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VOL. 71 | 2016 Whittaker functions, geometric crystals, and quantum Schubert calculus
Thomas Lam

Editor(s) Hiroshi Naruse, Takeshi Ikeda, Mikiya Masuda, Toshiyuki Tanisaki


This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective.

We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture. We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations.


Published: 1 January 2016
First available in Project Euclid: 4 October 2018

zbMATH: 1378.14058
MathSciNet: MR3644825

Digital Object Identifier: 10.2969/aspm/07110211

Rights: Copyright © 2016 Mathematical Society of Japan


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