An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field

TL;DR

This paper constructs analogs of irreducible cuspidal representations for PGL(2) over a two-dimensional local field, revealing similarities and differences with the classical case.

Contribution

It introduces a new construction of such representations over a two-dimensional local field, highlighting their properties and differences from classical representations.

Findings

Constructed representations from quadratic extensions and non-Galois invariant characters.

Restrictions to a Borel subgroup are irreducible but not isomorphic to classical cuspidal representations.

Proposed a notion of cuspidality for smooth representations of split reductive groups over local fields.

Abstract

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story turns out to be similar to the classical case, but also with some differences. We present a construction of such representations essentially (up to a small subtlety) starting from a quadratic extension $L$ of $K$ and a character $\theta:L^*/K^*\to \mathbb C^*$ which is not Galois invariant. We also show that the restriction of the representations we construct to the group $P(K)$ (here $P$ is a Borel subgroup of $PGL(2)$) is irreducible. However, contrary to the classical case it turns out that these restrictions are not isomorphic to the "standard" irreducible cuspidal representation of $P(K)$. In the Appendix we propose a notion of cuspidality for smooth representations of the group $H(K)$ for an arbitrary split reductive group $H$.