Alexander Stasinski

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GLn over a finite quotient ok of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups Gλ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GLn(F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GLn(ok) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups Gλ. A functional equation for zeta functions for representations of GLn(ok) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL4(o2) are constructed. Not all these representations are strongly cuspidal.

The Smooth Representations of GL2(𝔒)

We give a classification of the smooth (complex) representations of GL2(𝔬), where 𝔬 is the ring of integers in a non-Archimedean local field. The approach is based on Clifford theory of finite groups and a corresponding study of orbits and stabilizers. In terms of this classification, we identify the representations which are geometrically or infinitesimally induced, respectively.

The algebraisation of higher Deligne–Lusztig representations

In this paper we study higher Deligne–Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations, defined by Lusztig, coincide with certain explicit induced representations defined by Gérardin, thus giving a solution to a problem raised by Lusztig. In particular, we determine the dimensions of these representations. As an immediate application we verify a conjecture of Letellier for

Unramified representations of reductive groups over finite rings

\mathrm {GL}_2

The regular representations of

GL2 and

over finite local principal ideal rings

\mathrm {GL}_3