Theta lifting for discrete series representations of real unitary groups

Abstract

We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.Notation. For any representation π, we denote by π ∨ the contragredient of π and by π the complex conjugate of π. For any real reductive group G, we work with the category of (g, K)-modules unless otherwise specified, where g is the complexified Lie algebra of G and K is a maximal compact subgroup of G. Thus by abuse of terminology, we usually mean a (g, K)-module by a representation of G. Local theta liftingIn this section, we review the notion of local theta lifting. We follow the convention in [11,12], which is different from that in [24,15].2.1. Hermitian and skew-Hermitian spaces. Let F be a local field of characteristic zero. Let E be an étale quadratic algebra over F , so that E is either F × F or a quadratic extension of F . We denote by c the nontrivial automorphism of E over F . Let Tr E/F and N E/F be the trace and norm maps from E to F , respectively. Let ω E/F be the (possibly trivial) quadratic character of F × associated to E/F by local class field theory, so that Ker(ω) be the determinant of the matrixNote that ǫ(V ) depends on δ if ε = −1, E = F × F , and n is odd. We denote by U(V ) the unitary group of V , i.e. U(V ) = {g ∈ GL(V ) | gv, gw V = v, w V for all v, w ∈ V }.

Theta lifting for tempered representations of real unitary groups

“…Setup. We follow the convention in [7]. Let V be an m-dimensional Hermitian space over C and W an n-dimensional skew-Hermitian space over C. Let (r, s) and (p, q) be the signatures of V and W , respectively, so that r + s = m and p + q = n. As in §2.1, we identify U(V ) and U(W ) with U(r, s) and U(p, q), respectively.…”

“…In this paper, which is a sequel to [7], we continue to study the theta lifting introduced by Howe [6]. We focus on the case of real unitary groups and consider the theta lifting from U(p, q) to U(r, s).…”

“…When p + q = r + s or p + q = r + s ± 1, Paul [17,18] generalized the result of Li [13] and completely determined θ r,s (π) for arbitrary π. More generally, Atobe [1] gave a criterion for the nonvanishing of θ r,s (π) when π is tempered but for arbitrary p, q, r, s. In our previous paper [7], we described θ r,s (π) explicitly for any discrete series representation π. The purpose of this paper is to generalize it to the case of tempered representations.…”

“…The purpose of this paper is to generalize it to the case of tempered representations. For this, we first prove the following generalization of [7] to the case of (limits of) discrete series representations (see Theorem 3.1 for more details).…”

“…We now give some details of the proof of Theorem 1.1. As in our previous paper [7], the proof is global and relies on Arthur's multiplicity formula [16,8]. We refer the reader to [7, §1] for the explanation of the global-to-local argument and only mention the key ingredient here.…”