A combinatorial description of the Gindikin-Karpelevich formula

Authors

Benjamin T Salisbury, University of Connecticut

Date of Completion

January 2012

Keywords

Mathematics

Degree

Ph.D.

Abstract

The Gindikin-Karpelevich formula expresses the value of a certain p-adic integral over an algebraic group G as a product over the positive root system corresponding to the Langlands dual group G^v of G. We use the Young tableaux realization of the crystal basis B(∞) for the negative part of the quantum group corresponding to G^v to expand the latter product as a sum over the crystal B(∞). In other words, we define a statistic on tableaux which yields the appropriate co-efficient in the sum. This expansion is achieved when the crystal B(∞) is of non-exceptional finite type or type G ₂. We also interpret our statistic on tableaux in terms of Kamnitzer's MV polytopes and the Kashiwara-Saito geometric construction of B(∞) when the underlying Lie algebra is of type A_r. ^

Recommended Citation

Salisbury, Benjamin T, "A combinatorial description of the Gindikin-Karpelevich formula" (2012). Doctoral Dissertations. AAI3529383.
https://digitalcommons.lib.uconn.edu/dissertations/AAI3529383