Deep level Deligne--Lusztig induction for tamely ramified tori
Alexander B. Ivanov, Sian Nie
TL;DR
This work extends Deligne--Lusztig theory to tamely ramified tori over local fields by constructing deep level DL varieties in a Heisenberg-extended setting. It develops a Mackey-type framework and a concentration theorem to compare Green functions and traces with Yu–Kaletha supercuspidal data, yielding a cohomological realization of regular and general supercuspidals via $H_c^*(Z,\overline\mathbb{Q}_\ell)[\phi]$ and its extensions. The main contributions include a character formula for the DL-type modules, a detailed relation between $\mathcal{R}_{\mathbb{T}}^{\BK}(\phi)$ and its dagger counterpart, and an exhaustion result asserting that every irreducible supercuspidal arises (up to summands) from the cohomology of the constructed deep level DL varieties under suitable conditions. Overall, the paper provides a robust geometric framework for tamely ramified tori, tying together Moy–Prasad data, Heisenberg extensions, Green functions, and cohomological realizations of supercuspidals with potential for broader applicability in $p$-adic representation theory.
Abstract
Deep level Deligne--Lusztig representations, which are natural analogues of classical Deligne--Lusztig representations, recently play an important role in geometrization of irreducible supercuspidals of $p$-adic groups. In this paper, we propose a construction of deep level Deligne--Lusztig varieties/representations in the tamely ramified case, extending previous constructions in the unramified case. As an application, under a mild assumption on the residue field, we show that each regular irreducible supercuspidal is the compact induction of a deep level Deligne--Lusztig representation, and generally, each irreducible supercuspidal is a direct summand of the compact induction of the cohomology of a deep level Deligne--Lusztig variety.