2011-04-15

Polyhedral and path models for K-types of finite-dimensional G-reps

Allen Knutson, Cornell University

Let K be a symmetric subgroup of G, e.g. a maximal compact subgroup of a semisimple group. It is a basic problem to decompose irreps of G under K, called "branching". One can compute branching to arbitrary reductive subgroups via an alternating sum; the difficulty is to give a manifestly positive formula. If G = K x K (thinking of K itself as the diagonal), then this becomes the decomposition of tensor products, which was given a general answer by Littelmann in 1994, in terms of counting piecewise-linear paths in a Weyl chamber. I'll give a formula for the general case, counting some new "(G,K)-paths", and prove it in the asymptotic limit (of Duistermaat-Heckman measures rather than K-multiplicities), and exactly for the pairs (K x K, K), (GL(2n),Sp(n)), and (SL(3),SO(3)).