On the $p$-rationality of Deligne--Lusztig characters

Abstract: Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $χ:=R_{\mathbf{T}}<sup>{\mathbf{G}}(θ)$ is a Deligne--Lusztig character of a finite reductive group $\mathbf{G}<sup>F$, with $θ$ an irreducible character of a maximal torus $\mathbf{T}<sup>F$, and if $χ$ has degree prime to $p$, then the so-called $p$-rationality level of $χ$ coincides precisely with that of $θ$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p&#39;$-degree.

• The paper proves that a p′-degree Deligne–Lusztig character induced from a maximal torus has the same p-rationality level as its source character, except when both are almost p-rational.
• Its methods combine an ℓ-invariant, Sylow-subgroup restrictions, induction formulas, semisimple character values, and Green-function integrality to compare fields of values.
• The results determine maximal p-rationality levels in several Lusztig series of general and unitary linear groups, while establishing defining-characteristic and p-solvable cases of a broader Lusztig-induction conjecture.

Overview

The paper "On the $p$-rationality of Deligne–Lusztig characters" (2608.17871) studies how Lusztig induction affects the $p$-rationality level of character values of finite reductive groups $\mathbb{G}^F$, where $\mathbb{G}$ is a connected reductive linear algebraic group over $\overline{\mathbb{F}}_r$ and $F$ a Steinberg endomorphism. For a virtual character $\Xi$, the $p$-rationality level is $\nu_p(c(\Xi))$, where $c(\Xi)$ is the conductor of the field of values $p$0. The central result is that for a Deligne–Lusztig character $p$1 of $p$2-degree, the $p$3-rationality level of $p$4 equals that of $p$5, except in the degenerate case where both are almost $p$6-rational. This provides concrete evidence for the empirical observation that simple groups of Lie type can exhibit arbitrarily high $p$7-irrationality, in contrast with alternating and sporadic groups.

The $p$8-invariant and preparatory lemmas

The technical backbone is an $p$9-invariant, first appearing in Isaacs–Navarro's work on Sylow restrictions in $\mathbb{G}^F$0-solvable groups. For a virtual character $\mathbb{G}^F$1 of $\mathbb{G}^F$2, writing $\mathbb{G}^F$3 for the sum of constituents of $\mathbb{G}^F$4-rationality level $\mathbb{G}^F$5, one sets $\mathbb{G}^F$6. Two lemmas refine earlier results of Hung–Schaeffer Fry (2608.17871): first, $\mathbb{G}^F$7 implies $\mathbb{G}^F$8; second, for $\mathbb{G}^F$9 and $\mathbb{G}$0, the $\mathbb{G}$1-invariants of the Sylow restrictions $\mathbb{G}$2 and $\mathbb{G}$3 agree whenever either is at least $\mathbb{G}$4. Combining these with the character formula for induction yields the key comparison: if $\mathbb{G}$5 and $\mathbb{G}$6 has level at least $\mathbb{G}$7, then under either of two hypotheses — the level of $\mathbb{G}$8 coinciding with $\mathbb{G}$9, or $\overline{\mathbb{F}}_r$0 being linear — the levels of $\overline{\mathbb{F}}_r$1, $\overline{\mathbb{F}}_r$2, and the Sylow restriction all coincide.

The main theorem and its consequences

The main theorem states: if $\overline{\mathbb{F}}_r$3 has $\overline{\mathbb{F}}_r$4-degree, then $\overline{\mathbb{F}}_r$5 whenever $\overline{\mathbb{F}}_r$6; otherwise both $\overline{\mathbb{F}}_r$7 and $\overline{\mathbb{F}}_r$8 are almost $\overline{\mathbb{F}}_r$9-rational. The proof for $F$0 exploits the character formula for values at semisimple elements, $F$1 with $F$2, together with integrality of Green functions. The $F$3-degree hypothesis forces $F$4, and Lemma on linear characters applied to $F$5 gives $F$6, which is bounded above by the level of the semisimple values of $F$7; the reverse inequality is immediate from the character formula. Note the theorem is conditional on $F$8 being coprime to $F$9; the paper does not address characters of degree divisible by $\Xi$0.

A corollary transfers the statement to Lusztig series: if $\Xi$1 is semisimple with $\Xi$2 and $\Xi$3 for some $\Xi$4-stable maximal torus $\Xi$5 containing $\Xi$6, then the series $\Xi$7 contains a character of $\Xi$8-rationality level at least $\Xi$9. Combined with the known upper bound $p$0 from Hung–Tiep, this yields an exact determination of the maximal $p$1-rationality level in $p$2 for $p$3 or $p$4 under the stated torus-index hypotheses. This is a strong, quantitative conclusion: the $p$5-irrationality of characters in such series is governed exactly by the $p$6-part of the order of the labeling semisimple element.

Evidence for the Lusztig induction conjecture

The paper proposes a conjecture generalizing the main theorem to arbitrary Lusztig induced characters $p$7 of $p$8-degree: the level of the induction equals that of $p$9, unless both are almost $\nu_p(c(\Xi))$0-rational. Notably, the conjecture is new even for $\nu_p(c(\Xi))$1-stable parabolics, where it reduces to a question about ordinary induction preserving $\nu_p(c(\Xi))$2-rationality — a question the author explicitly leaves unanswered.

Two substantial cases are verified. First, in the defining characteristic $\nu_p(c(\Xi))$3, the conjecture holds: for $\nu_p(c(\Xi))$4 of $\nu_p(c(\Xi))$5-degree, either both $\nu_p(c(\Xi))$6 and $\nu_p(c(\Xi))$7 are almost $\nu_p(c(\Xi))$8-rational, or both have level exactly $\nu_p(c(\Xi))$9. The proof rests on a lemma, attributed to Malle's work, that irreducible characters of $c(\Xi)$0-degree of finite simple groups of Lie type in characteristic $c(\Xi)$1 have $c(\Xi)$2-rationality level at most $c(\Xi)$3, with the only exceptions to almost rationality being $c(\Xi)$4 and $c(\Xi)$5 in characteristic $c(\Xi)$6. The exceptional case analysis for $c(\Xi)$7 of type $c(\Xi)$8 with Levi containing $c(\Xi)$9 is carried out via the CHEVIE/GAP decomposition matrices of Lusztig induction and the character table of $p$00. A related remark records that Geck's character-sheaf methods and Tiep–Zalesskii's results imply all irreducible characters are almost $p$01-rational for $p$02, but this fails for $p$03 even among odd-degree characters.

Second, the conjecture holds when $p$04 is $p$05-stable and $p$06 is $p$07-solvable, via an Isaacs–Navarro result on conductors in $p$08-solvable groups. The paper also proves Navarro–Tiep's Conjecture C — that $p$09 for $p$10-degree $p$11 — for finite reductive groups in the defining characteristic. The proof is a Clifford-theoretic argument: a level-$p$12 character forces $p$13 and a constituent on $p$14 or $p$15 taking values in $p$16 at a suitable unipotent element, so $p$17 lies in the field of the Sylow restriction.

Limitations and open questions

The main theorem and its corollaries are restricted to characters of $p$18-degree; no statement is made for degrees divisible by $p$19. The general Conjecture on Lusztig induction remains open, including its $p$20-stable parabolic case, which would follow if the $p$21-solvable hypothesis in the Isaacs–Navarro-derived lemma could be removed — the author states this is not currently known. The exact maximality result for $p$22 and $p$23 requires the torus-index divisibility condition, and its extension to other types is not established. The $p$24 analysis depends on the classification of the small exceptional groups $p$25 and $p$26 and on explicit character tables, so it does not generalize verbatim.

Conclusion

The paper establishes that Deligne–Lusztig induction from a maximal torus preserves $p$27-rationality levels for $p$28-degree characters, derives exact maximal $p$29-rationality levels in certain Lusztig series of linear and unitary groups, and verifies the proposed generalization to arbitrary Lusztig induction in the defining characteristic and in the $p$30-solvable, $p$31-stable parabolic case, while also settling Navarro–Tiep's Sylow-restriction conjecture in defining characteristic. The remaining cases of the Lusztig induction conjecture, and the extension beyond $p$32-degree, constitute the natural open problems left by this work (2608.17871).