Affine Hecke and Schur algebras of type A without a square root of q
Rose Berry
TL;DR
The paper develops an integral affine cellular framework for the extended affine Hecke algebra and the affine $q$-Schur algebra of type $\tilde{A}_{n-1}$, defined over $\mathbb{Z}[q^{\pm1}]$ and avoiding $q^{1/2}$. By renormalising the Kazhdan-Lusztig data at each left and right cell, it constructs an integral asymptotic lattice and an integral version of the KL basis that closes under multiplication, enabling a robust affine cellular decomposition. It also proves integral left-cell isomorphisms via KL star operations and shows that the Schur algebra possesses idempotent two-sided ideals, leading to finite global dimension under localization. Consequently, the Hecke and Schur algebras acquire affine cellular structures over $\mathbb{Z}[q^{\pm1}]$, with explicit cell data and a pathway to derived-category stratifications relevant to $p$-adic representation theory of $GL_n$. The work integrates Kazhdan-Lusztig theory, asymptotic algebras, and Koenig’s affine cellular framework to extend cellular methods to non-square-root coefficient rings.
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.