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

  • Jeffrey Adams
  • 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 

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Jeffrey Adams
    • 1
  • Dan Barbasch
    • 2
  • David A. VoganJr.
    • 3
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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