On Doubling Construction for Real Unitary Dual Pairs

Konno, Takuya; Konno, Kazuko

doi:10.2206/kyushujm.61.35

Cited by 8 publications

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“…Given our choice of the data (χ V , χ W , ψ), the assertion follows from [23,Theorem 5.4]. We remark that the convention in [23] so that the tensor product representation µ 1 ⊗ µ 2 contains µ. Let µ ′ be the irreducible representation of K ′ corresponding to µ.…”

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confidence: 99%

Theta lifting for discrete series representations of real unitary groups

Ichino

2020

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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 }.

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confidence: 99%