Semisimple Picky Elements in Lie Type Groups

• Semisimple picky elements are defined as semisimple elements that lie in a unique Sylow ℓ-subgroup with centralisers determined by a unique Sylow d-torus.
• They underpin the Picky Conjecture by establishing bijections between irreducible characters of the group and its local subnormaliser, preserving key invariants.
• Their classification employs Deligne–Lusztig induction and Lusztig’s Jordan decomposition, offering actionable insights for modular representation theory.

A semisimple picky element is a semisimple element which lies in a unique Sylow ℓ-subgroup (for a prime ℓ ≠ the defining characteristic) of a finite group of Lie type or an associated algebraic group, a structure deeply intertwined with subnormalisers and the existence of tight character correspondences that generalize aspects of the McKay Conjecture. The concept and its representation-theoretic consequences have been formalized and classified in recent work, culminating in the verification of the Picky Conjecture for quasi-simple finite groups of Lie type outside the defining prime (Malle, 13 Mar 2025, Malle et al., 21 Oct 2025).

1. Definition and Characterisation of Semisimple Picky Elements

A semisimple element $x$ of a finite group $G$ (typically of Lie type, $G = \mathbb{G}^F$) is called picky with respect to a prime $\ell$ (distinct from the defining prime $p$) if $x$ lies in a unique Sylow $\ell$-subgroup of $G$ (Malle, 13 Mar 2025, Malle et al., 21 Oct 2025). Equivalently, the subnormaliser of $x$ in $G$ is the normaliser of the unique Sylow $\ell$-subgroup containing $x$: $$Sub_G(x) = N_G(P) \quad \text{for some unique } P \in Syl_\ell(G),\; x\in P$$ For semisimple $\ell$-elements (with $\ell$ an odd prime different from $p$), this pickiness is further sharpened by the existence of a unique Sylow $d$-torus $S$ (with $d = e_\ell(q)$, defined by the multiplicative order of $q$ modulo $\ell$) such that $C_G(x) = C_G(S)$ (Malle, 13 Mar 2025): $x \text{ picky} \iff C_G(x) = C_G(S)$ In the important case $d=1$ (e.g., when $\ell$ divides $q-1$), $x$ is picky if and only if it is regular semisimple: its centraliser is a maximal torus.

2. Subnormalisers and the Local Structure

The subnormaliser $Sub_G(x)$ of an element $x \in G$ is the subgroup generated by the normalisers of all Sylow subgroups containing $x$: $$Sub_G(x) = \langle N_G(P) : x \in P,\, P \in Syl_\ell(G) \rangle$$ For semisimple picky elements $x$, the structure simplifies significantly: there exists a unique Sylow $d$-torus $S$ with $x \in S$ and

$Sub_G(x) = N_G(S)$

This reduction is critical both for the tractability of the subnormaliser in group-theoretic terms and as a foundation for explicit character correspondences, as the normaliser of a torus is often much more accessible than general local subgroups.

3. The Picky Conjecture and Character Correspondences

The Picky Conjecture, as formulated by Moretó and Rizo and proven for groups of Lie type (outside the defining characteristic), posits a bijection between the irreducible complex characters of $G$ not vanishing on a picky element $x$ and the irreducible characters of $Sub_G(x)$ that do not vanish on $x$ (Malle et al., 21 Oct 2025, Malle, 13 Mar 2025). In favorable cases, this correspondence preserves more refined invariants—most notably degrees, character values (up to sign), and fields of values.

A particularly strong form, the Strong Picky Conjecture, asserts that for many classes of characters (notably those afforded by Deligne–Lusztig induction from suitable Levi subgroups), the nonzero values of the character on $x$ in $G$ and on $x$ in $N_G(P)$ coincide up to sign [(Malle et al., 21 Oct 2025), Prop. 3.9]: $$\chi(x) = \pm (\text{Induced value of } \theta \text{ on } x)$$ where $\chi = \pm R_L^G(\theta)$ is an irreducible character of $G$ from a minimal $d$-split Levi $L$ and $\theta$ a character of $L$.

4. Classification for Small Primes and Structural Criteria

The complete classification for $\ell = 2$ and $3$ in quasi-simple groups of Lie type is accomplished in [(Malle et al., 21 Oct 2025), §4]. For every simple, simply connected group $\mathbb{G}$ over a field of odd characteristic $p$, a picky $2$-element is characterised as follows:

• It lies in a unique $F$-stable maximal torus $T$ with $|T^F| = (q-1)^a(q + 1)^b$,
• Its order must be a $2$-power determined by the cyclotomic factors dividing $|T^F|$ [(Malle et al., 21 Oct 2025), Prop. 4.3]. The classification is case-by-case for small rank and exceptional types; see tables providing the centraliser structures and consequences for character degrees.

In many small rank groups (e.g., certain $^2B_2(q)$ and $^3D_4(q)$), picky elements of order $2$ or $3$ exist only for special $q$ (Fermat or Mersenne primes, $q=9$, etc.), and, except for these cases, pickiness is more common for primes $\ell$ with abelian but not cyclic Sylow subgroups.

5. Theoretical Framework: Lusztig Induction and Jordan Decomposition

The verification of the Picky Conjecture for semisimple picky elements utilises Deligne–Lusztig theory—specifically, that values of irreducible characters on $x$ are determined via Lusztig induction from tori or minimal $d$-split Levi subgroups: $$\chi(x) = \sum_\phi \langle \chi, R_L(\phi) \rangle\, \phi(x)$$ For picky $x$, almost all terms appearing in this expansion are conjugate to a single character, which enables the transfer of information to the local subgroup $N_G(S)$, since $x$ is locally as “generic” as possible.

This compatibility is strengthened by properties of Lusztig's Jordan decomposition, especially when it preserves sign and degree structures; under mild additional hypotheses, the Strong Picky Conjecture holds in all cases.

6. Broader Impact and Connections

The paper of semisimple picky elements not only supplies explicit local–global bridges (generalizing the Isaacs–Malle–Navarro reduction in the McKay conjecture) but also clarifies the role of regular semisimple elements and their centralisers in character theory and the representation theory of finite groups of Lie type.

The parametrization of semisimple classes and semisimple characters (in terms of tori, their component groups, and multi-faceted data from the Brauer complex and affine Weyl groups) is essential for this ongoing local–global analysis (Brunat, 2010). The combinatorial-geometric structure of these elements (e.g., having centraliser a maximal torus) is critical both structurally and algorithmically, since most character-theoretic invariants then reflect properties of very tractable local subgroups.

Table: Criteria and Consequences for Semisimple Picky Elements

Feature	Condition/Structure	Consequence
Definition	Unique Sylow ℓ-subgroup containing $x$	$Sub_G(x) = N_G(P)$
Centraliser	$C_G(x) = C_G(S)$ (unique Sylow $d$-torus $S$)	$x$ 'regular' ⇒ $C_G(x)$ toral
Character Correspondence	Bijection $Irr^{(0)}(G) \leftrightarrow Irr^{(0)}(Sub_G(x))$	Preserves ℓ-parts, values (sign)
Small primes ($\ell=2,3$)	Explicit case-by-case classification	Table of groups/types

7. Applications and Future Directions

The existence and classification of semisimple picky elements not only facilitates the verification of the Picky Conjecture (and its refinements)—thereby providing the sharpest bridges yet between global and local representation theory—but also heralds new computational techniques for explicit character evaluation, decomposition numbers, and automorphism group actions for finite groups of Lie type. These insights are anticipated to yield further progress in modular representation theory, block theory, and the algorithmic paper of finite simple and quasi-simple groups.

The extension of these correspondences beyond quasi-simple groups, analysis for more general primes or more intricate local subgroups, and the systematic paper of their exceptional behavior in covering and non-split cases present promising avenues for future research (Malle, 13 Mar 2025, Malle et al., 21 Oct 2025).