Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 4

Special Issue In Memory of Prof. Bertram Kostant

Guest Editors: Shrawan Kumar, Lizhen Ji, and Kefeng Liu

Schubert structure operators and $K^\ast_T (G/B)$

Pages: 1345 – 1385

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n4.a6

Authors

Rebecca Goldin (George Mason University, Fairfax, Virginia, U.S.A.)

Allen Knutson (Cornell University, Ithaca, New York, U.S.A.)

Abstract

We prove a formula for the structure constants of multiplication of equivariant Schubert classes in both equivariant cohomology and equivariant $K$-theory of Kac–Moody flag varieties $G/B$.We introduce new operators whose coefficients compute these (in a manifestly polynomial, but not positive, way), resulting in a formula much like and generalizing the positive Andersen–Jantzen–Soergel/Billey and Graham/Willems formulæ for the restriction of classes to fixed points.

Our proof involves Bott–Samelson manifolds, and in particular, the ($K$)‑cohomology basis dual to the ($K$)‑homology basis consisting of classes of sub-Bott–Samelson manifolds.

Keywords

Schubert calculus, equivariant cohomology, Bott–Samelson manifolds

2010 Mathematics Subject Classification

Primary 14M15, 14-xx. Secondary 55N91, 55-xx.

The full text of this article is unavailable through your IP address: 34.236.134.129

In loving memory of our friend Bert Kostant.

Allen Knutson was supported by National Science Foundation Award 1953948.

Received 31 August 2019

Accepted 21 June 2021

Published 22 December 2021