A geometric approach to characters of Hecke algebras

Abstract

To any element of a connected, simply connected, semisimple complex algebraic group G and a choice of an element of the corresponding Weyl group there is an associated Lusztig variety. When the element of G is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan-Lusztig basis of the corresponding Hecke algebra. In type A, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan-Lusztig basis element. The main idea is to find celular decompositions of desingularizations of these varieties and apply the Brosnan-Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan-Chow [BC18] solution to the Sharesian-Wachs [SW16] conjecture to non-codominant permutations, where singularities are involved. We also review the connections between Immanants, Hecke algebras, and Chromatic quasisymmetric functions of indifference graphs.

Chromatic symmetric functions from the modular law

“…With that in mind, we take the time now to explain how to construct the action of S n on the cohomology of regular semisimple Hessenberg varieties. We follow ) and [AN22a] (see also [Lus85]).…”

An update on Haiman's conjectures

“…For w a smooth permutation, so that Y w (X) is also smooth, this action can be explicitly characterized by a dot action on H * (Y w (X)) (as in [Tym08]). We have the following result due to Lusztig [Lus86], (see also [AN22]).…”

LLT Polynomials and Hecke Algebra Traces

“…In another context, the functions X inc(P),q appeared in the study of the space of diagonal harmonics. Let LLT inc(P),q := ∑ κ q asc(κ) x κ (1) x κ(2) • • • , where the sum is over arbitrary vertex colorings of inc(P). This is also a symmetric function called a unicellular LLT polynomial, a special case of a family of symmetric functions introduced by Lascoux-Leclerc-Thibon in 1997 in a different context.…”