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

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.