Number theory seminar
Speaker: Hariom Sharma (IIT Roorkee)
Host: Ravi Raghunathan
Title: On representations of $GL(n,D)$ with a symplectic model
Time, day and date: 10:00:00 AM - 11:00:00 AM, Thursday, August 07
Venue: online (https://meet.google.com/pqd-fyod-xqi)
Abstract: Let $F$ be a non-Archimedean local field of characteristic zero, and let $D$ be the unique quaternion division algebra over $F$. For $n \in \mathbb{N}$, let $G_n = GL(n,D)$. The subgroup $H_n = Sp(n,D)$ of $G_n$ denotes the unique non-split inner form of the symplectic group $Sp(2n, F)$.
A smooth admissible complex representation $(\pi,V)$ of $G_n$ is said to have a symplectic model (or to be $H_n$-distinguished) if there exists a non-zero linear functional $\phi$ on $V$ such that $\phi(\pi(h)v) = \phi(v)$ for all $h \in H_n$ and $v \in V$.
In this talk, we provide a complete list of irreducible admissible representations of $G_3$ and $G_4$ having a symplectic model. We demonstrate that induced representations from finite-length representations preserve the symplectic model. Furthermore, we classify those ladder representations of $G_n$ that admit a symplectic model. In addition, we prove a part of Prasad's conjecture which provides a family of irreducible unitary representations with a symplectic model.