My research interests lie in the representation theory of finite groups of Lie type, the groups over finite principal ideal local rings.
My research is in the representation theory of p-adic groups and their finite quotients. If F is a p-adic field with ring of integers O, the compact group GL_n(O) has finite quotients GL_n(O_ℓ). Here O_ℓ = O/ϖ^ℓ O is a finite local ring, such as Z/p^ℓZ or F_q[t]/(t^ℓ). Every irreducible smooth representation of GL_n(O) factors through one of these finite groups. They therefore form a bridge between the finite groups of Lie type GL_n(F_q), which is the case ℓ = 1, and the p-adic group GL_n(F). I study how representations of GL_n(O_ℓ) decompose, including their irreducible constituents, dimensions and multiplicities. This study tells us about representations of GL_n(F), for instance through the theory of types.
Recently I have worked on the following related questions:
the structure of principal series representations of GL_2(O_ℓ) and GL_3(O_2);
degenerate Whittaker spaces of GL_2n(O_ℓ), in connection with a conjecture of D. Prasad;
a correspondence between tensor products of irreducible representations and products of conjugacy classes of GL_2(F_q), with applications to trilinear forms.