Alladi Ramakrishnan Hall

Regular representaions and Whittaker models

Shiv Prakash Patel

IIT Delhi

Let  $O$ be the ring of integers of a non-archimedian local field $F$. Let
$G$ be a split reductive group defined over $O$. The uniqueness of Whittaker
models of the representations of $G(F)$ plays an important role in the
theory of automorphic forms. In this talk, we will discuss Whiitaker models
of representations of $G(O)$. On the other hand, the regular representations
of $G(O)$ have been useful in constructing the supercuspidal representations
of $G(F)$. For $G= GL_{n}$ or $SL_{n}$ we prove the uniqueness of the
Whittaker models for the representations of $G(O)$. In addition, we see that
only regular representations admit Whiitaker models. The proofs are based on
the explicit construction of regular representations.
This is a joint work with Pooja Singla.