A PROOF OF THE HOWE DUALITY CONJECTURE. 1. Introduction

A PROOF OF THE HOWE DUALITY CONJECTURE WEE TECK GAN AND SHUICHIRO TAKEDA to Professor Roger Howe who started it all on the occasion of his 70th birthday Abstract. We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic. 1. Introduction Let F be a nonarchimedean local field of characteristic not 2 and residue characteristic p. Let E be F itself or a quadratic field extension of F. For ɛ = ±, we consider a ɛ-hermitian space W over E of dimension n and an ɛ-hermitian space V of dimension m. We shall write W n or V m if there is a need to be specific about the dimension of the space in question. Set { ɛ if E = F ; ɛ 0 = 0 if E F. Let G(W ) and H(V ) denote the isometry group of W and V respectively. Then the group G(W ) H(V ) forms a dual reductive pair and possesses a Weil representation ω ψ which depends on a nontrivial additive character ψ of F (and some other auxiliary data which we shall suppress for now). To be precise, when E = F and one of the spaces, say V, is odd dimensional, one needs to consider the metaplectic double cover of G(W ); we shall simply denote this double cover by G(W ) as well. The various cases are tabulated in [GI, 3]. In the theory of local theta correspondence, one is interested in the decomposition of ω ψ into irreducible representations of G(W ) H(V ). More precisely, for any irreducible admissible representation π of G(W ), one may consider the maximal π-isotopic quotient of ω ψ. This has the form π Θ W,V,ψ (π) for some smooth representation Θ W,V,ψ (π) of H(V ); we shall frequently suppress (W, V, ψ) from the notation if there is no cause for confusion. It was shown by Kudla [K] that Θ(π) has finite length (possibly zero), so we may consider its maximal semisimple quotient θ(π). One has the following fundamental conjecture due to Howe [H]: Howe Duality Conjecture for G(W ) H(V ) (i) θ(π) is either 0 or irreducible. 1

2 WEE TECK GAN AND SHUICHIRO TAKEDA (ii) If θ(π) = θ(π ) 0, then π = π. A concise reformulation is: for any irreducible π and π, dim Hom H(V ) (θ(π), θ(π )) δ π,π := We take note of the following theorem: { 1, if π = π ; 0, if π π. Theorem 1.1. (i) If π is supercuspidal, then Θ(π) is either zero or irreducible (and thus is equal to θ(π)). Moreover, for any irreducible supercuspidal π and π, (ii) θ(π) is multiplicity-free. Θ(π) = Θ(π ) 0 = π = π. (iii) If p 2, the Howe duality conjecture holds. The statement (i) is a classic theorem of Kudla [K] (see also [MVW]), whereas (iii) is a well-known result of Waldspurger [W]. The statement (ii), on the other hand, is a recent result of Li-Sun-Tian [LST]. We note that the techniques for proving the three statements in the theorem are quite disjoint from each other. For example, the proof of (i) is based on arguments using the doubling see-saw and Jacquet modules of the Weil representation: these have become standard tools in the study of local theta correspondence. The proof of (iii) is based on K-type analysis and uses various lattice models of the Weil representation. Finally, the proof of (ii) is based on an argument using the Gelfand-Kazhdan criterion for the (non-)existence of equivariant distributions. In this paper, we shall not assume any of the statements in Theorem 1.1. Indeed, the purpose of this paper is to give a simple proof of the Howe duality conjecture using essentially the same tools in the proof of Theorem 1.1(i), as developed further in [MVW]. Thus, our main theorem is: Theorem 1.2. The Howe duality conjecture for G(W ) H(V ) holds. We end the introduction with a few remarks: The above setup makes sense even when E = F F is a split quadratic algebra, in which case the groups G(W ) and H(V ) are general linear groups. In that case, the Howe duality conjecture has been shown by Minguez in [M]. As we shall see, the proof of Theorem 1.2 is essentially analogous to the one given by Minguez. In an earlier paper [GT], we had extended Theorem 1.1(i) (with Θ(π) replaced by θ(π)) from supercuspidal to tempered representations. Using this, we had shown the Howe duality conjecture for almost equal rank dual pairs (where m (n + ɛ 0 ) 1). The argument in 2 of this paper (doubling see-saw) is the same as that in [GT, 2], but pushed to the limit beyond tempered representations. On the other hand, the

THE HOWE DUALITY CONJECTURE 3 argument in 3 (Kudla s filtration) is entirely different from that in [GT, 3] and uses a key technique of Minguez [M]. We remark that in the papers [M1, M2, M3, M4], Muić has conducted a detailed study of the local theta correspondence for symplectic-orthogonal dual pairs. In [M1], for example, he explicitly determined the theta lift of discrete series representations π in terms of the Moeglin-Tadić classification and observed as a consequence the irreducibility of θ(π). The Moeglin-Tadić classification was conditional at that point, and we are not sure where it stands today. In [M3, M4], Muić proved various general properties of the theta lifting of tempered representations (such as the issue of whether Θ(π) = θ(π)), and obtained very explicit information about the theta lifting under the assumption of the Howe duality conjecture. The main tools he used are Jacquet modules analysis and Kudla s filtration. Since the Howe duality conjecture is a simple statement without reference to classification, it seems desirable to have a classification-free proof. Indeed, our result renders most results in [M3, M4] unconditional. As is well-known, there is another family of dual pairs associated to quaternionic Hermitian and skew-hermtian spaces. Our proof does not apply to these quaternionic dual pairs, because we have made use of the MVW-involution π π MV W on the category of smooth representations of G(W ) and H(V ). For the same reason, the result of [LST] in Theorem 1.1(ii) is not known for these quaternionic dual pairs. Unlike the contragredient functor, which is contravariant in nature, the MVW-involution is covariant and has the property that π MV W = π if π is irreducible. It was shown in [LnST] that such an involution does not exist for quaternionic unitary groups. Nonetheless, even in this case, our proof gives a partial result which is often sufficient for global applications. Namely, if π is an irreducible unitary representation and we let θ unit (π) θ(π) denote the submodule generated by irreducible unitary summands, then the results of Theorem 1.2 hold for unitary π s and with θ(π) replaced by θ unit (π). This is because if π is unitary, then π = π, so that our proof can be applied. We leave the verification of the details of this case for the reader. Acknowledgments: This project was begun during the authors participation in the Oberwolfach workshop Modular Forms in April 2014 and completed while both authors were participating in the workshop The Gan-Gross-Prasad conjecture at Jussieu. We thank the Ecole Normale Superieur and the Institut des Hautes Études Scientifiques for hosting our respective visits. We also thank Goran Muić and Marcela Hanzer for several useful conversations and email exchanges about [M1, M2, M3, M4] and the geometric lemma respectively. We are extremely grateful to Alberto Minguez for explaining to us the key idea in his paper [M] for taking care of the representations on the boundary. Finally, we thank an anonymous referee who pointed out several embarrassing errors in a first version of this paper. The first author is partially supported by an MOE Tier Two grant R-146-000-175-112, whereas the second author is partially supported by NSF grant DMS-1215419.

4 WEE TECK GAN AND SHUICHIRO TAKEDA 2. Special Case of Theorem 1.2 In proving Theorem 1.2, there is no loss of generality in assuming that m n+ɛ 0, because otherwise one can switch the roles of G(W ) and H(V ). We shall assume this henceforth and set s m,n = m (n + ɛ 0) 0. 2 Moreover, we may argue by induction on n = dim W, with the base case with dim W = 0 being trivial. Hence, we may assume that Theorem 1.2 is known for dual pairs G(W ) H(V ) with dim V dim W + ɛ 0 < n + ɛ 0. In this section, we will give a proof for a special case, namely when π does not lie on the boundary. We note that this special case in fact covers almost all representations, and its proof does not require the use of the induction hypothesis. We begin by specifying the extra data needed to consider the Weil representation of G(W ) H(V ); these are needed to split the metaplectic cover Sp(W V ) over the dual pair (where the splitting exists). We shall follow the setup of [GI, 3.2-3.3] in fixing a pair of splitting characters χ = (χ V, χ W ), which are certain unitary characters of E, with associated Weil representation ω W,V,χ,ψ. If c denotes a generator of Gal(E/F ), then the character χ V satisfies c χ 1 V = χ V and likewise for χ W. We shall frequently suppress χ and ψ from the notation. We also set up some notations which will streamline the exposition. If ρ i are irreducible representations of GL ni for i = 1,..., a, we write ρ 1 ρ 2 ρ a for the representation of GL n1 + +n a which is obtained by normalized parabolic induction from the standard parabolic subgroup with Levi subgroup GL n1 GL na. For the group G(W n ), which we sometimes denote by G n, we let Q(X t ) = L(X t ) U(X t ) be the maximal parabolic subgroup which stabilizes a t-dimensional isotropic subspace, which has Levi factor L(X t ) = GL(X t ) G(W n 2t ). If ρ is a representation of GL(X t ) and σ is a representation of G(W n 2t ), we write ρ σ for the representation of G(W n ) obtained by normalized parabolic induction from the representation ρ σ of Q(X t ). Likewise, we have the analogous parabolic subgroup P (Y t ) = M(Y t ) N(Y t ) for H(V m ) when Y t is a t-dimensional isotropic subspace of V m. As alluded to in the introduction, Moeglin, Vigneras and Waldspurger have introduced in [MVW, p. 91] an involution π π MV W on the category of smooth representations of G(W ). It is a covariant functor such that π MV W = π if π is irreducible. It will be useful to observe that (ρ σ) MV W = c ρ σ MV W, where c ρ denotes the c-conjugate of ρ. We shall make use of this property frequently.

THE HOWE DUALITY CONJECTURE 5 For π Irr(G(W )), we shall say that π lies on the boundary of ω W,V if there exists t > 0 such that π χ V det GL(Xt) sm,n t 2 σ for some irreducible representation σ of G(W n 2t ). Dualizing and using the MVW involution [MVW, p.91] and Bernstein s form of Frobenius reciprocity, this is equivalent to Hom GL(Xt)(χ V det sm,n+ t 2, RQ(Xt) (π)) 0, where Q(X t ) stands for the parabolic subgroup opposite to Q(X t ). In this section, we shall prove: Theorem 2.1. Assume that m n + ɛ 0 and suppose that π Irr(G(W )) does not lie on the boundary of ω W,V. Then for any π Irr(G(W )), dim Hom H(V ) (θ(π), θ(π )) δ π,π. In particular, θ(π) is either zero or irreducible, and moreover for any irreducible π, 0 θ(π) θ(π ) = π = π. We consider the following see-saw diagram G(W W ) H(V ) H(V ) G(W ) G(W ) H(V ), where W denotes the space obtained from W by multiplying the form by 1, so that G(W ) = G(W ). Given irreducible representations π and π of G(W ), the see-saw identity [GI, 6.1] gives: Hom G(W ) G(W ) (Θ V,W +W (χ W ), π π χ V ) = Hom H(V ) (Θ(π ) Θ(π) MV W, C). Here Θ V,W +W (χ W ) denotes the big theta lift of the character χ W of H(V ) to G(W + W ). We analyze each side of the see-saw identity in turn. For the RHS, one has Hom H(V ) (Θ(π ) Θ(π) MV W, C) Hom H(V ) (θ(π ) θ(π), C) = Hom H(V ) (θ(π ), θ(π)). For the LHS of the see-saw identity, we need to understand Θ V,W +W (χ W ). It is known that Θ V,W +W (χ W ) is irreducible (see [GI, Prop. 7.2]). Moreover, it was shown by Rallis that (2.1) Θ V,W +W (χ W ) Ind G(W +W ) P ( W ) χ V det sm,n where W W + W is diagonally embedded and is a maximal isotropic subspace; P ( W ) is the maximal parabolic subgroup of G(W + W ) which stabilizes W and has Levi factor GL( W );

6 WEE TECK GAN AND SHUICHIRO TAKEDA Ind G(W +W ) P ( W ) χ V det s denotes the degenerate principal series representation induced from the character χ V det s of P ( W ) (normalized induction). Since s m,n 0, there is a surjective map (see [GI, Prop. 8.2]) Hence the see-saw identity gives: Ind G(W +W ) P ( W ) χ V det sm,n Θ V,W +W (χ W ). Hom G(W ) G(W ) (Ind G(W +W ) P ( W ) χ V det sm,n, π π χ V ) Hom H(V ) (θ(π ), θ(π)). To prove the theorem, it suffices to show that the LHS has dimension 1, with equality only if π = π. For this, we need the following lemma (see [KR]): Lemma 2.2. As a representation of G(W ) G(W ), Ind G(W +W ) P χ V det s possesses an equivariant filtration with successive quotients 0 I 0 I 1 I q = Ind G(W +W ) P χ V det s R t = I t /I t 1 (( ) ( )) = Ind G(W ) G(W ) Q t Q t χ V det Xt s+ t 2 χv det Xt s+ t 2 (χ V det W ) Cc (G(W n 2t )). n 2t Here, the induction is normalized and q is the Witt index of W ; Q t is the maximal parabolic subgroup of G(W ) stabilizing a t-dimensional isotropic subspace X t of W, with Levi subgroup GL(X t ) G(W n 2t ), where dim W n 2t = n 2t. G(W n 2t ) G(W n 2t ) acts on C c (G(W n 2t )) by left-right translation. In particular, R 0 = (χ V det W ) C c (G(W )). Applying the lemma, we claim that if π is not on the boundary, the natural restriction map Hom G(W ) G(W ) (Ind G(W +W ) P ( W ) χ V det sm,n, π π χ V ) Hom G(W ) G(W ) (R 0, π π χ V ) is injective. This will imply the theorem since the RHS has dimension 1, with equality if and only if π = π. To deduce the claim, it suffices to to show that for each 0 < t q, Hom G(W ) G(W ) (R t, π π χ V ) = 0.

THE HOWE DUALITY CONJECTURE 7 By Frobenius reciprocity (à la Bernstein), Hom G(W ) G(W ) (R t, π π χ V ) is equal to ( (χv Hom L(Xt) L(Xt) det sm,n+ t 2 χv det sm,n+ ) t 2 ( ) ) (χ V det W ) Cc (G(W n 2t )) n 2t, R Qt (π ) R Qt (π χ V ). Hence we deduce that the above Hom space is 0 if Hom GL(Xt)(χ V det sm,n+ t 2, RQt (π χ V )) = 0. By Bernstein s form of Frobenius reciprocity and dualizing, this is equivalent to Hom GL(Xt)(π (χ 1 V det W ), (χ 1 V det GL(X t)) det sm,n t 2 ) = 0 Now note that χ V det W GL(Xt) = χ 2 V det GL(Xt). Hence the above condition is equivalent to Hom GL(Xt)(π, χ V det sm,n t 2 ) = 0. Since this condition holds when π is not on the boundary of ω W,V, the theorem is proved. The above argument also gives Proposition 2.3. If π 1 π 2 Irr(G(W n )) (with n + ɛ 0 m) are such that then there exists t > 0 such that and for some τ 1 and τ 2 Irr(G(W n 2t )). Hom H(V ) (θ Wn,V m (π 1 ), θ Wn,V m (π 2 )) 0, π 1 χ V det GL(Xt) sm,n t 2 τ1 π 2 χ V det GL(Xt) sm,n t 2 τ2 Proof. If π 1 π 2 but Hom H(V ) (θ Wn,Vm (π 1 ), θ Wn,Vm (π 2 )) 0, then arguing as above one must have Hom G(W ) G(W ) (R t, π 1 π2 χ V ) 0 for some t > 0, which gives the conclusion of the proposition. 3. Proof of Theorem 1.2 In this section, we consider those representations π which lie on the boundary of ω W,V, so that π χ V det GL( X t) sm,n t 2 π0 for some π 0 Irr(G(W n 2t )) and some t > 0. By induction in stages, we have For simplicity, let us set π (χ V sm,n t+ 1 2 χv sm,n 1 2 ) π0. s m,n,t = s m,n t + 1 2 < 0.

8 WEE TECK GAN AND SHUICHIRO TAKEDA Let a > 0 be maximal such that where π Σ a := χ V sm,n,t 1 a σ 1 a = 1 1 }{{} a times and σ is some irreducible representation of G(W n 2a ). This is equivalent to saying that for any σ. Now suppose Then we have: 0 Hom Gn H m (ω W,V, π π ) Hom Gn H m (ω W,V, Σ a π ) σ χ V sm,n,t σ π θ(π). = Hom GL(Xa) G n 2a H m (R Q(Xa)(ω W,V ), χ V sm,n,t 1 a σ π ). The Jacquet module R Q(Xa)(ω W,V ) of the Weil representation has been computed by Kudla [K] (see also [MVW]). We state the result here: Lemma 3.1. The Jacquet module R Q(Xa)(ω Wn,V m ) has an equivariant filtration R Q(Xa)(ω Wn,V m ) = R 0 R 1 R a R a+1 = 0 whose successive quotient J k = R k /R k+1 is described in [GI, Lemma C.2]. More precisely, ) J k = Ind GL(Xa) G(W n 2a) H(V m) Q(X a k,x a) G(W n 2a ) P (Y k ) (χ V det Xa k λ a k Cc (GL k ) ω Wn 2a,V m 2k, where λ a k = s m,n + a k 2, V m = Y k + V m 2k + Yk with Y k a k-dimensional isotropic space, X a = X a k + X k and Q(X a k, X a ) is the maximal parabolic subgroup of GL(X a ) stabilizing X a k. GL(X k ) GL(Y k ) acts on Cc (GL k ) as ((b, c) f)(g) = χ V (det b)χ W (det c)f(c 1 gb) for (b, c) GL(X k ) GL(Y k ), f C c (GL k ) and g GL k. J k = 0 for k > min{a, q}, where q is the Witt index of V m. In particular, the bottom piece of the filtration (if nonzero) is: J a = Ind GL(X a) G(W n 2a ) H(V m) GL(X a) G(W n 2a ) P (Y a) (C c (GL a ) ω Wn 2a,V m 2a ).

The lemma implies that there is a natural restriction map THE HOWE DUALITY CONJECTURE 9 Hom GL(Xa) G n 2a H m (R Q(Xa)(ω W,V ), χ V sm,n,t 1 a σ π ) Hom GL(Xa) G n 2a H m (J a, χ V sm,n,t 1 a σ π ). We claim that this map is injective. To see this, it suffices to show that for all 0 k < a, Hom GL(Xa) G n 2a H m (J k, χ V sm,n,t 1 a σ π ) = 0. By the above lemma, this Hom space is equal to Hom M(Xa k,x a) G n 2a H m (Ind H(Vm) P (Y k ) χ V det Xa k λ a k C c (GL k ) ω Wn 2a,V m 2k, R Q(Xa k,x a) (χ V sm,n,t 1 a ) σ π ), where M(X a k, X a ) is the Levi factor of the parabolic subgroup of GL(X a ) stabilizing X a k. Because 1 a is generic, this Hom space can be nonzero only when a k = 1. In that case, one has: λ 1 = s m,n + 1 2 > s m,n t + 1 2 = s m,n,t, so that the above Hom space is zero even when a k = 1. Therefore we have J a 0 and 0 Hom GL(Xa) G n 2a H m (J a, χ V sm,n,t 1 a σ π ) = Hom Hm (χ W sm,n,t 1 a Θ Wn 2a,V m 2a (σ), π ). Dualizing and applying MVW, this shows that π χ W sm,n,t 1 a σ for some irreducible representation σ of H(V m 2a ) which is a subquotient of Θ Wn 2a,V m 2a (σ). Now let b a be maximal such that π χ W sm,n,t 1 b σ 0 for some irreducible representation σ 0 of H(V m 2b). We can now repeat the above argument, but going back from H(V m ) to G(W n ): (3.1) 0 Hom Gn H m (ω Wn,V m,ψ, π π ) Hom Gn H m (ω Wn,V m,ψ, π (χ W sm,n,t 1 b σ 0)) = Hom Gn GL(Y b ) H m 2b (R P (Yb )(ω Wn,V m,ψ), π χ W sm,n,t 1 b σ 0) Hom Gn GL(Y b ) H m 2b (J b, π χ W sm,n,t 1 b σ 0) = Hom GL(Xb ) G n 2b H m 2b (χ V sm,n,t 1 b ω Wn 2b,V m 2b, R Q(Xb ) (π) σ 0). Here the second injection follows by applying Lemma 3.1, with the roles of W n and V m switched and noting that the exponent λ b k (for k < b) in Lemma 3.1 satisfies: As before, this implies that λ b k = s m,n + b k 2 > 0 > s m,n,t. π χ V sm,n,t 1 b σ 0

10 WEE TECK GAN AND SHUICHIRO TAKEDA for some σ 0. By the maximality of a, we conclude that b = a. At this point, we have shown Proposition 3.2. Assume 0 π θ(π). If with a maximal (and for some σ), then π χ V sm,n,t 1 a σ π χ W sm,n,t 1 a σ for some σ and where a is also maximal for π. The main point is to prove: Proposition 3.3. Keeping the above notations, we have (i) The induced representation χ V sm,n,t 1 a σ (resp. χ W sm,n,t 1 a σ ) contains π (resp. π ) as a unique irreducible submodule. (ii) Moreover, 0 Hom Gn H m (ω Wn,V m,ψ, π π ) Hom Gn 2a H m 2a (ω Wn 2a,V m 2a, σ σ ). In particular, σ = θ Wn 2a,V m 2a (σ) by induction hypothesis. If one has this proposition, then (ii) and the induction hypothesis imply that σ is uniquely determined by σ. Then (i) and (ii) imply that θ(π) = π is irreducible, as desired. The proof of the proposition relies on the following key technical result, which we state in slightly greater generality as it may be useful for other purposes. Lemma 3.4. Let ρ be a supercuspidal representation of GL r (E) and consider the induced representation Σ = ρ a σ of G(W n ) where σ is an irreducible representation of G(W n ra ). Assume that (a) c ρ ρ; (b) σ ρ σ 0 for any σ 0. Then we have the following: (i) One has a natural short exact sequence 0 T R Q(Xra) Σ (c ρ ) a σ 0 and T does not contain any irreducible subquotient of the form ( c ρ ) a σ for any σ. In particular, R Q(Xra) Σ contains (c ρ ) a σ with multiplicity one, and R Q(Xra)Σ contains ρ a σ with multiplicity one, (ii) The induced representation Σ has a unique irreducible submodule. Proof. We shall use an explication of the Geometric Lemma of Bernstein-Zelevinsky due to Tadić [T, Lemmas 5.1 and 6.3]. (See [HM] for the metaplectic group.) Tadić s results imply that any irreducible subquotient δ σ of R Q(Xra) Σ is obtained in the following way.

THE HOWE DUALITY CONJECTURE 11 For any partition k 1 + k 2 + k 3 = ra, write the semisimplification of the normalized Jacquet module of ρ a to the Levi subgroup GL k1 GL k2 GL k3 as a sum of δ 1 δ 2 δ 3. Similarly, write the semisimplification of the normalized Jacquet module of σ to the Levi subgroup GL k2 G(W n 2ra 2k2 ) as a sum of δ 4 σ 5. Then δ is a subquotient of δ 3 c δ 1 c δ 4 whereas σ is a subquotient of δ 2 σ 5. For the case at hand, since ρ is supercuspidal, we can assume the partition of ra is of the form rk 1 + rk 2 + rk 3 = ra, and the (semisimplified) normalized Jacquet module of ρ a is the isotypic sum of ρ k 1 ρ k 2 ρ k 3. Hence we see that for any irreducible subquotient δ σ of R Q(Xra) Σ, δ is a subquotient of ρ k 3 ( c ρ k 1 ) c δ 4. Now the irreducible subquotients of T correspond to those partitions with k 2 > 0 or k 3 > 0. (Note that the case k 2 = k 3 = 0 corresponds to the closed cell in Q\G/Q, which gives the third term in the short exact sequence.) The conditions (a) and (b) then imply that δ ( c ρ ) a. This proves the statements about T in (i). Now Q(X ra ) and Q(X ra ) are conjugate in G(W ) by an element w which normalizes the Levi subgroup L(X ra ) = GL(X ra ) G(W n 2ra ), acting as the identity on G(W n 2ra ). Then one can see that w R Q(Xra) (Σ) = R Q(X ra)(σ) where the LHS is the representation of the Levi L(X ra ) obtained by twisting R Q(Xra) (Σ) by w. Hence, one deduces that R Q(Xra)(Σ) contains ρ a σ with multiplicity one. Finally, for (ii), let π Σ be any irreducible submodule. Then Frobenius reciprocity implies that the semisimplification of R Q(Xra)(π) contains ρ a σ. Thus, if Σ contains more than one irreducible submodule, the exactness of the Jacquet functor implies that R Q(Xra)(Σ) contains ρ a σ with multiplicity 2, which contradicts (i). Proof of Proposition 3.3. We shall apply this corollary with r = 1, ρ = χ V sm,n,t and Σ = χ V sm,n,t 1 a σ. Here, condition (a) holds since s m,n,t < 0, whereas condition (b) holds by the maximality of a. This proves Proposition 3.3(i). For (ii), we retake (3.1) where we now know that b = a: 0 Hom Gn H m ( ωwn,v m, π π ) Hom GL(Xb ) G n 2b H m 2b (χ V sm,n,t 1 b ω Wn 2b,V m 2b, R Q(Xb ) (π) σ ) Hom GL(Xa) G n 2a H m 2a ( χv sm,n,t 1 a ω Wn 2a,V m 2a, R Q(Xa) (χ V sm,n,t 1 a σ) σ ). Now to show Proposition 3.3 (ii), it suffices to show that the last Hom space embeds into Hom Gn 2a H m 2a (ω Wn 2a,V m 2a, σ σ ).

12 WEE TECK GAN AND SHUICHIRO TAKEDA To show the desired inclusion, set Σ = χ V sm,n,t 1 a σ in Lemma 3.4(i). Tensoring the short exact sequence with σ and then applying the functor Hom GL(Xa) G n 2a H m 2a (χ V sm,n,t 1 a ω Wn 2a,V m 2a, ), one sees that the desired injection follows from the assertion: Hom GL(Xa) G n 2a H m 2a (χ V sm,n,t 1 a ω Wn 2a,V m 2a, T σ ) = 0. But this follows from Lemma 3.4(ii) which asserts that T does not contain any irreducible subquotient of the form χ W sm,n,t 1 a σ for any σ. This finishes the proof of the proposition. It remains to prove that if θ(π 1 ) = θ(π 2 ) = π 0, then π 1 = π 2. By Proposition 2.3, this holds unless there exists t > 0 such that and This means that we may write and π 1 χ V det GL(Xt) sm,n t 2 τ1 π 2 χ V det GL(Xt) sm,n t 2 τ2. π 1 χ V sm,n,t 1 a 1 σ 1 π 2 χ V sm,n,t 1 a 2 σ 2 with a 1 and a 2 maximal (and for some σ 1 and σ 2 ). But then by Proposition 3.2 one must have a 1 = a 2 = a, where a is maximal such that π χ W sm,n,t 1 a σ for some σ. Moreover, we have seen in Proposition 3.3(ii) that θ Wn 2a,V m 2a (σ 1 ) = σ = θ Wn 2a,V m 2a (σ 2 ). By induction, we deduce that σ 1 = σ 2. We then deduce by Proposition 3.3(i) that π 1 = π 2 is the unique irreducible submodule of χ V sm,n,t 1 a 1 σ 1. This completes the proof of Theorem 1.2. References [GI] W. T. Gan and A. Ichino, Formal degrees and local theta correspondence, Invent. Math. 195 (2014), no. 3, 509-672. [GT] W. T. Gan and S. Takeda, On the Howe duality conjecture in classical theta correspondence, preprint (2014), arxiv:1405.2626. [HM] M. Hanzer, and G. Muić, Parabolic induction and Jacquet functors for metaplectic groups, J. Algebra 323 (2010), no. 1, 241 260. [H] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535 552. [K] S. S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), 229 255.

THE HOWE DUALITY CONJECTURE 13 [KR] S. S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter, Berlin, 2005, pp. 273 308. [LST] J.-S. Li, B. Sun, and Y. Tian, The multiplicity one conjecture for local theta correspondences, Invent. Math. 184 (2011), 117 124. [LnST] Y. N. Lin, B. Y. Sun and S. B. Tan, MVW-extensions of real quaternionic classical groups, Math. Z. 277 (2014), no. 1-2, 81-89. [M] A. Minguez, Correspondance de Howe explicite: paires duales de type II, Ann. Sci. Éc. Norm. Supér. 41 (2008), 717 741. [MVW] C. Mœglin, M.-F. Vignéras, and J.-L. Waldspurger, Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin, 1987. [M1] G. Muić, Howe correspondence for discrete series representations; the case of (Sp(n), O(V )), J. Reine Angew. Math. 567 (2004), 99-150. [M2] G. Muić, On the structure of the full lift for the Howe correspondence of (Sp(n), O(V )) for rank-one reducibilities, Canad. Math. Bull. 49 (2006), 578 591. [M3] G. Muić, On the structure of theta lifts of discrete series for dual pairs (Sp(n), O(V )), Israel J. Math. 164 (2008), 87 124 [M4] G. Muić, Theta lifts of tempered representations for dual pairs (Sp 2n, O(V )), Canadian J. Math. 60 No. 6 (2008), 1306-1335. [T] M. Tadić, Structure arising from induction and Jacquet modules of representations of classical p-adic groups, Journal of Algebra vol. 177 (1995), Pg. 1-33. [W] J.-L. Waldspurger, Démonstration d une conjecture de dualité de Howe dans le cas p-adique, p 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990, pp. 267 324. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 E-mail address: matgwt@nus.edu.sg Mathematics Department, University of Missouri, Columbia, 202 Math Sciences Building, Columbia, MO, 65211 E-mail address: takedas@missouri.edu