Extension of p-rationality preservation to Lusztig induction

Establish whether the p-rationality level of a Lusztig induced character from an F-stable Levi subgroup equals that of the inducing character when the induced character has p'-degree, with both characters almost p-rational in the remaining cases.

Background

The paper proves that for a Deligne–Lusztig character induced from an F-stable maximal torus, p-rationality is preserved whenever the relevant p-rationality level is at least 2, while both the inducing character and the induced character are almost p-rational otherwise. The authors ask whether the same conclusion extends to general Lusztig induction from an F-stable Levi subgroup of a parabolic subgroup. They state this as Conjecture A and note that it is new even when the parabolic subgroup is F-stable, where Lusztig induction becomes ordinary Harish–Chandra induction.

References

Can Theorem~\ref{thm:main} be extended to more general Lusztig induced characters $R_{ \le }{}(\psi)$ where $$ is an $F$-stable Levi subgroup of a parabolic subgroup $$ of $$? Some further cases we prove in Section~\ref{sec:ConjC} have convinced us to put forward the following. \begin{conjecturea} \label{conj:Lusztig-induction} Let $,F$, $r$, and $p$ be as above. Let $$ an $F$-stable Levi subgroup of a parabolic subgroup $$ of $$, and $\psi \in(F)$. Suppose that the Lusztig induced character $R_{ \le}{}(\psi)\in(F)$ has $p'$-degree. Then \left(R_{ \le }{}(\psi)\right) = (\psi) whenever $\max{(\psi),(R_{ \le }{}(\psi))}\ge2$. Otherwise, both $\psi$ and $R_{ \le }{}(\psi)$ are almost $p$-rational. \end{conjecturea}

— On the $p$-rationality of Deligne--Lusztig characters  (2608.17871 - Hung, 18 Aug 2026) in Conjecture A, Introduction, immediately before Section 2