Geometric Weil representations for star-analogues of SL(2,k)

TL;DR

This paper introduces a geometric method to construct Weil representations for star-analogues of SL(2,k), extending classical techniques and comparing them with algebraic approaches, including applications to finite truncated polynomial algebras.

Contribution

It develops an elementary geometric approach for Weil representations of star-analogues of SL(2,k) and compares it with generator-relation methods, including new results for finite truncated polynomial algebras.

Findings

Constructed Weil representations via geometric methods.

Compared geometric and algebraic constructions.

Derived the analogue of the Maslov Index for finite truncated polynomial algebras.

Abstract

We present here an elementary geometric approach to the construction of Weil representations of the star-analogues $SL_\ast(2,A), A$ a ring or algebra with involution $\ast$, of the group SL(2, k), k a field, reminiscent of the quantum groups $SL_q(2,A)$. We review as well the elementary construction of Weil representations for these groups via generators and relations, which uses the Bruhat presentation available in many cases. We compare the representations obtained by both methods in the non - classical case of the finite truncated polynomial algebra $A_m$ of degree $m$ with its canonical involution and obtain the analogue of the Maslov Index in this case.