### Multiplicities of representations of compact Lie groups, qualitative properties and some computations

Vergne, M *(Institut de Mathématiques de Jussieu)*

Thursday 17 October 2013, 11:30-12:30

Seminar Room 1, Newton Institute

#### Abstract

Co-author: Baldoni Velleda (Roma Tor Vergata-Italy)

Let V be a representation space for a compact connected Lie group G decomposing as a sum of irreductible representations pi of G with finite multiplicity m(pi,V).

When V is constructed as the geometric quantization of a symplectic manifold with proper moment map, the multiplicity function pi-> m(pi,V)$ is piecewise quasi polynomial on the cone of dominant weights. In particular, the function t-> m(t *pi,V) is a quasipolynomial,alonf the ray t*pi, when t runs over the non negative integers. We will explain how to compute effectively this quasi-polynomial (or the Duistermaat-Heckman measure) in some examples, including the function t-> c(t*lambda,t* mu,t*nu) for Clebsch-Gordan coefficients (in low rank) and the function t-> k(t*alpha,t*beta,t*gamma) for Kronecker-coefficients (with number of rows less or equal to 3). Our method is based on a multidimensional residue theorem (Jeffrey-Kirwan residues).

#### Video

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