Unitary Representations of Reductive Lie Groups. (AM-118)

Vogan, David A.

doi:10.1515/9781400882380

Cited by 170 publications

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“…For the group GL (2, R), the Speh representations δ (2, k) are precisely the discrete series. In this case the result δ (2, k) = δ (2, k) is well known (see 1.4.7 in [Vog81]). The general Speh representation δ (2m, k) is the unique irreducible quotient (Langlands quotient) of…”

Section: Computation Of Adduced Representations For (Almost) All Unitmentioning

confidence: 87%

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Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Gourevitch

Sahi

2012

Sel. Math. New Ser.

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Abstract. In this paper we study irreducible unitary representations of GLn (R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, for both smooth and K-finite vectors, thereby generalizing results of Kostant, Matumoto and others.Our second result relates the annihilator to the sequence of adduced representations, as defined in this setting by one of the authors. Based on those results, we suggest a new notion of rank of a smooth admissible representation of GLn(R), which for unitarizable representations refines Howe's notion of rank.Our third result computes the adduced representations for (almost) all irreducible unitary representations in terms of the Vogan classification.We also indicate briefly the analogous results over complex and p-adic fields.

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Section: Computation Of Adduced Representations For (Almost) All Unitmentioning

confidence: 87%

“…Since G n /P α is compact, one can define π 1 × · · · × π k analogously in the C ∞ category. We refer the reader to [Vog81] for details. We will occasionally need to consider this case especially in connection with complementary series construction in the next section and elsewhere.…”

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confidence: 99%

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

Gourevitch

Sahi

2012

Sel. Math. New Ser.

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“…They follow from Proposition 1.4, which describes the representation theory of nonconnected compact Lie groups. The first part of Proposition 1.4 -the bijection between irreducible G-representations and certain irreducible representations of N G (T, C) -was proven by Takeuchi [Ta,Theorem 4], and is also stated in [Vo,Theorem 1.17]. Since their notation is very different from that used here, we have found it simplest to keep our proof, rather than just refer to [Ta].…”

Section: Section 1 Detection Of Charactersmentioning

confidence: 92%

The representation ring of a compact Lie group revisited

Oliver

1998

Comment. Math. Helv.

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Abstract. We describe a new construction of the induction homomorphism for representation rings of compact Lie groups: a homomorphism first defined by Graeme Segal. The idea is to first define the induction homomorphism for class functions, and then show that this map sends characters to characters. This requires a detection theorem -a class function of G is a character if its restrictions to certain subgroups of G are characters -which in turn requires a review of the representation theory for nonconnected compact Lie groups.Mathematics Subject Classification (1991). Primary: 22E45; secondary: 55R40, 55R50.

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“…Suppose that G is connected and let À be the differential of x-If A + p is nonsingular and antidominant, the Dolbeault cohomology has the same underlying Harish-Chandra module as a certain discrete series module [5], so its underlying Harish-Chandra module is the standard Zuckerman module I q (G/H,E x ) [8]. One passes to general A by tensoring [1,10].…”

Section: Indication Of Proofmentioning

confidence: 99%

“…This is a tensoring argument. The point is that standard Zuckerman modules, and the maximal globalization functor, both behave well with respect to tensoring [7,8,10], so we need only show that the local cohomology Hp~r(X, 0(E X )) also behaves well. Arguing as in [6], that comes down to checking that the center Z(g) of the universal enveloping algebra of g acts on Hp~r (X, 0 (E x )) by the character that Harish-Chandra denotes X\+p-Local cohomology can be calculated from a relative covering using cocycles with coefficients that are holomorphic sections of E x , so this comes down to showing that Z(g) acts on 0(E X ) by XA+P-But a germ in 0(E X ) is represented by a germ in (C°°{Gc) <8>E x <g> An*) H ' n , and the Harish-Chandra homomorphism Z(g) -> U(f)) gives the result.…”

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confidence: 99%