On the unitary dual of the classical Lie groups, representations of \(Sp(p,q)\). (English) Zbl 0893.22008

Salamanca-Riba, Susana A.

Proc. Am. Math. Soc. 125, No. 10, 3107-3117 (1997).

Suppose \(G\) is a real reductive group with complexified Lie algebra \({\mathfrak g}\), \(T\) a Cartan subgroup

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in a maximal compact subgroup

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\(K\) of \(G\), whose complexified Lie algebra is \({\mathfrak t}\). The author is concerned with the following conjecture (Vogan-Zuckerman): Let \(X\) be a unitary Harish-Chandra module

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of \(G\) with infinitesimal character \(\gamma\) satisfying (1) \((\gamma-\rho,\alpha)\geq 0\) for all positive roots \(\alpha\) in \(\Delta ({\mathfrak G}, {\mathfrak t})\) and \(\rho\) half the sum of these positive roots. Then \(X\) is an \(A_q (\lambda)\)-module (here \(A_q(\lambda)\) are the modules with cohomology).
The author proves that if \(G=Sp(p,q)\) and \(X\) is a Harish-Chandra module with a Hermitian form

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\(\langle , \rangle\) and infinitesimal character satisfying (1), then either (a) \(X\) is an \(A_q (\lambda)\)-module or (b) there are two \(K\)-types \((\mu,V_\mu)\) and \((\eta, V_\eta)\) of \(X\) such that the Hermitian form \(\langle , \rangle\) on \(V_\mu \oplus V_\eta\) is indefinite.

Reviewer: T.Venkataramana (Bombay)

MSC:

22E46	Semisimple Lie groups and their representations
22D10	Unitary representations of locally compact groups
22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
20G05	Representation theory for linear algebraic groups

Keywords:

Real semisimple Lie groups; unitary representations; Zuckerman modules; real reductive group; Lie algebra

References:

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