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Groupoids, Geometric Induction and Gelfand Models

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 $2$.