Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
arxiv(2023)
摘要
We prove a Chevalley formula to multiply the motivic Chern classes of
Schubert cells in a generalized flag manifold G/P by the class of any line
bundle ℒ_λ. Our formula is given in terms of the
λ-chains of Lenart and Postnikov. Its proof relies on a change of basis
formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action
on torus-equivariant K-theory of the complete flag manifold G/B via left
Demazure–Lusztig operators. We revisit some wall-crossing formulae for the
stable envelopes in T^*(G/B). We use our Chevalley formula, and the
equivalence between motivic Chern classes of Schubert cells and K-theoretic
stable envelopes in T^*(G/B), to give formulae for the change of
polarization, and for the change of slope for stable envelopes. We prove
several additional applications, including Serre, star, and Dynkin, dualities
of the Chevalley coefficients, new formulae for the Whittaker functions, and
for the Hall–Littlewood polynomials. We also discuss (mostly conjectural)
positivity and log concavity properties of special cases of the Chevalley
coefficients.
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