The Theta Correspondence for Similitudes

In this paper we investigate the theta correspondence for similitudes over a nonarchimedean local field. We show that the two main approaches to a theta correspondence for similitudes from the literature are essentially the same, and we prove that a version of strong Howe duality holds for both constructions. Suppose that k is a nonarchimedean local field, X is a finite dimensional nondegenerate symmetric bilinear space over k, and Y is a finite dimensional nondegenerate symplectic bilinear space over k. Let p be the projection from the metaplectic cover of Sp(X ⊗k Y ) to Sp(X ⊗k Y ). Fix a nontrivial additive character of k, and let r be the corresponding smooth Weil representation of the metaplectic cover of Sp(X ⊗k Y ). Then the restriction of r to p−1(O(X))p−1(Sp(Y )) defines a correspondence between the smooth admissible duals of p−1(O(X)) and p−1(Sp(Y )). When the residual characteristic of k is odd, this correspondence satisfies strong Howe duality. Conceivably, r might be used to construct a representation that involves similitudes of X and Y , so that a correspondence between the smooth admissible duals of some covers of GO(X) and GSp(Y ) could be defined and analyzed. We consider two such constructions. The first extends the restriction of r to p−1(O(X))p−1(Sp(Y )) to a representation ω of a larger group involving similitudes. Apparently, ω was first implicitly introduced in [S], and first explicitly considered in [HK]. The second construction induces the restriction of r to p−1(Sp(Y )) to obtain a representation Ω that involves similitudes. As far as we know, Ω first appeared in [SA] in the case of finite fields, and in [PSS] in the case of nonarchimedean local fields. In this paper we show that the two approaches are essentially the same, and that when the residual characteristic of k is odd, a natural version of strong Howe duality holds for the associated correspondence. The two constructions of a correspondence for similitudes already have proven to be valuable tools in automorphic representation theory and its applications. Many examples have been considered. As a sample, see [JL], [S], [Co], [So], [HK], and [HST]. In particular, During the period of this work the author was a Research Associate with the NSF 1992–1994 special project Theta Functions, Dual Pairs, and Automorphic Forms at the University of Maryland, College Park. 1 the correspondence for similitudes should be useful in the investigation of Shimura varieties. It is similitudes that occur in the theory of Shimura varieties, not isometries. To give a detailed account of our results we need more notation. Let H = GO(X). If dimkX is even, let G ′ = GSp(Y ). If dimkX is odd, let G ′ be a certain two-fold cover of GSp(Y ). For g ∈ G′, let λ(g) be the similitude factor of the projection of g to GSp(Y ). Similarly, let λ(h) be the similitude factor of h ∈ H. Let G be the subgroup of g ∈ G′ such that λ(g) ∈ λ(H). Also, let G1 and H1 be the subgroups of g ∈ G and h ∈ H such that λ(g) = 1 and λ(h) = 1, respectively. Then ω is a representation of the group R = {(g, h) ∈ G′ ×H:λ(g) = λ(h)}, and Ω is a representation of G′ ×H. See sections 2 and 3 for precise definitions. In section 1, following [R], [MVW] and [KR], we define Howe duality, multiplicity preservation, and strong Howe duality. In Proposition 1.1 we show that, taken together, Howe duality and multiplicity preservation are equivalent to strong Howe duality. This result is well known to experts. We also state another result that is used in the last section. In sections 2 and 3 we carefully construct and relate ω and Ω. After recalling some basic facts about the metaplectic covers of similitude groups and splittings that follow from [Ra], [B] and [K], we show that the definitions of ω and Ω depend on the same fundamental identity in the metaplectic cover. Using the identity, we also prove that Ω is obtained from ω via compact induction: Ω ∼= c-Ind ′×H R ω. This is the first main result. In section 4 we study the correspondence defined by ω, assuming that strong Howe duality holds for the usual correspondence for isometries. After the key observation that R only involves G, we investigate whether the condition HomR(ω, π ⊗C τ) 6= 0 for π ∈ Irr(G) and τ ∈ Irr(H) gives rise to a correspondence satisfying the analogues of Howe duality and multiplicity preservation. Given that π and τ correspond, we show in Lemma 4.2 that the equivalence classes of irreducible constituents of π|G1 and τ |H1 are paired via the usual correspondence for isometries, so that in particular the numbers of equivalence classes are the same; we also show that π|G1 is multiplicity free if and only if τ |H1 is multiplicity free. This suggests that we restrict attention to representations with multiplicity free restrictions to G1 and H1, which we do. Then in Theorem 4.4 we prove that the analogues of Howe duality and multiplicity preservation hold. This is the second main result. In sections 5 and 6 we consider the consequences of section 4 for Ω. First, we define a natural G×H subrepresentation Ω of Ω such that Ω ∼= c-IndG×H R ω. 2 We prove that strong Howe duality holds for Ω with respect to the multiplicity free elements of Irr(G) and Irr(H) when strong Howe duality holds for the usual correspondence for isometries. Next, we consider whether the condition HomG′×H(Ω, π ⊗C τ) 6= 0 for π ∈ Irr(G′) and τ ∈ Irr(H) defines a correspondence satisfying Howe duality in the case G 6= G′. When G 6= G′, dimkX is even and the residual characteristic of k is odd, using that Ω satisfies Howe duality, we give an equivalent condition based on Proposition 1.2. This condition is called the theta dichotomy in [HKS]. Using the condition, we show in this case that Howe duality does not hold for Ω in the stable range. We also point out that when dimkX ≤ dimk Y , that is, when the theta dichotomy is expected to hold, strong Howe duality for Ω is expected to hold. This is the final main result. The main previous general work in this area is, as far as we know, [B]. However, the approaches we consider, and the approach of [B], are different. The paper [B] investigates whether the compactly induced representation of r to the metaplectic cover of GSp(X ⊗k Y ), and the inverse images of GO(X) and GSp(Y ) in the metaplectic cover are analogues of r, p−1(O(X)) and p−1(Sp(Y )). Complications arise since the inverse images of GO(X) and GSp(Y ) are poor analogues of p−1(O(X)) and p−1(Sp(Y )). In particular, the elements of the inverse images of GO(X) and GSp(Y ) do not always commute. As a consequence, [B] defines and considers a correspondence between sets of representations of the inverse images of GO(X) and GSp(Y ) rather than representations. What we call ω and Ω are not what are called ω and Ω in [B]. Still, we draw heavily on [B] for results about the metaplectic group for similitudes. We hope these results will be useful in several ways. The simple connection between ω and Ω should allow results about one representation to be applied to the other. As in [R], in the case of isometries, Howe duality and multiplicity preservation for ω will have fundamental consequences for the global correspondence of automorphic representations defined by ω. The results of section 4 are actually more general then stated above. The abstraction of section 4 may be useful in other contexts. These results may be useful in understanding examples in the literature. We believe that the hypotheses for these results are not too restrictive for many applications. Let us consider some of the hypotheses. First, Howe duality and multiplicity preservation for ω require that Howe duality and multiplicity preservation hold for the correspondence for isometries. By a theorem of J.-L. Waldspurger [W] this requirement is satisfied if the residual characteristic of k is odd. It is conjectured to hold when the residual characteristic even. Second, in the above description of Howe duality and multiplicity preservation for ω we need that representations have multiplicity free restrictions to G1 and H1. In applications, one often begins with a representation of G or H and assumes or proves that there exists a representation of the other group so that the two representations correspond with respect to ω. By Lemma 4.2, if the initial representation has multiplicity free restriction, then so will the corresponding representation: having multiplicity free restriction is contagious. Thus, in the case when the initial representation has multiplicity 3 free restriction, Howe duality and multiplicity preservation may be applied. Often, one can verify that the initial representation has multiplicity free restriction. For example, when dimkX is two or four, cases that have many important applications beginning with [JL] and [S], all elements of Irr(H) have multiplicity free restrictions [HPS]. Also, if the initial representation is generic then it has multiplicity free restriction. On the other hand, if the initial representation does not have multiplicity free restriction, then by Lemma 4.2 neither will the corresponding representation. In this way new examples of representations without multiplicity free restrictions might be constructed. Finally, we make some remarks about Ω. For the purposes of this paper, it seems that ω is more natural than Ω. In contrast to ω, the definition of Ω gives no indication that Howe duality should not always hold, or that it should hold for Ω. However, in other contexts it may be useful to consider Ω. One example might be seesaw reciprocity. Moreover, Ω has a definition independent of ω. This definition gives the so called extra variable Schrödinger model, which has been useful in some situations. The representations Ω and Ω are not artificial objects. The problem of Howe duality for Ω is not completely solved. This problem is fundamental and important. Many applications involve G′. See, for example, [HST], where the case dimkX = dimk Y = 4 is used. There the correspondence between Irr(G ′) and Irr(H) defined by ω is employed. By our results relating ω and Ω, this is equivalent to the consideration of the correspondence between Irr(G′) and Irr(H) defined by Ω. Also, the problem is deep. As mentioned above, in the case dimkX is even, G is a proper subgroup of G′ and dimkX ≤ dimk Y , Howe duality for Ω is equivalent to theta dichotomy. I would like to thank S.S. Kudla for his generous help and useful comments. In particular, he kindly allowed me access to his preprints [K], [KR] and [HKS], and it was his suggestion that Proposition 3.5 might be true. We use the following notation. Let J be a group of td-type, as in [C]. This means that J is a topological group and every neighborhood of the identity element of J contains a compact open subgroup. We will often explicitly assume that J has a countable basis. In this case, Shur’s lemma holds. See [C]. Let Irr(J) be the set of equivalence classes of smooth admissible irreducible representations of J . If π ∈ Irr(J) then π∨ ∈ Irr(J) is the contragredient representation of π. A character of J is a continuous homomorphism from J to C×. The notation for induction will be as in section 1.8 of [C]. In section 4 we will also use the notation of [GK]. In particular, if L is a closed subgroup of J , π ∈ Irr(L) and g ∈ J , then gπ ∈ Irr(L) is the representation with the same space as π and action defined by (gπ)(h) = π(g−1hg), and Jπ is the subgroup of g ∈ J such that gπ ∼= π. Throughout the paper k is a nonarchimedean local field of characteristic zero. Finally, let ( , )k denote the Hilbert symbol of k. 1. Howe duality and multiplicity preservation. For the convenience of the reader, we recall the statements of Howe duality and multiplicity preservation as in [R], [MVW] and [KR]. Let A and B be groups of td-type, with countable bases. Let (ρ,U) be a smooth 4 representation of A×B. Let π ∈ Irr(A). Define