Deligne–Lusztig theory

Summary

Deligne–Lusztig theory constructs representations of finite groups of Lie type, 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.