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We show that induced representations of $\mathrm{GL}_2({\mathbb F}_q)$ ofhigher dimensional representations can be written as cokernels of operatorsacting on symmetric power representations of $\mathrm{GL}_2({\mathbb F}_q)$.When the induction is from the Borel subgroup, respectively the anisotropictorus, the operators involve multiplication by newly defined twisted Dicksonpolynomials, respectively, twisted Serre operators. Our isomorphisms areexplicitly defined using differential operators. As a corollary, we improvesome periodicity results for quotients in the theta filtration.

Modular induced representations of $\mathrm{GL}_2(\mathbb F_q)$ as cokernels of symmetric power representations