A Polynomial Result for Dimensions of Irreducible Representations of Smooth Affine Group Schemes Over Principal Ideal Local Rings

Denote by o the valuation ring of a non-Archimedean local field with prime ideal p and finite residue field, and let r≥ 1 be an integer. We prove that for every smooth affine group scheme G over Z, the dimension of each irreducible representation of G(o/pʳ) is given by one of finitely many polynomials with coefficients in Q evaluated at q=|o/p|, provided that the residue characteristic p=char o/p is large and fixed.