# zbMATH — the first resource for mathematics

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

##### MSC:
 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