Quantization Operators and Invariants of Group Representations

Abstract

Let $G$ be a semi-simple Lie group and $\pi$ some representation of $G$ belonging to the discrete series. We give interpretations of the constant $\pi(g)$, for $g\in Z(G)$, in terms of geometric concepts associated with the flag manifold $M$ of $G$. In particular, when $G$ is compact this constant is related to the action integral around closed curves in $M$. As a consequence, we obtain a lower bound for de cardinal of the fundamental group of ${\rm Ham}(M)$, the Hamiltonian group of $M$. We also interpret geometrically the values of the infinitesimal character of $\pi$ in terms of quantization operators.