Wei et al. (2023)

[2311.09038] Skew Hecke algebras 
 
 
 
 
 
 
 
 
 
 
 

 
 
 

 
 
 
 
 
 
 Skew Hecke algebras

 
 
 James Waldron
 
 School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle-upon-Tyne, NE1 7RU
 
 james.waldron@ncl.ac.uk 
 
  and  
 Leon Deryck Loveridge
 
 Department of Science and Industry Systems, University of South-Eastern Norway, 3616 Kongsberg, Norway
 
 Okinawa Institute of Science and Technology Graduate University,
1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
 
 Leon.D.Loveridge@usn.no 
 
 

 
 Abstract.

 Let G 𝐺 G be a finite group, H ≤ G 𝐻 𝐺 H\leq G a subgroup, R 𝑅 R a commutative ring, A 𝐴 A an R 𝑅 R -algebra, and α 𝛼 \alpha an action of G 𝐺 G on A 𝐴 A by R 𝑅 R -algebra automorphisms.
Following Baker, we associate to this data the skew Hecke algebra ℋ R ​ ( G , H , A , α ) subscript ℋ 𝑅 𝐺 𝐻 𝐴 𝛼 \mathcal{H}_{R}(G,H,A,\alpha) , which is the convolution algebra of H 𝐻 H -invariant functions from G / H 𝐺 𝐻 G/H to A 𝐴 A .

 In this paper we study the basic structure of these algebras, proving 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 ℋ R ​ ( G , H , A , α ) subscript ℋ 𝑅 𝐺 𝐻 𝐴 𝛼 \mathcal{H}_{R}(G,H,A,\alpha) to the algebra of G 𝐺 G -invariants in the tensor product A ⊗ End R ​ ( Ind H G ​ R ) tensor-product 𝐴 subscript End 𝑅 superscript subscript Ind 𝐻 𝐺 𝑅 A\otimes\mathrm{End}_{R}(\mathrm{Ind}_{H}^{G}R) .
We show that if | G | 𝐺 |G| is a unit in A 𝐴 A , then ℋ R ​ ( G , H , A , α ) subscript ℋ 𝑅 𝐺 𝐻 𝐴 𝛼 \mathcal{H}_{R}(G,H,A,\alpha) is isomorphic to a corner ring inside the skew group algebra A ⋊ G right-normal-factor-semidirect-product 𝐴 𝐺 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 𝐺 subscript 𝑆 3 G=S_{3} , H = S 2 𝐻 subscript 𝑆 2 H=S_{2} , and α 𝛼 \alpha is the natural permutation action of S 3 subscript 𝑆 3 S_{3} on the polynomial algebra R ​ [ x 1 , x 2 , x 3 ] 𝑅 subscript 𝑥 1 subscript 𝑥 2 subscript 𝑥 3 R[x_{1},x_{2},x_{3}] .

 
 
 Key words and phrases: 

Skew group rings, Hecke algebras, finite groups.
 
 
 
 1. Introduction

 
 
 1.1. Skew Hecke algebras

 
 Let R 𝑅 R be a commutative ring, A 𝐴 A an R 𝑅 R -algebra, G 𝐺 G a finite group, H ≤ G 𝐻 𝐺 H\leq G a subgroup, and α : G → Aut R ​ -alg ​ ( A ) : 𝛼 → 𝐺 subscript Aut 𝑅 -alg 𝐴 \alpha:G\to\mathrm{Aut}_{R\text{-alg}}(A) a group homomorphism from G 𝐺 G to the group of R 𝑅 R -algebra automorphisms of A 𝐴 A .
Our main object of study in this paper is the associate

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