A geometric construction of types for the smooth representations of PGL(2) of a local field

TL;DR

This paper demonstrates that most types for smooth representations of PGL(2) over a non-Archimedean local field can be realized through the cohomology of finite graphs, providing a geometric perspective.

Contribution

It introduces a geometric construction linking types of PGL(2) representations to the cohomology of finite graphs, expanding understanding of their structure.

Findings

Most types of PGL(2,F) appear in graph cohomology

Provides a geometric interpretation of representation types

Connects representation theory with graph cohomology

Abstract

We show that almost all (Bushnell and Kutzko) types of PGL(2,F), F a non-Archimedean locally compact field of odd residue characteristic, naturally appear in the cohomology of finite graphs.