## Geometry Seminar - Abstracts

### Talk

Tuesday 6 June 2017, 16:00-17:00 in HG03.085

**Peter Hochs** (University of Adelaide)

*K-types of tempered representations*

### Abstract

Tempered representations of a semisimple Lie group G are
the irreducible unitary representations one needs in the Plancherel
decomposition of L^2(G). They are relevant to harmonic analysis
because of this, and also occur in the Langlands classification of
the larger class of admissible representations. If K< is a maximal
compact subgroup, then there is a considerable amount of information
in the restriction of a tempered representation to K. In joint work
with Yanli Song and Shilin Yu, we give a geometric realisation of
these restrictions as indices of Dirac operators on certain
homogeneous spaces/coadjoint orbits of G. As an application, we
find a geometric expression for the decomposition into irreducibles
of the restriction of the representation to K.

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