On cuspidal representations of general linear groups over discrete valuation rings

Abstract

We define a new notion of cuspidality for representations of GLn over a finite quotient o k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G λ 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(o k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G λ . A functional equation for zeta functions for representations of GLn(o k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL 4 (o 2 ) are constructed. Not all these representations are strongly cuspidal.

Representations of automorphism groups of finiteo-modules of rank two

“…In [1] it is further shown that there is a canonical bijection between cuspidal representations of G n and Galois orbits of strongly primitive characters ofõ × whenever n is prime.…”

“…Nevertheless, only in a very recent preprint [31] by Stasinski all the irreducible representations of GL 2 (o ) are constructed for general o. For higher dimensions there are some partial results of a more general nature by Hill [8][9][10][11], Lusztig [18] and Aubert, Onn, Prasad and Stasinski [1] however, the general case still remains out of reach.…”

“…In Section 5 we explain the inductive scheme in detail. In Section 6 we construct the cuspidal representationsĈ λ ⊂Ĝ λ for λ with 1 > 2 > 1 and briefly review the construction ofĈ 2 which appeared in [22] (but we shall follow the route taken in [1]). Section 7 is devoted to geometric and infinitesimal inductions.…”

“…In [1] it is shown that strongly cuspidal representations are cuspidal, with a reverse implication when n is prime. (b) Since the groups P M μ ,M λ and P M λ/μ ,M λ above depend only on μ and λ up to conjugation we shall denote them P μ →λ and P λ μ (respectively) whenever chances for confusion are slim.…”

The Smooth Representations of GL2(𝔒)

“…[1] and its references). The method we give here is similar to a more general construction used by Bushnell and Kutzko (cf.…”