Deep Level Deligne–Lusztig Reps

• Deep level Deligne–Lusztig representations are cohomologically-defined virtual representations that classify positive-depth supercuspidal representations of p-adic groups.
• The construction employs geometric analogues of classical Deligne–Lusztig theory, using the cohomology of varieties over finite-level quotients tied to elliptic maximal tori.
• The framework integrates techniques from affine Deligne–Lusztig theory and Yu–Kaletha constructions to decompose, analyze, and realize representations in varied parahoric settings.

Deep level Deligne–Lusztig representations are cohomologically-defined virtual representations of parahoric subgroups in $p$-adic groups, constructed as geometric analogues and deep-level analogues of the classical Deligne–Lusztig theory for finite groups of Lie type. These constructions realize, and in many cases classify, positive-depth irreducible (supercuspidal) representations of $p$-adic groups as compact inductions of geometric representations arising from the cohomology of certain infinite or finite-level varieties attached to elliptic maximal tori, often in the context of tamely ramified or unramified tori. The deep-level theory generalizes previous unramified cases and provides a geometric realization paralleling the analytic constructions of Yu and Kaletha for tame supercuspidals, with far-reaching implications for the local Langlands correspondence and the internal structure of representations of reductive $p$-adic groups (Ivanov et al., 11 Jan 2026, Nie, 2024, Ivanov et al., 17 Mar 2025).

1. The Deep Level Deligne–Lusztig Construction

Let $F$ be a non-archimedean local field with ring of integers $\mathcal{O}_F$, maximal ideal $p_F$, and residue field $k_F$ of characteristic $p$; let $G$ be a connected reductive group over $F$, splitting over a (fully general: tamely ramified) finite extension $E/F$. Fix an elliptic maximal torus $T \subset G$, split over $E$. For depth parameter $r \geq 1$, define:

• $G_r = G(\mathcal{O}_F/p_F^r)$, the "depth-$r$" smooth affine group scheme over $k_F$ (likewise for $T_r$).
• A distinguished pro-$p$ Iwahori subgroup $U_r \subset G_r$ (the "unipotent radical" of an $F$-stable Iwahori).

Given a character $$\theta : T_r^F \to \overline{\mathbb{Q}}_\ell^\times$$, define the deep level Deligne–Lusztig variety

$$X_{T, \theta}^r = \{ g \in G_r : g^{-1} F(g) \in U_r \},$$

with the $\ell$-adic rank-one local system $\mathcal{L}_\theta$ on $G_r$ inflated from $T_r$. The fundamental representation is the virtual $G_r^F$-module

$$R_{T, \theta}^r = \sum_{i=0}^{*} (-1)^i H_c^i(X_{T, \theta}^r, \mathcal{L}_\theta)$$

where $G_r^F$ and $T_r^F$ act by left and right translation, respectively (Ivanov et al., 11 Jan 2026).

2. Geometric and Cohomological Features

The cohomology $H_c^i(X_{T, \theta}^r, \mathcal{L}_\theta)$ is a virtual (and, under regularity/hypotheses, actual) representation of $G_r^F$, often influenced by the choice of torus $T$, the depth $r$, and the nature of $\theta$ (regular, generic, Howe admissible, etc.). Key facts:

• For tamely ramified elliptic $T$ and $p > 2$, the variety $X_{T, \theta}^r$ is smooth (perfect) over $k_F$, with dimension $\dim G_r - \dim T_r -$ (Moy–Prasad depth).
• The cohomology is frequently concentrated in a single degree when $\theta$ is sufficiently regular or generic (Ivanov et al., 17 Mar 2025, Ivanov et al., 2024).
• Cohomological analysis uses Mackey formulae, Heisenberg splittings, Lagrangian subgroups (via filtration), and reductions to additive-type Deligne–Lusztig calculations.

The deep-level construction extends seamlessly to parahoric subgroups via positive loop groups and uses the Greenberg functor to parametrize finite-level quotients, enabling the realization of "higher" Deligne–Lusztig varieties as finite-type or ind-perfect schemes (Ivanov et al., 11 Jan 2026, Ivanov et al., 17 Mar 2025).

3. Decomposition and Explicit Parametrization

A crucial advance is the explicit decomposition of deep-level Deligne–Lusztig representations. For each depth-$r$ character $\phi$ on $T_r^F$ admitting a Howe factorization, there is a compact open subgroup ("Yu-type subgroup") $K_{\phi, r}$ and a (finite group-theoretic) Weil–Heisenberg representation $\kappa_\phi$ such that:

$$R_{T,U,r}^G(\phi) = \operatorname{Ind}_{K_{\phi, r}^F}^{G_r^F} (\kappa_\phi \otimes R_{T, U, 0}^{G^0}(\phi_{-1})),$$

where $R_{T, U, 0}^{G^0}(\phi_{-1})$ is the classical depth-zero Deligne–Lusztig representation attached to the Levi quotient in the factorization (Ivanov et al., 17 Mar 2025, Nie, 2024). Each irreducible constituent is thus parametrized by a refined version of Yu's data:

• A Levi sequence $$G^{-1} = T \subset G^0 \subset \dots \subset G = G^d$$.
• Characters and splittings attached to progressively deeper level subgroups in the filtration.
• The geometric Weil–Heisenberg model replaces the classical, connecting $\ell$-adic cohomology to analytic induction.

This decomposition enables a direct link between the geometry of $X_{T, \theta}^r$ and the admissible representation-theoretic data utilized in tame supercuspidal induction.

4. Supercuspidal Realization and the Local Langlands Correspondence

Deep-level Deligne–Lusztig representations geometrically realize, or provide direct summands for, almost all irreducible supercuspidal representations of $G(F)$ in the tame (and even mildly wild) case. Explicitly:

• For any regular elliptic pair $(T, \theta)$ of depth $\leq r$, the compact induction

$$\operatorname{c-Ind}_{G_r^F}^{G(F)} R_{T, \theta}^r \cong \pi_{(T, \theta)}$$

recovers the corresponding Yu–Kaletha supercuspidal (Ivanov et al., 11 Jan 2026, Nie, 2024). More generally, all tame irreducible supercuspidals appear as direct summands in compact inductions from cohomology of deep-level Deligne–Lusztig varieties.

• The trace and character formulas mirror those from the classical Deligne–Lusztig theory but now apply in the context of parahoric subgroups and deeper congruence levels.
• Compactly induced deep-level representations match the expected local Langlands parameters via trace identities and compatibility with automorphic induction and Jacquet–Langlands transfer, as established in both the unramified and division-algebra (inner form) cases (Ivanov et al., 11 Jan 2026, Chan et al., 2018, Chan, 2015).

5. Cohomological and Orthogonality Properties

Much of the structure of deep-level Deligne–Lusztig representations is governed by:

• Orthogonality relations: For elliptic tori $T$, the induced representations satisfy strong scalar product formulas and Weyl-group-invariant orthogonality analogous to those in finite group Deligne–Lusztig theory (Chan, 2024, Dudas et al., 2020).
• Concentration in single degree: For sufficiently generic parameters, the cohomology is concentrated and gives rise to irreducible representations, often maximal in the sense of achieving the Weil bound on rational points (Ivanov et al., 2024, Ivanov et al., 17 Mar 2025).
• Stability phenomena: In Coxeter type and for certain torus settings, higher level unipotent representations degenerate to their level-one avatars for large $q$ (Chen, 2024).

The interplay of these properties secures the irreducibility, multiplicity-free decomposition, and necessary independence from technical choices (e.g., Borel subgroup) for the representations.

6. Connections to Other Geometric and Representation-Theoretic Frameworks

Deep-level Deligne–Lusztig theory intersects with several other frameworks:

• Kirillov's orbit method: For pro-$p$ and finite $p$-group quotients arising from Moy–Prasad theory, the deep-level representations coincide with those predicted by orbit method, with coadjoint orbits parametrizing irreducible modules (Ivanov et al., 2024).
• Affine and semi-infinite Deligne–Lusztig theory: The infinite-level and semi-infinite varieties realize local Langlands and Jacquet–Langlands correspondences, embedding finite-depth constructions as components of larger ind-schemes (Takamatsu, 2023, Chan et al., 2018).
• Parahoric and higher level constructions: The parahoric Deligne–Lusztig representations, and their orthogonality, generalize the finite-field and depth-zero cases to all positive depths, subsuming classical packets as special cases (Chan, 2024).
• Combinatorial models: For groups like $\mathrm{GL}_n$, flag model constructions and admissible flags afford explicit combinatorial parametrizations of nilpotent and regular orbits, further bridging algebraic and geometric perspectives (Chen, 2019).

7. Applications, Drinfeld Stratification, and Future Directions

Among the notable applications:

• The Chan–Oi Drinfeld stratification conjecture is resolved in the context of deep-level varieties: on each $\phi$-isotypic component, the cohomology localizes to a single "Drinfeld stratum," simplifying the decomposition (Ivanov et al., 17 Mar 2025).
• Explicit geometric construction of Fargues–Scholze parameters for epipelagic representations is facilitated by the computation of cohomological purity and Frobenius trace formulas in this setting (Ivanov et al., 17 Mar 2025).
• Explicit sign formulas govern the parity of the representations, with supporting evidence from Coxeter types and low rank cases (Chen, 2024).
• The theory provides a template for extending geometric methods to more general (e.g., wild, ramified) types—though full generalization beyond tamely ramified elliptic tori remains open (Chan, 2015, Nie, 2024).

Deep-level Deligne–Lusztig representations, via the interplay of geometry, group theory, and cohomology, are now central in the structural understanding and classification of irreducible supercuspidal representations of $p$-adic groups, and continue to drive connections between local harmonic analysis and arithmetic geometry (Ivanov et al., 11 Jan 2026, Ivanov et al., 17 Mar 2025, Nie, 2024).