The Steinberg Variety and Representations of Reductive Groups

TL;DR

This paper reviews how the Steinberg variety has been instrumental in advancing geometric representation theory, including proofs of key conjectures and the development of representations of reductive groups.

Contribution

It summarizes the main results achieved using the Steinberg variety, highlighting its role in proving conjectures and constructing representations in geometric representation theory.

Findings

Steinberg's variety was used to prove Grothendieck's conjectured formula.

It provided an alternative approach to Springer's representations.

Played a central role in the proof of the Deligne-Langlands conjecture.

Abstract

We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck. The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne-Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.