# The classification theorem and Harish-Chandra modules for the dual group

• Dan Barbasch
• David A. VoganJr.
Chapter
Part of the Progress in Mathematics book series (PM, volume 104)

## Abstract

We assemble here the results we have established linking the representation theory of real forms of G to Harish-Chandra modules for certain subgroups of G. This is for the most part a reformulation of the results of [56]. We work therefore in the setting of Proposition 6.24. Specifically, we fix always a weak E-group G Γ, and a semisimple orbit
$$\mathcal{O} \subset {}^{ \vee }\mathfrak{g}$$
(21.1)(a)
We fix a canonical flat Λ ⊂ O, or sometimes just a point λ ∈ O. In either case we get as in (6.6) a well-defined semisimple element
$$e\left( \Lambda \right) = \exp \left( {2\pi i} \right) \in {}^{ \vee }G\;\;\left( {\lambda \in \wedge } \right)$$
(21.1)(b)
and subgroups
$$P\left( \wedge \right) \subset {}^{ \vee }G{\left( \wedge \right)_0}$$
(21.1)(c)
The reductive group G(Λ)0 is the identity component of the centralizer of e(Λ). Its parabolic subgroup P(Λ) is the stabilizer of Λ in the action of ∨G on P(O) (cf. (6.10)). Define P (Λ)0 as in Proposition 6.24 to be the G(Λ)0-orbit of Λ. then
$$\mathcal{P}{\left( \wedge \right)^0} \simeq {}^{ \vee }G{\left( \wedge \right)_0}/P\left( \wedge \right),$$
(21.1)(d)
a flag manifold for ∨G(Λ)0. (This space will play the rôle of Y in Chapter 20.) Recall from (8.1) the ideal
$${I_{{\mathcal{P}{{\left( \wedge \right)}^{ \circ }}}}} \subset U\left( {{}^{ \vee }\mathfrak{g}\left( \wedge \right)} \right),$$
(21.1)(e)
and from Proposition 20.4 the nilpotent orbit
$${\mathcal{Z}_{{\mathcal{P}{{\left( \wedge \right)}^0}}}} \subset {}^{ \vee }\mathfrak{g}\left( \wedge \right)*.$$
(21.1)(f)

### Keywords

Filtration Manifold