Groupoids, Geometric Induction and Gelfand Models

TL;DR

This paper introduces a geometric induction method for group representations using action groupoids, providing new Gelfand models for symmetric and projective linear groups.

Contribution

It develops an intrinsic geometric induction approach for group representations, extending classical induction, and constructs Gelfand models for specific groups.

Findings

Geometric induction associates group representations to action groupoids.

Gelfand models are obtained for symmetric and projective linear groups.

The method applies to non-transitive G-sets and yields new representation models.

Abstract

In this paper we introduce an intrinsic version of the classical induction of representations for a subgroup $H$ of a (finite) group $G$, called here {\em geometric induction}, which associates to any, not necessarily transitive, $G$-set $X$ and any representation of the action groupoid $A(G,X)$ associated to $G$ and $X$, a representation of the group $G$. We show that geometric induction, applied to one dimensional characters of the action groupoid of a suitable $G$-set $X$ affords a Gelfand Model for $G$ in the case where $G$ is either the symmetric group or the projective general linear group of rank $2$.