The Kirillov orbit method, wrapping maps and \(e\)-functions

Abstract

Kirillov's character formula gives an expression for the character of an irreducible representation of a Lie group in terms of the (Euclidean) Fourier transform of its associated coadjoint orbit. Wildberger and I re-interpreted this using the wrapping map, which allows one to transfer Ad-invariant distributions from the Lie algebra to the Lie group, as a convolution homomorphism. In this talk, I will describe how the theory works for compact symmetric pairs \((G,K)\). The convolution of \(K\)-invariant distributions needs to be twisted by the so-called \(e\)-function, and one then retrieves the characters of \(G/K\) as limits of generalised Bessel functions.