Skew Hecke Algebras
James Waldron, Leon Deryck Loveridge
TL;DR
This work introduces and systematizes skew Hecke algebras $\\mathcal{H}_{R}(G,H,A,\\alpha)$, the convolution algebras of $H$-invariant maps from $G/H$ to $A$, unifying skew group algebras and Hecke algebras under a common framework. The main results establish a double coset decomposition giving an $A^{H}$-bimodule structure, and prove isomorphisms $\\mathcal{H}_{R}(G,H,A,\\alpha) \cong \mathrm{End}_{A\rtimes G}(A[G/H])^{\mathrm{op}} \cong (A \otimes \mathrm{End}_{R}(R[G/H])^{\mathrm{op}})^{G}$; when $|H|$ is a unit in $A$ there is a corner-ring description $\\mathcal{H}_{R}(G,H,A,\\alpha) \cong \mathbf{e}_{H} (A \rtimes G) \mathbf{e}_{H}$. The paper further shows compatibility with a broad array of operations (restriction/extension of scalars, cocycle perturbations, gradings, filtrations, opposite algebras) and analyzes fundamental group-theoretic constructions (factoring by normal subgroups, products, intermediate subgroups, conjugation, semidirect products). Concrete instances are worked out, notably $G=S_{3}$ with $H=S_{2}$ and $A=R[x_{1},x_{2},x_{3}]$, illustrating the module decompositions, product rules, fixed-point realizations, and corner algebra descriptions. The results extend classical identifications for Hecke algebras and skew group algebras, providing a unified, versatile framework with connections to topology, operator algebras, and geometry. This framework yields practical formulas and structural insights applicable to a range of algebraic and representation-theoretic problems.
Abstract
Let $G$ be a finite group, $H \le G$ a subgroup, $R$ a commutative ring, $A$ an $R$-algebra, and $α$ an action of $G$ on $A$ by $R$-algebra automorphisms. We study the associated \emph{skew Hecke algebra} $\mathcal{H}_{R}(G,H,A,α)$, which is the convolution algebra of $H$-invariant functions from $G/H$ to $A$. We prove for skew Hecke algebras a number of common generalisations of results about skew group algebras and results about Hecke algebras of finite groups. We show that skew Hecke algebras admit a certain double coset decomposition. We construct an isomorphism from $\mathcal{H}_{R}(G,H,A,α)$ to the algebra of $G$-invariants in the tensor product $A \otimes \mathrm{End}_{R} ( \mathrm{Ind}_{H}^{G} R )$. We show that if $|H|$ is a unit in $A$, then $\mathcal{H}_{R}(G,H,A,α)$ is isomorphic to a corner ring inside the skew group algebra $A \rtimes G$. Alongside our main results, we show that the construction of skew Hecke algebras is compatible with certain group-theoretic operations, restriction and extension of scalars, certain cocycle perturbations of the action, gradings and filtrations, and the formation of opposite algebras. The main results are illustrated in the case where $G = S_3$, $H = S_2$, and $α$ is the natural permutation action of $S_3$ on the polynomial algebra $R[x_1,x_2,x_3]$.