Zahi Hazan (Tel-Aviv University)
Degenerate Whittaker Models for Cuspidal Representations of GL(kn,F) 

We present a generalization of a theorem by D. Prasad (2000), where he computed the twisted Jacquet module of an irreducible cuspidal representation of GL(2n,F), with respect to the "Shalika" character of the unipotent radical of the parabolic subgroup of type (n,n). Prasad computed this module as a GL(n,F)-module. We studied the twisted Jacquet module of an irreducible cuspidal representation of GL(kn,F), with respect to a "regular" character of the unipotent radical of the parabolic subgroup of type (n,n,...,n) (k times) and analyzed it as a GL(n,F)-module.

We have obtained a compact description of the GL(n,F)-representation obtained from the twisted Jacquet module above. Our description of the module involves a power of the Steinberg representation of GL(n,F), which in particular, sheds new light on Prasad's description. We use techniques from diverse disciplines: such as q-Hypergeometric series, combinatorics and linear algebra. In particular, the calculation of the dimension of the module leads to a deep problem on matrices and q-Hypergeometric identities.

This is a joint work with Ofir Gorodetsky.