Multiplicity free induction for the pairs $({\mathrm{GL}}_2\times {\mathrm{GL}}_2,\mathrm{diag}({\mathrm{GL}}_2))$ and $({\mathrm{SL}}_3,{\mathrm{GL}}_2)$ over finite fields

Abstract

We classify the irredible representations of ${\mathrm{GL}}_2(q)$ for which the induction to the product group ${\mathrm{GL}}_2(q)\times {\mathrm{GL}}_2(q)$, under the diagonal embedding, decomposes multiplicity free. It turns out that only the irreducible representations of dimensions $1$ and $q-1$ have this property. We show that for ${\mathrm{GL}}_2(q)$ embedded into ${\mathrm{SL}}_3(q)$ via $g\mapsto \mathrm{diag}(g,\det g^{-1})$ none of the irreducible representations of ${\mathrm{GL}}_2(q)$ induce multiplicity free. In contrast, over the complex numbers, the holomorphic representation theory of these pairs is multiplicity free and the corresponding matrix coefficients are encoded by vector-valued Jacobi polynomials. We show that similar results cannot be expected in the context of finite fields for these examples.

32 pages