The Dimensions of the Smooth Representations of GL_n(o)

Let o be the valuation ring of a non-Archimedean local field with finite residue field and quotients o_r = o/p^r. We consider the problem of constructing the irreducible complex representations of GL_n(o_r). This is known to be a wild problem, however we prove certain new results about the dimensions of the representations of GL_n(o_r).
Our first approach involves studying automorphism groups of finite o-modules, whose representations can be used to build those of GL_n(o_2). We give a complete description of the dimensions of the representations of Aut_o(o_ℓ ⊕ o_1 ⊕ · · · ⊕ o_1) and show that they can be written as polynomials (independent of the choice of o) in q = |o_1|.
We also give a polynomial result for GL_n(o_r), namely that subject to the characteristic of o1 being large enough (possibly depending on n and r), there exists a finite set of polynomials R ⊆ Q[x], such that when evaluated at q = |o_1|, all of the dimensions of the irreducible representations of GL_n(o_r) appear in the set m(|o_1|)|m∈R. Our proof also holds when GL_n is replaced by an arbitrary smooth affine group scheme over Z.
Finally, we show that if o and o′ are valuation rings as before with |o_1| = |o′_1|, then C[GL_n(o_r)] = C[GL_n(o′_r)], provided the characteristic of o_1 is large enough. This generalises a result of Hadas, who proved the statement when o and o′ are restricted to be unramified extensions of Q_p if they have characteristic zero.