Multiplicity free induction for the pairs (GL ₂ × GL ₂ ,diag(GL ₂ )) and (SL ₃ ,GL ₂ ) over finite fields

We classify the irredible representations of GL 2(q) for which the induction to the product group GL 2(q) × GL2(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 GL 2(q) embedded into SL 3(q) via g → diag(g, det g −1 ) none of the irreducible representations of 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.