Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

TL;DR

This paper explores Kazhdan-Lusztig bases for parabolic Hecke algebras, focusing on type A, and applies these findings to classify representations and analyze Schur-Weyl duality kernels.

Contribution

It introduces two Kazhdan-Lusztig bases for parabolic Hecke algebras, describes their cells and representations, and connects these to Schur-Weyl duality kernels with new conjectures.

Findings

Classified irreducible representations of type A parabolic Hecke algebras.

Described cells via RSK correspondence, generalizing symmetric group results.

Formulated and proved conjectures on kernels in Schur-Weyl duality.

Abstract

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups, and for type A they coincide with the fused Hecke algebras appearing in a generalisation of the Schur-Weyl duality with the quantum group of GL(N). In this paper we investigate two different Kazhdan-Lusztig bases for parabolic Hecke algebras, together with the associated cells and the corresponding representations. We quickly specialise to type A, for which we describe the cells in terms of the RSK correspondence generalising thus the well-known description for the symmetric group. As a first application we recover the classification of irreducible representations of parabolic Hecke algebras of type A and provide a new construction of these representations. Next we turn to the Schur-Weyl duality and describe the kernel in terms of one the basis studied precedently. Moreover, we formulate some conjectures about a generator of these kernels in terms of Kazhdan-Lusztig basis elements, give some evidence and prove these conjectures in some special cases.