Deligne–Lusztig theory is a significant area in the field of representation theory of algebraic groups and finite groups of Lie type, named after Pierre Deligne and George Lusztig. This theory provides a way to construct and study representations of finite groups of Lie type via geometric methods, specifically by examining varieties over finite fields.
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Deligne-Lusztig theory constructs virtual characters of finite groups of Lie type from compactly supported étale cohomology of varieties associated with Frobenius and flags. Rational maximal tori and their finite-group characters provide induction data, while the geometry explains character orthogonality and cuspidality.