## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Quantization Operators and Invariants of Group Representations

Andrés Viña

#### 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 Ham$(M)$, the Hamiltonian group of $M$. We also interpret geometrically the values of the infinitesimal character of $\pi$ in terms of quantization operators.

#### Article information

Dates
First available in Project Euclid: 13 July 2015

https://projecteuclid.org/ euclid.pgiq/1436815531

Digital Object Identifier
doi:10.7546/giq-13-2012-265-277

Mathematical Reviews number (MathSciNet)
MR3087977

Zentralblatt MATH identifier
1382.53026

#### Citation

Viña, Andrés. Quantization Operators and Invariants of Group Representations. Proceedings of the Thirteenth International Conference on Geometry, Integrability and Quantization, 265--277, Avangard Prima, Sofia, Bulgaria, 2012. doi:10.7546/giq-13-2012-265-277. https://projecteuclid.org/euclid.pgiq/1436815531