tr(Y ), . . . , tr(Y n )) + R(Y )} where R(Y ) is a polynomial of degree less than the weight of Q.

84


Proof. It suffices to prove the lemma when Q is a monomial, say Q(x1 , . . . , xn ) = xd11 . . . xdnn of weight d1 + 2d2 + . . . + ndn . By the remark preceeding the lemma, without loss of generality we may replace all δij by 0 whenever i 6= j, to evaluate the effect of a polynomial in tr(Y ∂/∂Y ) on etr(Y ) . More precisely, let ∆ be the diagonal matrix operator   0 ∂11   .. ∆=  . . 0

∂nn

Then µ µ ¶ µµ ¶n ¶ ∂ ∂ etr(Y ) Q tr Y , . . . , tr Y ∂Y ∂Y = Q(tr(Y ∆), . . . , tr((Y ∆)n )etr(Y ) + R(Y )etr(Y ) , where R(Y ) has degree smaller than Q(tr(Y ), . . . , tr(Y n )). Thus we are reduced to proving the formula for (tr(Y ∆))d1 . . . (tr(Y ∆)n )dn . We do this by induction. Say d1 = 1. We have tr(Y ∆)etr(Y ) = tr(Y )etr(Y ) . Suppose, by induction, that the lemma is proved for xd11 −1 xd22 . . . xdnn . Applying Y ∆ to the inductive expression immediately yields the desired result. The general case follows the same way. Returning to the proof of algebraic independence, we see that the degree in (Y ) of P (tr(Y ), . . . , tr(Y n )) is equal to the weight of P , and hence by Lemma 2.3, the effect of µ µ ¶ µµ ¶n ¶¶ ∂ ∂ P tr Y , . . . , tr Y ∂Y ∂Y on etr(Y ) cannot be 0. This concludes the proof of Theorem 2.3.

3 The Lie Algebra Generators Here we describe the invariant differential operators via the exponential chart exp : Sym → Posn . To each K-invariant polynomial we associate a differential operator as follows.

3 The Lie Algebra Generators

85

Let P ∈ Pol(Sym)K be a K-invariant polynomial function on Sym. Let f be a function on Posn . Let fG be the function on G defined by fG (g) = f ([g]I) . We call fG the lift of f to G. With X denoting the coordinate matrix of elements in Sym as before, we define ¶ µ ¯ ∂ (DP fG )(g) = P f ([g][exp X]I) ¯X=O . ∂X

Observe that by definition of the Newton polynomial, one can also write this definition in the form µ µ ¶ µ ¶n ¶ ¯ ∂ ∂ # (DP fG )(g) = P tr f ([g][exp X]I) ¯X=O . , . . . , tr ∂X ∂X

Remark. The notation exp X implicitly involves an identification. Indeed, X is the P matrix of coordinates functions, and we identify it with the vector xij vij (sum taken over i 5 j). When writing exp X, we really mean vX = exp(vX ). Lemma 3.1. The function DP fG depends only on cosets G/K, and is therefore a function on Posn , denoted DP f . Proof. Replacing f by f ◦ [g], it suffices to prove that DP fG is K-invariant, that is (DP fG )(k) = DP fG (I) for all k ∈ K. But ¶ µ ¯ ∂ (DP fG )(k) = P f (k(exp 2X)k −1 ) ¯X=O ∂X ¶ µ ¯ ∂ =P f (exp 2kXk −1 )) ¯X=O ∂X

Let Z = kXk −1 . Then as at the beginning of Sect. 2,

∂ −1 ∂ ∂ −1 = t k −1 k =k k . ∂Z ∂X ∂X By the invariance of P under conjugation by k −1 , a change of variables concludes the proof Directly from the definition, it follows that DP ∈ DO(Posn )G , that is, DP is an invariant differential operator on Posn . Theorem 3.2. The association P 7→ DP is a linear isomorphism Pol(Sym)K → DO(Posn )G .

86


Proof. First we prove the injectivity of the map. It suffices to prove the injectivity at g = e (unit element of G). Let F (X) = f (exp 2X) be the pull back of f to Sym = TI Posn (tangent space at the origin). The function f locally near I, and so F locally near 0, can be chosen arbitrarily, for instance to be a monomial in the variables. If DP f = 0, then for every monomial F ((xij )) we have P (∂/∂X)F (xij )) = 0 whence P = 0, thus proving injectivity. As to the surjectivity, let D ∈ DO(Posn )G . Then for every function f on Posn , and g ∈ G, we have (Df )([g]I) = (D(f ◦ [g]))(I) . Thus it suffices to prove the existence of a polynomial P ∈ Pol(Sym)K such that for all f we have µ ¶ ¯ ∂ (Df )(I) = P f ([exp X]I) ¯X=O ∂X µ ¶ ¯ ∂ =P F (X) ¯X=O . ∂X

In the exponential chart, i.e., a neighborhood of O in Sym, there is a polynomial P satisfying this relation for all f , and the only question is whether this polynomial is K-invariant. But the conjugation action by K on Posn corresponds to the conjugation action of K in the chart, i.e. conjugation by an element k ∈ K commutes with the exponential, as we have already used in the first part of the proof. By the invariance of the operator, we have for all k ∈ K, ¶ ¶ µ µ ¯ ¯ ∂ ∂ −1 ¯ P F (kXk ) X=O = P F (X) ¯X=O . ∂X ∂X Put Z = kXk −1 . The same transformation discussed at the beginning of this section, with Y replaced by X, shows that ∂ ∂ = k −1 k. ∂X ∂Z Hence P

µ

k −1

¶ µ ¶ ¯ ¯ ∂ ∂ k F (Z) ¯Z=O = P F (Z) ¯Z=O ∂Z ∂Z

for all functions F in a neighborhood of O in Sym. This implies that P is K-invariant, and concludes the proof of the theorem. Now that Theorems 2.2 and 3.2 are proved essentially the same way, but the notation and context are sufficiently different so we reproduced the proofs separately. The key point is that in both charts, conjugation by elements of K commutes with the chart.

4 The Transpose of an Invariant Differential Operator

87

Remark. Here we have treated the relations between Pol(Sym)K and DO(Posn )G directly on the symmetric space Posn . Helgason treats the situation more elaborately on the group and an arbitrary reductive coset space, cf. [Hel 84], Chap. 2, Theorems 4.3 through 4.9. Given a polynomial P (X), we let P (∂) be the differential operator with constant coefficients obtained by substituting ∂ij for xij . Thus if we wish to suppress the variables, we could write µ ¶ ∂ P = P (∂) . ∂X This is a differential operator on the space of functions on Sym.

4 The Transpose of an Invariant Differential Operator Let M be a manifold with a volume form, and D a differential operator. As usual, we can deal with the hermitian integral scalar product, or the bilinear symmetric integral scalar product, given by the integral without the extra complex conjugate, with respect to the volume dµ. We let D∗ be the adjoint of D with respect to the hermitian product, and t D the transpose of D with respect to the symmetric scalar product. Thus for C ∞ functions ψ1 , ψ2 for which the following integrals are absolutely convergent, we have by definition Z Z (Dψ1 )ψ2 dµ = ψ1 (t Dψ2 )dµ . M

M

We shall denote the symmetric scalar product by [ψ1 , ψ2 ], to distinguish it from the hermitian one hψ1 , ψ2 i. Then the transpose formula reads [Dψ1 , ψ2 ] = [ψ1 , t Dψ2 ] . The existence of the transpose in general is a simple routine matter. Let Ω be a volume form on a Riemannian manifold. In local coordinates x1 , . . . , xN on a chart which in Euclidean space is a rectangle, say, we can write Ω(x) = β(x1 , . . . , xN )dx1 ∧ . . . ∧ dxN . We suppose the coordinates oriented so that the function θ is positive. Let D be a monomial differential operator, so in terms of the coordinates D = γ∂j1 . . . ∂jm , where γ is a function, and ∂j is the partial derivative with respect to the j-th variable. Then integrating over the manifold, if γ or ϕ or ψ has compact support in the chart, we can integrate by parts and the boundary terms will vanish, so we get

88


Z

Z

γψβ(∂j1 . . . ∂jm ϕ)dx1 ∧ . . . ∧ dxN Z = (−1)m ∂j1 . . . ∂jm (γψβ)ϕdx1 ∧ . . . ∧ dxN Z 1 m ∂j . . . ∂jm (γβψ)ϕΩ . = (−1) β 1

(Dϕ)ψΩ =

Thus we find Proposition 4.1. In local coordinates, suppose Ω(x) = β(x1 , . . . , xN )dx1 ∧ . . . ∧ dxN and D = γ∂j1 . . . ∂jm . If γ, or ϕ, or ψ has compact support in the chart, then t

1 Dψ = (−1)m ∂j1 . . . ∂jm (γβψ) . β

Using a partition of unity, this formula also applies under conditions of absolute convergence. Apply this to the volume form corresponding to the measure on Posn : dµn (Y ) = |Y |−(n+1)/2 dµeuc (Y )

whereby β(Y ) = |Y |−(n+1)/2 .

We get: Proposition 4.2. Let DY = α(Y )

Y µ ∂ ¶mij ∂yij

i5j

and m = t

P

mij . Then

DY = (−1)m |Y |(n+1)/2

Y µ ∂ ¶mij ◦ (α(Y )|Y |−(n+1)/2 ) . ∂yij

i5j

We shall relate convolutions and symmetries with the transpose. We note that the function (Z, Y ) 7→ tr(Y Z −1 ) is a point pair invariant on Posn . It follows that the function (Z, Y ) 7→ e−tr(Y Z

−1

)

= exp(−tr(Y Z −1 ))

is a point pair invariant, which goes to zero rapidly as tr(Y Z −1 ) goes to infinity. We define the gamma kernel or gamma point pair invariant to be −1 −1 ϕ(Z, Y ) = e−tr(Z Y ) = e−tr(Y Z ) .

4 The Transpose of an Invariant Differential Operator

89

Proposition 4.3. For a character ρ on Posn lying in a half space of convergence, the eigenvalue of the gamma operator is given by (ϕ ∗ ρ)(I) = Γn (ρ) = λΓ (ρ) , which was computed in Chap. 3, Proposition 2.1. In other words, (ϕ ∗ ρ)(Z) = Γn (ρ)ρ(Z) . Proof. This is just Proposition 2.2 of Chap. 3. As mentioned in Chap. 3, we emphasize the eigenvalue property of Γn (ρ) by writing Γn (ρ) = λΓ (ρ) . We can now prove a formula for the transpose of an invariant differential operator, as in Maass [Maa 71]. The result is stated without proof in [Sel 56], p. 53. Proposition 4.4. Let S(Y ) = Y −1 . Let D be an invariant differential operator, and let ˜ = [S]D . D Then t

˜ D=D

and

¯˜ = t D ¯. D∗ = D

Proof. By [JoL 01], Theorem 1.3 of Chap. 3, it suffices to verify the formula on the characters ρ. Actually, if we parametrize the characters as ρs with n complex variables s, it suffices to prove the relation for s lying in the half space of convergence of the gamma convolution. Let ϕ(Z, Y ) = exp(−tr(Y Z −1 )). Consider ρ = ρs lying in this half space. Let λρ = (ϕ ∗ ρ)(I) and let c(ρ) denote the eigenvalue of t D on ρ. Then by definition λρ t Dρ = λρ c(ρ)ρ . On the other hand, ˜ ˜Z λρ Dρ(Z) =D

Z

ϕ(Z, Y )ρ(Y )dµn (Y )

by Proposition 4.3

Posn

=

Z

˜ 1 ϕ(Z, Y )ρ(Y )dµn (Y ) D

Posn

=

Z

=

Z

ϕ(Z, Y )(t Dρ)(Y )dµn (Y ) by definition of transpose

=

Z

ϕ(Z, Y )c(ρ)ρ(Y )dµn (Y ) ,

D2 ϕ(Z, Y )ρ(Y )dµn (Y ) by [JoL 01], Chap. 4 Lemma 1.4

= λρ c(ρ)ρ .

90


˜ and t D have the same eigenvalues ρ for all ρ. Hence they are equal Hence D by [JoL 01], Theorem 1.3 of Chap. 3. This concludes the proof.

5 Invariant Differential Operators on A and the Normal Projection We start with invariant differential operators on A at the level of freshman calculus. We have A ≈ R+ × . . . × R+ with a = diag(a1 , . . . an ) ∈ A . Let a = Lie(A) as usual, so a is the vector space of diagonal matrices. For each i = 1, . . . , n let Di be the differential operator on the space Fu(A) of C ∞ functions on A, by ∂f . (Di f )(a) = ai ∂i f (a) = ai ∂ai Then a direct computation shows that each Di is invariant under multiplicative translations on A, that is Di ∈ IDO(A). The construction of invariant differential operators given previously on Posn can be reproduced directly on A. In other words, for a polynomial P (h1 , . . . , hn ) in the coordinates of an element of a with respect to the natural basis of R × . . . × R, one may define the differential operator DP on Fu(A) by the formula ¯ (DP f )(a) = P (∂/∂h)f (a exp h) ¯h=0 ¯ = P (∂/∂h1 , . . . , ∂/∂hn )f (a1 eh1 , . . . , an ehn ) ¯ . h=0

From the ordinary chain rule of calculus, it is immediate that DP = P (D1 , . . . Dn ) .

Proposition 5.1. The algebra of invariant differential operators on A is the polynomial algebra R[D1 , . . . Dn ]. The map P 7→ DP gives an algebra isomorphism from Pol(a) to IDO(A). The elements invariant under W (the Weyl group of permutations of the variables) correspond to each other under this isomorphism. Proof. This is a result in calculus, because of the special nature of the group A. Indeed, the ordinary exponential map gives a Lie group isomorphism of R × . . . × R with A. Then invariant differential operators on a = R × . . . × R are simply the differential operators with constant coefficients. One sees this at once from the invariance, namely if we let f ∈ Fu(a), then for v ∈ a, D(f (x + v)) = (Df )(x + v) .

5 The Normal Projection

91

This means that the coefficients in the expression D=

X

c(j) (x)

µ

∂ ∂x1

¶j1

...

µ

∂ ∂xn

¶jn

are translation invariant, and so constant. Note that the partial derivatives ∂/∂xi on a correspond to ai ∂/∂ai on A, with the change of variables ai = exi

or

xi = log ai .

For the general Harish-Chandra theory of polynomials and invariant differential operators, cf. [JoL 01], Theorem 1.1 of Chap. 2. Next, we describe systematically the relation of invariant differential operators on Posn with invariant differential operators on A via the normal projection, obtained by extending a function on A to a function on Posn by making it constant along orthogonal geodesics to A. The main theorem on Posn , Theorem 5.4 is a special case of Helgason’s general theorem on semisimple Lie groups [Hel 77] and [Hel 84], Chap. 2, Theorem 5.13. The treatment here in some sense follows Helgason, but he works by descending from the group, while we work directly on the symmetric space Posn . The present section uses some concepts from differential geometry. it will not be used in the sequel, and readers may omit it. Differential geometry will be essentially absent from our development. On the other hand, some readers may find it illuminating to see the present section provide an example of some general differential geometric considerations. As before, let a = Lie(A) be the algebra of diagonal matrices, and let a⊥ = Sym(0) be its orthogonal complement under the trace form, so a⊥ is the space (not algebra) of matrices with zero diagonal components. For a ∈ A, the multiplicative translate [a]Sym(0) is the normal space to the tangent space of [a]I, but in the present situation, [a]Sym(0) = Sym(0) . As we have seen previously, the variables X split naturally, so that we write X = (Xa , X (0) ) , where Xa = (x11 , . . . , xnn ) are the variables on a, and X (0) = (xij )i<j are the variables on Sym(0) = a⊥ . We may decompose a polynomial P ∈ Pol(Sym) a sum P = Pa + P (0) where Pa = Pa (Xa ) = Pa (x11 , . . . , xnn ) involves only the variables on a, and all monomials occurring in P (0) (meaning with non-zero coefficient) involve

92


a positive power of some coordinate xij (i < j) on a⊥ . We call this the aorthogonal decomposition of P , and we call Pa the projection of P on a. We call P (0) the a-orthogonal projection of P . We note the frequently used fact that P (0) = P (0) (X (0) ) vanishes on a. In particular, Pa (Xa ) is the restriction of P to a. Proposition 5.3. The map P 7→ Pa is a linear isomorphism ≈

Pol(Sym)K −→ Pol(a)W . Proof. This is essentially a repetition of Theorem 5.1, combined with the remarks preceding the proposition. The result essentially comes from the relation Sym = [K]a and Sym∨ = [K]a∨ . Associated to the action [a] : Posn → Posn by an element a ∈ A, we have the derivative [a]0 (I) : Sym → Posn

and

[a]0 (I) = [a] .

As for the exponential map, for v ∈ Sym, exp[a]I [a]v = [a] expI (v) . We view TI Posn = Sym as the tangent space of Posn at I, and T[a]I Posn is its image under [a]. So at each [a]I ∈ A we have the splitting T[a]I Posn = [a]Sym = [a]a + [a]a⊥ , which, by our choice of chart can be viewed as simply Sym again, but one has to be careful about the action in making identifications. Let NPosn A be the normal bundle of A in Posn . Then the fibers of NPosn A are simply the normal spaces [a]a⊥ , with a ∈ A, so may be identified with a⊥ . The exponential map (varying at each point) V 7→ exp[a]I (v) for v ∈ T[a]I Posn induces a differential isomorphism of a neighborhood of the zero section in the normal bundle with a neighborhood of A in Posn , called a tubular neighborhood. Actually, the tubular neighborhood exists globally. Theorem 5.4. The map eN A : A × a⊥ → Posn is a differential isomorphism.

given by

(a, v) 7→ [a] exp(v)

5 The Normal Projection

93

Proof. This is a special case of a result given by Loos [Loo 69], pp. 161–162, in the context of semisimple Lie groups and symmetric spaces. However, it is also a special case of a much more general theorem about Cartan-Hadamard spaces in differential geometry, see [La 99], Chap. X, Theorem 2.5 for an exposition and further historical comments. Helgason [Hel 78], Chap. 1, Theorem 14.6 and Chap. 6, Theorem 1.4 gives a topological version without differentiability, extending theorems of Mostow [Mos 53], also stated without differentiability. The product decomposition of Theorem 5.4 will be called the A-normal decomposition of Posn . The geometry is illustrated on the following figure, with w ∈ Sym(0) . [a] exp(tw) exp(tw) A [a] I I

We have drawn the exponential curves emanating from the unit matrix, and from some translation [a]I, in the geodesic normal direction, a situation we now discuss more extensively. Given a function f on A, we can define its normal extension to Posn , namely we let fPosn be the function on Posn which is constant on every translation [a] exp(a⊥ ), for all a ∈ A. From Theorem 5.4 we have two differential isomorphisms e

NA a + a⊥ = a + Sym(0) −→ A × a⊥ −→ Posn ,

where the left arrow is simply the exponential map on a and the identity on a⊥ . Thus the function f on A not only can be extended normally to Posn via eN A , but can be pulled back to a function F on a, and then be extended to FSym constant on each coset h + a⊥ , with h ∈ a. Thus for X ∈ a⊥ = Sym(0) , and a = exp h, we have by definition f (exp h) = F (h) For P (0) we then have (1)

and fPosn ([a] exp X) = FSym (h, X) . P (0) (∂)FSym = 0.

Next we shall relate these partial differential operators with Proposition 4.1 via a commutative diagram. Our goal is to prove Theorem 5.5. Note that each

94


W -invariant polynomial Pa on a gives rise to an invariant differential operator on A, in the obvious way, similar to the association P 7→ DP on Posn , namely µ ¶ ¯ ∂ (DPa f )(a) = P f ([a][exp h]I) ¯h=0 for f ∈ Fu(A) . ∂h Here h is the n-tuple of diagonal variables with hi = xii (i = 1, . . . , n), and Fu denotes the space of C ∞ functions. Let D be an invariant differential operator on Posn . We define its A⊥ , also written r⊥ normal (or radial) projection DA A (D), on the space of ⊥ functions Fu(A) as follows. Let fPosn be the normal extension of f to Posn . We ⊥ ⊥ apply D to fPos f , so by definition, and restrict this function to A to get DA n ⊥ ⊥ r⊥ A (D)f = DA f = (DfPosn )A .

⊥ Lemma 5.5. The map D 7→ DA is a linear isomorphism ≈

K W r⊥ . A : DO(Posn ) −→ IDO(A)

For D = DP we have

DPa = r⊥ A (DP ) .

Proof. Immediate from the definitions and formula (1). Theorem 5.6. We have a commutative diagram of linear isomorphisms: DO(Posn )G −−→ IDO(A)W x x     Pol(Sym)k −−→ Pol(a)W

given by

⊥ DP −−→ DPa = DA x x    

P

−−→

Pa

Proof. This just puts Proposition 5.1 and the previous lemma together.

5 Poisson Duality and Zeta Functions

This chapter recalls some standard facts about the Poisson summation formula on a euclidean space. It will be applied when the euclidean space is the space of matrices, with the trace scalar product, so we make all the standard formalism explicit in this case. We give two classical applications to the Epstein zeta function, which serve as prototypes. In the extension of the theory to Posn , we have to generalize the definition of the Bessel functions to this higher dimensional case, and this will be done in the next chapter. Then we can put all these results together in the study of Eisenstein series.

1 Poisson Duality Duality on Vector Spaces Over R Let V be a finite dimensional vector space over R, of dimension N . We suppose given a positive definite scalar product on V , denoted by hx, yi. We also suppose given a Lebesgue measure (any two such differ by a positive constant factor), denoted by µ. We normalize the Fourier transform of a function f by the formula Z FT 1. f ∨ (x) = f (y)e−2πihx,yi dµ(y) . V

We call f ∨ the Poisson dual, which is a normalization of the Fourier transform adapted for the summation formula. Let b > 0. Then dµ(by) = bN dµ(y) . Define f ◦ b to be the multiplicative translation, i.e. (f ◦ b)(x) = f (bx). Then FT 2.

(f ◦ b)∨ (x) = b−N f ∨ (x/b) .

Jay Jorgenson: Posn (R) and Eisenstein Series, Lect. Notes Math. 1868, 95–106 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com °

96


More generally, let A : V → V be an invertible linear map, and denote kAk the absolute value of the determinant of A. Then for the composite f ◦ A we have (f ◦ A)∨ = kAk−1 f ∨ ◦ t A−1 .

FT 3.

In particular, if A is symmetric (which will be the case in practice) we find (f ◦ A)∨ = kAk−1 f ∨ ◦ A−1 .

FT 4.

For example, if A is represented by a diagonal matrix dia(a1 , . . . , aN ) with respect to a basis, and aj > 0 for all j, then kAk = a1 . . . aN . Next, let z ∈ V , and define the additive translation fz by fz (x) = f (x − z) . Then FT 5.

(fz )∨ (x) = fz∨ (x) = e−2πihx,zi f ∨ (x) .

This comes at once from the invariance of µ under additive translations. FT 6. If the measure of the unit cube for an orthonormal basis is 1, and we define f − (x) = f (−x), then f ∨∨ = f − . One can either repeat the usual proof with the present normalization, or deduce it from the same formula for the otherwise normalized Fourier transform. That is, if we define Z √ f (x)e−ixy dν(y) with dν(y) = ( 2π)−N dy , f ∧ (x) = RN

and dy is the usual Lebesgue measure dy1 . . . dyN , then f ∧∧ = f − . A complete proof is found in basic texts, e.g. in Lang’s Real and Functional Analysis, Chap. 8, Theorem 5.1. FT 7. Assume again that the measure of the unit cube for an orthonormal basis is 1. Then the function f (x) = e−πhx,xi is self-dual, i.e. f ∨ = f . For the other normalization, the function g(x) = e−hx,xi/2 is self-dual, that is g ∧ = g. These relations are elementary from calculus.

2 The Matrix Scalar Product

97

Lattices. Let L be a lattice in V , that is a free Z-module of rank N , which is also R-free. We let L0 be the dual lattice, consisting of all x ∈ V such that hx, yi ∈ Z for all x ∈ L. Then L0 ≈ Hom(L, Z) in a natural way. For functions in the Schwartz space (at the very least) one has the Poisson Summation Formula X

X

f (α) = µ(V /L)−1

f ∨ (α0 ) .

α0 ∈L0

α∈L

Proof. The Poisson formula can be seen from our point of view as a special case of the heat kernel relation on a torus. However, we give the usual proof via Fourier inversion. Normalize µ so that µ(V /L) = 1. Let X f (x + α) . g(x) = α∈L

Then g is L-periodic, so g has a Fourier series expansion X 0 g(x) = cα0 (g)e2πihx,α i α0 ∈L0

with Fourier coefficients Z

cα0 (g) =

0

g(x)e−2πihx,α i dµ(x)

V /L

Z X

=

V /L

=

Z

0

f (x + α)e−2πihx,α i dµ(x)

α∈L 0

f (x)e−2πihx,α i dµ(x) = f ∨ (α0 ) .

V

Then

X

f (α) = g(0) =

α∈L

X

f ∨ (α0 ) ,

α0 ∈L0

which concludes the proof of the formula.

2 The Matrix Scalar Product We start with two positive integers p, q. Let: V = Rp×q = the vector space of p × q real matrices L = Zp×q = lattice of integral matrices.

98


For X, Y ∈ V define the trace scalar product, or scalar product for short, hX, Y i = tr( t XY ) . P 2 This defines a positive definite scalar product, since hX, Xi = xij . Then the lattice L is self dual, i.e. L = L0 . Note that the vectors Eij with (i, j)-component 1, and 0 otherwise, form an orthonormal basis, so we are dealing with a euclidean space of dimension N = pq, with N as in Sect. 1. We take the measure µ to be the ordinary Lebesgue measure, Y dµ(X) = dxij . i,j

The volume of the unit cube is 1. In particular, the function f (X) = e−πhX,Xi

is Poisson self dual, and formula FT 5 also applies that is for any function f in the Schwartz space, f ∨∨ = f − . Next, we look into some interesting linear automorphisms of V , to which we can apply FT 3, FT 4 and the Poisson formula. Let P ∈ Posp and Q ∈ Posq . These give rise to a linear map MP,Q = M (P, Q) : V → V

1

1

by MP,Q (X) = P 2 XQ 2 .

Lemma 2.1. The map MP,Q is symmetric positive definite, and satisfies (1)

M (P, Q)−1 = M (P −1 , Q−1 ) . Its determinant is given by

(2)

|M (P, Q)| = |P |q/2 |Q|p/2 . The lemma is immediate. We note that hP X, XQi = tr( t XP XQ) = tr(t XP 1/2 P 1/2 XQ1/2 Q1/2 ) = tr(Q1/2 t XP 1/2 P 1/2 XQ1/2 ) .

since tr(t XP XQ) = tr(P XQt X), we write tr(P [X]Q) without parentheses, and get (3)

tr(P [X]Q) = hMP,Q (X), MP,Q (X)i .

We now define the theta series by using the function f (X) = e−πhX,Xi , and its composite with the linear MP,Q , that is X X e−πtr(P [A]Q) = f (MP,Q (A)) . θ(P, Q) = A∈L

A∈L

The formalism of the Fourier transform and Poisson summation yields:

3 The Epstein Zeta Function: Riemann’s Expression

99

Theorem 2.2. The theta series satisfies the functional equation θ(P, Q) = |P |−q/2 |Q|−p/2 θ(P −1 , Q−1 ) . Remark. As a special case of the above situation, we can take q = 1, and V = Rn (space of column vectors). The scalar product is the usual one if we take Q = 1. We can also incorporate a translation in our tabulations. Let Z ∈ V . We define X X θ(P, Z, Q) = e−πtr(P [A+Z]Q) = f−Z (MP,Q (A)) . A∈L

A∈L

Then FT 5 yields (4)

θ(P, Z, Q) = |P |−q/2 |Q|−p/2

X

e−2πihA,Zi e−πtr(P

−1

[A]Q−1 )

.

A∈L

The sum on the right may be called a twist of the theta series by a character, namely the character X 7→ e−2πihX,Zi . It is the only change in the Poisson formula of Theorem 2.2, but has to be incorporated in the notation. Thus one may define the twisted theta function X θZ (P, Q) = e−2πihA,Zi e−πtr(P A]Q) . A∈L

Then we get:

Theorem 2.3. From the definitions and (4), θ(P, Z, Q) = |P |−q/2 |Q|−p/2 θZ (P −1 , Q−1 ) . Note that the sum over A ∈ L involves each non-zero element of L twice, namely A and −A. Thus one could write the contribution of the character in the series for θZ (P, Q) without the minus sign, since P −1 [A]Q−1 is even as a function of A.

3 The Epstein Zeta Function: Riemann’s Expression For Y ∈ Posn , we define the Epstein zeta function to be X X E(Y, s) = Y [a]−s = (t aY a)−s . a∈Zn −{0}

The vectors a ∈ Zn are viewed as column vectors, so t a is a row vector.

100


Lemma 3.1. The Epstein series converges absolutely for all s ∈ C with Re(s) > n/2. Proof. We estimate the number of lattice points a lying in a spherical annulus, that is k − 1 5 |a| 5 k with, say, the euclidean norm |a| .

The area of the (n − 1)-dimensional sphere in Rn of radius k is >>> |Y 1/2 a|2 >> |a|2 for |a| → ∞, so Y [a]s >> |a|2s . Hence the series converges whenever 2s − (n − 1) > 1, which is precisely when s > n/2, as asserted. Remark. Because of the way a vector a enters as a square in t aY a, we note that the terms in the Epstein series are really counted twice. Hence the Epstein zeta function is sometimes defined to be 1/2 of the expression we have used. However, we find that not dividing by 2 makes formulas come out more neatly later. We let the completed Epstein Λ-function be Λ(Y, s) = π −s Γ(s)E(Y, s) . For t > 0 we let θ(Y, t) =

(1)

X

e−πY [a]t .

a∈Zn

This is a special case of the theta series for matrices, with p = n, q = 1, P = Y and Q = t. For Re(s) > n/2 (the domain of absolute convergence of the zeta series), we have the expression of the Λ-function as a Mellin transform (2)

∞ Z∞ XZ dt s dt e−πY [a]t ts . = Λ(Y, s) = (θ(Y, t) − 1)t t t 0

a6=0 0

This simply comes by integrating term by term the theta series, taking the Mellin transform of each term. Having subtracted the term with a = 0 guarantees absolute convergence and the interchange of the series and Mellin integral.


101

We shall now give the Riemann type expression for the analytic continuation of the Epstein zeta series. We define the incomplete gamma integral for c > 0 by Γ∞ 1 (s, c)

=

Z∞

e−ct ts

dt . t

1

We note that

Γ∞ 1 (s, c)

is entire in s.

Theorem 3.2. The function s 7→ Λ(Y, s) has the meromorphic extension and ! Ã 1 1 |Y |− 2 − Λ(Y, s) − s − n/2 s ³ ´´ X³ − 12 ∞ n = Γ∞ Γ1 − s, πY −1 [a] 1 (s, πY [a]) + |Y | 2 a6=0

=

Z

1

∞

(θ(Y, t) − 1)t

s dt

t

+

Z∞ 1

1

|Y |− 2 (θ(Y −1 , t) − 1)tn/2−s

dt . t

The series and truncated integrals converge uniformly on every compact set to entire functions, and the other two terms exhibit the only poles of Λ(Y, s) with the residues. The function satisfies the functional equation ³ ´ 1 n Λ(Y, s) = |Y |− 2 Λ Y −1 , − s . 2

Proof. In (2), we write

Z∞

Z1

+

The integral over [1, ∞] yields the sum the other integral, we get

P

=

0

Z1 0

(θ(Y, t) − 1)t

s dt

t

=−

Z1

0

t

s−1

0

1 =− + s

Z∞

.

1

Γ∞ 1 (s, πY [a]) taken over a 6= 0. For

dt +

Z1

θ(Y, t)ts

dt t

0

Z1 0

|Y |−1/2 θ(Y −1 , t−1 )ts−n/2

dt . t

We subtract 1 and add 1 to θ(Y −1 , t−1 ). Integrating the term with 1 yields the second polar term |Y |−1/2 /(s − n/2). In the remaining integral with θ(Y −1 , t−1 )−1 , we change variables, putting u = t−1 , du/u = dt/t. Then the interval of integration changes from [0, 1] to [1, ∞], and the remaining terms in the desired formula comes out. This concludes the proof of the formula in the theorem.

102


By Theorem 2.2, we know the functional equation for θ(Y, t). The functional equation in Theorem 3.2 is then immediate, because except for the factor |Y |−1/2 , under the change s 7→ n/2 − s, the first two terms are interchanged, and the two terms in the sum are interchanged. The factor |Y |−1/2 is then verified to behave exactly as stated in the functional equation for ξ(Y, s). This concludes the proof. The integral expressions allow us to estimate ξ(Y, s) in vertical strips as is usually done in such situations, away from the poles at s = 0, s = n/2. Corollary 3.3. Let σ0 > 0, σ1 > n/2 and let S be that part of the strip −σ0 5 Re(s) 5 σ1 , with |s| = 1 and |s − n/2| = 1. Then for s ∈ S we have ´ ³ 1 1 n |Λ(Y, s)| 5 |Y |− 2 + 1 + Λ(Y, σ1 ) + |Y |− 2 Λ Y −1 , + σ0 . 2

Proof. We merely estimate the three terms in Theorem 3.2. The polar terms give the stated estimate since we are estimating outside the discs of radius 1 around the poles. We make the first integral larger by replacing s by σ1 , and then by replacing the limits of integration, making them from 0 to ∞, which gives ξ(Y, σ1 ) as an upper bound for the first integral. As to the second integral, we perform the similar change, but use the value s = −σ0 to end up with the stated estimate. This concludes the proof. Corollary 3.4. The function s(s − n/2)Λ(Y, s) is entire in s of order 1. The function Λ(Y, s) is bounded in every vertical strip outside a neighborhood of the poles. Proof. Routine argument from the functional equation, properties of the gamma function, and the boundedness of the Dirichlet series for ζ(Y, s) in a right half plane. We give some useful complements to the functional equation. Let Y˜ be the matrix such that Y Y˜ = |Y |In ,

so Y˜ is the matrix of minors of determinants. Then: Corollary 3.5. We have: (a) (b)

E(Y −1 , s) = |Y |s E(Y˜ , s)

and Λ(Y −1 , s) = |Y |s Λ(Y˜ , s) ´ ³ 1 n Λ(Y, s) = |Y |n/2−s− 2 Λ Y˜ , − s . 2

Proof. By definition

Y −1 = |Y |−1 Y˜ ,

so y −1 [a] = |Y |−1 Y˜ [a] and the first formula (a) drops out. Then by Theorem 3.2,


´ n Λ(Y, s) = |Y |−1/2 Λ Y −1 , − s ´ ³2 n = |Y |n/2−s−1/2 Λ Y˜ , − s 2 ³

103

by (a).

This concludes the proof.

The case n = 2 is important in subsequent applications, because it is used inductively to treat the case n = n. Hence we make some more comments here when n = 2. In this case, the functional equation can be formulated with Y on both sides (no Y −1 or Y˜ on one side is necessary). Proposition 3.6. Suppose n = 2. Then E(Y, s) = E(Y˜ , s) and so Λ(Y, s) = Λ(Y˜ , s) . Proof. Note that if Y =

µ

u v

v w

¶

then Y˜ =

µ

w −v

−v u

¶

.

Then for an integral vector (b, c) we have [b, c]Y˜ = wb2 − 2vbc + vc2 so that

[b, c]Y = [c, −b]Y˜ .

The map (b, c) 7→ (c, −b) permutes the non-zero elements of Z2 so the proposition follows. Corollary 3.7. For n = 2, we have the functional equation 1

Λ(Y, s) = |Y | 2 −s Λ(Y, 1 − s) . Proof. Corollary 3.5(b) and Proposition 3.6. For later use, we insert one more consequence, a special case of Corollary 3.3. Corollary 3.8. In the domain −2 < Re(s) < 3 and s outside the discs of radius 1 around 0, 1, we have the estimates: 1

|Λ(Y, s)| 5 1 + |Y |− 2 + Λ(Y, 3)|Y |5/2 + Λ(Y, 3) . Proof. Immediate from Corollary 3.3 and the functional equation.

104


4 Epstein Zeta Function: A Change of Variables For subsequent applications, we reformulate the functional equation of the Epstein zeta function in terms of new variables. To fit the notation to be used later, we write the Dirichlet series as X E(Y, z) = Y [a]−z a∈Z2 a6=0

and we introduce two complex variables s1 , s2 for which we set 1 z = s2 − s1 + . 2 We then use the notation ζ(Y, s1 , s2 ) = E(Y, z) . Theorem 4.1. (a) The function

in entire on C 2 . (b) The function

µ ¶ 1 (z − 1)E(Y, z) = s2 − s1 − ζ(Y ; s1 , s2 ) 2

π −z Γ(z)|Y |s2 E(Y, z)

with

z = s2 − s1 + 1/2 ,

is invariant under the permutations of s1 and s2 .

Proof. As to (a), from Theorem 3.2, with n = 2, we know that the function (z − 1)zΛ(Y, z) is entire, and (z − 1)zΛ(Y, z) = (z − 1)zπ −z Γ(z)E(Y, z)

so

= Γ(z + 1)π −z (z − 1)E(Y, z) ,

π −z (z − 1)E(Y, z) =

(z − 1)zΛ(Y, z) Γ(z + 1)

which is entire, and thus proves (a). As to (b), the invariance under z 7→ 1 − z means

π −z Γ(z)|Y |s2 E(Y, z) = π −(1−z) Γ(1 − z)|Y |s1 E(Y, 1 − z) .

But this relation is equivalent to

1

π −z Γ(z)E(Y, z) = Λ(Y, z) = |Y | 2 −z Λ(Y, 1 − z) ,

which is true by Corollary 3.7. This concludes the proof.

Note that we may further complete the function in Theorem 4.1 by defining η(Y ; s1 , s2 ) = z(1 − z)π −z Γ(z)|Y |s2 ζ(Y ; s1 , s2 ) .

The factor z(1 − z) is invariant under the transposition of s1 and s2 , and Theorem 4.1 may be alternatively stated by saying that η(Y ; s1 , s2 ) is entire in (s1 , s2 ) and invariant under the transposition of the two variables.

5 Epstein Zeta Function: Bessel-Fourier Series

105

5 Epstein Zeta Function: Bessel-Fourier Series The Bessel-Fourier series for the Epstein-Eisenstein function can still be done with the ordinary Bessel function, so we carry it out here separately, as an introduction to the more general result in the matrix case, when a generalization of the Bessel function will have to be taken into account. We write n = p + q with integers p, q = 1. We have a partial Iwasawa decomposition of Y ∈ Posn , given by W ∈ Posp , V ∈ Posq , X ∈ Rp×q with     W 0 Ip x  where u(X) =   . Y = [u(X)]  0 V 0 Iq

In Chap. 1, we took q = 1 with V = v, but there will be no additional difficulty here by taking arbitrary dimensions, so we might as well record the result in general. The Epstein zeta series (function) E(Y, s) is the same that we considered in the preceding section, with its completed Λ-function Λ(Y, s). We let Ks (u, v) be the usual Bessel integral Ks (u, v) =

Z∞

e−(ut+v/t) ts

dt . t

0

Theorem 5.1. Let Y ∈ Posn have the above partial Iwasawa decomposition. Then Λ(Y, s) = Λ(V, s) + |V |−1/2 Λ(W, s − q/2) XX t + |V |−1/2 e2πi bXc Ks−q/2 (πW [b], πV −1 [x]) , b6=0 c6=0

with the sums taken for b ∈ Zp , c ∈ Zq . Proof. The Λ-function is the Mellin transform of the theta function, Λ(Y, s) =

Z∞

θ(Y, t)ts

dt t

with

θ(Y, t) =

e−πY [a]t .

a6=0

0

We decompose a into two components µ ¶ b a= with b ∈ Zp c

X

and c ∈ Zq .

Then Y [a] = W [b] + V [t Xb + c] . We decompose the sum for θ(Y, t) accordingly:

106


X

=

a∈0

X b=0 c6=0

+

X X b6=0

.

c∈Zq

The sum with b = 0 gives the term Λq (V, s). The sum over all c for each b 6= 0 is then a theta series to which Poisson summation formula applies as in Theorem 2.3, to yield X t e−πW [b]t e−πV [ Xb+c]t c∈Zq

= e−πW [b]t |V |−1/2 t−q/2

X

t

e−2πi

bXc −πV −1 [c]t−1

e

.

c∈Zq

The term with c = 0 summed over all b 6= 0 yields |V |−1/2 Λ(W, s − q/2). The remaining sum is a double sum −1/2

|V |

X b=0 c6=0

2πit bXc

e

Z∞

e−π(W [b]t+V

−1

[c]t−1 ) s−q/2

t

dt . t

0

The theorem follows from the definition of the K-Bessel function.

6 Eisenstein Series First Part

In this chapter, we start systematically to investigate what happens when we take the trace over the discrete groups Γ = GLn (Z), for various objects. In the first section, we describe a universal adjointness relation which has many applications. One of them will be to the Fourier expansion of the Eisenstein series.

1 Adjointness Relations Let: U = Uni+ = group of upper unipotent n × n matrices. Γ = GLn (Z) Γ∞ = ΓU = Γ ∩ U . We let ρ be a character on Posn . The most classical Selberg primitive Eisenstein series is the series X ρ([γ]Y ) . E pr (Y, ρ) = γ∈ΓU \Γ

Since a character ρ satisfies ρ([u]Y ) = ρ(X) for u ∈ U , it follows that the value ρ([γ]Y ) depends only on the coset ΓU γ in ΓU \Γ, whence the sum was taken over such cosets to define the Eisenstein series. If ρ = ρ−s , then the sum is the usual X ρ−s ([γ]Y ) , EUpr (Y, ρ−s ) = E pr (Y, ρ−s ) =

depending on n complex variables s = (s1 , . . . , sn ), and ρ−s = ρ−1 s . The series converges absolutely in a half plane, as will be shown in Chap. 7, Sect. 2. Also note that for all γ ∈ Γ, we have


108


E pr ([γ]Y, ρ) = E pr (Y, ρ) . As usual, let d denote the determinant character. Let α be a complex number. Then E pr (Y, ρdα ) = |Y |α E pr (Y, ρ) . This is because |γ| = ±1 for all γ ∈ Γ, so |[γ]Y | = |Y | for all γ ∈ Γ. In some applications, it is convenient to carry along such an independent power of the determinant. Both groups Γ and U act on the space P = Posn . Under suitable absolute convergence conditions, there are two maps which we consider: The ΓU \Γ-trace (also called Einsenstein trace) TrΓU \Γ : functions on U \P → functions on Γ\P defined by TrΓU \Γ ϕ(Y ) =

X

ϕ([γ)]Y ) .

γ∈ΓU \Γ

The unipotent trace or ΓU \U -trace TrΓU \U : functions on Γ\P → functions on U \P defined by f (Y ) =

Z

f ([u]Y )du .

ΓU \U

Example The Eisenstein series is a ΓU \Γ-trace, namely E pr (Y, ϕ) = TrΓU \Γ ϕ(Y )

or E pr (ρ) = TrΓU \Γ (ρ) .

We shall give two essentially formal properties of the ΓU \Γ-trace. For the first one, see already SL2 (R), Chap. XIII, Sect. 1. Proposition 1.1. Under conditions of absolute convergence, the two maps TrΓU \Γ and TrΓU \U are adjoint to each other, that is more precisely hTrΓU \Γ ϕ, f iΓ\P = 2hϕ, TrΓU \U f iU \P where the scalar product is given by the usual hermitian integral, or also by the bilinear integral without the complex conjugation. Proof. For simplicity, we carry out the computation without the complex conjugation. We write formally dy instead of dµ(Y ). The factor 2 appears because Γ does not act faithfully on Posn , but with kernel ±I. So from the point of view of the measure, terms get counted twice in the third step of the following proof. We have:

1 Adjointness Relations

hTrΓU \Γ ϕ, f iΓ\P =

Z

TrΓU \Γ ϕ(y)f (y)dy

Z

X

109

Γ\P

=

ϕ([γ]y)f (y)dy

Γ\P ΓU \Γ

=2

Z

ϕ(y)f (y)dy

Z

Z

Z

ϕ(y)

ΓU \P

=2

ϕ([u]y)f ([u]y)dudy

U \P ΓU \U

=2

U \P

Z

f ([u]y)dudy

ΓU \U

= 2hϕ, TrΓU \Γ f iU \P . This concludes the proof. Next, we give a second adjointness relation, with a twist from left to right. Indeed, note how the ΓU \Γ-trace is a sum over γ ∈ ΓU \Γ, with ΓU on the left, whereas the sum on the right side of the equation in the next proposition is over γ ∈ Γ\ΓU , with ΓU on the right. Furthermore, the sum as written cannot be taken inside the integral sign, because the integral over Posn is needed to make the term involving γ independent of the coset γΓU . Cf. step (4) in the proof. Proposition 1.2. Suppose the function ϕ on Posn is U -invariant. Let f be a function on Posn . Under conditions of absolute convergence, we have the adjointness relation Z Z X TrΓU \Γ (ϕ)(Y )f (Y )dµ(Y ) = ϕ(Y )f ([γ]Y )dµ(Y ) . γ∈Γ/ΓU

Posn

Posn

Proof. The function TrΓU \Γ ϕ is Γ-invariant on the left by construction. The integral over Posn can then be decomposed by first summing the integrand over Γ, and then integrating on the quotient space Γ\Posn , so we obtain: Z (TrΓU \Γ ϕ)(Y )f (Y )dµ(Y ) Posn

(1)

=

Z

(TrΓU \Γ ϕ)(Y ) ·

Γ\Posn

(2)

=

Z

U \Posn



 ϕ(Y )

Z

ΓU \U

1X f ([γ]Y )dµ(Y ) 2

X

γ∈Γ

γ∈Γ



 f ([γu]Y )du dµ(Y˙ )

110


by applying Proposition 1.1 to the functions ϕ and Y 7→

P

f ([γ]Y ). Thus we

γ

are using the first adjointness relation to prove a second one. Now we consider separately the integrand: Z Z X X X f ([γu]Y )du = (3) f ([γ][η][u]Y )du ΓU \U

γ∈Γ

ΓU \U

(4)

=

X

γ∈Γ\ΓU

γ∈Γ/ΓU

Z

η∈ΓU

f ([γ][u]Y )du .

U

Substituting (4) in (2), we find that (2) becomes   Z X Z ϕ(Y ) (5) f ([γ][u]Y )du dµ(Y ) Γ/ΓU

U \Posn

=

(6)

Z

X

U

ϕ(Y )f ([γ]Y )dµ(Y ) ,

γ∈Γ/ΓU Pos n

which proves the proposition. As a special case, we find a relation of Terras [Ter 85], Proposition 3. Proposition 1.3. Let ρ be a character on Posn . Let A, B ∈ Posn . Then Z X −1 Kρ (A[γ], [γ −1 ]B) . E pr (Y, ρ)e−tr(AY +BY ) dµ(Y ) = γ∈Γ/ΓU

Posn

Proof. We simply let ϕ(Y ) = ρ(Y ) and f (Y ) = e−tr(AY +BY

−1

)

.

Then TrΓU \Γ (ρ) is the Eisenstein series. Furthermore f ([γ]Y ) = e−tr(A[γ]Y +BY = e−tr(A[γ]Y +γ

−1

−1

[γ −1 ])

B t γ −1 Y −1 )

,

so the desired formula falls out.

2 Fourier Expansion Determined by Partial Iwasawa Coordinates We want to imitate the Epstein zeta function with matrices rather than vectors (n-tuples). As far as is known, the general case has certain difficulties, so in

2 Fourier Expansion Determined by Partial Iwasawa Coordinates

111

the present section, we deal with a subcase where the proof for the Epstein zeta function has a direct analogue, the main difference lying in the use of the general Bessel function of Chap. 3, rather than the classical one-variable Bessel function. We fix positive integers p, q such that p + q = n, and p = q so that n = 2q. We then decompose an element A ∈ Zn×q in two components µ ¶ B A= with B ∈ Zp×q and C ∈ Zq×q . C We define the q-non-singular theta series for Y ∈ Posn and Z ∈ Posq : X ∗ (Y, Z) = e−πtr(Y [A]Z) . θn,q A∈Zp×q rk(B)=q

The sum is over all integral A ∈ Zn×q such that the B-component as above has rank q. Thus the sum can be taken separately over all such B, and for each B over all C ∈ Zq×q without any restriction on C. Thus the singular part of the theta series would correspond to the part with b = 0 in the Epstein zeta function, but the higher dimension complicates the simpler condition b = 0. We combine the above (p, q) splitting with corresponding partial Iwasawa coordinates, that is ! Ã W 0 Y = Iw+ (W, X, V ) = [u(X)] 0 V with W ∈ Posp , V ∈ Posq and x ∈ Rp×q . Matrix multiplication gives Y [A] = W [B] + V [t XB + C] .

(1)

At this point, we want to emphasize the formal aspects of the remaining arguments. What will be important is that B, C range over certain sets stable under the action of Γ on the right. Thus we abbreviate: Γ = Γq M = M(q) = Zq×q . Elements of M are denoted by C. M∗ = M∗ (p, q) = the set of elements B ∈ Zp×q of rank q. For the non-singular theta series, we find X X t ∗ θn,q (2) (Y, Z) = e−πtr(W [B]Z) e−πtr(V [ XB+C]Z) B∈M∗

=

X

B∈M∗

The Poisson formula yields

C∈M

−πtr(W [B]Z)

e

X

C∈M

e−πtr(V [

t

XB+C]Z)

.

112

(3)


X

e−πtr(V [

t

XB+C]Z)

C∈M

= |V |−q/2 |Z|−q/2

X

e−2πitr(

t

BXC) −πtr(V −1 [C]Z −1 )

e

.

C∈M

Define h1 (C, Z) =

X

e−πtr(W [B]Z) e−2πitr(

t

BXC)

B∈M∗

h2 (C, Z) = e−πtr(V

−1

[C]Z −1 )

|Z|−q/2 .

Then both h1 and h2 satisfy the equation h(Cγ, Z) = h(C, [t γ −1 ]Z)

for all γ ∈ Γq .

This is verified at once for h2 . For h1 , we move γ from the right of C to the left of t B. Then we use the fact that for given γ, the map B 7→ B t γ −1 permutes the elements of M∗ , so the sum over B ∈ M∗ is the same as the sum over B t γ −1 . The desired relationship drops out. With the above definitions, we obtain X ∗ θn,q (5) (Y, Z) = |V |−q/2 h1 (C, Z)h2 (C, Z) . C∈M

Note that each term in the sum satisfies the above equations. We then take ∗ with a test function Φ on Γq \Posq , namely the convolution of θp,q Z ∗ ∗ (6) ∗ Φ)(Y ) = θn,q (Y, Z)Φ(Z)dµq (Z) . (θn,q Γq \Posq

We disregard for the moment questions of absolute convergence, and work formally. Two of the more important applications are when Φ = 1 and when Φ is an Eisenstein series. For the moment, we just suppose Φ is a function on Γq \Posq . For simplicity of notation, and to emphasize some formal aspects, we abbreviate P = Posq . We shall need a formula for the integral over the quotient space Γ\P = Γq \Posq . Note that Γ acts on M on the right, and thus gives rise to right cosets of M. We shall deal with

2 Fourier Expansion Determined by Partial Iwasawa Coordinates

X

113

,

C∈M/Γ

and for any function f on M, we have the relation X X X f (C) = f (Cγ) . C∈M/Γ γ∈Γ

C∈M

Lemma 2.1. Let h = h(C, Z) be a function of two variables C ∈ M and Z ∈ P. Suppose that h(Cγ, Z) = h(C, [t γ −1 ]Z) Then

Z

Γ\P

X

h(C, Z)dµ(Z) =

C∈M

for all

X

2

C∈M/Γ

Z

γ∈Γ. h(C, Z)dµ(Z) .

P

Proof. Z

Γ\P

X

h(C, Z)dµ(Z) =

C∈M

Z

Γ\P

=

Z

Γ\P

=

X

X

X

h(Cγ, Z)dµ(Z)

X

X

h(C, [t γ −1 ]Z)dµ(Z)

C∈M/Γ γ∈Γ

C∈M/Γ γ∈Γ

C∈M/Γ

2

Z

h(C, Z)dµ(Z)

P

thus proving the lemma. Note that if the function h(C, Z) satisfies (4), then so does the function Φ(Z)h(C, Z), directly from the invariance of Φ. We shall now assume that Φ ∗ ∗ Φ. is a ΓU \Γ-trace to get a Fourier expansion for θn,q Proposition 2.2. Suppose Φ = TrΓU \Γ ϕ with a function ϕ on P. Then for Y = Iw+ (W, X, V ) in partial Iwasawa coordinates, under conditions of absolute convergence, we have X X t ∗ ∗ TrΓU \Γ (ϕ))(Y ) = (θn,q aB,C e−2πitr( BXC) C∈M/Γ B∈M∗

with coefficients aB,C = 2|V |−q/2

X

Γ∈Γ/ΓU

Kϕd−q/2 (πW [B][γ], π[γ −1 ]V −1 [C]) .

114


Proof. First remark that the expression on the right of the formula to be proved makes sense. Indeed, if we replace C by Cγ with some element γ ∈ Γ, then in tr(t BXC) we can move γ next to t B, and the sum over B ∈ M∗ then allows us to cancel γ. Hence the sum over B ∈ M∗ depends only on the coset of C in M/Γ. Next we recall that Φd−q/2 is also a ΓU \Γ -trace, namely trivially Φd−q/2 = TrΓU \Γ (ϕd−q/2 ) . Now: ∗ |V |q/2 (θn,q ∗ Φ)(Y ) Z ∗ Φ(Z)|V |q/2 θn,q (Y, Z)dµ(Z) = Γ\P

=

Z

Φ(Z)

=

C∈M/Γ

=

X

h1 h2 (C, Z)dµ(Z)

by equation (5)

Φ(Z)h1 h2 (Z)dµ(Z)

(by Lemma 2.1)

C∈M

Γ\P

X

X

2

Z

P

X

2e−2πitr(

t

BXC)

C∈M/Γ B∈M∗

Z

·

Φ(Z)|Z|−q/2 eπtr(W [B]Z+V

−1

[C]Z −1 )

dµ(Z)

P

=

X

X

C∈M/Γ B∈M∗

|V |q/2 aB,C e−2πitr(

t

BXC)

(by Proposition 1.2).

This concludes the proof. Remark. The preceding proposition applies to the special case when Φ(Z) = E pr (Z, ρ) is an Eisenstein series, in which case the Fourier expansion comse from Terras [Ter 85], Theorem 1.

3 Fourier Coefficients from Partial Iwasawa Coordinates We represent Y ∈ Posn in terms of partial Iwasawa coordinates Y = Iw+ (W, X, V ) with

W ∈ Posp , V ∈ Posq , X ∈ Rp×q .

Let f be a function on Posn , invariant under the group Γ = GLn (Z). What we shall actually need is invariance of f under the two subgroups:

3 Fourier Coefficients from Partial Iwasawa Coordinates

115

– the group of elements [u(N )] with N ∈ Zp×q ; – the group of elements µ ¶ γ 0 with γ ∈ GLp (Z) . 0 Iq The following theorem is valid under this weaker invariance, but we may as well assume the simpler hypothesis which implies both of these conditions. Under the invariance, f has a Fourier series expansion X aN (W, V )e2πihN,Xi , f (Y ) = N ∈Zp×q

with Fourier coefficients given by µ µ Z W aN (W, V ) = f [u(X)] 0

0 V

Rp×q /Zp×q

¶¶

e−2πihN,Xi dX ,

Q where dX = dxij is the standard euclidean measure on Rp×q . We then have the following lemma. Lemma 3.1. ([Gre 92]) For γ ∈ Γp the Fourier coefficients satisfy atγN (W, V ) = aN ([γ]W, V ) . Proof. First note that  γ  (2) 0

0 Iq

Now:

aN (V, [γ]W ) =

=

=

Z Z Z





Ip

X

0

Iq



Ip

X



γ

f 

f  f 

γX

0

Iq



[γ]W



0

Y = 



0

0



γ

 

γ

W

0

0

V



0

0 V  



 .



 e−2πihN,Xi dX W

0



 e−2πihN,Xi dX

0 Iq 0 V Iq   X W 0   e−2πihN,Xi dX . Iq 0 V

Now make the change of variables X 7→ γX so d(γX) = dX. Using (2) and the invariance of f under the actions of

116


 

γ

0

0

Iq

 

shows that the last expression obtained is equal to atγN (W, V ), because hN, γXi = ht γN, Xi . This concludes the proof.

4 A Fourier Expansion on SPosn(R) We consider the inductive Iwasawa decomposition of Chap. 1, taking p = n−1 and q = 1 on SLn (R). Let   In−1 x  for x ∈ Rn−1 (column vectors) . u(x) =  0 1 We have the corresponding decomposition for Y ∈ SPosn :  1/(n−1) (n−1)  v Y 0  Y = [u(x)]  0 v −1

Thus v ∈ R+ and Y (n−1) ∈ SPosn−1 . As in Sect. 1, we let U = Uni+ (R) ΓU = Γ∞ = Uni+ (Z). Let f be a function invariant under Γ∞ . Then in particular, f is invariant under Zn−1 . We may write f (Y ) = f (v, y (N −1) , X) and f has a Fourier series (1)

f (Y ) =

X

am,f (v, Y (n−1) )e2πihm,xi .

m∈Zn−1

The Fourier coefficients are given by the integrals am,f (v, Y (n−1) )  1/(n−1) (n−1)  Z v Y f [u(x)]  = 0 Rn−1 /Zn−1

0 v

−1



 e−2πihm,xi dx .

4 A Fourier Expansion on SPosn (R)

117

Example. We may use f (Y ) =

X

ρ([γ]Y )−1

Γ∞ \Γ

with a character ρ, so f is the standard Eisenstein series. Proposition 4.1. ( [Gre 92]) For γ ∈ Γn−1 , the Fourier coefficients satisfy at γm (v, Y (n−1) ) = am (v, [γ]Y (n−1) ) . Proof. Matrix multiplication gives " # " #Ã γ 0 γ γx v 1/(n−1) Y (n−1) Y = (2) 0 1 0 1 0

0 v −1

!

.

Then am (v, [γ]Y (n−1) ) !! #Ã Z Ã" In−1 x v 1/(n−1) [γ]Y (n−1) 0 e−2πihm,xi dx = f 0 1 0 v −1 #Ã !! #" Z Ã" In−1 x γ 0 v 1/(n−1) Y (n−1) 0 = f e−2πihm,xi dx 0 1 0 1 0 v −1 #Ã !! Z Ã" γ x v 1/(n−1) Y (n−1) 0 = f e−2πihm,xi dx. 0 1 0 v −1 We make the translation x 7→ γx, d(γx) = dx and use (2) to get #! " Z Ã γ 0 (n−1) (Y )e−2πihm,γxi dx f◦ )= am (v, [γ]Y 0 1 = at γm (v, Y (n−1) ) , thus proving the proposition. For m 6= 0 we may write m = d`, with ` primitive and d ∈ Z, d > 0 . Then putting e = en−1 = t (0, . . . , 0, 1), we have am (v, Y (n−1) ) = ad` (v, Y (n−1) ) = ade (v, [γ]Y (n−1) ) where ` = t γe and m = d t γe. Thus the Fourier series for f can be written (3) f (Y ) = a0 (v, Y (n−1) ) +

∞ X

X

d=1 Γn−1,1 \Γn−1

ade (v, [γ]Y (n−1) )e2πihde,xi .

118


∂ 5 The Regularizing Operator QY = |Y || ∂Y |

The above operator QY (Y variable in Posn ) was used especially by Maass [Maa 55], [Maa 71], and deserves its own section. We shall use it in Chap. XII as in Maass, as a regularizing operator. We have seen in Sect. 2 how to use terms of maximal rank in an Eisenstein series. The operator QY kills terms of lower rank because applied to exp(tr(BY )) a factor |B| comes out, and is equal to 0 if B is singular. First, plugging in formula (2) of Sect. 2, we get an example of the adjoint. ˜ Y = (−1)n |Y |(n+1)/2 | ∂ |◦|Y |1−(n+1)/2 . More generally, Proposition 5.1. Q ∂Y for a positive integer r, letting Qn,r = |Y |r | we find

∂ r | , ∂Y

¯r ¯ ¯ ¯ ˜ n,r = (−1)nr |Y |(n+1)/2 ¯ ∂ ¯ ◦ |Y |r−(n+1)/2 . Q ¯ ∂Y ¯

We note that QY is an invariant differential operator of degree n (the size of the determinant). Next we observe that for any n × n matrix M , ¯ ¯ ¯ ∂ ¯ tr(M Y ) ¯ ¯ = |M |etr(M Y ) . ¯ ∂Y ¯ e

This is immediate from the definitions. The gamma transform was used rather formally in Chap. 3, Proposition 2.5. We use more of its properties here, so we recall the notation Z −1 e−tr(Y Z ) f (Y )dµn (Y ) . (Γn #f )(Z) = Posn

If f is an eigenfunction (the absolute convergence of the integral then being assumed as part of the definition), we let λΓ (f ) be the corresponding eigenvalue. We recall that d is the determinant. Theorem 5.2. Suppose f, f d are eigenfunctions of the gamma transform. ˜ with eigenvalue Then f is an eigenfunction of Q λQ˜ (f ) = (−1)n λΓ (f d)/λΓ (f ), assuming λΓ (f ) 6= 0. Proof. We differentiate under the integral sign, namely:

∂ 5 The Regularizing Operator QY = |Y || ∂Y |

˜ Z λΓ (f )f (Z) = Q

Z

119

˜ Z e−tr(Y Z −1 ) f (Y )dµn (Y ) Q

Posn

Z

−1

(QY e−tr(Y Z ) )f (Y )dµn (Y ) by Lemma 1.4 ¯ µ¯ ¶ Z ¯ ∂ ¯ −tr(T Z −1 ) ¯ ¯ e f (Y )dµn (Y ) = |Y | ¯ ∂Y ¯ Z −1 = (−1)n |Z|−1 e−tr(T Z ) (f d)(Y )dµn (Y ) =

= (−1)n d(Z)−1 λΓ (f d)(f d)(Z).

Then d(Z)−1 and d(Z) cancel, and the formula of the theorem drops out. The first important special case is when f is a character ρs . In this case, by Proposition 2.1 of Chap. 3, λΓ (ρs ) = bn

n Y

i=1

with bn =

√

π

n(n−1)/2

Γ(si − αi )

and αi = (n − i)/2 .

˜ is a polynomial, namely Corollary 5.3. For f = ρs , the Q-eigenvalue λQ˜ (ρs ) = (−1)n

n Y

i=1

(si − αi ) .

Proof. We have f d = ρs+1 , and Γ(si − αi + 1) = (si − αi )Γ(s − αi ) for all i, so the corollary is immediate. (n−1)

Corollary 5.4. Let hs = q−z 3, Proposition 1.2. Then n

λQ˜ (hs ) = (−1)

n µ Y

i=1

be the Selberg power character as in Chap.

n−1 sn−i+1 − 4

¶

n

= (−1)

n µ Y

i=1

n−1 si − 4

¶

.

Proof. Use the value s# found in Chap. 3, Proposition 1.2, and plug in Corollary 5.3. We note that in this particular case, there is a symmetry and a cancelation which gets rid of the reversal of the variables s1 , . . . , sn . Using some duality formulas, we can then determine the eigenvalue of Q itself as follows.

120


Theorem 5.5. λQ (hs ) =

n ¡ Q

si +

i=1

n−1 4

Proof. We recall the involution

¢

.

s∗ = (−sn , . . . , −s1 ) from Chap. 3, Sect. 1. Since hs∗ = h∗s , we have λQ˜ (h∗s )

=

n µ Y

i=1

n−1 si + 4

¶

.

˜ = [S]Q and h∗ = [S][ω]hs . Directly from its definition, [ω]Q = Q. We have Q s The theorem is then immediate (canceling [S][ω], as it were).

7 Geometric and Analytic Estimates

In Chap. 1 and 2, we dealt at length with estimates concerning various coordinates on Posn and the volume on Posn . Here we come to deal with the metric itself, and the application of coordinate estimates to the convergence of certain Dirichlet series called Eisenstein series. Further properties of such series will then be treated in the next chapter. On the whole we follow the exposition in Maass [Maa 71], Sect. 3, Sect. 7 and especially Sect. 10, although we make somewhat more efficient use of the invariant measure in Iwasawa coordinates, thereby introducing some technical simplifications.

1 The Metric and Iwasawa Coordinates The basic differential geometry of the space Posn is given in Chap. XI of [La 99] and will not be reproduced here. We merely recall the basic definition. We view Symn (vector space of real symmetric n × n matrices) as the tangent space at every point Y of Posn . The Riemannian metric is defined at the point Y by the formula ds2 = tr((Y −1 dY )2 ) also written tr(Y −1 dY )2 . This means that if t 7→ Y (t) is a C 1 curve in Posn , then hY 0 (t), Y 0 (t)iY (t) = tr(Y (t)−1 Y 0 (t))2 , where Y 0 (t) is the naive derivative of the map of a real interval into Posn , viewed as an open subset of Symn . The two basic properties of this Riemannian metric are: Theorem 1.1. Let Symn have the positive definite scalar product given by hM, M1 i = tr(M M1 ). Then the exponential map exp : Symn → Posn is metric semi-increasing, and is metric preserving on lines from the origin.


122


Theorem 1.2. The Riemannian distance between any two points Y, Z ∈ Posn is given by the formula X dist(Y, Z) = (log ai )2 ,

where a1 , . . . , an are the roots of the polynomial det(tY − Z).

See [La 99], Chap. XI, Theorems 1.2, 1.3 and 1.4. In the present section, we shall consider the distance formula in the context of Iwasawa-Jacobi coordinates. As in Chap. 2, Sect. 2 the partial Iwasawa-Jacobi coordinates of an element Y ∈ Posn are given by the expression     W 0 Ip X  ,  , u(X) =  Y = [u(X)]  0 Iq 0 v with X ∈ Rp,q , W ∈ Posp and V ∈ Posq . Matrix multiplication shows that   W + [X]V XV  . (1) Y = V V tX In particular, by definition

V = Subq (Y ) is the lower right square submatrix which we have used in connection with the Selberg power function, cf. Chap. 3, Sect. 1. We shall need the matrix for Y −1 , given by  −1  W 0  [u(−X)] Y −1 =  0 v −1   W −1 −W −1 X  . = (2) −t XW −1 V −1 + [t X]W −1 Theorem 1.3. The metric on Posn admits the decomposition

tr(Y −1 dY )2 = tr(W −1 dW )2 + tr(V −1 dV )2 + 2tr(W −1 [dX]V ) . All three terms on the right are = 0. In particular, tr(V −1 dV )2 5 tr(Y −1 dY )2 , and the map Posn → Posq is metric decreasing.

given by

Y 7→ Subq (Y ) = V

1 The Metric and Iwasawa Coordinates

123

Proof. We copy Maass [Maa 71], Sect. 3. We start with   dW + [X]dV + dX · V t X + XV · d t X dX · V + XdV  . (3) dY =  t t dV · X + V · d X dV With the abbreviation



we have (4)

dY · Y −1 = 

L0

L1

L2

L3

 

tr(Y −1 dY )2 = tr(dY · Y −1 · dY · Y −1 ) = tr(L20 + L1 L2 ) + tr(L2 L1 + L23 ) .

A straightforward calculation yields (5)

L0 = dW · W −1 + XV · d t X · W −1 L1 = −dW · w−1 X − XV · d t X · W −1 X + dX + X · dV · V −1 L2 = V · dX · W −1 L3 = dV · V −1 − V · d t X · W −1 X .

The formula giving the decomposition of tr(Y −1 dY )2 as a sum of three terms then follows immediately from (4) and the values for the components in (5). As to the positivity, the only possible question is about the third term on the right of the formula. For this, we write W = A2 and V = B 2 with positive A, B. Let Z = B · d t X · A−1 . Then tr(W −1 [dX]V ) = tr(Z t Z) , which shows that the third quadratic form is positive definite and concludes the proof. Let G = GLn (R) as usual. It is easily verified that the action of G on Posn is metric preserving, so G has a representation as a group of Riemannian automorphisms of Posn . Again cf. [La 99] Chap. XI, Theorem 1.1. Here we are interested in the behavior of the determinant |Y | as a function of distance. Consider first a special case, taking distances from the origin I = In . By Theorem 1.2, we know that if Y ∈ Br (I) (Riemannian ball of radius r centered at I), then X (log ai )2 < r2 . dist(Y, I)2 = It then follows that there exists a number cn (r), such that for Y ∈ Br (I), we have 1 < |Y | < cn (r) . (6) cn (r)

124


Indeed, the determinant is equal to the product of the characteristic roots, |Y | = a1 . . . an . √

With the Schwarz inequality, we take cn (r) = e nr . Note that from an upper bound for |Y |, we get a lower bound automatically because Y 7→ Y −1 is an 2 isometry. From another point of view, we also have (log ai )2 = (log a−1 i ) . In the above estimate, we took a ball around I. But the transitive action of G on Posn gives us more uniformity. Indeed: Lemma 1.4. For any pair Y, Z ∈ Posn with dist(Y, Z) < r, we have cn (r)−1
cn (r)

2 Convergence Estimates for Eisenstein Series

125

Lemma 1.6. For all γ ∈ Γ = GLn (Z) we have Br ([γ]I) ⊂ Dr . Proof. Let Y ∈ Br ([γ]I). Then [γ −1 ]Y ∈ Br (I), and we can apply (6), as well as |[γ −1 ]I| = 1 to prove the inequality |Y | < cn (r). For the other inequality, by the distance decreasing property, we have dist(Subj [γ]I, Subj Y ) 5 dist([γ]I, Y ) < r . Hence by Lemma 1.4, |Subj Y | >

1 1 |Subj ([γ]I)| = cn (r) cn (r)

because [γ]I is an integral matrix, with determinant = 1. This concludes the proof. The set of elements [γ]I with γ ∈ Γ is discrete in Posn . We call r > 0 a radius of discreteness for Γ if dist([γ]I, I) < 2r implies γ = ±I, that is [γ] acts trivially on Posn . We shall need: Lemma 1.7. Let γ, γ 0 ∈ Γ, and let r be a radius of discreteness for Γ. If there is an element Y ∈ Posn in the intersection of the balls Br ([γ]I) and Br ([γ 0 ]I), then [γ] = [γ 0 ], that is γ = ±γ. Proof. By hypothesis, dist([γ]I, [γ 0 ]I) < 2r, so dist([γ −1 γ 0 ]I, I) < 2r , and the lemma follows.

2 Convergence Estimates for Eisenstein Series We shall need a little geometry concerning the action of the unipotent group on Posn , so we start with an independent discussion of this geometry. An element Y ∈ Posn can be written uniquely in the form Y = [u(X)]A and



a11  .. A= . 0

with u(X) = In + X , ... .. . ...

 0 ..  , a > 0 , .  ii

ann

and X = (xij ) is strictly upper triangular. We call (X, A) the full Iwasawa coordinates for Y on Posn .

126


Let Γ = GLn (Z) as usual, ΓU = subgroup of unipotent elements in Γ, so the upper triangular integral matrices with every diagonal element equal to 1. Thus γ ∈ ΓU can be written γ = In + X with an integral matrix X. It is easy to construct a fundamental domain for ΓU \Posn . First we note that a fundamental domain for the real unipotent group Uni+ (R) modulo the integral subgroup ΓU consists of all elements u(X) such that 0 5 xij < 1. We leave the proof to the reader. In an analogous discrete situation when all matrices are integral, we shall carry out the inductive argument in Lemma 1.2 of Chap. 8, using the euclidean algorithm. In the present real situation, one uses a “continuous” euclidean algorithm, as it were. Then we define: FU = set of elements [u(X)]A ∈ Posn with 0 5 xij < 1 . From the uniqueness of the Iwasawa coordinates, we conclude that FU is a strict fundamental domain for ΓU \Posn . The main purpose of this section is to prove the convergence of a certain series called an Eisenstein series. We shall prove it by an integral test, depending on the finiteness of a certain integral, which we now describe in a fairly general context. Let c > 0. We define the subset D(c) of Posn to be: D(c) = {Y ∈ Posn , |Y | < c and |Subj Y | > 1/c for all j = 1, . . . , n} . We recall the Selberg power function (n)

q−z (Y ) =

n Y

|Subj Y |−zj .

j=1

We are interested in the integral of this power function over a set D(c) ∩ FU . To test absolute convergence, it suffices to do so when all zj are real. The next lemma will prove absolute convergence when Re(zj ) > 1. Lemma 2.1. Let b > 1. Then Z n Y D(c)∩FU

|Subj Y |−b dµn (Y ) < ∞ .

j=1

Proof. In Chap. 2, Proposition 2.4, we computed the invariant measure dµn (Y ) in terms of the Iwasawa coordinates, and found (1)

dµn (Y ) =

n Y

i=1

i−(n+1)/2

aii

n Y daii Y

i=1

aii

i<j

dxij .


127

We note that |Subj Y | = an−j+1 . . . an , writing ai = aii . Hence, if we take ε > 0 and set b = 1 + ε, we have n Y

(2)

|Subj Y |−b =

j=1

n Y

a−i−εi . i

i=1

The effect of intersecting D(c) with Fu is to bound the xij -coordinates. Thus the convergence of the integral depends only on the ai -coordinates. To concentrate on them, we let dµn,A =

n Y

i−(n+1)/2

ai

i=1

n Y dai

i=1

ai

.

We let DA (c) be the region in the A-space defined by the inequalities 1 1 < a1 · · · an < c and an > , c c 1 1 an an−1 > , . . . , an an−1 · · · a1 > . c c Thus DA (c) is a region in the n-fold product of the positive multiplicative group, and the convergence of the integral in our lemma is reduced to the convergence of an integral in a euclidean region, so to calculus. Taking the product of the expressions in (1) and (2), and integrating over DA (c), we see that the finiteness of the integral in our lemma is reduced to proving the finiteness Z Y Y −(εi+1+(n+1)/2) aii daii < ∞ . (3) DA (c)

Just to see what’s going on, suppose n = 2 and the variables are a1 = u and a2 = v . The region is defined by the inequalities 1 1 < uv < c and v > . c c The integral can be rewritten as the repeated integral   Z∞ Zc/v   u−(1+ε+(n+1)/2) du v −(2ε+1+(n+1)/2) dv .  1/c

1/cv

The inner integral with respect to u can be evaluated, and up to a constant factor, it produces a term

128


v ε+(n+1)/2 which cancels the similar expression in the outer v-integral. Thus finally the convergence is reduced to Z∞ 1 dv < ∞ v 1+ε 1/c

which is true. Having n variables only complicates the notation but not the idea, which is to integrate successively with respect to dan , then dan−1 , and so forth until da1 , which we leave to the reader to conclude the proof of Lemma 2.1. Next we combine the metric estimates from the last section with the measure estimates which we have just considered. Let r be a radius of discreteness for Γ, defined at the end of the last section. Then Dr = D(cn (r)) , where D(c) is the set we considered in Lemma 2.1. Let {γm } (with m = 1, 2, . . .) be a family of coset representatives for ±ΓU \Γ. For each m we let τmk (k = 1, . . . , dm ) be a minimal number of elements of ±ΓU such that Br ([γm ]I) ⊂

dm [

[τmk ]FU .

k=1

In particular, the intersection Smk = Br ([γm ]I) ∩ [τmk ]FU is not empty for each m, k. The set D defined above is stable under the action of ΓU . Hence translating the sets Smk back into FU we conclude that −1 ]Smk ⊂ Dr ∩ FU for all m, k . [τmk

(4)

−1 By Lemma 1.7, the sets [τmk ]Smk are disjoint, for pairs (m, k) defined as above. We are now ready to apply the geometry to estimate certain series. Let ρ be a character. The primitive Eisenstein series is defined by X EUpr (Y, ρ) = ρ([γ]Y ) . γ∈ΓU \Γ

We shall be concerned with the character equal to the Selberg power function, (n−1) that is q−z , so that by definition, pr(n−1)

EU

(Y, z) =

X

n−1 Y

γ∈ΓU \Γ j=1

|Subj [γ]Y |−zj .


129

First, note that any Y ∈ Posn lies in some ball Br (I), and by Lemma 1.5, we see that the convergence of the series for any given Y is equivalent to the convergence with Y = I. We also have uniformity of convergence in a ball of fixed radius. In addition, we note that |Subn [γ]Y | = |[γ]Y | = |Y | for all γ ∈ Γ . Thus the convergence of the above Eisenstein series is equivalent with the convergence of pr(n)

EU

n X Y

(Y, z) =

|Subj [γ]Y |−zj .

γ∈ΓU \Γ j=1

Furthermore, zn has no effect on the convergence. The main theorem is: Theorem 2.2. The Eisenstein series converges absolutely for all zj with Re(zj ) > 1 for j = 1, . . . , n − 1. Proof. First we replace zj by a fixed number b > 1. We prove the convergence for Y = I, but we shall immediately take an average, namely we use the inequalities for Y ∈ Br (I), with r a radius of discreteness for Γ: (5)

X

E(I, b) =

n Y

|Subj ([γ]I)|−b

γ∈ΓU \Γ j=1

¿

Z

X

γ∈ΓU \Γ B (I) r

¿

n Y

Z

X

|Subj [γ]Y |−b dµ(Y )

j=1

γ∈ΓU \Γ B ([γ]I) r

n Y

|Subj Y |−b dµ(Y ) .

j=1

We combine the inclusion (4) with the estimate in (5). We use the fact that |Subj [τ ]Y | = |Subj Y |

for τ ∈ ΓU ,

and we translate each integral back into FU . We then obtain from (5) Z dm ∞ n X Y X |Subj Y |−b dµ(Y ) E(I, b) ¿n m=1 k=1

¿n

Z

Dr ∩FU

−1 [τmk ]Smk

n Y

j=1

|Subj Y |−b dµn (Y ) .

j=1

The sign ¿n means that the left side is less than the right side times a constant depending only on n. We have used here the fact already determined −1 that the sets [τmk ]Smk are disjoint and contained in Dr ∩ FU . The finiteness of the integral was proved in Lemma 2.1, which thereby concludes the proof of Theorem 2.2.

130


3 A Variation and Extension In the application of Chap. 8, one needs convergence of a modified Eisenstein series, specifically the following case. Theorem 3.1. The series n Y

X

|Subj Y |−zj

γ∈ΓU \Γ j=2

converges absolutely for Re(z2 ) > 3/2 and Re(zj ) > 1 with j = 3. The proof is the same as the proof of Theorem 2.2. One uses the same set D(c). Lemma 2.1 has its analogue for the product with one term omitted. The calculus computation comes out as stated. For instance, for n = 3, the region D(c) is defined by the inequalities 1 < uvw < c, c

vw >

1 , c

w>

1 . c

The series is dominated by the repeated integral Z∞

Z∞

c/vw Z

(vw)−3/2−ε u(n+1)/2 v 1−(n+1)/2 w2−(n+1)/2 dudvdw ,

1/c 1/wc 1/vwc

which comes out up to a constant factor to be Z∞

w−1−ε dw .

1/c

For various reasons, including the above specific application, Maass extends the convergence theorem still further as follows [Maa 71]. Let 0 = k0 < k1 < . . . < km < km+1 = n be a sequence of integers which we call an integral partition P of n. Let ni = ki − ki−1 , i = 1, . . . , m + 1 . Then n = n1 + . . . + nm+1 is a partition of n in the number theoretic sense. Matrices consisting of blocks of size ni (with i = 1, . . . , m + 1) on the diagonal generalize diagonal matrices. We let: ΓP = Subgroup of Γ consisting of elements which are upper diagonal over such block matrices, in other words, elements γ = (Cij ) Cii ∈ Γni for 1 5 i 5 m + 1 and Cij = 0 for 1 5 j < i 5 m + 1.

3 A Variation and Extension

131

In the previous cases, we have kj = j, nj = 1 for all j = 1, . . . , n, and m + 1 = n. The description of the groups associated with a partition as above is slightly more convenient than to impose further restriction, but we note that in this case the diagonal elements may be ±1, so we are dealing with the group T rather than the unipotent group U . A group such that ΓP above is also called a parabolic subgroup. We define the Eisenstein series as a function of variables z1 , . . . , zm by EP (Y, z) =

X

m Y

|Subki [γ]Y |−zi .

γ∈ΓP \Γ i=1

Theorem 3.2. ([Maa 71], Sect. 7) This Eisenstein series is absolutely convergent for Re(zi ) >

1 1 (ni+1 + ni ) = (ki+1 − ki−1 ), i = 1, . . . , m . 2 2

Proof. One has to go through the same steps as in the preceding section, with the added complications of the more elaborate partition. One needs the Iwasawa-Jacobi coordinates with blocks,     W1 . . . 0 In1 . . . Xij  ..  and u(X) =  .. ..  . .. .. Y = [u(X)]  ...  . . . .  .  0

...

Wm+1

0

...

Inm+1

The measure is given by dµn (Y ) =

m+1 Y

|Wi |(ki −ki−1 −n) dµ(Wi )

i=1

Y

dµeuc (Xij ) .

15i<j5m+1

The fundamental domain for ΓP consists of those Y whose coordinates satisfy: Wi ∈ Fundamental domain for Γni (i = 1, . . . , m + 1) in Posni . Xij has coordinates 0 5 xνµ < 1. The domain D(c) is now ½ DP (c) = Y such that Wi > 0 for all i = 1, . . . , m + 1; m+1 Y i=1

¾ 1 1 1 |Wi | < c, |Wm | > , |Wm ||Wm−1 | > , . . . , |Wm | . . . |W1 | > . c c c

thus we merely replace ai by |Wi | throughout the previous definition. Maass gives his proof right away with the more complicated notation, and readers can refer to it.

132


Note that Theorem 3.1 is a special case of Theorem 3.2. However, the notation of Theorem 3.1 is simpler, and we thought it worth while to state it and indicate its proof separately, using the easier notation for the Eisenstein series. The subgroup ΓP is usually called a parabolic subgroup. Such subgroups play an essential role in the compactification of Γn \Posn , and in the subsequent spectral eigenfunction decomposition.

8 Eisenstein Series Second Part

In Chap. 5, we already saw the Epstein zeta function, actually two zeta functions, one primitive and the other one completed by a Riemann zeta function. Indeed, let Y ∈ Posn . We may form the two series X X E pr (Y, s) = ([a]Y )−s and E(Y, s) = ([a]Y )−s a prim

a6=0

where the first sum is taken over a ∈ t Zn , a 6= 0 and a primitive; while the second sum is taken over all a ∈ t Zn , a 6= 0. Any a ∈ t Zn can be written uniquely in the form a = da1 with d ∈ Z+ and a1 primitive . Therefore E(Y, s) = ζQ (2s)E pr (Y, s) . We have to extend this property to the more general Selberg Eisenstein series on Posn . This involves a more involved combinatorial formalism, about integral matrices in Zj,j+1 with j = 1, . . . , n − 1. Thus the first section is devoted to the linear algebra formalism of such integral matrices and their decompositions. After that, we define the general Eisenstein series and obtain various expressions for them which are used subsequently in deriving the analytic continuation and functional equations. For all this, we will follow Maass from [Maa 71] after [Maa 55], [Maa 56]. He did a great service to the mathematical community in providing us with a careful and detailed account. However, we have had to rethink through all the formulas because we use left characters instead of right characters as in Maass-Selberg, and also we introduce the Selberg variables s = (s1 , . . . , sn ) as late as possible. Indeed, we work with more general functions than characters, for application to more general types of Eisenstein series constructed with automorphic forms, or beyond with the heat kernel. We note here one important feature about the structure of various fudge factors occurring in functional equations: they are eigenvalues of certain Jay Jorgenson: Posn (R) and Eisenstein Series, Lect. Notes Math. 1868, 133–162 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com °

134


operators, specifically three operators: a regularizing invariant differential operator, the gamma operator (convolution with the kernel of the gamma function on Posn ), a Hecke-zeta operator. To bring out more clearly the structure of these operators and their role, we separate the explicit computation of their eigenvalues from the position these eigenvalues occupy as fudge factors. When the eigenfunctions are characters, these eigenvalues are respectively polynomials, products of ordinary gamma functions, and products of Riemann zeta functions, with the appropriate complex variables. Such eigenvalues are those occurring in the theory of the Selberg Eisenstein series, which are the most basic ones. However, Eisenstein series like other invariants from spectral theory (including analytic number theory) have an inductive “ladder” structure, and on higher rungs of their ladder, the eigenvalues are of course more complicated and require more elaborate explicit computations, which will be carried out in their proper place. On the other hand, the general formulas given in the present chapter will be applicable to these more general cases.

1 Integral Matrices and Their Chains Throughout, we let: Γn = GLn (Z); M∗n = set of integral n × n matrices of rank n; M∗ (p, q) = set of integral p × q matrices of rank min(p, q); ∆n = set of upper triangular integral n × n matrices of rank n; Tn = Γn ∩ ∆n = group of upper triangular integral matrices of determinant ± 1 . We note that M∗n and ∆n are just sets of matrices, not groups. The diagonal components of an element in ∆n are arbitrary integers 6= 0, so elements of ∆n are not necessarily unipotent. On the other hand, the elements of Tn necessarily have ±1 on the diagonal, so differ from unipotent elements precisely by such diagonal elements. Note that ∆n is stable under the action of Tn on both sides, but we shall usually consider the left action. Thus we consider coset representatives in Γn for the coset space Tn \Γn and also coset representatives D ∈ ∆n of the coset Tn D, which is a subset of ∆n . Similarly, M∗n is stable under the action of Γn on both sides, and we can consider the coset space Γn \M∗n . Lemma 1.1. The natural inclusion ∆n ,→ M∗n induces a bijection Tn \∆n → Γn \M∗n of the coset spaces.

1 Integral Matrices and Their Chains

135

Proof. By induction, and left to the reader. We shall work out formally a more complicated variation below. The bijection of Lemma 1.1 is called triangularization. Next we determine a natural set of coset representatives for Tn \∆n . Lemma 1.2. A system of coset representatives of Tn \∆n consists of the matrices   d11 . . . d1n  ..  = (d ) .. D =  ... ij . .  0

...

dnn

satisfying dij = 0 if j < i (upper triangularity), djj > 0 all j, and 0 5 dij < djj

for

15i<j5n.

In other words, in a vertical column, the components are = 0 and strictly smaller than the diagonal component in this column. Proof. In the first place, multiplying an arbitrary element D ∈ ∆n by a diagonal matrix with ±1 diagonal components, we can make the diagonal elements djj , (j = 1, . . . , n) to be positive. Then we want to determine a nilpotent integral matrix X (upper triangular, zero on the diagonal) such that (I + X)D is among the prescribed representatives, and furthermore, such X is uniquely determined. This amounts to the euclidean algorithm, and is done by induction, starting with the top left. Pictorially, given an upper triangular integral matrix with positive diagonal elements dii and strictly upper triangular elements yij , we want to find X = (xij ) such that the product    d11 y12 . . . 1 x12 . . . x1n y1n  0 1 ... x2n  y2n    0 d22 . . .      .. . ..  . .. .. ..   ..  . . . .    0 . . . 1 xn−1,n   0 0 . . . dn−1,n−1 yn−1,n  0 0 ... 0 1 0 0 ... 0 dnn satisfies the inequalities in the lemma. We start at the top, so we first solve for x12 such that 0 5 y12 + x12 d22 < d22 . This inequality has a unique integral solution x12 . We then solve inductively for x13 , . . . , x1n ; then we go down the rows to conclude the proof. Lemma 1.3. Given integers djj > 0 (j = 1, . . . , n), the number of cosets Tn D with D having the given diagonal elements is n Y

j=1

j−1 djj .

136


Proof. Immediate. Remark. The previous lemmas have analogues for the right action of Γn on M∗n . First, Lemma 1.1 is valid without change for the right action of Γn on M∗n and the right action of Tn on ∆n . On the other hand, the inequalities defining coset representatives in Lemma 1.2 for the right have to read: 0 5 dij < dii for 1 5 i < j 5 n . Then the number of cosets DTn with D having given d11 , . . . , dnn > 0 is n Y

dn−j . jj

j=1

Next we deal with M∗ (n − 1, n) with n equal to a positive integer = 2. Lemma 1.4. Let C ∈ M∗ (n − 1, n). There exist γj ∈ Γj (j = n − 1, n) such that γn−1 Cγn−1 = (0, D) with D ∈ ∆n−1 , that is D is upper triangular. Proof. The proof is a routine induction. Let n = 1. Let C ∈ M∗ (1, 2), so C = (b, c) is a pair of integers, one of which is 6= 0. Let us write b = db1 , c = dc1 where (b1 , c1 ) is primitive, i.e. b1 , c1 are relatively prime, and d is a non-zero integer. We can complete a first column t (−c1 , b1 ) to an element of SL2 (Z) to complete the proof. The rest is done by induction, using blocks. A more detailed argument will be given in a similar situation, namely the proof of Lemma 1.6. We consider the coset space Tn−1 \M∗ (n − 1, n). Given a coset Tn−1 C, by Lemma 1.4 we can find a coset representative of the form (0, D)γ with γ ∈ Γn . We use such representatives to describe a fibration of Tn−1 \M∗ (n − 1, n) over Tn \Γn as follows. Lemma 1.5. Let π : Tn−1 \M∗ (n − 1, n) → Tn \Γn be the map which to each coset Tn−1 C with representative (0, D)γ associates the coset Tn γ. This map π is a surjection on Tn \Γn , and the fibers are Tn−1 \∆n−1 . Proof. Implicit in the statement of the lemma is that the association π as described is well defined, i.e. independent of the chosen representative. Suppose (0, D)γ = (0, D0 )γ 0

with D, D0 ∈ ∆n−1 .

Then (0, D) = (0, D0 )γ 0 γ −1 . Let τ = γ 0 γ −1 ∈ Γn . Then the above equation shows that actually τ is triangular, and so lies in Tn . This is done by an inductive argument, letting τ = (tij ) and starting with showing that t21 = 0, . . . tn+1,n+1 = 0, and then proceeding inductively to the right with


137

the second column, third column, etc. Thus γ, γ 0 are in the same coset of Tn \Γn , showing the map is well defined. We note that the surjectivity of π is immediate. As to the fibers, if τ ∈ Tn and D ∈ ∆n−1 , then (0, D)τ again has the form (0, D0 ) with D0 ∈ ∆n−1 . Thus by definition, the fiber above a coset Tn γ consists precisely of cosets Tn−1 (0, D)

with D ∈ ∆n−1 ,

which proves the lemma. In Lemma 1.5, we note that for each γ ∈ Γn we have a bijection Tn−1 \∆n−1 → fiber above Tn γ, induced by the representative map D 7→ (0, D)γ. The arguments of Lemma 1.4 and 1.5 will be pushed further inductively. The rest of this section follows the careful and elegant exposition in Maass [Maa 71]. Since we operate with the discrete group Γ on the left, we have to reverse the notation used in Selberg, Maass, and other authors, for example Langlands [Lgl 76], Appendix 1. Let Y ∈ Posn . If Yj = Subj (Y ) is the lower right j × j square submatrix of Y , then we can express Yj in the form µ ¶ 0 Yj = [(0, Ij )]Y = (0, Ij )Y (1) , Ij where Ij is the unit j × j matrix as usual. Note the operation on the left, and the fact that 0 denotes the j × (n − 1) zero matrix, so that (0, Ij ) is a j × n matrix. If Y = T t T with an upper triangular matrix T , then Yj = Tjt Tj , where Tj is the lower right j × j submatrix of T . From a given Y we obtain a sequence (Yn , Yn−1 , . . . , Y1 ) by the operation indicated in (1), starting with Yn = Y . We call this sequence the Selberg sequence of Y . Given γ ∈ Γn , we shall also form the Selberg sequence with Yn = [γ]Y . In some sense (to be formalized below) this procedure gives rise to “primitive” sequences. It will be necessary to deal with non-primitive sequences, and thus we are led to make more general definitions as follows. By an integral chain (more precisely n-chain) we mean a finite sequence C = (γ, Cn−1 , . . . , C1 ) with

γ ∈ Γn

and Cj ∈ M∗ (j, j + 1)

0 for j = 1, . . . , n − 1. Let C be such a chain. Let C 0 = (γ, Cn−1 , . . . , C10 ) be another chain. We define C equivalent to C 0 if either one of the following conditions are satisfied. EQU 1. There exist γj ∈ Γj (j = 1, . . . , n) such that

(2)

γ 0 = γn γ

−1 and Cj0 = γj Cj γj+1

for j = 1, . . . , n − 1 .

138


EQU 2. There exist γj ∈ Γj (j = 1, . . . , n − 1) such that (3)

0 Cj0 . . . Cn−1 γ 0 = γj Cj . . . Cn−1 γ

for

j = 1, . . . , n − 1 .

It’s obvious that (2) implies (3). Conversely, suppose EQU 2 and (3). We then let γn = γ 0 γ −1 , and it follows inductively that (2) is satisfied. A sequence (γ, Cn−1 , . . . , C1 ) will be said to be triangularized if we have that Cj = (0, Dj ) with Dj ∈ ∆j for j = 1, . . . , n − 1. Thus the first column of Cj is zero. The next lemmas give special representatives for equivalence classes. Lemma 1.6. Let Cj ∈ Zj,j+1 (j = 1, . . . , n − 1) be integral matrices. There exist elements γj ∈ Γj (j = 1, . . . , n) such that for j = 1, . . . , n − 1 we have   0 ∗ ... ∗   −1 γj Cj γj+1 = (0, T1 ) =  ... ... . . . ...  , 0

0

...

∗

that is, the first column on the right is 0, and the rest is upper triangular, with Tj ∈ Tri+ j . Thus every chain is equivalent to a triangularized one. Proof. Induction. For n = 2, the assertion is obvious, but we note how it illustrates the proof in general. We just have C1 = (b, c) with numbers b, c. We have γ1 = 1 and we write b = db1 , c = dc1 with (b1 , c1 ) relatively prime. Then we can complete a first column t (−c1 , b1 ) to an element of SL2 (Z) to complete the proof. Now by induction, suppose n = 3. There exist β2 , . . . , βn −1 with βj ∈ Γj such that the first column of Cj βj+1 is 0 for j = 1, . . . , n − 1. −1 Then βj Cj βj+1 also has first column equal to 0, and this also holds for j = 1. Hence without loss of generality, we may assume that Cj has first column equal to 0, that is   0 ∗∗∗   j−1,j Cj =  ... .  with Hj−1 ∈ Z 0

Hj−1

By induction, there exists ηj−1 ∈ Γj−1 (j = 2, . . . , n) such that   0 ∗...∗   ηj−1 Hj−1 ηj−1 =  ... ... . . . ...  0 0...∗

where the matrix on the right has first column 0, and the rest upper triangular. We let   1 0  for j = 1, . . . , n . γj =  0 ηj−1


Then γj ∈ Γj and matrix multiplication shows that    1 1 0 0 ∗ −1   γj Cj γj+1 = 0 0 ηj−1 0 Hj−1   0 ∗  ..  =. .

0 ηj−1

139

 

0 ηj−1 Hj−1 ηj−1

This last matrix has the desired form (0, Tj ), thereby concluding the proof. The next lemma will give a refinement by prescribing representatives even further. Lemma 1.7. For each coset of Tn \Γn , Tn−1 \∆n−1 , . . . , T1 \∆1 fix a coset representative. To each sequence (γ, Dn−1 , . . . , D1 ) whose components are among the fixed representatives, associate the chain (γ, (0, Dn−1 ), . . . , (0, D1 )) . Then this association gives a bijection from the set of representative sequences to equivalence classes of chains, i.e. every chain is equivalent to exactly one formed as above, with the fixed representatives. Proof. By Lemma 1.6, every equivalence class has a representative 0 ), . . . , (0, D10 )) (γ 0 , (0, Dn−1

with γ 0 ∈ Γn and Dj0 ∈ ∆j for j = n − 1, . . . , 1. There is one element τn ∈ Tn such that τn γ 0 is the fixed representative of the coset Tn γ 0 . Then we select the unique τn−1 such that if we put 0 )τn−1 (0, Dn−1 ) = τn−1 (0, Dn−1

then Dn−1 is the fixed representative of the coset Tn−1 Dn−1 . We can then continue by induction. This shows that the stated association maps bijectively on the families of equivalence classes and proves the lemma. A chain (γ, Cn−1 , . . . , C1 ) is called primitive if all the matrices Cj , with j = 1, . . . , n − 1, are primitive, that is, Cj can be completed to an element of Γj+1 by an additional row. The property of being primitive depends only on the equivalence class of the chain, namely if this property holds for C then it holds for every chain equivalent to C. Furthermore, if (γ, (0, Dn−1 ), . . . , (0, D1 )) is a triangularized representative of an equivalence

140


class, then it is primitive if and only if each Dj ∈ Γj . In the primitive case, we can choose the fixed coset representatives of Tj \Γj (j = 1, . . . , n − 1) to be the unit matrices Ij . The primitive chains of the form (γ, (0, In−1 ), . . . , (0, I1 ))

with γ ∈ Tn \Γn

will be called normalized primitive chains. Alternatively, one can select a fixed set of representatives {γ} for Tn \Γn , and the primitive chains formed with such γ are in bijection with the equivalence classes of all primitive chains. Formally, we state the result: Lemma 1.8. The map γ 7→ chains of (γ, (0, In−1 ), . . . , (0, I1 )) induces a bijection Tn \Γn → primitive equivalence classes of chains .

2 The ζQ Fudge Factor It will be convenient to put out of the way certain straightforward computations giving rise to the fudge factor involving the Riemann zeta function, so here goes. For a positive integer j we shall use the representatives of Tj \∆j from Lemma 1.2. We let n = 2. Let {z1 , z2 , . . .} be a sequence of complex variables. Let m = n. On Posm (n) we define the Selberg power function qz by the formula qz(n) (S) =

n Y

|Subj (S)|zj

with

S ∈ Posm .

j=1 (n−1)

(n)

In particular, we may work with qz on Posn , or also with qz on Posn , depending on circumstances. In any case, we see that we may also write qz(n) = dznn . . . dz11 , where dj is the partial determinant character, namely dj (S) = |Subj (S)| . In the next lemma, we consider both interpretations of qz . We shall look at values qz(n) ([(0, D)]S) where D ∈ ∆n is triangular, and S ∈ Posm . We note that this value is independent of the coset Tn D of D with respect to the triangular matrices with ±1 on the diagonal. We shall sum over such cosets. More precisely, let ϕ be a Tn -invariant function on Posn . Under conditions of absolute convergence, we define the Hecke-zeta operator on Posm by the formula

2 The ζQ Fudge Factor

X

HZn (ϕ) =

141

ϕ ◦ [(0, D)],

D∈Tn \∆n

that is for S ∈ Posm , X

HZn (ϕ)(S) =

ϕ([(0, D)]S) .

D∈Tn \∆n

We consider what is essentially an eigenfunction condition: EF HZ. There exists λHZ (ϕ) such that for all S ∈ Posm we have HZn (ϕ)(S) = λHZ (ϕ)ϕ(Subn S) . Implicit in this definition is the assumption that the series involved converges absolutely. The next lemma gives a first example. For any positive integer n, we make the general definition of the Riemann zeta fudge factor at level n, ΦQ,n (z) =

n Y

ζQ (2(zi + . . . + zn ) − (n − i)) .

i=1

Lemma 2.1. Let S ∈ Posm . Then X (n) (n) q−z ([(0, D)]S) = ΦQ,n (z)q−z (S) . D∈Tn \∆n

In other words, (n)

λHZ (q−z ) = ΦQ,n (z) . This relationship holds for Re(zi + . . . + zn ) > (n − i + 1)/2, i = 1, . . . , n, which (n) is the domain of absolute convergence of the Hecke-zeta operator on q−z . (n)

Proof. Directly from the definition of q−z , we find (1)

(n)

q−z ([(0, D)]S) = = =

n Y

i=1 n Y i=1 n Y

|[(0, Ii )(0, D)]S)|−zi |Subi (D)|−2zi |Subi (S)|−zi (n)

(dn−i+1 · · · dn )−2zi q−z (S) ,

i=1

where d1 , . . . , dn are the diagonal elements of D. Next we take the sum over all integral non-singular triangular D, from the set of representatives of Lemma 1.2, so

142




d1  .. D= . 0

 ∗ ..  . .  dn

... .. . ...

The sum over D can be replaced by a sum ∞ X

n Y

dk−1 k

d1 ,...,dn =1 k=1 (n)

by Lemma 1.3. With the substitution k = n − i + 1, the factor of q−z (S) in (1) can thus be expressed as n X Y D

(dn−i+1 . . . dn )−2zi

i=1

(2)

=

∞ X

...

d1 =1

n ∞ X Y

−2(zn−k+1 +...+zn )+k−1

dk

dn =1 k=1

= ΦQ,n (z) after reverting to indexing by i instead of n − k + 1. This proves the lemma. Next we deal with a similar but more involved situation, for which we make a general definition of the Riemann zeta fudge factors, namely ΦQ,j (z) = ΦQ,j (z1 , . . . , zj ) =

j Y

ζQ (2(zi + . . . + zj ) + j − i)

i=1

and (n)

ΦQ (z1 , . . . , zn ) =

n Y

ΦQ,j (z) .

j=1

These products will occur as factors in relations among Eisenstein series later. In the next lemma, we let {Dj } range over the representatives of Tj \∆j (j = (j) 1, . . . n) as given in Lemma 1.2. We let dνν denote the diagonal elements of Dj , with the indexing j − k + 1 5 ν 5 j, which will fit the indexing in the literature. The indexing also fits our viewing Dj as a lower right square submatrix. Lemma 2.3. X Dn

...

n n X Y Y D1

k=1 j=k

j Y

(n)

−2zk = ΦQ (z) (d(j) νν )

ν=j−k+1

=

Y

ζQ (2(zi + . . . + zj ) + j − i) .

15i5j5n

3 Eisenstein Series

143

Proof. For a fixed index j, we consider the sum on the left over the representatives {Dj }. The products inside the sum which are indexed by this value j then can be written j X Y Dj

j Y

−2zk . (d(j) νν )

k=1 ν=j−k+1

This is precisely the term evaluated in (2), and seen to be equal to ZQ,j (z). Taking the product over j = 1, . . . , n concludes the proof of the lemma.

3 Eisenstein Series Next we shall apply chains as in Sect. 1 to elements of Posn . Let Y ∈ Posn . Let C be a chain, C = (γ, Cn−1 , . . . , C1 ). For each j = 1, . . . , n − 1 define Cj (Y ) = [Cj · · · Cn−1 γ]Y,

Cn (Y ) = [γ](Y ) .

Thus Cj (Y ) = [Cj ]Cj+1 (Y ) for j = 1, . . . , n − 1. Let z1 , . . . , zn−1 be n−1 complex variables. We define the Selberg power (n−1) function qC = qC (depending on the chain) by the formula (n−1)

qC,z

(Y ) = |Cn−1 (Y )|zn−1 . . . |C1 (Y )|z1 .

(n)

One may also define qC

with one more variable, namely (n)

qC,z (Y ) =

n Y

|Cj (Y )|zj .

j=1

Let C be equivalent to C 0 . Then by (2) or (3) of Sect. 1 we have Cj0 (Y ) = [γj ]Cj (Y ) with γj having determinant ±1, so |Cj0 (Y )| = |Cj (Y )|. It follows that (n−1)

(n−1)

qC 0 ,z (Y ) = qC,z (n−1)

(Y ) ;

in other words, qC,z depends only on the equivalent class of C. Hence the power function can be determined by using the representatives given by Lemma 1.7. As in Sect. 1, we let Tn be the group of integral upper triangular n × n matrices with ±1 on the diagonal. We define the Selberg Eisenstein series X (n−1) (n−1) ET ,n (Y, z) = qC,−z (Y ) , C

144


where the sum is taken over all equivalence classes of chains. We define the primitive Selberg Eisenstein series by the same sum taken only over the primitive equivalence classes, that is X pr(n−1) (n−1) ET ,n (Y, z) = qC,−z (Y ) . C primitive

Furthermore, from Lemma 1.8, we know that a complete system of representatives for equivalence classes of primitive chains is given by (γ, (0, In−1 ), . . . , (0, I1 ))

with

γ ∈ Tn \Γn .

If C has the representative starting with γ, then we may write qC,z (Y ) = qz ([γ]Y ) . We may thus write the primitive Eisenstein series in the form X pr(n−1) (n−1) (1) (Y, z) = q−z ([γ]Y ) . ET ,n γ∈Tn \Γn

This is essentially the Eisenstein series we have defined previously, except that we are summing mod Tn instead of mod ΓU . However, we note that for any character ρ, and τ ∈ Tn we have the invariance property ρ([τ ]Y ) = ρ(Y ) for all Y ∈ Posn . Since (Tn : ΓU ) = 2n , denoting the old Eisenstein series by EUpr (Y, q−z ), we get EUpr (Y, z) = 2n ETpr (Y, z) .

(2) We recall explicitly that

EUpr (Y, ρ) = TrΓU \Γ (ρ)(Y ) =

X

ρ([γ]Y ) .

γ∈ΓU \Γ

To make easier the formal manipulations with non-primitive series, we list some relations. For given k = 1, . . . , n − 1 we consider the product (0, Dk ) . . . (0, Dn−1 ) = (0k,n−k , Tk ) where (γ, Dn−1 , . . . , D1 ) is a chain equivalent to C and Dj ∈ ∆j . Thus Tk is a triangular k × k matrix. To determine more explicitly the Eisenstein series, we may assume without loss of generality that C = (γ, (0, Dn−1 ), . . . , (0, D1 )) . Then

3 Eisenstein Series

(3)

145

Ck (Y ) = [(0, Tk )γ]Y = [Tk ][(0, Ik )γ]Y

and therefore |Ck (Y )| = |Tk |2 |Subk ([γ]Y )| .

(4) (k)

Let tνν denote the diagonal elements of Tk . Then of course |Tk |2 =

(5)

k Y

2 (t(k) νν ) .

ν=1

These products decomposition allow us to give a product expression for E in terms of E pr and the Riemann zeta function via the formula qC,−z (Y ) =

n−1 Y

|Ck (Y )|−zk

=

n−1 Y

|([γ]Y )k |−zk

(n−1)

(6)

k=1

k=1

n−1 Y j=k

j Y

−2zk , (d(j) νν )

ν=j−k+1

(j)

here dνν are the diagonal elements of Dj . (n−1)

Theorem 3.1. The Eisenstein series EU,n (Y, z) converges absolutely for Re(zj ) > 1 (j = 1, . . . , n − 1) and satisfies the relation (n−1)

(n−1)

EU,n (Y, z) = ΦQ

pr(n−1)

(z1 , . . . , zn−1 )EU

(Y, z) .

Proof. Both the relation and the convergence follow from (6) and Lemma 2.3 applied to n − 1 instead of n, and Theorem 2.2 of Chap. 7. Next, we have identities concerning the behavior of the Eisenstein series under the star involution. Recall that for any function ϕ on Posn , we define ϕ∗ (Y ) = ϕ([ω]Y −1 ) = ϕ(ωY −1 ω) . Proposition 3.2. Let ϕ be any U -invariant function such that its ΓU \Γ-trace converges absolutely. Then (TrΓU \Γ ϕ)(Y −1 ) = (TrΓU \Γ (ϕ∗ )(Y ) . In particular, if ρ is a left character, then EUpr (Y −1 , ρ) = EUpr (Y, ρ∗ ) . If {γ} is a family of coset representatives of ΓU \Γ, then {ω t γ −1 } is also such a family. Similarly for representatives of T \Γ.

146


S Proof. As to the second statement, write Γ = ΓU γ. Let ΓU¯ be the lower triangular subgroup. Then [ [ t Γ= γΓU¯ = ΓU¯ t γ −1 (taking the inverse) γ

= =

[

[

ωΓU¯ ωω t γ −1

(because Γ = ωΓ and ω 2 = I)

ΓU ω t γ −1

(because ωΓU¯ ω = ΓU ) .

This proves the second statement. Then the first formula comes out, namely: X ϕ([γ]Y −1 ) TrΓU \Γ ϕ(Y −1 ) = γ∈ΓU \Γ

=

X

ϕ(γY −1 t γ)

=

X

ϕ∗ (ω(t γ −1 Y γ −1 )ω)

γ

γ

= TrΓU \Γ ϕ∗ (Y ) by the preceding result, thus proving the proposition. The next two lemmas deal with similar identities with sums taken over cosets of matrices modulo the triangular group. Lemma 3.3. Let ϕ be a Tn -invariant function such that the following sums are absolutely convergent, i.e. a left character on Posn . Let S ∈ Posn+1 . Then X X ϕ([C]S) . ϕ∗ ((S[A])−1 ) = A∈M∗ (n+1,n)/Tn

C∈Tn \M∗ (n,n+1)

Proof. Inserting an ω inside the left side and using the definition of ϕ∗ , together with ϕ∗∗ = ϕ, we see that the left side is equal to X X ϕ(S[Aω]) . ϕ(S[A][ω]) = A

A∈M∗ (n+1,n)/Tn

By definition, M∗ (n + 1, n) =

S

ATn , with a family {A} of coset representa-

A ∗

tives. Since M∗ (n + 1, n) = M (n + 1, n)ω, we also have [ [ [ ATn = AωωTn ω = AωTn− A

Aω

where Tn− is the lower integral triangular group. Thus the family {Aω} is a family of coset representatives for M∗ (n + 1, n)/Tn− . Writing S[Aω] = [ω t A]S,

3 Eisenstein Series

147

we see that we can sum over the transposed matrices, and thus that the desired sum is equal to X ϕ([C]S) , C∈Tn \M∗ (n,n+1)

which proves the lemma. Instead of taking M∗ (n + 1, n)/Tn we could also take M∗ (n + 1, n)/ΓU . Since Tn /ΓU has order 2n , we see that we have a relation similar to (2), namely X X (7) ϕ∗ ([C]S) = 2n ϕ∗ ([C]S) . ΓU \M∗ (n,n+1)

Tn \M∗ (n,n+1)

Normalizing the series by taking sums mod Tn or mod ΓU only introduces the simple factor 2n each time. We shall now develop further the series on the right in Lemma 3.3, by using the eigenvalue property EF HZ stated in Sect. 2. Lemma 3.4. Suppose that ϕ is Tn U -invariant on Posn , and satisfies condition EF HZ (eigenfunction of Hecke-zeta operator). Then on Posn+1 , X ϕ ◦ [C] = λHZ (ϕ)TrTn+1 \Γn+1 (ϕ ◦ Subn ) . C∈Tn \M∗ (n,n+1)

Proof. By the invariance assumption on ϕ, we can use the fibration of Lemma 1.5, and write the sum on the left evaluated at S ∈ Posn+1 as X X ϕ([(0, D)][γ]S) . γ∈Tn+1 \Γn+1

D∈Tn \∆n

Then the inner sum is just the Hecke operator of ϕ, when evaluated at Subn [γ]S. The result then falls out. (n)

In particular, we may apply the lemma to the case when ϕ = q−z , and we obtain: Corollary 3.5. Let S ∈ Posn+1 . Then X (n) pr(n) (n) q−z ([C]S) = ΦQ,n (z)ET ,n+1 (S, q−z ) . C∈Tn \M∗ (n,n+1)

Proof. Special case of Lemma 3.4, after applying Lemma 2.1 which determines the eigenvalue of the Hecke-zeta operator.

148


4 Adjointness and the ΓU \Γ-trace We shall use differential operators introduced in Chap. 6. First, we observe that for c > 0, Y ∈ Posn , B ∈ Symn we have by direct computation ¯ ¯ ¯ ∂ ¯ −ctr(BY ) ¯ ¯ (1) = (−c)n |B| e−ctr(BY ) . ¯ ∂Y ¯ e

In particular, the above expression vanishes if B is singular. In the applications, B will be semipositive, and the effect of applying |∂/∂Y | will therefore be to eliminate such a term when B has rank < n. As in Chap. 6 let the (first and second) regularizing invariant differential operators be ¯ ¯ ¯ ∂ ¯ ˜ n |Y |−k Qn . ¯ and D = Dn = |Y |−k Q ¯ (2) Q = Qn = |Y | ¯ ∂Y ¯ Throughout we put k = (n + 1)/2 and D = Dn if we don’t need to mention n. We recall that (3)

˜ n = |Y |k Dn |Y |−k = Q|Y ˜ |k Q|Y |−k . D For S ∈ Posn+1 we let θ(S, Y ) =

X

e−πtr(S[A]Y )

A

where the sum is taken over A ∈ Zn+1,n . This is the standard theta series. We can differentiate term by term. By (1) and the subsequent remark, we note that X X DY θ(S, Y ) = DY e−π(S[A]Y ) = βA,S (Y )e−πtr(S[A]Y ) , rk(A)=n

rk(A)=n

where βA,S (S being now fixed) is a function of Y with only polynomial growth, and so not affecting the convergence of the series. Although its coefficients are complicated, there is one simplifying effect to having applied the differential operator D, namely we sum only over the matrices A of rank n. Thus we abbreviate as before, and for this section, we let: M∗ = M∗(n+1,n) = subset of elements in Z(n+1)×n of rank n . Then the sum expressing DY θ(S, Y ) is taken over A ∈ M∗ . Note that both θ and Dθ are functions of two variables, and thus will be viewed as kernels, which induce integral operators by convolution, provided they are applied to functions for which the convolution integral is absolutely convergent.

4 Adjointness and the ΓU \Γ-trace

149

We recall the functional equation for θ, θ(S −1 , Y −1 ) = |S|n/2 |Y |(n+1)/2 θ(S, Y ) .

(4)

From (3), we then see that Dθ satisfies the same functional equation, that is (Dθ)(S −1 , Y −1 ) = |S|n/2 |Y |(n+1)/2 (Dθ)(S, Y ) .

(5)

Here we have used the special value k = (n + 1)/2. We shall now derive an adjoint relation in the present context. For a U invariant function ϕ on Posn , we recall the ΓU \Γ-trace, defined by X TrΓU \Γ (ϕ)(Y ) = ϕ([γ]Y ) . γ∈ΓU \Γ

For functions ϕ such that the ΓU \Γ-trace and the following integral are absolutely convergent, we can form the convolution on Γn \Posn : Z (Dθ ∗ TrΓU \Γ ϕ)(S) = (DY θ)(S, Y )TrΓU \Γ (ϕ)(Y )dµn (Y ) . Γn \Posn

We abbreviate as before P = Posn ,

Γ = Γn

to make certain computations formally clearer. Lemma 4.1. For an arbitrary U -invariant function ϕ on Posn insuring absolute convergence of the series and integral, we have with k = (n + 1)/2: (Dθ ∗ TrΓU \Γ ϕ)(S) Z X = 2(−1)n |πS[A]| e−πtr(S[A]Y ) |Y |k+1 Q(ϕd−k )(Y )dµ(Y ) . A∈M∗ /ΓU

P

Thus the convolution on the left is a sum of gamma transforms. Proof. The proof is similar to those encountered before. We have: Z (Dy θ)(S, Y )TrΓU \Γ ϕ(Y )dµ(Y ) Γ\P

=

Z Γ\P

=

X

DY e−πtr(S[A]Y ) TrΓU \Γ ϕ(Y )dµ(Y )

A∈M∗

X

Z

A∈M∗ /Γ Γ\P

X

γ∈Γ

DY e−πtr(S[Aγ]Y ) TrΓU \Γ ϕ(Y )dµ(Y )

150


X

=

A∈M∗ /Γ

X

=

2

Z

DY e−πtr(S[A]Y )

X

ϕ([γ]Y )dµ(Y )

γ∈ΓU \Γ

P

X

A∈M∗ /Γ γ∈ΓU \Γ

2 =

¯ ¯ ¯ ¯ ˜ Y (|Y |k+1 ¯ ∂ ¯ e−πtr(S[A]Y ) )ϕ([γ](Y ))dµ(Y ) |Y |−k Q ¯ ∂Y ¯ P X X Z

A∈M∗ /Γ γ∈ΓU \Γ

2(−1)n |πS[A]|

Z

e−πtr(S[A]Y ) |Y |k+1 QY (|Y |−k ϕ([γ]Y ))dµ(Y ) ,

P

˜ Y from the exponential term to the using formula (2), and then transposing Q ϕ ◦ [γ](Y ) term. Now we make the translation Y 7→ [γ −1 ]Y in the integral over P. Under this change, ΓU \Γ 7→ Γ/ΓU , and the expression is equal to =

n X

A∈M∗ /Γ

X

2(−1)n |πS[A]|

γ −1 ∈Γ/ΓU

Z

e−πtr(S[Aγ

−1

]Y )

|Y |k+1 QY (|Y |−k ϕ(Y ))dµ(Y ) .

P

The two sums over Γ/ΓU and over M∗ /Γ can be combined into a single sum with A ∈ M∗ /ΓU , which yields the formula proving the lemma. Looking at the integral expression on the right in the lemma, we see at once that it is a gamma transform. Furthermore, if ϕd−k is an eigenfunction of Qn , then the integral can be further simplified, and this condition is satisfied in the case of immediate interest when ϕ is a character. However, it continues to be clearer to extract precisely what is being used of a more general function ϕ, which amounts to eigenfunction properties in addition to Tn U -invariance and the absolute convergence of the series and integral involved. Thus we list these properties as follows. EF Q. The function ϕd−(n+1)/2 is an eigenfunction of Qn . EF Γ. The function ϕd is an eigenfunction of the gamma transform, it being assumed that the integral defining this transform converges absolutely. We use λ to denote eigenvalues. Specifically, let D be an invariant differential operator. Let ϕ be a D-eigenfunction. We let λD (ϕ) be the eigenvalue so that Dϕ = λD (ϕ)ϕ .

4 Adjointness and the ΓU \Γ-trace

151

Similarly, we have the integral gamma operator, and for an eigenfunction ϕ, we let λΓ (ϕ) = Γn (ϕ) so that Γ#ϕ = λΓ (ϕ)ϕ . In addition, we define Λn (ϕ) = (−1)n λQ (ϕd−(n+1)/2 )λΓ (ϕd) . Theorem 4.2. Assume that ϕ is Tn U -invariant and satisfies the two properties EF Q and EF Γ. Then for S ∈ Posn+1 , under conditions of absolute convergence, X ϕ((πS[A])−1 ) . (Dθ ∗ TrΓU \Γ (ϕ)(S) = 2Λn (ϕ) A∈M∗ (n+1,n)/ΓU

Proof. By using the eigenfunction assumptions on the expression being summed on the right side of the equality in Lemma 4.1, and again if we set k = (n + 1)/2, we obtain: Z |πS[A]| e−πtr(S[A]Y ) |Y |k+1 λQ (ϕd−k )(ϕd−k )(Y )dµ(Y ) P

= λQ (ϕd−k )|πS[A]|

Z

e−πtr(S[A]Y ) (ϕd)(Y )dµ(Y )

P

= λQ (ϕd

−k

)|πS[A]|λΓ (ϕd)(ϕd)((πS[A])−1 )

by definition of the gamma transform and an eigenvalue, cf. Chap. 3, Proposition 2.2, = λQ (ϕd−k )λΓ (ϕd)ϕ((πS[A])−1 ) because the determinant cancels. This proves the theorem. Theorem 4.3. Let ϕ be Tn U -invariant, satisfying EF Q, EF Γ, and EF HZ. Then for S ∈ Posn+1 , when the series and integral are absolutely convergent, (Dθ ∗ TrΓU \Γ (ϕ∗ )(S) = Λn (ϕ∗ )λHZ (ϕ)TrΓUn+1 \Γn+1 (ϕ ◦ Subn )(πS) . Proof. We apply Theorem 4.2 to ϕ∗ instead of ϕ. The sum in Theorem 4.2 can be further simplified as follows: X ϕ∗ ((πS[A])−1 ) A∈M∗ (n+1,n)/ΓU

= 2n n

=2

X

ϕ∗ ((πS[A])−1 )

A∈M∗ (n+1,n)/Tn λHZ (ϕ)ETpr (πS, ϕ

◦ Subn )

by Lemmas 3.3 and 3.4.

The Eisenstein series here is on Posn+1 , and going back to ΓUn+1 instead of Tn+1 introduces the factor 1/2n+1 , which multiplied by 2n leaves 1/2. This 1/2 cancels the factor 2 occurring in Theorem 4.2. The relationship asserted in the theorem then falls out, thus concluding the proof.

152


Corollary 4.4. Let D = Dn be the invariant differential operator defined at the beginning of the section. Let ϕ be homogeneous of degree w, for instance ϕ is a character. Then for S ∈ Posn+1 , (Dθ ∗ TrΓU \Γn ϕ∗ )(S) = π w Λn (ϕ∗ )λHZ (ϕ)TrΓUn+1 \Γn+1 (ϕ ◦ Subn )(S) . Proof. We just pull out the homogeneity factor from inside the expression in Theorem 4.3. Remark. Remark Immediately from the definitions, one sees that for the Selberg power character, we have deg qz(n) = wn (z) =

n X

jzj .

j=1

This character may be viewed as a character on Posm for any m = n. The degree is the same in all cases. For application to the Eisenstein series, we (n) use of course q−z , which has degree −wn (z) = wn (−z). Actually, in the next section we shall change variables, and get another expression for the degree in terms of the new variables. The inductive formula of this section stems from the ideas presented by Maass [Maa 71], pp. 268–272, but we have seen how it is valid for much more general functions ϕ besides characters. Maass works only with the special characters coming from the Selberg power function, and normalizes these characters with s-variables. We carry out this normalization in the next section, as a preliminary to Maass’ proof of the functional equation.

5 Changing to the (s1 , . . . , sn)-variables We recall the Selberg power function of Chap. 3, Sect. 1, expressed in terms of two sets of complex variables z = (z1 , . . . , zn−1 ) and s = (s1 , . . . , sn ) , namely (1)

(n−1)

|Y |sn +(n−1)/4 q−z

(Y ) = hs (Y ) =

n Y

(tn−i+1 )2si +i−(n+1)/2 ,

i=1

where zj = sj+1 − sj +

1 2

for j = 1, . . . , n − 1,

or also (2)

(n−1)

q−z

(Y ) = |Y |−sn −(n−1)/4 hs (Y ) .

5 Changing to the (s1 , . . . , sn )-variables

153

To determine the degree of homogeneity of hs , we note that Y 7→ cY (c > 0) corresponds to t 7→ c1/2 t . Then we find immediately: deg hs =

(3)

n X

si

and

deg h∗s = −deg hs .

i=1

Throughout this section, we fix the notation. We let Γ = Γn , and (n−1)

ζ pr (Y, s) = EUpr (Y, q−z

(n−1)

) = TrΓU \Γ q−z

(Y )

= |Y |−sn −(n−1)/4 TrΓU \Γ hs (Y ) . Proposition 5.1. We have in the appropriate domain (see the remark below): ζ pr (Y −1 , s) = |Y |sn −s1 +(n−1)/2 ζ pr (Y, s∗ ) , where s∗ = (−sn , . . . , −s1 ), so s∗j = −sn−j+1 . Proof. We have ζ pr (Y −1 , s) = |Y |sn +(n−1)/4 TrΓU \Γ hs (Y −1 ) sn +(n−1)/4

= |Y |

TrΓU \Γ h∗s (Y

)

by Prop. 3.2

= |Y |sn +(n−1)/4 TrΓU \Γ hs∗ (Y ) sn −s1 +(n−1)/2 pr

= |Y |

by (2) by Chap. 3, Prop. 1.7

∗

ζ (Y, s )

by (2)

∗

because TrΓU \Γ hs∗ (Y ) = |Y |sn +(n−1)/4 ζ pr (Y, s∗ ) by (2). This concludes the proof. Remark. The domain of absolute convergence of the Eisenstein series (n−1) EUpr (Y, q−z ) was proved to be Re(zj ) > 1 for j = 1, . . . , n − 1, that is µ ¶ 1 Re sj+1 − sj + > 1 for j = 1, . . . , n − 1 . 2 From the relation s∗k = −sn−k+1 we see that s∗k+1 − s∗k +

1 1 = sj − sj−1 + 2 2

with

j =n−k+1.

Thus the domains of convergence in terms of the s∗ and s variables are “the same” half planes. We shall meet a systematic pattern as follows. Let ψ = ψ(u) be a function of one variable. For n = 2, we define

154


ψn (s) = ψn (s1 , . . . , sn ) =

n−1 Y

ψ(sn − si + 1/2)

i=1

ψ (n) (s) =

n Y

ψj (s1 , . . . , sj ) .

j=2

We note the completely general fact: Lemma 5.2. ψ (n) (s∗ ) = ψ (n) (s). This relation is independent of the function ψ, and is trivially verified from the definition of ψ (n) . It will apply to three important special cases below. We start with the function ψ(u) = ζQ (2u), where ζQ is the Riemann zeta function. Then we use a special letter ZQ and define ZQ,n (s) =

n−1 Y

ζQ (2(sn − si + 1/2))

i=1

(n)

Y

ZQ (s) =

ζQ (2(sj − si + 1/2)).

15i<j5n

Lemma 5.3. With ΦQ,n−1 as in Lemma 2.1, we have ΦQ,n−1 (z1 , . . . , zn−1 ) = ZQ,n (s1 , . . . , sn ) . Proof. By definition, ΦQ,n−1 (z) =

n−1 Y

ζQ (2(zi + . . . + zn−1 ) − (n − i − 1)) .

i=1

With the s-variables, we get a cancellation, namely (4)

zi + . . . + zn−1 = sn − si +

n−i . 2

This proves the lemma. (n−1)

The non-primitive Eisenstein series EU (Y, z) is defined to be the prod(n−1) uct of the primitive Eisenstein series times ΦQ (z). Hence from the transfer to the s-variables in Lemma 5.3 we have (n−1)

(5)

ζ(Y, s) = EU (n)

(n)

(n)

(Y, z) = ZQ (s)ζ pr (Y, s) .

Since ZQ (s∗ ) = ZQ (s) by Lemma 5.2, it follows that Proposition 5.1 is valid if we replace the primitive Eisenstein series ζ pr (Y, s) by ζ(Y, s). In connection with using Posn+1 via Theorem 4.3 and Corollary 4.4, it is (n) (n−1) natural to consider q−z as well as q−z .

5 Changing to the (s1 , . . . , sn )-variables

155

Lemma 5.4. Put zn = sn+1 − sn + 1/2, and let ϕs,sn+1 be the character on Posn defined by ϕs,sn+1 (Y ) = |Y |−sn+1 −(n+1)/4 hs (Y ) . In other words, ϕs,sn+1 = d−sn+1 −(n+1)/4 hs . Then on Posn , (n)

q−z = ϕs,sn+1 . Proof. By definition, (n)

(n−1)

q−z (Y ) = |Y |−zn q−z

(Y ) .

Substituting zn = sn+1 − sn + 12 and using (2) yields the desired relation. The Hecke-zeta eigenvalue is given by (6)

λHZ (ϕs,sn+1 ) = ZQ,n+1 (s1 , . . . , sn+1 ) = ZQ,n+1 (s, sn+1 ) .

This is just the formulation of Lemma 2.1 in the (s, sn+1 ) variables. Furthermore, ¶ n µ X n+1 wn (z) = deg ϕs,sn+1 = si − sn+1 − (7) . 4 i=1 This is immediate from (3) and the homogeneity degree of the determinant. We define various elementary functions from which we build others, and relate them to eigenvalues found in the preceding section. We let g(u) = π −u Γ(u)

and F (u) = u(1 − u)g(u) .

These are standard fudge factors in one variable u. Following the previous general pattern, we define gn (s) = gn (s1 , . . . , sn ) =

n−1 Y

g(sn − si + 1/2)

n−1 Y

F (sn − si + 1/2).

i=1

Fn (s) = Fn (s1 , . . . , sn ) =

i=1

Finally, we define g (n) (s) =

n Y

gj (s)

and F (n) (s) =

j=1

n Y

Fj (s) .

j=1

These definitions follow the same pattern that we used with the fudge factor (n) involving the Riemann zeta function, i.e. ZQ,n (s) and ZQ (s). In particular, Fn+1 (s1 , . . . , sn+1 ) =

F (n+1) (s1 , . . . , sn+1 ) F (n) (s1 , . . . , sn )

156


and F (n+1) (s, sn+1 ) =

n Y

Fj+1 (s1 , . . . , sj+1 ) .

j=1

The next lemma is the analogue of Proposition 5.3 for the fudge factor that we are now dealing with. Lemma 5.5. We have the explicit determination Fn+1 (s, sn+1 ) = π wn Λn (ϕ∗s,sn+1 ) . The exponent wn is the degree in (7), as a function of s1 , . . . , sn+1 . Proof. This is a tedious verification. (n)

We apply Corollary 4.4 to the character q−z = ϕs,sn+1 . We note that (n)∗

q−z = ϕ∗s,sn+1 = dsn+1 +(n+1)/4 hs∗ .

(8)

Lemma 5.6. For Y ∈ Posn , (u)∗

TrΓU \Γ (q−z (Y ) = TrΓU \Γ (ϕ∗s,sn+1 )(Y ) = |Y |sn+1 −s1 +n/2 ζ pr (Y, s∗ ) . Proof. By definition of s∗ = (−sn , . . . , s1 ) we have ζ pr (Y, s∗ ) = |Y |s1 −(n−1)/4 TrΓU \Γ h∗s (Y ) = TrΓU \Γ (ds1 −(n−1)/4 hs∗ )(Y ) . Multiplying by dsn+1 −s1 +n/2 and using (8) concludes the proof. For S ∈ Posn+1 , Γ = Γn , define ξ(S; s, sn+1 ) = (Dθ ∗ TrΓU \Γ (ϕ∗s,sn+1 ))(S) . Thus by definition of the convolution and Lemma 5.6, Z ξ(S; s, sn+1 ) = Dθ(S, Y )|Y |sn+1 −s1 +n/2 ζ(Y, s∗ )dµ(Y ) . Γn \Pn

Let B be the domain defined by the inequalities Re(sj+1 − sj + 1/2) > 1 for j = 1, . . . , n − 1 and sn+1 arbitrary, while B1 is defined by these inequalities together with Re(s1 − sn+1 + 1/2) > 1 . Estimates in Chap. 7 show that the integral for ξ converges absolutely in the domain B1 . In light of our definitions and Lemma 5.5, we may now reformulate Theorem 4.3 or rather Corollary 4.4 as follows. Theorem 5.7. In the domain defined by these inequalities, we have ξ(S; s, sn+1 ) = Fn+1 (s, sn+1 )ζ(S; s, sn+1 ) F n+1 (s1 , . . . , sn+1 ) = ζ(S; s, sn+1 ) . F (n) (s1 , . . . , sn )

6 Functional Equation: with Cyclic Permutations

157

6 Functional Equation: Invariance under Cyclic Permutations Here we follow Maass [Maa 71]. For the function ξ(S; s, sn+1 ) defined at the end of the preceding section, we first have Lemma 6.1. For S ∈ Posn+1 , ξ(S −1 ; s∗ , −sn+1 ) = |S|n/2 ξ(S; s, sn+1 ) . Proof. This result is proved by the Riemann method. The integral over Γn \Pn is decomposed into a sum Z Z Z + , = Γn \Pn

(Γn \Pn )(=1)

(Γn \Pn )(51)

where the parentheses (=1) and (51) signify the subdomain where the determinant is = 1 resp. 5 1. On the second integral, we make the change of variables Y 7→ Y −1 . Then letting Fn = Γn \Pn , we get: (1)

ξ(S; s, sn+1 ) Z © = Dθ(S, Y )|Y |sn+1 −s1 +n/2 ζ(Y, s∗ ) Fn (=1)

On the other hand, (2)

ª +Dθ(S, Y −1 )|Y |s1 −sn+1 −n/2 ζ(Y −1 , s∗ ) dµ(Y ) .

ξ(S −1 ; s∗ , −sn+1 ) Z © Dθ(S −1 , Y )|Y |−sn+1 +sn +n/2 ζ(Y, s) = Fn (=1)

ª +Dθ(S −1 , Y )|Y |sn+1 −sn −n/2 ζ(Y −1 , s) dµ(Y ) .

We now use two previous functional equations. One is the functional equation for the regularized theta functions, namely Sect. 4, formulas (4) and (5), which read: Dθ(S −1 , Y −1 ) = |S|n/2 |Y |(n+1)/2 Dθ(S, Y ) Dθ(S −1 , Y ) = |S|n/2 |Y |−(n+1)/2 Dθ(S, Y −1 ) The other equation is stated in Proposition 5.1, which is valid with ζ(Y, s) (n) (n) instead of ζ pr (Y, s), because ZQ (s∗ ) = ZQ (s) is the same factor needed to change the primitive Eisenstein series into the non-primitive one. Applying this proposition and the functional equation for the theta function

158


shows directly and immediately that the two terms under the integral for ξ(S −1 ; s∗ , sn+1 ) are changed precisely into the two terms which occur in the integral expression for ξ(S; s, sn+1 ) multiplied by |S|n/2 . This concludes the proof. Theorem 6.2. Let S ∈ Posn+1 and let η(S; s(n+1) ) = F (n+1) (s1 , . . . , sn+1 )|S|sn+1 ζ(S; s1 . . . , sn+1 ) . Then η(S; s1 , . . . , sn+1 ) is invariant under a cyclic permutation of the variables, that is η(S; s1 , . . . , sn+1 ) = F (n+1) (sn+1 , s1 , . . . , sn )|S|sn ζ(S; sn+1 , s1 , . . . , sn ) . Furthermore, η(S; s1 , . . . , sn+1 ) is holomorphic in the domain B. Proof. By Theorem 5.7 and F (n) (s∗ ) = F (n) (s), we have ξ(S −1 ; s∗ , −sn+1 ) = =

F (n+1) (s∗ ,−sn+1 ) ζ(S −1 ; s∗ , −sn+1 ) F (n) (s)

F (n+1) (sn+1 , s) |S|−sn+1 +sn +n/2 ζ(S; sn+1 , s1 , . . . , sn ) F (n) (s)

by Proposition 5.1, valid in the domain Re(Sj+1 − sj + 12 ) > 1 for each index j = 1, . . . , n − 1, that is in the domain B. On the other hand, |S|n/2 ξ(S; s, sn+1 ) = F (n+1) (s1 , . . . , sn , sn+1 ) n/2 |S| ζ(S, s1 , . . . , sn , sn+1 ) . F (n) (s1 , . . . , sn ) Using the definition of η(S; s1 , . . . , sn+1 ) and cross multiplying, we apply Lemma 6.2 to conclude the proof. Note. The three essential ingredients in the above proof are: EIS 1. For each integer n = 3 there is a fudge factor F (n) (s1 , . . . , sn ) such that for S ∈ Posn+1 we have ξ(S; s, sn+1 ) =

F (n+1) (s1 , . . . , sn , sn+1 ) ζ(S; s, sn+1 ) . F (n) (s1 , . . . , sn )

Furthermore, F (n) (s∗ ) = F (n) (s) (invariance under s 7→ s∗ ). See Lemma 5.2 and Theorem 5.7. EIS 2. ζ(Y −1 , s) = |Y |sn −s1 +(n−1)/2 ζ(Y, s∗ ) in the domain Re(sj+1 − sj + 1/2) > 1 . Ref: Proposition 5.1 and Lemma 5.2.

7 Invariance under All Permutations

159

EIS 3. ξ(S −1 ; s∗ , −sn+1 ) = |S|n/2 ξ(S; s, sn+1 ) Ref: Theorem 6.1. Finally, we prove the analytic continuation over all of Cn+1 by means of a theorem in several complex variables. That is, we want: Theorem 6.3. The function η(S, s1 , . . . , sn+1 ) is holomorphic on all of Cn+1 . Proof. We reduce the result to a basic theorem in several complex variables. Let σ be the cyclic permutation σ : (s1 , . . . , sn+1 ) 7→ (sn+1 , s1 , . . . , sn ) . By Theorem 6.2, we know that η is holomorphic in the domain D=

n [

σ j B ⊂ Cn+1 .

j=1

Let prRn+1 (D) = DR be the projection on the real part. Since the inequalities defining D involve only the real part, it follows that D = DR + iRn+1 , so D is what is commonly called a tube domain. By Theorem 2.5.10 in Hörmander [H¨ or 66], it follows that η is holomorphic on the convex closure of the tube. But DR contains a straight line parallel to the (n + 1)-th axis of Rn+1 . This line can be mapped on a line parallel to the j-th axis of Rn+1 for each j, by powers of σ. The convex closure of these lines in the real part ormander, it follows that the Rn+1 is all of Rn+1 , and by the theorem in H¨ convex closure of D is Cn+1 . This concludes the proof.

7 Invariance under All Permutations In light of the theorems in Sect. 6, all that remains to be done is to prove the invariance of the function η(Y ; s1 , . . . , sn ) = F (n) (s1 , . . . , sn )|Y |sn ζ(Y ; s1 , . . . , sn ) under a transposition, and even under the transposition between the special variables s1 and s2 . Then we shall obtain Selberg’s theorem: Theorem 7.1. For Y ∈ Posn , the function η(Y ; s1 , . . . , sn ) is invariant under all permutations of the variables.

160


Proof. The following proof follows Selberg’s lines and is the one given in (n−1) Maass [Maa 71]. We have ζ(Y ; s) = EU (Y, z) (the non-primitive Eisenstein series). The essential part of the proof will be to show that the function (n−1)

π −s1 Γ(z1 )EU

(Y, z) = π −(s2 −s1 +1/2) Γ(s2 − s1 + 1/2)ζ(Y, s)

is invariant under the transpositon of s1 and s2 . Before proving this, we show how it implies the theorem. As before, let g(u) = π −u Γ(u) . Then it follows that Y g(sj − si + 1/2)|Y |sn ζ(Y, s) = g (n) (s)|Y |sn ζ(Y, s) 15i<j5n

is invariant under the transposition of s1 and s2 . By Theorem 6.2 we conclude that this function is invariant under all permutations of (s1 , . . . , sn ). Theorem 7.1 follows by the factorization of F (n) (s1 , . . . , sn ) given in Sect. 5. (n−1) To prove the invariance of g(z1 )EU (Y, z) under the transposition of s1 and s2 , we go back to the definition of the Eisenstein series in terms of the z-variables, and we write this definition in the form (n−1)

ET

(Y, z) =

X

(Cn ,...,C2 )

n−1 Y

(1)

|Cj (Y )|−zj ET (C2 (Y ), z1 ) with

Cn = γ .

j=2

The sum over (Cn , . . . , C2 ) is over equivalence classes, whose definition for such truncated sequences is the same as for (Cn , . . . , C1 ), except for disregarding the condition on C1 . The theorem was proved in Chap. 5, Theorem 4.1 in the case n = 2, so we assume n = 3. We write the Eisenstein series with one further splitting, that is (n−1)

ET

(Y, z) =

X

(Cn ,...,C2 )

n−1 Y

(1)

|Cj (Y )|−zj |C2 (Y )|−z2 ET (C2 (Y ), z1 ) .

j=3

Although the notation with the chains was the clearest previously, it now becomes a little cumbersome, so we abbreviate Cj (Y ) = Yj

for j = 1, . . . , n .

Then we rewrite the above expressions for the Eisenstein series in the form (1)

(n−1)

ET

(Y, z) =

X

(Yn ,...,Y2 )

n−1 Y j=2

(1)

|Yj |−zj ET (Y2 , z1 )

7 Invariance under All Permutations

(2)

=

X

(Yn ,...,Y2 )

n−1 Y

161

(1)

|Yj |−zj |Y2 |−z2 ET (Y2 , z1 ) .

j=3

With the change of variables zj = sj+1 − sj + 1/2, we can also write (1)

|Y2 |−z2 ET (Y2 , z1 ) = 22 |Y2 |s2 −s3 +1/2 ζ(Y2 ; s1 , s2 ) . By Theorem 4.1 of Chap. 5, the function η2 (Y ; s1 , s2 ) = π −z1 Γ(z1 )|Y |s2 ζ(Y2 ; s1 , s2 ) is invariant under permutation of s1 and s2 , because in the notation of this reference, E(Y2 , z) = E (1) (Y2 , z) = ζ(Y2 ; s1 , s2 ) . Thus formally, we conclude that π −z1 Γ(z1 )ζ(Y ; s1 , . . . , s2 ) = π −z1 Γ(z1 )E (n−1) (Y, z) is invariant under the permutation of s1 and s2 . The only thing to watch for is that this permutation can be done while preserving the convergence of the series expression (2) for E (n−1) (Y, z). Thus one has to select an appropriate domain of absolute convergence, so that all the above expressions make sense. Maass does this as follows. We start with the inductive lowest dimensional piece, Λ2 (Y, z1 ) = π −z1 Γ(z1 )E (1) (Y, z1 ), which is the first case studied in Chap. 5, Sect. 3. We gave an estimate for this function in the strip Str(−2, 3), that is −2 < Re(z1 ) < 3 , away from 0 and 1, specifically outside the discs of radius 1 centered ar 0 and 1, as in Corollary 3.8 of Chap. 5. Next, we consider the series (3)

π −z1 Γ(z1 )E (n−1) (y, Z) =

X

n−1 Y

|Yj |−zj Λ2 (Y2 , z1 ) .

(Yn ,...,Y2 ) j=2

By Theorem 3.1, mostly Theorem 2.2 of Chap. 7, the series in (3) converges absolutely for Re(zj ) > 1, j = 1, . . . , n − 1. Similarly, by Chap. 7, Theorem 3.1, we also know that the series (4)

X

(Yn ,...,Y2 )

n−1 Y

|Yj |−zj

j=2

converges absolutely for Re(zj ) > 1 (3 5 j 5 n − 1) and Re(z2 ) > 3/2. By Chap. 5, Corollary 3.8 (put there for the present purpose), adding up the

162


power of |Y2 |, in the sbove strip outside the unit discs around 0, 1, it follows that the Eisenstein series from (1) converges absolutely in the domain D1 = points in Cn with z1 in the strip Str(−2, 3) outside the discs of radius 1 around 0, 1; and Re(z2 ) > 7/2;

Re(zj ) > 1

for j = 3, . . . , n − 1 .

Let D2 = subdomain of D1 satisfying the further inequality Re(z2 ) > 6. In terms of the variables z, we want to prove the functional equation X

(Yn ,...,Y2 )

n−1 Y

|Yj |−zj |Y2 |−z2 Λ2 (Y2 , z1 )

j=3

=

X

(Yn ,...,Y2 )

n−1 Y

|Yj |−zj |Y2 |−z1 −z2 +1/2 Λ2 (Y2 , 1 − z1 ) .

j=3

The series on both sides are convergent in D2 , so the formal argument is now justified, and we have proved that π −z1 Γ(z1 )E (n−1) (Y, z) is invariant under the equivalent transformations: z1 7→ 1 − z1 , z2 7→ z1 + z2 − 1/2, zj 7→ zj (j = 3, . . . , n − 1), sn 7→ sn , or transposition of s1 and s2 . This concludes the proof of Theorem 7.1. Remark. Just as Maass gave convergence criteria for Eisenstein series with more general parabolic groups [Maa 71], Sect. 7, he also gives the analytic continuation and functional equation for these more general groups at the end of Sect. 17, pp. 279–299.

Bibliography

[Ba 95] [Be 83] [Boc 52]

[Bor 69] [Bum 84] [Dri 97]

[Gin 64] [God 57] [Gre 88] [Gre 92] [Gre 93] [Gre 94] [Har 68] [Hel 62] [Hel 68] [Hel 77]

BALLMAN, W.: Lectures on Spaces of Nonpositive Curvature. Birkh¨ auser (1995). BENGSTON, T.: Bessel functions on Pn . Pacific J. Math. 108 (1983) 19–29. BOCHNER, S.: Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. S´ em. Math. Univ. Lund., Tome. suppl. dedicated to Marcel Riesz (1952), 12–20. BOREL, A.: Introduction aux groupes arithmétiques. Hermann (1969). BUMP, D.: Automorphic forms on GL(3, R). Lecture Notes in Math. 1083 Springer Verlag (1984). DRIVER, B. K.: Integration by parts and quasi-invariance for heat kernel measures on loop groups. J. Functional Analysis 149 (1997) 470– 547. GINDIKIN, S.: Analysis in homogeneous domains. Russian Math. Surveys 19 (1964) 1–90 GODEMENT, R.: Introduction aux travaux de Selberg. Séminaire Bourbaki (1957). GRENIER, D.: Fundamental domains for the general linear group. Pacific J. Math. 132 (1988) 293–317. GRENIER, D.: An analogue of Siegel’s phi-operator for automorphic forms for GLn (Z). Trans. AMS. 333 (1992) 463–477. GRENIER, D.: On the shape of fundamental domains in GL(n, R)/ O(n). Pacific J. Math. 160 (1993) 53–66. GRENIER, D.: Factoring L-functions as products of L-functions. Trans. AMS 345 (1994) 673–692. HARISH-CHANDRA.: Automorphic Forms on Semi-Simple Lie Groups: Notes by J. G. M. Mars. Lecture Notes in Math. 62 (1968). HELGASON, S.: Differential Geometry and Symmetric Spaces. Academic Press (1962). HELGASON, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press (1968). HELGASON, S.: Some results on eigenfunctions on symmetric spaces and eigenspace representations. Math. Scand. 41 (1977) 79–89.

164 [Hel 84]

Bibliography

HELGASON, S.: Groups and Geometric Analysis. Academic Press (1984). [Her 55] HERZ, C.: Bessel functions of matrix arguments. Ann. Math. 61 (1955) 474–523. [Hla 44] HLAWKA, E.: Zur Geometrie der Zahlen. Math. Zeitschr. 49 (1944) 285–312. ¨ [H¨ or 66] HORMANDER, L.: An introduction to complex analysis in several variables. VanNostrand, Princeton (1966). [ImT 82] IMAI, K., and TERRAS, A.: Fourier expansions of Eisenstein series for GL(3, Z). Trans. AMS 273 (1982) 679–694. [JoL 99] JORGENSON, J., and LANG, S.: Hilbert-Asai Eisenstein series, regularized products, and heat kernels. Nagoya Math. J. 153 (1999) 155– 188. [JoL 01] JORGENSON, J., and LANG, S.: Spherical Inversion on SLn (R). Springer-Verlag (2001). [kAR 65] KARPELEVIC, F. I.: The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trans. Moscow Math. Obsc. 14 (1965) 48–185; Trans. Moscow Math. Soc. (1965) 51– 199. [La 75/85] LANG, S.: SL2 (R). Addison-Wesley (1975); Springer-Verlag (1985). [La 93] LANG, S.: Real and Functional Analysis. Graduate Texts in Mathematics 142 Springer-Verlag (1993). [La 99] LANG, S.: Fundamentals of Differential Geometry. Springer-Verlag (1999). [Llds 76] LANGLANDS, R. P.: On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Math. 1083 Springer Verlag (1984). [Loo 69] LOOS, O.: Symmetric Spaces I and II. Benjamin (1969). [Maa 55] MAASS, H.: Die Bestimmung der Dirichletreihen mit Gr¨ ossen- charakteren zu den Modulformen n-ten Grades. J. Indian Math. Soc. 19 (1955) 1–23. [Maa 71] MAASS, H.: Siegel’s Modular Forms and Dirichlet Series. Lecture Notes in Math. 216 Springer Verlag (1971). [Min 1884] MINKOWSKI, H.: Grundlagen f¨ ur eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten. Mémoire Acadmie ´ des Sciences (1884). Collected Works I 3–144. ¨ [Min 05] MINKOWSKI, H.: Diskontinuit¨ atsbereich f¨ ur arithmetische Aquivalenz. J. reine angew. Math. 129 (1905) 270–274. Collected Works II 53–100. [Moo 64] MOORE, C.: Compactifications of symmetric spaces II: The Cartan domains. Amer. J. Math. 86 (1964) 358–378. [Mos 53] MOSTOW, D.: Some new decomposition theorems for semi-simple groups. Memoirs AMS (1953). [Nar 68] NARASIMHAN, R.: Analysis on Real and Complex Manifolds. North Holland (1968). [Sat 56] SATAKE, I.: Compaction des espaces quotients de Siegel I. Séminaire Cartan 1957–58, 3 March 1958, 12–01. [Sat 60] SATAKE, I.: On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. Math. 72 (1960) 555–580. [Sel 56] SELBERG, A.: Harmonic analysis and discontinuous groups. J. Indian Math. Soc. 20 (1956) 47–87.

Bibliography

165

¨ SIEGEL, C. L.: Uber die analytische theorie der quadratische Formen. [Sie 35] Ann. Math. 36 (1935) 527–606. [Sie 36] Ann. Math. 37 (1936) 230–263. [Sie 37] Ann. Math. 38 (1937) 212–291. ¨ [Sie 38] SIEGEL, C. L.: Uber die zeta funktionen indefiniter quadratischen Formen. [Sie 38] Ann. Math. 43 (1938) 682–708. [Sie 39] Ann. Math. 44 (1939) 398–426. [Sie 40] SIEGEL, C. L.: Einheiten quadratischer Formen. Abh. Math. Sem. Hansische Univ. 13 (1940) 209–239. [Sie 41] SIEGEL, C. L.: Equivalence of quadratic forms. Amer. J. Math. 63 (1941) 658–680. [Sie 43] SIEGEL, C. L.: Discontinuous groups. Ann. Math. 44 (1943) 674–689. [Sie 44a] SIEGEL, C. L.: On the theory of indefinite quadratic forms. Ann. Math. 45 (1944) 577–622. [Sie 44b] SIEGEL, C. L.: The average measure of quadratic forms with given determinant and signature. Ann. Math. 45 (1944) 667–685. [Sie 45] SIEGEL, C. L.: Some remarks on discontinuous groups. Ann. Math. 46 (1945) 708–718. [Sie 48] SIEGEL, C. L.: Indefinite quadratische Formen und Modulfunktionen. Courant Anniv. Volume (1948) 395–406. [Sie 51] SIEGEL, C. L.: Indefinite quadratische Formen und Funktionentheorie, I. Math. Ann. 124 (1951) 17–54; II, 364–387. [Sie 55/56] SIEGEL, C. L.: Lectures on Quadratic Forms. Tata Institute, Bombay (1955–56). [Sie 59] SIEGEL, C. L.: Zur Reduktionstheorie quadratischen Formen. Pub. Math. soc Japan (1959) Collected Papers #72, Volume III, 275–327. [Ter 80] TERRAS, A.: Integral formulas and integral tests for series of positive matrices Pacific J. Math. 89 (1980) 471–490. [Ter 85a] TERRAS, A.: The Chowla Selberg method for Fourier expansion of higher rank Eisenstein series. Canad. Math. Bull. 28 (1985) 280–294. [Ter 85b] TERRAS, A.: Harmonic Analysis on Symmetric Spaces and Applications, I. Springer-Verlag (1985). [Ter 88] TERRAS, A.: Harmonic Analysis on Symmetric Spaces and Applications, II. Springer-Verlag (1988). [ViT 82] VINOGRADOV, A., and TAKHTAZHAN, L.: Theory of Eisenstein series for the group SL(3, R) and its applications to a binary problem. J. Soviet Math. 18 (1982) 293–324. [Wal 73] WALLACH, N.: Harmonic Analysis on Homogeneous Spaces. Marcel Dekker (1973). [We 46] WEIL, A.: Sur quelques résultats de Siegel. Summa Braz. Math. 1 (1946) 21–39; Collected Papers I, Springer-Verlag (1979) 339–357.

[Sie 56]

Index

Adjointness formulas

102, 103

Fun conditions 7 Functional equation of Eisenstein 157, 160 Functional equation of theta 98, 149 Fundamental Domain 1, 6

Bengston Bessel function 58–73 Bessel function 58–73 Bessel-Fourier series 105 Chains of matrices 134 Changing variables 152 Character 50 Completed Lambda function 100 Convergence of Eisenstein series 129 d 50 D 148 Decomposition of Haar measure Determinant character 50 Dual lattice 97

Haar measure 25 Hecke zeta operator

25

EF HZ 141 Eigenfunction of Hecke zeta operator 140 Eigenvalue 89, 118, 119 Eisenstein series 107, 128, 133–135, 142–145, 151–154, 157, 160, 162 Eisenstein trace 108 Epstein zeta function 99–106 Equivalent chains of matrices 137 Estimate of Lambda function 107 First order Iwasawa decomposition 6 Fourier series 111–116 Fourier transform 70, 95 Full Iwasawa coordinates 125

Gamma function Γn 55 Gamma integral 55, 118 Gamma kernel 88 Gamma point pair invariant 88 Gamma transform 57 Grenier fundamental domain 6, 17

5,

140

Incomplete gamma integral 101 Inductive coordinates 14 Integral matrices 134 Invariant differential operators 90 Invariant differential operators and polynomials on A 90 Invariant polynomials 75, 91 Iwasawa coordinates 126 Iwasawa decomposition 2, 16 Jacobian

31

K-Bessel function

59

Λ-function 100–106 Lie algebra generators 84 Lower Bengston function 73 Maass Selberg generators

78

168

Index

Maass Selberg operators 80 Measure of fundamental domain 47 Measure on SPos 36 Mellin transform 55, 70 Metric 121, 122 Minkowski fundamental domain 7 Minkowski Measure of fundamental domain 47 Minkowski-Hlawka 44 Newton polynomials 77 Non-singular theta series 111 Normal decomposition 93 Normalized primitive chain 140 Parabolic subgroup 132 Partial determinant character 140 Partial Iwasawa decomposition 15, 114 Poisson formula 97 Polar coordinates 32 Polar Haar measure 33 Polynomial expression 81 Primitive chain 139 Primitive Eisenstein series 107, 128, 143 Projection on A 92 Radius of discreteness

125

Regularizing differential operator 148 Reversing matrix 51 Riemann zeta fudge factor 141

122,

Selberg Eisenstein series 143 Selberg power function 143 Siegel set 20, 23 Siegel’s formula 41 Standard coordinates 15 Strict fundamental domain 1 Subdeterminants 53 Theta series 102, 111 Trace 149 Trace scalar product 97 Transpose of differential operator Triangular coordinates 26 Triangularization 135, 138 Tubular neighborhood 92 Twists of theta series 99 Unipotent trace 108 Upper Bengston function Weight of polynomial Weyl group 75 Xi function

156–159

83

71

87

Posn̳(R) and Eisenstein series

Eisenstein Series and Applications

Eisenstein series and applications