DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
HIGHER ALGEBRAIC K -THEORY OF GROUP ACTIONS WITH FINITE STABILIZERS GABRIELE VEZZOSI and ANGELO VISTOLI
Abstract We prove a decomposition theorem for the equivariant K -theory of actions of affine group schemes G of finite type over a field on regular separated Noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition that holds, for example, for G abelian or smooth. We reduce the problem to the case of a GLn -action and finally to a split torus action. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. General constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Morphisms of actions and induced maps on K -theory . . . . . . . 2.2. The basic definitions and results . . . . . . . . . . . . . . . . . . 3. The main theorem: The split torus case . . . . . . . . . . . . . . . . . . 4. The main theorem: The case of G = GLn,k . . . . . . . . . . . . . . . . 5. The main theorem: The general case . . . . . . . . . . . . . . . . . . . 5.1. Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . 5.2. Hypotheses on G . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. Higher equivariant K -theory of Noetherian regular separated algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The purpose of this paper is to prove a decomposition theorem for the equivariant K -theory of actions of affine group schemes of finite type over a field on regular separated Noetherian algebraic spaces. Let X be a regular connected separated Noetherian scheme with an ample line bundle, and let K 0 (X ) be its Grothendieck ring of DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 27 April 2000. Revision received 13 June 2001. 2000 Mathematics Subject Classification. Primary 19E08, 14L30; Secondary 14A20. Authors’ work partially supported by the University of Bologna funds for selected research topics. 1
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vector bundles. Then the kernel of the rank morphism K 0 (X ) → Z is nilpotent (see [SGA6, Exp. VI, Th. 6.9]), so the ring K 0 (X ) is indecomposable and remains such after tensoring with any indecomposable Z-algebra. The situation is quite different when we consider the equivariant case. Let G be an algebraic group acting on a Noetherian separated regular scheme, or algebraic space, let X be over a field k, and consider the Grothendieck ring K 0 (X, G) of G-equivariant perfect complexes. This is the same as the Grothendieck group of G-equivariant coherent sheaves on X , and it coincides with the Grothendieck ring of G-equivariant vector bundles if all G-coherent sheaves are quotients of locally free coherent sheaves (which is the case, e.g., when G is finite or smooth and X is a scheme). Assume that the action of G on X is connected, that is, that there are no nontrivial invariant open and closed subschemes of X . Still, K 0 (X, G) usually decomposes, after inverting some primes; for example, if G is a finite group and X = Spec C, then K 0 (X, G) is the ring of complex representations of G, which becomes a product of fields after tensoring with Q. In [Vi2] the second author analyzes the case where the action of G on X has finite reduced geometric stabilizers. Consider the ring of representations R(G), and consider the kernel m of the rank morphism rk : K 0 (X, G) → Z. Then K 0 (X, G) is an R(G)-algebra; he shows that the localization morphism K 0 (X, G) ⊗ Q −→ K 0 (X, G)m is surjective and that the kernel of the rank morphism K 0 (X, G)m ⊗ Q → Q is nilpotent. Furthermore, he conjectures that K 0 (X, G) ⊗ Q splits as a product of the localization K 0 (X, G)m and some other ring, and he formulates a conjecture about what the other factor ring should be when G is abelian and the field is algebraically closed of characteristic zero. The proofs of the results in [Vi2] depend on an equivariant Riemann-Roch theorem, whose proof was never published by the author; however, all of the results have been proved and generalized in [EG]. The case where G is a finite group is studied in [Vi1]. Assume that k contains all nth roots of 1, where n is the order of the group G. Then the author shows that, after inverting the order of G, the K -theory ring K ∗ (X, G) of G-equivariant vector bundles on X (which is assumed to be a scheme in that paper) is canonically the product of a finite number of rings, expressible in terms of ordinary K -theory of L appropriate subschemes of fixed points of X . Here K ∗ (X, G) = i K i (X, G) is the graded higher K -theory ring. The precise formula is as follows. Let σ be a cyclic subgroup of G whose order is prime to the characteristic of k; then the subscheme X σ of fixed points of X under the actions of σ is also regular. The representation ring R(σ ) is isomorphic to the ring Z[t]/(t n −1), where t is a generator of the group of characters hom(σ, k ∗ ). We call e R(σ ) the quotient of the ring R(σ ) by the ideal generated by the element 8n (t), where 8n is the nth cyclotomic polynomial;
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this is independent of t. The ring e R(σ ) is isomorphic to the ring of integers in the nth cyclotomic field. Call NG (σ ) the normalizer of σ in G; the group NG (σ ) acts on the scheme X σ and, by conjugation, on the group σ. Consider the induced actions of NG (σ ) on the K -theory ring K ∗ (X σ ) and on the ring e R(σ ). Choose a set C (G) of representatives for the conjugacy classes of cyclic subgroups of G whose order is prime to the characteristic of the field. The statement of the main result of [Vi1] is as follows. THEOREM
There is a canonical ring isomorphism Y K ∗ (X, G) ⊗ Z[1/|G|] '
K ∗ (X σ ) ⊗ e R(σ )
NG (σ )
⊗ Z[1/|G|].
σ ∈C (G)
In the present paper we generalize this decomposition to the case in which G is an algebraic group scheme of finite type over a field k, acting with finite geometric stabilizers on a Noetherian regular separated algebraic space X over k. Of course, we cannot expect a statement exactly like the one for finite groups, expressing equivariant K -theory simply in terms of ordinary K -theory of the fixed point sets. For example, when X is the Stiefel variety of r -frames in n-space, then the quotient of X by the natural free action of GLr is the Grassmannian of r -planes in n-space, and K 0 (X, GLr ) = K 0 (X/ GLr ) is nontrivial, while K 0 (X ) = Z. Let X be a Noetherian regular algebraic space over k with an action of an affine group scheme G of finite type over k. We consider the Waldhausen K -theory group K ∗ (X, G) of complexes of quasi-coherent G-equivariant sheaves on X with coherent bounded cohomology. This coincides on the one hand with the Waldhausen K -theory group K ∗ (X, G) of the subcategory of complexes of quasi-coherent G-equivariant flat sheaves on X with coherent bounded cohomology (and hence has a natural ring structure given by the total tensor product) and on the other hand with the Quillen group K ∗0 (X, G) of coherent equivariant sheaves on X ; furthermore, if every coherent equivariant sheaf on X is the quotient of a locally free equivariant coherent sheaf, it also coincides with the Quillen group K ∗naive (X, G) of coherent locally free equivariant sheaves on X . These K -theories and their relationships are discussed in the appendix. Our result is as follows. First we have to see what plays the role of the cyclic subgroups of a finite group. This is easy; the group schemes whose rings of representations are of the form Z[t]/(t n − 1) are not the cyclic groups, in general, but their Cartier duals, that is, the group schemes that are isomorphic to the group scheme µn of nth roots of 1 for some n. We call these group schemes dual cyclic. If σ is a dual cyclic group, we can define e Rσ as before. A dual cyclic subgroup σ of G is called
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essential if X σ 6 = ∅. The correct substitute for the ordinary K -theory of the subspaces of invariants is the geometric equivariant K -theory K ∗ (X, G)geom , which is defined as follows. Call N the least common multiple of the orders of all the essential dual cyclic subgroups of G. Call S1 the multiplicative subset of the ring R(G) consisting of elements whose virtual rank is a power of N ; then K ∗ (X, G)geom is the localization S1−1 K ∗ (X, G). Notice that K ∗ (X, G)geom ⊗ Q = K ∗ (X, G)m , with the notation above. Moreover, if every coherent equivariant sheaf on X is the quotient of a locally free equivariant coherent sheaf, by [EG], we have an isomorphism of rings K 0 (X, G)geom ⊗ Q = A∗G (X ) ⊗ Q, where A∗G (X ) denotes the direct sum of G-equivariant Chow groups of X. We prove the following. Assume that the action of G on X is connected. Then the kernel of the rank morphism K 0 (X, G)geom → Z[1/N ] is nilpotent (see Cor. 5.2). This is remarkable; we have made what might look like a small step toward making the equivariant K -theory ring indecomposable, and we immediately get an indecomposable ring. Indeed, K ∗ (X, G)geom “feels like” the K -theory ring of a scheme; we want to think of K ∗ (X, G)geom as what the K -theory of the quotient X/G should be, if X/G were smooth, after inverting N (see Conj. 5.8). Furthermore, consider the centralizer CG (σ ) and the normalizer NG (σ ) of σ inside G. The quotient wG (σ ) = NG (σ )/CG (σ ) is contained inside the group scheme of automorphisms of σ , which is a discrete group, so it is also a discrete group. It acts on e R(σ ), by conjugation, and it also acts on the equivariant K -theory ring K ∗ (X σ , CG (σ )) and on the geometric equivariant K -theory ring K ∗ (X σ , CG (σ ))geom (see Cor. 2.5). We say that the action of G on X is sufficiently rational when the following two conditions are satisfied. Let k be the algebraic closure of k. (1) Each essential dual cyclic subgroup σ ⊆ G k is conjugate by an element of G( k ) to a dual cyclic subgroup of G. (2) If two essential dual cyclic subgroups of G are conjugate by an element of G( k ), they are also conjugate by an element of G(k). Obviously, every action over an algebraically closed field is sufficiently rational. Furthermore, if G is GLm , SLm , Spm , or a totally split torus, then any action of G is sufficiently rational over an arbitrary base field (see Prop. 2.3). If G is a finite group, then the action is sufficiently rational when k contains all nth roots of 1, where n is the least common multiple of the orders of the cyclic subgroups of k of order prime to the characteristic, whose fixed point subscheme is nonempty. Denote by C (G) a set of representatives for essential dual cyclic subgroup schemes, under conjugation by elements of the group G(k). Here is the statement of our result.
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MAIN THEOREM
Let G be an affine group scheme of finite type over a field k, acting on a Noetherian separated regular algebraic space X . Assume the following three conditions. (a) The action has finite geometric stabilizers. (b) The action is sufficiently rational. (c) For any essential cyclic subgroup σ of G, the quotient G/CG (σ ) is smooth. Then C (G) is finite, and there is a canonical isomorphism of R(G)-algebras Y w (σ ) K ∗ (X, G) ⊗ Z[1/N ] ' K ∗ (X σ , CG (σ ))geom ⊗ e R(σ ) G . σ ∈C (G)
Conditions (a) and (b) are clearly necessary for the theorem to hold. We are not sure about (c). It is rather mild, as it is satisfied, for example, when G is smooth (this is automatically true in characteristic zero) or when G is abelian. A weaker version of condition (c) is given in Section 5.2. In the case when G is abelian over an algebraically closed field of characteristic zero, the main theorem implies [Vi2, Conj. 3.6]. When G is a finite group, and the base field contains enough roots of 1, as in the statement of Theorem 1, then the conditions of the main theorem are satisfied; since the natural maps K ∗ (X σ , CG (σ ))geom → K ∗ (X σ )CG (σ ) become isomorphisms after inverting the order of G (see Prop. 5.7), the main theorem implies [Vi1, Th. 1]. However, the proof of the main theorem here is completely different from [Vi1, proof of Th. 1]. As B. Toen pointed out to us, a weaker version of our main theorem (with Qcoefficients and assuming G smooth, acting with finite reduced stabilizers) follows from his [To1, Th. 3.15]; the e´ tale techniques he uses in proving this result make it impossible to avoid tensoring with Q (see also [To2]). Here is an outline of the paper. First we define the homomorphism (see Sec. 2.2). Next, in Section 3, we prove the result when G is a totally split torus. Here the basic tool is the result of R. Thomason, which gives a generic description of the action of a torus on a Noetherian separated algebraic space, and we prove the result by Noetherian induction, using the localization sequence for the K -theory of equivariant coherent sheaves. As in [Vi1], the difficulty here is that the homomorphism is defined via pullbacks, and thus it does not commute with the pushforwards intervening in the localization sequence. This is solved by producing a different isomorphism between the two groups in question, using pushforwards instead of pullbacks, and then relating this to our map, via the self-intersection formula. The next step is to prove the result in the case when G = GLn ; here the key point is a result of A. Merkurjev which links the equivariant K -theory of a scheme with a GLn -action to the equivariant K -theory of the action of a maximal torus. This is carried out in Section 4. Finally (see Sec. 5), we reduce the general result to the case
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of GLn , by considering an embedding G ⊆ GLn , and the induced action of GLn on Y = GLn ×G X . It is at this point that condition (c) enters, allowing a clear description of Y σ where σ is an essential dual cyclic subgroup of G (see Prop. 5.6).
2. General constructions Notation. If S is a separated Noetherian scheme, X is a Noetherian separated Salgebraic space (which is most of the time assumed to be regular), and G is a flat affine group scheme over S acting on X , we denote by K ∗ (X, G) (resp., K 0 (X, G)) the Waldhausen K -theory of the complicial bi-Waldhausen (see [ThTr]) category W1,X,G of complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology (resp., the Quillen K -theory of G-equivariant coherent O X -modules). As shown in the appendix, if X is regular, K ∗ (X, G) is isomorphic to K ∗0 (X, G) and has a canonical graded ring structure. When X is regular, the isomorphism K ∗ (X, G) ' K ∗0 (X, G) then allows us to switch between the two theories when needed. 2.1. Morphisms of actions and induced maps on K -theory Let S be a scheme. By an action over S we mean a triple (X, G, ρ) where X is an S-algebraic space, G is a group scheme over S, and ρ : G × S X → X is an action of G on X over S. A morphism of actions ( f, φ) : (X, G, ρ) −→ (X 0 , G 0 , ρ 0 ) is a pair of S-morphisms f : X → X 0 and φ : G → G 0 , where φ is a morphism of S-group schemes, such that the following diagram commutes: ρ
G × S X −−−−→ φ× f y
X f y
G 0 × S X 0 −−−− → X0 0 ρ
Equivalently, f is required to be G-equivariant with respect to the given G-action on X and the G-action on X 0 induced by composition with φ. A morphism of actions ( f, φ) : (X, G, ρ) → (X 0 , G 0 , ρ 0 ) induces an exact functor ( f, φ)∗ : W3,X 0 ,G 0 → W3,X,G , where W3,Y,H denotes the complicial biWaldhausen category of complexes of H -equivariant flat quasi-coherent modules with bounded coherent cohomology on the H -algebraic space Y (see appendix). Let (E ∗ , ε∗ ) be an object of W3,X 0 ,G 0 ; that is, E ∗ is a complex of G 0 -equivariant flat quasicoherent O X 0 -modules with bounded coherent cohomology, and for any i, i εi : pr0∗ g ρ 0∗ E i 2 E −→
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is an isomorphism satisfying the usual cocycle condition. Here pr02 : G 0 × S X 0 → X 0 denotes the obvious projection, and similarly for pr2 : G × S X → X . Since ρ ∗ f ∗ E ∗ = ( fρ)∗ E ∗ = (φ × f )∗ ρ 0∗ E ∗ and ∗ pr∗2 f ∗ E ∗ =(φ × f )∗ pr0∗ 2E ,
. we define ( f, φ)∗ (E ∗ , ε∗ ) = ( f ∗ E ∗ , (φ × f )∗ (ε∗ )) ∈ W3,X,G (the cocycle condition for each (φ × f )∗ (εi ) following from the same condition for εi ); ( f, φ)∗ is then defined on morphisms in the only natural way. Now, ( f, φ)∗ is an exact functor and, if X and X 0 are regular so that the Waldhausen K -theory of W3,X,G (resp., of W3,X 0 ,G 0 ) coincides with K ∗ (X, G) (resp., K ∗ (X 0 , G 0 )) (see appendix), it defines a ring morphism ( f, φ)∗ : K ∗ (X 0 , G 0 ) −→ K ∗ (X, G). A similar argument shows that if f is flat, ( f, φ) induces a morphism ( f, φ)∗ : K ∗0 (X 0 , G 0 ) −→ K ∗0 (X, G). Example 2.1 Let G and H be group schemes over S, and let X be an S-algebraic space. Moreover, suppose that (1) G and H act on X ; (2) G acts on H by S-group scheme automorphisms (i.e., it is given a morphism G → Aut(GrSch)/S (H ) of group functors over S); (3) the two preceding actions are compatible; that is, for any S-scheme T , any g ∈ G(T ), h ∈ H (T ), and x ∈ X (T ), we have g · (h · x) = h g · (g · x), where (g, h) 7→ h g denotes the action of G(T ) on H (T ). If g ∈ G(S) and if gT denotes its image via G(S) → G(T ), let us define a morphism of actions ( f g , φg ) : (X, H ) → (X, H ) as f g (T ) : X (T ) −→ X (T ) : x 7 −→ gT · x, φg (T ) : H (T ) −→ H (T ) : h 7 −→ h gT . This is an isomorphism of actions and induces an action of the group G(S) on K ∗0 (X, H ) and on K ∗ (X, H ). This applies, in particular, to the case where X is an algebraic space with a G action and G B H , G acting on H by conjugation.
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2.2. The basic definitions and results Let G be a linear algebraic k-group scheme G acting with finite geometric stabilizers on a regular Noetherian separated algebraic space X over k. We denote by R(G) the representation ring of G. A (Cartier) dual cyclic subgroup of G over k is a k-subgroup scheme σ ⊆ G such that there exist an n > 0 and an isomorphism of k-groups σ ' µn,k . If σ, ρ are dual cyclic subgroups of G and if L is an extension of k, we say that σ and ρ are conjugate over L if there exists g ∈ G(L) such that gσ(L) g −1 = ρ(L) (where . H(L) = H ×Spec k Spec L, for any k-group scheme H ) as L-subgroup schemes of G (L) . A dual cyclic subgroup σ ⊆ G is said to be essential if X σ 6= ∅. We say that the action of G on X is sufficiently rational if (1) any two essential dual cyclic subgroups of G are conjugated over k if and only if they are conjugated over an algebraic closure k of k; (2) any essential dual cyclic subgroup ρ of G (k) is conjugated over k to a dual cyclic subgroup of the form σ(k) where σ ⊆ G is (essential) dual cyclic. We denote by C (G) a set of representatives for essential dual cyclic subgroups of G with respect to the relation of conjugacy over k. Remark 2.2 Note that if the action is sufficiently rational and if ρ, σ are essential dual cyclic subgroups of G which are conjugate over an algebraically closed extension of k, then they are also conjugate over k. 2.3 Any action of GLn , SLn , Sp2n , or of a split torus is sufficiently rational. PROPOSITION
Proof If G is a split torus, condition (1) is clear because G is abelian, while it follows from the rigidity of diagonalizable groups that any subgroup scheme of G k is in fact defined over k. Let σ ⊆ GLm be a dual cyclic subgroup. Since σ is diagonalizable, we have an eigenspace decomposition M V = km = Vχσ χ ∈b σ
such that the χ with Vχ 6 = 0 generate b σ . Conversely, given a cyclic group C and a decomposition M V = Vχ b χ ∈C
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b there is a corresponding embedding of such that the χ with Vχ 6= 0 generate C, the Cartier dual σ of C into GLn with Vχ = Vχσ for each χ ∈ C = b σ . Now, if σ ⊆ GLm,k is a dual cyclic subgroup defined over k, we can apply an element of GLm (k) to make the Vχ defined over k, and then gσ g −1 is defined over k. If σ ⊆ GLm and τ ⊆ GLm are dual cyclic subgroups that are conjugate over k, pick an element of GLm (k) sending σ to τ . This induces an isomorphism φ : σk ' τk , which by rigidity is defined over k. Then if χ and χ 0 are characters that correspond under the isomorphism of b σ and b τ induced by φ, then the dimension of Vχσ is equal to the τ dimension of Vχ 0 , so we can find an element g of GLm which carries each Vχσ onto the τ ; conjugation by this element carries σ onto τ . For SL , the proof corresponding Vχ0 m is very similar if we remark that to give a dual cyclic subgroup σ ⊆ SLm ⊆ GLm corresponds to giving a decomposition M V = km = Vχσ χ ∈b σ
Q dim Vχσ such that the χ with Vχσ 6 = 0 generate b σ , with the condition χ∈b =1∈b σ. σ χ For Spm ⊆ GL2m , a dual cyclic subgroup σ ⊆ Spm gives a decomposition M V = k 2m = Vχσ χ∈b σ
such that the χ with Vχσ 6 = 0 generate b σ , with the condition that for v ∈ Vχσ and 0 σ v ∈ Vχ0 the symplectic product of v and v 0 is always zero, unless χχ 0 = 1 ∈ b σ . Both conditions then follow rather easily from the fact that any two symplectic forms over a vector space are isomorphic. Let N(G,X ) denote the least common multiple of the orders of essential dual cyclic subgroups of G. Notice that N(G,X ) is finite: since the action has finite stabilizers, the group scheme of stabilizers is quasi-finite over X ; therefore the orders of the stabilizers of the geometric points of X are globally bounded. . We define 3(G,X ) = Z[1/N(G,X ) ]. If H ⊆ G is finite, we also write 3 H for Z[1/|H |]. Note that, if σ ⊆ G is dual cyclic, then 3σ = 3(σ,Spec k) , and if, moreover, σ is essential, 3σ ⊆ 3(G,X ) . If H ⊆ G is a subgroup scheme and if A is a ring, we write R(H ) A for R(H ) ⊗Z A. We denote by rk H : R(H ) −→ Z and by rk H,3(G,X ) : R(H )3(G,X ) −→ 3(G,X ) the rank ring homomorphisms. We let K ∗0 (X, G)3(G,X ) = K ∗0 (X, G) ⊗ 3G,X and K ∗ (X, G)3(G,X ) = K ∗ (X, G) ⊗ 3G,X
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(for the notation, see the beginning of this section). Recall that K ∗ (X, G)3(G,X ) is an R(G)-algebra via the pullback R(G) ' K 0 (Spec k, G) → K 0 (X, G) and that K ∗ (X, G) ' K ∗0 (X, G) since X is regular (see appendix). If σ is a dual cyclic subgroup of G of order n, the choice of a generator t for the . dual group b σ = HomGrSch/k (σ, Gm,k ) determines an isomorphism R(σ ) '
Z[t] . (t n − 1)
Let pσ be the canonical ring surjection Y Z[t], Z[t] −→ , (t n − 1) (8d ) d|n
and let f pσ be the induced surjection Z[t] Z[t] −→ , (t n − 1) (8n ) where 8d is the dth cyclotomic polynomial. If mσ is the kernel of the composition R(σ ) '
Z[t] Z[t] −→ , − 1) (8n )
(t n
the quotient ring R(σ )/mσ does not depend on the choice of the generator t for b σ. Notation. We denote by e R(σ ) the quotient R(σ )/mσ . We remark that if σ is dual cyclic of order n and if t is a generator of b σ , there are isomorphisms Y 3σ [t] 3σ [t] R(σ )3σ ' n ' . (1) (t − 1) (8d ) d|n
Let π f R(σ )3(G,X ) be the canonical ring homomorphism. The σ : R(G)3(G,X ) → e σ -localization K ∗0 (X, G)σ of K ∗0 (X, G)3(G,X ) is the localization of the R(G)3(G,X ) . −1 (1). The σ -localizaf module K ∗0 (X, G)3(G,X ) at the multiplicative subset Sσ = π σ tion K ∗ (X, G)σ is defined in the same way. If H ⊆ G is a subgroup scheme, we also write R(H )σ for Sσ−1 (R(H )3(G,X ) ). If σ is the trivial group, we denote by K ∗0 (X, G)geom the σ -localization of K ∗0 (X, G)3(G,X ) and call it the geometric part or geometric localization of K ∗0 (X, G)3(G,X ) . Note that πe1 coincides with the rank morphism rkG,3(G,X ) : R(G)3(G,X ) −→ 3(G,X ) . We have the same definition for K ∗ (X, G)geom . Let NG (σ ) (resp., CG (σ ) ⊆ NG (σ )) be the normalizer (resp., the centralizer) of σ in G; since Aut(σ ) is a finite constant group scheme, . NG (σ ) WG (σ ) = CG (σ )
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is also a constant group scheme over k associated to a finite group wG (σ ). 2.4 Let H be a k-linear algebraic group, let σ ' µn,k be a normal subgroup, and let Y be an algebraic space with an action of H/σ . Then there is a canonical action of w H (σ ) on K ∗0 (Y, C H (σ )). LEMMA
Proof Let us first assume that the natural map H (k) −→ w H (σ )
(2)
is surjective (which is true, e.g., if k is algebraically closed). Since C H (σ )(k) acts trivially on K ∗0 (Y, C H (σ )) and, by Example 2.1, H (k) acts naturally on K ∗0 (Y, C H (σ )), we may use (2) to define the desired action. In general, (2) is not surjective and we proceed as follows. Suppose that we can find a closed immersion of k-linear algebraic groups H ,→ H 0 such that (i) σ is normal in H 0 ; (ii) H 0 /C H 0 (σ ) ' W H (σ ); (iii) H 0 (k) → w H (σ ) is surjective. Consider the open and closed immersion Y × C H (σ ) ,→ Y × H ; this induces an open and closed immersion Y ×C H (σ ) C H 0 (σ ) ,→ Y ×C H (σ ) H 0 whose composition with the e´ tale cover Y ×C H (σ ) H 0 → Y × H H 0 is easily checked (e.g., on geometric points) to be an isomorphism. Therefore, K ∗0 Y × H H 0 , C H 0 (σ ) ' K ∗0 Y ×C H (σ ) C H 0 (σ ), C H 0 (σ ) ' K ∗0 Y, C H (σ ) , where the last isomorphism is given by the Morita equivalence theorem (see [Th3, Prop. 6.2]). By (i) and (iii) we can apply the argument at the beginning of the proof and get an action of w H (σ ) on K ∗0 (Y × H H 0 , C H 0 (σ )) and therefore on K ∗0 (Y, C H (σ )), as desired. It is not difficult to check that this action does not depend on the chosen immersion H ,→ H 0 . Finally, let us prove that there exists a closed immersion H ,→ H 0 satisfying conditions (i) – (iii) above. First choose a closed immersion j : H ,→ GLn,k for some n. Clearly, H/C H (σ ) ,→ GLn,k /CGLn,k (σ ), and, embedding σ in a maximal torus of GLn,k , it is easy to check that GLn,k (k) → GLn,k /CGLn,k (σ ) (k) is surjective. Now define H 0 as the inverse image of H/C H (σ ) in the normalizer NGLn,k (σ ).
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COROLLARY 2.5 There is a canonical action of wG (σ ) on K ∗0 (X σ , CG (σ )) which induces an action on K ∗0 (X σ , CG (σ ))geom .
Proof Since CG (σ ) = C NG (σ ) (σ ), Lemma 2.4, applied to Y = X σ (resp., Y = Spec k) and H = N G (σ ), yields an action of wG (σ ) on K ∗0 X σ , CG (σ ) resp., on K 0 (Spec k, CG (σ )) = R(CG (σ )) . The multiplicative system S1 = rk−1 (1) is preserved by this action so that there is an induced action on the ring S1−1 R(CG (σ )). The pullback K 0 Spec k, CG (σ ) → K 0 X σ , CG (σ ) is wG (σ )-equivariant, and then wG (σ ) acts on K ∗0 (X σ , CG (σ ))geom . Remark 2.6 If Y is regular, Lemma 2.4 also gives an action of w H (σ ) on K ∗ (Y, C H (σ )) since K ∗ (Y, C H (σ )) ' K ∗0 (Y, C H (σ )) (see appendix). In particular, since by [Th5, Prop. 3.1], X σ is regular, Corollary 2.5 still holds for K ∗ (X σ , CG (σ ))geom . Note also that the embedding of k-group schemes WG (σ ) ,→ Autk (σ ) induces, by Example 2.1, an action of wG (σ ) on K 0 (Spec k, σ ) = R(σ ). The product in σ induces a morphism of k-groups, σ × CG (σ ) −→ CG (σ ), which in its turn induces a morphism m σ : K ∗ X σ , CG (σ ) −→ K ∗ X σ , σ × CG (σ ) . LEMMA 2.7 If H ⊆ G is a subgroup scheme and if σ is contained in the center of H , there is a canonical ring isomorphism
K ∗ (X σ , σ × H ) ' K ∗ (X σ , H ) ⊗ R(σ ). Proof Since σ acts trivially on X σ , we have an equivalence (see [SGA3, Exp. I, par. 4.7.3]) M (σ × H ) − Coh X σ ' (H − Coh X σ ) (3) b σ
(where b σ is the character group of σ ) which induces an isomorphism K ∗0 (X σ , σ × H ) ' K ∗0 (X σ , H ) ⊗ R(σ ).
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
13
We conclude since K ∗ (Y, H ) ' K ∗0 (Y, H ) and K ∗ (X σ , σ × H ) ' K ∗0 (X σ , σ × H ) (see appendix). . For any essential dual cyclic subgroup σ ⊆ G, let 3 = 3(G,X ) , and consider the composition K ∗ (X, G)3 → K ∗ X, CG (σ ) 3 → K ∗ X σ , CG (σ ) 3 mσ −→ K ∗ X σ , CG (σ ) 3 ⊗3 R(σ )3 −→ K ∗ X σ , CG (σ ) geom ⊗3 e R(σ )3 , (4) where the first map is induced by group restriction, the last one is the geometric localization map tensored with the projection R(σ )3 → e R(σ )3 , and we have used Lemma 2.7 with H = CG (σ ); the second map is induced by restriction along X σ ,→ X which is a regular closed immersion (see [Th5, Prop. 3.1]) and therefore has finite Tor-dimension, so that the pullback on K -groups is well defined (see appendix). It is not difficult to show that the image of (4) is actually contained in the invariant submodule w (σ ) K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3 G , so that we get a map ψσ,X : K ∗ (X, G)3 −→ K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3
wG (σ )
.
Our basic map is . 9 X,G =
Y
ψσ,X : K ∗ (X, G)3
σ ∈C (G)
−→
Y
K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3
wG (σ )
. (5)
σ ∈C (G)
Note that 9 X,G is a morphism of R(G)-algebras as a composition of morphisms of R(G)-algebras. The following technical lemma is used in Propositions 3.5 and 4.6. LEMMA 2.8 Let G be a linear algebraic k-group acting with finite stabilizers on a Noetherian . separated k-algebraic space X , and let 3 = 3(G,X ) . Let H ⊆ G be a subgroup, and let σ be an essential dual cyclic subgroup contained in the center of H . Consider the composition
K ∗0 (Y σ , H )3 −→ K ∗0 (Y σ , H )3 ⊗3 R(σ )3 −→ K ∗0 (Y σ , H )geom ⊗3 e R(σ )3 , (6)
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VEZZOSI and VISTOLI
where the first morphism is induced by the product morphism σ × H → H (recall Lem. 2.7) and the second is the tensor product of the geometric localization morphism with the projection R(σ )3 → e R(σ )3 . Then (6) factors through K ∗0 (Y σ , H )3 → K ∗0 (Y σ , H )σ , yielding a morphism θ H,σ : K ∗0 (Y σ , H )σ −→ K ∗0 (Y σ , H )geom ⊗3 e R(σ )3 .
(7)
Proof Let S1 (resp., Sσ ) be the multiplicative subset in R(H )3 consisting of elements going to 1 via the homomorphism rk H,3 : R(H )3 → 3 (resp., R(H )3 → e R(σ )3 ). Ob0 σ 0 σ e serve that K ∗ (X , H )3 ⊗3 R(σ )3 (resp., K ∗ (X , H )geom ⊗3 R(σ )3 ) is canonically an R(H )3 ⊗ R(σ )3 -module (resp., an S1−1 R(H )3 ⊗ e R(σ )3 -module) and therefore an R(H )-module via the coproduct ring morphism 1σ : R(H )3 −→ R(H )3 ⊗ R(σ )3 resp., via the ring morphism 1σ f σ : R(H )3 −→ R(H )3 ⊗ R(σ )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 . If we denote by A0 the R(H )3 -algebra f σ : R(H )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 , it is enough to show that the localization homomorphism lσ0 : A0 −→ Sσ−1 (A0 ) is an isomorphism, because in this case the morphism (7) is induced by the Sσ localization of (6). Let A denote the R(H )3 -algebra λ1 ⊗ 1 : R(H )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 , where λ1 : R(H )3 → S1−1 R(H )3 denotes the localization homomorphism. It is a well-known fact that there is an isomorphism of R(H )3 -algebras ϕ : A0 → A; this is exactly the dual assertion to “the action H × σ → σ is isomorphic to the projection on the second factor H × σ → σ .” Therefore, we have a commutative diagram A0 lσ0 y
ϕ
−−−−→
A l yσ
Sσ−1 A0 −−−−→ Sσ−1 A Sσ−1 ϕ
where lσ denotes the localization homomorphism, and it is enough to prove that lσ is an isomorphism. To see this, note that the ring e R(σ )3 is a free 3-module of finite rank (equal to φ(|σ |), φ being the Euler function), and there is a norm homomorphism N:e R(σ )3 −→ 3
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
15
sending an element to the determinant of the 3-endomorphism of e R(σ )3 induced by multiplication by this element; obviously, we have ∗ N−1 (3∗ ) = e R(σ )3 . Analogously, there is a norm homomorphism N0 : A0 = S1−1 R(H )3 ⊗ e R(σ )3 −→ S1−1 R(H )3 , and ∗ N−1 (S1−1 R(H )3 )∗ = S1−1 R(H )3 ⊗ e R(σ )3 . There is a commutative diagram N0
S1−1 R(H )3 ⊗ e R(σ )3 −−−−→ S1−1 R(H )3 rk rk H,3 ⊗ idy y H,3 e R(σ )3
−−−−→ N
3
−1 ∗ ∗ and, by definition of S1 , we get rk−1 H,3 (3 ) = (S1 R(H )3 ) . Therefore, by definition of Sσ , Sσ /1 consist of units in A, and we conclude the proof of the lemma.
The following lemma, which is an easy consequence of a result of Merkurjev, is the main tool in reducing the proof of the main theorem from G = GLn,k to its maximal torus T . 2.9 Let X be a Noetherian separated algebraic space over k with an action of a split reductive group G over k such that π1 (G) (see [Me, Par. 1.1]) is torsion free. Then if T denotes a maximal torus in G, the canonical morphism LEMMA
K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Proof Let B ⊇ T be a Borel subgroup of G. Since R(B) ' R(T ) and K ∗0 (X, B) ' K ∗0 (X, T ) (see [Th4, proof of Th. 1.13, p. 594]), by [Me, Prop. 4.1], the canonical ring morphism K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism.
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Since Merkurjev states his theorem for a scheme, we briefly indicate how it extends to a Noetherian separated algebraic space X over k. By [Th1, Lem. 4.3], there exists an open dense G-invariant separated subscheme U ⊂ X . Since Merkurjev’s map commutes with localization, by the localization sequence and Noetherian induction it is enough to know the result for U . And this is given in [Me, Prop. 4.1]. Note that by [Me, Prop. 1.22], R(T ) is flat (actually free) over R(G), and therefore the localization sequence remains exact after tensoring with R(T ). The following is [Vi1, Lem. 3.2]. It is used frequently in the rest of the paper, and it is stated here for the convenience of the reader. LEMMA 2.10 Let W be a finite group acting on the left on a set A , and let B ⊆ A be a set of representatives for the orbits. Assume that W acts on the left on a product of abelian Q groups of the type α∈A Mα in such a way that
s Mα = Msα for any s ∈ W . For each α ∈ B , let us denote by Wα the stabilizer of α in W . Then the canonical projection Y Y Mα −→ Mα α∈A
α∈B
induces an isomorphism Y
Mα
W
−→
α∈A
Y
(Mα )Wα .
α∈B
3. The main theorem: The split torus case In this section, T is a split torus over k. PROPOSITION 3.1 Let T 0 ⊂ T be a closed
subgroup scheme (diagonalizable, by [SGA3, Exp. IX, par. 8.1]), finite over k. Then the canonical morphism Y e δ : R(T 0 )3T 0 −→ R(σ )3T 0 σ dual cyclic σ ⊆T 0
is a ring isomorphism. Proof Q Since both R(T 0 )3T 0 and e R(σ )3T 0 are free 3T 0 -modules of finite rank, it is enough
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
17
to prove that, for any nonzero prime p - |T 0 |, the induced morphism of F p -vector spaces Y e R(T 0 )3T 0 ⊗Z F p −→ R(σ )3T 0 ⊗Z F p (8) σ dual cyclic σ ⊆T 0
is an isomorphism. Now, for any finite abelian group A, we have an equality |A| = P AC ϕ(|C|), where ϕ denotes the Euler function, |H | denotes the order of the group H , and the sum is extended to all cyclic quotients of A; applying this to the group of characters Tb0 (so that the corresponding cyclic quotients C are exactly the group of characters b σ for σ dual cyclic subgroups of T 0 ), we see that the ranks of both sides in (8) coincide with |T 0 |, and it is then enough to prove that (8) is injective. Define a morphism Y Y e f : R(σ )3T 0 R(τ )3T 0 −→ τ dual cyclic τ ⊆T 0
σ dual cyclic σ ⊆T 0
of R(T 0 )3T 0 -modules by requiring, for any dual cyclic subgroup σ ⊆ T 0 , the commutativity of the following diagram: R(τ )3T 0 τ dual cyclic e τ ⊆T 0
Q
f
−−−−→
σ dual cyclic R(σ )3T 0 σ ⊆T 0
Q
Prσ y Q
τ ⊆σ
pr y σ
e R(τ )3T 0
R(σ )3T 0
]
←−−−− ϕ
where Prσ and prσ are the obvious projections and ϕ is the isomorphism Q
τ ⊆σ
resστ
R(σ )3T 0 −−−−−−→
Y τ ⊆σ
(e prτ )τ
R(τ )3T 0 −−−→
Y τ ⊆σ
e R(τ )3T 0
induced by (1). Obviously, f ◦ δ coincides with the map Y Y 0 resσT : R(T 0 )3T 0 −→ R(σ )3T 0 , σ dual cyclic σ ⊆T 0
σ dual cyclic σ ⊆T 0
so we are reduced to proving that R(T 0 )3T 0 ⊗Z F p −→
Y
R(σ )3T 0 ⊗Z F p
σ dual cyclic σ ⊆T 0
is injective, that is, that if A is a finite abelian group and p - |A|, then Y ϕ : F p [A] −→ F p [C] C∈{cyclic quotients ofA}
(9)
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VEZZOSI and VISTOLI
b = HomAbGrps (A, C∗ ) denotes the complex characters group of A, is injective. If A b then R( A) ' Z[A] and Y b Y A b −→ b ϕ= resCb : R( A) R(C). b C
b b C∈{cyclic subgroups of A}
b ⊗ ˙ Z Z[1/|A|] has image via Since p - |A|, it is enough to prove that if ξ ∈ R( A) A b ⊗Z Z[1/|A|] −→ R(C) b ⊗Z Z[1/|A|] resCb ⊗ Z[1/|A|] : R( A) b ⊗Z Z[1/|A|] for each cyclic C b ⊆ A, b then ξ ∈ contained in p R(C) b ⊗ ˙ Z Z[1/|A|] . p R( A) Q 0 ) ∈ b ⊗Z Z[1/|A|] By [Se, p. 73], there exists (θCb b b b R(C) C C∈{cyclic subgroups of A} such that X 0 b 1= indCAb ⊗ Z[1/|A|] (θCb ); b
b C
therefore ξ=
X
=
X
0 b ξ indCAb ⊗ Z[1/|A|] (θCb )
b C
0 b b A indCAb ⊗ Z[1/|A|] θCb (resCb ⊗ Z[1/|A|])(ξ )
b C
(by the projection formula), and we conclude the proof of the proposition. Remark 3.2 The proof of Proposition 3.1 is similar to [Vi1, proof of Prop. 1.5], which is, however, incomplete; that is why we have decided to give all the details here. COROLLARY 3.3 We have the following. (i) If σ 6 = σ 0 are dual cyclic subgroups of T , we have e R(σ )σ 0 = 0 and e R(σ )σ = e R(σ ). (ii) If T 0 ⊂ T is a closed subgroup scheme, finite over k, and if σ is a dual cyclic subgroup of T , we have R(T 0 )σ = 0 if σ * T 0 . (iii) If T 0 ⊂ T is a closed subgroup scheme, finite over k, the canonical morphism of R(T )-algebras Y R(T 0 )3T 0 −→ R(T 0 )σ σ dual cyclic σ ⊆T 0
is an isomorphism.
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
19
Proof (i) Suppose σ 6= σ 0 , and let T 0 ⊂ T be the closed subgroup scheme of T generated by σ and σ 0 . The obvious morphism π : R(T )3T 0 → e R(σ )3T 0 × e R(σ 0 )3T 0 factors through R(T 0 )3T 0 → e R(σ )3T 0 × e R(σ 0 )3T 0 , which is an epimorphism by Proposition 3.1. If ξ ∈ R(T )3T 0 with π(ξ ) = (0, 1) ⊗ 1, we have ξ ∈ Sσ 0 ∩ ker R(T )3T 0 → e R(σ )3T 0 . Then e R(σ )σ 0 = 0. The second assertion is obvious. (ii) and (iii) These follow immediately from (i) and Proposition 3.1. Now let X be a regular Noetherian separated k-algebraic space on which T acts with . finite stabilizers, and let 3 = 3(T,X ) . Obviously, C (T ) is just the set of essential dual cyclic subgroups of T since T is abelian. PROPOSITION 3.4 We have the following. (i) If jσ : X σ ,→ X denotes the inclusion, the pushforward ( jσ )∗ induces an isomorphism K ∗0 (X σ , T )σ −→ K ∗0 (X, T )σ .
(ii)
The canonical localization morphism K ∗0 (X, T )3 −→
Y
K ∗0 (X, T )σ
σ ∈C (T )
is an isomorphism, and the product on the left is finite. Proof (i) The proof is the same as that of [Th5, Th. 2.1], but we substitute Corollary 3.3(ii) for [Th5, Th. 2.1] since we use a localization different from Thomason’s. (ii) By the generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subspace U ⊂ X , a closed (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism 0
U ' T /T 0 × (U/T ) ' (U/T ) ×T T. Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗Z R(T 0 ), by Morita equivalence theorem (see [Th3, Prop. 6.2]) and [Th1, Lem. 5.6]. By Corollary 3.3(ii), the proposition for X = U follows from Corollary 3.3(iii). By Noetherian induction and the localization sequence for K 0 -groups (see [Th3, Th. 2.7]), the statement for U implies the same for X . Again using Noetherian induction, Thomason’s generic slice theorem for torus Q actions, and (i), one similarly shows that the product σ ∈C (T ) K ∗0 (X, T )σ is finite.
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VEZZOSI and VISTOLI
By Proposition 3.4, there is an induced isomorphism (of R(T )-modules, not a ring isomorphism due to the composition with pushforwards) Y K ∗0 (X σ , T )σ −→ K ∗0 (X, T )3 . (10) σ ∈C (T )
As shown in Lemma 2.8, the product morphism σ × T → T induces a morphism θT,σ : K ∗0 (X σ , T )σ −→ K ∗0 (X σ , T )geom ⊗ e R(σ )3 .
(11)
PROPOSITION 3.5 For any σ ∈ C (T ), θT,σ is an isomorphism.
Proof We write θ X,σ for θT,σ in order to emphasize the dependence of the map on the space. We proceed by Noetherian induction on X σ . Let X 0 ⊆ X σ be a T -invariant closed subspace, and let us suppose that (11) is an isomorphism with X replaced by any T invariant proper closed subspace Z of X 0 . By Thomason’s generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subscheme U ⊂ X 0 , a (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism 0 U σ ≡ U ' T /T 0 × (U/T ) ' (U/T ) ×T T. Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and, . obviously, 3T 0 ⊆ 3. Let Z σ ≡ Z = X 0 U . Since /
θ Z ,σ
/
/
K ∗0 (Z σ , T )σ
θY 0 ,σ
e )3 K ∗0 (Z σ , T )geom ⊗ R(σ
/
/
K ∗0 (X 0σ , T )σ
e )3 K ∗0 (X 0σ , T )geom ⊗ R(σ
θU,σ
/
/
K ∗0 (U σ , T )σ
e )3 K ∗0 (U σ , T )geom ⊗ R(σ
/
is commutative, by the induction hypothesis and the five-lemma it is enough to show that θU,σ is an isomorphism. By Morita equivalence theorem (see [Th3, Prop. 6.2]) and [Th1, Lem. 5.6], K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗Z R(T 0 ), so it is enough to prove that θSpec k,σ : K 00 (Spec k, T 0 )σ = R(T 0 )σ → K 00 (Spec k, T 0 )geom ⊗ e R(σ )3 0 = R(T )geom ⊗ e R(σ )3 is an isomorphism. But this follows immediately from Corollary 3.3(i) and (iii). Combining Proposition 3.5 with (10), we get an isomorphism Y 8 X,T : K ∗0 (X σ , T )geom ⊗ e R(σ )3 −→ K ∗0 (X, T )3 . σ ∈C (T )
(12)
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21
The following lemma is a variant of [Th5, Lem. 3.2], which already proves it after tensoring with Q. LEMMA 3.6 Let X be a Noetherian regular separated algebraic space over k on which a split k-torus acts with finite stabilizers, and let σ ∈ C (T ). Let X σ denote the regular σ -fixed subscheme, let jσ : X σ ,→ X be the regular closed immersion (see [Th5, Prop. 3.1]), and let N ( jσ ) be the corresponding locally free conormal sheaf. Then, for any T -invariant algebraic subspace Y of X σ , the cap-product λ−1 N ( jσ ) ∩ (−) : K ∗0 (Y, T )σ −→ K ∗0 (Y, T )σ
is an isomorphism. Proof We proceed by Noetherian induction on closed T -invariant subspaces Y of X σ . The statement is trivial for Y = ∅, so let us suppose Y nonempty and λ−1 N ( jσ ) ∩ (−) : K ∗0 (Z , T )σ −→ K ∗0 (Z , T )σ an isomorphism for any proper T -invariant closed subspace Z of Y . By Thomason’s generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subscheme U ⊂ Y , a closed (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism U σ ≡ U ' T /T 0 × (U/T ). Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k. Using the localization sequence and the five-lemma, we reduce ourselves to showing that λ−1 N ( jσ ) ∩ (−) : K ∗0 (U, T )σ −→ K ∗0 (U, T )σ is an isomorphism. For this, it is enough to show that (the restriction of) λ−1 (N ( jσ )) is a unit in K 0 (U, T )σ ' K 0 (U/T )3 ⊗ R(T 0 )σ (see [Th3, Prop. 6.2]). Decomposing N ( jσ ) according to the characters of T 0 , we may write, shrinking U if necessary, M rρ N ( jσ ) = OU/T ⊗ Lρ , ρ∈Tb0
where Lρ is the line bundle attached to the T 0 -character ρ and rρ ≥ 0, and thereQ fore λ−1 (N ( jσ )) = ρ∈Tb0 (1 − ρ)rρ in K 0 (U/T ) ⊗ R(T 0 ). The localization map R(T 0 )3 → R(T 0 )σ ' e R(σ )3 coincides with the composition πσ
pσ
R(T 0 )3 −→ R(σ )3 −→ e R(σ )3
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VEZZOSI and VISTOLI
of the restriction to σ followed by the projection (see Cor. 3.3), and then M rχ (id K 0 (U/T )3 ⊗πσ ) N ( jσ ) = OU/T ⊗ Lχ , χ∈b σ \{0}
in K 0 (U/T )3 ⊗ R(σ )3 , where the summand omits the trivial character since the decomposition of N ( jσ ) according to the characters of σ has vanishing fixed subsheaf N ( jσ )0 (see, e.g., [Th5, Prop. 3.1]). Therefore, Y λ−1 (id K 0 (U/T )3 ⊗πσ )(N ( jσ )) = (1 − χ )rχ , χ ∈b σ \{0}
and it is enough to show that the image of 1 − χ in e R(σ )3 via pσ is a unit for any nontrivial character χ of σ . Now, the image of such a χ in 3[t] e R(σ )3 ' (8|σ | ) (8|σ | being the |σ |th cyclotomic polynomial) is of the form [t l ] for some 1 ≤ l < |σ |, where [−] denotes the class mod8|σ | ; therefore the cokernel of the multiplication by 1 − [t l ] in 3[t](8|σ | ) is 3[t] =0 (8|σ | , 1 − t l ) since 8|σ | and (1 − t l ) are relatively prime in 3[t] for 1 ≤ l < |σ |. Thus 1 − [t l ] is a unit in 3[t](8|σ | ), and we conclude the proof of the lemma. We are now able to prove our main theorem for G = T . THEOREM 3.7 If X is a regular Noetherian separated k-algebraic space, then Y 9 X,T : K ∗ (X, T )3 −→ K ∗ (X σ , T )geom ⊗ e R(σ )3 σ ∈C (T )
is an isomorphism of R(T )-algebras. Proof Recall (see appendix) that K ∗ (X, T ) ' K ∗0 (X, T ) and K ∗ (X σ , T ) ' K ∗0 (X σ , T ), since both X and X σ are regular (see [Th5, Prop. 3.1]). Since 8 X,T is an isomorphism of R(T )-modules, it is enough to show that the composition 9 X,T ◦ 8 X,T is an isomorphism. A careful inspection of the definitions of 9 X,T and 8 X,T easily reduces the problem to proving that, for any σ ∈ C (T ), the composition jσ ∗
jσ∗
K ∗0 (X σ , T )σ −→ K ∗0 (X, T )σ −→ K ∗0 (X σ , T )σ
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
23
is an isomorphism, jσ : X σ ,→ X being the natural inclusion. Since jσ is regular, there is a self-intersection formula jσ∗ ◦ jσ ∗ (−) = λ−1 N ( jσ ) ∩ (−), (13) N ( jσ ) being the conormal sheaf associated to jσ , and we conclude by Lemma 3.6. To prove the self-intersection formula (13), we adapt [Th5, proof of Lem. 3.3]. First, by Proposition 3.4(i), jσ ∗ is an isomorphism, so it is enough to prove that jσ ∗ jσ∗ jσ ∗ (−) = jσ ∗ (λ−1 (N ( jσ )) ∩ (−)). By the projection formula (see Prop. A.5), we have
jσ ∗ jσ∗ jσ ∗ (−) = jσ ∗ jσ∗ (1) ∩ jσ ∗ (−) = jσ ∗ jσ∗ (O X ) ∩ jσ ∗ (−) = jσ ∗ (O X σ ) ∩ jσ ∗ (−) = jσ ∗ jσ∗ (O X σ ) ∩ (−) . Now, as explained in the appendix, to compute jσ∗ (O X σ ) we choose a complex F ∗ of flat quasi-coherent G-equivariant modules on X which is quasi-isomorphic to O X σ , and then X jσ∗ (O X σ ) = [ jσ∗ (F ∗ )] = [F ∗ ⊗ O X σ ] = (−1)i [H i (F ∗ ⊗ O X σ )]. i
But F ∗ is a flat resolution of O X σ , so H i (F ∗ ⊗ O X σ ) = ToriO X (O X σ , O X σ ) ' Vi N ( jσ ), where the last isomorphism (see [SGA6, Exp. VII, par. 2.5]) is natural and hence T -equivariant. Therefore, jσ∗ (O X σ ) = λ−1 (N ( jσ )), and we conclude the proof of the theorem. 4. The main theorem: The case of G = GLn,k In this section we use the result for 9 X,T to deduce the same result for 9 X,GLnk . THEOREM 4.1 Let X be a Noetherian regular separated algebraic space over a field k on which G = GLn,k acts with finite stabilizers. Then the map defined in (5),
9 X,G : K ∗ (X, G)3(G,X ) Y −→
K ∗ (X σ , C(σ ))geom ⊗3(G,X ) e R(σ )3(G,X )
wG (σ )
, (14)
σ ∈C (G)
is an isomorphism of R(G)-algebras and the product on the right is finite. Throughout this section, entirely devoted to the proof of Theorem 4.1, we simply write G for GLn,k , 3 for 3(G,X ) , and T for the maximal torus of diagonal matrices in GLn,k . First, let us observe that we can choose each σ ∈ C (G) contained in T . Moreover, 3(T,X ) = 3(G,X ) .
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VEZZOSI and VISTOLI
We need the following three preliminary lemmas (Lems. 4.2, 4.3, 4.4). If σ, τ ⊂ T are dual cyclic subgroups, they are conjugate under the G(k)-action if and only if they are conjugated via an element in the Weyl group Sn . For any group scheme H with a dual cyclic subgroup σ ⊆ H , we denote by mσH the kernel of \ R(H )3 → e R(σ )3 and by R(H )3,σ the completion of R(H )3 with respect to the H ideal mσ . The following lemma is essentially a variant of Lemma 2.9 for σ -localizations. LEMMA 4.2 Let G = GLn,k , let T be the maximal torus of G consisting of diagonal matrices, and let X be an algebraic space on which G acts with finite stabilizers. (i) For any essential dual cyclic subgroup σ ⊆ T , the morphisms ωσ,geom : K ∗0 X σ , CG (σ ) geom ⊗R(CG (σ ))3 R(T )3 −→ K ∗0 (X σ , T )geom , ωσ : K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3 −→ K ∗0 (X σ , T )σ
(ii)
induced by T ,→ CG (σ ) are isomorphisms. For any essential dual cyclic subgroup σ ⊆ T , G (σ ) ) N · K 0 X σ , C (σ ) (mC = 0, G σ ∗ σ
N 0,
and the morphism induced by T ,→ CG (σ ), \ ω cσ : K ∗0 X σ , CG (σ ) σ ⊗R(C\ R(T )3,σ −→ K ∗0 (X σ , T )σ , (σ )) G
3,σ
is an isomorphism. Proof (i) Since CG (σ ) is isomorphic to a product of general linear groups over k and since T is a maximal torus in CG (σ ), by Lemma 2.9 the canonical ring morphism K ∗0 X, CG (σ ) ⊗R(CG (σ )) R(T ) −→ K ∗0 (X, T ) (15) is an isomorphism. If H ⊆ G is a subgroup scheme, we denote by SσH the multiplicative subset of R(H )3 consisting of the elements sent to 1 by the canonical ring homomorphism R(H )3 → e R(σ )3 . By (15), ωσ coincides with the composition K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3 ' K ∗0 (X σ , T ) ⊗R(CG (σ ))3 (SσCG (σ ) )−1 R(CG (σ ))3 ⊗R(CG (σ ))3 R(T )3 id ⊗νσ −−−→ K ∗0 X σ , CG (σ ) ⊗R(CG (σ ))3 (SσT )−1 R(T )3 ' K ∗0 (X σ , T )σ ,
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
25
where νσ : (SσCG (σ ) )−1 R CG (σ )
3
⊗R(CG (σ ))3 R(T )3 → (SσT )−1 R(T )3
(16)
is induced by T ,→ CG (σ ) and the last isomorphism follows from (15); the same is true for ωσ,geom . Therefore, it is enough to prove that νσ and C (σ ) −1
νσ,geom : (S1 G
)
R CG (σ ) 3 ⊗R(CG (σ ))3 R(T )3 → (S1T )−1 R(T )3 C (σ )
are isomorphisms; that is, if Sτ denotes the image of S1 G R CG (σ ) 3 −→ R(T )3 ,
via the restriction map
then SτT /1 consists of units in (Sτ )−1 R(T )3 for τ = 1 and τ = σ . If 1σ denotes the Weyl group of CG (σ ), which is a product of symmetric groups, we have R(CG (σ )) ' R(T )1σ and therefore 1 (SτCG (σ ) )−1 R CG (σ ) 3 ' (Sτ )−1 R(T )3 σ since R(T ) is torsion free. Moreover, there is a commutative diagram
/ (Sτ )−1 R(T )3 j j j jjjj ψ jjj. j j j j =ϕ tjjj T −1 T )−1 R(T ) o e R(τ )3 ' (Sτ ) R(τ )3 (S 3 τ T −1 T C (σ ) −1 ) R
(Sτ G
CG (σ ) 3
(Sτ )
resτ
where ψ is induced by e πτ and the isomorphism e R(τ )3 ' (SτT )−1 R(τ )3 is obtained from Proposition 3.1 and Corollary 3.3. If we define the map 1 M : (Sτ )−1 R(T )3 −→ (Sτ )−1 R(T )3 σ , Y ξ 7 −→ g·ξ , g∈1σ
it is easily checked that for ξ ∈ (Sτ )−1 R(T )3 , ξ is a unit if M(ξ ) is a unit, and that ψ(M(ξ )) = 1 implies that ξ is a unit in ((Sτ )−1 R(T )3 )1σ . But ϕ is 1σ -equivariant, and therefore SτT /1 consists of units in (Sτ )−1 R(T )3 for τ = 1 or σ . (ii) Since R(CG (σ )) → R(T ) is faithfully flat, by (i) it is enough to prove that (mσT ) N K ∗0 (X σ , T )σ = 0
for N 0.
(17)
But (17) can be proved using the same technique as in the proof of, for example, Proposition 3.5, that is, Noetherian induction together with Thomason’s generic slice
26
VEZZOSI and VISTOLI
theorem for torus actions. The second part of (ii) follows, arguing as in (i), from the fact that (16) is an isomorphism since K ∗0 X σ , CG (σ )
σ
⊗R
CG
0 σ R C\ (σ ) ' K X , C (σ ) , G G ∗ 3,σ σ (σ ) 3
K ∗0 (X σ , T )σ
\ ⊗R(T )3 R(T )3,σ ' K ∗0 (X σ , T )σ .
If σ, τ ⊂ T are dual cyclic subgroups conjugated under G(k), they are conjugate through an element of the Weyl group Sn and we write τ ≈ Sn σ ; moreover, we have mσG = mτG because conjugation by an element in Sn (actually, by any element in G(k)) induces the identity morphism on K -theory and, in particular, on the representation ring. Then there are canonical maps Y \ \ R(T )3,τ , (18) R(G) 3,σ ⊗R(G)3 R(T )3 −→ τ dual cyclicτ ≈ Sn σ
\ R C\ )3,σ . G (σ ) 3,σ ⊗ R(CG (σ ))3 R(T )3 −→ R(T
(19)
4.3 Maps (18) and (19) are isomorphisms. LEMMA
Proof \ Since R(G) = R(T ) Sn → R(T ) is finite, the canonical map R(G) 3,σ ⊗R(G)3 mG
mG
σ σ \ \ R(T )3 → R(T )3 (where R(T )3 denotes the mσG -adic completion of the R(G)3 -module R(T )3 ) is an isomorphism. Moreover, R(G)3 = (R(T )3 ) Sn implies that q \ q \ G mσ R(T )3 = mτT = mτT , τ dual cyclic τ ≈ Sn σ
τ dual cyclic τ ≈ Sn σ
and, by Corollary 3.3(i), mτT + mτT0 = R(T )3 if τ 6= τ 0 . By the Chinese remainder lemma, we conclude that (18) is an isomorphism. Arguing in the same way, we get that the canonical map R C\ G (σ ) 3,σ ⊗R(CG (σ ))3 R(T )3 −→
Y
\ R(T )3,τ
τ dual cyclic τ ≈1σ σ
is an isomorphism, where 1σ = Sn ∩ CG (σ ) is the Weyl group of CG (σ ) with respect to T and we write τ ≈1σ σ to denote that τ and σ are conjugate through an element of 1σ . But 1σ ⊂ CG (σ ), so that τ ≈1σ σ if and only if τ = σ , and we conclude that (19) is an isomorphism.
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
27
LEMMA 4.4 For any essential dual cyclic subgroup σ ⊆ G, the canonical morphism \ \ R(G) 3,σ −→ R CG (σ ) 3,σ
is a finite e´ tale Galois cover (see [SGA1, Exp. V]) with Galois group wG (σ ). Proof Since R(T ) is flat over R(G) = R(T ) Sn , we have \ \ R(G) 3,σ ' R(G) 3,σ ⊗R(G)3 R(T )3
Sn
Sn \ ⊗ ' R(G) 3,σ R(G)3 R(T )3 Sn Y \ R(T )3·τ , ' τ dual cyclic τ ≈ Sn σ
the last isomorphism being given in Lemma 4.3. By Lemma 2.10, we get S \σ ' R(T \ R(G) )σ n,σ , where Sn acts on the set of dual cyclic subgroups of T which are Sn -conjugated to σ and where Sn,σ denotes the stabilizer of σ . Analogously, denoting by 1σ the Weyl group of CG (σ ), by Lemma 4.2(ii) we have \ R(T )3 1σ R C\ G (σ ) 3,σ ' R CG (σ ) 3,σ ⊗R C (σ ) G
3
' R(C\ G (σ ))3,σ ⊗R(CG (σ ))3 R(T )3 1 \ ' R(T )3,σ σ ,
1σ
where the last isomorphism is given by Lemma 4.3. From the exact sequence 1 → 1σ −→ Sn,σ −→ wG (σ ) → 1, we conclude that \ R(G) 3,σ ' R(C\ G (σ ))3,σ
wG (σ )
By [SGA1, Prop. 2.6, Exp. V], it is now enough to prove ometric points (i.e., the inertia groups of points) of Spec wG (σ )-action are trivial. First, let us observe that Spec(e R(σ )3 ) is a Spec(R(C\ G (σ ))3,σ ). This can be seen as follows. It show that if s denotes the order of σ , the map πσ : R CG (σ )
3
−→ R(σ )3 =
.
(20)
that the stabilizers of ge R(C\ G (σ ))3,σ under the closed subscheme of is obviously enough to
3[t] (t s − 1)
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VEZZOSI and VISTOLI
is surjective. First, consider the case where σ is contained in the center of G. Since R(σ )3 is of finite type over 3, we show that for any prime∗ p - s the induced map πσ, p : R CG (σ ) 3 ⊗ F p −→ R(σ )3 ⊗ F p is surjective. Note that if E denotes the standard n-dimensional representation of G, V πσ sends r E to nr t r . If p - n, then πσ, p is surjective (in fact, πσ (E) = nt and n is invertible in F p ). If p | n, let us write n = qm, with q = pi and p - m. Since (s, q) = 1, t q is a ring generator of R(σ )3 , and to prove πσ, p is injective, it is enough to show that p - qn . This is elementary since the binomial expansion of (1 + X )n = (1 + X q )m Q in F p [X ] yields qn = m in F p . For a general σ ⊆ T , let CG (σ ) = li=1 GLdi ,k , P where d = n, and let σi denote the image of σ in GLdi ,k , i = 1, . . . , l. Since Ql i σ ⊆ i=1 σi is an inclusion of diagonalizable groups, the induced map l l O Y σi = R(σi ) −→ R(σ ) R i=1
i=1
is surjective (e.g., see [SGA3, Vol. II]). But R(CG (σ ))3 → R(σ )3 factors as R CG (σ )
3
=
l O
R(GLdi ,k )3 −→
i=1
l O
R(σi )3 −→ R(σ )3 ,
i=1
and also the first map is surjective (by the previous case, since σi is contained in the center of GLdi ,k and |σi | divides |σ |). This proves that Spec(e R(σ )3 ) is a closed sub\ scheme of Spec(R(C\ (σ )) ). Since R(C (σ )) is the completion of R(CG (σ ))3 G 3,σ G 3,σ along the ideal ker R(CG (σ ))3 → e R(σ )3 , any nonempty closed subscheme of Spec R(C\ G (σ ))3,σ meets the closed subscheme Spec(e R(σ )3 ). To prove that wG (σ ) acts freely on the geometric points of \ Spec R(CG (σ ))3,σ , it is then enough to show that it acts freely on the geometric points of Spec(e R(σ )3 ). Actually, more is true: the map q : Spec(e R(σ )3 ) → Spec(3) is a (Z/sZ)∗ torsor† . In fact, if Spec() → Spec(3) is a geometric point, the corresponding geometric fiber of q is isomorphic to the spectrum of Y Y [t] ' (t − αi ) αi ∈e µs ()
∗ Recall † Recall
αi ∈e µs ()
that σ is essential; hence s is invertible in 3. that the constant group scheme associated to (Z/sZ)∗ is isomorphic to Autk (σ ).
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
29
and (Z/sZ)∗ acts by permutation on the primitive roots e µs (), by α 7→ α k , (k, s) = 1. In particular, the action of the subgroup wG (σ ) ⊂ (Z/sZ)∗ on Spec(e R(σ )3 ) is free. PROPOSITION 4.5 The canonical morphism
K ∗0 (X σ , CG (σ ))σ
Y
K ∗0 (X, G)3 −→
wG (σ )
σ ∈C (G)
is an isomorphism. Proof By Lemma 2.9, the canonical ring morphism K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Since R(G) → R(T ) is faithfully flat, it is enough to show that K ∗0 (X, T )3 ' K ∗0 (X, G)3 ⊗R(G)3 R(T )3 Y w (σ ) −→ K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 σ ∈C (G)
is an isomorphism. By Proposition 3.4(ii), we are left to prove that K ∗0 (X σ , CG (σ ))σ
Y
wG (σ )
⊗R(G)3 R(T )3 '
σ ∈C (G)
Y
K ∗0 (X, T )σ .
(21)
σ dual cyclic σ ⊂T
For any τ ∈ C (G) (τ ⊆ T , as usual), we have \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ 3,τ
'
K ∗0 (X τ , CG (τ ))τ
⊗R(C\ (τ ))
'
K ∗0 (X τ , CG (τ ))τ
⊗ R(G) \
G
3,τ
\ R(T )3,τ R(C\ G (τ ))3,τ ⊗R(G) \ 3,τ 3,τ \ R(C\ R(T )3,τ . G (τ ))3,τ ⊗R(C\ (τ )) 3,τ
G
\ By Lemma 4.4, for any R(G) 3,τ -module M, we have \ M ⊗R(G) \ R CG (τ ) 3,τ ' wG (τ ) × M 3,τ
since a torsor is trivial when base changed along itself. Therefore, \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ 3,τ
' wG (τ ) × K ∗0 (X τ , CG (τ ))τ ⊗R(C\ (τ )) G
3,τ
\ R(T )3,τ
(22)
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VEZZOSI and VISTOLI
with wG (τ ) acting on left-hand side by left multiplication on wG (τ ). Applying Lemma 4.2(ii) to the left-hand side, we get 0 τ \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ ' wG (τ ) × K ∗ (X , T )τ , 3,τ
and taking invariants with respect to wG (τ ), \ K ∗0 (X τ , CG (τ ))τ ⊗R(G) \ R(T )3,τ
wG (τ )
3,τ
' K ∗0 (X τ , T )τ .
(23)
Comparing (21) to (23), we are reduced to proving that for any σ ∈ C (G) there is an isomorphism w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 Y \ wG (τ ) . ' K ∗0 (X τ , CG (τ ))τ ⊗R(G) \ R(T )3,τ 3,τ
τ dual cyclic τ ≈ Sn σ
\ \ Since R(T )3,τ is flat over R(G) 3,τ and wG (τ ) acts trivially on it, we have (see [SGA1]) w (τ ) \ K ∗0 (X τ , CG (τ ))τ ⊗R(G) R(T )3,τ G \ 3,τ w (τ ) \ ' K ∗0 (X τ , CG (τ ))3,τ G ⊗R(G) \ R(T )3,τ . 3,τ
By Lemma 4.3, we have isomorphisms w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) R(G) 3,σ ⊗R(G)3 R(T )3 \ 3,σ Y w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) \ R(T )3,τ . 3,σ
τ dual cyclic τ ≈ Sn σ
G G \ \ (Recall that R(G) 3,σ = R(G) 3,τ for any τ ≈ Sn σ , since mσ = mτ .) For each −1 τ , choosing an element g ∈ Sn such that gσ g = τ determines an isomorphism K ∗0 (X σ , CG (σ ))σ ' K ∗0 (X τ , CG (τ ))τ whose restriction to invariants w (σ ) w (τ ) K ∗0 (X σ , CG (σ ))σ G ' K ∗0 (X τ , CG (τ ))τ G
is independent of the choice of g. Therefore, we have a canonical isomorphism w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 Y w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) \ R(T )3,τ 3,σ
τ dual cyclic τ ≈ Sn σ
'
Y τ dual cyclic τ ≈ Sn σ
K ∗0 (X τ , CG (τ ))τ
wG (τ )
\ ⊗R(G) \ R(T )3,τ , 3,τ
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
31
as desired. Since K ∗ (X, G) ' K ∗0 (X, G) and K ∗ (X σ , CG (σ )) ' K ∗0 (X σ , CG (σ )), comparing Proposition 4.5 with (14), we see that the proof of Theorem 4.1 can be completed by the following. PROPOSITION 4.6 For any σ ∈ C (G), the morphism given by Lemma 2.8 and induced by the product CG (σ ) × σ → CG (σ ), θCG (σ ),σ : K ∗0 X σ , CG (σ ) σ −→ K ∗0 X σ , CG (σ ) geom ⊗ e R(σ )3 ,
is an isomorphism. Proof To simplify the notation, we write θσ for θCG (σ ),σ . As usual, we may suppose σ contained in T . Since CG (σ ) is isomorphic to a product of general linear groups over k, we can take T as its maximal torus, and by Lemma 2.9, the canonical ring morphism K ∗0 X, CG (σ ) ⊗R(CG (σ )) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Moreover, R(CG (σ )) → R(T ) being faithfully flat, it is enough to prove that θσ ⊗ idR(T ) is an isomorphism. To prove this, let us consider the commutative diagram K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3
θσ ⊗id
/
K ∗0 (X, CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3
ωσ
K ∗0 (X σ , T )σ
γf σ
θT,σ
/
K ∗0 (X σ , T )geom ⊗ e R(σ )3
where •
•
• •
K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3 is an R(CG (σ ))3 -module via the coproduct ring morphism 1CG (σ ) : R(CG (σ ))3 → R(CG (σ ))3 ⊗ e R(σ )3 (induced by the product CG (σ ) × σ → CG (σ )); ωσ is the canonical map induced by the inclusion T ,→ CG (σ ) and is an isomorphism by Lemma 4.2; θT,σ is an isomorphism as shown in the proof of Theorem 3.7; γeσ sends (x ⊗ u) ⊗ t to (1T (t) · x|T ) ⊗ u, for x ∈ K ∗ (X σ , CG (σ ))geom , u ∈e R(σ )3 , t ∈ R(T )3 , 1T : R(T )3 → R(T )3 ⊗ e R(σ )3 being the coproduct induced by the product T × σ → T . So we are left to prove that γeσ is an isomorphism.
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VEZZOSI and VISTOLI
First, let us observe that if R is a ring, A → A0 is a ring morphism, and M is an A-module, there is a natural isomorphism (M ⊗Z R) ⊗ A⊗Z B (A0 ⊗Z R) −→ (M ⊗ A A0 ) ⊗Z R, (m ⊗ r1 ) ⊗ (a 0 ⊗ r2 ) 7 −→ (m ⊗ a 0 ) ⊗ r1r2 . Applying this to M = K ∗ (X σ , CG (σ ))geom , R = e R(σ )3 , A = R(CG (σ ))3 , A0 = R(T )3 and using Lemma 4.2, we get a canonical isomorphism f : K ∗0 (X σ , T )geom ⊗ e R(σ )3 0 e −→ K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 ⊗e R(σ )3 R(T )3 ⊗ R(σ )3 , (24) where we have denoted by (R(T )3 ⊗ e R(σ )3 )0 the R(CG (σ ))3 ⊗ e R(σ )3 -algebra res ⊗ id : R CG (σ ) 3 ⊗ e R(σ )3 −→ R(T )3 ⊗ e R(σ )3 . It is an elementary fact that there are mutually inverse isomorphisms αCG (σ ) , βCG (σ ) , and αT , βT fitting into the commutative diagrams R CG (σ ) 6 1C G (σ ) mmmmm m m mmm mmm
3
⊗e R(σ )3
(25)
αC G (σ ) R CG (σ ) 3 QQQ QQQ QQQ id⊗1 QQQQ( R CG (σ ) 3 ⊗ e R(σ )3 R(T 7)3 ⊗ e R(σ )3 o 1T oooo ooo ooo αT R(T )3 O OOO OOO O id⊗1 OOO' R(T )3 ⊗ e R(σ )3
(26)
and compatible with restriction maps induced by T ,→ CG (σ ). This is exactly the dual assertion to the general fact that “an action H × Y → Y is isomorphic over X to the projection on the second factor pr2 : H × Y → Y ,” for any group scheme H and any algebraic space Y . From (25) we get an isomorphism 0 e α : R(CG (σ ))3 ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 −→ R(T )3 ⊗ e R(σ )3 ,
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
33
0 where R(CG (σ ))3 ⊗ e R(σ )3 denotes the R(CG (σ ))3 -algebra 1CG (σ ) : R CG (σ ) 3 → R CG (σ ) 3 ⊗ e R(σ )3 . Therefore, if we denote by (R(T )3 ⊗ e R(σ )3 )00 the R(CG (σ ))3 ⊗ e R(σ )3 -algebra (res ⊗ id) ◦ αT : R CG (σ ) 3 ⊗ e R(σ )3 −→ R(T )3 ⊗ e R(σ )3 , the composition 0 K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 = K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 O 0 R(CG (σ ))3 ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 R(CG (σ ))3 ⊗e R(σ )3 id ⊗e α
R(σ )3 −→ K ∗0 (X σ , CG (σ ))geom ⊗ e O 00 R(T )3 ⊗ e R(σ )3 R(CG (σ ))3 ⊗e R(σ )3
id ⊗βT
−→
K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 O 0 R(T )3 ⊗ e R(σ )3 R(CG (σ ))3 ⊗e R(σ )3
'
K ∗0 (X σ , T )geom
⊗e R(σ )3
is an isomorphism and it can be easily checked to coincide with γeσ . 5. The main theorem: The general case In this section, we use Theorem 4.1 to deduce the same result for the action of a linear algebraic k-group G, having finite stabilizers, on a regular separated Noetherian k-algebraic space X . We write 3 for 3(G,X ) . We start with a general fact. PROPOSITION 5.1 Let X be a regular Noetherian separated k-algebraic space on which a linear algebraic k-group G acts with finite stabilizers. Then there exists an integer N > 0 such that if a1 , . . . , a N ∈ K 0 (X, G)geom have rank zero on each connected component of QN X , then the multiplication by i=1 ai on K ∗0 (X, G)geom is zero.
In particular, we have the following.
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VEZZOSI and VISTOLI
COROLLARY 5.2 Let X be a regular Noetherian separated k-algebraic space with a connected action of a linear algebraic k-group G having finite stabilizers. Then the geometric localization
rk0,geom : K 0 (X, G)geom −→ 3 of the rank morphism has a nilpotent kernel. Proof of Proposition 5.1 Let us choose a closed immersion G ,→ GLn,k (for some n > 0). By Morita equivalence, K ∗0 (X, G) ' K ∗0 (X ×G GLn,k , GLn,k ) and K 0 (X, G) ' K 0 (X ×G GLn,k , GLn,k ). Moreover, 3(X ×G GLn,k ,GLn,k ) = 3. Let ξ = x/s ∈ K ∗0 (X, G)geom with x ∈ K ∗0 (X, G)3 and s ∈ rk−1 (1) where rk : R(G) → 3 is the rank morphism, and let ai = αi /si with αi ∈ K 0 (X, G)3 and si ∈ rk−1 (1) for i = 1, . . . , N . Let us consider the elements x/1 in K ∗0 (X ×G GLn,k , GLn,k )geom , and αi /1 in K 0 (X ×G GLn,k , GLn,k )geom for i = 1, . . . , N . Since the canonical homomorphism K ∗0 (X ×G GLn,k , GLn,k )geom −→ K ∗0 (X, G)geom is a morphism of modules over the ring morphism K 0 (X ×G GLn,k , GLn,k )geom −→ K 0 (X, G)geom , if the theorem holds for G = GLn,k and if N is the corresponding integer, the product Q Q 0 i αi /1 in K 0 (X, G)geom annihilates x/1 ∈ K ∗ (X, G)geom and a fortiori i ai an0 nihilates ξ in K ∗ (X, G)geom . So, we may assume G = GLn,k . Let T be the maximal torus of diagonal matrices in G. By Lemma 4.2(i) with σ = 1, there are isomorphisms ω1,geom : K 0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 ' K 0 (X, T )geom , K ∗0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 ' K ∗0 (X, T )geom . Since R(GLn,k ) → R(T ) is faithfully flat and the diagram rk0,geom ⊗ id
K 0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 −−−−−−−→ 3 ⊗R(GLn,k ) R(T )3 id ⊗ rk ω1,geom y y 3 T K 0 (X, T )geom
−−−−→ rk0,geom
3
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
35
commutes, we reduce ourselves to proving the proposition for G = T , a split torus. To handle this case, we proceed by Noetherian induction on X . By [Th1, Prop. 4.10], there exist a T -invariant nonempty open subscheme j : U ,→ X , a closed diagonalizable subgroup T 0 of T , and a T -equivariant isomorphism U ' T /T 0 × (U/T ). Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗ R(T ) R(T 0 ), by Morita equivalence theorem (see [Th3, Prop. 6.2]). Let i : Z ,→ X be the closed complement of U in X , and let N 0 be an integer satisfying the proposition for both Z and U . Consider the geometric localization sequence i∗
j∗
K ∗0 (Z , T )geom −→ K ∗0 (X, T )geom −→ K ∗0 (U, T )geom , and let ξ ∈ K ∗0 (X, T )geom . Let a1 , . . . , a2N 0 ∈ K 0 (X, T )geom . By our choice of N 0 , j ∗ (a N 0 +1 · . . . · a2N 0 ∪ ξ ) = 0; thus a N 0 +1 · . . . · a2N 0 ∩ ξ = i ∗ (η) for some η in K ∗0 (Z , T )geom . By the projection formula, we get a1 · . . . · a2N 0 ∪ ξ = i ∗ i ∗ (a1 ) · . . . · i ∗ (a N 0 ) ∪ η , which is zero by our choice of N 0 and by the fact that rank morphisms commute with . pullbacks. Thus, N = 2N 0 satisfies our proposition. Remark 5.3 By Corollary 5.2, K ∗ (X, G)geom is isomorphic to the localization of K ∗ (X, G)3 at the multiplicative subset (rk0 )−1 (1), where rk0 : K 0 (X, G)3 → 3 is the rank morphism. Therefore, if X is regular, K ∗ (X, G)geom depends only on the quotient stack [X/G] (see [LMB]) and not on its presentation as a quotient. The main theorem of this paper is the following. 5.4 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group with a sufficiently rational action on X having finite stabilizers. Suppose, moreover, that for any essential dual cyclic k-subgroup scheme σ ⊆ G, the quotient algebraic space G/CG (σ ) is smooth over k (which is the case if, e.g., G is smooth or abelian). Then C (G) is finite, and the map defined in (5), Y w (σ ) 9 X,G : K ∗ (X, G)3 −→ K ∗ (X σ , C(σ ))geom ⊗ e R(σ )3 G , THEOREM
σ ∈C (G)
is an isomorphism of R(G)-algebras.
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VEZZOSI and VISTOLI
Remark 5.5 In Section 5.1 we also give less restrictive hypotheses on G under which Theorem 5.4 still holds. Note also that if X has the “G-equivariant resolution property” (i.e., any Gequivariant coherent sheaf is the G-equivariant epimorphic image of a G-equivariant locally free coherent sheaf), then in Theorem 5.4 one can replace our K ∗ with Quillen K -theory of G-equivariant locally free coherent sheaves. This happens, for example, if X is a scheme and G is smooth or finite (see [Th3]). 5.1. Proof of Theorem 5.4 Let us choose, for some n, a closed immersion G ,→ GLn,k and consider the algebraic space quotient . Y = GLn,k ×G X. We claim that if the theorem holds for Y with the induced GLn,k -action, then it holds for X with the given G-action. First, let us note that Y is separated. The action map ψ : G ×(GLn,k ×X ) → (GLn,k ×X )×(GLn,k ×X ) is proper (hence a closed immersion) since its composition with the separated map p123 : GLn,k ×X × GLn,k ×X → GLn,k ×X × GLn,k (here we use that X is separated) is just id X ×a, where a is the action map of G on GLn,k ; hence it is proper (see [EGAI, Rem. 5.1.7], which obviously carries over to . algebraic spaces). Let P = X × GLn,k . In the Cartesian diagram j
P ×Y P −−−−→ P × P π×π y y Y
−−−−→ Y × Y 1Y
P ×Y P ' G × P since π : P → Y is a G-torsor, and π is faithfully flat; therefore, 1Y is a closed immersion; that is, Y is separated. Note that 3(Y,GLn,k ) = 3. Consider the morphism defined in (5), Y w (σ ) 9 X,G : K ∗ (X, G)3 −→ K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3 G . (27) σ ∈C (G)
By Theorem 4.1, the map 9Y,GLn,k : K ∗ (Y, GLn,k )3 −→
Y ρ∈C (GLn,k )
K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3
wGL
n,k
(ρ)
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
37
is an isomorphism, and by the Morita equivalence theorem (see [Th3, Prop. 6.2]), K ∗ (Y, GLn,k )3 ' K ∗ (X, G)3 . We prove the theorem by constructing an isomorphism Y
w (ρ) K ∗ (Y ρ , C(ρ))geom ⊗ e R(ρ) GLn,k
ρ∈C (GLn,k )
−→
Y
K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3
wG (σ )
(28)
σ ∈C (G)
commuting with the 9’s and Morita isomorphisms. Let α : C (G) → C (GLn,k ) be the natural map. If Y ρ 6 = ∅, there exists a dual cyclic subgroup σ ⊆ G, GLn,k -conjugate to ρ (and X σ 6 = ∅); therefore, Y ρ = ∅ unless ρ ∈ im(α), and we may restrict the first product in (28) to those ρ in the image of α and suppose im(α) ⊆ C (G) as well. The following proposition describes the Y ρ ’s that appear. PROPOSITION 5.6 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group with a sufficiently rational action on X having finite stabilizers. Suppose, moreover, that for any essential dual cyclic k-subgroup scheme σ ⊆ G, the quotient algebraic space G/CG (σ ) is smooth over k. Let G ,→ GLn,k be a closed embedding, let ρ ∈ im(α) be an essential dual cyclic subgroup, and let . Y = GLn,k ×G X be the algebraic space quotient for the left diagonal action of G. If CGLn,k ,G (ρ) ⊆ C (G) denotes the fiber α −1 (ρ), then (i) choosing for each σ ∈ CGLn,k ,G (ρ) an element u ρ,σ ∈ GLn,k (k) such that u ρ,σ σ u −1 ρ,σ = ρ (in the obvious functor-theoretic sense) determines a unique isomorphism of algebraic spaces over k, a jρ : NGLn,k (σ ) ×NG (σ ) X σ −→ Y ρ ; σ ∈CGLn,k ,G (ρ)
(ii)
CGLn,k ,G (ρ) is finite.
Proof Part (ii) follows from (i) since Y ρ is quasi-compact. The proof of (i) requires several steps. (a) Definition of jρ . If σ ∈ CGLn,k ,G (ρ), let Nσ be the presheaf on the category Sch/k of k-schemes which associates to T → Spec k the set σ . NGLn,k (σ )(T ) × X (T ) ; Nσ (T ) = NG (σ )(T )
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VEZZOSI and VISTOLI
since NG (σ ) acts freely on NGLn,k (σ ) × X σ (on the left), the flat sheaf associated to bρ be the presheaf on Sch/k which associates to Nσ is NGLn,k (σ ) ×NG (σ ) X σ . Let Y T → Spec k the set n GLn,k (T ) × X (T ) 0 . bρ (T ) = Y [A, x] ∈ ∀T G(T ) o → T, ∀r ∈ ρ(T 0 ), [r A T 0 , x T 0 ] = [A T 0 , x T 0 ] ; bρ is Y ρ (e.g., see [DG, Chap. II, §1, n. 3]). If u ρ,σ ∈ the flat sheaf associated to Y GLn,k (k) is such that u ρ,σ σ u −1 ρ,σ = ρ (in the obvious functor-theoretic sense), the presheaf map b bρ , jρ,σ : Nσ −→ Y b bρ (T ) jρ,σ (T ) : Nσ (T ) 3 [B, x] −→ [u ρ,σ B, x] ∈ Y is easily checked to be well defined. Let jρ,σ : NGLn,k (σ ) ×NG (σ ) X σ → Y ρ denote . ` the associated sheaf map, and define jρ = σ ∈CGL ,G (ρ) jρ,σ . n,k (b) The map jρ induces a bijection on geometric points. This is an elementary check. Let ξ ∈ Y ρ () be a geometric point. Then there exist an fppf cover T0 → bρ (T0 ) representing ξ . Therefore, for each T → T0 Spec and an element [A, x] ∈ Y and each r ∈ ρ(T ) there exists g ∈ G(T ) such that r A T g −1 = A T , gx T = x T . Then A−1 ρ A defines (functorially over T0 ) a dual cyclic subgroup scheme σ00 of G (T0 ) over T0 . Since σ00 is isomorphic to some µn,T0 , it descends to a dual cyclic subgroup σ 0 of G over k which is GLn,k -conjugate to ρ since T0 → Spec has a section and GLn,k satisfies our rationality condition (RC) (see Rem. 2.2(i)). By definition of CGLn,k ,G (ρ), there exists a unique σ ∈ CGLn,k ,G (ρ) which is G-conjugated to σ 0 over k; that is, gσ 0 g −1 = σ (functorially) for some g ∈ G(k). Since σ ∈ CGLn,k ,G (ρ), there is an element u ∈ GLn,k (k) such that uσ u −1 = ρ. Therefore, u −1 Ag −1 restricted to T0 is in NGLn,k (σ )(T0 ), gx ∈ X σ (T0 ), and if [u −1 Ag −1 , gx]∼ denotes the element in (NGLn,k (σ ) ×NG (σ ) X σ )() represented by the element [u −1 Ag −1 , gx] in Nσ (T0 ), we have jρ,σ ()([u −1 Ag −1 , gx]∼ ) = ξ by definition of jρ,σ . Thus, jρ () is surjective. 0 Now, let η ∈ (NGLn,k (σ ) ×NG (σ ) X σ )() resp., η0 ∈ (NGLn,k (σ 0 ) ×NG (σ ) 0 X σ )() for σ and σ 0 in CGLn,k ,G (ρ). Choosing a common refinement, we can assume that there exists an fppf cover T0 → Spec such that η (resp., η0 ) is represented by an element [B, y] ∈ Nσ (T0 ) (resp., [B 0 , y 0 ] ∈ Nσ 0 (T0 )). If jρ ()(η) =
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
39
jρ ()(η0 ), there exists an fppf cover T1 → T0 such that [u ρ,σ B, y] = [u ρ,σ 0 B 0 , y 0 ] in GLn,k (T1 ) × X (T1 )/G(T1 ); that is, there is an element g ∈ G(T1 ) such that u ρ,σ Bg −1 = u ρ,σ 0 B 0 gy = y
0
in GLn,k (T1 ),
in X (T1 ).
Then it is easy to check that σ = g −1 σ 0 g over T1 and, as in the proof of surjectivity of jρ (), since T1 → Spec has a section and G satisfies our rationality condition (RC), σ and σ 0 are G-conjugated over k as well, and therefore σ = σ 0 as elements in CGLn,k ,G (ρ). In particular, g ∈ NG (σ )(T1 ) and [B, y] = [B 0 , y 0 ] in Nσ (T1 ). Since T1 → Spec is still an fppf cover, we have η = η0 and jρ () is injective. (c) Each jρ,σ is a closed and open immersion. It is enough to show that each jρ,σ is an open immersion because in this case it is also a closed immersion, Y ρ being ` quasi-compact. Since NGLn,k (ρ) acts on both σ ∈CGL ,G (ρ) NGLn,k (σ ) ×NG (σ ) X σ n,k and Y ρ and since jρ is equivariant, it will be enough to prove that jρ,ρ is an open immersion. We prove first that jρ,ρ is injective and unramified and then conclude the proof by showing that it is also flat (in fact, an e´ tale injective map is an open immersion). (c1 ) The map jρ,ρ is injective and unramified. It is enough to show that the inverse image under jρ,ρ of a geometric point is a (geometric) point. Consider the commutative diagram NGLn,k (ρ) × X ρ py
l
−−−−→ GLn,k ×X π y
NGLn,k (ρ) ×NG (ρ) X ρ −−−−→ i ρ ◦ jρ,ρ
Y
where l and i ρ : Y ρ ,→ Y are the natural inclusions and where p, π are the natural projections. Let y0 be a geometric point of Y in the image of i ρ ◦ jρ,ρ ; using the action of NGLn,k (ρ) on NGLn,k (ρ) ×NG (ρ) X ρ and Y ρ , we may suppose that y0 is of the form [1, x0 ] ∈ Y ρ (), with an algebraically closed field and x0 ∈ X ρ (). Obviously, −1 (y ), and, by faithful flatness (1, x0 ) ∈ NGLn,k (ρ) ×NG (ρ) X ρ () is contained in jρ,ρ 0 −1 of p, jρ,ρ (y0 ) = (1, x0 ) if p −1 (1, x0 ) = π −1 (y0 ) ∩ NGLn,k (ρ) ×NG (ρ) X ρ .
(29)
But G() ' π −1 (y0 ) via g 7→ (g −1 , gx0 ) and NG (ρ)() ' p −1 ((1, x0 )) via h 7→ (h −1 , hx0 ); therefore, (29) follows from NG (ρ) = NGLn,k (ρ) ∩ G. (c2 ) The map jρ,ρ is flat. This fact is proved in Section 5.2, where we also single out a more general technical hypothesis for the action of G on X under which Proposition 5.6 still holds.
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VEZZOSI and VISTOLI
The remaining part of this subsection is devoted to the conclusion of the proof of Theorem 5.4 using Proposition 5.6. First we show that Proposition 5.6(ii) allows one to define a canonical isomorphism Y w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k ρ∈C (GLn,k )
'
Y
R(σ )3 K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e
wGL
n,k
(σ )
;
σ ∈C (G)
next we show, using Lemma 2.10, that each factor in the right-hand side is isomorphic to w (σ ) K ∗ (CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 G . The conclusion (i.e., the isomorphism (28)) is then accomplished by establishing, for any regular Noetherian separated algebraic space Z on which G acts with finite stabilizers, a “geometric” Morita equivalence K ∗ (GLn,k ×G Z , GLn,k )geom ' K ∗ (Z , G)geom . First, note that the choice of a family {u ρ,σ | σ ∈ CGLn,k ,G (ρ)} of elements u ρ,σ ∈ GLn,k (k) such that u ρ,σ σ u −1 ρ,σ = ρ, which uniquely defines jρ in Proposition 5.6, also determines a unique family of isomorphisms int(u ρ,σ ) : CGLn,k (ρ) → CGLn,k (σ ) σ ∈ CGLn,k ,G (ρ) (where int(u ρ,σ ) denotes conjugation by u ρ,σ ), and this family gives us an action of CGLn,k (ρ) on a NGLn,k (σ ) ×NG (σ ) X σ σ ∈CGLn,k ,G (ρ)
(since NGLn,k (σ ), and then CGLn,k (σ ), acts naturally on NGLn,k (σ ) ×NG (σ ) X σ by left multiplication on NGLn,k (σ )). With this action, jρ becomes a CGLn,k (ρ)-equivariant isomorphism, and since int(u ρ,σ ) induces an isomorphism R(CGLn,k (ρ)) ' R(CGLn,k (σ )) commuting with rank morphisms, jσ induces an isomorphism K ∗ Y ρ , CGLn,k (ρ) geom ⊗ e R(ρ)3 Y K ∗ NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 ' σ ∈CGLn,k ,G (ρ)
which, by definition of the action of NGLn,k (ρ) on each NGLn,k (σ )×NG (σ ) X σ , induces
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
41
an isomorphism w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k Y w (σ ) ' K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 GLn,k . σ ∈CGLn,k ,G (ρ)
(30) Now, if jρ0 is induced, as in Proposition 5.6, by another choice of a family {vρ,σ | −1 = ρ, then σ ∈ CGLn,k ,G (ρ)} of elements vρ,σ ∈ GLn,k (k) such that vρ,σ σ vρ,σ −1 vρ,σ u ρ,σ ∈ NGLn,k (σ )(k) and there is a commutative diagram −1 u (vρ,σ ρ,σ )·
NGLn,k (σ ) ×NG (σ ) X σ PPP PPP PPP jρ PPPP (
Yρ
/
NGLn,k (σ ) ×NG (σ ) X σ nnn nnn n n nn jρ0 v nn n
Therefore, isomorphism (30) on the invariants is actually independent of the choice of the family {u ρ,σ | σ ∈ CGLn,k ,G (ρ)}. Since CGLn,k ,G (ρ) = α −1 (ρ) and, as already observed, Y ρ = ∅ unless ρ ∈ im(α), this gives us a canonical isomorphism Y w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k ρ∈C (GLn,k )
'
R(σ )3 K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e
Y
wGL
n,k
(σ )
.
σ ∈C (G)
Now, let us fix σ ∈ C (G), and let us choose a set A ⊂ NGLn,k (σ )(k) such that the classes in wGLn,k (σ ) of the elements in A constitute a set of representatives for the wG (σ )-orbits in wGLn,k (σ ); A is a finite set. Since CGLn,k (σ ) ×CG (σ ) X σ ,→ NGLn,k (σ ) ×NG (σ ) X σ is an open and closed immersion, the morphism a CGLn,k (σ ) ×CG (σ ) X σ −→ NGLn,k (σ ) ×NG (σ ) X σ , A
[C, x] Ai ∈A 7 −→ [Ai · C, x] (in the obvious functor-theoretic sense), which is easily checked to induce an isomorphism on geometric points, is an isomorphism. Therefore, there is an induced isomorphism Y K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 A
' K ∗ NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 .
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VEZZOSI and VISTOLI
Since wGLn,k (σ ) acts transitively on A with stabilizer wG (σ ), by Lemma 2.10 we get a canonical isomorphism w (σ ) K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 GLn,k w (σ ) ' K ∗ (CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 G . Since, by Morita equivalence (see [Th3, Prop. 6.2]), K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) ' K ∗ X σ , CG (σ ) ,
(31)
to conclude the proof of Theorem 5.4 we need only show that the natural morphism K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) geom ' K ∗ X σ , CG (σ ) geom (32) induced by (31) is still an isomorphism. Since the diagram K ∗ GLn,k ×CG (σ ) X σ , GLn,k geom
α
/ K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ )
WWWWW WWWWW WWWWW WWWWW γ W+
geom
β
K ∗ X σ , CG (σ ) geom
is commutative and, by Morita equivalence, K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) CGLn,k (σ )
CGLn,k (σ ) ×CG (σ ) X σ , GLn,k C (σ ) GLn,k × GLn,k X σ , GLn,k ,
' K ∗ GLn,k × ' K∗
to show that β is an isomorphism it is enough to prove that for any regular separated algebraic space Z on which G acts with finite stabilizers, Morita equivalence induces an isomorphism K ∗ (GLn,k ×G Z , GLn,k )geom ' K ∗ (Z , G)geom
(33)
since in this case both α and γ are isomorphisms. Let π : R(GLn,k ) → R(G) be the restriction morphism, let ρ : R(G) → K 0 (Z , G) be the pullback along Z → Spec k, let rk0 : R(GLn,k ) → 3 and . . rk : R(G) → 3 be the rank morphisms, and let S 0 = (rk0 )−1 (1), S = (rk)−1 (1), . and T = π(S 0 ) ⊆ S; the following diagram commutes: rk0,T
/ r9 3 rr r r rr rr rkgeom r r
T −1 K 0 (Z , G)3 K 0 (Z , G)geom
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
43
where rkgeom and rk0,T denote the localizations of the rank morphism rk0 : K 0 (Z , G)3 → 3. By Morita equivalence, the natural map (which commutes with the induced rank morphisms) K 0 (GLn,k ×G Z , GLn,k )geom −→ T −1 K 0 (Z , G)3 is an isomorphism, and then, by Proposition 5.1, ker(rk0,T : T −1 K 0 (Z , G)3 → 3) is nilpotent. Now, if s ∈ S, then rk0,T (ρ(s)/1) = rk(s) = 1, and therefore T −1 K 0 (Z , G)3 → K 0 (Z , G)geom and K 0 (GLn,k ×G Z , GLn,k )geom → K 0 (Z , G)geom are both isomorphisms. Since K ∗ (GLn,k ×G Z , GLn,k )3 is naturally a K 0 (GLn,k ×G Z , GLn,k )3 -module and an R(GLn,k )3 -module via the pullback ring morphism ρ 0 : R(GLn,k )3 → K 0 (GLn,k ×G Z , GLn,k )3 , we have K ∗0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (GLn,k ×G Z , GLn,k )3 ⊗ K 0 (GLn,k ×G Z ,GLn,k )3 K 0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (Z , G)3 ⊗ K 0 (Z ,G)3 K 0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (Z , G)3 ⊗ K 0 (Z ,G)3 K 0 (Z , G)geom ' K ∗0 (Z , G)geom , which proves (32), and we conclude the proof of Theorem 5.4. 5.2. Hypotheses on G In this subsection we conclude the proof of Proposition 5.6, showing that (this is part (c2 ) of the proof) jρ,ρ : NGLn,k (ρ) ×NG (ρ) X ρ −→ Y ρ is flat. This is the only step in the proof of Proposition 5.6 where we make use of the hypothesis that the quotient algebraic space G/CG (ρ) is smooth over k. Actually, our proof works under the following weaker hypothesis. Let S denote the spectrum of the dual numbers over k, S = Spec(k[ε]), 1
and for any k-group scheme H , let H (S, H ) denote the k-vector space of isomorphism classes of pairs (P → S, y), where P → S is an H -torsor and y is a k-rational point on the closed fiber of P. Then Proposition 5.6, and hence Theorem 5.4, still
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VEZZOSI and VISTOLI
holds with hypothesis (S) for any essential dual cyclic subgroup scheme σ ⊆ G, the quotient G/CG (σ ) is smooth over k replaced by the following: (S0 ) for any essential dual cyclic k-subgroup scheme σ ⊆ G, we have σ 1 1 dim H S, CG (σ ) = dim H (S, G) . First we prove that jρ,ρ is flat assuming (S0 ) holds. Then we show that (S) implies (S0 ). Since p : NGLn,k (ρ) × X ρ → NGLn,k (ρ) ×NG (ρ) X ρ is faithfully flat, it is enough . to prove that j ρ = jρ,ρ ◦ p is flat. Let π : GLn,k ×X → Y be the projection, and let f : GLn,k ×X × G −→ GLn,k ×X, (A, x, g) 7−→ (Ag −1 , gx). Consider the following Cartesian squares: U
/ π −1 (Y ρ )
uρ
NGLn,k (ρ) × X ρ
jρ
/ GLn,k ×X
(34)
π
/ Yρ
/Y
Since π is faithfully flat, it is enough to prove that u ρ is flat. But the squares U
NGLn,k (ρ) × X ρ
/ GLn,k ×X × G pr12
f
/ GLn,k ×X
/ GLn,k ×X
π
/Y
(35)
π
are Cartesian and (in the obvious functor-theoretic sense) U = (A, x, g) ∈ GLn,k ×X × G A−1 ρ A = ρ, x ∈ X ρ ' NGLn,k (ρ) × X ρ × G. . Moreover, if P = {A ∈ GLn,k | A−1 ρ A ⊆ G}, the map −1 π −1 (Y ρ ) = (A, x) ∈ NGLn,k (ρ) × X A−1 ρ A ⊆ G, x ∈ X A ρ A −→ P × X ρ , (A, x) 7−→ (A, Ax) is an isomorphism. Therefore, we are reduced to proving that the map vρ : NGLn,k (ρ) × X ρ × G −→ P × X ρ , (A, x, g) 7−→ (Ag −1 , Ax)
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
45
is flat. But since the diagram vρ
NGLn,k (ρ) × X ρ × G −−−−→ P × X ρ pr pr13 y y 1 NGLn,k (ρ) × G
P
−−−−→ 2ρ
. (where 2ρ (A, g) = (Ag −1 )) is easily checked to be Cartesian, it is enough to show that zρ is flat. To do this, let us observe that ρ acts by conjugation on GLn,k /G (quotient by the G-action on GLn,k by right multiplication), and we have a Cartesian diagram / GLn,k P τ
(GLn,k /G)ρ
/ GLn,k /G
Then τ is a G-torsor and 2ρ is G-equivariant. Thus, the following commutative diagram, in which the vertical arrows are G-torsors, 2ρ
NGLn,k (ρ) × G −−−−→ pr1 y NGLn,k (ρ)
P τ y
−−−−→ (GLn,k /G)ρ χρ
. (where χρ (A) = [A] ∈ GLn,k /G) is Cartesian, and we may reduce ourselves to prove that χρ is flat. Now, observe that NGLn,k (ρ) acts on the left of both NGLn,k (ρ) and (GLn,k /G)ρ in such a way that χρ is NGLn,k (ρ)-equivariant. Therefore, it is enough to prove that χρ is flat when restricted to the connected component of the identity in NGLn,k (ρ), that is, that the map χρ0 : CGLn,k (ρ) −→ (GLn,k /G)ρ is flat. Now, CGLn,k (ρ) = (GLn,k )ρ , where ρ acts by conjugation; both (GLn,k )ρ and (GLn,k /G)ρ are smooth by [Th5, Prop. 3.1] (since GLn,k and GLn,k /G are smooth); each fiber of χρ0 has dimension equal to dim(CG (ρ)) because χρ0 is CGLn,k (ρ)equivariant for the natural actions; and all the fibers are obtained from (χρ0 )−1 ([1]) = CG (ρ) by the CGLn,k (ρ)-action. Therefore, χρ0 is flat if dim CGLn,k (ρ) = dim CG (ρ) + dim (GLn,k /G)ρ . (36) Note that, in any case, dim CGLn,k (ρ) ≤ dim CG (ρ) + dim (GLn,k /G)ρ .
(37)
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Since GLn,k is smooth, dim((GLn,k /G)ρ ) = dimk (T1 (GLn,k /G)ρ ), where T1 de1 notes the tangent space at the class of 1 ∈ GLn,k . Moreover, since H (S, GLn,k ) = 0, there is an exact sequence of k-vector spaces 1
0 → Lie(G) −→ Lie(GLn,k ) −→ T1 (GLn,k /G) −→ H (S, G) → 0
(38)
which, ρ being linearly reductive over k, yields an exact sequence of ρ-invariants 0 → Lie(G)ρ −→ Lie(GLn,k )ρ −→ T1 (GLn,k /G)ρ −→ H (S, G)ρ → 0. (39) 1
But GLn,k is smooth, so dimk Lie(GLn,k )ρ = dim(GLn,k )ρ = dim CGLn,k (ρ) , and, since Lie(CG (ρ)) = Lie(G) ∩ Lie(CGLn,k (ρ)), we get dimk Lie(G)ρ = dimk Lie(CG (ρ)) . By (39), we get 1 dimk H (S, G)ρ = dimk T1 (GLn,k /G)ρ − dim CGLn,k (ρ) + dimk Lie(CG (ρ)) = dim (GLn,k /G)ρ − dim CGLn,k (ρ) + dim CG (ρ) + dimk Lie(CG (ρ)) − dim CG (ρ) ; (40) hence (36) is satisfied if 1 dimk H (S, G)ρ = dimk Lie(CG (ρ)) − dim CG (ρ) . But
(41)
1 dimk H (S, CG (ρ)) = dimk Lie(CG (ρ)) − dim CG (ρ)
by the exact sequence (analogous to (38) with G replaced by CG (ρ)) 0 → Lie CG (ρ) −→ Lie(GLn,k ) −→ T1 GLn,k /CG (ρ) 1 −→ H S, CG (ρ) → 0; hence (41) holds by hypothesis (S0 ). We complete the proof of Proposition 5.6 by showing that (S) implies (S0 ). Since CG (ρ) ⊆ G, we have a natural map 1 1 : H S, CG (ρ) −→ H (S, G)ρ , and by (40) and (37), we get 1 1 dimk H (S, G)ρ ≥ dimk H (S, CG (ρ)) .
(42)
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Now, if (S) holds, that is, if G/CG (ρ) is smooth, and if [P → S, y] is a class in 1 H (S, G)ρ , P/CG (ρ) → S is smooth and y induces a point in the closed fiber of P/CG (ρ) → S; we may reduce the structure group to CG (ρ), thus showing that is surjective. By (42), we conclude that is an isomorphism, and this implies (S0 ).
5.3. Final remarks PROPOSITION 5.7 Let X be a Noetherian regular separated algebraic space over k, and let G be a finite group acting on X . There is a canonical isomorphism of R(G)-algebras K ∗ (X, G)geom ⊗ Z 1/|G| ' K ∗ (X )G ⊗ Z 1/|G| .
Proof Since ker rk : K 0 (X ) → H0 (X, Z[1/|G|]) is nilpotent by Corollary 5.2, the canonical homomorphism π ∗ : K ∗ (X, G) −→ K ∗ (X )G induces a ring homomorphism (still denoted by π ∗ ) π ∗ : K ∗ (X, G)geom ⊗ Z 1/|G| −→ K ∗ (X )G ⊗ Z 1/|G| . Moreover, the functor π∗ : F 7 −→
M
g∗F ,
g∈G
defined on coherent O X -modules, induces a homomorphism π∗ : K ∗0 (X )G ⊗ Z[1/|G|] −→ K ∗0 (X, G)geom ⊗ Z[1/|G|], and (recalling that K ∗ (X, G) ' K ∗0 (X, G)) we obviously get π ∗ π∗ (F ) = |G| · F . On the other hand, we have π∗ π ∗ (F ) ' F ⊗ π∗ O X . But rk(π∗ O X ) = |G|, and therefore π∗ π ∗ is an isomorphism, too, because of Corollary 5.2. As a corollary of this result and of Theorem 5.4, we recover [Vi1, Th. 1], which was proved there in a completely different way.
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We conclude the paper with a conjecture expressing the fact that K ∗ (X, G)geom should be the K -theory of the quotient X/G, if X/G is regular, after inverting the orders of all the essential dual cyclic subgroups of G. CONJECTURE 5.8 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group acting on X with finite stabilizers in such a way that the quotient X/G exists as a regular algebraic space. Let N denote the least common multiple of the orders of all the essential dual cyclic subgroups of G, and let 3 = Z[1/N ]. If p : X → X/G is the quotient map, the composition p∗
K ∗ (X/G)3 −→ K ∗ (X, G)3 −→ K ∗ (X, G)geom is an isomorphism. Remark 5.9 Bertrand Toen pointed out to us that if X/G is smooth, it follows from the results of [EG] that the composition K 0 (X/G) ⊗ Q −→ K 0 (X, G) ⊗ Q −→ K 0 (X, G)geom ⊗ Q is an isomorphism.
Appendix. Higher equivariant K -theory of Noetherian regular separated algebraic spaces In this appendix we describe the K -theories we use in the paper and their relationships. We essentially follow the example of [ThTr, Sec. 3]. We also adopt the language of [ThTr]. Let us remark that it is strongly probable that there exist equivariant versions of most of the results in [ThTr, Sec. 3]. In particular, there should exist a higher K theory of G-equivariant cohomologically bounded pseudocoherent complexes on Z (resp., of G-equivariant perfect complexes on Z ) for any quasi-compact algebraic space Z having most of the alternative models described in [ThTr, Pars. 3.5 – 3.12]. The arguments below can also be considered as a first step toward an extension of [ThTr, Lems. 3.11, 3.12] to the equivariant case on algebraic spaces. However, to keep the paper to a reasonable size, we have decided to give only the results we need, and, moreover, we have made almost no attempt to optimize the hypotheses. We would also like to mention the paper [J] (in particular, Section 1) in which, among many other results, the general techniques of [ThTr] are used as guidelines for the K -theory of arbitrary Artin stacks.
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49
We work in a slightly more general situation than required in the rest of the paper. Let S be a separated Noetherian scheme, and let G be a group scheme affine over S which is finitely presented, separated, and flat over S. We denote by G-AlgSpreg the category of regular Noetherian algebraic spaces separated over S with an action of G over S and equivariant maps. Definition A.1 If X ∈ G-AlgSpreg , we denote by K ∗ (X, G) (resp., K ∗0 (X, G), resp., K ∗naive (X, G)) the Waldhausen K -theory of the complicial bi-Waldhausen category (see [ThTr]) W1,X of complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology (resp., the Quillen K -theory of the abelian category of Gequivariant coherent O X -modules, resp., the Quillen K -theory of the exact category of G-equivariant locally free coherent O X -modules). PROPOSITION A.2 Let Z → S be a morphism of Noetherian algebraic spaces such that the diagonal Z → Z × S Z is affine, and let H → S be an affine group space acting on Z . Let F be an equivariant quasi-coherent sheaf on Z of finite flat dimension; then there exists a flat equivariant quasi-coherent sheaf F 0 on Z together with a surjective H equivariant homomorphism F 0 → F . In particular, if Z is regular, this holds for all equivariant quasi-coherent sheaves F on Z .
The hypotheses of Proposition A.2 ensure that the usual morphism Z × S H → Z × S Z is affine. In fact, the projection Z × S H → Z is obviously affine, the projection Z × S Z → Z has affine diagonal, so this follows from the elementary fact that if Z → U → V are morphisms of algebraic spaces, Z → V is affine, and U → V has affine diagonal, then Z → U is affine. Consider the quotient stack Z = [Z /H ] (see [LMB]); the argument above implies that the diagonal Z → Z × S Z is affine. Since an H -equivariant quasi-coherent O Z -module is the same as a quasi-coherent module over Z , now Proposition 5.3 follows from the more general result below. A.3 Let S be a Noetherian algebraic space, and let X be a Noetherian algebraic stack over S with affine diagonal. Let F be a quasi-coherent sheaf of finite flat dimension on X ; then there exists a flat quasi-coherent sheaf F 0 on X together with a surjective homomorphism F 0 → F . PROPOSITION
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Proof Take an affine scheme U with a flat morphism f : U → X ; then f is affine, and, in particular, the pushforward f ∗ on quasi-coherent sheaves is exact. Consider a quasicoherent sheaf F on X of finite flat dimension, with the adjunction map F → f ∗ f ∗ F . This map is injective; call Q its cokernel. Clearly, the flat dimension of f ∗ f ∗ F is the same as the flat dimension of F ; we claim that the flat dimension of Q is at most equal to the flat dimension of F . Now, if there were a section X → U of f , then the sequence 0 → F → f∗ f ∗F → Q → 0 would split and this would be clear. However, to compute the flat dimension of Q , we can pull back to any flat surjective map to X ; in particular, after pulling back to U , we see that f acquires a section, and the statement is checked. Now U is an affine scheme, so we can take a flat quasi-coherent sheaf P on U with a surjective map u : P → f ∗ F . Call F 0 the kernel of the composition f ∗ P → f ∗ f ∗ F → Q ; then F 0 surjects onto F , and it fits into an exact sequence 0 → F 0 → f ∗ P → Q → 0. But f ∗ P is flat over X , so the flat dimension of F 0 is less than the flat dimension of Q , unless Q is flat. But since the flat dimension of Q is at most equal to the flat dimension of F , we see that the flat dimension of F 0 is less than the flat dimension of F , unless F is flat. The proof is completed with a straightforward induction on the flat dimension of F . THEOREM A.4 Let X be an object in G-AlgSpreg . The obvious inclusions of the following complicial biWaldhausen categories induce homotopy equivalences on the Waldhausen K . theory spectra K (i) (X ) = K (Wi,X ), i = 1, 2, 3. In particular, the corresponding (i) Waldhausen K -theories K ∗ (X, G) coincide. (i) W1,X = (complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology). (ii) W2,X = (bounded complexes in G − Coh X ). (iii) W3,X = (complexes of flat quasi-coherent G-equivariant O X -modules with bounded coherent cohomology). Moreover, the Waldhausen K -theory of any of the categories above coincides with Quillen K -theory K ∗0 (X, G) of G-equivariant coherent O X -modules.
Proof By [Th2, Par. 1.13], the inclusion of W2,X in W1,X induces an equivalence of K -theory spectra. Proposition 5.3, together with [ThTr, Lem. 1.9.5] (applied to D = (flat Gequivariant O X -modules) and A = (G-equivariant O X -modules)), implies that for any object E ∗ in W1,X there exist an object F ∗ in W3,X and a quasi-isomorphism
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
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F ∗ −→ g E ∗ . Therefore, by [ThTr, Par. 1.9.7 and Th. 1.9.8], the inclusion of W3,X in W1,X induces an equivalence of K -theory spectra. The last statement of the theorem follows immediately from [Th2, Par. 1.13, p. 518]. Since any complex in W3,X is degreewise flat and X is regular (hence boundedness of cohomology is preserved under tensor product∗ ), the tensor product of complexes makes the Waldhausen K -theory spectrum of W3,X into a functor K (3) from GAlgSpreg to ring spectra, with product K (3) ∧ K (3) −→ K (3) , exactly as described in [ThTr, Par. 3.15]. In particular, by Theorem 5.3, K ∗ is a functor from G-AlgSpreg to graded rings. In the same way, the tensor product with complexes in W3,X gives a pairing K (3) ∧ K (1) −→ K (1) (1)
between the corresponding functors from G-AlgSpreg to spectra, so that K ∗ (X, G) (3)
becomes a module over the ring K ∗ (X, G) functorially in (X, G) ∈ G-AlgSpreg . We denote the corresponding cap-product by ∩ : K ∗(3) (X, G) ⊗ K ∗(1) (X, G) −→ K ∗(1) (X, G), which becomes the ring product in K ∗ (X, G) with the identifications allowed by Theorem 5.3. Note that there is an obvious ring morphism η : K ∗naive (X, G) → (3) K ∗ (X, G), and if 0
∩naive : K ∗naive (X, G) ⊗ K ∗ (X, G) −→ K ∗0 (X, G) denotes the usual “naive” cap-product on Quillen K -theories, there is a commutative diagram ∩naive
K ∗naive (X, G) ⊗ K ∗0 (X, G) −−−−→ K ∗0 (X, G) u η⊗u y y (3)
(1)
(1)
K ∗ (X, G) ⊗ K ∗ (X, G) −−−−→ K ∗ (X, G) ∩
where u is the isomorphism of Theorem 5.3. Because of that, we simply write ∩ for both the naive and nonnaive cap-products. Note that, as shown in [Th2, Par. 1.13, p. 519], K ∗0 (−, G) (and therefore K ∗ (X, G) under our hypotheses) is a covariant functor ∗ In
fact, this is a nonequivariant statement and a local property in the flat topology, so it reduces to the same statement for regular affine schemes, which is elementary (see also [SGA6]).
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for proper maps in G-AlgSpreg ; on the other hand, since any map in G-AlgSpreg has finite Tor-dimension, K ∗ (−, G) is a controvariant functor from G-AlgSpreg to (graded) rings. In fact, if f : X −→ Y is a morphism in G-AlgSpreg , the same argument in [ThTr, Par. 3.14.1] shows that there is an induced pullback exact functor (3) f ∗ : W3,Y → W3,X , and then we use Theorem A.4 to identify K ∗ (−, G) with K ∗ (−, G). PROPOSITION A.5 (Projection formula) Let j : Z −→ X be a closed immersion in G-AlgSpreg . Then, if α is in K ∗ (X, G) and β in K ∗0 (Z , G), we have j∗ j ∗ (α) ∩ β = α ∩ j∗ (β)
in K ∗0 (X, G). Proof Since j is affine, j∗ is exact on quasi-coherent modules and therefore induces an exact functor of complicial bi-Waldhausen categories j∗ : W1,Z → W1,X (the condition of bounded coherent cohomology being preserved by regularity of Z and X ). Therefore, the maps (α, β) 7 −→ j∗ j ∗ (α) ∩ β , (α, β) 7 −→ α ∩ j∗ (β) from K ∗ (X, G) × K ∗ (Z , G) to K ∗0 (X, G) ' K ∗ (X, G) are induced by the exact functors W3,X × W1,Z −→ W1,X , (F ∗ , E ∗ ) 7 −→ j∗ j ∗ (F ∗ ) ⊗ E ∗ , (F ∗ , E ∗ ) 7 −→ F ∗ ⊗ j∗ (E ∗ ).
(43)
But for any equivariant quasi-coherent sheaf F on X and G on Z , there is a natural (hence, equivariant) isomorphism j∗ ( j ∗ F ⊗ G ) ' F ⊗ j∗ G which, again by naturality, induces an isomorphism between the two functors in (43); therefore, we conclude by [ThTr, Par. 1.5.4]. Remark A.6 Since we need the projection formula only for (regular) closed immersion in this paper, we have decided to state the result only in this case. However, since, by [Th2,
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Par. 1.13 p. 519], K ∗ (X, G) coincides also with Waldhausen K -theory of the category W4,X of complexes of G-equivariant quasi-coherent injective modules on X with bounded coherent cohomology, therefore, by Theorem 5.3, it also coincides with Waldhausen K -theory of the category W5,X complexes of G-equivariant quasicoherent flasque modules on X with bounded coherent cohomology. For any proper map f : X → Y in G-AlgSpreg , we have an exact functor f ∗ : W5,X → W5,Y , which therefore gives a “model” for the pushforward f ∗ : K ∗ (X, G) → K ∗ (Y, G) (cf. [ThTr, Par. 3.16]). Now, the proof of [ThTr, Prop. 3.17] should also give a proof of Proposition 3 with j replaced by any proper map in G-AlgSpreg because it only uses [ThTr, Th. 2.5.5], which obviously holds for X and Y Noetherian algebraic spaces, and [SGA4, Exp. XVII, par. 4.2], which should give a canonical G-equivariant Godement flasque resolution of any complex of G-equivariant modules on any algebraic space in G-AlgSpreg since it is developed in a general topos. It is very probable that Theorem 5.3 and therefore the functoriality with respect to morphisms of finite Tor-dimension still hold without the regularity assumption on the algebraic spaces. On the other hand, it should also be true that with G and X as above (therefore, X regular), the Waldhausen K -theory of the category of G-equivariant perfect complexes on X coincides with K ∗0 (X, G). This last statement should follow (with a bit of work to identify K ∗0 (X, G) with the Waldhausen K -theory of Gequivariant pseudocoherent complexes with bounded cohomology on X ) from [J, Th. 1.6.2]. Acknowledgments. We wish to thank the referee for useful and precise remarks. We also thank Bertrand Toen, who pointed out the content of Remark 5.9 to us. References [SGA4] M. ARTIN, A. GROTHENDIECK, and J. L. VERDIER, Th´eorie des topos et cohomologie e´ tale des sch´emas, Vol. 3, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973. MR 50:7132 53 [SGA6] P. BERTHELOT, A. GROTHENDIECK, and L. ILLUSIE, Th´eorie des intersections et th´eor`eme de Riemann-Roch, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 6), Lecture Notes in Math. 225, Springer, Berlin, 1971. MR 50:7133 2, 23, 51 [DG] M. DEMAZURE and P. GABRIEL, Groupes alg´ebriques, Vol. 1: G´eom´etrie alg´ebrique, g´en´eralit´es, groupes commutatifs, Masson, Paris, 1970. MR 46:1800 38 [SGA3] M. DEMAZURE and A. GROTHENDIECK, Sch´emas en groupes, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 3), Lecture Notes in Math. 151, 152, 153, Springer, Berlin, 1970. MR 43:223a, MR 43:223b, MR 43:223c 12, 16, 28 [EG] D. EDIDIN and W. GRAHAM, Riemann-Roch for equivariant Chow groups, Duke
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Math. J. 102 (2000), 567 – 594. MR 2001f:14018 2, 4, 48 [SGA1] A. GROTHENDIECK, Revˆetements e´ tales et groupe fondamental, S´eminaire de G´eom´etrie alg´ebrique du Bois-Marie (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971. 27, 30 [EGAI] A. GROTHENDIECK and J. A. DIEUDONNE´ , El´ements de g´eom´etrie alg´ebrique, I: Le langage des sch´emas, Springer, Berlin, 1971. 36 [J] R. JOSHUA, Higher intersection theory on algebraic stacks, II, preprint, 2000, http://math.ohio-state.edu/˜joshua/pub.html 48, 53 [LMB] G. LAUMON and L. MORET-BAILLY, Champs alg´ebriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin, 2000. MR 2001f:14006 35, 49 [Me] A. S. MERKURJEV, Comparison of the equivariant and the standard K -theory of algebraic varieties, St. Petersburg Math. J. 9 (1998), 815 – 850. MR 99d:19003 15, 16 [Q] D. QUILLEN, “Higher algebraic K -theory, I” in Algebraic K -Theory, I: Higher K -Theories (Seattle, 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973. MR 49:2895 [Se] J.-P. SERRE, Linear representations of finite groups, Grad. Texts in Math. 42, Springer, New York, 1977. MR 56:8675 18 [Sr] V. SRINIVAS, Algebraic K -Theory, 2d ed., Progr. Math. 90, Birkh¨auser, Boston, 1996. MR 97c:19001 [Th1] R. W. THOMASON, Comparison of equivariant algebraic and topological K -theory, Duke Math. J. 53 (1986), 795 – 825. MR 88h:18011 16, 19, 20, 21, 35 [Th2] , Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515 – 543. MR 87j:14028 50, 51, 53 [Th3] , “Algebraic K -theory of group scheme actions” in Algebraic Topology and Algebraic K -Theory (Princeton, 1983), Ann. of Math. Stud. 113, Princeton Univ. Press, Princeton, 1987, 539 – 563. MR 89c:18016 11, 19, 20, 21, 35, 36, 37, 42 [Th4] , Equivariant algebraic vs. topological K -homology Atiyah-Segal-style, Duke Math. J. 56 (1988), 589 – 636. MR 89f:14015 15 [Th5] , Une formule de Lefschetz en K -th´eorie e´ quivariante alg´ebrique, Duke Math. J. 68 (1992), 447 – 462. MR 93m:19007 12, 13, 19, 21, 22, 23, 45 [Th6] , Les K -groupes d’un sch´ema e´ clat´e et une formule d’intersection exc´edentaire, Invent. Math. 112 (1993), 195 – 215. MR 93k:19005 [ThTr] R. W. THOMASON and T. TROBAUGH, “Higher algebraic K -theory of schemes and of derived categories” in The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkh¨auser, Boston, 1990, 247 – 435. MR 92f:19001 6, 48, 49, 50, 51, 52, 53 [To1] B. TOEN, Th´eor`emes de Riemann-Roch pour les champs de Deligne-Mumford, K -Theory 18 (1999), 33 – 76. MR 2000h:14010 5 [To2] , Notes on G-theory of Deligne-Mumford stacks, preprint, arXiv:math.AG/9912172 5 [Vi1] A. VISTOLI, Higher equivariant K -theory for finite group actions, Duke Math. J. 63 (1991), 399 – 419. MR 92d:19005 2, 3, 5, 16, 18, 47 [Vi2] , “Equivariant Grothendieck groups and equivariant Chow groups” in Classification of Irregular Varieties (Trento, Italy, 1990), ed. E. Ballico,
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F. Catanese, and C. Ciliberto, Lecture Notes in Math. 1515, Springer, Berlin, 1992, 112 – 133. MR 93j:14008 2, 5
Vezzosi Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, Bologna 40127, Italy;
[email protected] Vistoli Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, Bologna 40127, Italy;
[email protected] 55
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
FINITENESS OF DE RHAM COHOMOLOGY IN RIGID ANALYSIS ¨ ELMAR GROSSE-KLONNE
Abstract For a large class of smooth dagger spaces—rigid spaces with overconvergent structure sheaf—we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot’s rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction. Introduction Let R be a complete discrete valuation ring of mixed characteristic, let π ∈ R be a uniformizer, and let k = Frac(R), k¯ = R/(π). It is a simple observation that the de Rham cohomology Hd∗R (X ) of a positive-dimensional smooth affinoid k-rigid space X computed with respect to its (usual) structure sheaf is not finite-dimensional. The idea of instead using an overconvergent structure sheaf arises naturally from the paper of P. Monsky and G. Washnitzer [25]. The Monsky-Washnitzer cohomology of a ¯ smooth affine k-scheme Spec(A) is the de Rham cohomology of A˜ † ⊗ R k, where A˜ † is a weakly complete formal lift of A. Monsky-Washnitzer cohomology has recently been shown to be finite-dimensional (independently by Berthelot [2] and, based on common work with G. Christol, by Z. Mebkhout [24]). The algebra A˜ † ⊗ R k can be geometrically interpreted as a k-algebra of overconvergent functions on the rigid space Sp( A˜ ⊗ R k), where A˜ is a lifting of A to a formally smooth π-adically complete R-algebra. In [11] we introduce a category of k-rigid spaces with overconvergent structure sheaf, which we call k-dagger spaces, and we study a functor X 7 → X 0 from this category to the category of k-rigid spaces which is not far from being an equivalence. For example, X and X 0 have the same underlying G-topological space and the same stalks of structure sheaves. Finiteness of Monsky-Washnitzer cohomology implies finiteness of de Rham cohomology for affinoid k-dagger spaces with good reduction; in the above notation, the algebra A˜ † gives rise to the affinoid k-dagger space X with 0(X, O X ) = A˜ † ⊗ R k. Our main result generalizes this statement as follows.
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 4 July 2000. Revision received 15 May 2001. 2000 Mathematics Subject Classification. Primary 14F30, 14G22. 57
58
¨ ELMAR GROSSE-KLONNE
THEOREM A (Corollary 3.5 plus Theorem 3.6) Let X be a quasi-compact smooth k-dagger space, let U ⊂ X be a quasi-compact open subset, and let Z → X be a closed immersion. Then T = X − (U ∪ Z ) has finite-dimensional de Rham cohomology Hd∗R (T ).
By [11, Theorem 3.2], this implies finiteness of de Rham cohomology also for certain smooth k-rigid spaces Y , for example, if Y admits a closed immersion i into a polydisk without boundary (at least if i extends to a closed immersion with bigger radius) or if Y is the complement of a quasi-compact open subspace in a smooth proper k-rigid space. But our main corollary is of course the following. COROLLARY B (Corollary 3.8) ∗ (X/k) (see [2]) are finite¯ For a k-scheme X of finite type, the k-vector spaces Hrig dimensional.
We do not re-prove finiteness of Monsky-Washnitzer cohomology; rather, we reduce our Theorem A to it. A big part of this paper is devoted to the study of de Rham cohomology in situations of semistable reduction. We need and prove the following theorem. THEOREM C (Theorem 2.3) S Let X be a strictly semistable formal R-scheme, and let Xk¯ = i∈I Yi be the decomposition of the closed fibre into irreducible components. For K ⊂ I , set Y K = T † i∈K Yi . Let X be a k-dagger space such that its associated rigid space is identified with Xk . For a subscheme Y ⊂ Xk¯ , let ]Y [†X be the open dagger subspace of X † corresponding to the open rigid subspace ]Y [X of Xk . Then for any ∅ 6= J ⊂ I , the S canonical map Hd∗R (]Y J [†X ) → Hd∗R ]Y J − (Y J ∩ ( i∈I −J Yi ))[†X is bijective.
Another important tool is A. de Jong’s theorem on alterations by strictly semistable pairs, in its strongest sense. We proceed as follows. After recalling some facts on dagger spaces in Section 0, we formulate in Section 1 some basic concepts about D -modules on rigid and dagger spaces. This follows the complex analytic case (see, e.g., [23]). Instead of reproducing well-known arguments, we focus only on what is specific to the nonarchimedean case. Then we construct a long exact sequence for de Rham cohomology with supports in blowing-up situations. As in [18] (for algebraic k-schemes), this results from the existence of certain trace maps for proper morphisms; we define such trace maps based on constructions from [8], [3], and [26]. Finally, we prove the important technical fact that the de Rham cohomology Hd∗R (X ) of a smooth dagger
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space X depends only on its associated rigid space X 0 ; hence knowledge of X 0 (e.g., a decomposition into a fibre product) gives information about Hd∗R (X ). In Section 2, we begin to look at R-models of the associated rigid spaces; more specifically, we consider the case of semistable reduction. Its main result is Theorem C. It enables us to reduce Theorem A, in the case where U = ∅ and X has semistable reduction, to the finiteness of Monsky-Washnitzer cohomology. In Section 3, we first prove Theorem A in the case when U = ∅ = Z . After reduction to the case where X is affinoid and defined by polynomials, we apply de Jong’s theorem to an R-model of a projective compactification of X to reduce to the finiteness result of Section 2. The case of general Z is handled by a resolution of singularities (see [4]). Then we treat the case of general U by another application of de Jong’s theorem. The formal appearence of these last arguments bears some resemblance to the finiteness proofs in [2] and [18]. But there are also distinctive features, in the simultaneous control of special and generic fibre, and in particular in our second application of de Jong’s theorem. We apply it to a certain closed immersion of R-schemes X¯ k¯ ∪ Y¯ → X¯ , where the space X − U we are interested in is realized in (the tube ]Y¯ [ of) the compactifying divisor—not in its open complement. 0. Dagger spaces Let k be a field of characteristic zero, complete with respect to a nonarchimedean valuation |.|, with algebraic closure ka , and let 0 ∗ = |ka∗ | = |k ∗ | ⊗ Q. We gather some facts from [11]. For ρ ∈ 0 ∗ , the k-affinoid algebra Tn (ρ) consists P of all series aν X ν ∈ k[[X 1 , . . . , X n ]] such that |aν |ρ |ν| tends to zero if |ν| → ∞. S ∗ The algebra Wn is defined to be Wn = ρ>1 Tn (ρ). A k-dagger algebra A is a ρ∈0 ∗
quotient of some Wn ; a surjection Wn → A endows it with a norm that is the quotient seminorm of the Gauss norm on Wn . All k-algebra morphisms between k-dagger algebras are continuous with respect to these norms, and the completion of a k-dagger algebra A is a k-affinoid algebra A0 in the sense of [6]. There is a tensor product ⊗†k in the category of k-dagger algebras. As for k-affinoid algebras, one has for the set Sp(A) of maximal ideals of A the notions of rational and affinoid subdomains, and for these the analogue of Tate’s acyclicity theorem (see [6, Theorem 8.2.1]) holds. The natural map Sp(A0 ) → Sp(A) of sets is bijective, and via this map the affinoid subdomains of Sp(A) form a basis for the strong G-topology on Sp(A0 ) from [6]. Imposing this G-topology on Sp(A), one gets a locally G-ringed space, an affinoid k-dagger space. (Global) k-dagger spaces are built from affinoid ones precisely as in [6]. The fundamental concepts and properties from [6] translate to k-dagger spaces. ∗ The
notation Wn is taken from [16]. There the author assigns only the name of Washnitzer to this algebra. However, the referee pointed out that the name Monsky-Washnitzer algebra is the usual one.
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There is a faithful functor from the category of k-dagger spaces to the category of k-rigid spaces, assigning to a k-dagger space X a k-rigid space X 0 (to which we refer as the associated rigid space; however, we use the notation (?)0 not only for this functor). X and X 0 have the same underlying G-topological space and the same stalks of structure sheaf. A smooth k-rigid space Y admits an admissible open affinoid S covering Y = Vi such that Vi = Ui0 for uniquely determined (up to noncanonical isomorphisms) affinoid k-dagger spaces Ui . Furthermore, this functor induces an equivalence between the respective subcategories formed by partially proper spaces as defined below. In particular, there is an analytification functor from k-schemes of finite type to k-dagger spaces. For a smooth partially proper k-dagger space X with associated k-rigid space X 0 , the canonical map Hd∗R (X ) → Hd∗R (X 0 ) between the de Rham cohomology groups is an isomorphism. This follows from applying [11, Theorem 3.2] to the morphism between the respective Hodge – de Rham spectral sequences. By a dagger space not specified otherwise, we mean a k-dagger space; we use similarly the terms dagger algebras, rigid spaces, and so on. In the sequel, all dagger spaces and rigid spaces are assumed to be quasiseparated. We denote by D = {x ∈ k; |x| ≤ 1} (resp., D0 = {x ∈ k; |x| < 1}) the unit disk with (resp., without) boundary, with its canonical structure of k-dagger or k-rigid space, depending on the context. For ∈ 0 ∗ , the ring of global functions on the polydisk {x ∈ k n ; all |xi | ≤ }, endowed with its canonical structure of kdagger space, is denoted by kh −1 · X 1 , . . . , −1 · X n i† . The dimension dim(X ) of a dagger space X is the maximum of all dim(O X,x ) for x ∈ X . We say that X is pure-dimensional if dim(X ) = dim(O X,x ) for all x ∈ X . A morphism f : X → Y of rigid or dagger spaces is called partially proper (cf. [19, p. 59]) if f is separated, if there is an admissible open affinoid covering S Y = i Yi , and if for all i there are admissible open affinoid coverings f −1 (Yi ) = S S 0 0 j∈Ji X i j = j∈Ji X i j with X i j ⊂⊂Yi X i j for every j ∈ Ji (where ⊂⊂Yi is defined as in [6]). 1 Let Z → X be a closed immersion into an affinoid smooth dagger space. There is a proper surjective morphism g : X˜ → X with X˜ smooth, g −1 (Z ) a divisor with normal crossings on X˜ , and g −1 (X − Z ) → (X − Z ) an isomorphism. LEMMA
Proof Write X = Sp(Wn /I ), Z = Sp(Wn /J ) with ideals I ⊂ J ⊂ Wn . Since these ideals are finitely generated, there are a ρ > 1 and ideals Iρ ⊂ Jρ ⊂ Tn (ρ) such that I = Iρ · Wn and J = Jρ · Wn , and such that the rigid space X ρ = Sp(Tn (ρ)/Iρ ) is smooth.
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Apply [4, Theorem 1.10] to the closed immersion Z ρ = Sp(Tn (ρ)/Jρ ) → X ρ to get a morphism of rigid spaces X˜ ρ → X ρ with the desired properties. Its restriction S to the partially proper open subspace ρ 0 n, all F ; see [12, p. 47].) If Z and Y are smooth and pure-dimensional and if M ∈ D(D Z ), we define f + (M ) = R f ! (DY ←Z ⊗L D Z M ) ∈ D(DY ). If M ∈ D− (D X ), we get f + (M ) ∈ D− (DY ). PROPOSITION 1 f g Let X → Y → Z be partially proper morphisms between smooth pure-dimensional dagger (resp., rigid) spaces. We assume that g is a projection or a closed immersion. For M ∈ D− (D X ), there is a canonical isomorphism g+ ( f + M ) ∼ = (g ◦ f )+ M in D− (D Z ).
Proof We have to show the projection formula L −1 ∼ D Z ←Y ⊗L D Z ←Y ⊗Lf −1 D DY ←X ⊗L DY R f ! (DY ←X ⊗D X M ) = R f ! ( f D X M ). Y
If g is a projection Y = W × Z → Z , we see that D Z ←Y = ωY/Z ⊗O Z D Z is a coherent (DY = DW ⊗k D Z )-right module (because ωY/Z = ωW ⊗k O Z is a coherent (DW ⊗k O Z )-right module). Therefore we obtain the above projection formula using the way-out lemma (see [17, Chapter I, Section 7]), since for finite free DY -right modules (instead of the DY -right module D Z ←Y ) the projection formula is evident. If g is a closed immersion, D Z ←Y is a locally free DY -right module. This can be shown, as in the complex analytic case, by using the fact that, locally for an admissible covering of Z , there are isomorphisms Y × Dcodim Z (Y ) ∼ = Z such that g corresponds to the zero section; this is [21, Theorem 1.18] in the rigid case but holds true also in the dagger case, as one observes by examining the proof in [21]. By the commutation of R f ! with pseudo-filtered limits (cf. [19, Section 5.3.7] or [12, Lemma 4.8]), again one reduces the proof of the projection formula to the case of finite free DY -right modules.
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1.7 Let Y → X be a closed immersion into a smooth dagger (resp., rigid) space, defined by the coherent ideal J ⊂ O X . Then 0 ∗Y (E ) = lim HomO X (O X /J n , E ) →
n
and E (∗Y ) = lim HomO X (J n , E ) →
n
for a D X -module E are again D X -modules. We get right-derived functors R0 ∗Y (−) and R(−)(∗Y ) as functors D+ (D X ) → D+ (D X ), but also as functors D(D X ) → D(D X ) and D− (D X ) → D− (D X ). We have distinguished triangles +1
R0 ∗Y (K ) → K → R K (∗Y ) → for all K ∈ D(D X ). 1.8 We list some properties of the above functors. The proofs are similar to those in [23]; the projection formulas needed can be justified as in Proposition 1. (a) If f : Z → Y is a partially proper morphism between smooth puredimensional dagger (resp., rigid) spaces and if M ∈ D− (D Z ), there is a canonical isomorphism DR( f + M )[dim(Y )] ∼ = R f ! DR(M )[dim(Z )]. (b) If in addition T → Y is a closed immersion and if TZ = T ×Y Z , there is a canonical isomorphism R0 ∗T ( p+ M ) ∼ = p+ (R0 ∗TZ M ). (c) Let X be smooth, let Yi → X be closed immersions (i = 1, 2), and let M ∈ D(D X ). There are a canonical isomorphism R0 ∗Y1 R0 ∗Y2 (M ) ∼ = R0 ∗(Y1 ∩Y2 ) (M ) and a distinguished triangle +1
R0 ∗(Y1 ∩Y2 ) (M ) → R0 ∗Y1 (M ) ⊕ R0 ∗Y2 (M ) → R0 ∗(Y1 ∪Y2 ) (M ) → . (d) (e)
Let X be smooth, let Y → X be a closed immersion, and let M ∈ D(D X ). ∼ There is a canonical isomorphism R0 ∗Y (O X ) ⊗L O X M = R0 ∗Y (M ). Let X be smooth and affinoid, let Sp(B) = Y → X be a closed immersion of pure codimension d such that all local rings Bx (for x ∈ Sp(B) = Y ) are locally complete intersections, and let F be a D X -module that is locally free as an O X -module. Then R i 0 ∗Y (F ) = 0 for all i 6 = d (a well-known algebraic fact).
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PROPOSITION 2 s Let Z → Y → X be a chain of closed immersions, where Y and X are smooth and pure-dimensional. If d = codim(s), there is a canonical isomorphism s+ R0 ∗Z OY ∼ = R0 ∗Z O X [d].
Proof (cf. [24, Lemma 3.3-1]) First, Section 1.8(b), (c) allows us to assume that Z = Y . Denote by I ⊂ O X the ideal of Y in X . Observe that DY →X = D X /I.D X and that this, as well as D X ←Y , is locally free over DY (cf. the proof of Proposition 1). We claim that there is a canonical map of D X -right modules ExtdO X (O X /I, ω X ) ⊗DY DY →X → lim ExtdO X (O X /I k , ω X ). →
k
Indeed, choose an injective resolution J • of the D X -right module ω X , and define the morphism of complexes HomO X (O X /I, J • ) ⊗DY DY →X → lim HomO X (O X /I k , J • ) →
k
as follows. Let g be a local section of HomO X (O X /I, J m ), and let P be a local section of DY →X , represented by the local section P˜ of D X . The O X -linear map O X → J m which sends 1O X to g(1O X ) · P˜ actually induces an element of lim→ HomO X (O X /I k , J m ). Indeed, if P˜ is of order n, then g(1O X ) · P˜ is ank
nihilated by I n+1 . The promised map is the one that sends g ⊗ P to the local section of lim→ HomO X (O X /I k , J m ) just described. Now since we have as usual k ωY ∼ = Extd (O X /I, ω X ), we get a map OX
ωY ⊗DY DY →X → lim ExtdO X (O X /I k , ω X ) →
k
of D X -right modules. Applying HomO X (ω X , .) (cf. Section 1.3), it becomes the map D X ←Y ⊗DY OY → lim ExtdO X (O X /I k , O X ) = R d 0 ∗Y O X →
(∗)
k
of D X -left modules. We claim that (∗) is an isomorphism. Indeed, if x1 , . . . , xn are local coordinates on X such that Y is defined by x1 , . . . , xd , and if δ1 , . . . , δn is the basis of HomO X (1X , O X ) dual to d x1 , . . . , d xn , one verifies that both sides in (∗) are identified with D X /(x1 , . . . , xd , δd+1 , . . . , δn ). Since the right-hand side in (∗) is already all of R0 ∗Y O X [d] (due to Section 1.8(e)), and since on the left-hand side we may write ⊗L instead of ⊗, we are done.
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PROPOSITION 3 Let X be smooth, proper, and of pure dimension n, let Y be smooth and purep dimensional, and let Z = X × Y → Y be the canonical projection. Then there is a canonical trace map p+ O Z [n] → OY . If, furthermore, T → Y and S → Z ×Y T = X × T are closed immersions, there are canonical trace maps p+ R0 ∗S O Z [n] → R0 ∗T OY and R0 Z , DR(R0 ∗S O Z ) [2n] → R0 Y, DR(R0 ∗T OY ) ,
which are isomorphisms if the composition S → X × T → T is an isomorphism. Proof We begin with the rigid case. In [3] P. Beyer described a finite admissible open coverS T ing X = i Ui such that all U J = i∈J Ui (for J ⊂ I ) have the following properties: there is a closed immersion U J → (D0 )n J for some n J ∈ N, and for all affinoid Yˆ , all coherent OU J ×Yˆ -modules F , and all j > n, we have R j ( p J,Yˆ )! F = 0, where p ˆ : U J × Yˆ → Yˆ denotes the projection. By means of Mayer-Vietoris sequences, J,Y
we get R j p! F = 0 for all coherent O Z -modules F , all j > n; hence R m p! •Z /Y = 0 for m > 2n and R
2n
p! •Z /Y
=
R n p! nZ /Y n n Im(R n p! n−1 Z /Y → R p! Z /Y )
.
In view of the fact that p+ O Z [n] ∼ = Rp! •Z /Y (cf. Section 1.5), to define p+ O Z [n] → OY it is therefore enough to define a map t p : R n p! ω Z /Y → OY vanishing on n n Im(R n p! n−1 Z /Y → R p! Z /Y ). We take the following map (cf. [26], [3]). For U J → D J = (D0 )n J ⊂ Dn J = Sp khT1 , . . . , Tn J i as above and projection p D J ,Yˆ : D J × Yˆ → Yˆ , there is a canonical identification R n J ( p D J ,Yˆ )! ω D J ×Yˆ /Yˆ (Yˆ ) n X = ω= aµ T µ dT1 ∧ · · · ∧ dTn J ; aµ ∈ OYˆ (Yˆ ) µ∈Zn J
µ n, and the composition of the canonical map R n p! ω Z /Y → R n p!0 ω Z 0 /Y 0 with t p0 : R n p!0 ω Z 0 /Y 0 → OY 0 has its image in OY ⊂ OY 0 ; hence we obtain a map t p : R n p! ω Z /Y → OY . (This can be checked locally on Y ; if Y is affinoid, then this t p is the direct limit of the maps t p for the morphisms of rigid spaces p : X 0 × Y0 → Y0 , for appropriate extensions Y 0 ⊂ Y0 .) If now in addition T and S are given, we can derive from t p the other promised maps using the isomorphisms from Section 1.8; note that R0 Z , DR(R0 ∗S O Z ) ∼ = R0 Y, Rp! DR(R0 ∗S O Z ) because S → T is quasi-compact. Finally, suppose that S → T is an isomorphism. Our additional statement in this situation is seen to be local on X . By the definition of t p : R n p! ω Z /Y → OY , we may replace our X with X = (D0 )n (dropping the assumption on properness). Passing to an admissible covering of Y , we may assume that there is a section s : Y → Z of p : Z → Y inducing the inverse of S → T . It comes with an isomorphism R0 ∗T OY ∼ = R0 ∗T p+ R0 ∗Y O Z [n] ∼ = p+ R0 ∗S O Z [n] by Section 1.8(b). It is enough to show that its composition with the map in question, p+ R0 ∗S O Z [n] → R0 ∗T OY , is an isomorphism. Of course, this will follow once we know that the underlying map OY ∼ = p+ s+ OY ∼ = p+ R0 ∗Y O Z [n] → p+ O Z [n] → OY
is the identity. Since OY (Y ) is Jacobson, we may assume that Y = Sp(k) for this. The definition of t p as above does not depend on the choice of the closed embedding into some (D0 )n . This tells us that the map tidY we get for X = Y = Sp(k), computed by means of the embedding s, is the identity; but on the other hand, it is precisely the map we are interested in, by the compatibility of the Gysin map in the definition of t p with the Gysin map in Proposition 2.
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COROLLARY 1.9 Let gi : Z → Yi (i = 1, 2) be closed immersions into smooth affinoid dagger spaces Yi of pure dimension n i . Then we have R0 Y1 , DR(R0 ∗Z OY1 ) [2n 1 ] ∼ = R0 Y2 , DR(R0 ∗Z OY2 ) [2n 2 ].
Proof Because of Section 1.8(a) and Proposition 2, applied to closed immersions Yi → Dn i for appropriate m i ∈ N, we may assume that Yi = Dn i . If l : Z → X = Y1 × Y2 is the diagonal embedding, it is enough to give isomorphisms R0 Yi , DR(R0 ∗Z OYi ) ∼ = R0 X, DR(R0 ∗Z O X ) [2n 3−i ]. Let i = 1. The open embedding j : X → W = Y1 × Pnk 2 induced by the open embedding into projective space Y2 = Dn 2 → Pnk 2 induces a closed immersion ( j ◦ l) : Z → W , and we have R0(X, DR(R0 ∗Z O X )) ∼ = R0(W, DR(R0 ∗Z OW )). Now apply Proposition 3. For i = 2 the argument is the same. 1.10 Let Z be an affinoid k-dagger space, q ∈ N. The definition h qd R (Z , k) = h qd R (Z ) = dimk H 2n−q (Y, DR(R0 ∗Z OY )) , where Z → Y is a closed embedding into a smooth affinoid k-dagger space Y , is justified by Corollary 1.9. For a finite field extension k ⊂ k1 , let (?)1 = (?) ×Sp(k) Sp(k1 ). Then h qd R (Z , k) = h qd R (Z 1 , k1 ). Indeed, clearly dimk (Hd∗R (X/k)) = dimk1 (Hd∗R (X 1 /k1 )) for any smooth k-dagger space X ; hence dimk H ∗ (X, DR(R0 Z O X )) = dimk1 H ∗ (X 1 , DR(R0 Z 1 O X 1 )) for closed subspaces Z of smooth k-dagger spaces X . Now use Proposition 4. 1.11 Let f : X → Y be a finite e´ tale morphism of smooth dagger (or rigid) spaces, Y S irreducible, X = X i the decomposition into connected components. Assume all maps f | X i : X i → Y to be surjective. Then there are an l = deg( f ) ∈ N and a trace map t : f ∗ •X → •Y t
such that the composition •Y → f ∗ •X → •Y is multiplication by l. In particular, Hdi R (Y ) → Hdi R (X ) is injective for all i ∈ N.
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Indeed, for admissible open connected U = Sp(A) ⊂ Y with decomposition S f −1 (U ) = j Sp(B j ) such that each B j is free over A, let t j : B j → A be the trace q q map, and for q ∈ Z, let f ∗ X (X ) → Y (Y ) be the A-linear map q
q
q
q
q
f ∗ X (X ) = B j = A ⊗ A B j → A = Y (Y ) which sends ω ⊗ b to t j (b) · ω. By the same computation as in the proof of [18, Proposition 2.2], we see that for varying q it commutes with the differentials. Clearly, P it glues for varying U , and the number l = j l j , where l j denotes the rank of B j over A, is independent of U and fulfills our requirement. In fact, the e´ taleness of f is not really needed; f is flat in any case, by regularity of X and Y . If U is as above, let L j = Frac(B j ), K = Frac(A). Then f induces P finite separable field extensions K ⊂ L j . Let l = j [L j : K ]. The trace maps q q q σ j : L j → K give rise to L j /k = K /k ⊗ K L j → K /k , ω ⊗ f 7→ σ j ( f ) · ω, q
q
restricting to σ j : B j → A . Compare with the discussion in [25, Theorem 8.3]. We do not need this. 1.12 Let X be a smooth dagger (or rigid) space, and let j : U → X be an open immersion with complement Y = X − j (U ). We do not put a structure of dagger (or rigid) space on Y . By R0 Y (.) : D+ (D X ) → D+ (D X ) we denote the right-derived functor of the left exact functor F 7 → Ker(F → j∗ j −1 F ) on abelian sheaves on X , and by R j∗ : D+ (DU ) → D+ (D X ) we denote the rightderived functor of j∗ . Note that R j∗ DR(L ) ∼ = DR(R j∗ L ) for L ∈ D+ (DU ). If 0 0 j : U → X is another open immersion with complement Y 0 = X − j 0 (U 0 ) such that U 0 ∪ U is an admissible covering of an admissible open subset of X , there is a distinguished triangle +1
R0 Y ∩Y 0 (K ) → R0 Y (K ) ⊕ R0 Y 0 (K ) → R0 Y ∪Y 0 (K ) → . 4 We have the following. (a) Let Y → X be a closed immersion into a smooth dagger (or rigid) space X . The canonical map R0 X, DR(R0 ∗Y O X ) → R0 X, DR(R0 Y O X ) (∗) PROPOSITION
is an isomorphism.
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(b)
Let Z → Y be another closed immersion. There is a long exact sequence · · · → H i X, DR(R0 ∗Z O X ) → H i X, DR(R0 ∗Y O X ) → H i X − Z , DR(R0 ∗Y −Z O X −Z ) → H i+1 X, DR(R0 ∗Z O X ) → · · · .
Proof Step 1: The rigid case. First assume that Y is locally defined by a single equation. Then the inclusion j : U = (X − Y ) → X of the complement is a quasi-Stein morphism, hence (see [22]) acyclic for coherent OU -modules. (Since we do not know if the analogue in the dagger case holds, we are forced to distinguish.) It follows • . On the other hand, R O (∗Y ) = O (∗Y ), and by [21, that DR(R j∗ OU ) = j∗ U X X • is an isomorphism. We get Theorem 2.3], the canonical map D R(O X (∗Y )) → j∗ U (a) for this type of Y . For general Y , assertion (a) is now deduced by an induction on the number of defining local equations, using the Mayer-Vietoris sequences from Sections 1.8 and 1.12. Assertion (b) follows from (a) and the fact that for every sheaf F on X we have a natural distinguished triangle +1
R0 Z F → R0 Y F → R j∗ R0 Y −Z j −1 F →, where j : (X − Z ) → X is the open immersion. Take an injective resolution I • of F ; then j −1 I • is an injective resolution of j −1 F , and 0 → 0 Z I • → 0 Y I • → j∗ 0 Y −Z j −1 I • → 0 is exact. Step 2: The dagger case. Again (b) follows from (a). For (a), first assume that Y is also smooth. As in the proof of Proposition 1, we find an affinoid admissible open S covering X = i∈I Ui such that for each i ∈ I either Ui ∩ Y is empty or there exists an isomorphism φi : Ui ∼ = Dm × (Ui ∩ Y ) such that Ui ∩ Y → Ui is the zero section. By a Cech argument, one sees that it is enough to prove that for all finite and T nonempty subsets J of I , if we set U J = i∈J Ui , the canonical map R0 U J , DR(R0 ∗Y O X ) → R0 U J , DR(R0 Y O X ) is an isomorphism. If U J ∩ Y is empty, this is trivial, so we assume that U J ∩ Y is nonempty. Choose one j ∈ J . For ∈ |k ∗ | ∩ ]0, 1], let D() be the closed disk of radius (with its dagger structure), and let m U j,J, = φ −1 j D () × (U J ∩ Y ) .
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The set of open subspaces U j,J, is cofinal in the set of all open neighbourhoods of U J ∩ Y in the affinoid space U j,J,1 . Since U J ∩ U j,J,1 is such a neighbourhood, we find an 0 such that U j,J,0 ⊂ U J . Now observe that the canonical restriction maps R0 U J , DR(R0 ∗Y O X ) → R0 U j,J,0 , DR(R0 ∗Y O X ) , R0 U J , DR(R0 Y O X ) → R0 U j,J,0 , DR(R0 Y O X ) are isomorphisms. Therefore we need to show that R0 U j,J,0 , DR(R0 ∗Y O X ) → R0 U j,J,0 , DR(R0 Y O X ) is an isomorphism. In other words, we may assume from the beginning that X = Dm × Y and that Y → X is the zero section. Let D = Dm ⊂ Pm k = (projective space) = P, let 0 be its origin, let W = P × Y , V = P − {0}, and think of Y = {0} × Y as embedded into W . Since the natural restriction maps R0 W, DR(R0 ∗Y OW ) → R0 X, DR(R0 ∗Y O X ) , R0 W, DR(R0 Y OW ) → R0 X, DR(R0 Y O X ) are isomorphisms, it suffices to show that R0 W, DR(R0 ∗Y OW ) → R0 W, DR(R0 Y OW ) is an isomorphism. The dagger spaces {0}, P, and V are partially proper; therefore Step 1(b) applies to give us the long exact Gysin sequence i−2m+1 i i · · · → Hdi−2m ({0}) → · · · . R ({0}) → Hd R (P) → Hd R (V ) → Hd R
By the K¨unneth formulas (in this case easily derived from Lemma 3), we thus obtain the long exact sequence i−2m+1 i i · · · → Hdi−2m (Y ) → · · · . R (Y ) → Hd R (W ) → Hd R (W − Y ) → Hd R
Because of Hd∗−2m (Y ) ∼ = H ∗ (W, DR(R0 ∗Y OW )), this implies what we want. R Now for arbitrary Y , we may as in Step 1 suppose that Y is defined by a single equation and that X is affinoid. Then we can reduce to the case where Y is a divisor with normal crossings as in [14], considering a proper surjective morphism g : X 0 → X with X 0 smooth, U 0 = g −1 (U ) → U an isomorphism, and g −1 (Y ) a divisor with normal crossings on X 0 (such a g exists by Lemma 1). But in view of the MayerVietoris sequences from Sections 1.8 and 1.12, the normal crossings divisor case is equivalent to the case where Y is smooth, which has been treated above.
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COROLLARY 1.13 Let f : X 0 → X be a proper morphism of dagger (or rigid) spaces, and let Y → X be a closed immersion such that f | X 0 −Y 0 : (X 0 − Y 0 ) → (X − Y ) is an isomorphism, where Y 0 = X 0 × X Y . Let b : X → Z be a closed immersion into a smooth space, and let a : X 0 → W 0 be a locally closed immersion into a smooth proper space of pure dimension n. Then (a, b ◦ f ) : X 0 → W 0 × Z = Z 0 is a closed embedding, and there is a long exact sequence · · · → H i Z 0 , DR(R0 ∗Y 0 O Z 0 ) → H i−2n Z , DR(R0 ∗Y O Z ) ⊕ H i Z 0 , DR(R0 ∗X 0 O Z 0 ) → H i−2n Z , DR(R0 ∗X O Z ) → H i+1 Z 0 , DR(R0 ∗Y 0 O Z 0 ) → · · · .
Proof By Proposition 3, there is a morphism between the acyclic complexes that we get when we apply Proposition 4(b) to Y 0 → X 0 → Z 0 and to Y → X → Z ; observe that H i Z 0 − Y 0 , DR(R0 ∗X 0 O Z 0 ) ∼ = H i W 0 × (Z − Y ), DR(R0 ∗X 0 O Z 0 ) for this, and that by the construction of the trace map in Proposition 3, it is indeed a morphism of complexes; that is, the resulting diagrams commute. Every third rung of this morphism of complexes is bijective (also by Proposition 3); therefore we can perform a diagram chase according to the pattern in the proof of [18, Proposition 4.3]. LEMMA 2 Let X 1 and X 2 be smooth dagger spaces, and let φ : X 10 → X 20 be an isomorphism of the associated rigid spaces. Then φ gives rise to an isomorphism φ † : Hd∗R (X 2 ) ∼ = Hd∗R (X 1 ).
Proof ˜ is Set X = X 1 × X 2 , X 0 = X 10 × X 20 , and φ˜ = (id, φ) : X 10 → X 0 . Then 1 = Im(φ) 0 0 a Zariski closed subspace of X isomorphic to X 1 . The canonical projections X 1 ← X → X 2 induce maps a1
a2
Hd∗R (X 1 ) −→ H ∗ (X, lim jV ∗ •V ) ←− Hd∗R (X 2 ), →
V
where in the middle the term V runs through the open immersions jV : V → X of dagger spaces with 1 ⊂ V 0 and where V 0 is the rigid space associated with V , regarded as an open subspace of X 0 . We claim that the ai are isomorphisms. The
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claim is local, so we may assume that X 1 and X 2 are affinoid and connected, that there are elements t1 , . . . , tm ∈ O X 1 (X 1 ) = A1 such that dt1 , . . . , dtm is a basis of 1X 1 (X 1 ) over A1 , and that there exist an open affinoid subspace U ⊂ X such that 1 ⊂ U 0 , where U 0 ⊂ X 0 is the associated rigid space, an element δ ∈ 0 ∗ , and an isomorphism of rigid spaces ρ : U 0 → Sp khδ −1 · Z 1 , . . . , δ −1 · Z m i × 1, where δ −1 · Z i is sent to δ −1 · (ti ⊗ 1 − 1 ⊗ (φ ∗ )−1 (ti )) ∈ OU 0 (U 0 ) (cf. [21, Theorem 1.18]). For 0 < ≤ δ, let U0 = ρ −1 Sp(kh −1 · Z 1 , . . . , −1 · Z m i) × 1 . This is an open subspace of X 0 , and we let U be the corresponding open subspace of X . Since U0 is a Weierstrass domain in X 0 , the same is true for U in X . (If necessary, modify the defining functions slightly to get overconvergent ones.) In particular, U is • = Rj • affinoid, so jU ∗ U U ∗ U . Since X is quasi-compact, lim → commutes with V formation of cohomology, and since the U are cofinal in {V }, it is now enough to show that for arbitrary 0 < ≤ δ the maps bi, : Hd∗R (X i ) → Hd∗R (U ) are isomorphisms (i = 1, 2). By [5], we can find an isomorphism σ : X 1 → X 2 such that the induced map σ 0 : X 10 → X 20 is close to φ, in particular, so close that for σ˜ = (id, σ ) : X 1 → X we ρ have Im(σ˜ ) ⊂ U/2 . Similarly, we can approximate the map O X 0 (X 10 ) ∼ = O1 (1) → 1 OU 0 (U 0 ) → OU0 (U0 ) by a map O X 1 (X 1 ) → OU (U ). Its extension to the map ∗
O X 1 (X 1 ) ⊗†k kh −1 · Z 1 , . . . , −1 · Z m i† → OU (U )
which sends −1 · Z i to −1 · (ti ⊗ 1 − 1 ⊗ (σ ∗ )−1 (ti )) is an isomorphism since its completion is close to the isomorphism obtained from ρ. So we have an isomorphism U ∼ = Sp(kh −1 · Z 1 , . . . , −1 · Z m i† ) × X 1 , where the closed immersion σ˜ : X 1 → U corresponds to the zero section. Hence the maps Hd∗R (U ) → Hd∗R (X 1 ) induced by σ˜ are isomorphisms, by Lemma 3. Since σ˜ is a section for the canonical map U → X 1 that gives rise to b1, , we derive the bijectivity of b1, . That b2, is bijective is seen symmetrically. Now we define φ † = a1−1 ◦ a2 . One can show that this construction is compatible with compositions: If X 3 is a third dagger space with associated rigid space X 30 , and if γ : X 20 → X 30 is an isomorphism, then φ † ◦ γ † = (γ ◦ φ)† (see [13]). We do not need this fact here.
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2. De Rham cohomology of tubes of a semistable reduction From now on, let R be a complete discrete valuation ring of mixed characteristic (0, p), and let π ∈ R be a uniformizer, k its fraction field, and k¯ = R/(π ) its residue field. 3 We have the following. (a) Let r, n ∈ Z, 0 ≤ r ≤ n, and let µ and γi for 1 ≤ i ≤ r and δi for 1 ≤ i ≤ n Q be elements of 0 ∗ such that γi ≤ δi for all 1 ≤ i ≤ r , and ri=1 δi ≥ µ. Define the open dagger subspace V of the dagger affine space Ank by LEMMA
r n Y V = (x1 , . . . , xn ) ∈ Ank ; |xi | ≥ µ, |xi | ≤ δi i=1
o for all 1 ≤ i ≤ n, and |xi | ≥ γi for all 1 ≤ i ≤ r . q
Let X 1 , . . . , X r be the first r standard coordinates on Ank . Then Hd R (V ) is the kvector space generated by the classes of the q-forms d X i1 / X i1 ∧ · · · ∧ d X iq / X iq q
with 1 ≤ i 1 < · · · < i q ≤ r . In particular, if r = 0, we have Hd R (V ) = 0 for all q > 0. If X is another smooth dagger space, the canonical maps M
q
q
β
q
Hd R1 (X ) ⊗k Hd R2 (V ) → Hd R (X × V )
q1 +q2 =q
are bijective. Q (b) Suppose even γi < δi for all 1 ≤ i ≤ r , and ri=1 δi < µ. Define the open dagger (resp., rigid) subspace V of the dagger (resp., rigid) affine space Ank by r n Y n V = (x1 , . . . , xn ) ∈ Ak ; |xi | > µ, |xi | < δi i=1
o for all 1 ≤ i ≤ n, and |xi | > γi for all 1 ≤ i ≤ r . Then the same assertions as in (a) hold. (Of course, if V is the dagger (resp., rigid) space, then X should be a dagger (resp., rigid) space, too.) Proof (a) Note that V is affinoid. We may assume that X is also affinoid and connected, X = Sp(B). After a finite extension of k, we may assume that there are δ i , γ i , and
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µ in k such that |δ i | = δ i , |γ i | = γ i , and |µ| = µ. We regard O X ×V (X × V ) as a subring of
−1 −1 −1 −1 −1 . B δ −1 1 · X 1 , (δ 1 · X 1 ) , . . . , δ n · X n , (δ n · X n ) (Computing in this ring was suggested by the referee.) (i) We begin with the following observation. Let a=
X
θv
v∈Zn
n Y
v
X j j ∈ O X ×V (X × V ),
j=1
θv ∈ B, θv = 0 whenever there is an r < j ≤ n with v j < 0. Fix 1 ≤ l ≤ n. We claim that X
b=
vl−1 θv
v∈Zn
n Y
v
X j j ∈ O X ×V (X × V ),
j=1
vl 6 =0
that is, that this sum also converges in O X ×V (X × V ). Indeed, consider the surjection of dagger algebras τ
D = B ⊗†k khX 1 , . . . , X n , Y1 , . . . , Yr , Z i† −→ O X ×V (X × V ), r −1 Y −1 −1 −1 −1 X i 7→ δi · X i , Yi 7→ (γ i · X i ) , Z 7→ µ · Xi . i=1
By the definition of ⊗†k , we have ˆ k khδ −1 · X 1 , . . . , δ −1 · X n , δ −1 · Y1 , . . . , δ −1 · Yr , δ −1 · Z i, D = lim O X (X )⊗ →
X ,δ
where the X run through the strict neighbourhoods of X 0 , the rigid space associated with X , in an appropriate affinoid rigid space that contains X 0 as a relatively compact open subset, and where δ runs through all δ > 1. Hence, a given c=
X w∈Zn+m+1 ≥0
βw
n Y j=1
wj
Xj
r Y
w j+n
Yj
Z wn+r +1 ∈ D
j=1
(βw ∈ B) is an element of ˆ k khδ −1 · X 1 , . . . , δ −1 · X n , δ −1 · Y1 , . . . , δ −1 · Yr , δ −1 · Z i O X (X )⊗
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for some X , δ, and one easily sees that d=
X
(wl − wl+n − wn+r +1 )−1 βw
n Y
wj
Xj
r Y
j=1
w∈Zn+m+1 ≥0
w j+n
Yj
Z wn+r +1
j=1
wl −wl+n −wn+r +1 6 =0
is then an element of ˆ k khδ1−1 · X 1 , . . . , δ1−1 · X n , δ1−1 · Y1 , . . . , δ1−1 · Yr , δ1−1 · Z i O X (X )⊗ for any 1 < δ1 < δ; in particular, d ∈ D, too. Clearly, if τ (c) = a, then τ (d) = b, and the claim follows. (ii) For 1 ≤ i 1 < · · · < i t ≤ n, we write d X i / X i = d X i1 / X i1 ∧ · · · ∧ d X it / X it , ˆ
(d X i / X i )it = d X i1 / X i1 ∧ · · · ∧ d X it−1 / X it−1 . q
Every ω ∈ X ×V (X × V ) can be uniquely written as a convergent series ω=
X
X
X
σt,i,v
0≤t≤q 1≤i 1 r . Indeed, if ω is represented as in (∗), then by (i) the series X
η=
X
0≤t≤q 1≤i 1 0 almost everywhere in X . In [HSu] a collection of forms as above is called a CartanWhitney presentation of the metric gauge determined by X . The existence of such a presentation is thus guaranteed by Proposition 8.16, and Semmes’s result about the nonexistence of local bi-Lipschitz parametrizations implies that the residue,
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Res(ρ, p), of no Cartan-Whitney presentation ρ = (ρ1 , ρ2 , ρ3 ) can be 1 at a point p ∈ F (for the terminology here, see [HSu]). Similar remarks apply to the examples based on the Whitehead continuum, Bing’s dogbone space, and other similar constructions, for example, those mentioned in Remark 8.14. The beautiful book of R. Daverman [Da] offers a thorough discussion of various decomposition spaces (see, in particular, [Da, Sec. II.9]). We leave it to the reader to visualize the particular examples of BLD-maps that arise in this manner. 9.2. Nonexistence of radial limits Recall that a continuous map f from a domain in R n into R n is quasi-regular if f is 1,n in the local Sobolev space Wloc and if the (formal) differential d f (x) of f satisfies n |d f (x)| ≤ K det d f (x) for a.e. x, for some finite K ≥ 1 (see [Re2], [Ri2] for the basic theory of these maps). BLD-maps are particular examples of quasi-regular maps by Section 6.13. The boundary behavior of BLD-maps is often easy to describe by using only definition (0.2) (cf. [MV¨a]). In contrast, the boundary behavior of bounded quasi-regular maps of the unit ball B n , say, is not well understood. For instance, it is not known whether a quasi-regular map f : B n → B n has a single radial limit if n ≥ 3. (Dimension n = 2 is special here because quasi-regular maps then factor as quasi-conformal maps composed with holomorphic functions by Bojarski’s measurable Riemann mapping theorem. In particular, radial limits exist in a set of positive Hausdorff dimension.) Martio and U. Srebro [MS] have recently shown, by an explicit construction, that there exist bounded quasi-regular maps of B n with no radial limits in an exceptional set of any prescribed Hausdorff dimension less than n − 1 on the boundary ∂ B n . The exceptional sets provided by Martio and Srebro are all totally disconnected. Next we indicate how the generalized Berstein-Edmonds Theorem 0.3 can be used to give a large family of similar examples. In our examples, the exceptional sets can be complicated topologically. The examples of Martio and Srebro correspond, in a sense, to the case where p = 0 in the proof of Theorem 9.3. One should note, however, that the maps constructed in [MS] are locally injective; our examples are necessarily branched. 9.3 Let 0 be a geometrically finite torsion free Kleinian group without parabolic elements acting on the 2-sphere S 2 with the limit set 30 not the whole sphere. Then there exists a bounded quasi-regular map f : B 3 → B 3 such that f has no radial limit at points in 30 . THEOREM
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Proof The group 0 acts in the hyperbolic 3-space, in its ball model B 3 , with the quotient M 3 = B 3 / 0 a hyperbolic 3-manifold. The action of 0 on S 2 \ 30 uniformizes finitely many Riemann surfaces 60 , . . . , 6 p , p ≥ 0, and we can compactify M by adding these surfaces to it. The resulting manifold W is a compact 3-manifold with boundary, and the assumption of geometric finiteness guarantees that the interior of W , in some fixed PL-metric, is quasi-conformally equivalent to the hyperbolic metric in M. Thus the map f = F ◦ π, where π : B 3 → M is a locally isometric covering projection and F : W → R 3 is a bounded BLD-map provided by Theorem 0.3, is a bounded quasi-regular map. (Note that the case of one boundary component, p = 0, can be reduced to Theorem 0.3 as well; we leave the details to the reader.) Clearly, the limit f (r w), r → 1, cannot exist for points w ∈ 30 . The theorem follows. It is well known that the limit set 30 , as in Theorem 9.3, must have Hausdorff dimension less than 2, but it can be arbitrarily close to 2 (see, e.g., [BJ] or [Ca] for further discussion and references). 9.4. Lipschitz maps with strange fibers In nonlinear elasticity, one is interested in finding topological properties in a mapping ˇ [MTY], for example). Reshetnyak from its analytic data only (see [Ba], [HK1], [IS], proved in 1966 (see [Re1]) that quasi-regular maps, as defined in Section 9.2, are branched covers or constants. One way to express Reshetnyak’s theorem is to consider the dilatation function |d f (x)|n K f (x) = det d f (x) 1,n for a map f ∈ Wloc , where it is understood that K (x, f ) = 1 at points x where the formal differential does not exist, or where det d f (x) = 0. Reshetnyak’s theorem states that if K f is in L ∞ , then f is a discrete and open map or constant (locally). ˇ ak [IS] ˇ proved that in dimension n = 2, it suffices to assume T. Iwaniec and V. Sver´ 1 K f ∈ L loc for the same conclusion. They also conjectured that K f ∈ L n−1 loc suffices p in all dimensions. It is now known that K f ∈ L loc for some p > n − 1 suffices (see [MVi], [ViM]). Earlier, J. Ball [Ba, Exam. 1, p. 317] had exhibited examples of nonconstant p maps f : R n → R n that take a line segment to a point and that have K f ∈ L loc for all p < n − 1. In the next theorem, we point out how the geometric decomposition space theory, as in Section 8, and the generalized Berstein-Edmonds Theorem 0.3 can be
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used to give examples of Lipschitz maps f : S 3 → S 3 , of nonzero degree, with more exotic fibers than line segments, and with K f still in some local Lebesgue space. THEOREM 9.5 For each excellent package as defined in Section 8.1 and for each 0 < p < 2, there exist an equivalent excellent package with defining sequence (Ci ) and a sensepreserving Lipschitz map f : S 3 → S 3 such that (a) f |S 3 \ ∩i Ci is a finite-to-one branched cover; T (b) each component of i Ci is a fiber of f ; p (c) K f ∈ L loc .
We call two excellent packages equivalent if there is a diffeomorphism of R 4 that conjugates the pertinent maps in the packages. In (c), we use Lebesgue 3-measure and define K f in Euclidean terms in R 3 . Note that f has nonzero degree. The map f to be constructed depends on the particular choice of exponent p. Proof Fix a defining sequence (Ci ) and 0 < p < 2; we have the freedom to choose (Ci ) up to a diffeomorphism. Here and below we use the notation from Section 8. Each domain D, D1 , . . . , Dk in the package is a handle-body in R 3 with smooth boundary. We assume that D is a smooth regular neighborhood of its core, where by a core we mean a 1-polygon inside D. We assume that D is essentially a 1neighborhood of γ , up to a small but fixed error. We also assume that the length `0 of the core is so large that we can choose D to satisfy |D| ≤ 4π`0 , where | · | denotes Lebesgue 3-measure. Now fix ε > 0 small depending only on p and the package, to be defined later. The cores γi of Di lie inside D subject to some topological constraints; in any case, we can choose sense-preserving diffeomorphisms ϕi : D → Di , ϕi (γ ) = γi , such that det dϕi ≤ ηε 2 and that Lip ϕi−1 ≤ ε−1 , where η > 1 is a fixed constant and Lip stands for the Lipschitz constant. Thus the Di ’s are essentially ε-neighborhoods of the γi ’s. The dilation factor ρ of the similarities ψ1 , . . . , ψk is at our disposal, and we choose ρ = ε. It follows that Lip(ψα ◦ θ ◦ ϕα−1 ) ≤ C ρ n ε−n = C
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and that det dψα det dϕα−1 ≥ (ρ 3 /ηε2 )n = (ε/η)n for |α| = n. Here and below, C denotes any positive constant independent of α. Now define f = F ◦ ψα ◦ θ ◦ ϕα−1 on E α , where F is a BLD-map from the decomposition space M(= R 3 /G) to R 3 . The existence of F is guaranteed by the study in Section 8 (see Rems. 8.19). We claim that f has the desired properties. To this end, first observe that f is globally Lipschitz. Second we note that det d f ≥ C(ε/η)n on E α if |α| = n, and then we compute Z XZ (det d f )− p d x = D
(det d f )− p d x Eα
α
≤C
X
≤C
X
|E α |ε− p|α| η p|α|
α
k n ηn ε2n− pn η pn < ∞,
n
provided we choose ε small enough so that kε2− p η p+1 < 1. Finally, outside the domain D the integrability requirement is clearly satisfied. This completes the proof of the theorem. 9.6. BLD-maps and Poincar´e inequalities Spaces that admit a Poincar´e inequality have turned out to be a proper environment for abstract first-order calculi. This fact was implicitly used in the proof of our Theorem 6.18. The validity of such an inequality is a bi-Lipschitz invariant property of a space. We next demonstrate that, to some extent, this invariance remains true when BLDmaps are used. (For calculus in spaces with a Poincar´e inequality, see [Che], [HaK], [HK3], [S5], [Sh], for example.) Recall (see [HK3]) that a metric measure space (X, µ) admits a (weak) (1, 1)Poincar´e inequality if there exist constants C ≥ 1 and τ ≥ 1 such that Z Z |u − u B | dµ ≤ C diam B ρ dµ (9.7) τB
B
for all balls B in X , for all bounded measurable functions u in the enlarged ball τ B, and for all Borel functions ρ : X → [0, ∞] with the property that Z |u(a) − u(b)| ≤ ρ ds γab
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whenever γab is a rectifiable curve joining two points a and b in τ B. Thus, ρ is an upper gradient of u in τ B (cf. the proof of Lem. 6.19). In (9.7), u B denotes the mean value of u in B. THEOREM 9.8 Let X be a complete, quasi-convex oriented generalized n-manifold, and assume that there exists a BLD-map of finite multiplicity either from X onto R n or from R n onto X . Then X admits a weak (1, 1)-Poincar´e inequality with constants depending only on the data associated with X and f .
The harder case in Theorem 9.8 is to establish the Poincar´e inequality on X if X admits a BLD-map onto R n . To this end, we rely on the work of Semmes [S2], in addition to the results from Part I of this paper. It is particularly crucial here to have the path-lifting property of BLD-maps, as explained in Section 3.3. The case where R n is the receiving space is also more interesting in light of the existing examples (cf. Section 8). Proof Assume first that there exists a surjective BLD-map f : X → R n of finite multiplicity. Then X is Ahlfors n-regular by Proposition 6.3. Under these conditions, the validity of a weak (1, 1)-Poincar´e inequality essentially follows from the work [S2], but a few explanatory remarks are due. Semmes shows in [S2] that a weak (1, 1)-Poincar´e inequality holds in every complete Ahlfors n-regular metric space X , provided there exists a constant C > 1 so that the following assertion is true: given x, y ∈ X , there exists a Lipschitz map F : X × (0, |x − y|) → S n with Lipschitz constant satisfying Lip F|X × (ε, |x − y| − ε) ≤ C ε−1 ,
0 < ε < |x − y|/2,
such that the maps Ft = F(·, t) : X → S n , 0 < t < |x − y|, all have nonzero degree and satisfy Ft |X \ B(x, Ct) ≡ σ, 0 < t < |x − y|/2, and Ft |X \ B(y, C(|x − y| − t)) ≡ σ,
|x − y|/2 < t < |x − y|,
for some fixed point σ ∈ S n (see [S2, Sec. 12 and App. B]). To show the existence of a map F as above, we use Proposition 4.13, and postcompose our BLD-map f : X → R n with an appropriate Lipschitz map R n → S n . Indeed, if B(x, r ) is a ball in X , then there is a ball B 0 = B( f (x), r 0 ) in R n ,
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with r 0 comparable to r , such that B 0 ⊂ f (B) by Proposition 4.13; moreover, the x-component U of f −1 (B 0 ) lies in B(x, r1 ), and it contains B(x, r2 ), with both r1 and r2 comparable to r . Choose a homotopically nontrivial C/r 0 -Lipschitz map h B 0 : R n → S n such that h B 0 |R n \ B 0 ≡ σ for some fixed point σ ∈ S n . Then define a map FB by FB = h B 0 ◦ f in U , and FB ≡ σ in X \ U . This map FB : X → S n is homotopically nontrivial, and it takes the value σ outside U ; in particular, F|X \ B(x, r1 ) ≡ σ . Because the radii r 0 , r1 , and r2 are all comparable to r , it is easy to see how, given a pair of points x and y in X , the above construction of the map FB provides a continuous family of Lipschitz maps Ft , as required by Semmes’s criterion. We leave the details to the interested reader. This proves the first part of the theorem. Now assume that there exists a surjective BLD-map f : R n → X of finite multiplicity. The validity of the Poincar´e inequality in X is a rather straightforward application of the change-of-variables formula (5.5) (see also Sec. 6.13) and Prop. 4.13. We leave the details to the reader. This completes the proof of the theorem. Remarks 9.9 (a) The proof of Theorem 9.8 shows that R n can often be replaced by a more general space in order to get similar results. On the other hand, the other hypotheses such as completeness, quasiconvexity, and finite multiplicity are generally needed (cf. the examples in Rem. 4.6). (b) If we assume in Theorem 9.8, in addition, that f : X → R n is Lipschitz, then quasiconvexity of X need not be assumed; it follows as a consequence. Indeed, Proposition 4.13 remains valid as pointed out in Remark 4.16(a). Next one shows without difficulty that X is pathwise connected and (as in the proof of Theorem 9.8) that X admits a Poincar´e inequality. Quasiconvexity of X now follows, as demonstrated in [HaK]. Acknowledgments. We thank Frank Raymond and Dennis Sullivan for valuable discussions. We are most grateful to the referees of this paper for their extraordinarily careful reading of the entire manuscript and for their many useful comments. References [A]
J. W. ALEXANDER, Note on Riemannian spaces, Bull. Amer. Math. Soc. 26 (1920),
[AK]
L. AMBROSIO and B. KIRCHHEIM, Rectifiable sets in metric and Banach spaces,
[Ba]
J. M. BALL, Global invertibility of Sobolev functions and the interpenetration of
370 – 372. 468 Math. Ann. 318 (2000), 527 – 555. CMP 1 800 768 483 matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 315 – 328. MR 83f:73017
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Heinonen Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109, USA;
[email protected] Rickman Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 Helsinki, Finland;
[email protected] DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3,
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE SHU-YEN PAN
Abstract In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field. 1. Introduction Let F be a nonarchimedean local field, and let W be a finite-dimensional nondegenerate symplectic space over F. Let (G, G 0 ) be a (type I) reductive dual pair in the symplectic group Sp(W ). It is known that G, G 0 and G · G 0 are naturally embedded ^ e as subgroups of Sp(W ). Let Sp( W ) be the metaplectic cover of Sp(W ), and let G 0 0 e ^ (resp., G ) be the inverse image of G (resp., G ) in Sp(W ). By restricting the Weil e0 and pulling back to G e0 via the ^ e·G e×G representation of Sp( W ) to the subgroup G 0 0 e e e e homomorphism G × G → G · G , we obtain a correspondence called local theta correspondence or Howe duality between some irreducible admissible representations e0 . It has been proved by e and some irreducible admissible representations of G of G R. Howe (see [MVW]) and J.-L. Waldspurger [W] that this correspondence is one-toone when the residual characteristic of F is odd. Determining the explicit local theta correspondence is an important but extremely difficult problem. Only a few special examples are known so far. Instead of determining the explicit correspondence, a more accessible task is to investigate what kind of properties of the representations are preserved by the correspondence. Several results in this direction are known. For example, Howe [H] shows that spherical representations are preserved by the correspondence for unramified reductive dual pairs, and A.-M. Aubert [A] shows that irreducible admissible representations with nontrivial vectors fixed by an Iwahori subgroup are also preserved for unramified reductive dual pairs. In this paper, we study the (unrefined) minimal K -types of the irreducible admissible representations paired by the theta correspondence. In particular, we prove DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3, Received 28 March 2000. Revision received 24 July 2001. 2000 Mathematics Subject Classification. Primary 22E50; Secondary 1F27, 20C33.
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that the depths of irreducible admissible representations are preserved by the local theta correspondence. The concept of minimal K -types of irreducible admissible representations of a padic reductive group was introduced in [MP1] and [MP2] by A. Moy and G. Prasad. Associated to each point x in the Bruhat-Tits building of a reductive group G over F, an increasing sequence {ti }i≥0 of nonnegative real numbers with t0 = 0, a (decreasing) filtration G x,ti of the parahoric subgroup G x , and a filtration g∗x,−ti of the dual space g∗ of the Lie algebra g of G are defined. It is known that G x,t j is a normal subgroup of G x,ti for j ≥ i. The quotient group G x,ti /G x,ti+1 is a finite reductive group for i = 0 and is a finite abelian group for i > 0. It is also known that there is a natural isomorphism between the Pontrjagin dual (G x,ti /G x,ti+1 )∧ and g∗x,−ti /g∗x,−ti−1 for ti > 0. An (unrefined) minimal K -type for G is a pair (G x,ti , ζ ), where ζ is an irreducible representation of G x,ti /G x,ti+1 such that either ζ is cuspidal when ti = 0, or ζ , realized as a residue class in g∗x,−ti /g∗x,−ti−1 , contains no nilpotent element when ti > 0 (cf. Section 3.2). We say that an irreducible admissible representation (π, V ) of G contains a minimal K -type (G x,ti , ζ ) if the restriction of π to G x,ti contains ζ . Moy and Prasad prove that every irreducible admissible representation of a reductive group G contains a minimal K -type (G x,ti , ζ ) with x satisfying a certain optimal property, called an optimal point, and any two minimal K -types (G x,ti , ζ ), (G x 0 ,t 00 , ζ 0 ) coni tained in an irreducible admissible representation must be associated in some sense, in particular, ti = ti00 . Therefore, they define the common number ti to be the depth of that representation. It is known that the depth of any irreducible admissible representation is always a nonnegative rational number. With little modification the concept of minimal K -types can be extended to metaplectic covers of p-adic classical groups. In particular, the depth of an irreducible ade is defined and is again a nonnegative rational number. missible representation of G The following is the major result (see Theorem 6.6) of this work. THEOREM
Let (G, G 0 ) be a reductive dual pair. Let (π, V ) (resp., (π 0 , V 0 )) be an irreducible e0 ). Suppose that π and π 0 are paired in the e (resp., G admissible representation of G theta correspondence. Then the depth of π is equal to the depth of π 0 . e → G and A reductive dual pair (G, G 0 ) is said to be split if both extensions G 0 0 0 e G → G split. Suppose that (G, G ) is a split reductive dual pair. When we fix e0 , an irreducible admissible representation of G e and G 0 → G e (resp., splittings G → G 0 e G ) can be regarded as an irreducible admissible representation of G (resp., G 0 ) via the splittings. Therefore, we have a one-to-one correspondence between some irreducible admissible representations of G and some irreducible admissible representations of
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
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G 0 . It is clear that we need to specify nice splittings of the metaplectic covers to have a nice description of this correspondence. A result about the splittings is in [P1]. Suppose that (G, G 0 ) is a reductive dual pair of unitary groups or of symplectic and e0 be the splittings obtained in e and β 0 : G 0 → G orthogonal groups. Let β : G → G [P1]. It is not difficult to see that the depth of an irreducible admissible representation e is equal to the depth of the irreducible admissible representation π ◦ β of G. π of G Hence, we have the following corollary (see Theorem 12.2). COROLLARY Let (G, G 0 ) be
a split reductive dual pair of unitary groups or symplectic-orthogonal e0 be the splittings obtained in [P1]. Let π e and β 0 : G 0 → G groups. Let β : G → G e0 ) such that π e (resp., G (resp., π 0 ) be an irreducible admissible representation of G 0 e is equal and π correspond in the local theta correspondence. Then the depth of π ◦ β 0 0 e. to the depth of π ◦ β The idea of the proof of our main result is in fact very simple. From Waldspurger’s result, [W, cor. III.2], we know that the vectors of a (generalized) lattice model of the e modulo the action of Weil representation fixed by certain compact subgroups of G 0 e the other group G are functions of support bounded by some lattice in W . And we know that functions with support in some lattice in W must fixed by some compact e0 . Therefore, the depth of one representation, π, does give us an upper subgroup of G bound for the depth of the other representation, π 0 . However, two difficulties must be conquered to ensure that the depths are really equal. The first difficulty is to refine Moy and Prasad’s result on minimal K -types, at least for classical groups. In particular, we need the result that every irreducible admissible representation of a p-adic classical group has a minimal K -type (G x,ti , ζ ) such that x is a generalized barycenter in a fixed chamber of the building. This part, which is in Sections 3 and 4, mostly relies on or is directly from J.-K. Yu’s work in [Y2] and [Y1]. The second difficulty is to sharpen Waldspurger’s theorem. This is the major content of Sections 5 – 10. The occurrences of nongenuine barycenters in minimal K -types do make the situation more complicated. However, Waldspurger has done the first and most difficult step in [W]. And we just follow his strategy. The content of this paper is as follows. In Section 2, we establish the basic setting of this work and define the notation that is used throughout this paper. In Section 3, we recall Moy and Prasad’s and Yu’s results on minimal K -types and give the modification for metaplectic groups. In Section 4, we have another description of minimal K -types for p-adic classical groups in terms of regular small admissible lattice chains. This is the form that is needed for proving our main theorem. In Section 5, we recall the generalized lattice model of the Weil representation and Waldspurger’s
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result in [W]. In Section 6, we state the main propositions (Propositions 6.3 and 6.5), which can be regarded as a generalization of Waldspurger’s result. Our main theorem (Theorem 6.6) follows this proposition. In Sections 7 and 8, we prove several lemmas that are needed for the proof of Proposition 6.3 in Section 10. Results on Cayley transforms and generalized lattice models of the Weil representation are introduced in these two sections. In Section 9, we prove Proposition 6.2. The proofs of Proposition 6.3 and Corollary 6.4 are in Section 10. In Section 11, we prove Proposition 6.5. The material of this section is also in [P2]. We put it here for completeness. In the last section, we give the result on depth preservation for split reductive dual pairs by using the admissible splitting defined in [P1]. 2. Local theta correspondence In this section, we introduce the notation and the basic setting of this paper. The material is well known, and the references are [K1], [MVW], [Pr], and [W]. 2.1. Notation Let F be a nonarchimedean local field, let O F be the ring of integers of F, let p F be the prime ideal, let $ F be a uniformizer of O F , let f := O F /p F be the residue field, and let τ F be the identity automorphism of F. Let 5O F denote the quotient map from O F to f. We always assume that the characteristic of f is odd. Let ψ be a fixed nontrivial (additive) character of F. Let F 0 be a quadratic extension of F, let O F 0 be the ring of integers of F 0 , let $ F 0 be a uniformizer of O F 0 , let f0 be the residue field of F 0 , and let τ F 0 be the nontrivial automorphism of F 0 over F. We make the choice such that $ F 0 is equal to $ F if F 0 is unramified, and τ F 0 ($ F 0 ) is equal to −$ F 0 if F 0 is ramified. Let D 0 be a central quaternion algebra over F, let $ D 0 be a uniformizer, and let τ D 0 be the canonical involution of D 0 over F. We make the choice such that τ D 0 ($ D 0 ) is equal to −$ D 0 . Let (D, $, τ ) be one of the triples (F, $ F , τ F ), (F 0 , $ F 0 , τ F 0 ), or (D 0 , $ D 0 , τ D 0 ). Let O be the ring of integers of D, let p be the prime ideal, and let d be the residue field of D. Let 5O denote the quotient map from O to d. Let V be a finite-dimensional right vector space over D, and let be 1 or −1. A map h, i : V × V → D is called an -hermitian form if it satisfies the following conditions: hx + y, zi = hx, zi + hy, zi, hxa, ybi = τ (a)hx, yib,
hx, y + zi = hx, yi + hx, zi, hx, yi = τ hy, xi
(2.1.1)
for any x, y ∈ V and a, b ∈ D. The form is nondegenerate if hx, yi = 0 for all y ∈ V implies x = 0. The pair (V , h, i) is called a (nondegenerate) -hermitian space when h, i is a nondegenerate -hermitian form on V . In particular, V is called a symplectic
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
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space if D = F and = −1. A vector v ∈ V is called isotropic if hv, vi = 0. The space V is called anisotropic if it contains no nontrivial isotropic vectors. If a is a real number, let bac denote the largest integer not greater than a, and let dae denote the smallest integer not less than a. 2.2. Heisenberg groups and the Weil representations Let W , hh, ii be a symplectic space over F. Define the Heisenberg group H(W ) associated to W , hh, ii to be the group with underlying set W × F and multiplication defined by 1 (w1 , t1 ) · (w2 , t2 ) := w1 + w2 , t1 + t2 + hhw1 , w2 ii 2
(2.2.1)
for w1 , w2 ∈ W and t1 , t2 ∈ F. It is known that the center of H(W ) is {0} × F, which is identified with F, and it is also known that there is only one (up to equivalence) irreducible representation ρψ of H(W ) with central character ψ by the Stone–von Neumann theorem. Let (ρψ , S ) be an irreducible representation of H(W ) with central character ψ. The symplectic group Sp(W ) acts on H(W ) by the formula g · (w, t) := (g · w, t) ^ for g ∈ Sp(W ) and (w, t) ∈ H(W ). Define the metaplectic cover Sp( W ) of Sp(W ) to be the topological subgroup of Sp(W ) × Aut(S ) consisting of the pairs (g, Mg ) satisfying Mg ◦ ρψ (h) = ρψ (g · h) ◦ Mg (2.2.2) ^ for g ∈ Sp(W ), Mg ∈ Aut(S ), and any h ∈ H(W ). It is clear that (g, Mg ) ∈ Sp( W) × ^ implies (g, z Mg ) ∈ Sp( W ) for any z ∈ C . We have a short exact sequence of group homomorphisms β α ^ 1 −→ C× −→ Sp( W ) −→ Sp(W ) −→ 1
(2.2.3)
^ given by α : z 7 → (1, z1) and β : (g, Mg ) 7 → g. The metaplectic group Sp( W ) comes equipped with a representation ωψ on the space S given by ωψ (g, Mg ) := Mg .
(2.2.4)
This representation (ωψ , S ) is called the Weil representation or the oscillator repre^ sentation of Sp( W ). 2.3. Reductive dual pairs and local theta correspondence Let (G, G 0 ) be a (type I irreducible) reductive dual pair in Sp(W ); that is, there exist a division algebra D with involution τ , a right -hermitian D-space (V , h, i), and a right 0 -hermitian D-space (V 0 , h, i0 ) such that 0 = −1, and W = V ⊗ D V 0 in such a
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SHU-YEN PAN
way that G = U (V ), G 0 = U (V 0 ), the groups of isometries, and the skew-symmetric bilinear form hh, ii := Trd D/F h, i ⊗ τ ◦ h, i0 , where Trd D/F denotes the reduced trace from D to F, and , 0 are 1 or −1. From the definition of the form hh, ii, we know that there are embeddings ιV 0 : U (V ) → Sp(W )
and
ιV : U (V 0 ) → Sp(W ).
^ ^ ^ Let U (V ) be the inverse image of ιV 0 (U (V )) in Sp( W ). The group U (V ) is called 0 ^ the metaplectic cover of U (V ). Let U (V ) be defined similarly. One can check that 0 ^ ^ \ U (V ) and U (V ) commute with each other. Let U (V ) be the two-fold cover of U (V ) ^ \ in U (V ). We know that U (V ) is a totally disconnected group. A representation (π, V ) ^ of U (V ) is called admissible if π|C× (z) is multiplication by z and π|U[ is an ad(V ) missible representation of a totally disconnected group. ^ It is known that (ωψ , S ) is an admissible representation of Sp( W ). Then (ωψ , 0 ^ ^ ^ S ) can be regarded as a representation of U (V )× U (V ) via the restriction to U (V )· 0 0 0 ^ ^ ^ ^ ^ U (V ) and the homomorphism U (V ) × U (V ) → U (V ) · U (V ). Let (π, V ) (resp., ^ (π 0 , V 0 )) be an irreducible admissible representation of the metaplectic group U (V ) 0 ^ (resp., U (V )). The representation (π, V ) is said to correspond to the representation ^ ^ (π 0 , V 0 ) if there is a nontrivial (U (V ) × U (V 0 ))-map 5 : S −→ V ⊗C V 0 .
(2.3.1)
This establishes a correspondence, called the local theta correspondence or Howe ^ duality, between some irreducible admissible representations of U (V ) and some ir0 ^ reducible admissible representations of U (V ). It has been proved by Howe (see [MVW, chap. 5]) and Waldspurger [W] that the local theta correspondence is oneto-one when the residue characteristic of F is odd. 2.4. Induction principle A two-dimensional -hermitian space is called a hyperbolic plane if it admits a onedimensional subspace of isotropic vectors. Let V00 be an anisotropic 0 -hermitian space over D, and let Vk0 be an 0 -hermitian space isomorphic to the direct sum of V00 and k copies of hyperbolic planes. The Witt index of Vk0 is k. There is an embed0 ding Vk0 ⊂ Vk+1 given by v 7→ (v, 0, 0) for v ∈ Vk0 . The chain V00 ⊂ V10 ⊂ V20 ⊂ · · · 0 of -hermitian spaces is called a Witt tower or a Witt series. If n > k, then U (Vn0 ) has a parabolic subgroup Pn,k whose Levi component is isomorphic to U (Vk0 ) × T , where ^ en,k (resp., T e) be the inverse image of Pn,k (resp., T ) in U T is a split torus. Let P (V 0 ). n
If
π0
^ is an irreducible admissible representation of U (Vk0 ) and ξ is a character of T ,
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en,k . We have then π 0 ⊗ ξ can be lifted as an irreducible admissible representation of P the following result, which is called the induction principle, from [K1]. PROPOSITION
^ Let (π, V ) be an irreducible admissible representation of U (V ). Suppose that π occurs in the theta correspondence for the dual pair (U (V ), U (Vk0 )) and is paired with ^ an irreducible admissible representation (π 0 , V 0 ) of U (V 0 ). Then the representation k
π occurs in the theta correspondence for the dual pair (U (V ), U (Vn0 )) for any n > k and is paired with an irreducible component of the parabolic induced representation ^ U (Vn0 )
Ind Pe
n,k
(π 0 ⊗ ξ ), where ξ is an unramified character of T .
2.5. Stable range A reductive dual pair (U (V ), U (V 0 )) is in the stable range if the dimension of one of the spaces V , V 0 is less than or equal to the Witt index of the other space. The following proposition is well known. PROPOSITION
Let (U (V ), U (V 0 )) be a reductive dual pair. Suppose that the dimension of V is less than or equal to the Witt index of V 0 . Then every irreducible admissible representation ^ of U (V ) occurs in the theta correspondence for (U (V ), U (V 0 )). 3. Unrefined minimal K -types for p-adic classical groups, I In the first two sections, we review the important result of minimal K -types from [MP1] and [MP2]. The definition of generalized barycenters in Section 3.4 is from [Y2]. 3.1. Moy-Prasad filtrations Let G be (the F-rational points of) a connected classical group over F, let g be the Lie algebra of G, and let g∗ be the dual space of g. Let x be a point in the Bruhat-Tits building B (G) of G. Let {G x,t }t∈R+ ∪{0} , {gx,t }t∈R , and {g∗x,t }t∈R be the decreasing S filtrations associated to x defined in [MP1] and [MP2]. Define G x,t + := s>t G x,s , S S gx,t + := s>t gx,s , and g∗x,t + := s>t g∗x,s . We know that filtrations {G x,t }t∈R+ ∪{0} , {gx,t }t∈R , and {g∗x,t }t∈R depend on the discrete valuation of D. Here we normalize the discrete valuation such that the value group of D × is Z. Therefore, we have gx,t $ = gx,t+1 and $ g∗x,t = g∗x,t+1 . This normalization is different from the normalization used in [MP1]. However, gx,t here is just gx,lt in [MP1], where l := e0 /e and e (resp., e0 ) is the ramification index of D (resp., the splitting field of G) over F. Let G x be the subgroup of G of elements that fix x. Let {ti | i ≥ 0} be the sequence (depending on x) such that t0 := 0, G x,ti 6= G x,t + , and G x,t + = G x,ti+1 . It is i
i
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known that G x,t j is a normal subgroup of G x,ti for j ≥ i and that the quotient group G x,ti /G x,ti+1 is a finite classical group when i = 0 and a finite abelian group when i > 0. There is a nondegenerate f-bilinear (G x /G x,0+ )-invariant pairing g∗x,−ti /g∗x,−ti−1 × gx,ti /gx,ti+1 −→ f
(3.1.1)
by ( X¯ , Y¯ ) 7→ X (Y ) (mod p F ) for ti > 0, where X¯ (resp., Y¯ ) is the residue class of X ∈ g∗x,−ti (resp., Y ∈ gx,ti ) in g∗x,−ti /g∗x,−ti−1 (resp., gx,ti /gx,ti+1 ). Therefore, we have a natural isomorphism between the Pontrjagin dual (G x,ti /G x,ti+1 )∧ of G x,ti / G x,ti+1 and the quotient g∗x,−ti /g∗x,−ti−1 for ti > 0. 3.2. Unrefined minimal K -types An element X of g∗x,−ti is called a nilpotent element for the coadjoint action of G x on g∗x,−ti if there is a one-parameter subgroup λ : F × → G such that limt→0 Ad(λ(t)) · X = 0, where Ad denotes the coadjoint action of G on g∗ . An irreducible representation ζ of G x,ti /G x,t + is called a nondegenerate representation if i (i) ζ is a cuspidal representation of G x,ti /G x,t + when ti = 0; i (ii) ζ , viewed as a residue class in g∗x,−ti /g∗x,(−t )+ , contains no nilpotent elements i when ti > 0. An (unrefined) minimal K -type of an irreducible admissible representation (π, V ) of G is a pair (G x,t , ζ ), where ζ is a nondegenerate representation of G x,t /G x,t + , and V G x,t + , the subspace of vectors fixed by G x,t + , is nontrivial and, as a representation of G x,t /G x,t + , contains ζ . The following is the fundamental result of Moy and Prasad (see [MP1, Th. 5.2], [MP2, Th. 3.5]) on minimal K -types. PROPOSITION
Let (π, V ) be an irreducible admissible representation of group G. Then there is a unique nonnegative rational number ρ such that ρ is the smallest number t such that there exists an optimal point y with V G y,t + nontrivial. Moreover, if ρ is positive and y is a point such that V G y,ρ + is nontrivial, then the space V G y,ρ + decomposes into nondegenerate representations of G y,ρ /G y,ρ + . Conversely, if t is positive and V G x,t + contains a nondegenerate representation of G x,t /G x,t + for some x, then t is equal to ρ. This unique rational number ρ is called the depth of the representation (π, V ). 3.3. Nonconnected classical groups The definition of filtration subgroups G x,t can be extended to a nonconnected classical group as follows. Let G be a classical group, and let G ◦ be the connected component of G. We define the building of G to be the building of G ◦ . We have G x,t = G ◦x,t
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for any x and positive t. The groups G and G ◦ have the same Lie algebra g. Hence, they have the same filtrations of the Lie algebras and of the dual spaces of the Lie algebras. Therefore, we still have the natural isomorphism (G x,ti /G x,ti+1 )∧ ' g∗x,−ti /g∗x,−ti−1
(3.3.1)
for any classical group G and positive ti . The details can be found in [Y1]. It is not difficult to see that Proposition 3.2 is also true for a nonconnected classical group. The proof of the existence of minimal K -types of positive depths is analogous to the proof of Proposition 3.2. For the existence of minimal K -types of depth zero, the proof depends only on Harish-Chandra’s theory of Eisenstein series for reductive groups over a finite field, which is also valid for nonconnected finite classical groups. 3.4. Generalized barycenters Let G := U (V ) be a classical group. Let δ be the smallest positive rational number such that φ + tδ is an affine root for any affine root φ and any integer t. Let 1af := {α0 , . . . , αn } be the system of simple affine roots of G. There are unique integers ci Pn such that i=0 ci αi = δ. Let be the subgroup of the extended Weyl group of G defined in [IM]. It is known that acts on the local Dynkin diagram of G. A simple affine root αi is called terminal if there is at most one other simple affine root joined to αi . A simple affine root αi is called exceptional if it is terminal and ci = 2. A pair of simple affine roots {αi , α j } is called an exceptional pair if αi 6 = α j and if αi and α j form an orbit under the action of on the local Dynkin diagram. Let νi be the vertex corresponding to αi in a fixed Weyl chamber C0 . Let 4 be a nonempty subset of 1af . Then 4 determines a subsimplex C4 of C0 . Assign weights u i to αi ∈ 4 in the following way: if {αi , α j } is an exceptional pair such that both αi and α j are in 4, then (u i , u j ) := (1, 1), (1, 2), or (2, 1); if αi is an exceptional simple affine root, then u i := 1 or 2; if αi is not exceptional and there is no α j such that {αi , α j } is an exceptional pair contained in 4, then u i := 2. The weighted barycenter .X X x4 := u i νi ui (3.4.1) αi ∈4
αi ∈4
of the subsimplex determined by 4 with the weights {u i } is called a generalized barycenter of C4 . If u i = ci for all αi ∈ 4, then the corresponding generalized barycenter is called the (genuine) barycenter. Example Let D be an unramified quadratic extension of F, and let V be a three-dimensional hermitian space over D such that the Witt index of V is 1. Assume that we have the decomposition V = v1 D⊕v2 D⊕v3 D such that v1 D is a totally isotropic subspace of
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V with the dual space v3 D and hv1 , v3 i = hv2 , v2 i = 1. Let G := U (V ) = U2,1 (F) be the group of isometries. We know that G has two simple affine roots α0 , α1 with the relation α0 + 2α1 = 1; that is, c0 = 1, c1 = 2, and δ = 1. An apartment of the building of G is one-dimensional, and a closed chamber is a (closed) segment. Let ν1 , ν2 be the vertices of the Weyl chamber such that νi is related to αi . There are three nonempty subsets of the set of simple affine roots, namely, 41 := {α0 }, 42 := {α1 }, and 43 := {α0 , α1 }. Now α1 is exceptional and α0 is not. Therefore, both 41 and 42 have unique barycenters x41 and x42 , respectively, but 43 has two generalized barycenters, namely, the genuine barycenter x43 corresponding to the weights u 0 = 1, 0 corresponding to the weights u 1 = 2, and the nongenuine generalized barycenter x4 3 u 0 = 1, u 1 = 1. We can identify the Weyl chamber with the closed segment [0, 1] such that ν0 = 0, ν1 = 1. Then it is clear that x41 = 0, x42 = 1, x43 = 2/3, and 0 = 1/2. x4 3
3.5. A refinement of Proposition 3.2 The following result, which can be viewed as a refinement of Proposition 3.2 for classical groups, is from [Y2]. PROPOSITION
The point y in Proposition 3.2 can be chosen from the set of generalized barycenters in a fixed Weyl chamber. 3.6. Minimal K -types for metaplectic covers Let IG be a fixed Iwahori subgroup of G. We know that IG = G x0 , where x0 is the barycenter of a Weyl chamber C0 in the Bruhat-Tits building of G. Since IG is compact, it is known that the extension e IG → IG splits, where e IG is the inverse image e e e of IG in G. Fix a splitting β IG : IG → IG . It is known that G x,r ⊆ IG for x ∈ C0 and eIG (G x,0+ ); that is, we regard r > 0. Therefore, we can and do identify G x,0+ with β e e G x,0+ as a subgroup of G via the splitting β IG . Those subgroups G x,r for x ∈ C0 are e And it is not difficult to enough to define minimal K -types for metaplectic groups G. see that Proposition 3.5 still holds for metaplectic groups when we define the building e to be the building of G. (It was suggested by Jiu-Kang Yu that we need to fix a of G splitting of an Iwahori subgroup to define unrefined minimal K -types for metaplectic groups.) 3.7. Depth and parabolic induction In [MP2], Moy and Prasad prove that the depth is preserved by parabolic induction. More precisely, the depth of an irreducible subquotient of IndGP π is equal to the depth of π, where π is (the lifting to P of) an irreducible admissible representation of the
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Levi component M of a parabolic subgroup P of G. Clearly, Moy and Prasad’s result can be extended to a metaplectic cover of G. More precisely, we know that I M := IG ∩ M is an Iwahori subgroup of M, where IG is a fixed Iwahori subgroup of G as in Section 3.6. Let the depth of an irreducible e be defined with respect to I M , where M e is the inverse admissible representation of M e e Then the depth of an irreducible component of IndG e image of M in G. eπ, where P P e is the inverse image of P in G, is equal to the depth of π , where π is an irreducible e admissible representation of M. 4. Unrefined minimal K -types for p-adic classical groups, II Section 4.1 is from [W]. Some terminologies such as regular small admissible lattice chain and numerical invariant were suggested by Jiu-Kang Yu. Also, the definition (4.8.1) is due to him. 4.1. Good lattices in a hermitian space, I Let D be defined as in Section 2.1. Let V , h, i be a finite-dimensional right hermitian space over D. Let L be a lattice in V , that is, a free right O -module whose rank is equal to the dimension of V . Two lattices L 1 , L 2 are said to be similar if L 1 = L 2 $ k for some k. Two lattices L 1 , L 2 are said to be equivalent if there is an element g ∈ U (V ) such that g · L 1 = L 2 . A decomposition V = X ⊕ V ◦ ⊕ Y of vector spaces is called L-admissible if X, Y are totally isotropic, in duality, and orthogonal to V ◦ , and if L = (L ∩ X ) ⊕ (L ∩ V ◦ ) ⊕ (L ∩ Y ). Fix an integer κ for V (more specific information about κ is given in (5.2.1)). Define L ∗ := v ∈ V hv, l i ∈ pκ for all l ∈ L . (4.1.1) It is clear that L ∗ is also a lattice in V . The lattice L ∗ is called the dual lattice of L (with respect to the integer κ). The lattice L is self-dual if L ∗ is equal to L. A lattice L is called a good lattice if it satisfies the condition L ∗$ ⊆ L ⊆ L ∗.
(4.1.2)
Let L be a good lattice in V . Then l∗ := L ∗/L and l := L/L ∗ $ are vector spaces over d. Let 5 L ∗ : L ∗ → l∗ , 5 L : L → l be the quotient maps. Define sesquilinear forms h, il∗ , h, il on l∗ and l, respectively, by
5 L ∗ (w), 5 L ∗ (w0 ) l∗ := 5O hw, w0 i$ 1−κ ,
5 L (v), 5 L (v 0 ) l := 5O hv, v 0 i$ −κ (4.1.3)
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for w, w0 ∈ L ∗ , v, v 0 ∈ L, and 5O is the quotient map O → d. Note that the forms h, il∗ and h, il are nondegenerate and depend on the choice of a prime element $ . A good lattice is said to be maximal (resp., minimal) if it is a maximal element (resp., a minimal element) in the set of all good lattices with the partial order defined by inclusion. It is easy to see that a good lattice L is maximal (resp., minimal) if and only if the space (l∗ , h, il∗ ) (resp., (l, h, il )) is anisotropic. Clearly, any lattice L satisfying the condition L ⊆ L ∗ is contained in a maximal good lattice. That is, if L ⊆ L ∗ , then there is a maximal good lattice 0 in V such that L ⊆ 0 ⊆ 0 ∗ ⊆ L ∗ . The following lemma is from [W]. LEMMA
Let 0 be a maximal good lattice in V . (i) There is a 0-admissible decomposition V = X ⊕ V ◦ ⊕ Y such that V ◦ is anisotropic; that is, 0 = (0 ∩ X ) ⊕ (0 ∩ V ◦ ) ⊕ (0 ∩ Y ) and 0 ∗ = (0 ∩ X ) ⊕ (0 ∩ V ◦ )∗ ⊕ (0 ∩ Y ), where (0 ∩ V ◦ )∗ is the dual lattice of 0 ∩ V ◦ in V ◦ . (ii) Suppose that 0 0 is another maximal good lattice in V . Then there is an element g ∈ U (V ) such that g · 0 = 0 0 . (iii) Suppose that { z i | i ∈ I } for some index set I is a finite subset of 0 and that l is a positive integer such that the set {50 (z i ) | i ∈ I } is linearly independent in 0/ 0 ∗ $ and hz i , z j i ≡ 0 (mod pl+κ ). Then there exists a subset {vi | i ∈ I } of 0 and a 0-admissible decomposition V = X 1 ⊕ V ◦ ⊕ Y1 such that the set {vi | i ∈ I } is a basis of X 1 and vi − z i ∈ 0$ l for every i ∈ I . Proof Part (i) is [W, cor. I.9], (ii) is [W, lem. I.10], and (iii) is [W, cor. I.8]. 4.2. Good lattices in a hermitian space, II We know that V can be written as an orthogonal direct sum V ◦ ⊕ V 1 , where V ◦ is anisotropic and V 1 is a direct sum of hyperbolic planes. There is only one good lattice A◦ in V ◦ . Let V 1 = X ⊕ Y be a complete polarization, and let m be the Witt index of V . Suppose that x1 , . . . , xm is a basis of X and y1 , . . . , ym is a basis of Y such that hxi , x j i = hyi , y j i = 0 and hxi , y j i = δi j $ κ . Then it is clear that Bi := x1 p + · · · + xi p + xi+1 O + · · · + xm O + ym O + · · · + y1 O
(4.2.1)
is a good lattice in V1 for each 0 ≤ i ≤ m. LEMMA
The lattice Ni := A◦ + Bi is a good lattice in V , and every good lattice in V is equivalent to Ni for some i = 0, . . . , m.
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Proof It is obvious that Ni is good lattice because Ni∗ = A◦∗ + x1 O + · · · + xm O + ym O + · · · + yi+1 O + yi p−1 + · · · + y1 p−1 . Let L be any good lattice in V . We know from Lemma 4.1(i) that V has an Ladmissible decomposition X 0 ⊕ V 0◦ ⊕ Y 0 such that V 0◦ is anisotropic, X 0 is totally isotropic, and Y 0 is dual to X 0 . Now L ∩ Y 0 is a lattice in Y 0 because the decomposition is L-admissible. Therefore, there exists a basis {y10 , . . . , ym0 } of Y 0 such that L ∩ Y 0 = y10 O + · · · + ym0 O . Let {x10 , . . . , xm0 } be the basis of X 0 dual to {y10 , . . . , ym0 } (i.e., hxi0 , x 0j i = hyi0 , y 0j i = 0 and hxi0 , y 0j i = δi j $ κ ). Then after reordering the index, we may conclude that 0 0 O O + · · · + xm L ∩ X 0 = x10 p + · · · + xi0 p + xi+1
for some i. Now let g be the element in U (V ) such that g · x 0j = x j , g · y 0j = y j , and g|V 0 ◦ is the identity. Therefore, it is clear that g · L = Ni for some i. 4.3. Admissible decompositions Let L be a lattice in V . A basis {vi }i∈I of V for some finite index set I is called L L-admissible if we have L = i∈I (L ∩ vi D). Let L n ⊆ L n−1 ⊆ · · · ⊆ L 1 ⊆ L 0 = L n $ −1 be lattices in V . Then from [W, lem. I.14] we know that there exists a basis of V which is L i -admissible for each i = 1, . . . , n. Therefore, we have the following lemma. LEMMA
Let L n ⊆ L n−1 ⊆ · · · ⊆ L 1 ⊆ L 0 be a set of good lattices in V . Then there exist a decomposition V = X ⊕ V ◦ ⊕ Y , a finite set {xi , v j , yi }i∈I, j∈J , and a disjoint union `n+1 I = i=0 Ii satisfying the following conditions: (i) the space V ◦ is anisotropic, and X, Y are dual to each other; (ii) {xi , yi }i∈I is a self-dual basis of X ⊕ Y (i.e., hxi , x j i = hyi , y j i = 0 and hxi , y j i = δi j $ κ ); S S (iii) the set l≤k {xi $ }i∈Il l>k {xi }i∈Il ∪ {yi }i∈I ∪ {v j } j∈J is an O -basis of L k . 4.4. Lattice chains A nonempty collection of lattices L := {L i }i∈Z in V is called a lattice chain in V if it satisfies the following conditions: (i) L is totally ordered by inclusion; that is, L i ⊆ L i+1 for each i; (ii) each lattice L i is similar to a good lattice or the dual lattice of a good lattice; (iii) there exists a number n such that L i+n = L i $ for all i.
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The number n is called the period of a lattice chain L . A lattice chain L is said to be regular if L i 6= L j whenever i 6 = j. Two lattice chains L := {L i }i∈Z , L 0 := {L i0 }i∈Z are said to be equivalent if there exist an element g ∈ U (V ) and a number k such that 0 g · L i = L i+k for all i. 4.5. Small admissible lattice chains A lattice chain L is called a small admissible lattice chain with numerical invariant (n, n 0 ), where n is a positive integer and n 0 = 0 or 1, if it satisfies the following conditions: (i) the period of L is n; (ii) L i∗ = L −i−n 0 for all i when n is even and n 0 = 1, L i∗ = L −i−n 0 for all i 6≡ 0 or n/2 (mod n) when n is even and n 0 = 0, L i∗ = L −i−n 0 for all i 6≡ (n − 1)/2 (mod n) when n is odd and n 0 = 1, L i∗ = L −i−n 0 for all i 6≡ 0 (mod n) when n is odd and n 0 = 0; (iii) L ∗b(n−1−n 0 )/2c $ ⊆ L b(n−1−n 0 )/2c ⊆ · · · ⊆ L 1 ⊆ L 0 ⊆ L ∗0 and L ∗−1 ⊆ L −1 ⊆ L −2 ⊆ · · · ⊆ L −b(n+n 0 )/2c ⊆ L ∗−b(n+n 0 )/2c $ −1 ⊆ L −b(n+n 0 )/2c−1 . A lattice chain L is called a big admissible lattice chain with numerical invariant (n, n 0 ) if it satisfies (i), (ii), and (iii0 ) L b(n−1−n 0 )/2c+2 ⊆ L ∗b(n−1−n 0 )/2c+1 $ ⊆ L b(n−1−n 0 )/2c+1 ⊆ · · · ⊆ L 2 ⊆ L 1 ⊆ L ∗1 and L ∗0 ⊆ L 0 ⊆ L −1 ⊆ · · · ⊆ L −b(n+n 0 )/2c+1 ⊆ L ∗−b(n+n 0 )/2c+1 $ −1 . A lattice chain is called admissible if it is small or big admissible. Let L := {L i }i∈Z be a lattice chain. Define ]
]
L ] := {L i | L i := L ∗−i−n 0 , i ∈ Z}. ]
]
]
(4.5.1)
]
Clearly, L i ⊂ L i−1 , L i+n = L i $ for any i, and L ] is a lattice chain. Moreover, ]
]
we have (L i )] = L i . From the definition it is easy to check that if L i 6= L i , then ] L i = L i∗ $ k for some integer k. It is not difficult to see that L is small (resp., big) admissible if and only if L ] is big (resp., small) admissible. It is clear from (4.5.1) that ] hL i , L j i ⊆ pκ+b(i+ j+n 0 )/nc (4.5.2) ]
for all i, j. A lattice chain L is called self-dual if L i = L i for all i. LEMMA
Suppose that L := {L i }i∈Z is a small admissible lattice chain of period n. Then ] L i ⊆ L i ⊆ L i−1 for any i.
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Proof First, suppose that (n, n 0 ) = (1, 0). From (iii) we know that L 0 ⊆ L ∗0 ⊆ L −1 . Now ] ] we know that L 0 = L ∗0 . Therefore, we have L 0 ⊆ L 0 ⊆ L −1 . Next, suppose that (n, n 0 ) = (1, 1). From (iii) we know that L −1 ⊆ L ∗−1 $ −1 ⊆ ]
]
L −2 . Now we know that L −1 = L ∗0 = L ∗−1 $ −1 . Therefore, we have L −1 ⊆ L −1 ⊆ L −2 . Now suppose that (n, n 0 ) = (2, 0). From (iii) we know that L 0 ⊆ L ∗0 ⊆ L −1 ⊆ ] ] ∗ L −1 $ −1 ⊆ L −2 . Now we know that L 0 = L ∗0 , L −1 = L ∗1 = L ∗−1 $ −1 . Hence, we ]
]
have L 0 ⊂ L 0 ⊂ L −1 ⊂ L −1 ⊂ L −2 . Finally, suppose that (n, n 0 ) 6 = (1, 0), (1, 1), or (2, 0). Then we know from the ] ] ] definition that L i = L i or L i−1 = L i−1 for any i. Because L i ⊆ L i−1 and L i ⊆ ]
L i−1 , the lemma is clearly true. 4.6. An example Keep the notation as in Example 3.4, and assume that κ = 0. Then we have the following two (equivalence classes of) good lattices in V : B0 := v1 O + v2 O + v3 O , B1 := v1 p + v2 O + v3 O . Let B2 := v1 p + v2 p + v3 O . We have B0∗ = B0 , B1∗ = B2 $ −1 , and B2 ⊂ B1 ⊂ B0 . We know that B0 is a maximal good lattice while B1 is minimal. Then we have five (equivalence classes of) regular small admissible lattice chains: L1 : · · · ⊂ B0 $ 3 ⊂ B0 $ 2 ⊂ B0 $ ⊂ B0 ⊂ B0 $ −1 ⊂ B0 $ −2 ⊂ · · · , L2 : · · · ⊂ B2 $ ⊂ B1 $ ⊂ B2 ⊂ B1 ⊂ B2 $ −1 ⊂ B1 $ −1 ⊂ · · · , L20 : · · · ⊂ B2 $ 3 ⊂ B2 $ 2 ⊂ B2 $ ⊂ B2 ⊂ B2 $ −1 ⊂ B2 $ −2 ⊂ · · · , L3 : · · · ⊂ B0 $ ⊂ B2 ⊂ B1 ⊂ B0 ⊂ B2 $ −1 ⊂ B1 $ −1 ⊂ · · · , L30 : · · · ⊂ B2 $ ⊂ B0 $ ⊂ B2 ⊂ B0 ⊂ B2 $ −1 ⊂ B0 $ −1 ⊂ · · · .
We know that L1 , L2 , L3 are regular self-dual lattice chains with numerical invariant (1, 0), (2, 1), (3, 0), respectively. The lattice chains L20 , L30 are not self-dual, and we have 0]
L2 : · · · ⊂ B1 $ 3 ⊂ B1 $ 2 ⊂ B1 $ ⊂ B1 ⊂ B1 $ −1 ⊂ B1 $ −2 ⊂ · · · , 0]
L3 : · · · ⊂ B1 $ ⊂ B0 $ ⊂ B1 ⊂ B0 ⊂ B1 $ −1 ⊂ B0 $ −1 ⊂ · · · .
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The numerical invariants of L20 , L30 are (1, 1), (2, 0), respectively. Let L := {L i }i∈Z be one of the above lattice chains. The index of L is normalized such that 0 B0 when L is L1 , L3 , or L3 , L 0 = B1 when L is L2 , B2 when L is L20 . 4.7. Bruhat-Tits buildings Let G := U (V ) be a classical group, and let B be the Bruhat-Tits building of G. In [BT], F. Bruhat and J. Tits give a realization of B as the set of all admissible norms α : V → R ∪ {+∞} which are maximally subordinate to the -hermitian form h, i. Based on their result, Yu in [Y1] gives another realization of the building as the set of all admissible filtrations of lattices in V . Under the realization by Yu, the admissible filtrations of lattices corresponding to a generalized barycenter can be rewritten as a regular small admissible lattice chain. We give an example to illustrate this relation. More detail about this can be found in [Y1]. Example Keep the notation as in Sections 3.4 and 4.6. The correspondence of the generalized barycenters and the regular small admissible lattice chains described in Section 4.6 0 ↔ L 0 . Note that the two lattice chains L , L 0 is x4i ↔ Li , x42 ↔ L20 , and x4 2 3 2 3 correspond to the same vertex x42 . 4.8. Open compact subgroups Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n. For any nonnegative integer d, we define G L := g ∈ G g · L i = L i for all i , ] ] G L ,(d/n)+ := g ∈ G (g − 1) · L i ⊆ L i+d+1 , (g − 1) · L i ⊆ L i+d for all i , ] ] G L ,(d+1)/n := g ∈ G (g − 1) · L i ⊆ L i+d+1 , (g − 1) · L i ⊆ L i+d+1 for all i . (4.8.1) It is easy to check that G L ,d/n = {g ∈ G | (g − 1) · L i ⊆ L i+d for all i} and G L ,(d/n)+ = G L ,(d+1)/n when L is self-dual. It is known that G L ,(d+1)/n and G L ,(d/n)+ are open compact subgroups of G. For any d ∈ Z, we define ] ] gL ,(d+1)/n := X ∈ g X · L i ⊆ L i+d+1 , X · L i ⊆ L i+d+1 for all i , ] ] gL ,(d/n)+ := X ∈ g X · L i ⊆ L i+d+1 , X · L i ⊆ L i+d for all i . (4.8.2) The following proposition is from [Y2].
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PROPOSITION
Let L be a regular small admissible lattice chain of period n. Let x be a the corresponding point in the building of G. Then G L ,(i+1)/n = G x,(i+1)/n and G L ,(i/n)+ = G x,(i/n)+ for any i ≥ 0. If L is a regular self-dual lattice chain with numerical invariant (1, 0), (1, 1), or (2, 1), then L is determined by its unique good lattice L = L 0 . In this case, we denote the group G L ,(0/n)+ (resp., G L ,n/n , G L ) by G L ,0+ (resp., G L ,1 , G L ); that is, we have G L := g ∈ G g · L = L , G L ,0+ := g ∈ G (g − 1) · L ∗ ⊆ L , (g − 1) · L ⊆ L ∗ $ , G L ,1 := g ∈ G (g − 1) · L ⊆ L$ . (4.8.3) 4.9. Minimal K -types of an irreducible admissible representation As in Section 3.6, a splitting IG → e IG of a fixed Iwahori subgroup of G has been e via fixed. Then G L ,d/n , G L ,(d/n)+ can be regarded as open compact subgroups of G the splitting IG → e IG when G L ,d/n and G L ,(d/n)+ are contained in IG . Suppose that d is a positive integer. Then from Propositions 3.2 and 4.8, an irreducible admissible e of positive depth has an unrefined minimal K -type of the representation (π, V ) of G form (G L ,d/n , ζ ) for some regular small admissible lattice chain L with numerical invariant (n, n 0 ). In particular, (π, V ) with a nonzero vector fixed by G L ,(d/n)+ must have depth less than or equal to d/n. Now we consider irreducible admissible representations of depth zero. It is known that if x is a point in the building of G, then there exists a vertex ν of a chamber such that G ν,0+ ⊆ G x,0+ . Therefore, an irreducible admissible representation of G is of depth zero if and only if it has nontrivial vectors fixed by G ν,0+ for some vertex ν. Moreover, a vertex ν in the building corresponds to a regular self-dual lattice chain generated by a single good lattice L in V under the correspondence. Therefore, we e is of depth zero if and know that an irreducible admissible representation (π, V ) of G only if V has a nontrivial vector fixed by G L ,0+ for some good lattice L such that G L ,0+ is contained in a fixed Iwahori subgroup IG of G. 5. A result of Waldspurger The material in this section is mostly from [W]. 5.1. Generalized lattice model of the Weil representation In this section we define a special realization of the Weil representation with respect to a good lattice in W .
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Let pλFF be the conductor of a character ψ of F, and let A be a good lattice in W with respect to the number λ F (i.e., the number κ in (4.1.1) is λ F here). Let a∗ denote the quotient A∗/A, and let 5 A∗ be the projection from A∗ to a∗ . We know that a∗ is a vector space over f. Define hh, iia∗ : a∗ × a∗ → f by hh5 A∗ (a1 ), 5 A∗ (a2 )iia∗ := 5O F $ F1−λ F hha1 , a2 ii , (5.1.1) where a1 , a2 ∈ A∗ and 5O F is the quotient map O F → f. It is easy to check that hh, iia∗ is a nondegenerate skew-symmetric f-bilinear form. We remark here that the form hh, iia∗ depends on the choice of a prime element $ F . Let H(a∗ ) be the Heisen berg group associated to the symplectic space a∗ , hh, iia∗ . Let ψ¯ be the character of f ¯ O (t)) := ψ(t$ λ F −1 ), where t ∈ O F . It is clear that ψ¯ is nontrivial. defined by ψ(5 F F Let (ω¯ ψ¯ , S) be the Weil representation of the finite symplectic group Sp(a∗ ) associ ated to the triple a∗ , hh, iia∗ , ψ¯ . Although the skew-symmetric form hh, iia∗ and the character ψ¯ depend on the choice of $ F , the Weil representation ω¯ ψ¯ does not. This can be seen easily from the Schr¨odinger model of (ω¯ ψ¯ , S). Let ρ¯ψ¯ denote the representation of H(a∗ ) corresponding to the character ψ¯ on the space S given by the Stone–von Neumann theorem. Now H(A∗ ) := A∗ × pλFF −1 is a subgroup of the Heisenberg group H(W ). We have a homomorphism 5H(A∗ ) : H(A∗ ) → H(a∗ )
by (a, t) 7 → 5 A∗ (a), 5O F (t$ F1−λ F )
for a ∈ A∗ and t ∈ pλFF −1 . Let K A be the stabilizer in Sp(W ) of the good lattice A, and let K A0 be the subgroup of K A defined by K A0 := g ∈ K A (g − 1) · A∗ ⊆ A . (5.1.2) It is clear that K A0 is a normal subgroup of K A and that the quotient K A /K A0 is isomorphic to Sp(a∗ ). Let ρ eψ be the representation of H(A∗ ) inflated from ρ¯ψ¯ by the projection 5H(A∗ ) , and let e ωψ be the representation of K A inflated from ω¯ ψ¯ by the 0 projection K A → K A /K A . Let S (A) be the space of locally constant, compactly supported maps f : H(W ) → S such that f ι(t)h 1 h 2 = ψ(t)e ρψ (h 1 ) · f (h 2 ), (5.1.3) where t ∈ F, h 1 ∈ H(A∗ ), h 2 ∈ H(W ), and ι denotes the embedding F → H(W ) by sending t to (0, t). Now H(W ) acts on S (A) by the right translation; that is, (ρψ (h) · f )(h 0 ) := f (h 0 h), where h, h 0 ∈ H(W ), f ∈ S (A), and ρψ denotes the action. This realization of the Weil representation (ωψ , S (A)) is called a generalized lattice model.
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To simplify the notation, we identify W with the subset W ×{0} of H(W ). Hence, A∗ is identified with A∗ × {0} ⊂ H(A∗ ). Then we have ρ eψ (a) · f (w) = f (aw) = f ((a, 0)(w, 0)) by (5.1.3) for a ∈ A∗ and w ∈ W . Now 1 1 (a, 0)(w, 0) = a + w, hha, wii = ι hha, wii (a + w, 0) 2 2 by the group law of the Heisenberg group defined in (2.2.1). Finally, we have f ι((1/2)hha, wii)(a + w, 0) = ψ (1/2)hha, wii f (a + w) by (5.1.3) again. Therefore, we can identify S (A) with the space of locally constant, compactly supported maps f : W → S such that f (a + w) = ψ
1 2
hhw, aii ρ eψ (a) · f (w)
(5.1.4)
for any a ∈ A∗ , w ∈ W . For a union of A∗ -cosets R, we define S (A) R := f ∈ S (A) the support of f is in R , S (A)w := S (A) A∗ +w
for w ∈ W .
(5.1.5)
Clearly, S (A)w is a vector space of dimension equal to the dimension of S. From now on, let S (A) be the space of locally constant, compactly supported maps f : W → S satisfying (5.1.4). For g ∈ Sp(W ), we define M[g] ∈ Aut(S (A)) by Z 1 (M[g] · f )(w) := ψ hha, wii ρ eψ (a −1 ) · f g −1 · (a + w) da, (5.1.6) 2 A∗ where g ∈ Sp(W ), f ∈ S (A), w ∈ W , and da is a Haar measure on A∗ . It is easy to ^ check that (g, M[g]) belongs to Sp( W ); that is, it satisfies the identity M[g]◦ρψ (h) = ρψ (g · h) ◦ M[g]. Then we can normalize the Haar measure da so that (M[k] · f )(w) = e ωψ (k) · f (k −1 · w)
(5.1.7)
eA be the inverse image of K A in for k ∈ K A , f ∈ S (A), and w ∈ W . Let K ^ ^ eA given by Sp( W ) under the extension Sp( W ) → Sp(W ). The map K A → K e k 7→ (k, M[k]) defines a splitting of the extension K A → K A . Therefore, if we ^ identify K A as a subgroup of Sp( W ) by the splitting k 7 → (k, M[k]), then ωψ (k) is equal to ωψ (k, M[k]) = M[k]; that is, the action of the Weil representation restricted K A is just given by M[k]. 5.2. Basic setting of reductive dual pairs Let (D, O , $, τ ) be as defined in Section 1.1. Let κ (resp., κ 0 ) be the integer κ in (4.1.1) defining the duality of lattices in V (resp., V 0 ). Let ψ be a fixed character of F
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of conductor pλFF for an integer λ F . Then ψ ◦ Trd D/F is a character of D of conductor pλ , where λ = λ F if D = F or an unramified extension of F, and λ = 2λ F − 1 otherwise. We assume that λ = κ + κ0 (5.2.1) throughout this paper. As in Section 5.1, we also assume that the duality of lattices in W is defined with respect to λ F . ^ Let H denote the Hecke algebra of U (V ); that is, let H be the space of com^ pactly supported locally constant functions f : U (V ) → C such that f (α(z)g) ˜ = −1 × × ^ ^ z f (g), ˜ where z ∈ C , α : C → U (V ) (cf. (2.2.3)), g˜ ∈ U (V ) with the multiplication defined by convolution (cf. [MVW, chap. 5, sec. I.1] ). We note here that the category of nondegenerate left H -modules is equivalent to the category of admissible ^ representations of U (V ). If M (resp., M 0 ) is a lattice in V (resp., V 0 ), then M ⊗O M 0 is a lattice in W . From now on, for simplicity, we write M ⊗ M 0 instead of M ⊗O M 0 to denote the tensor product of M and M 0 as O -modules. Let 0 (resp., 0 0 ) be a fixed maximal good lattice in V (resp., V 0 ). Define A := 0 ∗ ⊗ 0 0 ∩ 0 ⊗ 0 0∗ or, equivalently, A = 0 ∗ ⊗ 0 0∗ $ + 0 ⊗ 0 0 . Then A is a lattice in W and A∗ $ F ⊆ A ⊆ A∗ ; that is, A is a good lattice. We can check that A∗ = 0 ∗ ⊗ 0 0 + 0 ⊗ 0 0∗ = 0 ∗ ⊗ 0 0∗ ∩ 0 ⊗ 0 0 $ −1 . Let R (0) denote the set of pairs of lattices (M, N ) in V such that N$ |∩ 0∗$
M ⊆ N |∩ |∩ ⊆ 0 ⊆ 0∗ ⊆
Suppose that a pair of lattices (M, N ) is in R (0). We define J M,N := g ∈ U (V ) (g − 1) · N ∗ ⊆ M .
(5.2.2)
(5.2.3)
We can check that J M,N is an open compact subgroup of U (V ). Define B M,N := M ∗ ⊗ 0 0 + N ∗ ⊗ 0 0∗ = M ∗ ⊗ 0 0∗ ∩ N ∗ ⊗ 0 0 $ −1 .
(5.2.4)
Then B M,N is a lattice in W and contains A∗ . For convenience we freely identify the following pairs of isomorphic objects: W ' Hom D (V , V 0 ),
A ' HomO (0, 0 0 $ κ ) ∩ HomO (0 ∗ , 0 0∗ $ κ ), A∗ ' HomO (0, 0 0∗ $ κ ) ∩ HomO (0 ∗ , 0 0 $ κ−1 ), B M,N ' HomO (M, 0 0∗ $ κ ) ∩ HomO (N , 0 0 $ κ−1 ).
(5.2.5)
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5.3. A result of Waldspurger In this section we record a deep result by Waldspurger which plays a crucial role for ^ our main result. Suppose that K is a compact subgroup of U (V ) and that (π, V ) is a representation of K . Suppose that (η, E) is an irreducible representation of K . Let V (η) denote the isotypic component of π of type η. If η is the trivial representation, then V (η) is the subspace fixed by K and is also denoted by V K . The following important result is in [W, cor. III.2]. PROPOSITION
Let (U (V ), U (V 0 )) be a (type I) reductive dual pair, let 0 (resp., 0 0 ) be a maximal good lattice in V (resp., V 0 ), let A be the lattice defined in Section 5.2, let (M, N ) ^ be in R (0), let K be a compact subgroup of U (V ), and let η be an irreducible representation of K . Suppose that J M,N is contained in K , S (A) B M,N is stable by K , and S (A) B M,N (η) 6= {0}. Then S (A)(η) = ωψ (H 0 ) · S (A) B M,N (η),
(5.3.1)
^ where H 0 is the Hecke algebra of U (V 0 ). 6. The main theorem The statement of Proposition 6.3 was modified from a suggestion of Jiu-Kang Yu. Let (G, G 0 ) := (U (V ), U (V 0 )) be a reductive dual pair. 6.1. A lattice in the symplectic space Let L := {L i | i ∈ Z} (resp., L 0 := {L 0j | j ∈ Z}) be an admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). For any integer s, we define \ ] \ 0] Bs (L , L 0 ) := L i ⊗ L 0j ∩ Li ⊗ L j . (6.1.1) i+ j=s
i+ j=s
It is clear that Bs (L , L 0 ) is a lattice in W . For any nonnegative integer d such that the number n + n 0 + n 00 + d is even, we define d B L , L 0, := B(−n−n 0 −n 0 −d)/2 (L , L 0 ). (6.1.2) 0 n LEMMA
Let L (resp., L 0 ) be an admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). (i) If s ≥ (−n − n 0 − n 00 + 1)/2, then Bs (L , L 0 ) ⊆ Bs (L , L 0 )∗ . (ii) If s ≤ (−n − n 0 − n 00 )/2, then Bs (L , L 0 )∗ ⊆ Bs (L , L 0 ). In particular, d ∗ d B L , L 0, ⊆ B L , L 0, n n
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for any nonnegative integer d. The proof of this lemma is postponed to Section 9. If d = 0 and L is a regular small admissible lattice generated by a single good lattice L, then B(L , L 0 , 0/n) is denoted by B(L , L 0 ), where L 0 is the good lattice in V 0 which generates L 0 . Hence, B(L , L 0 ) := L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ .
(6.1.3)
It is not difficult to check that B(L , L 0 ) = L ⊗ L 0 + L ∗ $ ⊗ L 0∗ . Hence, we have B(L , L 0 )∗ = L ∗ ⊗ L 0 + L ⊗ L 0∗ = L$ −1 ⊗ L 0 ∩ L ∗ ⊗ L 0∗ . It is also easy to see that B(L , L 0 ) is a good lattice in W ; that is, B(L , L 0 )∗ $ F ⊆ B(L , L 0 ) ⊆ B(L , L 0 )∗ . 6.2. The vectors fixed by open compact subgroups Let (ρψ , S ) be an irreducible smooth representation of the Heisenberg group H(W ) associated to the character ψ of F of conductor pλFF . Let B be a lattice in W such that B ⊆ B ∗ . Then we have hhb, b0 ii ∈ pλFF for any b, b0 ∈ B. Therefore, B × pλFF is a subgroup of H(W ). The representation (ρψ | B×pλ F , S ) factors through the projection F
B×pλFF → B because pλFF is in the kernel of ψ. The action of B on the space S is also denoted by ρψ . By Lemma 6.1 and the above remark, we know that B(L , L 0 , d/n)∗ acts on S when d is nonnegative. PROPOSITION
^ Let (ωψ , S ) be the Weil representation of Sp( W ). (i) Suppose that d is a positive integer, L is a small admissible lattice chain in V with numerical invariant (n, n 0 ), and L 0 is a small admissible lattice chain in V 0 with numerical invariant (n, n 00 ) such that n + n 0 + n 00 + d is even. Then 0 ∗ the subspace S B(L ,L ,d/n) is fixed pointwise by G L ,(d/n)+ and G 0L 0 ,(d/n)+ . (ii) Suppose that L (resp., L 0 ) is a good lattice in V (resp., V 0 ). Then the subspace 0 S B(L ,L ) is fixed pointwise by G L ,0+ and G 0L 0 ,0+ . We postpone the proof of this proposition to Section 9. 6.3. Key proposition for positive depths Let L be a small admissible lattice chain in V with numerical invariant (n, n 0 ). Let Q (d) denote the set of small admissible lattice chains L 0 in V 0 with numerical invariant (n, n 00 ) such that n + n 0 + n 00 + d is even. The following proposition, which may be regarded as a generalization of Proposition 5.3, plays a crucial role for our main theorem.
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PROPOSITION
Let (U (V ), U (V 0 )) be a reductive dual pair in Sp(W ). Let d be a positive integer, and let L be a regular small admissible lattice chain in V with numerical invariant ^ (n, n 0 ). Let (ωψ , S ) be a model of the Weil representation of Sp( W ). Then X 0 ∗ S G L ,(d/n)+ = ωψ (H 0 ) · S B(L ,L ,d/n) , (6.3.1) L 0 ∈Q (d)
^ where H 0 is the Hecke algebra of U (V 0 ). The proof of this proposition is postponed to Section 10. 6.4. The case when the Witt index is large COROLLARY
Keep the assumptions in the previous proposition. If we also assume that the Witt P index of V 0 is large enough, then the sum L 0 ∈Q(d) in Proposition 6.3 can be taken over all regular small admissible lattice chains in Q (d). The proof of this corollary is also in Section 10. 6.5. Key proposition for depth zero The following proposition, whose proof is postponed to Section 11, is an analogue of Proposition 6.3 for depth zero. PROPOSITION
Let (G, G 0 ) := (U (V ), U (V 0 )) be a reductive dual pair, and let ψ be a nontrivial character of F. Suppose that L is a good lattice in V . Then X 0 S G L ,0+ = ωψ (H 0 ) · S B(L ,L ) , (6.5.1) L0
^ where L 0 is a good lattice in V 0 , H 0 is the Hecke algebra of U (V 0 ), and (ωψ , S ) is ^ the Weil representation of Sp( W ). 6.6. The main theorem THEOREM
Let (U (V ), U (V 0 )) be a reductive dual pair. Let (π, V ) (resp., (π 0 , V 0 )) be an ir^ ^ reducible admissible representation of U (V ) (resp., U (V 0 )) such that π and π 0 are paired in the theta correspondence. Then the depth of π is equal to the depth of π 0 .
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SHU-YEN PAN
Proof Let G := U (V ) and G 0 := U (V 0 ). First, we suppose that the depth of π is zero. As ^ in Section 4.9, we know that an irreducible admissible representation (π, V ) of U (V ) G L ,0+ is of depth zero if and only if V is nontrivial for some good lattice L in V . Let 5 be the projection S → V ⊗C V 0 , where S is the Weil representation. We have a nontrivial surjective map S G L ,0+ → V G L ,0+ ⊗C V 0 . Because we assume that the 0 space V G L ,0+ is nontrivial, by Proposition 6.5 there is an element f ∈ S B(L ,L ) for some good lattice L 0 in V 0 such that 5( f ) is not zero. But 5( f ) is fixed by G L ,0+ and G 0L 0 ,0+ by Proposition 6.2(ii). Therefore, 5( f ) belongs to V G L ,0+ ⊗C V
0G 0
L 0 ,0+
. Hence,
0G 0
V L 0 ,0+ is also nontrivial. Therefore, the depth of π 0 is also zero. By symmetry, if π 0 is of depth zero, then π is also of depth zero. Next, we suppose that the depth of π is positive. From Section 4.9 we may assume that the depth of π is d/n for some positive integers d, n and that (π, V ) has a minimal K -type (G L ,d/n , ζ ) for some regular small admissible lattice chain L in V with numerical invariant (n, n 0 ). Therefore, we know that V G L ,(d/n)+ is nontrivial. Let V 00 be an 0 -hermitian space in the same Witt tower of V 0 with a large Witt index, and let G 00 := U (V 00 ). Then (π, V ) still occurs in the theta correspondence for the reductive dual pair (U (V ), U (V 00 )) by the induction principle of the theta correspondence (see Proposition 2.4). Suppose that (π, V ) is paired with an irreducible admissible representation (π 00 , V 00 ) of U^ (V 00 ). Let (ωψ , S ) be the Weil representa^ tion of Sp(V ⊗ V 00 ). By Proposition 6.3, we have S G L ,(d/n)+ = ωψ (H 00 ) ·
X
S B(L ,L
0 ,d/n)∗
,
(6.6.1)
L 0 ∈Q (d)
where H 00 is the Hecke algebra of U^ (V 00 ). Because we have the map 5 : S G L ,(d/n)+ −→ V G L ,(d/n)+ ⊗C V 00 and V G L ,(d/n)+ is nontrivial, there must be a small admissible lattice chain L 0 in V 00 0 ∗ such that n + n 0 + n 00 + d is even and 5 S B(L ,L ,d/n) is nontrivial. Because the Witt index of V 00 is very large, we can arrange L 0 to be a regular small admissible lattice chain in V 00 by Corollary 6.4. Then V 00
G 00
L 0 ,(d/n)+
is nontrivial by Proposition
00 00 G L 0 ,(d/n)+
6.2(i). Therefore, V is also nontrivial. Hence, by Proposition 3.2 and the remark in Section 4.9, the depth of π 00 is less than or equal to d/n, which is the depth of π. By the induction principle (see Proposition 2.4), we know that π 00 is a ^ U (V 00 )
(π 0 ⊗ ξ ), where P is a parabolic subquotient of the induced representation Ind Pe subgroup of U (V 00 ) whose Levi component is isomorphic to U (V 0 ) × T for a split e is the inverse image of P in U^ torus T in U (V 00 ), P (V 00 ), and ξ is an unramified
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character of T . Then by the result of Moy and Prasad (see [MP2, Th. 5.2]) and the remark in Section 3.7, we know that the depths are preserved by parabolic induction; that is, the depth of π 0 ⊗ ξ is equal to the depth of π 00 . It is obvious that the depth of π 0 ⊗ ξ is equal to the depth of π 0 since ξ is an unramified character. Hence, we have proved that the depth of π is greater than or equal to the depth of π 0 . But the depth of π 0 is not zero; otherwise, the depth of π is also zero, as proved in the first paragraph. Hence, by symmetry, the depth of π 0 is also greater than or equal to the depth of π . Therefore, depths of π and π 0 must be equal when one of the representations has positive depth. The proof is complete.
7. Cayley transform 7.1. Cayley transforms, I As usual, let U (V ) denote the group of isometries of an -hermitian space V , h, i . Let u(V ) be the Lie algebra of U (V ), that is, the space of elements c ∈ End D (V ) such that hc · v, v 0 i + hv, c · v 0 i = 0 for all v, v 0 ∈ V . If c belongs to u(V ) and 1 + c is invertible, then we define u(c) := (1 − c)(1 + c)−1 . Similarly, if u belongs to U (V ) and 1 + u is invertible, then we define c(u) := (1 − u)(1 + u)−1 . It is easy to check that u(c) (resp., c(u)) belongs to U (V ) (resp., u(V )) when it is defined. For x, y ∈ V , define an element cx,y in End D (V ) by the formula cx,y · v := xhy, vi − yhx, vi.
(7.1.1)
It is easy to check that cx,y belongs to u(V ). Let u x,y := u(cx,y ) if 1 + cx,y is invertible. If u x,y is defined for some x, y, then it is easy to check that 1 + u x,y is invertible and c(u x,y ) = cx,y . LEMMA
We have cx,ya = cxτ (a),y and cx,y = c y,−x for any x, y ∈ V and any a ∈ D. Proof From the definition in (7.1.1), we have cx,ya · v = xhya, vi − yahx, vi for any v, x, y ∈ V . Hence, cx,ya · v is equal to xτ (a)hy, vi − yhxτ (a), vi, which is just cxτ (a),y · v. This proves the first equality. For the second equality, we have cx,y · v = xhy, vi − yhx, vi = yh−x, vi − (−x)hy, vi = c y,−x · v.
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7.2. Cayley transforms, II Let L be a regular small admissible lattice chain in V with numerical invariant (n, n 0 ). LEMMA
Suppose that d is a nonnegative integer and that g is an element in G L ,(d+1)/n (resp., G L ,(d/n)+ ). Then the element c(g) is well defined and belongs to gL ,(d+1)/n (resp., gL ,(d/n)+ ). Proof Suppose that g is an element in G L ,(d+1)/n . Then (g − 1) · L i ⊆ L i+d+1 and (g − ] ] 1) · L i ⊆ L i+d+1 for all i. Therefore, it is clear that g + 1 is invertible. Hence, the ]
]
element c(g) is well defined. It is clear that (1 + g) · L i = L i and (1 + g) · L i = L i , so ] ] (1− g)(1+ g)−1 · L i = (1− g)· L i and (1− g)(1+ g)−1 · L i = (1− g)· L i . Therefore, ] ] we have (1 − g)(1 + g)−1 · L i ⊆ L i+d+1 and (1 − g)(1 + g)−1 · L i ⊆ L i+d+1 . Hence, c(g) belongs to gL ,(d+1)/n . The proof for the other case is similar, so we omit it. 7.3. A valuation In the remaining part of this section, we want to express an element g ∈ G L ,(d/n)+ in terms of u x,y with x, y satisfying certain conditions determined by L and d. This result is used in Section 10. Suppose that L is a lattice in V . Define a valuation ord L : V → Z
(7.3.1) ]
by ord L (v) := m if v ∈ L$ m − L$ m+1 . Recall that L i := L ∗−i−n 0 from Section 4.5. PROPOSITION
Let d be a nonnegative integer, let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, and let κ be the integer in (5.2.1). Suppose that x, y are elements in V satisfying the conditions ord L ∗j (x) + ord L j+d (y) ≥ −κ, ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, j
j+d
ord L ∗j (y) + ord L j+d (x) ≥ −κ, ord(L ] )∗ (y) + ord L ] (x) ≥ −κ, j
(7.3.2)
j+d
for all j ∈ Z. Then cx,y is in gL ,d/n , and gL ,d/n is generated by those elements cx,y where x, y satisfy the above conditions and are multiples of elements in a given L -admissible basis of V . Proof Let {vi }i∈I , for some finite index set I , be an L -admissible basis of V given by
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Lemma 4.3, and let l be an element in L j . Because of the assumption in (7.3.2), we have ord L j+d (yhx, li) ≥ 0 and ord L j+d (xhy, li) ≥ 0, for all j; that is, yhx, li ∈ L j+d and xhy, li ∈ L j+d , for all j. Therefore, cx,y (l) belongs to L j+d for all j because cx,y · l is just xhy, li − yhx, li. Then cx,y · L j is contained in L j+d for all j. By this ] ] argument we can prove that cx,y · L j ⊆ L j+d for all j. Therefore, cx,y is an element of gL ,d/n . This proves the first statement of the proposition. On the other hand, suppose that c is an element in gL ,d/n . Then we can find βi, j ∈ D for i, j ∈ I such that c is equal to the map c0 ∈ End D (V ) defined by P c0 · v := i, j∈I v j βi, j hvi , vi for any v ∈ V . Define c00 ∈ End D (V ) by the formula P c00 · v := − i, j∈I vi τ (βi, j )hv j , vi for any v ∈ V . Then it is not difficult to check that hc0 · v, v 0 i + hv, c00 · v 0 i is equal to zero for any v, v 0 ∈ V . Because c0 ∈ u(V ), that is, hc0 · v, v 0 i + hv, c0 · v 0 i = 0 for all v, v 0 ∈ V , we know that c0 = c00 , c00 ∈ u(V ), and c = (1/2)(c0 + c00 ). Therefore, it is easy to check that c · v = (1/2)(c0 + c00 ) · v = P i, j∈I cv j βi, j ,vi /2 · v for v ∈ V , that is, that c=
X
cv j βi, j ,vi /2 .
i, j∈I
Now the only thing left is to prove that each pair (v j βi, j , vi /2) does satisfy the conditions in (7.3.2) for any i, j. But this is an easy consequence of the assumption that the basis {vi }i∈I is L -admissible. 7.4 PROPOSITION
Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, and let d be a positive integer. Suppose that x, y are elements in V satisfying (7.3.2) for all j ∈ Z. Then u x,y is defined and u x,y ∈ G L ,d/n . Moreover, G L ,d/n is topologically generated by those u x,y where x, y satisfy the above conditions and are multiples of elements in a given L -admissible basis of V . Proof Let {vi }i∈I be an L -admissible basis of V as given by Lemma 4.3. From (7.3.2) and ] the proof of Proposition 7.3, we know that cx,y · L j is contained in L j+d for all j. P k Therefore, we know that limk→∞ ckx,y = 0. Then ∞ k=0 (−cx,y ) converges and the limit is the inverse of 1 + cx,y . Therefore, 1 + cx,y is invertible and u x,y is defined. ] It is easy to check that (u x,y − 1) = −2cx,y (1 + cx,y )−1 . Now (u x,y − 1) · L j = ]
]
]
−2cx,y (1 + cx,y )−1 · L j is equal to −2cx,y · L j . Finally, −2cx,y · L j is contained in L j+d for all j from the proof of the previous proposition. Therefore, we have u x,y ∈ G L ,d/n . Now we begin to prove the second statement. Let g be an element in G L ,d/n for
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SHU-YEN PAN ]
]
some d ≥ 1. Then we have (g − 1) · L j ⊆ L j+d and (g − 1) · L j ⊆ L j+d for all j from (4.8.1). Therefore, it is easy to see that 1 + g is invertible. Let c := c(g) be as defined in Section 7.1. Then c belongs to gL ,d/n by Lemma 7.2. As in the proof of P Proposition 7.3, we can find βi, j ∈ D for i, j ∈ I such that c = i, j∈I cv j βi, j ,vi /2 and each pair (v j βi, j , vi /2) also satisfies (7.3.2). Hence, u v j βi, j ,vi /2 is defined and belongs to G L ,d/n for each (i, j) ∈ I × I by the first part of Proposition 7.4. Define an element Y g1 := u v j βi, j ,vi /2 , i, j∈I
where the product can be taken in any order. Then it is not difficult to see that (c(g) − c(g1 )) · l is contained in L j+2d for any l ∈ L j and that (c(g) − c(g1 )) · l ] is contained ] ] in L j+2d for any l ] ∈ L j . Therefore, we have ((g − 1) + (g1−1 − 1)) · L j ⊆ L j+2d and ]
]
((g − 1) + (g1−1 − 1)) · L j ⊆ L j+2d for all j. Let g2 denote the element gg1−1 . Then g2 belongs to G L ,d/n , and g2 − 1 is equal to (g − 1)(g1−1 − 1) + (g1−1 − 1) + (g − 1). Therefore, (g2 − 1) · L j ⊆ (g − 1) · L j+d + L j+2d , which is contained in L j+2d ] ] for all j and also (g2 − 1) · L j ⊆ L j+2d for all j. Therefore, we have proved that for any element g ∈ G L ,d/n , we can find an element g1 which is generated by those u x,y for x, y satisfying (7.3.2) such that gg1−1 belongs to G L ,2d/n . By mathematical induction, we can prove that for any element g ∈ G L ,d/n and any positive integer i, there exists an element gi generated by those u x,y for x, y satisfying (7.3.2) such that ggi−1 belongs to G L ,id/n . But we know that the set {G L ,id/n }i∈N forms a system of open neighborhoods of the identity. Therefore, g can be topologically generated by those u x,y for x, y satisfying (7.3.2). Note that x, y can always be chosen from multiples of elements in a given L -admissible basis. 7.5 COROLLARY
Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, let d be a positive integer, and let κ be the integer in (5.2.1). Suppose that x, y ∈ V satisfy the conditions ord L ∗j (x) + ord L ]
j+d+1
(y) ≥ −κ,
ord(L ] )∗ (x) + ord L j+d (y) ≥ −κ, j
ord L ∗j (y) + ord L ]
j+d+1
(x) ≥ −κ,
ord(L ] )∗ (y) + ord L j+d (x) ≥ −κ,
(7.5.1)
j
for all j ∈ Z. Then u x,y is defined and u x,y ∈ G L ,(d/n)+ . Moreover, G L ,(d/n)+ is topologically generated by those u x,y where x, y satisfy (7.5.1) and are multiples of elements in a given L -admissible basis. Proof The proof is almost the same as that of Proposition 7.4. We therefore omit it.
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559
7.6 COROLLARY
Suppose that L is a good lattice in V and that x, y are elements in V . (i) If ord L (x) + ord L (y) ≥ −κ and ord L ∗ (x) + ord L ∗ (y) ≥ 1 − κ, then u x,y is defined and belongs to G L ,0+ . (ii) If ord L (x) + ord L ∗ (y) ≥ 1 − κ and ord L (y) + ord L ∗ (x) ≥ 1 − κ, then u x,y is defined and belongs to G L ,1 . Proof The proof is again similar to the proof of Proposition 7.4. In fact, (i) is [W, lem. I.17].
7.7 COROLLARY
Let L be a regular small admissible lattice chain of period n in V , let d be a positive integer, and let κ be the integer in (5.2.1). (i) The group G L ,d/n is topologically generated by those u x,y with x, y ∈ V satisfying the following two conditions: ] (1) x ∈ L i , y ∈ L j for some i, j such that i + j ≥ d − 1 + n − n 0 − κn; ] (2) x ∈ L i 0 , y ∈ L j 0 for some i 0 , j 0 such that i 0 + j 0 ≥ d − 1 + n − n 0 − κn. (ii) The group G L ,(d/n)+ is topologically generated by those u x,y with with x, y ∈ V satisfying the following two conditions: ] ] (1) x ∈ L i , y ∈ L j for some i, j such that i + j ≥ d + n − n 0 − κn; (2) x ∈ L i 0 , y ∈ L j 0 for some i 0 , j 0 such that i 0 + j 0 ≥ d − 1 + n − n 0 − κn. Proof We prove the statement only for the group G L ,d/n because the proof of (ii) is similar. By Proposition 7.4, it suffices to prove that x, y satisfy (7.3.2); that is, ord L ∗j (x) + ord L j+d (y) ≥ −κ, ord L ∗j (y) + ord L j+d (x) ≥ −κ, ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, j
j+d
and ord(L ] )∗ (y) + ord L ] (x) ≥ −κ, for all j if and only if there exist i 0 , j0 , i 1 , j1 j
j+d
]
]
such that x ∈ L i0 , y ∈ L j0 with i 0 + j0 ≥ d − 1 + n − n 0 − r n and x ∈ L i1 , y ∈ L j1 with i 1 + j1 ≥ d − 1 + n − n 0 − r n. ] ] Note that (L j )∗ = (L ∗− j−n 0 )∗ = L − j−n 0 and L ∗j = L − j−n 0 . Therefore, if x is in ]
L i0 and y is in L j0 for some i 0 , j0 such that i 0 + j0 ≥ d − 1 + n − n 0 − κn, and if x ]
is in L i1 and y is in L j1 for some i 1 , j1 such that i 1 + j1 ≥ d − 1 + n − n 0 − κn, then it is clear that x, y satisfy (7.3.2). On the other hand, suppose that x, y satisfy (7.3.2). Suppose also that x is in ] ] L i0 − L i0 +1 for some i 0 and in L i1 − L i1 +1 for some i 1 . Then ord L ] (x) is zero − j−n 0
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for j such that i 0 − n < − j − n 0 ≤ i 0 ; that is, ord L ]
− j−n 0
(x) is zero for j such that ]
−i 0 − n 0 + n + d > j + d ≥ −i 0 − n 0 + d. Since L ∗j = L − j−n 0 and ord L ∗j (x) + ord L j+d (y) ≥ −κ, we have ord L j+d (y) ≥ −κ for any j such that j + d < −i 0 − n 0 +n +d. In particular, y belongs to L −i0 −n 0 +n+d−1 $ −κ = L −i0 −n 0 +n+d−1−κn . Let j0 := −i 0 −n 0 +n+d−1−κn. Then we have y ∈ L j0 and i 0 + j0 ≥ d−1+n−n 0 −κn. Also, we have that ord L − j−n0 (x) is zero for j such that i 1 − n < − j − n 0 ≤ i 1 ; that is, ord L − j−n0 (x) is zero for j such that −i 1 − n 0 + n + d > j + d ≥ −i 1 − n 0 + d. ]
Since (L j )∗ = L − j−n 0 and ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, we have ord L ] (y) ≥ j
j+d
j+d
−κ for any j such that j + d < −i 1 − n 0 + n + d. In particular, y belongs to ] ] L −i1 −n 0 +n+d−1 $ −κ = L −i1 −n 0 +n+d−1−κn . Let j1 := −i 1 − n 0 + n + d − 1 − κn. ]
Then we have y ∈ L j1 and i 1 + j1 ≥ d − 1 + n − n 0 − κn. So (i) is proved. 8. A few lemmas In this section, we provide several lemmas needed for the proof of our major result. Statements of Lemmas 8.3 and 8.4 were due to Jiu-Kang Yu. Material in Sections 8.1 and 8.6 is from [MVW]. 8.1 Let A be a good lattice in W , and let K A0 be as defined in (5.1.2). If g is an element in K A0 , then it is clear that 1 + g is invertible. Hence, c(g) (cf. Section 7.1) is well defined. As in Section 5.1, let (ωψ , S (A)) denote the generalized lattice model of the Weil representation with respect to a good lattice A in W . The following lemma is [W, lem. II.4]. LEMMA
Let g be an element in K A0 , let w be in W , and let c := c(g) belong to sp(W ). (1) Suppose that c · w is in A∗ . Then ωψ (g) · f (w) = ψ hhw, c · wii ρ eψ (2c · w) · f (w) (2)
for f ∈ S (A). Suppose that c = cx,y for some x, y ∈ V and that c · w is in A∗ . Then hhw, cx,y · wii = −2 Trd D/F hw · x, w · yi0 , where Trd D/F denotes the reduced trace from D to F.
8.2 Suppose that Q is a lattice in W and is contained in the good lattice A. Then we have Q ⊆ Q ∗ . Hence, Q acts on S (A), as remarked in Section 6.2.
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561
LEMMA
Let Q be a lattice in W contained in a good lattice A. Then S (A) Q = S (A) Q ∗ . Proof Let a be an element in Q, let f be an element in S (A), and let ρψ denote the action of Q on S (A). Then we have (ρψ (a)· f )(w) = f (w +a) = f (a +w). We know that f (a + w) = ψ (1/2)hhw, aii ρ eψ (a) · f (w) from (5.1.4). But ρ eψ (a) is trivial because Q is contained in A. Therefore, we have 1 ρψ (a) · f (w) = ψ hhw, aii f (w) 2 for w ∈ W . Then f is fixed by Q if and only if ψ (1/2)hhw, aii is equal to 1 for all a ∈ Q and w ∈ W with f (w) 6 = 0. Therefore, f is fixed by Q if and only if f has support in Q ∗ . By this lemma, for K = G L ,(d/n)+ such that J M,N ⊆ G L ,(d/n)+ and η trivial, (5.3.1) can be rewritten as G ∗ S (A)G L ,(d/n)+ = ωψ (H 0 ) · S (A) B M,N L ,(d/n)+ ∗ because B M,N ⊆ A ⊆ B M,N . Clearly, the lattice A in the above expression is not essential. Hence, we have ∗
S G L ,(d/n)+ = ωψ (H 0 ) · (S B M,N )G L ,(d/n)+ ,
where S is any model of the Weil representation. In particular, we have G S (A0 )G L ,(d/n)+ = ωψ (H 0 ) · S (A0 ) B M,N L ,(d/n)+
(8.2.1)
(8.2.2)
for any good lattice A0 in W such that A0 ⊆ B M,N . 8.3 Let A be a good lattice in W , and let K be a compact subgroup of Sp(W ) such that K is contained in K A , the stabilizer of A in Sp(W ). We know that the action M[g] of K on S (A) is a representation (not just a projective representation). LEMMA
Let A be a good lattice in W , and let B be another lattice so that A∗ ⊆ B. Let K be a compact subgroup of Sp(W ) such that K ⊆ K A . If K stabilizes B, then K also stabilizes the space S (A) B . Proof Let w be an element in B, and let g be an element in the subgroup K . Then
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SHU-YEN PAN
M[g](S (A)w ) is clearly contained in S (A)g·(A∗ +w) because of (5.1.7) and the assumption that K is contained in the stabilizer of B. Now A is contained in B, and w belongs to B. Hence, g · (A∗ + w) is contained in g · B = B. Because S (A) B is a sum of the subspaces S (A)w for w ∈ B, we see that S (A) B is stabilized by K . 8.4 Let (U (V ), U (V 0 )) be a reductive dual pair in Sp(W ), and let (ωψ , S ) be a model of ^ the Weil representation of Sp( W ). Suppose that K is a compact subgroup of U (V ). ^ We can regard K as a subgroup of Sp( W ) via some splitting. Hence, K acts on S . LEMMA
Let H ⊆ K be compact subgroups of U (V ), and let S0 be a subspace of S such that ^ S H = ωψ (H 0 ) · S0H , where H 0 is the Hecke algebra of U (V 0 ). Suppose that S0 K K 0 is K -stable. Then S = ωψ (H ) · S0 . Proof ^ ^ It is clear that ωψ (H 0 ) · S0K is contained in S K because U (V ) and U (V 0 ) commute ^ e be the inverse image of K under the extension U with each other. Let K (V ) → × e with K × C . Let χ K be the element in H U (V ). As a set, we can identify K e. Then we know that with constant value 1 on K × {1} and with value 0 outside K
ωψ (χ K ) is a projection of S onto S K . Now S K = ωψ (χ K ) · S K is contained in ωψ (χ K )ωψ (H 0 ) · S0H because S K is contained in S H = ωψ (H 0 ) · S0H . Then ωψ (χ K )ωψ (H 0 ) · S0H is equal to ωψ (H 0 )ωψ (χ K ) · S0H by the commutativity of ^ ^ U (V ) and U (V 0 ). Now ωψ (χ K ) · S H is equal to (S H ) K = S K because S0 is 0
0
0
K -stable. Therefore, we obtain the inequality S K ⊆ ωψ (H 0 ) · S0K . Hence, we conclude that S K = ωψ (H 0 ) · S0K . 8.5 LEMMA
Let A be a good lattice in W , and let w be an element in W . Let K be the subgroup of K A0 of elements g such that (g − 1) · w belongs to A. Then the map ψw : K → C× defined by g 7→ ψ((1/2)hh(g − 1) · w, wii) is a character of K . Proof Let g1 , g2 be two elements in K . Then both (g1−1 − 1) · w and (g2 − 1) · w belong to A. It is clear that hhg · w, wii = hh(g − 1) · w, wii because hhw, wii = 0. Now g1 g2 − 1
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563
is equal to (g1 − 1)(g2 − 1) + (g1 − 1) + (g2 − 1). Therefore, ψw (g1 g2 ) = ψ
1 2 1
hh(g1 g2 − 1) · w, wii
1 hh(g1 − 1)(g2 − 1) · w, wii ψ hh(g1 − 1) · w, wii 2 2 1 · ψ hh(g2 − 1) · w, wii . 2
=ψ
It is easy to compute that hh(g1 − 1)(g2 − 1) · w, wii = hh(g2 − 1) · w, (g1−1 − 1) · wii. Hence, ψ (1/2)hh(g1 − 1)(g2 − 1) · w, wii = 1 because both elements (g2 − 1) · w, (g1−1 − 1) · w are in A and A is a good lattice. Therefore, we have ψ
1
1 1 hh(g1 g2 − 1) · w, wii = ψ hh(g1 − 1) · w, wii ψ hh(g2 − 1) · w, wii , 2 2 2
that is, ψw (g1 g2 ) = ψw (g1 )ψw (g2 ). It is clear that the map ψw is continuous. Hence, ψw is a character of K . 8.6 Let K be a compact subgroup of Sp(W ) contained in K A0 for some good lattice A in W . If w is an element in W and f is a vector in S (A)w , then we define Z f [w, K ] := ωψ (k) · f dk, (8.6.1) K
where dk is a Haar measure on K . Then it is clear that f [w, K ] belongs to S (A) A∗ +K ·w . If f [w, K ] is not the zero vector, then f [w, K ] is fixed by K . Moreover, those f [w, K ] when f runs over a basis of S (A)w span the subspace of S (A) K of functions with support in A∗ + K · w; that is, we have X K S (A) A∗ +K ·w = C f [w, K ] (8.6.2) f ∈S (A)w
(cf. [MVW, chap. 5, sec. III.1]). The following lemma, which is from [MVW, chap. 5, sec. III.3], plays an important role in the proofs of the main results in Section 11. LEMMA
Let w be an element in W , and let K be a compact subgroup of K A0 . Suppose that f is a nonzero vector in S (A)w . Then f [w, K ] is nonzero if and only if f is fixed by the subgroup K 1 := {g ∈ K | g −1 · w ∈ A + w}. Proof First, we prove that f [w, K ] is nonzero if and only if f [w, K ](w) is nonzero. Now
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SHU-YEN PAN
f [w, K ] is a mapping with support in A∗ + K · w. Suppose that f [w, K ] is nonzero, that is, that f [w, K ](w0 ) 6= 0 for some element w0 in A∗ + K ·w. Write w0 = a +k ·w for some a ∈ A∗ , k ∈ K . Now we have 1 f [w, K ](a + k · w) = ψ hhk · w, aii ρ eψ (a) · f [w, K ](k · w) 2 by (5.1.4) and (5.1.7). Hence, f [w, K ](w0 ) 6 = 0 if and only if f [w, K ](k · w) 6 = 0. Moreover, from (8.6.1) it is clear that f [w, K ](k·w) = f [w, K ](w). Hence, f [w, K ] is nonzero if and only if f [w, K ](w) is nonzero. R R Now f [w, K ](w) = K (ωψ (k) · f )(w) dk is equal to K f (k −1 · w) dk because K is contained in K A0 . Thus, we have Z Z f (k −1 · w) dk = f (k −1 · w) dk K
K1
because f is supported in A∗ + w. Now f (k −1 · w) = ψ (1/2)hh(k − 1) · w, wii f (w) for k ∈ K 1 because (k − 1) · w belongs to A. Therefore, we conclude that Z Z f [w, K ](w) = ψw (k) f (w) dk = ψw (k) dk f (w), (8.6.3) K1
K1
where ψw is defined as in Lemma 8.5. By Lemma 8.5, we know that ψw is a character R of K 1 . The integral K 1 ψw (k) dk in (8.6.3) is essentially equal to the sum of values of a character over all elements of a finite group. Therefore, the last integral in (8.6.3) is nonzero if and only if ψw is trivial on K 1 . Hence, f [w, K ] 6 = 0 if and only if ψw is trivial on K 1 .
9. Proofs of Lemma 6.1 and Proposition 6.2 9.1 Here we have an easy lemma describing some basic properties of the lattice Bs (L , L 0 ) defined in Section 6.1, where L := {L i | i ∈ Z}, L 0 := {L j | j ∈ Z} are admissible lattice chains of period n in V , V 0 , respectively. Recall that \ ] \ 0] Bs (L , L 0 ) := L i ⊗ L 0j ∩ Li ⊗ L j . i+ j=s
i+ j=s
Part (ii) of the following lemma is due to Jiu-Kang Yu. LEMMA
Let s, s1 , s2 be integers:
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
(i) (ii) (iii)
565
B (L , L 0 ) ⊆ Bs2 (L , L 0 ) if s2 ≤ s1 ; Ts1 P 0 0 i+ j=s L i ⊗ L j = i 0 + j 0 =s+n−1 L i 0 ⊗ L j 0 ; T T ] 0] Bs (L , L 0 )∗ = i+ j=−s−n 0 −n 0 −n+1 L i ⊗L 0j + i+ j=−s−n 0 −n 0 −n+1 L i ⊗L j . 0
0
Proof Part (i) is trivial. Part (ii) is easy to check by Lemma 4.3. Now we prove (iii). The dual lattice Bs (L , L 0 )∗ of Bs (L , L 0 ) is \ \ X ] X ] 0] ∗ 0] L i ⊗ L 0j ∩ Li ⊗ L j = (L i )∗ ⊗ L 0∗j + L i∗ ⊗ (L j )∗ . i+ j=s
i+ j=s
i+ j=s
]
i+ j=s ]
0]
0]
We know that (L i )∗ = L −i−n 0 , L 0∗j = L − j−n 0 , L i∗ = L −i−n 0 , and (L j )∗ = L 0− j−n 0 0 0 from Section 4.5. Therefore, X X ] 0] Bs (L , L 0 )∗ = L −i−n 0 ⊗ L − j−n 0 + L −i−n 0 ⊗ L 0− j−n 0 0
i+ j=s
X
=
0]
]
X
Li ⊗ L j +
i+ j=−s−n 0 −n 00
0
i+ j=s
L i ⊗ L 0j .
(9.1.1)
i+ j=−s−n 0 −n 00
Similarly to (ii) of this lemma, we have X 0] Li ⊗ L j = i+ j=−s−n 0 −n 00
0]
\
Li ⊗ L j ,
i+ j=−s−n 0 −n 00 −n+1 ]
X i+ j=−s−n 0 −n 00
]
\
L i ⊗ L 0j =
L i ⊗ L 0j .
(9.1.2)
i+ j=−s−n 0 −n 00 −n+1
Hence, we conclude that Bs (L , L 0 )∗ =
\ i+ j=−s−n 0 −n 00 −n+1
]
L i ⊗ L 0j +
\
0]
Li ⊗ L j .
i+ j=−s−n 0 −n 00 −n+1
9.2. Proof of Lemma 6.1 By Lemma 9.1(iii) it is clear that if s ≥ −s − n 0 − n 00 − n + 1 i.e., if s ≥ (−n − n 0 − n 00 + 1)/2 , then Bs (L , L 0 ) is contained in Bs (L , L 0 )∗ . Hence, (i) is proved. Because of the assumption s ≤ (−n − n 0 − n 00 )/2, we have s ≤ −s −n−n 0 −n 00 . Therefore, we have Bs (L , L 0 )∗ ⊆ Bs (L , L 0 ) for s ≤ (−n − n 0 − n 00 )/2. Hence, (ii) is proved.
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SHU-YEN PAN
9.3 LEMMA
Let s be an integer. Then \ \ L i ⊗ L 0j ∩ i+ j=s
]
0]
i+ j=s+1
]
\
Li ⊗ L j ⊆
L i ⊗ L 0j +
i+ j=s+1
0]
\
Li ⊗ L j .
i+ j=s+1
Proof ] 0] ] 0] 0] ] 0] ] If L i = L i or L 0j = L j , then it is clear that L i ⊗L j = L i ⊗L j or L i ⊗L j = L i ⊗L 0j . ]
0]
We know that L i ⊆ L i ⊆ L i−1 and L 0j ⊆ L j ⊆ L 0j−1 . Hence, it suffices to prove that ]
0]
]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L i ⊗ L 0j + L i ⊗ L j ]
(9.3.1)
]
0]
when L i 6= L i and L 0j 6= L j . Then from Section 4.5 we have L i = L i∗ $ k and 0]
L j = L 0∗j $ l for some k, l. Because L i+n = L i $ and L 0j+n = L 0j $ for any i, j, we only need to consider the cases for k = 0, 1 and l = 0, 1. Define L := L i when k = 0, L := L i∗ $ when k = 1; and define L 0 := L 0j when l = 0, L 0 := L 0∗j $ when l = 1. It is clear that L , L 0 are good lattices in V , V 0 , respectively. First, suppose that k = l = 0. Clearly, L i−1 ⊆ L$ −1 and L 0j−1 ⊆ L 0 $ −1 . Then we have ]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L ⊗ L 0 $ −1 ∩ L ∗ ⊗ L 0∗ , ]
0]
L i ⊗ L 0j + L i ⊗ L j = L ∗ ⊗ L 0 + L ⊗ L 0∗ .
(9.3.2)
]
Next, suppose that k = 1 and l = 0. Clearly, L i = L ∗ $ , L i = L i∗ $ = L, L i−1 ⊆ L ∗ , and L 0j−1 ⊆ L 0 $ −1 . Then we have ]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ , ]
0]
L i ⊗ L 0j + L i ⊗ L j = L ⊗ L 0 + L ∗ ⊗ L 0∗ $.
(9.3.3)
The equality of the two terms in the right-hand side of (9.3.2) or (9.3.3) follows easily from Lemma 9.1(ii). Hence, (9.3.1) holds for the two cases. The proofs for the cases (k, l) = (0, 1) or (1, 1) are similar and omitted. 9.4 Let L (resp., L 0 ) be a small admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). Define As (L , L 0 ) :=
\ i+ j=s
]
L i ⊗ L 0j +
\ i+ j=s
0]
Li ⊗ L j .
(9.4.1)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
As in the proof of Lemma 9.1, it is easy to check that \ ] As (L , L 0 )∗ = L i ⊗ L 0j ∩ i+ j=−s−n−n 0 −n 00 +1
We define 0 L0 ⊗ L0 A := L −1 ⊗ L 0−1 $ Aν (L , L 0 )
567
0]
\
L i ⊗ L j . (9.4.2)
i+ j=−s−n−n 0 −n 00 +1
if n = 1, L is self-dual, and n 0 = n 00 = 0; if n = 1, L is self-dual, and n 0 = n 00 = 1; if either n ≥ 2 or n = 1,
n 0 + n 00
(9.4.3)
= 1,
where ν := d(−n − n 0 − n 00 + 2)/2e. LEMMA
The lattice A defined in (9.4.3) is a good lattice in W . Proof First, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Now A∗ = L ∗0 ⊗ L 0∗ 0. We know that L 0 = L ∗0 because n 0 = 0 and L is self-dual. We also know that 0 0∗ 0 0 L 0∗ 0 $ ⊆ L 0 ⊆ L 0 from Section 4.5(iii), that is, that L 0 is a good lattice in V . Hence, A is a good lattice in W . Next, suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Now A∗ = L ∗−1 ⊗ 0∗ L −1 $ −1 . Now L ∗−1 = L 0 = L −1 $ because L is self-dual and n 0 = 1. So A∗ = 0∗ 0 0∗ −1 from Section 4.5(iii). Hence, A is a L −1 ⊗ L 0∗ −1 . We have L −1 ⊆ L −1 ⊆ L −1 $ good lattice in W . ] 0] Now we consider the remaining cases. We know that L i ⊆ L i−1 and L j ⊆ L 0j−1 T ] for any i, j. So As (L , L 0 ) ⊆ i+ j=s−1 L i ⊗ L 0j . We know that L i ⊆ L i and 0]
L 0j ⊆ L j for any i, j. Hence, \ \ L i ⊗ L 0j ⊆
]
i+ j=−s−n−n 0 −n 00 +1
i+ j=s−1
0]
\
L i ⊗ L 0j ∩
Li ⊗ L j
i+ j=−s−n−n 0 −n 00 +1
if s − 1 ≥ −s − n − n 0 − n 00 + 1. Then we have, by (9.4.2), As (L , L 0 ) ⊆ As (L , L )∗ for s ≥ (−n − n 0 − n 00 + 2)/2. In particular, Aν (L , L 0 ) ⊆ Aν (L , L )∗ . Moreover, we have Aν (L , L 0 )∗ $ F ⊆ Aν (L , L 0 )∗ $ \ =
]
i+ j=−ν−n 0 −n 00 −n+1
=
\ i+ j=−ν−n 0 −n 00 +1
]
L i ⊗ L 0j ∩
0]
\
L i ⊗ L 0j $ ∩
Li ⊗ L j $
i+ j=−ν−n 0 −n 00 −n+1
\ i+ j=−ν−n 0 −n 00 +1
0]
Li ⊗ L j .
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SHU-YEN PAN
Because now either n ≥ 2 or n = 1 and n 0 + n 00 = 1, we have ν ≤ −ν − n 0 − n 00 + 1. So Aν (L , L 0 )∗ $ F ⊆ Aν (L , L 0 ). Hence, Aν (L , L 0 ) is a good lattice in W . 9.5 LEMMA A∗ ⊆ B(L , L 0 , d/n)
for any nonnegative integer d.
Proof First, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Then A∗ = L 0 ⊗ L 0∗ 0. 0 0 Now d must be odd, and it is not difficult to see that B(L , L , 1/1) = L 0 ⊗ L −1 = 0 −1 , as in the proof of Lemma 9.4. L 0 ⊗ L 00 $ −1 . We know that L 0 ⊗ L 0∗ 0 ⊆ L 0 ⊗ L 0$ Next, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Then A∗ = L ∗−1 ⊗ 0∗ L −1 $ −1 . Now d is odd again, and it is not difficult to check that B(L , L 0 , 1/1) = L −1 ⊗ L 0−1 . We know that now L ∗−1 = L 0 = L −1 $ . Moreover, we also know that 0 ∗ 0 L 0∗ −1 ⊆ L −1 from the proof of Lemma 9.4. Then it is clear that A ⊆ B(L , L , 1/1). Next, we consider the other situation. Recall that now \ \ ] 0] A∗ = L i ⊗ L 0j ∩ Li ⊗ L j i+ j=−ν−n−n 0 −n 00 +1
i+ j=−ν−n−n 0 −n 00 +1
from (9.4.2), where ν = d(−n − n 0 − n 00 + 2)/2e, and that \ ] \ d 0] B L , L 0, = L i ⊗ L 0j ∩ Li ⊗ L j n i+ j=µ
i+ j=µ
from (6.1.2), where µ = (−n − n 0 − n 00 − d)/2. Suppose that −n − n 0 − n 00 is even. Then −ν − n 0 − n 00 − n + 1 = (−n − n 0 − n 00 )/2 ≥ (−n − n 0 − n 00 − d)/2 for any nonnegative integer d. Suppose that −n − n 0 − n 00 is odd. In this case, d has to be odd because we always assume that −n −n 0 −n 00 −d is even. Then −ν −n 0 −n 00 −n +1 = (−n − n 0 − n 00 − 1)/2 ≥ (−n − n 0 − n 00 − d)/2 for any nonnegative odd integer d. Hence, the lemma is proved. 9.6 Let A be as defined in Section 9.4. Recall that K A0 := {g ∈ Sp(W ) | (g −1)· A∗ ⊆ A} from Section 5.1. LEMMA
Let d be a positive integer, and let g be an element in G L ,(d/n)+ . Then d ∗ d (g − 1) · B L , L 0 , ⊆ B L , L 0, . n n In particular, G L ,(d/n)+ is a subgroup of K A0 .
(9.6.1)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
569
Proof Recall that we identify G L ,d/n and ιV 0 (G L ,d/n ); so G L ,d/n is regarded as a subgroup of Sp(W ). Let g be an element in G L ,(d/n)+ . Then \ \ d 0] ] (g − 1) · B L , L 0 , = (g − 1) · L i ⊗ L j ∩ (g − 1) · L i ⊗ L 0j . n i+ j=µ
i+ j=µ
]
]
But we know that (g − 1) · L i ⊆ L i+d+1 and (g − 1) · L i ⊆ L i+d from (4.8.1). Therefore, we have \ ] \ d 0] ⊆ L i+d+1 ⊗ L j ∩ L i+d ⊗ L 0j (g − 1) · B L , L 0 , n i+ j=µ i+ j=µ \ \ ] 0] Li ⊗ L j ∩ ⊆ L i ⊗ L 0j . i+ j=µ+d
i+ j=µ+d+1
From Lemma 9.3, we know that \ \ ] 0] L i ⊗ L 0j ∩ Li ⊗ L j ⊆ i+ j=µ+d
i+ j=µ+d+1
\
]
L i ⊗ L 0j +
i+ j=µ+d+1
\
0]
Li ⊗ L j .
i+ j=µ+d+1
Now µ + d + 1 = −µ − n 0 − n 00 − n + 1. So from Lemma 9.1(iii) we have proved d ∗ d (g − 1) · B L , L 0 , ⊆ B L , L 0, n n
(9.6.2)
for any g ∈ G L ,(d/n)+ . We know that A∗ ⊆ B(L , L 0 , d/n). Then B(L , L 0 , d/n)∗ ⊆ A. Therefore, we have (g − 1) · A∗ ⊆ A for any g ∈ G L ,(d/n)+ . Hence, G L ,(d/n)+ is a subgroup of K A0 . 9.7 Let L (resp., L 0 ) be a good lattice in V (resp., V 0 ), and let B := B(L , L 0 ) as in (6.1.3). We know that B is a good lattice in W . LEMMA
G L ,0+ is a subgroup of K B0 . Proof Recall that B = L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ , B ∗ = L ∗ ⊗ L 0∗ ∩ L$ −1 ⊗ L 0 , and G L ,0+ = {g ∈ G | (g − 1) · L ∗ ⊆ L , (g − 1) · L ⊆ L ∗ $ }. Let g be an element in G L ,0+ . Then (g − 1) · B ∗ = (g − 1) · L ∗ ⊗ L 0∗ ∩ (g − 1) · L$ −1 ⊗ L 0 ⊆ L ⊗ L 0∗ ∩ L ∗ $ ⊗ L 0 = B. Hence, G L ,0+ is a subgroup of K B0 .
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SHU-YEN PAN
9.8. Proof of Proposition 6.2. Let A be the lattice defined in (9.4.3). From Lemma 9.4 we know that A is a good lattice in W . Let (ωψ , S (A)) be the generalized lattice model of the Weil representation with respect to the good lattice A. Let φ : S → S (A) be an equivalence of the two models of the Weil representation. Clearly, we have φ(S G L ,(d/n)+ ) = S (A)G L ,(d/n)+ ,
φ(S B(L ,L
0 ,d/n)∗
) = S (A) B(L ,L
B(L ,L 0 ,d/n)∗
Therefore, it suffices to prove that the subspace S (A) by G L ,(d/n)+ . Let g be an element in G L ,(d/n)+ . Then we have
d ∗ d ⊆ B L , L 0, (g − 1) · B L , L 0 , n n
0 ,d/n)∗
.
is fixed pointwise
(9.8.1)
from Lemma 9.6. Since A∗ ⊆ B(L , L 0 , d/n) for any positive integer d from 0 ∗ Lemma 9.5, we know that S (A) B(L ,L ,d/n) = S (A) B(L ,L 0 ,d/n) by Lemma 8.2. From (9.8.1) it is clear that B(L , L 0 , d/n) is stabilized by G L ,(d/n)+ . Hence, S (A) B(L ,L 0 ,d/n) is stabilized by G L ,(d/n)+ by Lemma 8.3. Suppose that w is an element in B(L , L 0 , d/n) and that f is a nonzero element in S (A)w . Then ωψ (g) · f (w) = f (g −1 · w) = f (g −1 − 1) · w + w 1 = ψ hhw, (g −1 − 1) · wii f (w) 2 because f satisfies (5.1.4) and (g −1 − 1) · w is in A. But (g −1 − 1) · w is also in B(L , L 0 , d/n)∗ , so ψ (1/2)hhw, (g −1 − 1) · wii is equal to 1. Hence, we have proved that (ωψ (g) · f )(w) = f (w) for any w ∈ B(L , L 0 , d/n). Therefore, S (A) B(L ,L 0 ,d/n) is fixed pointwisely by G L ,(d/n)+ . Similarly, S (A) B(L ,L 0 ,d/n) is also fixed pointwisely by G 0L 0 ,(d/n)+ . This is the proof of (i). Now we prove (ii) of the proposition. Let B denote the lattice B(L , L 0 ). Suppose f ∈ S (B) B ∗ and g ∈ G L ,0+ . We know that g is in K B0 by Lemma 9.6. We have (ωψ (g) · f )(x) = (M B [g] · f )(x) = e ωψ (g) · ( f (g −1 · x)) from (5.1.7) for any x ∈ W . Now e ωψ (g) is trivial because g is in K B0 . Hence, (ωψ (g) · f )(x) is not zero only if −1 g · x belongs to B ∗ . But B ∗ is stable by g, so ωψ (g) · f belongs to S (B) B ∗ . Now suppose that w is in B ∗ . Hence, (g −1 − 1) · w is in B. Then ωψ (g) · f (w) = f (g −1 − 1) · w + w 1 e (g −1 − 1) · w · f (w). = ψ hhw, (g −1 − 1) · wii ρ 2 Because (g −1 − 1) · w is in B, ρ e((g −1 − 1) · w) becomes trivial. Moreover, w ∈ B ∗ and (g −1 − 1) · w ∈ B imply that ψ (1/2)hh(g − 1) · w, wii = 1. Hence,
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
571
(ωψ (g) · f )(w) = f (w) for any w ∈ B ∗ . Therefore, S (B) B ∗ is fixed pointwise 0 by G L ,0+ . By Lemma 8.2, we conclude that S B(L ,L ) is fixed pointwise by G L ,0+ . 0 Similarly, S B(L ,L ) is also fixed pointwise by G 0L 0 ,0+ . 10. Proofs of Proposition 6.3 and Corollary 6.4 We prove Proposition 6.3 and Corollary 6.4 in this section. Jiu-Kang Yu provided several ideas to simplify the original proof. In particular, the statements of Lemma 10.1 and Propositions 10.5 and 10.7 are modified from his suggestion. 10.1 We fix a maximal good lattice 0 (resp., 0 0 ) in V (resp., V 0 ). LEMMA
Let n be a positive integer, let n 00 be 0 or 1, and let {Mi }i∈Z and {Ni }i∈Z be two decreasing chains of free O -modules in V 0 satisfying the following conditions: (i) Mi+n = Mi $ and Ni+n = Ni $ for all i; (ii) Mi ⊆ Ni ⊆ Mi−1 for all i; 0 0 (iii) hMi , N j i0 ⊆ pκ +b(i+ j+n 0 )/nc for all i, j. Then there exists a small admissible lattice chain {L i0 }i∈Z in V 0 with numerical in0] variant (n, n 00 ) such that Mi ⊂ L i0 and Ni ⊂ L i for all i. Proof We can always find two decreasing chains {Mi0 }i∈Z and {Ni0 }i∈Z of lattices in V 0 satisfying conditions (i), (ii), (iii) in the lemma and Mi ⊆ Mi0 , Ni ⊆ Ni0 for all i. Therefore, without loss of generality, we may assume that all Mi and Ni are in 0 fact lattices in V 0 . If n 00 = 1, then hN0 , N0 i0 ⊆ hM−1 , N0 i0 ⊆ pκ from (iii). Then N0 ⊆ N0∗ . Hence, N0 is contained in some maximal good lattice in V 0 , as explained 0 in Section 4.1. If n 00 = 0, then hM0 , M0 i0 ⊆ pκ . Hence, M0 is contained in some maximal good lattice in V 0 . Let 0 00 be a maximal good lattice in V 0 such that N0 ⊆ 0 00 if n 00 = 1 and M0 ⊆ 0 00 if n 00 = 0. Define L 0−k 0 := N−k 0 + 0 00∗ , ( N−b(n+n 0 )/2c + 0 00∗ 0 0 L −b(n+n 0 )/2c := 0 M−b(n+n 0 )/2c + 0 00∗ 0
L 0k := (N−k−n 0 + 0 00∗ )∗ , 0
if n + n 00 is odd, if n + n 00 is even, (10.1.1)
for k 0 = 1, . . . , b(n + n 00 )/2c − 1 and k = 0, . . . , b(n − 1 − n 00 )/2c. Note that b(n + n 00 )/2c + b(n − 1 − n 00 )/2c = n − 1. Therefore, the lattices L i0 for i such that −b(n + n 00 )/2c ≤ i ≤ −b(n + n 00 )/2c + n − 1 have been defined. Let L 0 := {L i0 }i∈Z
572
SHU-YEN PAN
be the chain of lattices generated by the set {L i0 }−b(n+n 0 )/2c≤i≤−b(n+n 0 )/2c+n−1 via 0 0 0 the formula L i+nl = L i0 $ l for any l. Now we want to check that L 0 satisfies all requirements in the lemma. It is clear from (10.1.1) that M−k 0 ⊆ L 0−k 0 for k 0 = 1, . . . , b(n + n 00 )/2c. Suppose that k = 0, . . . , b(n − 1 − n 00 )/2c. Because hMk , N−k−n 0 i0 ⊆ pκ +b(k−k−n 0 +n 0 )/nc ⊆ 0
0
0
0
∗ 00 pκ , we have Mk ⊆ N−k−n 0 . We also know that Mk ⊆ M0 ⊆ 0 . Therefore, Mk ⊆ 0
0
∗ 0 0 0 l 00 00∗ ∗ l N−k−n 0 ∩ 0 = (N−k−n 0 + 0 ) = L k . Since Mi+nl = Mi $ and L i+nl = L i $ , 0
0
we have Mi ⊆ L i0 for all i. 0] We know that L −i−n 0 := L i0∗ . Suppose that k = 0, . . . , b(n − 1 − n 00 )/2c. Then 0
0]
0]
L −k−n 0 = N−k−n 0 + 0 00∗ from (10.1.1). Hence, it is clear that N−k−n 0 ⊆ L −k−n 0 0
0
0
0
for k = 0, . . . , b(n − 1 − n 00 )/2c. Suppose that k 0 = 1, . . . , b(n + n 00 )/2c − 1. Then 0] L k 0 −n 0 = (Nk 0 −n 0 + 0 00∗ )∗ = Nk∗0 −n 0 ∩ 0 00 from (10.1.1). It is clear that Nk 0 −n 0 ⊆ 0
0
0
0
Nk∗0 −n 0 . If n 0 = 0, then k 0 − n 00 ≥ 1. Hence, Nk 0 −n 0 ⊆ M0 ⊆ 0 00 . If n 0 = 1, 0
0
then
k0
−
n 00
≥ 0. Hence, Nk 0 −n 0 ⊆ N0 ⊆ 0
0]
0 00 .
Hence, Nk 0 −n 0 ⊆ L k 0 −n 0 for 0
0
k 0 = 1, . . . , b(n + n 00 )/2c − 1. Suppose now that α := b(n + n 00 )/2c ≥ 1. We have 0] ∗ 00 ⊆ M ∗ Nα−n 0 ⊆ Nα−n ∩ 0 00 = L α−n 0 . Therefore, we conclude that 0 ∩ 0 α−n 0 0
0
0]
0
0
Ni ⊆ L i for all i. Now we want to check that L 0 is a small admissible lattice chain with numerical invariant (n, n 00 ). From (10.1.1), it is clear that L 0−1 ⊆ L 0−2 ⊆ · · · ⊆ L 0−b(n+n 0 )/2c , 0
L 0b(n−1−n 0 )/2c ⊆ L 0b(n−1−n 0 )/2c−1 ⊆ · · · ⊆ L 00 . 0
(10.1.2)
0
∗ 00 and L 0 00∗ from (10.1.1). Therefore, it Now L 00 = N−n 0 ∩ 0 −1 = N−1 + 0 0
is clear that L 00 ⊆ L 0−1 . Let α := b(n + n 00 )/2c and β := b(n − 1 − n 00 )/2c. Then we want to check that L 0−α $ ⊆ L 0β . Now L 0−α $ = M−α $ + 0 00∗ $ and ∗ 00 from (10.1.1). It is clear that 0 00∗ $ ⊆ 0 00 and M $ = L 0β = N−β−n 0 ∩ 0 −α 0
Md(n−n 0 )/2e ⊆ M0 ⊆ 0 00 . And N−β−n 0 ⊆ M−β−n 0 −1 ⊆ M−n ⊆ 0 00 $ −1 . Then 0 0 0 ∗ ∗ 0 00∗ $ = (0 00 $ −1 )∗ ⊆ N−β−n 0 . From Lemma 10.1(iii), we know that Mi ⊆ N−i−n 0 . 0
0
∗ ∗ ∗ ∗ So M−α+n ⊆ Nα−n−n 0 = N−d(n+n 0 )/2e ⊆ N−b(n−1+n 0 )/2c = N−β−n 0 . Hence, we 0 0 0 0 have proved that L 0−b(n+n 0 )/2c $ ⊆ L 0b(n−1−n 0 )/2c . 0
L0
0
Therefore, is a decreasing chain of lattices in V It is obvious that the period of L is n. In this paragraph we assume that n + n 00 is even. Let α := b(n + n 00 )/2c. Then we have −α ≥ −n. Therefore, M−α ⊆ M−n = M0 $ −1 ⊆ 0 00 $ −1 . Hence, M−α + 0.
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
573
0 00∗ ⊆ 0 00 $ −1 . Therefore, we may regard M−α /(M−α ∩ 0 00∗ ) ' (M−α + 0 00∗ )/ 0 00∗
(10.1.3)
as a d-subspace of 0 00 $ −1/ 0 00∗ , where d := O /p. Let {z i }i∈I , for some finite index set I , be a subset of M−α such that the images of these z i in M−α /(M−α ∩ 0 00∗ ) are linearly independent over d. We have −2α + n 00 ≥ −n. Thus, hM−α , M−α i0 ⊆ pκ +b(−2α+n 0 )/nc ⊆ pκ +b−n/nc ⊆ pκ −1 . 0
0
0
0
(10.1.4)
Therefore, hz i $, z j $ i0 ⊆ pκ +1 for any i, j ∈ I . Then the set {z i $ }i∈I as subset of 0 00 satisfies the conditions in Lemma 4.1(iii) with l = 1. Then by Lemma 4.1, there exist a 0 00 -admissible decomposition V 0 = X ⊕ V 0◦ ⊕ Y , where X, Y are totally isotropic and in duality, and a basis {vi }i∈I of X such that vi − z i $ ∈ 0 00 $ for every i ∈ I . Note that a decomposition of V 0 is 0 00 -admissible if and only if it is 0 00∗ -admissible. Therefore, from the choice of the set {z i }i∈I , we have 0
M−α + 0 00∗ = 0 00∗ + (0 00∗ ∩ X )$ −1 = (0 00∗ ∩ X ) ⊕ (0 00∗ ∩ V 0◦ ) ⊕ (0 00∗ ∩ Y ) + (0 00∗ ∩ X )$ −1 = (0 00∗ ∩ X )$ −1 ⊕ (0 00∗ ∩ V 0◦ ) ⊕ (0 00∗ ∩ Y ).
(10.1.5)
Therefore, we have (M−α + 0 00∗ )∗ = (0 00 ∩ X ) ⊕ (0 00 ∩ V 0◦ ) ⊕ (0 00 ∩ Y )$.
(10.1.6)
Note that 0 00 ∩ X = 0 00∗ ∩ X and 0 00 ∩ Y = 0 00∗ ∩ Y . Therefore, (M−α + 0 00∗ )∗ is a good lattice in V 0 . Now (M−α + 0 00∗ )∗ ⊆ L 0k ⊆ 0 00 for k = 0, . . . , b(n − 1 − n 00 )/2c. Hence, L 0k is a good lattice for k = 0, . . . , b(n − 1 − n 00 )/2c. Thus, each L i0 is similar to a good lattice or the dual lattice of a good lattice. The proof for the case when n + n 00 is odd is similar. It is not difficult to see from (10.1.1) that L i0∗ = L 0−i−n 0 for i 6≡ 0
0 0 or bn/2c (mod n). If n 00 = 1, then we also have L 0∗ 0 = L −1 . If n +n 0 is odd, then we also have b(n + n 00 )/2c = b(n − 1 − n 00 )/2c + n 00 and L 0∗ = L 0b(n−1−n 0 )/2c . −b(n+n 0 )/2c 0
0
Note that −b(n + n 00 )/2c ≡ bn/2c (mod n) when n + n 00 is odd. So Section 4.5(ii) is satisfied. Now we show that Section 4.5(iii) is also satisfied. If β ≥ 0, it is easy to check 0 0 0∗ that L 0β and L 00 are good lattices in V 0 . So we have L 0∗ β $ ⊆ L β and L 0 ⊆ L 0 . From 0∗ 0 0∗ 00∗ 00∗ (10.1.1.) we know that L 0 = N−n 0 + 0 and L −1 = N−1 + 0 . Hence, L 0 ⊆ L 0−1 . 0 −1 = (M ∗ ∩ Suppose that n + n 00 is even. Now L 0−α = M−α + 0 00∗ and L 0∗ −α $ −α 00 −1 00∗ 00 −1 0 )$ . It is obvious that 0 ⊆ 0 $ . We already know that M−α ⊆ 0 00 $ −1 , so ∗ $ −1 . From (10.1.4), we know that M ∗ −1 . Hence, we have 0 00∗ ⊆ M−α −α ⊆ M−α $
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SHU-YEN PAN
0 0 −1 . Now L 0 −1 = (N ∗ L 0−α ⊆ L 0∗ ∩ 0 00 )$ −1 . Since −α $ −α−1 = L β−n = L β $ −β−n 0 0
now n + n 00 is even, we have −β − n 00 = −α + 1. So we have N−β−n 0 ⊆ M−α . 0 0 ∗ ⊆ N∗ −1 ⊆ L 0 Then M−α . Hence, L 0∗ −α $ −α−1 . The proof when n + n 0 is odd is −β−n 0 0
similar. Hence, Section 4.5(iii) is satisfied. Therefore, we conclude that L 0 is a small admissible lattice chain in V 0 with numerical invariant (n, n 00 ). 10.2 LEMMA
Let 0 0 be a maximal good lattice in V 0 . Let {Mi }i∈Z and {Ni }i∈Z be two decreasing chains of free O -modules in V 0 satisfying all conditions in Lemma 10.1. Suppose also that M0 ⊆ 0 0∗ when n 00 = 0, N0 ⊆ 0 0∗ when n 00 = 1. Then we can also require that L 00 ⊆ 0 0 . Proof 0 Suppose that n 0 = 0. Because M0 ⊆ 0 0∗ and hM0 , M0 i0 ⊆ pκ , we have hM0 + 0 0 , M0 + 0 0 i0 ⊆ pκ . 0
(10.2.1)
If z+0 0 is a nontrivial coset in (M0 +0 0 )/ 0 0 , then z+0 0 must be isotropic by (10.2.1). But we know that 0 0∗/ 0 0 is anisotropic because 0 0 is maximal. Hence, M0 + 0 0 must be equal to 0 0 ; that is, M0 is contained in 0 0 . Similarly, if n 00 = 1, then we have N0 ⊆ 0 0 . Hence, the maximal good lattice 0 0 can be used as 0 00 in (10.1.1). Then from (10.1.1) we see that L 00 ⊆ 0 0 . 10.3 Let L := {L i | i ∈ Z} be a small admissible lattice chain in V with numerical invariant (n, n 0 ), and let d be a positive integer. Put n 00 := 0 or 1, so that n+n 0 +n 00 +d is even. Let µ denote the integer (−n − n 0 − n 00 − d)/2. Fix a maximal good lattice 0 (resp., 0 0 ) in V (resp., V 0 ). We assume that every good lattice in L is contained in 0. Define ( L −µ−n 0 if n 00 = 0, M := ] L −µ−n 0 = L ∗µ if n 00 = 1, L] = L ∗µ+n 0 if n 00 = 0, −µ−n 0 −n 00 0 N := (10.3.1) L 0 if n 00 = 1. −µ−n 0 −n 0
Clearly, (M, N ) belongs to the set R (0) defined in Section 5.2. LEMMA
Let M and N be defined as above. Then the group JM,N defined in (5.2.3) is contained
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
575
in G L ,(d/n)+ . Proof Suppose that n 00 = 0. Then we know that N ∗ = L µ+n 0 and M = L −µ−n 0 . Therefore, 0 we have J M,N = g ∈ U (V ) (g − 1) · L µ+n 0 ⊆ L −µ−n 0 . 0
Let g be an element in J M,N . Because −µ − n 0 − (µ + n 00 ) = n + d, we have (g − 1) · L µ+n 0 ⊆ L µ+n 0 +d+n . Let i be an integer such that µ + n 00 ≤ i < µ + n 00 + n. 0
0
]
Then L i ⊆ L µ+n 0 and L µ+n 0 +d+n ⊆ L i+d+1 ⊆ L i+d+1 . Hence, (g − 1) · L i ⊆ 0
]
0
]
]
L i+d+1 for any µ+n 00 ≤ i < µ+n 00 +n. Because L i+kn = L i $ k and L i+kn = L i $ k , we have (g − 1) · L i ⊆ ] (g−1)· L i+1
] L i+d+1
for all i. We also have
] L i+1
⊆ L i . Thus, we have
⊆ L i+d+1 for all i. Hence, J M,N is contained in G L ,(d/n)+ from (4.8.1). The proof for n 00 = 1 is similar, so we omit it. 10.4 LEMMA
Suppose that d ≥ 3n + 2 and that L 0 is a small admissible lattice chain in V 0 such that every good lattice in L 0 is contained in a fixed maximal good lattice 0 0 . Then (i) (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ ⊆ B M,N , (ii) (g − 1) · B M,N ⊆ 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 for any g ∈ G L ,(d/n)+ , (iii) (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ ⊆ B(L , L 0 , d/n), (iv) G L ,(d/n)+ ⊆ K A0 , where A := 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 . Proof As in Section 6.1, we know that (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ = 0$ −1 ⊗ 0 0 ∩ 0 ∗ ⊗ 0 0∗ . Suppose first that n 00 = 0. Then M = L −µ−n 0 and N = L ∗µ+n 0 . Hence, M ∗ = ]
0
]
L µ , N ∗ = L µ+n 0 , and B M,N = L µ ⊗ 0 0∗ ∩ L µ+n 0 ⊗ 0 0 $ −1 . Recall that µ := 0
0
]
(−n − n 0 − n 00 − d)/2; so µ < −2n. Hence, 0 ∗ ⊆ L µ and 0$ −1 ⊆ L µ+n 0 $ −1 . 0 Therefore, 0$ −1 ⊗ 0 0 ∩ 0 ∗ ⊗ 0 0∗ ⊆ B M,N . The proof for n 00 = 1 is similar. Let g be an element in G L ,(d/n)+ . Suppose that n 00 = 0. Then we know that ] B M,N = L µ ⊗ 0 0∗ ∩ L µ+n 0 ⊗ 0 0 $ −1 . Therefore, (g − 1) · B M,N ⊆ L µ+d ⊗ 0 0∗ ∩ 0
]
L µ+n 0 +d+1 $ −1 ⊗ 0 0 . Clearly, if d is large enough (in particular, if d ≥ 3n + 2), 0
]
then we have L µ+d ⊆ 0 and L µ+n 0 +d+1 $ −1 ⊆ 0 ∗ . Hence, (g − 1) · B M,N ⊆ 0
0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 . The proof for n 00 = 1 is similar.
576
SHU-YEN PAN
Recall that \ ] \ d 0] B L , L 0, = L i ⊗ L 0j ∩ Li ⊗ L j . n i+ j=µ
i+ j=µ
]
0]
Now µ < −2n. We know that 0 ∗ ⊆ L i ⊆ L i and 0 0∗ ⊆ L 0j ⊆ L j when i < 0 and j < 0. Hence, (iii) is obvious. From (i) we have A∗ ⊆ B M,N . From (ii) we have (g − 1) · B M,N ⊆ A for any g ∈ G L ,(d/n)+ . Hence, (g − 1) · A∗ ⊆ (g − 1) · B M,N ⊆ A. Therefore, G L ,(d/n)+ is a subgroup of K A0 . 10.5 Recall that for a positive integer d and a small admissible lattice chain L in V , we let Q (d) denote the set of small admissible lattice chains L 0 in V 0 with numerical invariant (n, n 00 ) such that −n − n 0 − n 00 − d is even. PROPOSITION
Let L be a regular small admissible lattice chain in V of period n, and let d be a positive integer. Assume that d ≥ 3n + 2. Then we have X 0 ∗ S G L ,(d/n)+ ⊆ ωψ (H 0 ) · S B(L ,L ,d/n) . (10.5.1) L 0 ∈Q (d)
Proof Let (n, n 0 ) denote the numerical invariant of L . Put n 00 := 0 or 1 so that n+n 0 +n 00 +d is even. Let Q (0 0 , d) denote the subset of small admissible lattice chains L 0 in Q (d) such that every good lattice L 0 ∈ L 0 is contained in a fixed maximal good lattice 0 0 . Obviously, it suffices to prove that X 0 ∗ S G L ,(d/n)+ ⊆ ωψ (H 0 ) · S B(L ,L ,d/n) . L 0 ∈Q (0 0 ,d)
Let M, N be defined as in Section 10.2. Let A be the good lattice 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 in W . By Lemma 10.4 we have A∗ ⊆ B M,N and A∗ ⊆ B(L , L 0 , d/n) for any L 0 ∈ Q (0 0 , d). Hence, by Lemma 8.2, it suffices to prove that X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) . L 0 ∈Q (0 0 ,d)
Now we identify B M,N with HomO (L −µ−n 0 $ −κ , 0 0∗ ) ∩ HomO (L ] −µ−n
0 0 −n 0
$ 1−κ , 0 0 ) if n 00 = 0,
−κ Hom (L ] , 0 0∗ ) ∩ HomO (L −µ−n 0 −n 0 $ 1−κ , 0 0 ) if n 00 = 1 O −µ−n 0 $ 0 (10.5.2)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
577
as in (5.2.5), where µ := (−n − n 0 − n 00 − d)/2. Assume that w is an element in B M,N and that f [w, G L ,(d/n)+ ] is not zero, where f [w, G L ,(d/n)+ ] is defined in (8.6.1) with a nonzero element f ∈ S (A)w . By Lemma 10.4(iv), we know that G L ,(d/n)+ is a subgroup of K A0 , so f [w, G L ,(d/n)+ ] is defined. ] Let x 0 be an element in L i , let y 0 be an element in L j , and let x be x 0 , y be y 0 $ 1−κ for some i, j. Assume that i, j are two integers such that d − n 0 ≤ i + j < d + n − n 0 . ] ] We know that L i ⊆ L i and L j ⊆ L j ⊆ L j−1 . Then the pair x, y satisfies the conditions in Corollary 7.7(ii). Therefore, u x,y is defined and belongs to G L ,(d/n)+ . By Lemma 10.4(ii), we have (u x,y − 1) · B M,N ⊆ A. Therefore, by Lemma 8.1(i), we have ωψ (u x,y )· f = ψ hhw, cx,y ·wii f . Since we assume that f [w, G L ,(d/n)+ ] 6= 0, ψ hhw, cx,y · wii must be 1 by Lemma 8.6. Therefore, hhw, cx,y · wii is in pλFF , which is the kernel of ψ, so we have Trd D/F (hw · x, w · yi0 ) ∈ pλFF by Lemma 8.1(ii). Then 0 we have hw · x, w · yi0 ∈ pλ = pκ+κ from (5.2.1). Hence, we have hw · x 0 $ −κ , w · y 0 $ −κ i0 ∈ pκ −1 . 0
]
Because x 0 and y 0 can be arbitrary in L i and L j , respectively, we have ]
hw · L i $ −κ , w · L j $ −κ i0 ⊆ pκ −1 . 0
(10.5.3)
Define Mi := w(L −µ−n 0 +i $ −κ ), ]
N j := w(L −µ−n 0 + j $ −κ ),
(10.5.4)
for any i, j. Therefore, (10.5.3) becomes hMi+µ+n 0 , N j+µ+n 0 i0 ⊆ pκ −1 . 0
Let i 0 := i + µ + n 0 , j 0 := j + µ + n 0 . The assumption d − n 0 ≤ i + j < d + n − n 0 implies −n −n 00 ≤ i + j −(n −n 0 +n 00 +d) < −n 00 . Then we have −n ≤ i 0 + j 0 +n 00 < 0. Hence, we have 0 0 0 0 hMi 0 , N j 0 i0 ⊆ pκ +b(i + j +n 0 )/nc for all i 0 , j 0 such that −n ≤ i 0 + j 0 + n 00 < 0. Because Mi+nk = Mi $ k and N j+nk = N j $ k for any k, we have hMi , N j i0 ⊆ pκ +b(i+ j+n 0 )/nc 0
0
for all i, j. It is clear that M j and N j are free O -modules in V 0 , M j1 (resp., N j1 ) is contained in M j2 (resp., N j2 ) whenever j2 ≤ j1 , and M j+n (resp., N j+n ) is equal to M j $ (resp., N j $ ) for each j. Moreover, we have N j+1 ⊆ M j ⊆ N j because
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SHU-YEN PAN
]
]
L −µ−n 0 + j+1 ⊆ L −µ−n 0 + j ⊆ L −µ−n 0 + j . Then by Lemma 10.1 there is a small admissible lattice chain L 0 := {L 0j } j∈Z in V 0 with numerical invariant (n, n 00 ) such 0]
that M j (resp., N j ) is contained in L 0j (resp., L j ) for all j. From the definition, we T ] know that w is an element in i+ j=µ L i ⊗ M j ∩ L i ⊗ N j under the identification ] ] L i ⊗ M j = HomO (L i )∗ $ −κ , M j = HomO (L −i−n 0 $ −κ , M j ), ]
L i ⊗ N j = HomO (L i∗ $ −κ , N j ) = HomO (L −i−n 0 $ −κ , N j ). T T ] 0] Therefore, w is in the lattice i+ j=µ L i ⊗ L 0j ∩ i+ j=µ L i ⊗ L j ; that is, w is in B(L , L 0 , d/n). By (10.5.2) and (10.5.4), we know that M0 is contained in 0 0∗ if n 00 = 0 and that N0 is contained in 0 0∗ if n 00 = 1. So we can require that L 00 be contained in 0 0 by Lemma 10.2. Therefore, all good lattices in L 0 are contained in 0 0 . Then we have A∗ ⊆ B(L , L 0 , d/n) by Lemma 10.4(iii). We know that X X G S (A) B M,N L ,(d/n)+ = C f [w, G L ,(d/n)+ ] w∈B M,N f ∈S (A)w
from (8.6.2). The support of f [w, G L ,(d/n)+ ] is contained in B(L , L 0 , d/n) because A∗ + w ⊆ B(L , L 0 , d/n) and B(L , L 0 , d/n) is stabilized by G L ,(d/n)+ . Therefore, we have proved that if f [w, G L ,(d/n)+ ] is nonzero, then f [w, G L ,(d/n)+ ] must be in S (A) B(L ,L 0 ,d/n) for some small admissible lattice chain L 0 ∈ Q (0 0 , d). Hence, we have X G S (A) B M,N L ,(d/n)+ ⊆ S (A) B(L ,L 0 ,d/n) . (10.5.5) L 0 ∈Q (0 0 ,d)
It is clear that G L ,(d/n)+ stabilizes B M,N . Hence, G L ,(d/n)+ stabilizes S (A) B M,N by Lemma 8.3. It is also obvious that (S (A) B M,N )G L ,(d/n)+ is not trivial. Therefore, we have G S (A)G L ,(d/n)+ = ωψ (H 0 ) · S (A) B M,N L ,(d/n)+ (10.5.6) by (8.2.2). Combining (10.5.5) and (10.5.6), we have proved that G L ,(d/n)+ X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) .
(10.5.7)
L 0 ∈Q (0 0 ,d)
We know that each space S (A) B(L ,L 0 ,d/n) is fixed by G L ,(d/n)+ by Proposition 6.2(i). So we have X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) . (10.5.8) L 0 ∈Q (0 0 ,d)
Hence, the proposition is proved.
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
579
10.6 Let A be the good lattice defined in (9.4.3). LEMMA
Let d be an integer greater than 1. Then d (g − 1) · B L , L 0 , ⊆A n for any g ∈ G L ,((d−1)/n)+ .
G L ,((d−1)/n)+ . In particular, B(L , L 0 , d/n) is stabilized by
Proof Let g be an element in G L ,((d−1)/n)+ . We have \ \ d ] 0] (g − 1) · B L , L 0 , = (g − 1) · L i ⊗ L 0j ∩ (g − 1) · L i ⊗ L j . n i+ j=µ
i+ j=µ
]
]
We know that (g − 1) · L i ⊆ L i+d and (g − 1) · L i ⊆ L i+d−1 by (4.8.1). Hence, we have \ \ d ] 0] (g − 1) · B L , L 0 , ⊆ L i ⊗ L 0j ∩ Li ⊗ L j . n i+ j=µ+d
i+ j=µ+d−1
From Lemma 9.3, we know that d (g − 1) · B L , L 0 , ⊆ n
]
\
L i ⊗ L 0j +
i+ j=µ+d
\
0]
Li ⊗ L j .
i+ j=µ+d
We have µ + d = (−n − n 0 − n 00 + d)/2. First, suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Now µ + d = (−1 + d)/2 ≥ 0. So it is clear that \ i+ j=µ+d
]
\
L i ⊗ L 0j +
0]
L i ⊗ L j ⊆ L 0 ⊗ L 00 = A.
i+ j=µ+d
Next, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Now µ + d = (−3 + d)/2 ≥ −1. So \ \ ] 0] L i ⊗ L 0j + L i ⊗ L j ⊆ L 0 ⊗ L 0−1 = L −1 ⊗ L 0−1 $. i+ j=µ+d
i+ j=µ+d
0 We know that L 0−1 $ ⊆ L 0∗ −1 from the proof of Lemma 9.4. Hence, L −1 ⊗ L −1 $ ⊆ L −1 ⊗ L 0∗ −1 = A. Now we consider the remaining cases. Because we now assume that
580
SHU-YEN PAN
d is greater than 1 and −n − n 0 − n 00 + d is even, we have (−n − n 0 − n 00 + d)/2 ≥ d(−n − n 0 − n 00 + 2)/2e = ν. Hence, we conclude that \ \ ] 0] L i ⊗ L 0j + L i ⊗ L j ⊆ Aν (L , L 0 ). i+ j=µ+d
i+ j=µ+d
10.7 PROPOSITION
Suppose that t ≥ 2 and that L (resp., L 0 ) is a small admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )) such that n + n 0 + n 00 + t is even. Then X 0 ∗ 00 ∗ (S B(L ,L ,t/n) )G L ,((t−1)/n)+ ⊆ S B(L ,L ,(t−1)/n) . (10.7.1) L 00 ∈Q (t−1)
Proof Let A be the good lattice defined in Section 9.4 for given L and L 0 . From Lemma 9.5 we know that A∗ ⊆ B(L , L 0 , t/n) for any nonnegative t, so the subspace S (A) B(L ,L 0 ,t/n) is defined. Let Q (A, t − 1) denote the subset of small admissible lattice chains in Q (t − 1) such that A∗ ⊆ B(L , L 00 , (t − 1)/n). Obviously, it suffices to prove that X 00 ∗ 0 ∗ (S B(L ,L ,t/n) )G L ,((t−1)/n)+ ⊆ S B(L ,L ,(t−1)/n) . L 00 ∈Q (A,t−1)
By the remark in Section 8.2, we need to prove that X G S (A) B(L ,L 0 ,t/n) L ,((t−1)/n)+ ⊆
S (A) B(L ,L 00 ,(t−1)/n) .
L 00 ∈Q (A,t−1)
Let w be an element in B(L , L 0 , t/n), and let µ be the integer (−n − n 0 − n 00 − t)/2. As in Section 10.5, we define Mi := w(L −µ−n 0 +i $ −κ ), ]
N j := w(L −µ−n 0 + j $ −κ ),
(10.7.2)
for any i, j. By an argument similar to the proof of Proposition 10.5, we can show that 0 0 hMi , N j i0 ⊆ pκ +b(i+ j+1+n 0 )/nc (10.7.3) for all i, j. (Note that G L ,(d/n)+ in the proof of Proposition 10.5 is replaced by G L ,((t−1)/n)+ here.) Let Mi0 := Mi−n 0 and N 0j := N j−n 0 . Then (10.7.3) becomes 0
0
hMi0 , N 0j i0 ⊆ pκ +b(i+ j+1−n 0 )/nc 0
0
(10.7.4)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
581
for all i, j. Then by Lemma 10.1 there is a small admissible lattice chain L 00 := {L 00j } j∈Z in V 0 with numerical invariant (n, 1 − n 00 ) such that M 0j is contained in L 00j 00]
and N 0j is contained in L j for all j. Now we have \ ] \ ] \ ] Li ⊗ M j = L i ⊗ M 0j+n 0 ⊆ L i ⊗ L 00j+n 0 = i+ j=µ
and \
0
i+ j=µ
\
Li ⊗ N j =
i+ j=µ
i+ j=µ
\
L i ⊗ N 0j+n 0 ⊆ 0
i+ j=µ
0
00]
i+ j=µ
L i ⊗ L 00j
i+ j=µ+n 00
00]
\
L i ⊗ L j+n 0 = 0
]
\
Li ⊗ L j .
i+ j=µ+n 00
But we know that µ + n 00 =
−n − t − n 0 − n 00 −n − (t − 1) − n 0 − (1 − n 00 ) + n 00 = . 2 2
Hence, ]
\
\
L i ⊗ L 00j ∩
i+ j=µ+n 00
i+ j=µ+n 00
t − 1 00] L i ⊗ L j = B L , L 00 , . n
(10.7.5)
T ] From the definition in (10.7.2), we know that w belongs to i+ j=µ L i ⊗ M j ∩ T 00 i+ j=µ L i ⊗ N j . So we have shown that w is in B(L , L , (t − 1)/n) for some L 00 . From (10.7.4) we have hM 0j , N−0 j−1+n 0 i0 ⊆ pκ . 0
0
Identify B(L , L 0 , t/n) with \ \ ] 0] Hom(L −i−n 0 $ −κ , L 0j ) ∩ Hom(L −i−n 0 $ −κ , L j ). i+ j=µ
(10.7.6)
i+ j=µ 0]
Then we know that N−0 j−1+n 0 is contained in L − j−1 . Thus, we may choose L 00j to 0
0]
contain (L − j−1 )∗ = L 0j+1−n 0 . Therefore, 0
\
] Li
⊗
L 00j
⊇
i+ j=µ+n 00
]
\
0
i+ j=µ+n 00
]
\
L i ⊗ L 0j+1−n 0 =
L i ⊗ L 0j .
i+ j=µ+1
Moreover, since M 0j ⊆ L 0j−n 0 , we may also choose L 00j to be contained in L 0j−n 0 . Then 0
\
00]
Li ⊗ L j =
i+ j=µ+n 00
0
\
L i ⊗ L 00∗ − j−(1−n 0 ) ⊇ 0
i+ j=µ+n 00
=
\ i+ j=µ+n 00
\
0]
L i ⊗ L j+1−n 0 = 0
L i ⊗ L 0∗ − j−1
i+ j=µ+n 00
\ i+ j=µ+1
0]
Li ⊗ L j .
582
SHU-YEN PAN
We know that t − 2 B L , L 0, = n
]
\
L i ⊗ L 0j ∩
i+ j=µ+1
\
0]
Li ⊗ L j .
(10.7.7)
i+ j=µ+1
Hence, the lattice B(L , L 00 , (t − 1)/n) contains the lattice B(L , L 0 , (t − 2)/n). Hence, B(L , L 00 , (t − 1)/n) contains the lattice A∗ by Lemma 9.5 because now t ≥ 2. Then the support of f [w, G L ,((t−1)/n)+ ] is contained in B(L , L 00 , (t − 1)/n). Therefore, we have proved that if w ∈ B(L , L 0 , t/n) and f [w, G L ,((t−1)/n)+ ] is nonzero, then f [w, G L ,((t−1)/n)+ ] must be in S (A) B(L ,L 00 ,(t−1)/n) for some small admissible lattice chain L 00 with numerical invariant (n, 1 − n 00 ). Therefore, we conclude that X G S (A) B(L ,L 0 ,t/n) L ,((t−1)/n)+ ⊆ S (A) B(L ,L 00 ,(t−1)/n) . (10.7.8) L 00 ∈Q (t−1)
10.8. Proof of Proposition 6.3. Let k be an integer such that d + k ≥ 3n + 2. Then we have X ∗ S G L ,((d+k)/n)+ ⊆ ωψ (H 0 ) · S B(L ,M ,(d+k)/n)
(10.8.1)
M ∈Q (d+k)
by Proposition 10.5. Then we have S G L ,((d+k−1)/n)+ = (S G L ,((d+k)/n)+ )G L ,((d+k−1)/n)+ X ∗ G L ,((d+k−1)/n)+ = ωψ (H 0 ) · S B(L ,M ,(d+k)/n) . M ∈Q (d+k)
By Lemma 10.6, each B(L , M , (d + k)/n) is stabilized by G L ,((d+k−1)/n)+ . Hence, ∗ each space S B(L ,M ,(d+k)/n) is stabilized by G L ,((d+k−1)/n)+ by Lemmas 8.2 and 8.3. Hence, X ∗ G L ,((d+k−1)/n)+ S G L ,((d+k−1)/n)+ = ωψ (H 0 ) · S B(L ,M ,(d+k)/n) M ∈Q (d+k)
by Lemma 8.4. Then we have
X
S B(L ,M ,(d+k)/n)
∗
G L ,((d+k−1)/n)+
M ∈Q (d+k)
=
X M ∈Q (d+k)
(S B(L ,M ,(d+k)/n) )G L ,((d+k−1)/n)+ . ∗
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
583
By Proposition 10.7, we have (S B(L ,M ,(d+k)/n) )G L ,((d+k−1)/n)+ ⊆ ∗
S B(L ,M
X
0 ,(d+k−1)/n)∗
.
M 0 ∈Q (d+k−1)
(10.8.2) Therefore, we have S G L ,((d+k−1)/n)+ ⊆ ωψ (H 0 ) ·
(S B(L ,M
X
0 ,(d+k)/n)∗
)G L ,((d+k−1)/n)+
M 0 ∈Q (d+k)
S B(L ,M
X
⊆ ωψ (H 0 ) ·
0 ,(d+k−1)/n)∗
.
M 0 ∈Q (d+k−1)
Repeating the same process, we conclude that X S G L ,(d/n)+ ⊆ ωψ (H 0 ) ·
S B(L ,L
0 ,d/n)∗
.
(10.8.3)
L 0 ∈Q (d)
The opposite inclusion ωψ (H 0 ) ·
X
S B(L ,L
0 ,d/n)∗
⊆ S G L ,(d/n)+
(10.8.4)
L 0 ∈Q (d)
^ ^ follows from Proposition 6.2 and the commutativity of U (V ) and U (V 0 ). The proof is complete. 10.9. Proof of Corollary 6.4. Let Q 0 (d) denote the subset of regular small admissible lattice chains in Q (d). It is clear that X 0 ∗ S G L ,(d/n)+ ⊃ ωψ (H 0 ) · S B(L ,L ,d/n) . L 0 ∈Q 0 (d)
So now we prove the opposite inclusion. Because the Witt index of V 0 is now large, we may assume that we have a decomposition V 0 = V10 ⊕ X ⊕ Y such that X, Y are totally isotropic and dual to each other, M j ⊆ V10 , N j ⊆ V10 for all j, and dim D (X ) = n 0 + 1 for some number n 0 ≥ n − 1, where M j , N j are defined in (10.5.3) or (10.7.2). Moreover, we can assume that the decomposition V 0 = V10 ⊕ X ⊕ Y satisfies the condition M M xk O ⊕ yk O , 0 0 = (0 0 ∩ V10 ) ⊕ 0≤k≤n 0
0≤k≤n 0
where {xi , yi }0≤i≤n 0 is a self-dual basis of X ⊕ Y (i.e., hxi , x j i = hyi , y j i = 0 and 0 hxi , y j i = δi j $ κ ). Then 0 0 ∩ V10 is a maximal good lattice in V10 . From the proofs of Proposition 10.5 and Proposition 10.7, we know that M j , N j are free O -modules
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in V10 , and that M j+n = M j $ , N j+n = N j $ , M j ⊆ N j ⊆ M j−1 , and hMi , N j i0 ⊆ 0 0 pκ +b(i+ j+n 0 )/nc for all i, j. Suppose that L i00 are the lattices L i0 in V10 constructed in the proof of Lemma 10.1, and let L 00 be the small admissible lattice chain {L i00 }i∈Z (in V10 ). Two lattices in L 00 are not necessarily distinct. Let M M M L i0 := L i00 ⊕ xk p ⊕ xk O ⊕ yk O 0≤k≤i
i+1≤k≤n 0
0≤k≤n 0
for i = 1 − n 0 , . . . , b(n − n 0 − 1)/2c. We regard as a subset of L i0 for each i. Let L 0 be the lattice chain as constructed in the proof of Lemma 10.1 for the new L i0 . Then clearly L 0 is a regular small admissible lattice chain in V 0 . As in Sections 10.5 and 10.7, we show that if f [w, G L ,(d/n)+ ] is nonzero, then w is in B(L , L 00 , d/n) for some L 00 given as above. Hence, w is in B(L , L 0 , d/n) because L i00 ⊂ L i0 for each i. Thus, by an argument similar to that in Section 10.5, we prove that X 0 ∗ S G L ,(d/n)+ ⊂ ωψ (H 0 ) · S B(L ,L ,d/n) . L i00
L 0 ∈Q 0 (d)
The proof is complete.
11. Proof of Proposition 6.5 11.1 0 in V 0 such that We fix a maximal good lattice 0 0 and a minimal good lattice 0m 0 ⊆ 00. 0m LEMMA 0∗ is a good lattice in W . Let L be a good lattice in V . Then L ∗ ⊗ 0 0 ∩ L ⊗ 0m
Proof From [W, sec. I.15] we know that there exists a decomposition V = X 1 ⊕ X 2 such that L = (L ∩ X 1 ) ⊕ (L ∩ X 2 ) and L ∗ = (L ∩ X 1 ) ⊕ (L ∩ X 2 )$ −1 . There also exists a decomposition V 0 = Y1 ⊕ Y2 ⊕ Y3 ⊕ Y4 such that 0 0m = (0 0 ∩ Y1 )$ ⊕ (0 0 ∩ Y2 )
⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
0 0∗ = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )$ −1 ⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
0∗ 0m = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )$ −1 ⊕ (0 0 ∩ Y3 )$ −1 ⊕ (0 0 ∩ Y4 ).
0 0 = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )
From the above decompositions it is easy to check that 0∗ 0 0∗ L ⊗ 0m ∩ L ∗ ⊗ 0 0 ⊆ L ∗ ⊗ 0m + L ⊗ 0 0∗ = (L ⊗ 0m ∩ L ∗ ⊗ 0)0∗ .
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
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Moreover, we have 0 L ∗ ⊗ 0m + L ⊗ 0 0∗ ⊆ L ∗ ⊗ 0 0 + L ⊗ 0 0∗ = L$ −1 ⊗ 0 0 ∩ L ∗ ⊗ 0 0∗ 0∗ ⊆ (L ⊗ 0m ∩ L ∗ ⊗ 0 0 )$ −1 . 0∗ is a good lattice in W . Hence, L ∗ ⊗ 0 0 ∩ L ⊗ 0m
From the decompositions in the proof of the lemma, it is easy to check that 0∗ 0∗ L ∗ ⊗ 0 0 ∩ L ⊗ 0m = L ⊗ 0 0 + L ∗ $ ⊗ 0m .
(11.1.1)
11.2 LEMMA Let 0 0 be
(i)
(ii)
(iii)
a fixed maximal good lattice in V 0 . Suppose that M0 , M−1 are two O -modules in V 0 such that M0 ⊆ M−1 ⊆ 0 M0 $ −1 , M0 ⊆ 0 0 , and hM0 , M−1 i0 ⊆ pκ . Then there exists a good lattice L 0 in V 0 such that L 0 ⊆ 0 0 , M0 ⊆ L 0 , and M−1 ⊆ L 0∗ . 0 be a minimal good lattice in V 0 such that 0 0 ⊆ 0 0 . Suppose that Let 0m m M0 , M−1 are two O -modules in V 0 such that M0 ⊆ M−1 ⊆ M0 $ −1 , M0 ⊆ 0∗ , and hM , M i0 ⊆ pκ 0 . Then there exists a good lattice L 0 in 0 0 , M−1 ⊆ 0m 0 −1 0 ⊆ L 0 ⊆ 0 0 , M ⊆ L 0 , and M 0∗ V 0 such that 0m 0 −1 ⊆ L . 0 Suppose that M is an O -module in V 0 such that hM, Mi0 ⊆ pκ and M ⊆ 0 0∗ . Then M is contained in 0 0 .
Proof Without loss of generality, we may assume that M0 , M−1 are lattices in V 0 . From the assumption, we have M−1 + 0 0∗ ⊆ M0 $ −1 + 0 0∗ ⊆ 0 0 $ −1 . So we may regard M−1 /(M−1 ∩ 0 0∗ ) ' (M−1 + 0 0∗ )/ 0 0∗ as a d-subspace of 0 0 $ −1/ 0 0∗ . Let {z i }i∈J (for some finite index set J ) be a subset of M−1 such that the images of these z i in M−1 /(M−1 ∩ 0 0∗ ) are linearly independent over d. From the assumption, we have 0 0 hM−1 , M−1 i0 ⊆ pκ −1 . Therefore, hz i $, z j $ i0 ⊆ pκ +1 for any i, j ∈ J . Therefore, the set {z i $ }i∈J as subset of 0 0 satisfies the condition in Lemma 4.1(iii) with l = 1. Thus, there exists a 0 0 -admissible decomposition V 0 = X ⊕ V 0◦ ⊕ Y , where X , Y are totally isotropic and in duality, and a basis {vi }i∈J of X such that vi − z i $ ∈ 0 0 $ for every i ∈ J . Note that a decomposition of V 0 is 0 0 -admissible if and only if it is 0 0∗ -admissible. Therefore, from the choice of the set {z i }i∈J , we have M−1 + 0 0∗ = (0 0∗ ∩ X )$ −1 ⊕ (0 0∗ ∩ V 0◦ ) ⊕ (0 0∗ ∩ Y ).
(11.2.1)
From (11.2.1), we see that (M−1 + 0 0∗ )∗ is a good lattice in V 0 . Define L 0 := (M−1 + ∗ ∩ 0 0 . Therefore, L 0∗ = M 0∗ 0 0 0 0∗ )∗ = M−1 −1 + 0 . Hence, we have L ⊆ 0 and
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∗ . Therefore, we have M ∗ 0∗ 0 L 0 ⊆ M−1 −1 ⊆ L . Since M0 ⊆ M−1 and M0 ⊆ 0 , we have 0 0 M0 ⊆ L . Then L satisfies all requirements. 0 ⊆ L 0. For (ii), let L 0 be given as in the proof of (i). We only need to check that 0m 0∗ because As in the previous paragraph, we have L 0∗ = M−1 + 0 0∗ . Hence, L 0∗ ⊆ 0m 0∗ 0∗ 0∗ 0 0 M−1 ⊆ 0m and 0 ⊆ 0m . Therefore, we conclude that 0m ⊆ L . Part (iii) has already been proved in Lemma 10.2.
11.3 LEMMA 0∗ . Then G Let L be a good lattice in V , and let A := L ∗ ⊗0 0 ∩ L ⊗0m L ,0+ is contained 0 in K A .
Proof From (11.1.1) it is easy to see that 0 A∗ = L ∗ ⊗ 0 0∗ ∩ L ⊗ 0m $ −1 .
(11.3.1)
0 ⊆ A. Hence, Let g be an element in G L ,0+ . Then (g − 1) · A∗ ⊆ L ⊗ 0 0∗ ∩ L ∗ ⊗ 0m 0 g is in K A .
11.4 PROPOSITION 0∗ ∩ L$ −1 ⊗ 0 0 and that Let L be a good lattice in V . Suppose that w is in L ∗ ⊗ 0m 0∗ . Then f [w, G L ,0+ ] is nonzero for some f ∈ S (A)w , where A := L ∗ ⊗ 0 0 ∩ L ⊗ 0m 0 ∗ 0 0 0 0 w belongs to B(L , L ) for some good lattice L such that 0m ⊆ L ⊆ 0 .
Proof We know that A is a good lattice in W by Lemma 11.1. By Lemma 11.3, G L ,0+ 0∗ ∩ L$ −1 ⊗ 0 0 is a subgroup of K A0 , so f [w, G L ,0+ ] is defined. Identify L ∗ ⊗ 0m −κ 0∗ ∗ 1−κ 0 with HomO (L$ , 0m ) ∩ HomO (L $ , 0 ). Let x be an element in L, and let y be an element in L ∗ $ 1−κ . Then ord L ∗ (x) ≥ ord L (x) ≥ 0, ord L (y) ≥ −κ, and ord L ∗ (y) ≥ 1 − κ. Then x, y satisfy the condition in Corollary 7.6(i). Hence, u x,y is defined and belongs to G L ,0+ . Now cx,y · w ⊆ A as in the proof of Lemma 11.3, so ωψ (u x,y ) · f (w) = ψ hhw, cx,y · wii f (w) by Lemma 8.1(i). Because we have (u −1 −1)·w ∈ A and we assume that f [w, G L ,0+ ] x,y is nonzero, we have ψ hhw, cx,y · wii = 1 by Lemma 8.6. Hence, we have hhw, cx,y · wii ∈ pλFF . Now Trd D/F hw · x, w · yi0 is in pλFF by Lemma 8.1(ii). Therefore, 0 hw · x, w · yi0 is in pλ = pκ+κ from the definition in Section 5.2. Therefore, hw · 0 x$ −κ , w · yi0 ∈ pκ . Because x (resp., y) is arbitrary in L (resp., L ∗ $ 1−κ ), we have hw · L$ −κ , w · L ∗ $ 1−κ i0 ⊆ pκ . 0
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
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Now we have w · L ∗ $ 1−κ ⊆ w · L$ −κ ⊆ w · L ∗ $ −κ since L is a good lattice. 0∗ ∩ L$ −1 ⊗ 0 0 , we have w · L ∗ $ 1−κ ⊆ 0 0 and w · Because w is in L ∗ ⊗ 0m −κ 0∗ L$ ⊆ 0m . Hence, by Lemma 11.2(ii) there exists a good lattice L 0 in V 0 such 0 ⊆ L 0 ⊆ 0 0 , w · L ∗ $ 1−κ ⊆ L 0 , and w · L$ −κ ⊆ L 0∗ . Therefore, w is in that 0m HomO (L$ −κ , L 0∗ )∩HomO (L ∗ $ 1−κ , L 0 ), which is exactly L ∗ ⊗ L 0∗ ∩ L ⊗ L 0 $ −1 = B(L , L 0 )∗ . 11.5. Proof of Proposition 6.5. Now we begin to prove Proposition 6.5. The inclusion X 0 ωψ (H 0 ) · S B(L ,L ) ⊆ S G L ,0+ L0
^ is an easy consequence of Proposition 6.2(ii) and the commutativity of U (V ) and 0 ^ ^ U (V ) in Sp(W ). We prove the opposite inclusion by discussion according to the following three separate cases: (1) L = L ∗, (2) L = L ∗$ , (3) L ∗ $ 6= L 6= L ∗ . First, suppose that we are in the first case, that is, that L is self-dual. Hence, L is a maximal good lattice in V . Then G L ,0+ = {g ∈ G | (g − 1) · L ⊆ L$ }. Let M := L$ , N := L, and let R (L), B M,N , J M,N be as defined in Section 5.2. Clearly, the pair (M, N ) belongs to R (L). So B M,N = L$ −1 ⊗ 0 0 + L ⊗ 0 0∗ = L$ −1 ⊗ 0 0 for a fixed maximal good lattice 0 0 in V 0 . It is easy to check that J M,N = G L ,0+ in this case. Now we have (g − 1) · B M,N ⊆ L ⊗ 0 0 for all g ∈ G L ,0+ . Clearly, A := L ⊗ 0 0 = B(L , 0 0 ) is a good lattice in W , and G L ,0+ is a subgroup of K A0 . Let x be an element in L, and let y be in L$ 1−κ . Then we have ord L (x) ≥ 0 and ord L (y) ≥ 1 − κ. Therefore, x, y satisfy the condition in Corollary 7.6(i). Hence, u x,y is defined and belongs G L ,0+ . Let w be an element in B M,N . Then cx,y · w belongs to A. Therefore, ωψ (u x,y ) · f = ψ hhw, cx,y · wii f by Lemma 8.1(i) for f ∈ S (A)w . Suppose that f [w, G L ,0+ ] is nonzero. Then ψ hhw, cx,y · wii = 1 by Lemma 8.6. Hence, we have hhw, cx,y · wii ∈ pλFF . Now Trd D/F hw · x, w · yi0 ∈ pλFF 0 by Lemma 8.1(ii). From Section 5.2 we know that hw · x, w · yi0 ∈ pλ = pκ+κ . 0 Hence, we have hw · x$ −κ , w · yi0 ∈ pκ . Therefore, hw · L$ −κ , w · L$ 1−κ i0 is 0 contained in pκ . Since w is in L$ −1 ⊗ 0 0 , we have w · L$ 1−κ ⊆ 0 0 . Hence, by Lemma 11.2(i), we know that there exists a good lattice L 0 in V 0 such that L 0 ⊆ 0 0 and w · L$ −κ ⊆ L 0∗ . Hence, w ∈ L ⊗ L 0∗ = B(L , L 0 )∗ because L = L ∗ in this case. Therefore, we have proved that if w is in B M,N and f [w, G L ,0+ ] is nonzero for some f ∈ S (A)w , then w must be in B(L , L 0 )∗ for some good lattice L 0 ⊆ 0 0 . So
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we conclude that S (A) B M,N
G L ,0+
=
X
X
f [w, G L ,0+ ] ⊆
w∈B M,N f ∈S (A)w
X
S (A) B(L ,L 0 )∗ .
L0
Therefore, by (8.2.1) and Proposition 5.3, we have X ∗ 0 S G L ,0+ = ωψ (H 0 ) · (S B M,N )G L ,0+ ⊆ ωψ (H 0 ) · S B(L ,L ) . L0
Secondly, suppose that L = L ∗ $ . The proof for this case is very similar to the proof for the first case. Now again G L ,0+ = {g ∈ G | (g − 1) · L ⊆ L$ } in this case. Let M = N := L. We have (M, N ) ∈ R (0), where 0 is a fixed maximal good lattice in V containing L. Now we have B M,N = L ∗ ⊗ 0 0 + L ∗ ⊗ 0 0∗ = L ∗ ⊗ 0 0∗ for a fixed maximal good lattice 0 0 in V 0 . And we also have J M,N = G L ,0+ . Clearly, A := L ⊗ 0 0∗ is a good lattice in W . Then G L ,0+ is a subgroup of K A0 . Let x be an element in L, and let y be an element in L$ −κ . Then x, y satisfy the conditions in Corollary 7.6(i). Hence, u x,y is defined and belongs to G L ,0+ . By the same argument 0 as in the first case, we can prove that hw · L$ −κ , w · L$ −κ i0 ⊆ pκ . Now w · L$ −κ is contained in 0 0∗ because w ∈ B M,N . Therefore, we get w · L$ −κ ⊆ 0 0 by Lemma 11.2(iii). Let L 0 := 0 0 , which is of course a good lattice in V 0 . Hence, w is in L ∗ ⊗ L 0 = B(L , L 0 )∗ because L ∗ = L$ −1 in this case. Therefore, we have P B(L ,L 0 ) by the same argument as in the first case. S G L ,0+ ⊆ ωψ (H 0 ) · L0 S Finally, we suppose that L ∗ $ 6 = L 6 = L ∗ . Now we let M := L ∗ $ , N := L. Then we know that (M, N ) ∈ R (0), B M,N = L$ −1 ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 $ −1 ,
(11.5.1)
and J M,N = G L ,1 . Suppose that g is an element in G L ,1 . Then from (11.5.1) we have (g − 1) · B M,N ⊆ L ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 . Let A be the good lattice L ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 . Hence, (g − 1) · B M,N ⊆ A ⊆ A∗ ⊆ B M,N for all g ∈ G L ,1 . Thus, G L ,1 is a subgroup of K A0 . Let w be an element in B M,N . We identify B M,N with HomO (L ∗ $ 1−κ , 0 0∗ ) ∩ HomO (L$ 1−κ , 0 0 ). Let x be an element in L, and let y be an element in L$ 1−κ . Then x, y satisfy the condition in Corollary 7.6(ii). Hence, u x,y is defined and belongs to G L ,1 . By the 0 same argument as in the first case, we can prove that hw · L$ 1−κ , w · L$ −κ i0 ⊆ pκ . We know that w · L$ 1−κ ⊆ 0 0 . Therefore, by Lemma 11.2(i), we have that w · L$ −κ 0 be a is contained in L 00∗ for some good lattice L 00 in V 0 such that L 00 ⊆ 0 0 . Let 0m
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
589
0 ⊆ L 00 . Hence, w · L$ −κ is contained in minimal good lattice in V 0 such that 0m 0∗ ∗ 0m . Let x be an element in L $ , and let y be an element in L ∗ $ 1−κ . Then again x, y satisfy the condition in Corollary 7.6(ii). Hence, u x,y is defined and belongs to G L ,1 . By the same argument as in the first case, we can prove that hw · L ∗ $ 1−κ , w · 0 L ∗ $ 1−κ i0 ⊆ pκ . Since w is in B M,N , we have w · L ∗ $ 1−κ ⊆ 0 0∗ . Therefore, we have w · L ∗ $ 1−κ ⊆ 0 0 by Lemma 11.2(iii). Therefore, we have proved that if w ∈ B M,N and f [w, G L ,1 ] is nonzero for some f ∈ S (A)w , then w must be in 0∗ ∩ L ⊗ 0 0 $ −1 for some minimal good lattice 0 0 ⊆ 0 0 . Therefore, we have L ∗ ⊗ 0m m G L ,1 X ⊆ S (A) R , S (A) B M,N 0 0m
0∗ ∩ L ⊗ 0 0 $ −1 . Hence, by Lemma 8.2, we have where R := L ∗ ⊗ 0m X ∗ ∗ (S B M,N )G L ,1 ⊆ SR , 0 0m
^ where S is any model of the Weil representation of Sp( W ). Define A0 := L ∗ ⊗ 0 0 ∩ 0∗ . Now by Proposition 11.4 we know that if w ∈ R and f [w, G L ⊗0m L ,0+ ] is nonzero for some f ∈ S (A0 )w , then w ∈ B(L , L 0 )∗ for some good lattice L 0 in V 0 such that L 0 ⊆ 0 0 . Hence, we have proved X X 0 ∗ G L ,0+ SR ⊆ S B(L ,L ) . 0 0m
L0
Therefore, by (8.2.1) and Proposition 5.3, we have ∗
S G L ,0+ ⊆ ωψ (H 0 ) · (S B M,N )G L ,0+ ⊆ ωψ (H 0 ) ·
X
0 S B(L ,L ) .
L0
The proof is complete.
12. Admissible splitting and depth preservation 12.1. Reductive dual pairs and splittings ^ A reductive dual pair (U (V ), U (V 0 )) is said to be split if both extensions U (V ) → 0 0 ^ U (V ) and U (V ) → U (V ) split. It is known that all reductive dual pairs are split except the cases when D = F and the orthogonal space is odd dimensional. Suppose ^ now that (U (V ), U (V 0 )) is a split reductive dual pair. Fix splittings U (V ) → U (V ) 0 0 ^ ^ and U (V ) → U (V ). Then an irreducible admissible representation of U (V ) (resp., ^ U (V 0 )) can be regarded as an irreducible admissible representation of U (V ) (resp.,
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U (V 0 )) via the splittings. Hence, we establish a one-to-one correspondence between some irreducible admissible representations of U (V ) and some irreducible admissible representations of U (V 0 ). We emphasize here that this correspondence depends on the choice of splittings as well as the forms h, i, h, i0 and the character ψ. An explicit formula of a splitting for each metaplectic cover is given in [K2] when the splitting exists. Another splitting of the metaplectic cover is given in [P1] whenever D is com^ eL 0 : U (V ) → U mutative. From now on we assume that D is commutative. Let β (V ) V L e be the splitting given in [P1]. As indicated in the notation, the splitting βV 0 depends on a good lattice L in V . Recall that when we define the minimal K -types for the ^ metaplectic group U (V ), we need to choose an Iwahori subgroup I of U (V ). We choose a good lattice L such that I ⊆ G L , where G L is the stabilizer of L in U (V ). Then we have the following result. PROPOSITION
^ Let π be an irreducible admissible representation of U (V ). Then the depth of π is L e equal to the depth of the representation π ◦ βV 0 of U (V ). Proof eL 0 has nontrivial fixed points by G x,r + for some point x in the Suppose that π ◦ β V building of G. There exists an element g ∈ G such that gG x,r + g −1 = G g·x,r + ⊆ I . eL 0 has nontrivial fixed points by G g·x,r + . Therefore, π has nontrivial Hence, π ◦ β V eL 0 (G g·x,r + ). Because the depth of π (resp., π ◦ β eL 0 ) is the minimal fixed points by β V V eL 0 ) has nontrivial points fixed by β eL 0 (G y,r + ) real number r such that π (resp., π ◦ β V V (resp., G y,r + ) for all y such that G y,r + ⊆ I , the proposition is proved. 12.2. Depth preservation for split reductive dual pairs From Proposition 12.1 and Theorem 6.6, we have the following result on depth preservation for split reductive dual pairs. THEOREM
Let (U (V ), U (V 0 )) be a split reductive dual pair such that D is commutative. Fix an Iwahori subgroup I (resp., I 0 ) of U (V ) (resp., U (V 0 )) and a good lattice L (resp., L 0 ) in V (resp., V 0 ) such that I ⊆ G L (resp., I 0 ⊆ G 0L 0 ). Let π (resp., π 0 ) be an ^ ^ irreducible admissible representation of U (V ) (resp., U (V 0 )) such that π and π 0 eL 0 is equal to correspond in the local theta correspondence. Then the depth of π ◦ β V 0 L 0 e . the depth of π ◦ β V
Acknowledgments. This work is the major part of the author’s dissertation at Cor-
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nell University. The author would like to thank his advisor, Professor Dan Barbasch, wholeheartedly for his help, encouragement, and constant support during these years. Professor Jiu-Kang Yu carefully read the early drafts, eliminated many mistakes, and provided numerous improvements. The details of his contributions are in the main body of this paper. Finally, the author would like to thank the referee for pointing out a few errors in the first draft of the paper. References [A]
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J.-K. YU, Descent mapping in Bruhat-Tits theory, preprint, 1998. 533, 539, 546
, Unrefined minimal K -types for p-adic classical groups, preprint, 1998. 533, 537, 540, 546
Department of Mathematics, National Cheng Kung University, 1 Ta Hsuen Road, Tainan City 701, Taiwan, Republic of China;
[email protected]