Abstract
AbstractLet G be a linear semisimple algebraic group and B its Borel subgroup. Let $${\mathbb {T}}\subset B$$
T
⊂
B
be the maximal torus. We study the inductive construction of Bott–Samelson varieties to obtain recursive formulas for the twisted motivic Chern classes of Schubert cells in G/B. To this end we introduce two families of operators acting on the equivariant K-theory $${\text {K}}_{\mathbb {T}}(G/B)[y]$$
K
T
(
G
/
B
)
[
y
]
, the right and left Demazure–Lusztig operators depending on a parameter. The twisted motivic Chern classes coincide (up to normalization) with the K-theoretic stable envelopes. Our results imply wall-crossing formulas for a change of the weight chamber and slope parameters. The right and left operators generate a twisted double Hecke algebra. We show that in the type A this algebra acts on the Laurent polynomials. This action is a natural lift of the action on $${\text {K}}_{\mathbb {T}}(G/B)[y]$$
K
T
(
G
/
B
)
[
y
]
with respect to the Kirwan map. We show that the left and right twisted Demazure–Lusztig operators provide a recursion for twisted motivic Chern classes of matrix Schubert varieties.
Publisher
Springer Science and Business Media LLC
Reference64 articles.
1. Aganagic, M., Okounkov, A.: Elliptic stable envelopes. J. Am. Math. Soc. 34(1), 79–133 (2021)
2. Aluffi, P., Mihalcea, L.C.: Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds. Compos. Math. 152(12), 2603–2625 (2016)
3. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes (2022). arXiv:2212.12509
4. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells. Duke Math. J. 172(17), 3257–3320 (2023)
5. Aluffi, P., Mihalcea, L.C., Schürmann, J., Su, C.: Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem. Ann. Sci. Éc. Norm. Supér. (4) 57(1), 87–141 (2024)