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MODEL TRANSITION UNDER LOCAL THETA CORRESPONDENCE 6 BAIYING LIU 1 0 2 n a Abstract. We study model transition for representations occur- J ring in the local theta correspondence between split even special 7 orthogonalgroupsandsymplectic groups,overa non-archimedean local field of characteristic zero. ] T R 1. Introduction . h Let F be a non-archimedean local field of characteristic zero. Let G t a be a reductive group defined over F. An important way to character- m ize irreducible admissible representations of G(F) is to consider vari- [ ous kinds of models. For example, non-degenerate Whittaker models 1 of generic representations play significant roles in the theory of local v 9 factors of representations of G(F) and in the theory of automorphic 6 forms. Given a reductive dual pair (G,H), and consider the local theta 6 1 correspondence between representations of G(F) and of H(F), an in- 0 teresting question is that how models of representations of G(F) and . 1 of H(F) are related under local theta correspondence. 0 In [JNQ10a], Jiang, Nien and Qin proved that under the local theta 6 1 correspondence, for certain representations, the generalized Shalika : model on SO (F) is corresponding to the symplectic linear model v 4n i on Sp (F), and conjectured that it is true for general representations X 4n (see [JNQ10b, p. 542]). In this paper, we prove some results related to r a this conjecture (see Theorem 2.1). More precisely, we introduce generalized symplectic linear models on Sp (F), which generalizes the 4m symplectic linear models, and study the relations between the generalized Shalika models on SO (F) and the generalized symplectic linear 4n models on Sp (F) under the local theta correspondence. A special 4m case (m = n) of this result is proved by Hanzer ([H15]) independently. Date: January 8, 2016. 2000 Mathematics Subject Classification. Primary: 11F70. Secondary: 22E50. Keywords andphrases. LocalThetaCorrespondence,GeneralizedShalikaMod- els, Symplectic Linear Models. The authoris supportedin partby NSF GrantsDMS-1302122,andin partby a postdoc research fund from Department of Mathematics, University of Utah. 1 2 BAIYINGLIU We also introduce generalized Shalika models on Sp (F) and gener- 4n alized orthogonal linear models on SO (F), and study the relations 4m between them under the local theta correspondence. Mœglin ([M98]), Gomez and Zhu ([GZ14]) studied the local theta lifting of generalized Whittaker models associated to nilpotent orbits. Note that the generalized Shalika models are indeed generalized Whit- taker models associated to certain nilpotent orbits, but the generalized symplectic/orthogonal linear models are not. For example, the gener- alizedShalika modelsonSO (F)aregeneralized Whittaker models as- 4n sociated to the nilpotent orbits parametrized by the partition [22n]. By [M98]and[GZ14], thefull localtheta liftonSp (F)hasa nonzero gen- 2k eralized Whittaker model associated to a nilpotent orbit parametrized by the partition [32n1ℓ], if 6n + ℓ = 2k. In general it is not known whether the small theta lift on Sp (F) (if nonzero) would also carry 2k this model. This paper is organized as follows. In Section 2, we give the defi- nitions for various models for representations of split even special orthogonal groups and symplectic groups, and introduce the main result Theorem 2.1. In Section 3 and 4, we prove Part (1) and Part (2) of Theorem 2.1 respectively. In Section 5, we consider the converse of Theorem 2.1, and discuss some related results. Acknowledgments. The main results of this paper were worked outwhentheauthorwasagraduatestudentatUniversityofMinnesota. The author would like to thank his advisor Prof. Dihua Jiang for constant support, encouragement and helpful discussions. The author also would like to thank Prof. Gordan Savin, Prof. David Soudry and Prof. Chen-bo Zhu for helpful conversations, and Raul Gomez for helpful communications. The author also would like to thank the referee for helpful comments and suggestions on this paper. 2. Models of representations In this section, we define various models for representations of split even special orthogonal groups and symplectic groups. For any positive integer k, let v be the k×k matrix with 1’s in the k second diagonal and zero’s elsewhere. Let SO = {g ∈ GL | tgv g = v } 2ℓ 2ℓ 2ℓ 2ℓ be the split even special orthogonal group. And let SO = SO (F). 2ℓ 2ℓ Let Sp = {g ∈ GL |tgJ g = J } 2ℓ 2ℓ 2ℓ 2ℓ MODEL TRANSITION UNDER LOCAL THETA CORRESPONDENCE 3 0 v be the symplectic group, where J = ℓ . And let Sp = 2ℓ (cid:18)−vℓ 0(cid:19) 2ℓ Sp (F). 2ℓ Let P = M N be the Siegel parabolic subgroup of SO . Then 2n 2n 2n 4n elements in P have the following form 2n g 0 I X (g,X) = m(g)n(X) = n (cid:18) 0 g∗ (cid:19)(cid:18) 0 In (cid:19) where g ∈ GL (F), g∗ = v tg−1v , and X ∈ M (F) satisfying 2n 2n 2n (2n)×(2n) tX = −v Xv . The generalized Shalika group of SO is defined to 2n 2n 4n be H = H = M′ N = {(g,X) ∈ P | g ∈ Sp }. 2n 2n 2n 2n 2n Define a character of H as follows 1 ψ ((g,X)) = ψ( tr(J Xv )) H 2n 2n 2 1 −I 0 = ψ( tr( n X)). 2 (cid:18) 0 In(cid:19) Then an irreducible admissible representation (σ,V ) of SO has a σ 4n generalized Shalika model if Hom (V ,IndSO4n(ψ )) = Hom (V ,ψ ) 6= 0. SO4n σ H H H σ H The nonzero elements in the Hom space are called generalized Shalika functionals or ψ -functionals of σ. In [N10], Nien proved the unique- H ness of the generalized Shalika models. Let Q = L V be the Siegel parabolic subgroup of Sp . Then 2n 2n 2n 4n elements in Q have the following form 2n g 0 I X (g,X) = m(g)n(X) = n (cid:18) 0 g∗ (cid:19)(cid:18) 0 In (cid:19) where g ∈ GL (F), g∗ = v tg−1v , and X ∈ M (F) satisfying 2n 2n 2n (2n)×(2n) tX = v Xv . The generalized Shalika group of Sp is defined to be 2n 2n 4n H = H = L′ V = {(g,X) ∈ P | g ∈ SO }. 2n 2n 2n 2n 2n Let e e 1 ψe((g,X)) = ψ( tr(X)) H 2 be a character of H. An irreducible admissible representation (π,V ) π of Sp has a generalized Shalika model if 4n e HomSp4n(Vπ,IndSHep4n(ψHe)) = HomHe(Vπ,ψHe) 6= 0. The nonzero elements in the Hom space are called generalized Shalika functionals or ψe-functionals of π. H 4 BAIYINGLIU Consider SO ×SO as a subgroup of SO (2m > n) via the 2n 4m−2n 4m embedding A 0 0 B 1 1 A B A B  0 A B 0  ( 1 1 , 2 2 ) 7→ 2 2 . (cid:18)C D (cid:19) (cid:18)C D (cid:19) 0 C D 0 1 1 2 2 2 2   C 0 0 D  1 2   An irreducible admissible representation (σ,V ) of SO (2m ≥ n) has σ 4m a generalized orthogonal linear model if Hom (V ,IndSO4m(F) (1) SO4m(F) σ SO2n(F)×SO4m−2n(F) = Hom (V ,1) SO2n(F)×SO4m−2n(F) σ 6= 0. The nonzero elements in the Hom space are called generalized orthogonal linear functionals of π. Similarly, consider Sp ×Sp as a subgroup of Sp (2m > n) 2n 4m−2n 4m via the embedding A 0 0 B 1 1 A B A B  0 A B 0  ( 1 1 , 2 2 ) 7→ 2 2 . (cid:18)C D (cid:19) (cid:18)C D (cid:19) 0 C D 0 1 1 2 2 2 2   C 0 0 D  1 2   An irreducible admissible representation (π,V ) of Sp (2m ≥ n) has π 4m a generalized symplectic linear model if Hom (V ,IndSp4m (1)) Sp4m π Sp2n×Sp4m−2n = Hom (V ,1) Sp2n×Sp4m−2n π 6= 0 The nonzero elements in the Hom space are called generalized symplectic linear functionals of π. When m = n, the generalized symplectic linear model is usually called the symplectic linear model. In [Zh10], Zhang proves the uniqueness of the symplectic linear models for the cases of n = 1,2. The uniqueness of generalized symplectic linear models on Sp is unknown in general. 4m Let ω be the Weil representation corresponding to the reductive ψ dual pair (O ,Sp ) with O being split. Following is the main result 2k 2ℓ 2k of this paper. Theorem 2.1. (1) Assume that σ is an irreducible admissible representation of SO with a nonzero generalized Shalika model, 4n and π is an irreducible admissible representation of Sp (2m ≥ 4m MODEL TRANSITION UNDER LOCAL THETA CORRESPONDENCE 5 n) which is corresponding to σ under the local theta correspondence, that is, Hom (ω ,σ ⊗π) 6= 0. SO4n×Sp4m ψ Then π has a nonzero generalized symplectic linear model. (2) Assume that π is an irreducible admissible representation of Sp with a nonzero generalized Shalika model, and σ is an 4n irreducible admissible representation of SO (2m ≥ n) which 4m is corresponding to π under the local theta correspondence, that is, Hom (ω ,σ ⊗π) 6= 0. SO4m×Sp4n ψ Then, σ has a nonzero generalized orthogonal linear model. In [H15], Hanzer proved the case of m = n for Theorem 2.1, Part (1), independently. Some results related to the converse of Theorem 2.1 will be discussed in Section 5. Mœglin ([M98]), Gomez and Zhu ([GZ14]) considered the local theta lifting of generalized Whittaker models associated to nilpotent orbits. It turns out a nonzero generalized Shalika model for an irreducible admissible representation σ of SO is indeed a generalized Whittaker 4n model associated to a nilpotent orbit parametrized by the partition [22n]. Then they showed that the full local theta lift of σ on Sp 2k has a nonzero generalized Whittaker model associated to a nilpotent orbit parametrized by the partition [32n1ℓ], if 6n+ℓ = 2k. Note that this later model is not a generalized symplectic linear model. And, in general it is not known that whether the small theta lift would also carry this model. 3. Proof of Theorem 2.1, Part (1) In this section, we prove Theorem 2.1, Part (1), using a method similar to that in [JS03, Section 2.1]. Assume that σ is an irreducible admissible representation of SO 4n with a nonzero generalized Shalika model, and π is an irreducible admissible representation of Sp (2m ≥ n) which is corresponding to σ 4m under the local theta correspondence, i.e., Hom (ω ,σ ⊗π) 6= 0. SO4n×Sp4m ψ Sinceσ hasanonzerogeneralizedShalikamodel,i.e., Hom (V ,ψ ) 6= H σ H 0, we have that Hom (ω ,ψ ⊗π) 6= 0. H×Sp4m(F) ψ H Let V be a 4n-dimensional vector space over F, with the nondegen- erate symmetric from v . Fix a basis {e ,...,e ,e ,...,e } of V 4n 1 2n −2n −1 6 BAIYINGLIU over F, such that (e ,e ) = (e ,e ) = 0,(e ,e ) = δ , for i,j = i j −i −j i −j ij 1,...,2n. Let V+ = Span {e ,...,e }, V− = Span {e ,...,e }, F 1 2n F −1 −2n then V = V+ +V− is a complete polarization of V. Let W be a 4m- dimensional symplectic vector space over F, with the symplectic from 0 v J = 2m . Fix a basis 4m (cid:18) −v2m 0 (cid:19) {f ,...,f ,f ,...,f } 1 2m −2m −1 ofW over F, suchthat(f ,f ) = (f ,f ) = 0,(f ,f ) = δ , fori,j = i j −i −j i −j ij 1,...,2m. LetW+ = Span {f ,...,f },W− = Span {f ,...,f }, F 1 2m F −1 −2m then W = W++W− is a complete polarization of W. Then, we realize the Weil representation ω in the space S(V− ⊗W) ∼= S(W2n). ψ First, we compute the Jacquet module J (S(W2n)). By N2n,ψH|N2n [MVW87, Section 2.7], for φ ∈ S(W2n), (3.1) 1 ω (n(X),1)φ(y ,...,y ) = ψ( tr(Gr(y ,...,y )v X))φ(y ,...,y ), ψ 1 2n 1 2n 2n 1 2n 2 where Gr(y ,...,y ) is the Gram matrix of vectors y ,...,y . Since 1 2n 1 2n tX = −v Xv , 2n 2n 1 ψ( tr(Gr(y ,...,y )v X)) 1 2n 2n (3.2) 2 = ψ(x (y ,y )+···+x (y ,y )+x (y ,y )+···), 11 1 2n nn n n+1 12 2 2n where 1tr(Gr(y ,...,y )v X)involvesallx termsexceptx ,x , 2 1 2n 2n ij 1,2n 2,2n−1 ...,x , since they are zeros. And, 2n,1 1 −I 0 (3.3) ψ (n(X)) = ψ( tr( n X)) = ψ(−(x +···+x )). H 2 (cid:18) 0 In (cid:19) 11 nn By comparing (3.2), (3.3), let C = {(y ,...,y ) ∈ W2n|Gr(y ,...,y ) = −J }. 0 1 2n 1 2n 2n Then, by (3.1), (3.2) and (3.3), we claim that (3.4) J (S(W2n)) ∼= S(C ). N2n,ψH|N2n 0 Indeed, let C = W2n\C , which is an open set, since C is closed in 0 0 W2n. By [BZ76, Section 1.8], there is an exact sequence 0 → S(C) −→i S(W2n) −→r S(C ) → 0, 0 whereiisthecanonicalembeddingthatextendsanyfunctionsupported inC by zero tothewhole space W2n, andr isthe restrictionmap toC . 0 By the exactness of Jacquet functors, there is also an exact sequence 0 → J (S(C)) −→i∗ J (S(W2n)) −r→∗ J (S(C )) → 0. N2n,ψH|N2n N2n,ψH|N2n N2n,ψH|N2n 0 MODEL TRANSITION UNDER LOCAL THETA CORRESPONDENCE 7 By (3.1), (3.2) and (3.3), for any (y ,...,y ) ∈ C, the character 1 2n 1 n(X) 7→ ψ( tr(Gr(y ,...,y )v X))−ψ (n(X)) 1 2n 2n H 2 is nontrivial. Hence, for φ ∈ S(C), there exists a large enough compact subgroup N of N , such that φ 2n (ω (n(X),1)−ψ (n(X)))φ(y ,...,y )dX ψ H 1 2n Z N φ 1 = (ψ( tr(Gr(y ,...,y )v X))−ψ (n(X)))φ(y ,...,y )dX 1 2n 2n H 1 2n Z 2 N φ = 0. So, J (S(C)) = 0, and N2n,ψH|N2n J (S(W2n)) ∼= J (S(C )) ∼= S(C ). N2n,ψH|N2n N2n,ψH|N2n 0 0 This proves the claim in (3.4). Therefore, (3.5) HomM2′n×Sp4m(S(C0),1⊗π) 6= 0. Note that M′ = {m(g) ∈ M |g ∈ Sp }. 2n 2n 2n By [MVW87, Section 2.7], for φ ∈ S(C ), m(g) ∈ M′ , h ∈ Sp , 0 2n 4m ω (m(g),h)φ(y ,...,y ) = φ((y h,...,y h)v gv ). ψ 1 2n 1 2n 2n 2n Note that by Witt Theorem, M′ ×Sp acts transitively on C , and 2n 4m 0 (f ,...,f ,f ,...,f ) −2m+n−1 −2m 2m 2m−n+1 is a representative. Let R be the stabilizer of this representative. Then a 0 b a b R = {(m(g), 0 g−1 0 ) | g ∈ Sp , ∈ Sp }. 2n (cid:18) c d (cid:19) 4m−2n c 0 d   Hence, S(C ) is isomorphic to the compactly induced representation 0 c−IndM2′n×Sp4m(1). R Therefore, by (3.5), HomM2′n×Sp4m(c−IndRM2′n×Sp4m(1),1⊗π) 6= 0, that is, HomM2′n×Sp4m(1⊗π,IndRM2′n×Sp4m(1)) 6= 0, e 8 BAIYINGLIU where π is the contragredient of π. Note that in general, given admissible representations σ and τ of a connected reductive group G, one has e Hom (σ,τ) ∼= Hom (τ,σ). G G And note that e e ^ c−IndM2′n×Sp4m(1) ∼= IndM2′n×Sp4m(1), R R since both M′ ×Sp and R are unimodular. 2n 4m Then, by Frobenius Reciprocity, we have Hom (1⊗π,1) 6= 0, R that is, e Hom (π,1) 6= 0, Sp2n(F)×Sp4m−2n(F) which means that π has a nonzero generalized symplectic linear model. e It follows from an argument as in the proof of [GRS99a, Theorem 17] that π also heas a nonzero generalized symplectic linear model. Indeed, by [MVW87, pages 91-92], π ∼= πδ, where δ = v , and 4m πδ(g) = π(δgδ−1) = π(δgδ). Hence, πδ has a nonzero generalized symplectic linear model. Assume that l is aenonzero generalized symplectic linear functional of πδ. Then, for any (g ,g ) ∈ Sp ×Sp , v ∈ V , 1 2 2n 4m−2n π l(π(g ,g )v) = l(πδ(g ,g )δv) = l(πδ(v g v ,v g v )v) = l(v), 1 2 1 2 2n 1 2n 4m−2n 2 4m−2n noting that (v g v ,v g v ) ∈ Sp ×Sp . So, l is also 2n 1 2n 4m−2n 2 4m−2n 2n 4m−2n a nonzero generalized symplectic linear functional of π. Therefore, π has a nonzero generalized symplectic linear model. This completes the proof of Theorem 2.1, Part (1). 4. Proof of Theorem 2.1, Part (2) In this section, we prove Theorem 2.1, Part (2), using a similar argument as in Section 3. Assume that π is an irreducible admissible representation of Sp 4n with a nonzero generalized Shalika model, and σ is an irreducible admissible representation of SO (2m ≥ n) which is corresponding to π 4m under the local theta correspondence, i.e., Hom (ω ,σ ⊗π) 6= 0. SO4m×Sp4n ψ Since π has a nonzero generalized Shalika model, that is, HomHe(Vπ,ψHe) 6= 0, we have that HomSO4m×He(ωψ,σ ⊗ψHe) 6= 0. MODEL TRANSITION UNDER LOCAL THETA CORRESPONDENCE 9 Let V be a 4m-dimensional vector space over F, with the nondegen- erate symmetric from v . Fix a basis {e ,...,e ,e ,...,e } of 4m 1 2m −2m −1 V over F, such that (e ,e ) = (e ,e ) = 0,(e ,e ) = δ , for i,j = i j −i −j i −j ij 1,...,2m. LetV+ = Span {e ,...,e },V− = Span {e ,...,e }, F 1 2m F −1 −2m then V = V+ +V− is a complete polarization of V. Let W be a 4n- dimensional symplectic vector space over F, with the symplectic from 0 v J = 2n . Fix a basis 4n (cid:18) −v2n 0 (cid:19) {f ,...,f ,f ,...,f } 1 2n −2n −1 ofW over F, suchthat(f ,f ) = (f ,f ) = 0,(f ,f ) = δ , fori,j = i j −i −j i −j ij 1,...,2n. LetW+ = Span {f ,...,f },W− = Span {f ,...,f }, F 1 2n F −1 −2n then W = W++W− is a complete polarization of W. Realize the Weil representation ω in the space S(V ⊗W+) ∼= S(V2n). ψ First, as in Section 3, we need to compute the Jacquet module of the Weil representation J (S(V2n)). N2n,ψH|N2n By [MVW87, Section 2.6], for φ ∈ S(V2n), (4.1) 1 ω (n(X),1)φ(x ,...,x ) = ψ( tr(Gr(x ,...,x )Xv ))φ(x ,...,x ), ψ 1 2n 1 2n 2n 1 2n 2 where Gr(x ,...,x ) is the Gram matrix of vectors x ,...,x . Since 1 2n 1 2n tX = v Xv , 2n 2n 1 (4.2) ψ( tr(Gr(x ,...,x )Xv )) 1 2n 2n 2 1 (4.3) = ψ(x (x ,x )+···+x (x ,x )+ x (x ,x )+···), 11 1 2n nn n n+1 1,2n 1 1 2 where 1tr(Gr(x ,...,x )Xv ) involves all x terms. And, 2 1 2n 2n ij 1 (4.4) ψ (n(X)) = ψ( tr(X)) = ψ(x +···+x ). H 11 nn 2 By comparing (4.1), (4.2), let C = {(x ,...,x ) ∈ W2n|Gr(x ,...,x ) = v }. 0 1 2n 1 2n 2n Then, by (4.1), (4.2) and (4.4), a similar argument (details omitted) as in Section 3 shows that J (S(V2n)) ∼= S(C ). N2n,ψH|N2n 0 Therefore, (4.5) HomSO4m×M2′n(S(C0),σ ⊗1) 6= 0. Note that M′ = {m(g) ∈ M |g ∈ SO }. 2n 2n 2n 10 BAIYINGLIU By [MVW87, Section 2.6], for φ ∈ S(C ), m(g) ∈ M′ , h ∈ SO , 0 2n 4m ω (h,m(g))φ(x ,...,x ) = φ((h−1x ,...,h−1x )g). ψ 1 2n 1 2n Note that by Witt Theorem, SO ×M′ acts transitively on C , and 4m 2n 0 (e ,...,e ,e ,...,e ) 2m−n+1 2m −2m −2m+n−1 is a representative. Let R be the stabilizer of this representative. Then a 0 b a b R = { 0 g−1 0 )|g ∈ SO , ∈ SO }. 2n (cid:18) c d (cid:19) 4m−2n c 0 d   Hence, S(C ) is isomorphic to the compactly induced representation 0 c−IndSO4m×M2′n(F)(1). R Hence, by (4.5), HomSO4m×M2′n(c−IndRSO4m×M2′n(1),σ ⊗1) 6= 0, that is, HomSO4m×M2′n(σ ⊗1,IndSRO4m×M2′n(1)) 6= 0, as in Section 3. By Frobenius Reciprocity, we have e Hom (σ ⊗1,1) 6= 0, R that is, e Hom (σ,1) 6= 0. SO2n×SO4m−2n So, σ has a nonzero generalized orthogonal linear model. It follows e that σ also has a generalized orthogonal linear model, as in Section 3. Indeeed, by [N10, Lemma 4.2], σ ∼= σδ2m, where I 0 0 0 2me−1  0 0 1 0  δ = , 4m 0 1 0 0    0 0 0 I2m−1    and σδ4m(g) = σ(δ gδ−1) = σ(δ gδ ). So, σδ4m has a nonzero 4m 4m 4m 4m generalized orthogonal linear model. Assume that ℓ is a nonzero generalized orthogonal linear functional of σδ2m. Then, for any (g ,g ) ∈ 1 2 SO ×SO , v ∈ V , 2n 4m−2n σ ℓ(σ(g ,g )v) = ℓ(σδ4m(g ,g )δ4mv) = ℓ(σδ4m(δ g δ ,g )v) = ℓ(v), 1 2 1 2 2n 1 2n 2 noting that δ g δ ∈ SO . Hence, ℓ is also a nonzero generalized or- 2n 1 2n 2n thogonal linear functional of σ. Therefore, σ has a nonzero generalized orthogonal linear model. This completes the proof of Theorem 2.1, Part (2).