A Fixed Point Formula and Harish-Chandra’s Character Formula

Peter Hochs  (University of Adelaide)

14:45-15:45, Nov 1, 2017   Science Building A1510

Abstract:

The fixed point formula of Atiyah, Segal and Singer is an expression for the equivariant index of an elliptic operator on a compact manifold, in terms of the fixed point set of the group action on the manifold. Atiyah and Bott showed that a special case of this fixed point formula is Weyl’s character formula in the representation theory of compact Lie groups. In recent work with Hang Wang, we obtained a generalisation of the fixed point formula to operators on noncompact manifolds acted on cocompactly by noncompact semisimple Lie groups. A special case of this formula is Harish-Chandra’s character formula for the discrete series, which is a generalisation of Weyl’s character formula to noncompact Lie groups. The link between K-theory and representation theory used here is a trace map on a dense subalgebra of the reduced C^*-algebra of a semisimple Lie group, generalising the von Neumann trace.

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