Thursday 25 October 2018, 16:00-17:00 in HG03.085
Peter Hochs (Adelaide)
Orbital integrals in index theory and K-theory
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.