Skew Hecke Algebras

Let $G$ be a finite group, $H \le G$ a subgroup, $R$ a commutative ring, $A$ an $R$-algebra, and $\alpha$ an action of $G$ on $A$ by $R$-algebra automorphisms. We study the associated \emph{skew Hecke algebra} $\mathcal{H}_{R}(G,H,A,\alpha)$, 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 $\mathcal{H}_{R}(G,H,A,\alpha)$ 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 $\mathcal{H}_{R}(G,H,A,\alpha)$ 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 $\alpha$ is the natural permutation action of $S_3$ on the polynomial algebra $R[x_1,x_2,x_3]$.