Cells in affine q-Schur algebras

TL;DR

This paper develops algebraic and geometric methods to construct and analyze canonical bases for affine q-Schur algebras of any type, establishing dualities, positivity, and cell structures similar to affine Hecke algebras.

Contribution

It introduces a new framework for affine q-Schur algebras, including duality, inner product, and cell theory, extending Lusztig's results to arbitrary types.

Findings

Canonical bases are positive and almost orthonormal.

Duality between affine q-Schur and affine Hecke algebras is established.

Cell and asymptotic algebra structures are formulated and analyzed.

Abstract

We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.