The Gindikin-Karpelevich Formula and Combinatorics of Crystals
| Speaker: | Ben Salisbury |
|---|---|
| Affiliation: | Central Michigan University |
| Room: | Mathematics 3 (M3) 3103 |
Abstract:
The Gindikin-Karpelevich formula computes the constant of proportionality for the intertwining integral between two induced spherical representation of a p-adic reductive group G. The right-hand side of this formula is a product over positive roots (with respect to the Langlands dual of G) which may also be interpreted as a sum over a crystal graph. The original interpretation as a sum is due to Brubaker-Bump-Friedberg and Bump-Nakasuji where G=GL_{r+1}, in which vertices of the crystal graph are parametrized by paths to a specific vector in the graph according to a pattern prescribed by a reduced expression of the longest element of the Weyl group of G. I will explain this rule, explain how it may be translated into a statistic on the Young tableaux realization of the same crystal graph, and discuss its generalization to the affine Kac-Moody setting where the notion of the longest element of the Weyl group no longer makes sense. This is joint work with Kyu-Hwan Lee, Seok-Jin Kang, and Hansol Ryu.