Character sheaves geometrize character theory on reductive groups. Instead of recording a representation by a class function on conjugacy classes, they place conjugation-equivariant perverse sheaves on the group itself and recover functions over finite fields by taking Frobenius traces.
This route connects Character Theory, Springer Correspondence, Geometric Representation Theory and Geometric Langlands.
From class functions to sheaves
For a finite group, a character is constant on conjugacy classes. For an algebraic group G, conjugation is the action g:x↦gxg^{-1}. A geometric analogue of a class function should therefore carry conjugation symmetry.
Character sheaves are special simple perverse sheaves, or complexes in the appropriate equivariant derived category, selected by Lusztig’s geometric constructions and equipped with conjugation equivariance.
Why perverse sheaves
Ordinary local systems work well on smooth strata but do not behave robustly under singular closure. Perverse sheaves combine local systems on strata with controlled extension across singularities.
Intersection cohomology complexes are fundamental examples. They preserve duality and allow geometry on singular orbit closures to participate in representation theory.
Conjugacy and centralizers
The geometry of conjugacy classes is controlled by centralizers. Semisimple and unipotent parts of an element lead to different strata, while local systems on conjugacy classes encode representations of component groups of centralizers.
This mirrors the Springer correspondence, where component groups and Weyl groups organize local systems on nilpotent orbits.
Induction
Parabolic induction has a geometric counterpart. One uses correspondences involving a parabolic subgroup P, its Levi quotient L and the ambient group G to transport suitable sheaf data from L to G.
Cuspidal character sheaves are the building blocks that do not arise by induction from proper Levi subgroups in the relevant sense.
Weyl groups enter again
Endomorphism algebras of induced complexes often carry Weyl-group or relative Weyl-group actions. Decomposition into simple summands is therefore indexed by representation-theoretic data attached to these finite reflection groups.
The same Weyl combinatorics appearing in Springer theory and Hecke algebras reappears inside sheaf decomposition.
Sheaf–function dictionary
When G and the sheaf are defined over a finite field, a Frobenius structure on a complex K produces a function on G(F_q) by taking alternating traces of Frobenius on stalk cohomology.
This converts geometric objects into class functions. Characteristic functions of character sheaves are closely related to irreducible characters and almost characters, with normalization and exceptional-family issues requiring care.
Deligne–Lusztig geometry
Deligne–Lusztig varieties construct representations of finite groups of Lie type from ℓ-adic cohomology. Character sheaves extend the same philosophy from specially constructed varieties to a geometric character theory living directly on G.
The relation is not that every character sheaf is simply one Deligne–Lusztig variety; rather, both theories use Frobenius, Weyl data and sheaf cohomology to produce representation-theoretic functions.
Character sheaves and character tables
For finite reductive groups, character sheaves help organize families of irreducible characters and Fourier-transform matrices relating natural bases of class functions.
Geometry therefore explains blocks of the character table through strata, local systems, induction and duality rather than treating the table as an isolated numerical object.
Verification workflow
- Fix whether G is over C, an algebraic closure, or a finite field.
- Specify the sheaf theory and coefficient field.
- Distinguish conjugation equivariance from ordinary equivariance.
- Separate character sheaves from arbitrary perverse sheaves.
- Track Frobenius structures before forming trace functions.
- State normalizations when comparing trace functions with irreducible characters.
- Distinguish parabolic induction of sheaves from ordinary group induction.
Practice
1. Why is conjugation symmetry natural? 2. Why use perverse sheaves? 3. What does Frobenius produce? 4. How do Weyl groups enter? 5. What is a cuspidal character sheaf? 6. How does this relate to Deligne–Lusztig theory?
Answers. Characters are class functions; perverse sheaves extend local data across singular strata; Frobenius traces produce functions on finite rational points; Weyl groups act through induction and endomorphism algebras; cuspidal objects are not induced from proper Levi data; both geometrize representations through sheaf cohomology and Frobenius.
Representation Mathematics — Batch 14
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