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

TL;DR

This paper classifies when irreducible representations of certain finite groups induce multiplicity free representations in larger groups, revealing specific cases where this property holds or fails, and contrasting finite field and complex number scenarios.

Contribution

It provides a classification of irreducible representations of $ ext{GL}_2(q)$ with multiplicity free induction to $ ext{GL}_2(q) imes ext{GL}_2(q)$ and analyzes the absence of this property in embeddings into $ ext{SL}_3(q)$, contrasting with complex cases.

Findings

Only 1-dimensional and $(q-1)$-dimensional irreducible representations induce multiplicity free in $ ext{GL}_2(q)$.

No irreducible representations of $ ext{GL}_2(q)$ induce multiplicity free when embedded into $ ext{SL}_3(q)$.

Multiplicity free representations over complex numbers relate to vector-valued Jacobi polynomials.

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.