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)?
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)? 


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

