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Unipotent representations and unitarity. (English) Zbl 0629.22007
Non-commutative harmonic analysis and Lie groups, Proc. Int. Conf., Marseille-Luminy 1985, Lect. Notes Math. 1243, 73-85 (1987).
[For the entire collection see Zbl 0607.00006.]
This paper describes a classification of the unitary dual of a complex reductive Lie group of classical type. By results of Harish-Chandra, this problem is equivalent to a classification of irreducible (\({\mathfrak g},K)\)- modules admitting a \({\mathfrak g}\)-invariant positive Hermitian form. Suppose that an irreducible module (\(\pi\),V) admits a non-degenerate form \(<, >\). Consider, for each K-type \(\gamma\), the \(\pm\)-eigenspaces \([\gamma:V]_{\pm}\) of \(<, >\). A formal K-character \([V]_{\pm}=\sum \dim [\gamma:V]_{\pm} \chi_{\gamma}\) is called a signature, and V is unitary if and only if \([V]_-\) is 0.
Let \(P=MN\) be a real parabolic subgroup. Two standard ways of constructing unitary representations are unitary induction from P, for which there is a natural unitary structure given via \(L^ 2(K)\), and formation of complementary series, i.e., representations for which a positive-definite inner product may be defined by mapping into the contragredient representation by standard Kunze-Stein intertwining operators. Signatures are well-behaved under unitary induction and complementary series. One would like to construct a finite set \(U_ 0\) of unitary representations from which the unitary spectrum of G may be obtained by these two methods.
A set \(U_ 1\) of special unipotent representations is defined for integral infinitesimal character by the author and D. A. Vogan [Ann. Math., II. Ser. 121, 41-110 (1985; Zbl 0582.22007)], and an analogous set \(U_{1/2}\) is defined here for half-integral infinitesimal character. The families of representations introduced by Barbasch and Vogan generalize families introduced by J. Arthur [Lect. Notes Math. 1041, 1-49 (1984; Zbl 0541.22011)] and are described in terms of dual group data.
A difficulty not present in previously know cases is to show that the representations in \(U_ 0\) are unitary. These representations in general are not unitarily induced, or complementary series, or endpoints of complementary series.
Let G be a complex classical group. The main result of this paper is the following: A (\({\mathfrak g},K)\)-module (\(\pi\),V) is unitary if and only if V is obtained by unitary induction and/or complementary series from a representation in \(U_ 0=U_ 1\cup U_{1/2}\).
Reviewer: C.D.Keys

22E30 Analysis on real and complex Lie groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22D10 Unitary representations of locally compact groups