## Quantization operators and invariants of group representations.(English)Zbl 1248.53076

Summary: Let $$G$$ be a semisimple 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 the cardinal of the fundamental group of $$\text{Ham}(M)$$, the Hamiltonian group of $$M$$. We also interpret geometrically the values of the infinitesimal character of $$\pi$$ in terms of quantization operators.

### MSC:

 53D50 Geometric quantization 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods