Affine Hecke and Schur algebras of type A without a square root of q

TL;DR

This paper constructs an affine cellular structure for affine Hecke and Schur algebras of type A over integers without requiring a square root of q, facilitating applications in p-adic representation theory.

Contribution

It introduces a new affine cellular structure over algebra without algebra with a square root of q, enabling broader applications in p-adic group representations.

Findings

Affine cellular structure over algebra without algebra with algebra with a square root of q.

Finite global dimension of the Schur algebra due to idempotence properties.

Application potential in the representation theory of algebra over rings where p is invertible but lacks a square root.

Abstract

We provide an affine cellular structure on the extended affine Hecke algebra and affine $q$-Schur algebra of type $A_{n-1}$ that is defined over $\mathbb{Z}\left[q^{\pm1}\right]$, that is, without an adjoined $q^{\frac{1}{2}}$. This is with an eye to applications in the representation theory of $\mathrm{GL}_n(F)$ for a $p$-adic field $F$ over coefficient rings in which $p$ is invertible but does not have a square root, which have been a topic of recent interest. This is achieved via a renormalisation of the known affine cellular structure over $\mathbb{Z}\left[q^{\pm\frac{1}{2}}\right]$ at each left and right cell, which is chosen to ensure that the diagonal intersections remain subalgebras and that the left and right cells remain isomorphic. We furthermore show that the affine cellular structure on the Schur algebra has idempotence properties which imply finite global dimension, an important ingredient for the applications to representations of $p$-adic groups.