On Generalized Weil Representations over Involutive Rings

TL;DR

This paper constructs generalized Weil representations for groups over involutive rings, extending classical representations to a broader algebraic setting with applications to symplectic and non-classical groups.

Contribution

It introduces a generator-and-relation approach to define Weil representations over involutive rings, encompassing classical and novel group structures.

Findings

Constructed generalized Weil representations for involutive rings.

Included examples like symplectic groups and non-classical groups.

Demonstrated the applicability to finite modular analogues of jet algebras.

Abstract

We construct via generators and relations, generalized Weil representations for analogues of classical $SL(2,k), k$ a field, over involutive base rings $(A, \ast).$ This family of groups covers different kinds of groups, classical and non classical. We give some examples that include symplectic groups as well as non classical groups like $SL_\ast(2,A_m), $ where $A_m$ is the finite modular analogue of the algebra of real m-jets in one dimension with its canonical involutive symmetry.