The Steinberg representation is irreducible

geometry
cohomology
number theory
Steinberg representation

Appears in collection : Summer School 2022 - Cohomology, Geometry and Explicit number theory (COGENT)

The Steinberg representation is a topologically-defined representation of groups like GL_n(k) that plays a fundamental role in the cohomology of arithmetic groups. The main theorem I will discuss says that for infinite fields k, the Steinberg representation is irreducible. For finite fields, this is a classical theorem of Steinberg. No background in representation theory will be assumed. This is joint work with Andrew Snowden.