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.


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