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Skew Hecke Algebras by Waldron, James; Loveridge, Leon Deryck is a scholarly article available to read on EtoBox.

What is Skew Hecke Algebras about?

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 illustrat

Author

Waldron, James; Loveridge, Leon Deryck

Published

2023

Language

EN