Reductive Dual Pair Research Articles

Splittings of the metaplectic covers of some reductive dual pairs

Splittings of the metaplectic covers of some reductive dual pairs

A Cauchy Harish-Chandra integral, for a real reductive dual pair

A Cauchy Harish-Chandra integral, for a real reductive dual pair

The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups

The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups

Capelli identities for classical Lie algebras

We extend the Capelli identity from the Lie algebra \(\frak{gl}_N\) to the other classical Lie algebras \(\frak{so}_N\) and \(\frak{sp}_N\). We employ the theory of reductive dual pairs due to Howe.

Quadratic base change for 𝑝-adic 𝑆𝐿(2) as a theta correspondence I: Occurrence

The local theta correspondence is considered for reductive dual pairs ( S L 2 ( F ) , O ( F ) ) \left ( \mathrm {SL}_{2}\left ( F\right ) ,\mathrm {O}\left ( F\right ) \right ) where F F is a p p -adic field of characteristic zero and O \mathrm {O} is the orthogonal group attached to a quaternary quadratic form with coefficients in F F and of Witt rank one over F F . It is shown that certain representations of S L 2 ( F ) \mathrm {SL}_{2}\left ( F\right ) occur in the correspondence.

Exceptional Howe Correspondences Over Finite Fields

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We consider the restriction of the reflection representation to various reductive dual pairs in exceptional groups, and determine the correspondence of generic representations.

Howe Correspondence for Real Unitary Groups

Roger Howe proved that for any reductive dual pair (G,G′) in the symplectic groupSp(2n,R), there is a one-to-one correspondence of irreducible admissible representations of some two-fold covers ofGandG′. We determine this correspondence explicitly for dual pairs of the form (U(p,q),U(r,s)) withr+s=p+q, and describe it in terms of Langlands parameters. In this case, the correspondence may be understood in a natural way as a correspondence of representations of the linear groups, rather than the appropriate covers. We show that every irreducible admissible representation ofU(p,q) occurs in the correspondence with precisely one unitary group of equal rank. This result verifies a conjecture of Harris, Kudla, and Sweet, who investigated the correspondence for unitary groups of equal size overp-adic fields. The correspondence of discrete series representations was determined by J.-S. Li. For induced representations, the correspondence is obtained in a natural way from the corresponding discrete series on the respective Levi factors of the parabolic subgroups ofU(p,q) andU(r,s). Generalizing a result of Li, we show that under the correspondence representations with nonzero cohomology are matched in an interesting way, with unitarity not necessarily preserved. The proof uses the induction principle which is due to Kudla, and an argument involvingK-types and the space of joint harmonics (Howe).

A result on the Gelfand-Kirillov dimension of representations of classical groups

Let ( G , G ′ ) (G,G’) be the reductive dual pair ( O ( p , q ) , S p ( 2 n , R ) ) (O(p,q),Sp(2n,\mathbb {R})) . We show that if π \pi is a representation of S p ( 2 n , R ) Sp(2n,\mathbb {R}) (respectively O ( p , q ) O(p,q) ) obtained from duality correspondence with some representation of O ( p , q ) O(p,q) (respectively S p ( 2 n , R ) Sp(2n,\mathbb {R}) ), then its Gelfand-Kirillov dimension is less than or equal to ( p + q ) ( 2 n − p + q − 1 2 ) (p+q)(2n-\frac {p+q-1}{2}) (respectively 2 n ( p + q − 2 n + 1 2 ) 2n(p+q-\frac {2n+1}{2}) ).

Eigenfunctions of the Weil representation of unitary groups of one variable

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the p-adic case when p is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke L-functions in separate papers. We also recover Moen’s results on when a local theta lifting from U(1) to itself does not vanish. 0. Introduction and Notation Let E/F be a quadratic extension of local fields. If (V, ( , )) is an Hermitian space over E, and (W, 〈 , 〉) is a skew-Hermitian space over E, the unitary groups G = G(W ) and G′ = G(V ) form a reductive dual pair in Sp(W), where W = V ⊗W has the symplectic form 12 trE/F ( , ) ⊗ 〈 , 〉 over F . According to a well-known result, this dual pair splits in the metaplectic cover of Sp(W), and thus has a Weil representation ω. We consider the very special case in which dimE V = dimEW = 1. In this case, G ∼= G′ = U(1) = E is compact and abelian, where E is the kernel of the norm map N : E∗ −→ F ∗. So irreducible representations of G are just characters, and the Weil representation has a direct sum decomposition:

Two reductive dual pairs in groups of typeE

Two reductive dual pairs in groups of typeE

The Oscillator Character Formula, for isometry groups of split forms in deep stable range

Let G˜ and G˜ ′ be a reductive dual pair of the type mentioned in the title, with G˜ the smaller member. Let Π and Π′ be unitary representations of G˜,G˜ ′ which occur in Howe’s correspondence. We express the distribution character of Π′ in terms of the character of Π via an explicit integral kernel operator.

Tilting modules for classical groups and howe duality in positive characteristics

We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote by G1 (m) the group of rank m from the first series and by G2 (n) the group of rank n from the second series. For any pair (Ga(m),G2(n)) we construct the Gl(m) x G2(n)-module M(m,n). The construction of M(m, n) is independent of characteristic; for char k -~ 0, the actions of Gl(m) and G2(n) on M(m,n) form a reductive dual pair in the sense of Howe. We prove that M(m,n) is a tilting Gl(m)- and G2(n)-module and that Endvl(m) M(m, n) is generated by G2(n) and vice versa. The existence of such a module provides much information about the relations between the category /(:l(m, n) of rational G1 (rn)-modules with highest weights bounded in a certain sense by n and the category /C2(m,n) of rational G2(n)-modules with highest weights bounded in the same sense by rn. More specifically, we prove that there is a bijection of the set of dominant weights of Gl(m)-modules from/Cl(m, n) to the set of dominant weights of G2(m)-modules from K:2(m, n) such that Ext groups for induced G)(m)-modules from/Cl(m, n) are isomorphic to Ext groups for corre- sponding Weyl modules over G2(n). Moreover, the derived categories DblCI (m, n) and DblC2(m, n) appear to be equivalent. We also use our study of the modules M(m, n) to find generators and relations for the algebra of all G-invariants in A'(V "1 @(V*)'2), where G =GL,,, Sp2,~, Om and V is the natural G-module.

The duality correspondence of infinitesimal characters

We determine the correspondence of infinitesimal characters of representations which occur in Howe’s Duality Theorem. In the appendix we identify the lowest K-types, in the sense of Vogan, of the unitary highest weight representations of real reductive dual pairs with at least one member compact.

Reductive Dual Pair Correspondence for Complex Groups

Reductive Dual Pair Correspondence for Complex Groups

Unipotent Representations and Reductive Dual Pairs over Finite Fields

Unipotent Representations and Reductive Dual Pairs over Finite Fields

Unipotent representations and reductive dual pairs over finite fields

Consider the representation correspondence for a reductive dual pair ( G 1 , G 2 ) ({G_1},{G_2}) over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of "first occurrence", the representation correspondence takes a cuspidal unipotent representation of G 1 {G_1} to one of G 2 {G_2} . This should serve as a fundamental case in studying the correspondence in general over both finite and local fields.

Supercuspidal representations and the theta correspondence. II. 𝑆𝐿(2) and the anisotropic 𝑂(3)

A parametrization is given of the local theta correspondence attached to the reductive dual pair ( SL 2 ( F ) , O ( F ) ) ({\text {SL}}_2(F),\;{\text {O}}(F)) where F F is a nonarchimedean local field of odd residual characteristic and O {\text {O}} is the orthogonal group of a ternary quadratic form which is anisotropic over F F . The parametrization is in terms of inducing data. Various lattice models of the oscillator representation are used.

On the Singular Rank of a Representation

Consider the reductive dual pair $({\text {S}}{{\text {p}}_{2n}},{{\text {O}}_{p,q}})$. We prove that if $\pi$ is a representation of ${\text {S}}{{\text {p}}_{2n}}$ coming from duality correspondence with some representation of ${{\text {O}}_{p,q}}$ then the wave front set of $\pi$ has rank $\leq p + q$. For $p + q < n$ this implies a result stated (but not proved) by Howe.

On the singular rank of a representation

Consider the reductive dual pair ( S p 2 n , O p , q ) ({\text {S}}{{\text {p}}_{2n}},{{\text {O}}_{p,q}}) . We prove that if π \pi is a representation of S p 2 n {\text {S}}{{\text {p}}_{2n}} coming from duality correspondence with some representation of O p , q {{\text {O}}_{p,q}} then the wave front set of π \pi has rank ≤ p + q \leq p + q . For p + q &gt; n p + q &gt; n this implies a result stated (but not proved) by Howe.

Transcending classical invariant theory

Let Sp2, (R) = Sp2, = Sp be the real symplectic group of rank n. Let Sp denote the two-fold cover of Sp. There is a unitary representation, constructed by Shale [Sh] and Weil [WA], of Sp that is of considerable interest in the theory of automorphic forms. We denote this representation by co and call it the oscillator representation. The purpose of this paper is to establish about C? a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co. We begin by formulating the result. Let E be a reductive subgroup of Sp, and let E denote the inverse image of E in Sp. Denote by R(E) the set of infinitesimal equivalence classes of continuous irreducible admissible representations of E on locally convex spaces. Let co' be the smooth representation of Sp associated to co. Let co be realized on a Hilbert space % and let the subspace of smooth vectors, on which space co' is defined, be written . Denote by R9(E , co) the set of elements of (E) which are realized as quotients by co' (E)-invariant closed subspaces of t'g . Consider a reductive dual pair (G, G') C Sp [H2]. It is not hard to show that G and G' commute with one another. Hence, we may regard co'I G. G' as a representation of Gx G' . It is well known [F] that R (Gx G') S9 '(G) xR9(G') . The identification associates to p E R(G) and p' E i(G') the tensor product p X p'. (We note that p X p' is not defined as a topological vector space; nevertheless, the infinitesimal equivalence class of p op' is well defined.) Select p X p' E (G x G') . Suppose that, in fact, p? p' E 9(G_ G', c) . Then clearly p E R (G, c) and p' E ,9(G', c) . Hence, ( G' , co) defines the graph of a correspondence between certain subsets of R9(G, cv) and ?J(G', co) . In fact, the situation is quite precise.