Hecke Algebra Correspondences for the metaplectic group
aa r X i v : . [ m a t h . R T ] A p r HECKE ALGEBRA CORRESPONDENCESFOR THE METAPLECTIC GROUP
SHUICHIRO TAKEDA AND AARON WOOD
Abstract.
Over a p -adic field of odd residual characteristic, Gan and Savin proved a cor-respondence between the Bernstein components of the even and odd Weil representations ofthe metaplectic group and the components of the trivial representation of the equal rank oddorthogonal groups. In this paper, we extend their result to the case of even residual charac-teristic. Introduction
Fix a nonarchimedean local field k of residue characteristic p and characteristic different from2. Let W be a non-degenerate symplectic space over k of dimension 2 n and f Sp( W ) the 2-foldmetaplectic cover of Sp( W ). For an additive character ψ of k , let ω ψ be the Weil representationof f Sp( W ), which decomposes into its even and odd constituents, ω ψ = ω + ψ ⊕ ω − ψ . In the categoryof genuine, smooth representations of f Sp( W ), let G ± ψ be the Bernstein component containing ω ± ψ .Consider quadratic spaces V ± of dimension 2 n + 1 with trivial discriminant, where V + has thetrivial Hasse invariant and V − the non-trivial one. Then SO( V + ) is the split adjoint group oftype B n and SO( V − ) is its unique non-split inner form. In the category of smooth representationsof SO( V ± ), let S ± be the Bernstein component containing the trivial representation of SO( V ± ).Let ǫ be + or –. In [GS2], Gan and Savin proved an equivalence of categories between G ǫψ and S ǫ assuming that p = 2. The aim of this paper is to extend their result to the case of evenresidual characteristic. We follow their general strategy of exploiting minimal types of the Weilrepresentation to define a Hecke algebra H ǫψ , showing that the category G ǫψ is equivalent to thecategory of H ǫψ -modules, and giving an isomorphism between H ǫψ and the standard Iwahori-Heckealgebra of SO( V ǫ ).A key ingredient for extending their result is an analysis of the K -types of the Weil represen-tation in arbitrary residual characteristic which was carried out by Savin and the second-namedauthor in [SW]. We also employ the machinery of Bushnell, Henniart, and Kutzko in [BHK] tocompare the Plancherel measures induced from the respective Hecke algebras. More explicitly,the layout of the paper is as follows. §
1. We introduce notation and summarize some relevant background material. §
2. We describe a minimal type for an open compact subgroup and compute the correspondingspherical Hecke algebra H ǫψ . We give an isomorphism between H ǫψ and the standard Iwahori-Hecke algebra H ǫ of SO( V ǫ ). We show that the isomorphism H ǫψ ∼ = H ǫ is, in fact, an iso-morphism of Hilbert algebras with involution, thus giving a coincidence of induced Plancherelmeasures under suitable normalization. A corollary of this result is that the correspondenceof Hecke algebra modules preserves formal degree. §
3. We prove that the category of H ǫψ -modules is equivalent to the category G ǫψ , thus giving thedesired equivalence of categories G ǫψ ∼ = S ǫ . From the theory of Plancherel measures, we deducethat this equivalence preserves the temperedness and square-integrability of representations. Acknowledgements
The authors would like to thank Gordan Savin for his insight and suggestions. In particular,the idea to use vector-valued Hecke algebras arose in conversations between him and the second-named author while participating at the Oberwolfach workshop on “Spherical Spaces and HeckeAlgebras.” The second-named author would like to thank the Oberwolfach Research Institutefor Mathematics (MFO) for fostering a stimulating research environment. The authors wouldalso like to thank Wee Teck Gan for answering several questions regarding [GS2].The first-named author was partially supported by NSF grant DMS-1215419.Finally, the authors would like to thank the anonymous referee to make various useful sugges-tions. 1.
Preliminaries
Throughout the paper, k is a nonarchimedean local field with residual characteristic p ; weallow for arbitrary residual characteristic but assume that the characteristic of k is different from2. Let O be the ring of integers and ̟ a chosen uniformizer. Denote by q the cardinality of theresidue field and by e the valuation of 2 in k . If p = 2, then e is the ramification index of 2;otherwise e = 0. Let ψ be a non-trivial additive character of k ; for convenience, we assume that ψ has conductor 2 e , i.e., that 4 O is the largest additive subgroup of O on which ψ acts trivially.For a vector space V over k , we denote by S ( V ) the Schwartz space of smooth, compactlysupported, C -valued functions on V . We denote the subspaces of even and odd functions in S ( V )by S ( V ) + and S ( V ) − , respectively.1.1. The symplectic group
Sp( W ) . Let W be a non-degenerate symplectic space over k ofdimension 2 n with basis { e , . . . , e n , f , . . . , f n } , where h e i , e j i = 0 = h f i , f j i for all i, j and h e i , f j i = δ i,j . The symplectic group Sp( W ) is the group of invertible transformations of W which preserve the symplectic form. The decomposition W = X + Y , where X is the span of the e i and Y is the span of the f i , is a polarization of W .Let W C be the C -span of the symplectic basis and sp ( W C ) the symplectic Lie algebra, consistingof endomorphisms T : W C → W C such that h T u, v i + h u, T v i = 0 for all u, v ∈ W C . Let h be thediagonal Cartan subalgebra relative to the symplectic basis and h ∗ = Hom C ( h , C ) its linear dual.The roots of sp ( W C ) form a root system of type C n , defined byΣ = {± ǫ i ± ǫ j : 1 ≤ i < j ≤ n } ∪ {± ǫ i : 1 ≤ i ≤ n } ⊂ h ∗ , where ǫ i : h → C is given by H = (cid:18) a − a (cid:19) ǫ i ( H ) = a i . We take ∆ = { α , . . . , α n } as the set of simple roots, where α n = 2 ǫ n and α i = ǫ i − ǫ i +1 otherwise.This choice of simple roots decomposes Σ into positive roots Σ + and negative roots Σ − .Each root α ∈ h ∗ has a corresponding coroot ˇ α ∈ h such that α (ˇ α ) = 2; the coroots form aroot system of type B n . Denote by h R the real span of the coroots.Let Σ a = { α + m : α ∈ Σ , m ∈ Z } be the set of affine roots, where α + m is the affine functionalon h R given by ( α + m )( H ) = α ( H ) + m . We take ∆ a = ∆ ∪ { α } to be the set of simple affineroots, where α = − ǫ + 1.For each affine root α + m , define s α + m to be the reflection across the affine hyperplane P α + m = { x ∈ h R : α ( x ) + m = 0 } . We write s i = s α i for the simple affine reflections across ECKE ALGEBRA CORRESPONDENCES 3 the affine hyperplanes P i = P α i . The affine space h R is an apartment for Sp( W ). The chambersof the apartment are the connected components of h R r S P α + m . For n = 2, the root system,coroot system, and apartment are sketched below. h ∗ α α P P P h R ˇ α ˇ α The Weyl group Ω is the group generated by the simple reflections s , . . . , s n . The affine Weylgroup Ω a is the group generated by the affine simple reflections s , s , . . . , s n ; it is the semi-directproduct Ω a = D Ω of a translation group D and the Weyl group. Both Ω and Ω a are Coxetergroups whose braid relations are given according to the following Coxeter diagram. s s s s n · · · For each root α ∈ Σ, we fix a map Φ α : SL ( k ) → Sp( W ) such that the images of the unipotentupper and lower triangular matrices in SL ( k ) are the root subgroups of Sp( W ) correspondingto α and − α , respectively. For α + m ∈ Σ a , we define the map Φ α + m : SL ( k ) → Sp( W ) byΦ α + m (cid:18) a bc d (cid:19) = Φ α (cid:18) a ̟ m b̟ − m c d (cid:19) ;we write x α + m ( t ) = Φ α + m (cid:18) t (cid:19) = x α ( ̟ m t ) ( t ∈ k ) ,w α + m ( t ) = Φ α + m (cid:18) t − t − (cid:19) = w α ( ̟ m t ) ( t ∈ k × ) ,h α + m ( t ) = Φ α + m (cid:18) t t − (cid:19) = h α ( t ) ( t ∈ k × ) . We take the element w α i (1) as a representative in Sp( W ) of the simple affine reflection s i .We will frequently use the same notation to refer to an element w = s i · · · s i r in Ω a and itsrepresentative w = w α i (1) · · · w α ir (1) in Sp( W ).1.2. Open compact subgroups of
Sp( W ) . For 0 ≤ i ≤ n , we define the lattice L i = Span O { e , . . . , e n , ̟ f , . . . , ̟ f i , f i +1 , . . . , f n } . The stabilizer K i = { g ∈ Sp( W ) : g L i ⊆ L i } is a maximal open compact subgroup of Sp( W ); itis the group generated by those Φ α j ( O ) for which j = i ; it is also the stabilizer of the point z i inthe apartment, where z = (0 , , . . . , , z = ( , , . . . , , . . . , z n = ( , , . . . , ) . In this way, each K i corresponds to the vertex z i in the ‘standard’ apartment; hence, everymaximal open compact subgroup of Sp( W ) is conjugate to one of the K i , cf. [Ti, § K , . . . , K n is an Iwahori subgroup I . The unipotent radical of I is gen-erated by the simple affine root groups Φ α i (SL ( O )). In this way, the Iwahori subgroup I SHUICHIRO TAKEDA AND AARON WOOD corresponds to the chamber in the apartment which is bounded by the hyperplanes P , . . . , P n ;in addition, the vertices of the chamber are precisely z , . . . , z n . The rank 2 picture is as follows. z z z P P P The double cosets in I \ Sp( W ) /I are parameterized by the affine Weyl group; namely, each I -double coset is of the form IwI for some w ∈ Ω a . The number of I -single cosets in IwI is[
IwI : I ] = q ℓ ( w ) , where ℓ is the length function on Ω a .1.3. Metaplectic group and the Weil representation.
For a polarization W = X + Y , theSchwartz space S ( Y ) realizes the unique (up to isomorphism) representation ρ ψ of the Heisenberggroup with the central character ψ . Via the action of Sp( W ) on the Heisenberg group, ρ ψ givesa projective representation of Sp( W ) which lifts to a linear representation ω ψ , called the Weilrepresentation, of the central extension Sp ′ ( W ) of Sp( W ) given by1 → C × → Sp ′ ( W ) → Sp( W ) → . It is a theorem of Weil that the derived group of Sp ′ ( W ) is a 2-fold cover f Sp( W ) of Sp( W ) andthat ω ψ is a faithful representation of f Sp( W ), cf. [We, IV.42-43], [MVW, 2.II.1].For a subgroup H ⊆ Sp( W ), we denote its preimage in f Sp( W ) by e H . For each root α , theelement x α ( t ) canonically lifts to an element e x α ( t ) in f Sp( W ). We may therefore define lifts of w α ( t ) and h α ( t ) via the formulas e w α ( t ) = e x α ( t ) e x − α ( − t − ) e x α ( t ) , e h α ( t ) = e w α ( t ) e w α ( − . We will take e w α i (1) for a representative in f Sp( W ) of the affine simple reflection s i . We willcontinue to abuse notation when referring to an element of Ω a or its representatives in eitherSp( W ) or f Sp( W ).Explicitly, the Weil representation on S ( Y ) is given by (cid:2)e x ( a ) φ (cid:3) ( y ) = ψ ( t yay ) φ ( y ) , (cid:2)e h ( a ) φ (cid:3) ( y ) = β a | det a | / φ ( t y ) , (cid:2) e wφ (cid:3) ( y ) = γ b φ ( y ) . Here, e x ( a ), e h ( a ), and e w are respective lifts of (cid:18) a (cid:19) , (cid:18) a t a − (cid:19) , and (cid:18) − (cid:19) ; ECKE ALGEBRA CORRESPONDENCES 5 the Fourier transform of φ is defined by b φ ( y ) = Z Y ψ (2 t uy ) φ ( u ) du ;and β a and γ are specific 8th roots of unity, whose precise value plays no role in the currentinvestigation.Under the Weil representation ω ψ , the (positive) root groups act as follows: (cid:2)e x ǫ i − ǫ j ( t ) φ (cid:3) ( y ) = φ ( y + ty i f j ) , (cid:2)e x ǫ i + ǫ j ( t ) φ (cid:3) ( y ) = ψ (2 ty i y j ) φ ( y ) , (cid:2)e x ǫ i ( t ) φ (cid:3) ( y ) = ψ ( ty i ) φ ( y ) . Minimal types of the Weil representation.
Realized as a representation of S ( Y ), theWeil representation ω ψ decomposes into the sum of even and odd functions, ω + ψ ⊕ ω − ψ . We considerthe lattices L i = L i ∩ Y . As computed in [SW], e K i acts on τ i = S ( L / L i ) , viewed naturallyas a subspace of S ( Y ). The space τ consists entirely of even functions and is an irreducible e K -module. Otherwise, as a e K i -module, τ i decomposes as τ + i ⊕ τ − i . Each τ ± i admits a tensorproduct structure, τ ± i = S ( O f / ̟ O f ) ± ⊗ · · · ⊗ S ( O f i / ̟ O f i ) ± ⊗ S ( O f i +1 / O f i +1 ) ⊗ · · · ⊗ S ( O f n / O f n ) , hence the dimension of τ ± i is q en ( q i ± τ i ⊆ τ i +1 for 0 ≤ i < n .We also note that, for 1 ≤ i ≤ n −
1, the simple affine reflection s i essentially acts on φ ∈ S ( Y )by interchanging the i th and ( i + 1)th components; the reflection s n acts essentially via Fouriertransform on the n th component; the reflection s acts on the first component φ of φ as (cid:2) s φ (cid:3) ( y ) = c b φ ( ̟ − y ) , for some constant c .Lastly, we note that the Iwahori group e I = e K ∩ · · · ∩ e K n is contained in each e K i , hence itpreserves each of the minimal types τ ± i . Similarly, the group e J = e K ∩ · · · ∩ e K n is contained in e K i for i = 0, so it preserves the minimal type τ ± i for i = 0.We record these observations in the following lemma. Lemma 1.1.
The Iwahori group e I preserves τ ± i for ≤ i ≤ n , and the group e J preserves τ ± i for ≤ i ≤ n . Moreover,1. the elements s , . . . , s n preserve τ while s inflates τ to τ +1 ;2. the elements s , . . . , s n preserve τ − while s inflates τ − to τ − . Spherical Hecke algebras.
We summarize some generalities on Hecke algebras, most ofwhich may be found in [GS2].Let G be a totally disconnected topological group and K ⊆ G an open compact subgroup; fixa Haar measure dg on G . For an irreducible, finite-dimensional representation ( σ, V σ ) of K , let( σ ∗ , V ∗ σ ) be its contragredient representation and define the σ -spherical Hecke algebra by H ( G (cid:12) K ; σ ) = (cid:26) f : G → End( V ∗ σ ) : f is smooth and compactly supported ,f ( k gk ) = σ ∗ ( k ) f ( g ) σ ∗ ( k ) , for k i ∈ K, g ∈ G (cid:27) ;it is an algebra under convolution with an identity element which we denote 1 σ .For a smooth representation ( π, V π ) of G , consider the space ( V π ⊗ V ∗ σ ) K of K -fixed vectorsin V π ⊗ V ∗ σ ; this space admits a natural action of H ( G (cid:12) K ; σ ) by π ( f )( v ⊗ e ) = Z G π ( g ) v ⊗ f ( g ) e dg, SHUICHIRO TAKEDA AND AARON WOOD where v ⊗ e ∈ ( V π ⊗ V ∗ σ ) K and f ∈ H ( G (cid:12) K ; σ ).Let Γ be an open compact subgroup of G containing K ; assume that the index [Γ : K ] is finite.We consider H (Γ (cid:12) K ; σ ) as a finite-dimensional subalgebra of H ( G (cid:12) K ; σ ) via H (Γ (cid:12) K ; σ ) = { f ∈ H ( G (cid:12) K ; σ ) : supp( f ) ⊆ Γ } . We have a natural isomorphism L : H (Γ (cid:12) K ; σ ) ∼ / / End Γ (Ind Γ K ( σ ∗ )) given by( L ( f ) φ )( g ) = Z Γ f ( h ) φ ( h − g ) dh for f ∈ H (Γ (cid:12) K ; σ ), φ ∈ Ind Γ K ( σ ∗ ), and g ∈ Γ.Suppose that ( π, V π ) is an irreducible, smooth, finite dimensional representation of Γ such that( V π ⊗ V ∗ σ ) K = 0. Then ( V π ⊗ V ∗ σ ) K is a simple H (Γ (cid:12) K ; σ )-module via the action of H ( G (cid:12) K ; σ ).Now assume H (Γ (cid:12) K ; σ ) is commutative. Then ( V π ⊗ V ∗ σ ) K is one dimensional, and the actionof H (Γ (cid:12) K ; σ ) factors through a maximal ideal m ⊆ H (Γ (cid:12) K ; σ ). MoreoverInd Γ K ( σ ∗ ) / (cid:16) L ( m ) · Ind Γ K ( σ ∗ ) (cid:17) ∼ = π ∗ . Therefore if ( π , V ) , . . . , ( π l , V r ) are the irreducible representations (up to isomorphism) of Γsuch that ( V i ⊗ V ∗ σ ) K = 0, then we haveInd Γ K ( σ ∗ ) ∼ = π ∗ ⊕ · · · ⊕ π ∗ r . For each f ∈ H (Γ (cid:12) K ; σ ), the trace of L ( f ) is λ d + · · · + λ r d r , where d i = dim V i and λ i = π i ( f ).If f is not supported on K , then the trace of L ( f ) is 0. The case of r = 2 is summarized by thefollowing lemma. Lemma 1.2.
Suppose that dim H (Γ (cid:12) K ; σ ) = 2 with T ∈ H (Γ (cid:12) K ; σ ) not supported on K . Let ( π i , V i ) , for i = 1 , , be the two irreducible representations (up to isomorphism) of Γ such that ( V i ⊗ V ∗ σ ) K = 0 . Write d i = dim V i and λ i = π i ( T ) . Then1. λ d + λ d = 0 ;2. the dimension of Ind Γ K ( σ ∗ ) is d = d + d ;3. the minimal polynomial of T is ( T − λ )( T − λ ) = 0 . For the groups and representations we will consider (specifically, representations on C -vectorspaces of connected reductive k -groups and their central extensions), there is additional structureon H ( G (cid:12) K ; σ ), namely the ∗ -operation, f ∗ ( g ) = f ( g − ), and the trace operation, tr( f ) = f (1).Following [BHK, § H ( G (cid:12) K ; σ ) is a normalized Hilbert algebra with involution f f ∗ andscalar product [ f , f ] = vol( K )dim σ tr( f ∗ f ) . This structure yields a Plancherel formula on H ( G (cid:12) K ; σ ): there is a positive Borel measure µ σ on the C ∗ -algebra completion C ∗ ( K, σ ) of H ( G (cid:12) K ; σ ) such that[ f, σ ] = Z c C ∗ ( K,σ ) tr π ( f ) dˆ µ σ ( π ) . Note that µ σ depends on the chosen Haar measure of G .We now consider this situation for two such groups, G , G . For i = 1 ,
2, fix an open compactsubgroup K i ⊆ G i , an irreducible smooth representation σ i of K i , and a Haar measure µ i of G i . Let b µ i be the Plancherel measure on b G i with respect to the Haar measure µ i ; following thenotation of [BHK] we denote by r b G i the support of b µ i . We write r b G i ( σ i ) for the subspace of r b G i consisting of the representations π for which ( π ⊗ σ ∗ i ) K i = 0. ECKE ALGEBRA CORRESPONDENCES 7
From [BHK, § α : H ( G (cid:12) K ; σ ) → H ( G (cid:12) K ; σ )such that, for all f ∈ H ( G (cid:12) K ; σ ),1. α ( f ∗ ) = α ( f ) ∗ , and2. tr( f ) = 0 implies tr (cid:0) α ( f ) (cid:1) = 0,then it is an isomorphism of Hilbert algebras. We then apply [BHK, Cor. C, p.57]. Lemma 1.3.
An isomorphism α : H ( G (cid:12) K ; σ ) → H ( G (cid:12) K ; σ ) of Hilbert algebras induces a homeomorphsim b α : r b G ( σ ) → r b G ( σ ) such that µ ( K )dim σ b µ ( b α ( S )) = µ ( K )dim σ b µ ( S ) for any Borel subset S of r b G ( σ ) . In the latter sections, we will apply this lemma with G = f Sp( W ). Strictly speaking, thegroups considered in [BHK] are connected, reductive k -groups; however, there is no obstructionin extending this result to the metaplectic group.2. Hecke algebra isomorphisms
In this section, we define our Hecke algebras H ± ψ of f Sp( W ) and show that they are isomorphicto the affine Hecke algebras H ± of SO( V ± ).2.1. Hecke algebra of
SO( V + ) . Let V + be a quadratic space of dimension 2 n + 1 with trivialdiscriminant and trivial Hasse invariant; then SO( V + ) is a split, adjoint, orthogonal group oftype B n . Let I + and Ω + a denote its Iwahori subgroup and affine Weyl group, respectively.The standard Iwahori-Hecke algebra is the set of smooth, compactly-supported I + -bi-invariantfunctions on SO( V + ), H + = H (SO( V + ) (cid:12) I + ; ) . For each w ∈ Ω + a , take U w to be the characteristic function on the double coset I + wI + . Thecollection { U w } forms a basis of H + as a vector space. As an algebra, H + is generated byelements U , . . . , U n , and σ , where U i = U w i for w i a simple affine reflection in Ω + a , and σ is theouter automorphism which exchanges the nodes on the Coxeter diagram corresponding to U and U . The quadratic relations for the U i are( U i + 1)( U i − q ) = 0 , and the braid relations are given by the affine diagram of type B n . U U U U n · · · σ SHUICHIRO TAKEDA AND AARON WOOD
For details, see [IM, § σ = 1 and σU σ = U , we see that U is abstractly unnecessary as a generator.Hence, H + is generated by σ , U , . . . , U n subject to the quadratic relations,( σ + 1)( σ −
1) = 0 and ( U i + 1)( U i − q ) = 0 , and the braid relations given by the affine diagram of type C n . σ U U U n · · · τ -spherical Hecke algebra of f Sp( W ) . The restriction of the minimal type τ from e K to the Iwahori subgroup e I remains irreducible, as shown in [SW]. In this section, we computethe τ -spherical Hecke algebra H + ψ = H ( f Sp( W ) (cid:12) e I ; τ ) . Theorem 2.1.
The Hecke algebra H + ψ is generated by invertible elements T , T , . . . , T n , satis-fying the quadratic relations ( T + 1)( T −
1) = 0 and ( T i + 1)( T i − q ) = 0 for i = 0 , and the braid relations of affine diagram of type C n . T T T T n · · · In particular, H + ψ is abstractly isomorphic to H + .Furthermore, this isomorphism is an isomorphism of Hilbert algebras; and, if the Haar mea-sures on f Sp( W ) and SO( V + ) are respectively normalized by vol( e I ) = dim( τ ) = q en = | | − n and vol( I + ) = 1 , then the Plancherel measures on H + ψ and H + coincide.Proof. We prove this theorem by investigating the structure of some 2-dimensional Hecke subal-gebras. For 0 ≤ i ≤ n , we define e I i = e I ∪ e Is i e I = \ j = i e K j ;In the apartment, e I i corresponds to the wall which separates the fundamental chamber I fromthe chamber s i Is − i or, equivalently, to the wall whose vertices are K j with j = i . The rank 2picture is as follows. s Is − s Is − s Is − IK K K We take H + ψ,i to be the subalgebra consisting of elements supported on e I i ; that is, H + ψ,i = H ( e I i (cid:12) e I ; τ ) . This subalgebra is at most 2-dimensional and is isomorphic to End e I i (cid:0) Ind e I i e I ( τ ∗ ) (cid:1) ; it is exactly2-dimensional if and only if the induced representation is reducible.We define τ ,i to be the subspace of S ( Y ) generated by the action of e I i on τ ; by Lemma 1.1, τ ,i = ( τ if i = 0 τ +1 if i = 0 . Working in the dual setting, Frobenius reciprocity guarantees that τ ,i may be realized as asubmodule of Ind e I i e I ( τ ), so it suffices to verify that it is a submodule of strictly smaller dimension.We note that d = dim (cid:0) Ind e I i e I ( τ ) (cid:1) = dim( τ ) · [ e I i : e I ] = q en ( q + 1)and d = dim( τ ,i ) = ( q en if i = 0 , q en ( q + 1) if i = 0 , hence Ind e I i e I ( τ ∗ ) is indeed reducible.Since H + ψ,i is 2-dimensional, it contains an element T i which is supported precisely on e Is i e I .In order to normalize T i and to compute its quadratic relation, we consider the decompositionInd e I i e I ( τ ∗ ) = π ∗ ⊕ π ∗ , where π ∗ = τ ∗ ,i has dimension d and π ∗ has dimension d = d − d = ( q en +1 if i = 0 , q en ( q + 1) if i = 0 . We normalize T i to act by λ = − π ∗ and by λ on π ∗ . Using Lemma 1.2, we have λ = d d = ( q if i = 0 , i = 0 , giving the desired quadratic relation ( T i + 1)( T i − λ ) = 0. The invertibility of T i follows fromits quadratic relation; explicitly, T − = T and T − i = q − ( T i − q + 1) for i = 0 . Suppose that we have a braid relation s i s j · · · = s j s i · · · in Ω a . Then each of the Hecke algebra elements T i T j . . . and T j T i . . . is supported on the same e I -double coset. From the normalization of the T i , each of these elements must act on ( τ ⊗ τ ∗ ) e I in the same way. Whence T i T j · · · = T j T i · · · . Therefore, the braid relations for the T i are the same as those for the s i , so any minimal expression w = s i · · · s i r defines a Hecke algebra element T w = T i · · · T i r supported on e Iw e I . From thequadratic and braid relations, we have an explicit isomorphism H + ψ → H + given by T σ and T i U i for i = 0 . We now show that H + ψ ∼ = H + is an isomorphism of Hilbert algebras. As each Hecke algebrais supported on its respective affine Weyl group, we have thattr( T w ) = ( w = 1 , w = 1 and tr( U w ) = ( w = 1 , w = 1 , so the trace-zero property is clearly preserved. For w ∈ Ω + a , the I + -double cosets of w and w − are equal, so the ∗ -operation in H + satisfies U ∗ i = U i , and hence, U ∗ w = U w − . In H + ψ , wehave that T ∗ i and T i are both supported on e Is i e I , so T ∗ i acts on τ ∗ by a constant. For φ ∈ τ ∗ ,[ φ, T ∗ i φ ] = [ T i φ, φ ], so T ∗ i and T i must act by the same constant. Thus, T ∗ i = T i and T ∗ w = T w − ,so the ∗ -operation is also preserved.From the normalization given in the statement of the theorem, the preservation of the Plancherelmeasures follows immediately from Lemma 1.3. (cid:3) Corollary 2.2.
The isomorphism H + ψ ∼ = H + preserves the formal degree of the Steinberg repre-sentations of the respective Hecke algebras. Remark. If p = 2, then the proof for the isomorphism H + ψ ∼ = H + is essentially the one given in[GS2]. One notable difference is in the specific normalization of Hecke operators, which is alwaysa delicate issue. In [GS2], they work in the central extension f Sp( W ) of Sp( W ) by the 8th rootsof unity and normalize the generating Hecke operators to act on certain lifts of affine reflectionsin a specified way. We have opted to normalize the generating Hecke operator T i to act by − τ ∗ . Remark.
The Steinberg representation of a Hecke algebra is defined by having each of thegenerating Hecke operators act by −
1. In [GS2], they show directly that the formal degreesof the respective Steinberg representations coincide. This computation is avoided here becauseit follows from the more general coincidence of the induced Plancherel measures. Indeed, theimplementation of the theory of induced Plancherel measures is the other notable differencebetween this proof and that of [GS2].
Remark.
Assuming k = Q , an isomorphic Hecke algebra is constructed in [Wo] by finding a1-dimensional type for a subgroup of e I . This construction extends to the case where k is anunramified extension of Q but does not appear to work for ramified extensions.2.3. Hecke algebra of
SO( V − ) . For the remainder of the section, we suppose that n ≥ V − be a quadratic space of dimension 2 n + 1 with trivial discriminant and non-trivial Hasseinvariant; then SO( V − ) is the non-split inner form of SO( V + ). Let I − be the Iwahori subgroup ofSO( V − ), which is the pointwise stabilizer of a fundamental chamber in its Bruhat-Tits building,and Ω − a its affine Weyl group, which is generated by reflections s − , . . . , s − n subject to the braidrelations of the affine diagram of type C n − . s − s − s − s − n · · · The standard Iwahori-Hecke algebra is the set of smooth, compactly-supported I − -bi-invariantfunctions on SO( V − ), H − = H (SO( V − ) (cid:12) I − ; ) . For each w ∈ Ω − a , let U w be the characteristic function on the double coset I − wI − . Thecollection { U w } forms a basis of H − as a vector space. As an algebra, H − is generated by U , . . . , U n , where U i = U s − i . These generators satisfy the quadratic relations,( U + 1)( U − q ) = 0 and ( U i + 1)( U i − q ) = 0 for i = 1 , ECKE ALGEBRA CORRESPONDENCES 11 and the same braid relations as the s − i . See [GS2] or [Ti] for details.2.4. τ − -spherical Hecke algebra of f Sp( W ) . We define the open compact subgroup e J ⊆ f Sp( W )to be the full inverse image of J = \ j =0 K j = I ∪ Is I, and consider the restriction of τ − to e J . The group e J contains the metaplectic preimage of thesubgroup Φ − ǫ +1 (cid:0) SL ( O ) (cid:1) × I n − , where I n − is an Iwahori subgroup of the symplectic group of type C n − . From [SW], eachcomponent of this direct product acts irreducibly on the corresponding component of the tensorproduct τ − = S ( O / ̟ O ) − ⊗ S ( O n − / O n − ) , hence the restriction of τ − to e J must remain irreducible. In this section, we compute the τ − -spherical Hecke algebra H − ψ = H ( f Sp( W ) (cid:12) e J ; τ − ) . We define Ω ′ a = h s ′ , . . . , s ′ n i ⊆ Ω a , where s ′ i = ( s i if i = 1 s s s if i = 1 . The reflection s ′ corresponds to the affine reflection s − ǫ +1 , hence Ω ′ a is isomorphic to the affineWeyl group of type C n − , i.e., to Ω − a ; explicitly, Ω ′ a acts as the affine Weyl group of type C n − on the hyperplane P .The proof of the following lemma is a slight variation on that of [GS2, Lemma 10]. Lemma 2.3.
The support of H − ψ is contained in e J Ω ′ a e J .Proof. Fix f ∈ H − ψ and σ ∈ Ω a . Write σ = as , where s ∈ Ω and a is translation by ( a , . . . , a n ).As J = I ∪ Is I , we have that σ and s σ represent the same e J -double coset, so it suffices to showthat f ( σ ) = 0 implies that either σ or s σ is in Ω ′ a .The element σ conjugates the root group of α to the root group of β + m , where β = s − ( − ǫ )is a long root and m = 1 − a ; in particular, σ − e x α (4) σ = e x β + m ( t ) for some t ∈ O .From the description of the Weil representation in Section 1.3, we derive the following criteriafor long roots α : e x α ( u ) ∈ ker τ − if and only if ( u ∈ O if α = − ǫ ,u ∈ ̟ O if α = − ǫ . Therefore, σ or s σ is in Ω ′ a if and only if β + m = ± α .Let t ∈ O be such that σ − e x α (4) σ = e x β + m ( t ) . First suppose that m > − β = − ǫ or that m > β = − ǫ . Then e x β + m ( t ) ∈ ker τ − and e x α (4) / ∈ ker τ − , hence f ( σ ) = f ( σ )( τ − ) ∗ (cid:0)e x β + m ( t ) (cid:1) = f (cid:0) σ e x β + m ( t ) (cid:1) = f (cid:0)e x α (4) σ (cid:1) = ( τ − ) ∗ (cid:0)e x α (4) (cid:1) f ( σ ) , giving that f ( σ ) = 0. Now suppose that m < β = 2 ǫ or that m < − β = 2 ǫ . Then e x − β − m ( t ) ∈ ker τ − and e x − α (4) / ∈ ker τ − , hence f ( σ ) = f ( σ )( τ − ) ∗ (cid:0)e x − β − m ( t ) (cid:1) = f (cid:0) σ e x − β − m ( t ) (cid:1) = f (cid:0)e x − α (4) σ (cid:1) = ( τ − ) ∗ (cid:0)e x − α (4) (cid:1) f ( σ ) , giving that f ( σ ) = 0.In sum, if f ( w ) = 0, then β + m must equal ± α , and the lemma is proved. (cid:3) Theorem 2.4.
The Hecke algebra H − ψ is generated by invertible elements T , . . . , T n , satisfyingthe quadratic relations ( T + 1)( T − q ) = 0 and ( T n + 1)( T n − q ) = 0 for i = 0 , and the braid relations of the affine diagram of type C n − . T T T T n · · · In particular, H − ψ is abstractly isomorphic to H − .Furthermore, this isomorphism is an isomorphism of Hilbert algebras and, if the Haar measureson f Sp( W ) and SO( V − ) are respectively normalized by vol( e J ) = dim τ − = q en ( q − and vol( I − ) = 1 , then the Plancherel measures on H − ψ and H − coincide.Proof. This proof is similar to the proof of Theorem 2.1. We investigate the structure of some2-dimensional Hecke subalgebras in order to see that H − ψ is supported exactly on e J Ω ′ a e J . For1 ≤ i ≤ n , we define e J i to be the group generated by e J and e Js ′ i e J ; in particular, e J i is the fullinverse image of J i = \ j =0 ,i K j . The group J corresponds to the facet of the fundamental chamber with vertices K , . . . , K n , i.e.,the facet that lies in the hyperplane P . The conjugate s ′ i J ( s ′ i ) − corresponds to a facet in thesame hyperplane. The figure on the left depicts the apartment in rank 2; the figure on the rightdepicts the hyperplane P in the rank 3 case. K K K Js Js − s ′ J ( s ′ ) − P s Js − s Js − s ′ J ( s ′ ) − JK K K If i = 1, then e J i = e J ∪ e Js i e J , hence [ e J i : e J ] = q + 1; for the case i = 1, we note that e J is theunion of those e Iw e I for which w is in the group generated by s and s , hence[ e J : e J ] = [ e J : e I ][ e J : e I ] = 1 + 2 q + 2 q + 2 q + q q = 1 + q + q + q . We take H − ψ,i to be the subalgebra of H − ψ consisting of elements supported on e J i ; that is, H − ψ,i = H ( e J i (cid:12) e J ; τ − ) . This subalgebra is at most 2-dimensional and is isomorphic to End e J i (cid:0) Ind e J i e J ( τ − ) ∗ (cid:1) .Let τ − ,i be the subspace of S ( Y ) generated by the action of e J i on τ − . We use Lemma 1.1repeatedly to compute τ ,i . First, we note that e J preserves τ − i for 1 ≤ i ≤ n , so it suffices toconsider the action of s ′ i on τ − . Next, if i ≥
2, then s ′ i = s i preserves τ − , hence τ − ,i = τ − .Lastly, we claim that τ − , = τ − . Since s ′ = s s s , we need to consider the action of s and s . By the same lemma, s inflates τ − to τ − . Hence, it remains to show that, if φ ∈ τ − , then s φ ∈ τ − . For such φ , we recall from Section 1.3 that the first component φ is in S ( O / ̟ O ) − and that s acts on φ via (cid:2) s φ (cid:3) ( y ) = c b φ ( ̟ − y ) . Since the Fourier transform maps S ( O / ̟ O ) − to S ( ̟ − O / O ) − , the first component of s φ remains in S ( O / ̟ O ) − and the claim is proved.To see that H − ψ,i is exactly 2-dimensional, we again work in the dual setting and note that d = dim τ − ,i = dim( τ − ) if i = 1 , dim( τ − ) if i = 1 = q en ( q −
1) if i = 1 , q en ( q −
1) if i = 1is strictly smaller than d = dim (cid:0) Ind e J i e J ( τ − ) (cid:1) = dim( τ − ) · [ e J i : e J ] = q en ( q −
1) if i = 1 , q en ( q −
1) if i = 1 . Hence, for 1 ≤ i ≤ n , there exists T i ∈ H − ψ supported precisely on e Js i e J . We consider thedecomposition Ind e J i e J ( τ − ) ∗ = π ∗ ⊕ π ∗ , where π ∗ = ( τ − ,i ) ∗ has dimension d and π ∗ has dimension d = d − d = q en ( q − q ) if i = 1 , q en ( q − q ) if i = 1 . We normalize T i to act by λ = − π ∗ and by λ on π ∗ . Using Lemma 1.2, we have λ = d d = ( q if i = 1 ,q if i = 1 , giving the desired quadratic relation ( T i + 1)( T i − λ ) = 0. The invertibility of T i follows fromits quadratic relation; explicitly, T − i = λ − ( T − λ + 1).The proof of the braid relations mimics the proof of Theorem 2.1 with e J instead of e I , τ − instead of τ , and Ω ′ a instead of Ω a . We note that computations involving e J -double cosetsinvolve a weighted length function ℓ ′ on Ω ′ a , defined by setting ℓ ′ ( s ′ ) = 3 and ℓ ′ ( s ′ i ) = 1 if i = 1. The details of this length function are contained in [GS2, Prop. 1]; it suffices to mention herethat 1. [
JwJ : J ] = q ℓ ′ ( w ) for all w ∈ Ω ′ a .2. If w , w ∈ Ω ′ a satisfy ℓ ′ ( w ) + ℓ ′ ( w ) = ℓ ′ ( w w ), then Jw J · Jw J = Jw w J .For a minimal expression w = s ′ i · · · s ′ i r in Ω ′ a , the braid relations in H − ψ allow us to define acanonical Hecke operator T w = T i · · · T i r supported precisely on the double coset e Jw e J . Fromthe quadratic and braid relations, we have the explicit isomorphism H − ψ → H − given by T i U i .As in the proof of Theorem 2.1, one can show that H − ψ ∼ = H − as Hilbert algebras, hence Lemma1.3 will give the coincidence of Plancherel measures, under the prescribed normalization. (cid:3) Corollary 2.5.
The isomorphism H − ψ ∼ = H − preserves the formal degree of the Steinberg repre-sentations of the respective Hecke algebras. Remark.
Let ǫ be + or − . Even for a fixed character ψ , the isomorphism H ǫψ ∼ = H ǫ constructedin these two sections is far from unique: the Hecke algebras admit many inner automorphismswhich preserve the quadratic and braid relations of the generators. Remark.
These two Hecke algebra isomorphims may be constructed using additive characters ofany conductor; we chose the conductor 2 e for its convenience. For an even conductor, the methodof construction would essentially go unchanged. For an odd conductor, different minimal typesfrom [SW] would be employed. What follows is a very brief summary of the relevant details.Consider an additive character ψ ′ given by ψ ′ ( t ) = ψ ( ̟ − c t ). This new character has conductor2 e + c ; that is, 4 ̟ c O is the largest subgroup of k on which ψ ′ acts trivially. The parity of c isimportant, so we write c = 2 k + δ , where δ is 0 or 1, and we define the diagonal matrix g c tohave ̟ k + δ in the first n entries and ̟ − k in the last n entries.The conjugation, x x c = g − c xg c , is an automorphism of Sp( W ). If δ = 0, g c is an elementof the affine Weyl group and conjugation by g c is an inner automorphism. If δ = 1, g c is not anelement of the affine Weyl group (or even of the symplectic group!) and conjugation by g c is anouter automorphism. We write G c = g − c Gg c for any subgroup G of Sp( W ) and e G c for its inverseimage in f Sp( W ). In the apartment, conjugation by g c corresponds to translation by cz n , so thefundamental chamber I with vertices K , . . . , K n is translated to the chamber I c with vertices K ,c , . . . , K n,c . If δ = 0, then K i,c is conjugate in Sp( W ) to K i . If δ = 1, then K i,c is conjugatein Sp( W ) to K n − i . For any δ , I c is conjugate in Sp( W ) to I . The rank 2 picture is as follows. I I I K K K K , K , K , K , K , K , If we follow conjugation by the Weil representation ω ψ , we get a Weil representation ω ψ,c ; inparticular, ω ψ,c (cid:0)e x ( a ) (cid:1) = ω ψ (cid:0)e x ( ̟ − c a ) (cid:1) ,ω ψ,c (cid:0)e h ( a ) (cid:1) = ω ψ (cid:0)e h ( a ) (cid:1) ,ω ψ,c (cid:0) e w (cid:1) = ω ψ (cid:0) e w ( ̟ c ) (cid:1) , where e w ( ̟ c ) is a lift of g − c wg c . Some straight-forward computations reveal that ω ψ,c = ω ψ ′ . f Sp( W ) f Sp( W )GL( S ( Y )) g c ω ψ ′ ω ψ,c We note that x ∈ Sp( W ) stabilizes a lattice L if and only if x c stabilizes g − c L . For convenience,we write L i,c = g − c L i and L i,c = L i,c ∩ Y .For δ = 0, the space τ i,c = S ( L ,c / L i,c ) is a type for e K i,c . It is easily checked that τ i,c isisomorphic to τ i . The group J c = K ,c ∩ · · · ∩ K n,c is isomorphic to J = K ∩ · · · ∩ K n .For δ = 1, the space τ i,c = S ( L i,c / ̟L ,c ) is a type for e K i,c . It is easily checked that τ i,c isisomorphic to τ n − i . The group J c = K ,c ∩ · · · ∩ K n,c is isomorphic to K ∩ · · · ∩ K n − .In either case, we build the two Hecke algebras H + ψ,c = H ( f Sp( W ) (cid:12) e I c ; τ ,c ) and H − ψ,c = H ( f Sp( W ) (cid:12) e J c ; τ − ,c ) . The same sort of geometry of the apartment employed in the previous sections will yield theexistence of generators of H ± ψ,c with the same quadratic and braid relations as H ± ψ .The curious reader is referred to [GS2], where Gan and Savin use an odd conductor in theircomputation of H − ψ under the assumption that p = 2.3. Equivalence of categories between G ± ψ and S ± In the category of smooth genuine representations of f Sp( W ), let G ± ψ be the Bernstein compo-nent containing the even/odd Weil representation ω ± ψ . In the category of smooth representationsof SO( V ± ), let S ± be the Bernstein component containing the trivial representation.We will prove our main theorem, namely that there is an equivalence of categories between G ǫψ and S ǫ , where ǫ is + or − . Our proof essentially follows that of [GS2].3.1. Equivalence between G + ψ and S +0 . Let U (resp. U − ) be the unipotent radical in Sp( W )generated by positive (resp. negative) root groups. Let e B = e T U ⊆ f Sp( W ) be the preimage ofthe Borel subgroup B = T U of Sp( W ). (Recall that the unipotent radical U splits in f Sp( W ).)An element t of the maximal torus T may be expressed uniquely as t = ( t , . . . , t n ) = h ǫ ( t ) · · · h ǫ n ( t n ) , hence we have a canonical lift of t given by e t = e h ǫ ( t ) · · · e h ǫ n ( t n ) . With this convention, multiplication in e T is given by e t · e u = ( t, u ) e tu = n Y i =1 ( t i , u i ) e h ǫ i ( t i u i ) , where the cocycle ( t, u ) ∈ {± } is the product of Hilbert symbols ( t i , u i ) on k . Note thatmultiplication in T is commutative.Recalling the action of T on Y by ty = ( t y , . . . , t n y n ), the action of e t on S ( Y ) is given by e tφ ( y ) = β t | det t | / φ ( ty ) , where β t is a 4th root of unity satisfying β t β u = ( t, u ) β tu .Given a character χ = ( χ , . . . , χ n ) on T , we define a genuine character e χ on e T by e χ ( e t ) = χ ( t ) β t . We extend this character trivially to all of e B and define I ( e χ ) to be the normalized inducedrepresentation Ind f Sp( W ) e B e χ . By Frobenius reciprocity,(3.1) Hom f Sp( W ) ( π, I ( e χ )) ∼ = Hom e T ( π U , e χ ) , where π is any smooth representation of f Sp( W ) and π U is the normalized Jacquet module withrespect to the Borel e B . Lemma 3.2.
The Bernstein component G + ψ is precisely the component whose irreducible repre-sentations are submodules of I ( e χ ) for some unramified character χ .Proof. The functional l : S ( Y ) + → C defined by l ( φ ) = φ (0) factors through the Jacquet module( ω + ψ ) U and gives a non-trivial element in Hom e T (( ω + ψ ) U , e χ ) for some unramified χ , which in turngives an embedding ω + ψ ⊆ I ( e χ ) via Frobenius reciprocity. (cid:3) The Iwahori subgroup e I admits a factorizatioin e I = I U − e I T I U , where I U − = e I ∩ U − , e I T = e I ∩ e T , and I U = e I ∩ U . (Note that I U and I U − split in the centralextension f Sp( W ).)Now let us define the “Jacquet module” ( τ ) U of τ with respect to I U ; that is, ( τ ) U is thequotient of τ = S ( L / L ) by (cid:10) τ ( u ) φ − φ : u ∈ I U , φ ∈ τ (cid:11) , which may be viewed as a representation of e I T . Lemma 3.3.
The space ( τ ) U is one dimensional and spanned by the image of the characteristicfunction of L . Moreover each element e t ∈ e I T acts by β t on ( τ ) U .Proof. Suppose that φ ∈ τ is supported on a + 2 L for a ∈ L r L , and let i be such that a i ∈ O × . The element e x ǫ i (1) acts on φ by the constant ψ ( a i ) = 1. Therefore, the image of φ in( τ ) U is trivial.On the other hand, let φ be the characteristic function on 2 L . Using the formulas in Section1.3 for the action of the positive root groups, it is simple to check that I U acts trivially on φ .Moreover, we know that e t ∈ e I T acts by e tφ ( y ) = β t φ ( ty ) = β t φ ( y ). (cid:3) Theorem 3.4.
The functor from the category G + ψ to the category of H + ψ -modules, given by π ( π ⊗ τ ∗ ) e I , is an equivalence of categories. In particular, there is an equivalence of categories between G + ψ and S +0 given by the isomorphism H + ψ ∼ = H + of Hecke algebras. Furthermore, this equivalencepreserves the temperedness and square integrability of representations. ECKE ALGEBRA CORRESPONDENCES 17
Proof.
We have the natural surjection r : ( π ⊗ τ ∗ ) e I → ( π U ⊗ ( τ ∗ ) U ) e I T , which is a slight variant of what is called “Jacquet’s Lemma” in [B2, 64-65]. (To prove ourversion, one can follow the argument there. Also, see [B1, Prop. 3.5.2].)We first show that r is an isomorphism. Suppose that v ∈ ker r , i.e., that there exists an opencompact subgroup U v of U such that R U v π ( u ) v du = 0.For a translation λ = ( λ , . . . , λ n ) in D ⊆ Ω a , we write λ = e h ǫ ( ̟ λ ) · · · e h ǫ n ( ̟ λ n )as its representative in e T . Take λ ∈ D such that λ ≥ . . . ≥ λ n ≥ λ − I U λ ⊇ U v . Then e Iλ e I = q ℓ ( λ ) [ i =1 λu i e I where the u i are representatives of the e I U -cosets in λ − I U λ . Let T λ be the Hecke algebra elementsupported on e Iλ e I obtained using a minimal expression for λ ∈ Ω a as in Theorem 2.1. Then π ( T λ ) v = π ( λ ) q ℓ ( λ ) X i =1 π ( u i ) v = π ( λ ) Z λ − e I U λ π ( u ) v du = 0 . The element T λ is invertible, as it is the product of invertible elements, hence v = 0 and r isinjective.Let π be an irreducible representation of f Sp( W ) such that ( π ⊗ τ ∗ ) e I = 0. As r is an iso-morphism, ( π U ⊗ ( τ ∗ ) U ) e I T = 0. This implies that Hom e T ( π U , e χ ) = 0 for some unramified χ ,since e t ∈ e I T acts by β t on ( τ ) U . Therefore, by Frobenius reciprocity, we have that π is asubrepresentation of I ( e χ ).Conversely, let π be an irreducible submodule of I ( e χ ) for some unramified χ . By Frobeniusreciprocity, we have that 0 = ( π U ⊗ ( τ ∗ ) U ) e I T ∼ = ( π ⊗ τ ∗ ) e I . Thus, condition (iii) of [BK, 3.11] is satisified, which proves the equivalence of categories.To complete the proof, we note that, as categories, S +0 ∼ = H + -modules ∼ = H + ψ -modules ∼ = G + ψ ;moreover, the trivial representation of SO( V + ) corresponds to the trivial module of H + ∼ = H + ψ ,and hence to the even Weil represntation ω + ψ .Finally, to show that the equivalence preserves temperedness and square integrability, let usnote that the equivalence H + ψ -modules ∼ = G + ψ implies that ( e I, τ ) is an s -type in the sense of[BHK, 1.6]. (To see this, let e ∈ H + ψ be the idempotent corresponding to τ and R e ( f Sp( W ))the full subcategory of the category of smooth genuine representations of f Sp( W ) as defined in[BHK, 1.4]. Then we have the functor R e ( f Sp( W )) → H + ψ -modules given by π ( π ⊗ τ ∗ ) e I .Clearly this functor composed with the equivalence H + ψ -modules ∼ = G + ψ is the identity, whichimplies R e ( f Sp( W )) = G + ψ . Hence ( e I, τ ) is an s -type with s being the inertial equivalence classrepresentanted by ( e B, ).) Hence by the first paragraph of [BHK, 0.6], one can see that all theirreducible tempered representations in G + ψ are in r b G ( τ ) with G = f Sp( W ). The same appliesto the group SO( V + ). Furthermore, from [BHK, 5.1], the equivalence S +0 ∼ = G + ψ restricts to the homeomorphism b α of Lemma 1.3 with G = f Sp( W ) , σ = τ , G = SO( V + ) , σ = . Hence, wesee that the equivalence preserves temperedness and square integrability. (cid:3) Remark.
In [GS2], the preservation of temperedness and square integrability is shown by usingCasselman’s criterion. In this paper, however, we invoke the theory of [BHK], which can beapplied once the Hecke algebra isomorphism H + ψ ∼ = H + is shown to be an isomorphism of Hilbertalgebras. Indeed, this is one of the benefits of showing that H + ψ ∼ = H + is not just an algebraisomorphism but a Hilbert algebra isomorphism.3.2. Equivalence between G − ψ and S − . Consider the partial flag X n ⊆ X n − ⊆ · · · ⊆ X where X i is the k -span of e i , . . . , e n . Let P = M N be the parabolic subgroup which is thestabilizer of this partial flag. Let W be the symplectic subspace spanned by { e , f } so thatSp( W ) = SL ( k ). Define ω ψ, to be the Weil representation of f Sp( W ), realized as a representa-tion in the space S ( k f ). This representation decomposes into even and odd parts; the odd part ω − ψ, is supercuspidal.Let e P = f M N be the preimage of P in f Sp( W ). Each element m ∈ f M is uniquely written as m = m · e h ǫ ( t ) · · · e h ǫ n ( t n )where t i ∈ k × and m ∈ f Sp( W ). Given a character χ = ( χ , . . . , χ n ), we define a genuinerepresentation ω − ψ, ⊗ e χ of f M by (cid:2) ω − ψ, ⊗ e χ (cid:3) ( m ) = ω − ψ, ( m ) n Y i =2 χ i ( t i ) β t i . We set I ( ω − ψ, ⊗ e χ ) = Ind f Sp( W ) e P ( ω − ψ, ⊗ e χ )to be the normalized induced representation. For a smooth representation π of f Sp( W ) and π N its normalized Jacquet module, we have Frobenius reciprocity:Hom f Sp( W ) ( π, I ( ω − ψ, ⊗ e χ )) ∼ = Hom f M ( π N , ω − ψ, ⊗ e χ ) . Lemma 3.5.
The Bernstein component G − ψ is precisely the component whose irreducible repre-sentations are submodules of I ( ω − ψ, ⊗ e χ ) for an unramified character χ .Proof. The functional l : S ( Y ) − → S ( k f ) − , defined by restriction of functions from Y to k f , factors through the Jacquet module ( ω − ψ ) N . Therefore, there is a non-trivial element inHom f M (( ω − ψ ) N , ω − ψ, ⊗ e χ ) for some unramified χ , which gives an embedding ω − ψ ⊆ I ( ω − ψ ⊗ e χ ) viaFrobenius reciprocity. (cid:3) Theorem 3.6.
The functor from the category G − ψ to the category of H − ψ -modules, given by π ( π ⊗ ( τ − ) ∗ ) e J , is an equivalence of categories. In particular, there is an equivalence of categories between G − ψ and S − given by the isomorphism H − ψ ∼ = H − of Hecke algebras. Furthermore, this equivalencepreserves the temperedness and square integrability of representations. ECKE ALGEBRA CORRESPONDENCES 19
Proof.
Let e J M = e J ∩ f M and e J N = e J ∩ N . As in the previous subsection, we define the “Jacquetmodule” ( τ − ) N with respect to e J N , which we view as a representation of e J M . Recall that τ − = S ( O / ̟ O ) − ⊗ S ( O n − / O n − ) , and hence, ( τ − ) N = S ( O / ̟ O ) − ⊗ ( τ ,n − ) U n − , where the second factor is the Jacquet module from the previous subsection in rank n − τ − ) N is an irreducible representation of e J M .We have the natural surjection r : ( π ⊗ ( τ − ) ∗ ) e J → ( π N ⊗ ( τ − ) ∗ N ) e J M . Just as in Proposition 3.6, one can show that r is injective, which together with Frobeniusreciprocity shows that ( π ⊗ ( τ − ) ∗ ) e J = 0 if and only if π is a submodule of I ( ω − ψ, ⊗ e χ ) for someunramified χ . Hence [BK, (3.11)] implies the equivalence of the categories.Finally, this equivalence implies that ( e J, τ − ) is an s -type in the sense of [BHK], from whichone can deduce the preservation of temperedness and square integrability just as in the previoussection. (cid:3) Remark.
As a final remark, let us mention that in [GS2, Sec. 15 and 16] it is shown that thetheta correspondence preserves unramified Langlands parameters, which relies on another work[GS1] of Gan and Savin. The only obstruction to remove the p = 2 assumption from [GS1],however, is the Howe duality conjecture, which was recently proven by Gan and the first-namedauthor in [GT] for the case at hand. Hence, everything discussed in [GS2, Sec. 15 and 16] holdswithout the assumption p = 2. References [B1] J. N. Bernstein (r´edig´e par P. Deligne),
Le “centre” de Bernstein. Repr´esentations des groupes r´eductifsur un corps local (Hermann, Paris, 1984) 1–32.[B2] J. N. Bernstein, Draft of:
Representations of p -adic groups , available online.[BHK] C. J. Bushnell, G. Henniart, and P. C. Kutzko, Types and explicit Plancherel formulae for reductive p -adic groups , On certain L -functions, Conference proceedings on the occasion of Freydoon Shahidi’s60th birthday, Clay Math. Proc. 13, AMS Providence, R.I. (2011), 55–80.[BK] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p-adic groups: structure theoryvia types
Proc. London Math. Soc. (3) 77 (1998) 582–634.[GS1] W. T. Gan and G. Savin,
Representations of metaplectic groups I: epsilon dichotomy and local Langlandscorrespondence
Compos. Math. 148 (2012), no. 6, 1655–1694.[GS2] W. T. Gan and G. Savin,
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