ANNALS OF MATHEMATICS
Multiplicity one theorems
By Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis, and Gerard´ Schiffmann
SECOND SERIES, VOL. 172, NO. 2 September, 2010
anmaah Annals of Mathematics, 172 (2010), 1407–1434
Multiplicity one theorems
By AVRAHAM AIZENBUD, DMITRY GOUREVITCH, STEPHEN RALLIS, and GÉRARD SCHIFFMANN
Abstract
In the local, characteristic 0, non-Archimedean case, we consider distributions on GL.n 1/ which are invariant under the adjoint action of GL.n/. We prove that C such distributions are invariant by transposition. This implies multiplicity at most one for restrictions from GL.n 1/ to GL.n/. Similar theorems are obtained for C orthogonal or unitary groups.
Introduction
Let ކ be a local field non-Archimedean and of characteristic 0. Let W be a vector space over ކ of finite dimension n 1 > 1, and let W V U be a direct C D ˚ sum decomposition with dim V n. Then we have an imbedding of GL.V / into D GL.W /. Our goal is to prove the following theorem:
THEOREM 1. If (resp. ) is an irreducible admissible representation of GL.W / (resp. of GL.V /), then dim HomGL.V /. GL.V / ; / 6 1: j We choose a basis of V and a nonzero vector in U thus getting a basis of W . We can identify GL.W / with GL.n 1; ކ/ and GL.V / with GL.n; ކ/. The trans- C position map is an involutive anti-automorphism of GL.n 1; ކ/ which leaves C GL.n; ކ/ stable. It acts on the space of distributions on GL.n 1; ކ/. C Theorem 1 is a corollary of:
THEOREM 2. A distribution on GL.W / which is invariant under the adjoint action of GL.V / is invariant by transposition.
One can raise a similar question for orthogonal and unitary groups. Let ބ be either ކ or a quadratic extension of ކ. If x ބ, then x is the conjugate of x if 2 x ބ ކ, and is equal to x if ބ ކ. ¤ D 1407 1408 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
Let W be a vector space over ބ of finite dimension n 1 > 1. Let :;: be a C h i nondegenerate hermitian form on W . This form is bi-additive and dw; d w d d w; w ; w ; w w; w : h 0 0i D 0h 0i h 0 i D h 0i Given a ބ-linear map u from W into itself, its adjoint u is defined by the usual formula u.w/; w w; u .w / : h 0i D h 0 i Choose a vector e in W such that e; e 0; let U ބe and V U h i ¤ D D ? the orthogonal complement. Then V has dimension n and the restriction of the hermitian form to V is nondegenerate. Let M be the unitary group of W that is to say the group of all ބ-linear maps m of W into itself which preserve the hermitian form or equivalently such that mm 1. Let G be the unitary group of V . With the p-adic topology both groups D are of type lctd (locally compact, totally disconnected) and countable at infinity. They are reductive groups of classical type. The group G is naturally imbedded into M .
THEOREM 10. If (resp ) is an irreducible admissible representation of M (resp of G), then dim HomG. G; / 1: j Ä Choose a basis e1; : : : en of V such that ei ; ej ކ. For h i 2 n X w x0e xi ei ; D C 1 put n X w x0e xi ei : x D x C 1 If u is a ބ-linear map from W into itself, then let u be defined by x u.w/ u.w/: x D x Let be the anti-involution .m/ m 1 of M ; Theorem 1 is a consequence D x 0 of
THEOREM 20. A distribution on M which is invariant under the adjoint action of G is invariant under .
Let us describe briefly our proof. In Section 1 we recall why Theorem 2.20/ implies Theorem 1.10/. For the proofs of Theorems 2 and 20 we systematically use two classical results: Bernstein’s localization principle and a variant of Frobenius reciprocity which we call Frobenius descent. For the convenience of the reader they are both recalled in Section 2. MULTIPLICITYONETHEOREMS 1409
Then we proceed with GL.n/. The proof is by induction on n; the case n 0 D is trivial. In general we first linearize the problem by replacing the action of G on GL.W / by the action on the Lie algebra of GL.W /. As a G-module this Lie algebra is isomorphic to a direct sum g V V ކ with g the Lie algebra of G and ˚ ˚ ˚ V the dual space of V . The group G GL.V / acts trivially on ކ, by the adjoint D action on its Lie algebra and the natural actions on V and V . The component ކ plays no role. Let u be a linear bijection of V onto V which transforms some basis of V into its dual basis. The involution may be taken as .X; v; v / .u 1 t X u; u 1.v /; u.v//: 7! We have to show that a distribution T on g V V which is invariant under G ˚ ˚ and skew relative to the involution is 0. In Section 3 we prove that the support of such a distribution must be contained in a certain singular set. On the g side, using Harish-Chandra descent we get that the support of T must be contained in .z ᏺ/ .V V /, where z is the center of C ˚ g and ᏺ the cone of nilpotent elements in g. On the V V side we show that the ˚ support must be contained in g , where is the cone v; v 0 in V V . h i D ˚ On z the action is trivial so we are reduced to the case of a distribution on ᏺ . In Section 4 we consider such distributions. The end of the proof is based on two remarks. First, viewing the distribution as a distribution on ᏺ .V V /, its ˚ partial Fourier transform relative to V V has the same invariance properties and ˚ hence must also be supported on ᏺ . This implies, in particular, a homogeneity condition on V V . The idea of using Fourier transform in this kind of situation ˚ goes back at least to Harish-Chandra ([HC99]) and it is conveniently expressed using a particular case of the Weil or oscillator representation. For .v; v / , let Xv;v be the map x x; v v of V into itself. The 2 7! h i second remark is that the one parameter group of transformations
.X; v; v/ .X Xv;v ; v; v/ 7! C is a group of (nonlinear) homeomorphisms of Œg; g which commute with G and the involution. It follows that the image of the support of our distribution must also be singular. This allows us to replace the condition v; v 0 by the stricter h i D condition Xv;v Im adX. 2 Using the stratification of ᏺ we proceed one nilpotent orbit at a time, transfer- ring the problem to V V and a fixed nilpotent matrix X. The support condition ˚ turns out to be compatible with direct sum so that it is enough to consider the case of a principal nilpotent element. In this last situation the key is the homogeneity condition coupled with an easy induction. The orthogonal and unitary cases are proved roughly in the same way. In Section 5 we reduce the support to the singular set. Here the main difference is that we use Harish-Chandra descent directly on the group. Note that some sub- groups that occur in the Harish-Chandra descent have components of type GL so 1410 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN that Theorem 2 has to be used. Finally, in Section 6, we consider the case of a distribution supported in the singular set; the proof follows the same line as in Section 4. As for the archimedean case, partial analogs of the results of this paper were obtained in [AGS08], [AGS09], and [vD09]. Recently, the full analogs were ob- tained in [AG09] and [SZ]. Recently, it was shown that Theorem 1 implies an analogous theorem for general linear groups over local fields of positive characteristic (see [AAG]). Let us add some comments on the theorems themselves. First note that The- orem 2 gives an independent proof of a well-known theorem of Bernstein: choose a basis e1; : : : ; en of V , add some vector e0 of W to obtain a basis of W , and let P be the isotropy of e0 in GL.W /. Then Theorem B of [Ber84] says that a distribution on GL.W / which is invariant under the action of P is invariant under the action of GL.W /. Now, by Theorem 2 such a distribution is invariant under the adjoint action of the transpose of P and the group of inner automorphisms is generated by the images of P and its transpose. We thus get an independent proof of Kirillov conjecture in the characteristic 0, non-Archimedean case. The occurrence of involutions in multiplicity at most one problems is of course nothing new. The situation is fairly simple when all the orbits are stable by the involution thanks to Bernstein’s localization principle and constructivity theorem ([BZ76], [GK75]). In our case this is not true: only generic orbits are stable. Non- stable orbits may carry invariant measures but they do not extend to the ambient space (a similar situation is already present in [Ber84]). An illustrative example is the case n 1 for GL. It reduces to ކ acting on D ކ2 as .x; y/ .tx; t 1y/. On the x axis the measure d x dx= x is invariant 7! D j j but does not extend invariantly. However the symmetric measure
Z Z f f .x; 0/d x f .0; y/d y 7! C ކ ކ does extend. As in similar cases (for example [JR96]) our proof does not give a simple ex- planation of why all invariant distributions are symmetric. The situation would be much better if we had some kind of density theorem. For example, in the GL case, n 1 let us say that an element .X; v; v/ of g V V is regular if .v; Xv; : : : X v/ t n 1 ˚ ˚ is a basis of V and .v;:::; X v/ is basis of V . The set of regular elements is a nonempty Zariski open subset; regular elements have trivial isotropy subgroups. The regular orbits are the orbits of the regular elements; they are closed, separated by the invariant polynomials, and stable by the involution (see [RS]). In particular they carry invariant measures which, the orbits being closed, do extend and are invariant by the involution. It is tempting to conjecture that the subspace of the space of invariant distributions generated by these measures is weakly dense. This MULTIPLICITYONETHEOREMS 1411 would provide a better understanding of Theorem 2. Unfortunately, if true at all, such a density theorem is likely to be much harder to prove. Assuming multiplicity at most one, a more difficult question is to find when it is one. Some partial results are known. For the orthogonal group (in fact the special orthogonal group) this question has been studied by B. Gross and D. Prasad ([GP92], [Pra93]) who formulated a precise conjecture. An up to date account is given by B. Gross and M. Reeder ([GR06]). For an even more recent account see [GGP], [MW], [Wala], [Walb]. In a different setup, in their work on “Shintani” functions, A. Murase and T. Sugano obtained complete results for GL.n/ and the split orthogonal case but only for spherical representations ([KMS03], [MS96]). Finally, we should mention that Hakim’s publication [Hak03], which, at least for the discrete series, could perhaps lead to a different kind of proof. Recently, Waldspurger showed how to adapt our proofs of Theorems 10 and 20 to include the case of special orthogonal groups (see [Walc]). Also, [GGP] and [Sun] showed that our results imply the uniqueness of general Bessel and Fourier- Jacobi models over p-adic local fields in almost all cases (see [GGP, Cors. 15.2, 16.2, and 16.3]). “Multiplicity at most one” theorems have important applications to the relative trace formula, to automorphic descent, to local and global liftings of automorphic representations, and to determinations of L-functions. In particular, multiplicity at most one is used as a hypothesis in the work [GPSR97] on the study of automorphic L-functions on classical groups. At least for the last two authors, the original motivation for this work came in fact from [GPSR97].
1. Theorem 2.20/ implies Theorem 1.10/ A group of type lctd is a locally compact, totally disconnected group which is countable at infinity. We consider smooth representations of such groups. If
.; E / is such a representation, then .;E/ is the smooth contragredient. We denote smooth induction by Ind and compact induction by ind. For any topological space T of type lctd, .T / is the space of functions locally constant, complex valued, defined on T , and with compact support. The space 0.T / of distributions on T is the dual space to .T /. PROPOSITION 1.1. Let M be a lctd group and N a closed subgroup, both unimodular. Suppose that there exists an involutive anti-automorphism of M such that .N / N and such that any distribution on M , biinvariant under N , is D fixed by . Then, for any irreducible admissible representation of M M M dim HomM ind .1/; dim HomM ind .1/; 1: N N Ä This is well known (see for example [Pra90]). Remark. There is a variant for the nonunimodular case; we will not need it. 1412 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
COROLLARY 1.1. Let M be a lctd group and N a closed subgroup, both uni- modular. Suppose that there exists an involutive anti-automorphism of M such that .N / N and such that any distribution on M , invariant under the adjoint D action of N , is fixed by . Then, for any irreducible admissible representation of M and any irreducible admissible representation of N dim HomN . N ; / dim HomN ../ N ; / 1: j j Ä Proof. Let M M N and N be the closed subgroup of M which is the 0 D 0 0 image of the homomorphism n .n; n/ of N into M . The map .m; n/ mn 1 7! 7! of M 0 onto M defines a homeomorphism of M 0=N 0 onto M . The inverse map is m .m; 1/N . On M =N left translations by N correspond to the adjoint 7! 0 0 0 0 action of N on M . We have a bijection between the space of distributions T on M invariant under the adjoint action of N and the space of distributions S on M 0 which are biinvariant under N 0. Explicitly, D Z E S; f .m; n/ T; f .mn; n/dn : h i D N Suppose that T is invariant under and consider the involutive anti-automorphism of M given by .m; n/ ..m/; .n//. Then 0 0 0 D D Z E S; f 0 T; f ..n/.m/; .n//dn : h ı i D N Using the invariance under and for the adjoint action of N , we get D Z E D Z E T; f ..n/.m/; .n//dn T; f ..n/m; .n//dn N D N D Z E T; f .mn; n/dn D N D E S; f : D
Hence S is invariant under 0. Conversely, if S is invariant under 0, then the same computation shows that T is invariant under . Under the assumption of the corollary we can now apply Proposition 1.1 and we obtain the inequality M 0 M 0 dim HomM indN .1/; dim HomM indN .1/; 1: 0 0 ˝ 0 0 ˝ Ä We know that IndM 0 .1/ is the smooth contragredient representation of indM 0 .1/; N 0 N 0 hence M 0 M 0 HomM indN .1/; HomM . ; IndN .1//: 0 0 ˝ 0 ˝ 0 Frobenius reciprocity tells us that
M 0 HomM ; IndN .1/ HomN . / N ; 1 : 0 ˝ 0 0 ˝ j 0 Clearly, HomN . / N ; 1 HomN ; . N / HomN . N ; /: 0 ˝ j 0 j j MULTIPLICITYONETHEOREMS 1413
Using again Frobenius reciprocity we get
M HomN ; . N / HomM indN ./; : j
In the above computations we may replace by and by . Finally,
M 0 HomM .indN .1/; / HomN .; . N // 0 0 ˝ j HomN . N ; / j M HomM .ind ./; /; N M 0 HomM .indN .1/; / HomN .; ../ N // 0 0 ˝ j HomN ../ N ; / j M HomM .ind . /; /: N Going back to our situation and keeping the notation of the introduction, we consider first the case of the general linear group. We take M GL.W / and D N GL.V /. Let E be the space of the representation and let E be the smooth D dual (relative to the action of GL.W //. Let E be the space of and E be the smooth dual for the action of GL.V /. We know ([BZ76, 7]) that the contragredient t 1 representation in E is isomorphic to the representation g . g / in E . 7! The same is true for . Therefore an element of HomN . N ; / may be described j as a linear map A from E into E such that, for g N , 2 A.g/ .t g 1/A: D
An element of HomN ../ N ; / may be described as a linear map A0 from E j into E such that, for g N , 2 A .t g 1/ .g/A : 0 D 0 We have obtained the same set of linear maps: HomN N ; HomN . N ; /: j j We are left with two possibilities: either both spaces have dimension 0 or they both have dimension 1 which is exactly what we want. From now on we forget Theorem 1 and prove Theorem 2. Consider the orthogonal/unitary case, with the notation of the introduction. In Chapter 4 of [MVW87] the following result is proved. Choose ı GLކ.W / such 2 that ıw; ıw0 w0; w . If is an irreducible admissible representation of M , h i D h i ı let be its smooth contragredient and define by ı .x/ .ıxı 1/: D Then ı and are equivalent. We choose ı 1 in the orthogonal case ބ ކ. D D In the unitary case, fix an orthogonal basis of W , say e1; : : : ; en 1, such that C 1414 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN e2; : : : ; en 1 is a basis of V ; put ei ; ei ai . Then C h i D D X X E X xi ei ; yj ej ai xi yi : D Define ı by X X ı xi ei xi ei : D Note that ı2 1. D Let E be the space of . Then, up to equivalence, is the representation 1 m .ımı /. If is an admissible irreducible representation of G in a vector 7! space E, then an element A of Hom G; is a linear map from E into E such that j A.ıgı 1/ .g/A; g G: D 2 1 In turn the contragredient of is equivalent to the representation g .ıgı / 7! in E. Then an element B of Hom G; is a linear map from E into E such that j B.g/ .ıgı 1/B; g G: D 2 As ı2 1 the conditions on A and B are the same: D Hom G; Hom G; : j j However, assuming Theorem 20, by Corollary 1.1, we have dim Hom G; dim Hom G; 1; j j Ä so that both dimensions are 0 or 1. Replacing by we get Theorem 10. From now on we forget about Theorem 10.
2. Some tools
We shall state two theorems which are systematically used in our proof. If X is a Hausdorff totally disconnected locally compact topological space (lctd space in short) we denote by .X/ the vector space of locally constant functions with compact support of X into the field of complex numbers ރ. The dual space 0.X/ of .X/ is the space of distributions on X. All the lctd spaces that we introduce are countable at infinity. If an lctd topological group G acts continuously on a lctd space X, then it acts on .X/ by .gf /.x/ f .g 1x/ D and on distributions by .gT /.f / T .g 1f /: D G The space of invariant distributions is denoted by 0.X/ . More generally, if G; is a character of G we denote by 0.X/ the space of distributions T which transform according to that is to say gT .g/T . D MULTIPLICITYONETHEOREMS 1415
The following result is due to Bernstein [Ber84, 1.4].
THEOREM 2.1 (Localization principle). Let q Z T be a continuous map W ! 1 between two topological spaces of type lctd. Denote Zt q .t/. Consider WD 0.Z/ as .T /-module. Let M be a closed subspace of 0.Z/ which is an .T /- L submodule. Then M t T .M 0.Zt //. D 2 \ COROLLARY 2.1. Let q Z T be a continuous map between topological W ! spaces of type lctd. Let an lctd group H act on Z preserving the fibers of q. Let be a character of H . Suppose that for any t T , .q 1.t//H; 0. Then 2 0 D .Z/H; 0. 0 D The second theorem is a variant of Frobenius reciprocity ([Ber84, 1.5] and [BZ76, 2.21–2.36]).
THEOREM 2.2 (Frobenius descent). Let a unimodular lctd topological group H act transitively on an lctd topological space Z. Let ' E Z be an H-equi- W ! variant map of lctd topological spaces. Let x Z. Suppose that its stabilizer 2 StabH .x/ is unimodular. Let W be the fiber of x. Let be a character of H . Then (i) there exists a canonical isomorphism Fr .E/H; .W /StabH .x/; . W 0 ! 0 (ii) for any distribution .E/H; , Supp.Fr.// Supp./ W . 2 0 D \ Using the explicit formulas for Fr (see the above references) one checks that Frobenius descent commutes with Fourier transform. Namely, keeping our notation, let W be a finite-dimensional linear space over ކ with a nondegenerate bilinear form B. Let H act on W linearly preserving B and let ᏲB be the Fourier transform on W . H; PROPOSITION 2.1. For any .Z W/ , we have ᏲB .Fr.// Fr.ᏲB .// 2 0 D where Fr is taken with respect to the projection Z W Z. ! This last proposition will be used in Sections 4 and 6. Finally, as ކ is non-Archimedean, a distribution which is 0 on some open set may be identified with a distribution on the (closed) complement. This will be used throughout this work.
3. Reduction to the singular set: the GL.n/ case
Consider the case of the general linear group. From the decomposition W D V ކe we get, with obvious identifications ˚ End.W / End.V / V V ކ: D ˚ ˚ ˚ Note that End.V / is the Lie algebra g of G. The group G acts on End.W / by 1 t 1 g.X; v; v ; t/ .gXg ; gv; g v ; t/. As before choose a basis .e1; : : : ; en/ of D V and let .e1; : : : ; en/ be the dual basis of V . Define an isomorphism u of V 1t 1 onto V by u.ei / e . On GL.W / the involution is h u h u. It depends D i 7! 1416 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN upon the choice of the basis but the action on the space of invariant distributions does not depend upon this choice. It will be convenient to introduce an extension G of G. Let Iso.V; V / be the z set of isomorphisms of V onto V . We define G G Iso.V; V /. The group z D [ law for g; g G and u; u Iso.V; V / is 0 2 0 2 g g gg ; u g ug; g u tg 1u; u u tu 1u : 0 D 0 D D 0 D 0 Now from W V ކe we obtain an identification of the dual space W with D ˚ V ކe , with e ;V .0/ and e ; e 1. Any u as above extends to an ˚ h i D h i D isomorphism of W onto W by defining u.e/ e . The group G acts on GL.W /: D z h ghg 1; h t.uhu 1/; g G; h GL.W /; u Iso.V; V / 7! 7! 2 2 2 and also on End.W / with the same formulas. Let be the character of G which is 1 on G and 1 on Iso.V; V /. Our goal z G; is to prove that .GL.W // z .0/. 0 D G; G; PROPOSITION 3.1. If .g V V / z .0/, then .GL.W // z .0/. 0 ˚ ˚ D 0 D Proof. We have End.W / End.V / V V ކ and the action of G on ކ z DG; ˚ ˚ ˚ G; is trivial. Thus 0.g V V / z .0/ implies that 0.End.W // z .0/: Let G; ˚ ˚ D D T .GL.W // z . Let h GL.W / and choose a compact open neighborhood 2 0 2 K of Det h such that 0 K. For x End.W / define '.x/ 1 if Detx K … 2 D 2 and '.x/ 0 otherwise. Then ' is a locally constant function. The distribution D .' GL.W //T has a support which is closed in End.W / hence may be viewed as a j G; distribution on End.W /. This distribution belongs to 0.End.W // z so it must be equal to 0. It follows that T is 0 in the neighborhood of h. As h is arbitrary we conclude that T 0. D G; Our task is now to prove that .g V V / z .0/. We shall use induction 0 ˚ ˚ D on the dimension n of V . The action of G is, for X g, v V , v V , g G, z 2 2 2 2 and u Iso.V; V /, 2 .X; v; v / .gXg 1; gv;tg 1v /; .X; v; v / .t.uXu 1/;tu 1v ; uv/: 7! 7! The case n 0 is trivial. D We suppose that V is of dimension n 1, assuming the result up to dimension G; n 1 and for all ކ. If T .g V V / z we are going to show that its support 2 0 ˚ ˚ is contained in the “singular set”. This will be done in two stages. On V V let be the cone v ; v 0. It is stable under G. ˚ h i D z LEMMA 3.1. The support of T is contained in g . Proof. For .X; v; v/ g V V put q.X; v; v/ v; v . Let be 2 ˚ ˚ G; D h i the open subset q 0. We have to show that ./ z .0/. By Bernstein’s ¤ 0 D localization principle (Corollary 2.1) it is enough to prove that, for any fiber t 1 G; D q .t/; t 0, one has .t / z .0/. ¤ 0 D MULTIPLICITYONETHEOREMS 1417
G acts transitively on the quadric v ; v t. Fix a decomposition V h i D D ކ" V1 and identify V ކ" V with " ;" 1. Then .X; "; t" / t ˚ D ˚ 1 h i D 2 and the isotropy subgroup of ."; t" / in G is, with an obvious notation, G . By z zn 1 G; Frobenius descent (Theorem 2.2) there is a linear bijection between 0.t / z and G1; 1 the space 0.g/ z and this last space is .0/ by induction. Let z be the center of g that is to say the space of scalar matrices. Let ᏺ Œg; g be the nilpotent cone in g. G; PROPOSITION 3.2. If T 0.g V V / z , then the support of T is con- 2 G; ˚ ˚ G; tained in .z ᏺ/ . If .ᏺ / z .0/, then .g V V / z .0/. C 0 D 0 ˚ ˚ D Proof. Let us prove that the support of such a distribution T is contained in .z ᏺ/ .V V /. We use Harish-Chandra’s descent method. For X g let C ˚ 2 X Xs Xn be the Jordan decomposition of X with Xs semi-simple and Xn D C nilpotent. This decomposition commutes with the action of G. The centralizer z ZG.X/ of an element X g is unimodular ([SS70, p. 235]) and there exists an 2 isomorphism u of V onto V such that t X uXu 1 (any matrix is conjugate to its D transpose). It follows that the centralizer ZG.X/ of X in G, which is a semi-direct z z product of ZG.X/ and S2, is also unimodular. Let E be the space of monic polynomials of degree n with coefficients in ކ. For p E, let gp be the set of all X g with characteristic polynomial p. Note that 2 2 g is fixed by G. By the Bernstein localization principle (Theorem 2.1) it is enough p z n G; to prove that if p is not .T / for some , then .gp V V / z .0/. 0 D Fix p. We claim that the map X Xs restricted to gp is continuous. Indeed 7! let ކ be a finite Galois extension of ކ containing all the roots of p. Let z s Y ni p./ . i / D 1 n be the decomposition of p. Recall that if X gp and Vi Ker.X I / i , then 2 D V Vi and the restriction of Xs to Vi is the multiplication by i . Then choose D ˚ a polynomial R with coefficients in ކ such that for all i, R is congruent to i n z modulo . i / i and R.0/ 0. Clearly Xs R.X/. As the Galois group of ކ D D z over ކ permutes the i , we may even choose R ކŒ. This implies the required 2 continuity. There is only one semi-simple orbit p in gp and it is closed. We use Frobenius descent (Theorem 2.2) for the map .X; v; v / Xs from gp V V to p. 7! Fix a p; its fiber is the product of V V by the set of nilpotent elements 2 ˚ which commute with a. It is a closed subset of the centralizer m Zg.a/ of a in g. D Let M ZG.a/ and M ZG.a/. D D z Following [SS70], let us describe these centralizers. Let P be the minimal polynomial of a; all its roots are simple. Let P P1 :::Pr be the decomposition D of P into irreducible factors over ކ. Then the Pi are two-by-two relatively prime. If Vi KerPi .a/, then V Vi and V V . An element x of G which D D ˚ D ˚ i 1418 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN commutes with a is given by a family x1; : : : ; xr where each xi is a linear map f g from Vi to Vi , commuting with the restriction of a to Vi . Now ކŒ acts on Vi , by specializing to a Vi , and Pi acts trivially so that, if ކi ކŒ=.Pi /, then Vi j D becomes a vector space over ކi . The ކ-linear map xi commutes with a if and only if it is ކi -linear. Fix i. Let ` be a nonzero ކ-linear form on ކi . If vi Vi and v V , 2 i0 2 i then vi ; v0 is an ކ-linear form on ކi ; hence there exists a unique element 7! h i i S.vi ; v / of ކi such that vi ; v ` S.vi ; v / . One checks trivially that S is i0 h i0 i D i0 ކi -linear with respect to each variable and defines a nondegenerate duality, over ކi between Vi and Vi. Here ކi acts on Vi by transposition, relative to the ކ-duality :;: of the action on Vi . Finally, if xi Endކ Vi , then its transpose, relative to the h i 2 i duality S.: ; :/, is the same as its transpose relative to the duality :;: . h i Thus M is a product of linear groups and the situation .M; V; V / is a com- posite case, each component being a linear case (over various extensions of ކ). Let u be an isomorphism of V onto V such that t a uau 1 and that, for D each i, u.Vi / V . Then u M and M M uM . D i 2 D [ Suppose that a does not belong to the center of g. Then each Vi has dimension strictly smaller than n and we can use the inductive assumption. Therefore .m V V /M ; .0/: 0 ˚ ˚ D However the nilpotent cone ᏺm in m is a closed subset so M ; .ᏺm V V / z .0/: 0 D Together with Lemma 3.1 this proves the first assertion of the proposition. If a belongs to the center, then M G and the fiber is .a ᏺ/ V V . This D z C implies the second assertion.
Remark. Strictly speaking the singular set is defined as the set of all .X; v; v/ such that for any polynomial P invariant under G one has P.X; v; v / P.0/. So z D we should take care of the invariants P.X; v; v / v ;X pv for all p and not D h i only for p 0. It can be proved, a priori, that the support of the distribution T D has to satisfy these extra conditions. As this is not needed in the sequel we omit the proof.
4. End of the proof for GL.n/ G; In this section we consider a distribution T .ᏺ / z and prove that 2 0 T 0. The following observation will play a crucial role. Choose a nontrivial D additive character of ކ. On V V we have the bilinear form ˚ .v1; v /; .v2; v / v ; v2 v ; v1 1 2 7! h 1 i C h 2 i Define the Fourier transform by Z '.v2; v2/ '.v1; v1/ . v1; v2 v2; v1 / dv1dv1; y D V V h i C h i ˚ MULTIPLICITYONETHEOREMS 1419 with dv1dv1 normalized so that there is no constant factor appearing in the inver- sion formula. This Fourier transform commutes with the action of G; hence the (partial) z Fourier transform Tb of our distribution T has the same invariance properties and the same support conditions as T itself. Let ᏺi be the union of nilpotent orbits of dimension at most i. We will prove, by descending induction on i, that the support of any .G; /-equivariant distribu- z tion on Œg; g must be contained in ᏺi . Suppose we already know that, for some i, the support must be contained in ᏺi . We must show that, for any nilpotent orbit ᏻ of dimension i, the restriction of the distribution to ᏻ is 0. If v V and v V , then we call Xv;v the rank one map x v ; x v. 2 2 7! h i Let
.X; v; v/ .X Xv;v ; v; v/; .X; v; v/ g ; ކ: D C 2 2 Then is a one parameter group of homeomorphisms of g , and note that Œg; g is invariant. The key observation is that commutes with the action of G . Therefore the image of T by transforms according to the character of G. z z Its support is contained in Œg; g and hence must be contained in ᏺ and in fact in ᏺi . This means that if .X; v; v / belongs to the support of T , then, for all , .X Xv;v ; v; v / must belong to ᏺi . C The orbit ᏻ is open in ᏺi . Thus if X ᏻ the condition X Xv;v ᏺi 2 C 2 implies that, at least for small enough, X Xv;v ᏻ. It follows that Xv;v j j C 2 belongs to the tangent space to ᏻ at the point X; this tangent space is the image of adX. Define Q.X/ to be the set of all pairs .v; v / such Xv;v Im adX. Therefore 2 it is enough to prove the following lemma:
G; LEMMA 4.1. Let T .ᏻ V V / z . Suppose that the support of T and 2 0 of Tb are contained in the set of triplets .X; v; v / such that .v; v / Q.X/. Then 2 T 0. D Note that the trace of Xv;v is v ; v and that Xv;v Im ad X implies that h i 2 its trace is 0. Therefore Q.X/ is contained in . We proceed in three steps. First we transfer the problem to V V and a fixed ˚ nilpotent endomorphism X. Then we show that if Lemma 4.1 is true for .V1;X1/ and .V2;X2/ then it is true for the direct sum .V1 V2;X1 ;X2/. Finally, using ˚ ˚ the decomposition of X in Jordan blocks, we are left with the case of a principal nilpotent element for which we give a direct proof, using Weil representation. Consider the map .X; v; v / X from ᏻ V V onto ᏻ. Choose X ᏻ 7! 2 and let C (resp C ) be the stabilizer in G (resp. in G) of an element X of ; both z z ᏻ groups are unimodular; hence we may use Frobenius descent (Theorem 2.2). Now we have to deal with a distribution, which we still call T , which belongs C ; to .V V / z such that both T and its Fourier transform are supported by 0 ˚ 1420 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
Q.X/ (Proposition 2.1). Let us say that X is nice if the only such distribution is 0. We want to prove that all nilpotent endomorphisms are nice.
LEMMA 4.2. Suppose that we have a decomposition V V1 V2 such that D ˚ X.Vi / Vi . Let Xi be the restriction of X to Vi . Then if X1 and X2 are nice, so is X.
Proof. Let .v; v / Q.X/ and choose A g such that Xv;v ŒA; X. 2 2 D Decompose v v1 v2 and v v v , and put D C D 1 C 2 A A A 1;1 1;2 : D A2;1 A2;2 Writing X as a two-by-two matrix and looking at the diagonal blocks one gets v;v that Xv ;v ŒAi;i ;Xi . This means that i i D Q.X/ Q.X1/ Q.X2/: For i 1; 2 let Ci be the centralizer of Xi in GL.Vi / and Ci the corresponding D z extension by S2. Let T be a distribution as above and let '2 .V2 V /. Let 2 ˚ 2 T1 be the distribution on V1 V defined by '1 T;'1 '2 . The support of ˚ 1 7! h ˝ i T1 is contained in Q.X1/, and T1 is invariant under the action of C1. We have
Tb1;'1 T1; '1 T; '1 '2 T;b '1 '2 : h i D h y i D h b ˝ i D h } ˝ b i Here '1.v1; v/ '1. v1; v/. By assumption the support of Tb is contained in { 1 D 1 Q.X/ so that the support of Tb1 is supported in Q.X1/ Q.X1/. Because .X1/ D is nice this implies that T is invariant under C . 1 z1 Extend the action of C1 to V V trivially. We obtain that T is invariant z ˚ with respect to C . Similarly it is invariant under C . Since the actions of C and z1 z2 z1 C together with the action of C generate the action of C we obtain that T must z2 z be invariant under C and hence must be 0. z Decomposing X into Jordan blocks we still have to prove Lemma 4.1 for a principal nilpotent element. We need some preliminary results.
LEMMA 4.3. The distribution T satisfies the following homogeneity condi- tion: T; f .tv; tv / t n T; f .v; v / : h i D j j h i Proof. We use a particular case of Weil or oscillator representation. Let E be a vector space over ކ of finite-dimension m. To simplify assume that m is even. Let q be a nondegenerate quadratic form on E and let b be the bilinear form b.e; e / q.e e / q.e/ q.e /: 0 D C 0 0 Fix a continuous nontrivial additive character of ކ. We define the Fourier trans- form on E by Z f .e0/ f .e/ .b.e; e0//de y D E where de is the self-dual Haar measure. MULTIPLICITYONETHEOREMS 1421
There exists ([RS]) a representation of SL.2; ކ/ in .E/ such that: 1 u f .e/ .uq.e//f .e/; 0 1 D t 0 .q/ m=2 1 f .e/ t f .te/; 0 t D .tq/j j 0 1 f .e/ .q/f .e/: 1 0 D y
The .tq/ are complex numbers of modulus 1. In particular if .E; q/ is a sum of hyperbolic planes, these numbers are all equal to 1. We have a contragredient action in the dual space 0.E/. Suppose that T is a distribution on E such that T and Tb are supported on the isotropic cone q.e/ 0. D This means that 1 u 1 u T; f T; f and Tb ; f Tb ; f : 0 1 D h i 0 1 D h i Using the relation 0 1 T;'b T; .q/ f ; h i D 1 0 the second relation is equivalent to 1 0 T; f T; f : u 1 D h i The matrices 1 u 1 0 and u ކ 0 1 u 1 2 generate the group SL.2; ކ/. Therefore the distribution T is invariant by SL.2; ކ/. In particular, .tq/ T; f .te/ t m=2 T; f h i D .q/ j j h i and T .q/Tb. D Remark. For even m, .tq/= .q/ is a character and there do exist nonzero distributions invariant under SL.2; ކ/. In odd dimensions we get a representation of the 2-fold covering of SL.2; ކ/ and we obtain the same homogeneity condition. However .tq/= .q/ is not a character; hence the distribution T must be 0.
In our situation we take E V V and q.v; v / v ; v . Then D ˚ D h i b .v1; v /; .v2; v / v ; v2 v ; v1 : 1 2 D h 1 i C h 2 i 1422 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
The Fourier transform commutes with the action of G. Both T and T are supported z b on Q.X/ which is contained in . As .tq/ 1 this proves the lemma and also D that T Tb. D Remark. The same type of argument could have been used for the quadratic form T r.XY / on sl.V / Œg; g. This would have given a short proof for even n D and a homogeneity condition for odd n. Now we find Q.X/.
LEMMA 4.4. If X is principal, then Q.X/ is the set of pairs .v; v/ such that for 0 k < n, v ;X kv 0. Ä h i D Proof. Choose a basis .e1; : : : ; en/ of V such that Xe1 0 and Xej ej 1 D D for j 2. Consider the map A XA AX from the space of n by n matrices 7! into itself. This map is anti-symmetric with respect to the Killing form and hence its image is the orthogonal complement to its kernel. A simple computation shows that the kernel of this map, that is to say the Lie algebra c of the centralizer C , is the space of polynomials (of degree at most n 1 ) in X. Therefore, k Q.X/ .v; v/ Xv;v Im adX .v; v/ 0 k < n; tr.Xv;v X / 0 D f j 2 g D f j8 Ä D g k .v; v/ 0 k < n; v;X v 0 : D f j8 Ä h i D g End of the proof of Lemma 4.1. For principal X, we proceed by induction on n. Keep the above notation. The centralizer C of X is the space of polynomials (of degree at most n 1) in X with nonzero constant term. In particular the orbit of en is the open subset xn 0. We shall prove that the restriction of T to V is 0. ¤ Note that the centralizer of en in C is trivial. By Frobenius descent (Theorem 2.2), to the restriction of T corresponds a distribution R on V with support in the set of v such that .en; v / Q.X/. By the last lemma this means that R is a multiple aı 2 of the Dirac measure at the origin. The distribution T satisfies the two conditions T; f .v; v / T; f .tv; t 1v / t n T; f .tv; tv / h i D h i D j j h iI therefore, T; f .v; t 2v / t n T; f .v; v / : h D j j h i Now T is recovered from R by the formula Z Z t 1 T; f .v; v/ R; f .cen; c v/ dc a f .cen; 0/dc; f . V /: h i D C h i D C 2 Unless a 0 this is not compatible with this last homogeneity condition. D Exactly in the same way one proves that T is 0 on V , where is the open orbit x 0 of C in V . The same argument is valid for Tb (which is even 1 ¤ equal to T . . . ). If n 1, then T is obviously 0. If n 2, then there exists a distribution T on D 0 M ކej ކe ˚ j 1
We now turn our attention to the unitary case. We keep the notation of the introduction. In particular W V ބe is a vector space over ބ of dimension D ˚ n 1 with a nondegenerate hermitian form :;: such that e is orthogonal to V . C h i The unitary group G of V is imbedded into the unitary group M of W . Let A be the set of all bijective maps u from V to V such that
u.v1 v2/ u.v1/ u.v2/; u.v/ u.v/; and u.v1/; u.v2/ v1; v2 : C D C D x h i D h i An example of such a map is obtained by choosing a basis e1; : : : ; en of V such that ei ; ej ކ and defining h i 2 X X u xi ei xi ei : D x Any u A is extended to W by the rule u.v e/ u.v/ e and we define an 2 C D C x action on GL.W / by m um 1u 1. The group G acts on GL.W / by the adjoint 7! action. Let G be the group of bijections of GL.W / onto itself generated by the actions z of G and A. It is a semi-direct product of G and S2. We identify G to a subgroup of G and A to a subset. When a confusion is possible we denote the product in G z z with a . We define a character of G by .g/ 1 for g G, and .u/ 1 for z D G; 2 D u G G. Our overall goal is to prove that .M / z .0/. 2 z n 0 D Let G act on G V as follows: z g.x; v/ .gxg 1; g.v//; D g G; u A; x G; and v V: u.x; v/ .ux 1u 1; u.v//; 2 2 2 2 D Our first step is to replace M by G V . 1424 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
G; PROPOSITION 5.1. Suppose that 0.G V/ z .0/ for any V and any G; D hermitian form on V . Then we have .M / z .0/. 0 D Proof. We have in particular .M W/M ; .0/. Let Y be the set of all 0 D .m; w/ such that w; w e; e ; it is a closed subset, invariant under M, hence h i D h i .Y /M ; .0/. By Witt’s theorem M acts transitively on w w; w 0 D D f jh i D e; e . We can apply Frobenius descent (Theorem 2.2) to the map .m; w/ w h ig 7! of Y onto . The centralizer of e in M is isomorphic to G acting as before on z G: M ; the fiber M e . We have a linear bijection between 0.M / z and 0.Y / ; f G: g therefore .M / z .0/. 0 D G; The proof that 0.G V/ z .0/ is by induction on n. If g is the Lie algebra D G; of G, we shall prove simultaneously that .g V/ z .0/. In this case G acts 0 D on its Lie algebra by the adjoint action, and for u G G one puts, for X g; 1 2 z n 2 u.X/ uXu . D G; The case n 0 is trivial, so we may assume that n 1. If T .G V/ z D 2 0 in this section, then we will prove that the support of T must be contained in the “singular set”. Let Z (resp. z) be the center of G (resp. g) and ᐁ (resp. ᏺ) the (closed) set of all unipotent (resp. nilpotent) elements of G (resp. g).
G; G; LEMMA 5.1. If T .G V/ z (resp. T .g V/ z ), then the support 2 0 2 0 of T is contained in Zᐁ V (resp. .z ᏺ/ V/. C This is Harish-Chandra’s descent. We first review some facts about the cen- tralizers of semi-simple elements, following [SS70]. Let a G, semi-simple; we want to describe its centralizer M (resp. M) in 2 G (resp. in G) and to show that .M V/M ; .0/. z 0 D View a as a ބ-linear endomorphism of V and call P its minimal polynomial. Then, as a is semi-simple, P decomposes into irreducible factors P P1 :::Pr D two-by-two relatively prime. Let Vi Ker Pi .a/ so that V Vi . Any element D D ˚ x which commutes with a will satisfy xVi Vi for each i. For m R./ d0 dm ; d0dm 0 D C C ¤ let m R ./ d0 dm: D C C Then, from aa 1 we obtain, if m is the degree of P , D P.a/v; v v; a mP .a/v : h 0i D h 0i (Note that the constant term of P cannot be 0 because a is invertible.) It follows that P .a/ 0 so that P is proportional to P . Now P P :::P ; hence D D 1 r there exists a bijection from 1; 2; : : : ; r onto itself such that P is proportional f g i MULTIPLICITYONETHEOREMS 1425 to P .i/. Let mi be the degree of Pi . Then, for some nonzero constant c
mi 0 Pi .a/vi ; vj vi ; a P .a/vj D h i D h i i mi c vi ; a P .a/vj ; vi Vi ; vj Vj : D h .i/ i 2 2 We have two possibilities.
Case 1: .i/ i. The space Vi is orthogonal to Vj for j i; the restriction D ¤ of the hermitian form to Vi is nondegenerate. Let ބi ބŒ=.Pi / and consider D Vi as a vector space over ބi through the action .R./; v/ R.a/v. As a Vi is 7! j invertible, is invertible modulo .Pi /; choose such that 1 modulo .Pi /. Let D i be the semi-linear involution of ބi , as an algebra over ބ: X j X j dj dj modulo .Pi /: 7! Let ކi be the subfield of fixed points for i . It is a finite extension of ކ, and ބi is either a quadratic extension of ކi or equal to ކi . There exists a ބ-linear form ` 0 on ބi such that `.i .d// `.d/ for all d ބi . Then any ބ-linear form L ¤ D 2 on ބi may be written as d `.d/ for some unique ބi . 7! 2 If v; v Vi , then d d.a/v; v is ބ-linear map on ބi ; hence there exists 0 2 7! h 0i S.v; v / ބi such that 0 2 d.a/v; v `.dS.v; v //: h 0i D 0 One checks that S is a nondegenerate hermitian form on Vi as a vector space over ބi . Also a ބ-linear map xi from Vi into itself commutes with ai if and only if it is ބi -linear, and it is unitary with respect to our original hermitian form if and only if it is unitary with respect to S. So in this case we call Gi the unitary group of S. It does not depend upon the choice of `. As no confusion may arise, for ބi 2 we define i ./. x D We choose an ކi -linear map ui from Vi onto itself, such that ui .v/ u.v/ D x and S.ui .v/; ui .v0// S.v; v0/. Then because of our original choice of ` we also D 1 1 have ui .v/; ui .v0/ v; v0 . Note that u.a Vi / u a Vi . h i D h i j D j Case 2. Suppose now that j .i/ i. Then Vi Vj is orthogonal to V D ¤ ˚ k for k i; j , and the restriction of the hermitian form to Vi Vj is nondegenerate, ¤ ˚ both Vi and Vj being totally isotropic subspaces. Choose an inverse of modulo Pj . Then for any P ބŒ 2 P.a/vi ; vj vi ; P ..a//vj ; vi Vi ; vj Vj ; h i D h x i 2 2 where P is the polynomial deduced from P by changing its coefficients into their x conjugate. This defines a map, which we call i from ބi onto ބj . In a similar way we have a map j which is the inverse of i . Then, for ބi we have 2 vi ; vj vi ; i ./vj . h i D h i View Vi as a vector space over ބi . The action
.; vj / i ./vj 7! 1426 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN defines a structure of ބi -vector space on Vj . However note that for ބ we have 2 i ./ so that i ./vj may be different from vj . To avoid confusion we shall D x write, for ބi 2 vi vi and i ./vj vj : D D As in the first case choose a nonzero ބ-linear form ` on ބi . For vi Vi and 2 vj Vj the map vi ; vj is a ބ-linear form on ބi ; hence there exists a 2 7! h i unique element S.vi ; vj / ބi such that, for all 2 vi ; vj `.S.vi ; vj //: h i D The form S is ބi -bilinear and nondegenerate so that we can view Vj as the dual space over ބi of the ބi -vector space Vi . Let .xi ; xj / Endބ.Vi / Endބ.Vj /. They commute with .ai ; aj / if and only 2 if they are ބi -linear. The original hermitian form will be preserved, if and only if S.xi vi ; xj vj / S.vi ; vj / for all vi ; vj . This means that xj is the inverse of the D transpose of xi . In this situation we define Gi as the linear group of the ބi -vector space Vi . 1 Let ui be a ބi -linear bijection of Vi onto Vj . Then ui .avi / a ui .vi / and 1 1 1 D u .avj / a u .vj /. i D i Recall that M is the centralizer of a in G. Then .M; V / decomposes as a “product”, each “factor” being either of type .Gi ;Vi / with Gi a unitary group (Case 1) or .Gi ;Vi Vj / with Gi a general linear group (Case 2). Gluing together 1 the ui (Case 1) and the .ui ; u / (Case 2), we get an element u G G such that i 2 z n ua 1u 1 a which means that it belongs to the centralizer of a in G. Finally, if M D z is the centralizer of a in G, then .M;V/ is imbedded into a product each “factor” z being either of type .Gi ;Vi / with Gi a unitary group (Case 1) or .Gi ;Vi Vj / with z z Gi a general linear group (Case 2). If a is not central, then for each i the dimension of Vi is strictly smaller than n and from the result for the general linear group and the inductive assumption in the orthogonal or unitary case, we conclude that .M V/M ; .0/. 0 D Proof of Lemma 5.1. In the group case, consider the map g Pg where Pg 7! is the characteristic polynomial of g. It is a continuous map from G into the set of polynomials of degree at most n. Each nonempty fiber Ᏺ is stable under G but also under G G. Bernstein’s localization principle tells us that it is enough to prove z n G; that .Ᏺ V/ z .0/. 0 D Now it follows from [SS70, Chap. IV] that Ᏺ contains only a finite number of semi-simple orbits; in particular the set of semi-simple elements Ᏺs in Ᏺ is closed. Let us use the multiplicative Jordan decomposition into a product of a semi-simple and a unipotent element. Consider the map from Ᏺ V onto Ᏺs which associates to .g; v/ the semi-simple part gs of g. This map is continuous (see the corresponding proof for GL) and commutes with the action of G. In each z Ᏺs MULTIPLICITYONETHEOREMS 1427
1 orbit is both open and closed therefore . / is open and closed and invariant 1 G; under G. It is enough to prove that for each such orbit . . // z .0/. By z 0 D Frobenius descent (Theorem 2.2), if a and is not central, this follows from 2 1 the above considerations on the centralizer of such an a and the fact that .a/ is a closed subset of the centralizer of a in G, the product of the set of unipotent z element commuting with a by V . Now gs is central if and only if g belongs to Zᐁ, hence the lemma. For the Lie algebra the proof is similar, using the additive Jordan decomposition. Going back to the group, if a is central, then we see that it suffices to prove G; that 0.ᐁ V/ z .0/, and, similarly for the Lie algebra, it is enough to prove G; D that .ᏺ V/ z .0/. 0 D Now the exponential map (or the Cayley transform) is a homeomorphism of onto commuting with the action of G. Therefore it is enough to consider the ᏺ ᐁ z Lie algebra case. We now turn our attention to V . Let v V v; v 0 . D f 2 jh i D g G; PROPOSITION 5.2. If T .ᏺ V/ z , then the support of T is contained 2 0 in ᏺ . Proof. Let t v V v; v 0 : D f 2 j h i D g Each is stable by G, hence, by Bernstein’s localization principle, to prove that t z the support of T is contained in ᏺ 0 it is enough to prove that, for t 0, G; ¤ .ᏺ t / z .0/. 0 D By Witt’s theorem the group G acts transitively on t . We can apply Frobe- nius descent to the projection from ᏺ t onto t . Fix a point v0 t . The 2 fiber is v . Let G be the centralizer of v in G. We have to show that ᏺ 0 z1 0 z G ; f g G ; .ᏺ/ z1 .0/, and it is enough to prove that .g/ z1 .0/. 0 D 0 D The vector v0 is not isotropic so we have an orthogonal decomposition
V ބv0 V1 D ˚ with V1 orthogonal to v0. The restriction of the hermitian form to V1 is nonde- generate and G is identified with the unitary group of this restriction, and G is 1 z1 the expected semi-direct product with S . As a G -module the Lie algebra g is 2 z1 isomorphic to a direct sum g g1 V1 W ˚ ˚ where g1 is the Lie algebra of G1 and W a vector space over ކ of dimension 0 or 1 and on which the action of G is trivial. The action on g V is the usual one z1 1 1 G1; ˚ so that, by induction, we know that 0.g1 V1/ z .0/. This readily implies G ; ˚ D that .g/ z1 .0/. 0 D G; To summarize: it remains to prove that .ᏺ / z .0/. 0 D 1428 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
6. End of the proof in the orthogonal and unitary cases
We keep our general notation. We have to show that a distribution on ᏺ , which is invariant under G, is invariant under G. To some extent the proof will be z similar to the one we gave for the general linear group. In particular we will use the fact that if T is such a distribution, then its partial Fourier transform on V is also invariant under G. The Fourier transform on V is defined using the bilinear form
.v1; v2/ v1; v2 v2; v1 7! h i C h i which is invariant under G. z For v V put 2 'v.x/ x; v v; x V: D h i 2 It is a rank-one endomorphism of V and 'v.x/; y x; 'v.y/ . h i D h i LEMMA 6.1. i) In the unitary case, for ބ such that the map 2 D x .X; v/ .X 'v; v/ W 7! C is a homeomorphism of Œg; g onto itself which commutes with G. z ii) In the orthogonal case, for ކ the map 2 .X; v/ .X X'v 'vX; v/ W 7! C C is a homeomorphism of Œg; g onto itself which commutes with G. z The proof is a trivial verification. We now use the stratification of ᏺ. Let us first check that an adjoint orbit is stable not only by G but by G. z Choose a basis e1; : : : ; en of V such that ei ; ej ކ; this gives a conjugation P P h i 2 u v xi ei v xi ei on V . If A is any endomorphism of V , then A is the W D 7! x D x endomorphism v A.v/. The conjugation u is an element of G G and, as such, 7! x z n it acts on g V by .X; v/ . uXu 1; u.v// . X; v/. In [MVW87, Chap. 7! D S x 4, Prop. 1.2] it is shown that for X g there exists an ކ-linear automorphism a 2 of V such that a.x/; a.y/ x; y (this implies that a.x/ x) and such that h i D h i D x aXa 1 X. Then g ua G and gXg 1 X so that X belongs to the D D 2 D S S adjoint orbit of X. Note that a G G and as such acts as a.X; v/ .X; a.v//; 2 z n D it is an element of the centralizer of X in G G. z n Remark. We need to check this only for nilpotent orbits and this will be done later in an explicit way, using the canonical form of nilpotent matrices.
Let ᏺi be the union of all nilpotent orbits of dimension at most i. We shall prove, by descending induction on i, that the support of a distribution T 0.ᏺ G; 2 / z must be contained in ᏺi . MULTIPLICITYONETHEOREMS 1429
So now assume that i 0 and that we already know that the support of any G; T .ᏺ / z must be contained in ᏺi . Let ᏻ be a nilpotent orbit of 2 0 dimension i; we have to show that the restriction of T to ᏻ is 0. In the unitary case fix ބ such that x and consider, for every t ކ, the 2 D G; 2 homeomorphism ; the image of T belongs to .ᏺ / z so that the image t 0 of the support of T must be contained in ᏺi . If .X; v/ belongs to this support this means that X t'v ᏺi . C 2 If i 0 so that ᏺi 0 , then this implies that v 0 so that T must be a D D f g D multiple of the Dirac measure at the point .0; 0/ and hence is invariant under G, so z it must be 0. If i > 0 and X ᏻ, then as ᏻ is open in ᏺi , we get that, at least for t small 2 j j enough, X t'v ᏻ and therefore 'v belongs to the tangent space Im ad.X/ C 2 of ᏻ at the point X. Define
Q.X/ v V 'v Im ad.X/ ;X ᏺ;.unitary case/: D f 2 j 2 g 2 Then we know that the support of the restriction of T to ᏻ is contained in .X; v/ X ᏻ; v Q.X/ f j 2 2 g and the same is true for the partial Fourier transform of T on V . In the orthogonal case for i 0, the distribution T is the product of the Dirac D measure at the origin of g by a distribution T 0 on V . The distribution T 0 is invariant under G but the image of G in End.V / is the same as the image of G so that T is z 0 invariant under G hence must be 0. z If i > 0 we proceed as in the unitary case, using . We define
Q.X/ v V X'v 'vX Im ad.X/ ;X ᏺ;.orthogonal case/ D f 2 j C 2 g 2 and we have the same conclusion. In both cases, for i > 0, fix X ᏻ. We use Frobenius descent for the projection 2 map .Y; v/ Y of V onto . Let C (resp. C ) be the stabilizer of X in G ᏻ ᏻ z 7! G; C ; (resp. G). We have a linear bijection of .ᏻ / z onto .V / z . z 0 0 C ; LEMMA 6.2. Let T .V / z . If T and its Fourier transform are supported 2 0 in Q.X/, then T 0. D Let us say that a nilpotent element X is nice if the above lemma is true. Sup- pose that we have a direct sum decomposition V V1 V2 such that V1 and V2 D ˚ are orthogonal. By restriction we get nondegenerate hermitian forms :;: i on Vi . h i We call Gi the unitary group of :;: i , gi its Lie algebra, and so on. Suppose that h i X.Vi / Vi so that Xi X Vi is a nilpotent element of gi . D j LEMMA 6.3. If X1 and X2 are nice so is X.
Proof. We claim that Q.X/ Q.X1/ Q.X2/. Indeed if A A A 1;1 1;2 g; D A2;1 A2;2 2 1430 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN then from x y x y A 1 ; 1 1 ;A 1 0 x2 y2 C x2 y2 D we get in particular Ai;i xi ; yi xi ;Ai;i yi 0 h i C h i D so that Ai;i gi . Note that 2 ŒX1;A1;1 ŒX; A : D ŒX2;A2;2 If vi Vi and vj Vj , then we define 'v ;v Vi Vj by 'v ;v .xi / 2 2 i j W 7! i j D xi ; vi vj . Then, for v v1 v2 h i D C 'v1;v1 'v2;v1 'v : D 'v1;v2 'v1;v2
Therefore if, for A g, we have 'v ŒX; A, then 'v ;v ŒXi ;Ai;i . This proves 2 D i i D the assertion for the unitary case. The orthogonal case is similar. The end of the proof is the same as the end of the proof of Lemma 4.2 Now in both orthogonal and unitary cases nilpotent elements have normal forms which are orthogonal direct sums of “simple” nilpotent matrices. This is precisely described in [SS70, IV 2.19, p. 259]. By the above lemma it is enough to prove that each “simple” matrix is nice.
Unitary case. There is only one type to consider. There exists a basis e1; : : : ; en of V such that Xe1 0 and Xei ei 1; i 2. The hermitian form is given by D D n i ei ; ej 0 if i j n 1; ei ; en 1 i . 1/ ˛ h i D C ¤ C h C i D with ˛ 0. Note that ˛ . 1/n 1˛. Suppose that v Q.X/; for some A g we ¤ D 2 2 have 'v XA AX. For any integer p 0 D p p p 1 Tr.'vX / Tr.XAX AX / 0: D C D p p P Now Tr.'vX / X v; v . Let v xi ei . Hence D h i D n p n p p X X n i X v; v xi p ei ; v . 1/ ˛xi pxn 1 i 0: h i D C h i D C x C D 1 1
For p n 1 this gives xnxn 0. For p n 2 we get nothing new but for D x D D p n 3 we obtain xn 1 0. Going on, by an easy induction, we conclude that D D xi 0 if i .n 1/=2. D C p 2p 1 If n 2p 1 is odd, put V1 1 ބei , V0 ބep 1, and V2 p C2 ބei . D C p D ˚ D C 2p D ˚ C If n 2p is even, put V1 1 ބei , V0 .0/, and V2 p 1ބei . In both cases D D ˚ D D ˚ C we have V V1 V0 V2. We use the notation v v2 v0 v1. D ˚ ˚ D C C The distribution T is supported by V1. Call ıi the Dirac measure at 0 on Vi . Then we may write T U ı0 ı2 with U .V1/. The same thing must be D ˝ ˝ 2 0 MULTIPLICITYONETHEOREMS 1431 true of the Fourier transform of T . Note that U is a distribution on V , ı is a Haar y 2 y2 measure dv1 on V1, and that, for n odd, ıy0 is a Haar measure dv0 on V0. So we have Tb dv1 U if n is even and Tb dv1 dv0 U if n is odd. In the odd case D ˝ y D ˝ ˝ y this forces T 0. In the even case, up to a scalar multiple, the only possibility is D T dv1 ı2. D ˝ Let X X i a xi ei . 1/ xi ei : W 7! x Then a G G. It acts on g by Y aYa 1 and in particular aXa 1 X so 2 z n 7! D that a C C . The action on V is given by v a.v/. It is an involution. The 2 z n 7! subspace V1 is invariant and so dv1 is invariant. This implies that T is invariant under C , so it must be 0. z Orthogonal case. There are two different types of “simple” nilpotent matrices. The first type is the same as the unitary case, with ˛ 1 and thus n odd, but now D our condition is that X'v 'vX ŒX; A for some A g. As before this implies q C D 2 that Tr.'vX / 0 but only for q 1. Put n 2p 1; we get xj 0 for j > p 1. D D C D C Decompose V as before: V V1 V0 V2. Our distribution T is supported by D ˚ ˚ the subspace v2 0, so we write it T U ı2 with U .V1 V0/. This is also D D ˝ 2 0 ˚ true for the distribution Tb, so we must have U dv1 R with R a distribution on D ˝ V0. Finally, T dv1 R ı2. Now Id C and T is invariant under C so that R D ˝ ˝ 2 must be an even distribution. On the other end the endomorphism a of V defined by i p 1 1 a.ei / . 1/ ei belongs to C and aXa X and u .X; v/ . X; v/ D D W 7! belongs to G G. The product a u of a and u in G belongs to C C . Clearly T z n z z n is invariant under a u so that T is invariant under C , so it must be 0. z The second type is as follows. We have n 2m, an even integer and a de- D composition V E F with both E and F of dimension m. We have a basis D ˚ e1; : : : ; em of E and a basis f1; : : : ; fm of F such that
ei ; ej fi ; fj 0 h i D h i D and m i ei ; fj 0 if i j m 1 and ei ; fm 1 i . 1/ : h i D C ¤ C h C i D Finally, X is such that Xei ei 1; Xfi fi 1. D D Let be the matrix of the restriction of X to E or to F . Write an element A g as two-by-two matrix A .ai;j /. Then 2 D Œ; a Œ; a ŒX; A 1;1 1;2 : D Œ; a2;1 Œ; a2;2 Suppose that v Q.X/, and let 2 X X v e f with e xi ei ; f yi fi : D C D D 1432 A. AIZENBUD, D. GOUREVITCH, S. RALLIS, and G. SCHIFFMANN
We get 'f;e 'f;e 'f;f 'f;f X'v 'vX C C C D 'e;e 'e;e ' ' C e;f C e;f where, for example, 'e;e is the map f f ; e e from F into E. Thus, for some A, 0 7!h 0 i 'e;e 'e;e a2;1 a2;1: C D In this formula, using the basis .ei / and .fi /, replace all the maps by their matrices. q P Then, as before, we have Tr.'e;e / 0 for 1 q m 1. If e0 xi fi qD Äq Ä D (the xi are the coordinates of e), then Tr. 'e;e/ is e; e . Thus, as in the other h 0i cases, we have xj 0 for j > m=2 if m is even and j > .m 1/=2 if m is odd. D C The same thing is true for the yi . If m 2p is even, let V1 i p.ކei ކfi / and V2 i>p.ކei ކfi /; write D D ˚ Ä ˚ D ˚ ˚ v v1 v2 the corresponding decomposition of an arbitrary element of V . Let D C ı2 be the Dirac measure at the origin in V2 and dv1 a Haar measure on V1. Then, as in the unitary case, using the Fourier transform, we see that the distribution T must be a multiple of dv1 ı2. ˝ i i 1 The endomorphism a of V defined by a.ei / . 1/ ei and a.fi / . 1/ fi D D C belongs to G and aXa 1 X. The map u .Y; v/ . Y; v/ belongs to G G D W 7! z n so that the product a u in G belongs to C C . It clearly leaves T invariant so z z n that T 0. D Finally, if m 2p 1 is odd, then we put V1 i p.ކei ކfi /, V0 D C D ˚ Ä ˚ D ކep 1 ކfp 1, V2 i p 2.ކei ކfi /. As in the unitary case we find that C ˚ C D ˚ C ˚ T dv1 R ı2 with R a distribution on V0. As Id C we see that R must D ˝ ˝ i 2 i be even. Then again, define a G by a.ei / . 1/ ei and a.fi / . 1/ fi , and 2 D D consider a u with u.Y; v/ . Y; v/. As before a u C C and leaves T D 2 z n invariant so we have to take T 0. D Acknowledgements. The first two authors would like to thank their teacher Joseph Bernstein for teaching them most of the mathematics they know. They cordially thank Joseph Bernstein and Eitan Sayag for guiding them through this project. They would also like to thank Vladimir Berkovich, Yuval Flicker, Erez Lapid, Omer Offen, and Yiannis Sakellaridis for useful remarks. The first two authors worked on this project while participating in the program Representation theory, complex analysis, and integral geometry of the Hausdorff Institute of Mathematics (HIM) at Bonn joint with Max Planck Institute fur¨ Math- ematik. They wish to thank the organizers of the activity and the director of HIM for inspiring environment and perfect working conditions. Finally, the first two authors wish to thank Vladimir Berkovich, Stephen Gel- bart, Maria Gorelik, and Sergei Yakovenko from the Weizmann Institute of Science for their encouragement, guidance, and support. The first two authors were partially supported by Binational Science Foun- dation grant 2006254, German-Israeli Foundation grant 861-32.6/2005, and Israel MULTIPLICITYONETHEOREMS 1433
Science Foundation grant 1438/06. The last author thanks the Math Research In- stitute of Ohio State University in Columbus for several invitations which allowed him to work with the third author.
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(Received September 28, 2007)
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Copyright © 2010 by Princeton University (Mathematics Department) Printed in U.S.A. by Sheridan Printing Company, Inc., Alpha, NJ table of contents o.12 o 2 No. 172, Vol.
Gonzalo Contreras. Geodesic flows with positive topological entropy, twist maps and hyperbolicity ...... 761–808 Eckart Viehweg. Compactifications of smooth families and of moduli spaces of polarized manifolds ...... 809–910 Chang-Shou Lin and Chin-Lung Wang. Elliptic functions, Green functions and the mean field equations on tori ...... 911–954 Yichao Tian. Canonical subgroups of Barsotti-Tate groups...... 955–988 Akshay Venkatesh. Sparse equidistribution problems, period bounds and subconvexity ...... 989–1094 Thomas Geisser. Duality via cycle complexes ...... 1095–1126 Viktor L. Ginzburg. The Conley conjecture ...... 1127–1180 Christopher Voll. Functional equations for zeta functions of groups and rings...... 1181–1218 mathematics of annals Monika Ludwig and Matthias Reitzner. A classification of SL.n/ invariant valuations ...... 1219–1267 Isaac Goldbring. Hilbert’s fifth problem for local groups...... 1269–1314 Robert M. Guralnick and Michael E. Zieve. Polynomials with PSL.2/ monodromy...... 1315–1359 Robert M. Guralnick, Joel Rosenberg and Michael E. Zieve.A new family of exceptional polynomials in characteristic two ...... 1361–1390 Raman Parimala and V. Suresh. The u-invariant of the function fields of p-adic curves ...... 1391–1405 Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis and Gerard´ Schiffmann. Multiplicity one theorems ...... 1407–1434 Stanislav Smirnov. Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model ...... 1435–1467 Kannan Soundararajan. Weak subconvexity for central values of L-functions ...... 1469–1498 Roman Holowinsky. Sieving for mass equidistribution ...... 1499–1516 Roman Holowinsky and Kannan Soundararajan. Mass equidistribution for Hecke eigenforms ...... 1517–1528 Kannan Soundararajan. Quantum unique ergodicity for
SL2.ޚ/ ވ ...... 1529–1538 2010 September, n