Question

Here, G is a compact lie group. V is a finite dimensional irrepn of G.

By Atiyah, every element in K_{G}(T_{G}V) is a symbol of a transversally elliptic operator on V.

Of course, K_{G}(T_{G}V) is a R(G)-module.What is K_{G}(T_{G}V)?

Answer 1

With De Concini+Procesi+vergne, we deermined this R(G) module when G is a torus. See our series of article on ArXiv (Dahmen-Micchelli spaces, etc..)

When G is compact connected, there are natural conjectural answer, in terms of the maxial torus of G, but we do not know yot the answer.