Restriction of a cuspidal module for finite general linear groups. (Q1427379)

Let \(G=G_n(q)\) be the general linear group of degree \(n\) over the finite field \(\mathbb{F}_q\). Let \(A\) be the affine linear subgroup consisting of those matrices whose last row agrees with the identity matrix. Let \(A'\) be the affine linear subgroup consisting of those matrices whose first column agrees with the identity matrix. Let \(U\) be the subgroup consisting of unipotent upper triangular matrices. It is known that an irreducible cuspidal \(G\)-module \(M\) has an irreducible restriction to \(A\) that may be identified with \(\text{Ind}_U^A\langle b\rangle\), where \(b\) is an eigenvector for the Gel'fand-Graev character \(\Psi\colon U\to\mathbb{C}\). As the pair \((A,G)\) is isomorphic to the pair \((A',G)\), one has the analogous result for the restriction to \(A'\). But now the author studies the restriction of \(M\) to \(C=A\cap A'\). He finds its constituents explicitly in terms of certain left ideals in the group ring \(\mathbb{C} G\) and constituents of the Gel'fand-Graev representation of \(G\). The answer shows that \(\text{Res}_C^GM\) is multiplicity free.