---
tag: ZH21D5B62C
kind: concept
name: Deligne–Lusztig theory
msc: [20C33, 20G40]
sources:
  - url: https://en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory
same_as:
  wikidata: Q5253966
  wikipedia: Deligne–Lusztig theory
---

## Summary

Deligne–Lusztig theory constructs representations of finite [[groups of Lie type|NPEBABQ49M]], such as $GL_n(\F_q)$, from the $\ell$-adic cohomology of certain algebraic varieties.
For each Frobenius-stable maximal torus $T$ and each character $\theta$ of its finite group of points it gives a virtual character $R_T^\theta$.
Every irreducible character of the finite group occurs in some $R_T^\theta$.
The theory, introduced by Deligne and Lusztig in 1976, is the basis of the classification of the irreducible characters of finite reductive groups by Lusztig.

## Explanations

### L1 human

Finite groups like the invertible matrices over a finite field have many representations, and most of them cannot be built by elementary methods such as induction from subgroups.
Deligne and Lusztig found them inside the geometry of special varieties on which the group acts.
The cohomology of these varieties, a tool from algebraic topology adapted to algebraic geometry, carries representations of the finite group.
Taking alternating sums of these gives virtual characters that together contain all irreducible characters.