---
title: Deep Level Deligne–Lusztig Reps
url: https://www.emergentmind.com/topics/deep-level-deligne-lusztig-representations
type: topic
---

# Deep Level Deligne–Lusztig Reps

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 [2601.06929; 2406.06430; 2503.13412].

## 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 [2601.06929].

## 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 [2503.13412; 2402.09017].
- 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 [2601.06929; 2503.13412].

## 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 [2503.13412; 2406.06430]. 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 [2601.06929; 2406.06430]. 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 [2601.06929; 1811.11204; 1505.07185].

## 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 [2405.00671; 2010.15489].
- 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 [2402.09017; 2503.13412].
- Stability phenomena: In Coxeter type and for certain torus settings, higher level unipotent representations degenerate to their level-one avatars for large $q$ [2404.00233].

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 [2402.09017].
- **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 [2306.17382; 1811.11204].
- **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 [2405.00671].
- **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 [1904.00769].

## 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 [2503.13412].
- 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 [2503.13412].
- Explicit sign formulas govern the parity of the representations, with supporting evidence from Coxeter types and low rank cases [2404.00233].
- 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 [1505.07185; 2406.06430].

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 [2601.06929; 2503.13412; 2406.06430].

Source: https://www.emergentmind.com/topics/deep-level-deligne-lusztig-representations