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.

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