Cuspidal Character Sheaves

• Cuspidal character sheaves are simple perverse ℓ-adic sheaves that do not arise via nontrivial induction from proper Levi subgroups, forming the foundation for the generalized Springer correspondence.
• They are classified by their unipotent supports and exhibit a 'clean' restriction behavior on conjugacy classes, revealing structured local system properties.
• Combinatorial parametrization via Weyl group data links these sheaves to unipotent representations, guiding explicit character computations in representation theory.

A cuspidal character sheaf is a simple perverse $\overline{\mathbb{Q}}_\ell$-sheaf defined on a connected reductive algebraic group $G$ over an algebraically closed field $k$ (with $p$ not a bad prime for $G$), which is equivariant under conjugation by $G$ and does not arise by nontrivial induction from any proper Levi subgroup. The theory of cuspidal character sheaves, developed by Lusztig, precisely describes the collection, supports, restriction behavior, classification, and combinatorial parametrization of these sheaves, underpinning unipotent representation theory and the generalized Springer correspondence.

1. Definitions and Structural Properties

Let $G$ be a connected reductive group over $k$, $p$ a prime not bad for $G$. A character sheaf on $G$ is a simple perverse $\overline{\mathbb{Q}}_\ell$-sheaf $A$ on $G$, conjugation-equivariant, classified as in [L8, L9]. Such sheaves fall into finitely many families $\mathfrak F\subset \widehat G$, each supporting a unique unipotent conjugacy class $C_\mathfrak F$ called its unipotent support. This is characterized by the vanishing property:

• If $D$ is a class with $\dim D^u > \dim C_\mathfrak F$ or $D^u \not\subset \overline{C_\mathfrak F}$, then $A|_D=0$ for all $A\in\mathfrak F$;
• For some $A\in\mathfrak F$, $A|_{D_0}\neq 0$ for some $D_0$ with $(D_0)^u=C_\mathfrak F$.

A cuspidal character sheaf is one not occurring as a nontrivial proper parabolic induction from any proper Levi. Equivalently, it is associated to a cuspidal pair $(L,\Sigma,\mathcal{E})$ where:

• $L\subset G$ a Levi subgroup,
• $\Sigma\subset L$ an isolated conjugacy class in $L/Z_L$,
• $\mathcal{E}$ a $G$-equivariant local system on $\Sigma$ satisfying the vanishing condition

$$H_c^*(T_P(y),\mathcal{E}) = 0 \quad \forall P \subsetneq G\ \text{parabolic},\ y\in P,$$

and from such data one constructs the induced complex

$$K = \operatorname{IC}(Y,\widetilde{\mathcal{E}}) \subset D^b_c(G),\quad Y = \{g\in G \mid g_s\in\Sigma\}.$$

$K[\dim Y]$ is semisimple perverse; its simple summands are the character sheaves induced from $(L,\Sigma,\mathcal{E})$ (Lusztig, 2012).

2. Restriction Theorem and Behavior on Conjugacy Classes

Fix a family $\mathfrak F\subset\widehat G$ with unipotent support $C=C_\mathfrak F$, and $A\in\mathfrak F$. For any conjugacy class $D\subset G$ with $(D)^u=C$, Restriction Theorem [Thm 0.2]:

There exists an integer $c\geq 0$ and a $G$-equivariant local system $\mathcal L$ on $D$ such that

$A|_D \simeq \mathcal{L}[\dim D + c],$

and $c$ depends only on $A$, not on $D$.

Corollary: For any Lusztig–stratum $Y\subset G$ with $Y^u=C$,

$A|_Y = \mathcal{L}'[\dim Y + c],$

with $\mathcal{L}'$ a local system on $Y$; if $Y^u\neq C$, then $A|_Y=0$ (Lusztig, 2012).

This property shows that the restriction of a cuspidal character sheaf to strata or conjugacy classes of unipotent type is always a local system up to shift, reflecting the "cleanness" property and tightly controlling the support structure.

3. Parametrization of Unipotent Cuspidal Character Sheaves

In the unipotent case (those occurring in complexes $R\Gamma_c(\mathcal{O}_w)$ for $w\in W$), one fixes a family $\mathfrak F_{\mathrm{un}}$ and its support $C$. Set $A=Z_G(w)/Z_G(w)^\circ$ for $w$ a semisimple element with unipotent part in $C$. For each $D$ with $D^u=C$, the semisimple part defines a class $\phi(D)\subset A$. For each class $\gamma\subset A$, define

$$\mathcal{S}_{C,\gamma} = \{ D \mid D^u = C,\ \phi(D) = \gamma\}.$$

Choosing $x\in\gamma$, $\operatorname{Irr} Z_A(x)$ the irreducible characters, each $(\gamma,\rho)$ with $\rho\in\operatorname{Irr} Z_A(x)$ yields a unique $G$-equivariant local system $\mathcal{E}_{\gamma,\rho}$ on each $D\in\mathcal{S}_{C,\gamma}$:

$$\mathcal{E}_{\gamma,\rho} = G \times_{Z_G(g)} \rho.$$

Parametrization Theorem [2.4]:

Each unipotent character sheaf $A$ arises uniquely from such $(\gamma,\rho)$:

• If $D\in\mathcal{S}_{C,\gamma}$, $A|_D \simeq \mathcal{E}_{\gamma,\rho}[\dim D + c]$;
• If $D\in\mathcal{S}_{C,\gamma'}$ with $\gamma'\neq \gamma$, $A|_D=0$.

Thus, $A\mapsto(\gamma,\rho)$ is a bijection between sheaves in the family $\mathfrak F_{\mathrm{un}}$ and pairs

$$\mathcal{M}(A) = \{ (g_s,\rho) \mid g_s\in A\ \text{up to conjugacy},\ \rho\in\operatorname{Irr} Z_A(g_s)\}.$$

This mirrors the parametrization of irreducible representations in terms of centralizer components and local systems (Lusztig, 2012).

4. Combinatorial Parametrization via Weyl Groups

A central insight is the equivalence between combinatorial data from the Weyl group and the set of unipotent cuspidal character sheaves. For the Coxeter system $(W,S)$ of $G$, define:

• For $J\subset S$, $W_J$ the parabolic subgroup, $W^J=N_W(W_J)/W_J$ the relative Weyl group.
• For each irreducible factor, a finite set $\mathfrak{S}_{W_J}$ of “cuspidal roots of unity”.

Set

$$\mathfrak{S}_W = \{ (J,\varepsilon,\zeta) \mid J\subset S,\ \varepsilon\in\operatorname{Irr}(W^J),\ \zeta\in\mathfrak{S}_{W_J} \}.$$

Main Correspondence:

$$\mathfrak{S}_W \;\longleftrightarrow\; \{\text{unipotent representations of }G(\mathbb{F}_q)\} \;\longleftrightarrow\; \{\text{unipotent character sheaves on }G\}$$

The bijections are characterized by:

• Frobenius eigenvalues on cohomology in the first exchange,
• The scalar by which central elements act on stalks (“shift-eigenvalue” $X_A$) in the second (Lusztig, 2012).

This is the rigorous realization of Springer–Lusztig theory in terms of combinatorial invariants, essential for explicit classification in types $A$, $B$, $C$, $D$, $E$, $F$, $G$.

5. Examples in Classical Types

• Type $A_{n-1}$: No nontrivial cuspidal pairs; cuspidal character sheaves are the Kummer local systems induced from the maximal torus, restricting to regular semisimple classes as rank-$1$ local systems.
• Type $B_n$, $C_n$: Cuspidal local systems occur precisely when $n=k^2+k$. Here, a unique cuspidal Levi of type $A_{k-1}\times A_{k-2}\times\cdots$ supports a unique cuspidal local system. Its induction gives two cuspidal character sheaves, splitting on the subregular unipotent class as the sum of trivial and nontrivial rank-$1$ local systems (up to shift) (Lusztig, 2012).

6. Methodological and Geometric Techniques

• Restriction to conjugacy classes: Analyzes the geometry of the induction diagram, the semisimplicity of perverse sheaves on centralizers, and stratifications by fibers over semisimple classes.
• Parametrization and endomorphism algebras: The classification of unipotent cuspidal character sheaves uses calculations of endomorphism algebras and restrictions to conjugacy classes, giving direct sums of intersection–cohomology complexes with local–system coefficients indexed by irreducible component group characters.
• Correspondence with representations: The linkage with unipotent representations is established via parabolic induction controlled by Hecke algebras attached to $W^J$ (Deligne–Lusztig, Lusztig), matching eigenvalue invariants on both sides.

7. Consequences for Representation Theory

Cuspidal character sheaves serve as the minimal building blocks for the construction of all character sheaves via parabolic induction. Their parametrization via Weyl group data and component groups allows for explicit expressions of character values at unipotent elements, ties with the almost-characters of $G(\mathbb{F}_q)$, and forms the foundation for the generalized Springer correspondence.

This structure underpins explicit character computations, classifications in exceptional and classical types, and deepens the combinatorial and geometric understanding of the representation theory of reductive groups over algebraically closed fields in non-bad characteristic (Lusztig, 2012).

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References:

• G. Lusztig, "Restriction of a character sheaf to conjugacy classes" (Lusztig, 2012)
• G. Lusztig, "Introduction to character sheaves", Proc. Symp. Pure Math. 47 (1987)
• G. Lusztig, "Families and Springer's correspondence" (Lusztig, 2012)
• G. Lusztig, "Intersection cohomology complexes on a reductive group", Invent. Math. 75 (1984)
• G. Lusztig, "A unipotent support for irreducible representations", Adv. Math. 94 (1992)

All core results and constructions above are adopted verbatim from the cited sources.