Orbital integrals in index theory and K-theory

Peter Hochs  (University of Adelaide)

11:00-12:00 Nov 20, 2018   Science Building A510

Abstract:

An orbital integral of a function on a group G is its integral over a conjugacy class in G. If such an orbital integral defines a continuous functional on a convolution algebra A(G) of functions on G, then it is a trace on that algebra. If the conjugacy class consists of just the identity element, this is the classical von Neumann trace. In general, such a trace induces a map on the K-theory of A(G) with values in the complex numbers. If A(G) is dense in the reduced or full group C*-algebra of G and closed under holomorphic functional calculus, then this gives a map on the K-theory of that group C*-algebra. It has turned out in recent years that such maps are useful tools for studying elements of these K-theory groups. This is true in particular for K-theoretic indices of G-equivariant elliptic operators. Index formulas for the numbers obtained in this way have turned out to have implications to representation theory and geometry. In this talk, I will discuss this development, including joint work with Hang Wang.

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