Lusztig's Jordan Decomposition in SU(n, q) Groups

• Lusztig's Jordan decomposition is a framework that parametrizes irreducible characters of finite unitary groups by associating semisimple elements with unipotent characters of their centralizers.
• It employs combinatorial techniques and wreath product constructions to extend classical methods, ensuring compatibility with Lusztig's twisted induction for accurate representation labeling.
• The approach adapts Bonnafè’s methods to handle non-connected centers in SU(n, q), using explicit induction and cohomological analyses to control character theory.

Lusztig's Jordan decomposition is a parametrization of the irreducible characters of finite groups of Lie type, particularly special unitary groups SU(n, q), in terms of semisimple elements and the unipotent characters of their centralizers. The decomposition extends the classical results for groups with connected center to groups with non-connected center, such as SU(n, q), and is constructed via explicit techniques paralleling and adapting Bonnafé's methods for SL(n, q). Its defining feature is compatibility with Lusztig's twisted induction—a functor between representations—so that the parametrization commutes with induction from Levi subgroups, which is essential for preserving the structure under representation-theoretic operations. Unipotent characters of centralizers are described using wreath product constructions, and the parametrization explicitly incorporates non-connectedness and quasi-central automorphisms. The development relies on combinatorial and cohomological analysis via Deligne–Lusztig theory, Clifford theory, and the Mellin transform, providing a bijection between irreducible characters in Lusztig series and unipotent characters of the corresponding centralizers.

1. The Jordan Decomposition Framework for SU(n, q)

For finite special unitary groups SU(n, q), which have non-connected centers, each irreducible character is labeled by a semisimple element s (in the dual group, up to G-conjugacy) together with a unipotent character of its centralizer. This yields a bijection: $E(G, [s]) \leftrightarrow E(C^*_G(s), 1)$ where $E(G, [s])$ denotes the Lusztig series (rational series) of characters associated to the conjugacy class of $s$, and $E(C^*_G(s), 1)$ denotes the set of unipotent characters of the centralizer $C^*_G(s)$.

The central difficulty in SU(n, q) versus SL(n, q) arises from the non-connectedness of the center, necessitating adapted parametrization techniques and explicit calculation of centralizers and their unipotent characters.

2. Parametrization of Unipotent Characters via Wreath Products

The centralizers $C^*_G(s)$ can be expressed as generalized wreath products

$H = H^\circ \rtimes A$

where $H^\circ$ is a connected reductive group (typically a direct product of general/unitary linear groups) and $A$ a finite group acting via quasi-central automorphisms. The Weyl group $W$ associated to $H$ decomposes as

$W = W^\circ \rtimes A$

with $W^\circ = N_{H^\circ}(T)/T$ for a maximal torus $T$.

The parametrization in the connected case (GL(n, q), GU(n, q)) employs the Lusztig–Srinivasan construction. For irreducible $\eta \in \text{Irr}(W^\circ)$,

$R_\eta = \sum_{w \in W^\circ} \eta(w) R_{T_w}$

where $R_{T_w}$ is a virtual character from the cohomology of the maximal quasi-torus $T$ of type $w$. There exists a sign $\varepsilon_\eta \in \{-1, 1\}$ such that $\varepsilon_\eta R_\eta$ is irreducible and unipotent.

In the non-connected case, one lifts this bijection by defining: $$R_{\eta*\xi} := \operatorname{Ind}_{H^\circ . A_\eta}^H(\xi R_\eta)$$ where $(\eta * \xi)$ encodes both a character $\eta$ of $W^\circ$ and an irreducible character $\xi$ of the stabilizer $A_\eta \leq A$. This induction procedure and explicit combinatorics provide a complete parametrization of unipotent characters in such wreath product groups.

3. Centralizers of Semisimple Elements in SU(n, q)

Given $s$ a semisimple element in the projective unitary group, Proposition 2.5 demonstrates its preimage in the covering group has a centralizer group isomorphic to a form $H = H^\circ \rtimes A$ with $A$ cyclic of order coprime to $p$. Thus, the full machinery of wreath product parametrization applies to these centralizers.

The bijection

$\aleph_{G,s}: E(G, [s]) \rightarrow E(C^*_G(s), 1)$

is realized using the explicit methods for $H = H^\circ \rtimes A$ above. This is central for labeling the rational series and controlling the character theory via centralizer data.

4. Compatibility with Lusztig’s Twisted Induction

A critical property is that the Jordan decomposition “commutes” with Lusztig’s twisted induction functor

$R_{L \subseteq P}$

for Levi subgroups $L \leq G$. The paper establishes this compatibility via commutative diagrams: $$\begin{matrix} ZE(L, [s]) & \xrightarrow{R_{L \subseteq P}} & ZE(G, [s]) \ \downarrow & & \downarrow \ ZE(C_L^*(s), 1) & \xrightarrow{R_{C(s)}} & ZE(C_G^*(s), 1) \end{matrix}$$ The effect of twisted induction on the parameters $\eta$ and its extensions is computed explicitly (see Theorems 4.4, 4.8), thus ensuring induced representations retain the correct “Jordan label.” The use of the Mellin transform on the character parametrizations (\S4) allows analysis of the effect of induction operators and verifies full compatibility.

5. Adaptation and Extension of Bonnafè’s Methods

Bonnafé’s approach for SL(n, q) treated centralizers as non-connected reductive groups described by wreath products. Adapting to SU(n, q) introduces unitary complications and sign factors in degree formulas. Key points of adaptation include:

• A modified definition of character extensions from $H^\circ$ to $H$ (see Definition 1.3, Remark 2.3).
• Explicit use of irreducible characters of wreath products (Proposition 1.7, Definition 1.6), replacing connected extensions $\chi \rtimes A$.
• Careful handling of Frobenius endomorphism and quasi-central automorphisms (Proposition 1.8).

This nuanced extension employs Clifford theory, Mellin transforms, and explicit combinatorial constructions to fully realize the Jordan decomposition for SU(n, q), respecting all twisted induction compatibilities.

6. Essential Formulas and Structural Summary

The main formulas encapsulating the decomposition are: $R_\eta = \sum_{w \in W^\circ} \eta(w) R_{T_w}$ and

$$R_{\eta*\xi} = \operatorname{Ind}_{H^\circ . A_\eta}^H (\xi R_\eta)$$

for the parametrization of unipotent characters. Compatibility with twisted induction is encoded in commutative diagrams, and the entire construction is anchored to explicit combinatorial and group-theoretic arguments.

The outcome is a bijective parametrization of characters in SU(n, q) rational series by pairs $(s, \text{unipotent character})$ of centralizers, with full compatibility under representation-theoretic operations, and precise adaptation of Bonnafè’s SL(n, q) techniques to the unitary setting.

7. Implications for Representation Theory and Beyond

The developed Jordan decomposition for SU(n, q) is structurally foundational for:

• Classification of irreducible representations via semisimple and unipotent data.
• Analysis of character degrees, defect groups, and block theoretical properties.
• Compatibility with functorial constructions such as Lusztig’s twisted induction, ensuring parabolic and Levi subgroup representations inherit correct parametrization.
• Algorithmic and combinatorial approaches to modular representation theory in finite groups of Lie type, especially where centers are non-connected and classical techniques are insufficient.

By encoding the representation theory in terms of explicit centralizer data and unipotent character labels, this decomposition provides the backbone for structural, computational, and theoretical advances in finite group character theory.