Parametrizing Representations of K after David Vogan

Computing K-type multiplicities in standard representations (after Vogan)

The first part of these notes is an updating (and correction) of [Khat] and is devoted to a paremetrization of the irreducible representations of the (generally disconnected) maximal compact subgroup of a real group in Harish-Chandra’s class. The second part describes how to use that paremetrization of the first part to compute K-type multiplicities in standards modules. (By Frobenius reciproci...

A parametrization of K̂ (after Vogan)

Let G be a real reductive group in Harish-Chandra’s class. It may be instructive and useful to weaken that hypothesis, but we content ourselves with it here. Let K be the maximal compact subgroup of G. The point of these notes is to recall a parametrization of K̂ (i.e. equivalence classes of irreducible representations of K) due to David Vogan. Note that even if G is algebraic, the description o...

Dirac Cohomology, Unitary Representations and a Proof of a Conjecture of Vogan

The main result in this paper is a proof of Vogan’s conjecture on Dirac cohomology. In the fall of 1997, David Vogan gave a series of talks on the Dirac operator and unitary representations at the MIT Lie groups seminar. In these talks he explained a conjecture which can be stated as follows. Let G be a connected semisimple Lie group with finite center. Let K be the maximal compact subgroup of ...

Representations with Scalar K-types and Applications

We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representations to representations with scalar K-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan.

Cohomology and Group Representations