Pure and Applied Mathematics Quarterly
Volume 17 (2021)
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
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.
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