Deep Level Deligne–Lusztig Varieties

• Deep level Deligne–Lusztig varieties are geometric objects that extend classical constructions to p-adic groups using higher congruence subgroups.
• Their construction employs Bruhat–Tits theory and Moy–Prasad filtrations, linking explicit geometric stratifications with advanced cohomological methods.
• The resulting cohomological frameworks yield inner product, orthogonality, and induction formulas that underpin the classification of supercuspidal representations.

Deep level Deligne–Lusztig varieties are a family of varieties that generalize the foundational constructions of Deligne and Lusztig to higher congruence level (deep parahoric) subgroups in the study of representations of $p$-adic groups. Their geometry and cohomology provide a bridge from the classical, finite-field theory of Deligne–Lusztig to the construction and analysis of irreducible supercuspidal representations of $p$-adic groups. The development of deep level Deligne--Lusztig theory encompasses new geometric structures, fundamental inner-product and orthogonality relations, connections to advanced representation-theoretic constructs such as the local Langlands correspondence, and explicit realization of expected correspondences and conjectures in the context of both tame and wild ramification.

1. Definition and Construction of Deep Level Deligne–Lusztig Varieties

Classic Deligne–Lusztig varieties are constructed for reductive groups $G$ over finite fields, using data of a maximal torus $T$ and a Borel subgroup $B$ containing it, and are related to representations of $G(\mathbb F_q)$. Deep level Deligne–Lusztig varieties generalize this by incorporating filtrations from integral models arising from Bruhat–Tits theory for $p$-adic groups $G$ defined over a non-Archimedean local field $k$ with ring of integers $\mathcal O_k$, uniformizer $\varpi$, and residue field $\mathbb F_q$.

For $G$ split over the completion $\breve k$ of its maximal unramified extension, let $x$ be a (special) point in the Bruhat–Tits building. The parahoric group scheme $\mathcal G = \mathcal G_x$ leads to a system of "level-$r$" congruence quotient groups: $G_r = \mathcal G(\mathcal O_k / \varpi^r)$ together with truncated unipotent subgroups $U_r$ arising from parahoric models of $U$ (the unipotent radical of a Bruhat–Tits–rational Borel).

The deep level Deligne–Lusztig variety attached to a torus $T$ (with associated depth-$r$ Moy–Prasad filtration $T_r$) and a Borel $B=TU$ is then defined as the perfectly smooth $\mathbb F_q$-scheme: $$X_{w,r} = \{ g \in G_r : g^{-1} F(g) \in U_r^- \cap F(U_r) \}$$ where $U_r^-$ denotes the opposite unipotent, $F$ is the Frobenius, and $w$ is typically a (twisted) Coxeter element in the Weyl group specifying the "type". Left-multiplication by $G_r$ and right translation by $T_r$ yield commuting group actions.

For tamely ramified tori, the construction incorporates subtler filtrations and centralizer structure; an explicit description of $X_{T,r}$ in terms of elements lying in prescribed Moy–Prasad cosets is used, together with additional structures such as Lagrangian splittings in associated Heisenberg extensions (Ivanov et al., 11 Jan 2026).

These varieties interpolate between the classical Deligne–Lusztig theory ($r=1$), higher-level unipotent varieties, and the semi-infinite or infinite-level theory of Lusztig–Feigin–Frenkel. As $r \to \infty$, the towers of these varieties converge, in an appropriate sense, to Lusztig’s semi-infinite objects (Chan et al., 2018, Takamatsu, 2023).

2. Cohomological Representation Theory and Induction

The primary mechanism linking deep level Deligne–Lusztig varieties to representation theory is through their $\ell$-adic cohomology, especially compactly supported cohomology with coefficients in $\overline{\mathbf Q}_\ell$. For a character $\theta: T(k) \to \overline{\mathbf Q}_\ell^\times$, one considers the $\theta$-isotypic part of the cohomology: $$H_c^*(X_{w,r})[\theta] = \bigoplus_{i \geq 0} H_c^i(X_{w,r}, \overline{\mathbf Q}_\ell)[\theta]$$ which is a virtual representation of the finite parahoric group $G_r$ (or $P_r$).

The alternating sum over degrees defines virtual representations, e.g.,

$$R_{T,\theta}^r := \sum_{i=0}^{2 \dim X_{T,r}} (-1)^i H_c^i(X_{T,r}, \overline{\mathbb Q}_\ell)[\theta],$$

which are then functorially inflated and compactly induced to yield admissible representations of $G(k)$. In settings of tamely ramified tori, these constructed representations are shown to be irreducible and supercuspidal when $\theta$ is regular and the residue characteristic is suitably large (Ivanov et al., 11 Jan 2026).

A central structural result is that, for regular $\theta$ (trivial Weyl stabilizer), $R_{T,\theta}^r$ is irreducible, and every tame supercuspidal is a direct summand in the compact induction of some such $R_{T,\theta}^r$, generalizing Yu's construction (Ivanov et al., 11 Jan 2026). In the unramified context, the variety of types and explicit parameterization of irreducible parahoric representations in terms of tori and characters is achieved, especially in settings of Coxeter type (Ivanov et al., 2024).

3. Inner Product, Orthogonality, and Decomposition Theorems

A key breakthrough in deep level Deligne–Lusztig theory is the generalization of Mackey-type inner product and orthogonality relations to the deep congruence level context. For characters $\theta, \theta'$ of associated tori,

$$\dim_{\overline{\mathbf Q}_\ell} \operatorname{Hom}_{P_r}\bigl( H_c^*(X_{w,r})[\theta], H_c^*(X_{w,r})[\theta'] \bigr) = \#\{ u \in W_e^F : u(\theta) = \theta' \},$$

where $W_e^F$ is the Weyl group of the reductive quotient fixed by Frobenius (Ivanov et al., 2024, Dudas et al., 2020, Chen, 2017).

This result implies that for regular (i.e., in general position) characters, the associated cohomological representations are irreducible, and the system of such representations forms an orthonormal basis under the inner product on class functions of $P_r$. For Coxeter data, the orthogonality relations specialize to unipotent representation theory, mirroring the role of classical Deligne–Lusztig characters (Dudas et al., 2020).

Explicit branching rules and induction formulas analogous to those in Yu’s construction (involving compact “Yu-type” subgroups and Heisenberg representations) are established, with consequences for the structure and degree-concentration of cohomology (Ivanov et al., 17 Mar 2025).

4. Geometry, Perfection, and Product Decompositions

The fine geometric structure of deep level Deligne–Lusztig varieties underlies their representational properties. For groups such as $\mathrm{GSp}_{2n}$, connected components of affine Deligne–Lusztig varieties at large level decompose (after perfection) as direct products: $$X_{r,m}(b)^{\circ,\mathrm{perf}} \simeq X_B^{\mathrm{perf}} \times \mathbb A^{\mathrm{perf}},$$ where $X_B$ is a classical Deligne–Lusztig variety for the finite reductive group and $\mathbb A^{\mathrm{perf}}$ is a perfect affine space (Takamatsu, 2023). The explicit description of the Schubert cell stratification, coordinate charts from Iwahori decompositions, and the separation of nonlinear equations (classical DL) from affine constraints (affine space) yield strong geometric and representation-theoretic consequences.

Chan–Ivanov-type varieties $X_h$ for congruence quotient data provide finite-type analogues, and their inverse limits can be identified with components of infinite-level or semi-infinite varieties, establishing the compatibility of finite-level and semi-infinite constructions (Chan et al., 2018, Takamatsu, 2023).

In Coxeter-type settings, the key geometric input is the choice of minimal-length "convex elements" in the Weyl group (minimal $\sigma$-elliptic representatives), leading to affine fibration properties in the stratification, cohomological vanishing off the main stratum, and comparison isomorphisms with simpler varieties (Ivanov et al., 17 Mar 2025).

5. Applications to Supercuspidal Representations and the Local Langlands Correspondence

The primary representation-theoretic application is the construction and exhaustion of supercuspidal representations of $p$-adic reductive groups. The cohomology of deep level Deligne–Lusztig varieties with regular or Howe-factorizable characters realizes the inducing data of all (tame) supercuspidal representations, generalizing and geometricizing the construction of Yu (Ivanov et al., 11 Jan 2026, Ivanov et al., 17 Mar 2025).

At infinite level, these varieties and their cohomological constructions coincide with Lusztig's semi-infinite varieties, realizing Lusztig’s original conjectural link between geometry and cuspidal representation theory (Takamatsu, 2023, Chan et al., 2018).

Cohomological constructions yield representations compatible with the local Langlands and Jacquet–Langlands correspondences. For instance, for $G=\mathrm{GL}_n$, cohomology with characters induced from unramified field extensions yields isomorphism classes of irreducible supercuspidals and matches the expected character-theoretic formulas (Chan et al., 2018).

In the context of Newton stratification in Shimura varieties, local affine Deligne–Lusztig varieties precisely match the geometry and dimensions of Newton strata, confirming parts of the Grothendieck conjecture in parahoric level (Hamacher, 2020).

6. Stratifications, Induction, and Further Directions

The Drinfeld stratification organizes deep level Deligne–Lusztig varieties according to Levi subgroups, with strata corresponding to intermediate induction steps and cohomological purity results (Chan et al., 2020). Results confirm concentration of cohomology in a single degree under appropriate depth and regularity constraints, and provide comparison isomorphisms across Drinfeld strata.

A salient open problem is the detailed analysis of singularities and the development of perverse sheaf frameworks for these higher level varieties, especially beyond classical and Coxeter-type cases (Chen, 2017). Ongoing work connects geometric parameters arising from Fargues–Scholze theory to cohomological constructions from deep level Deligne–Lusztig geometry (Ivanov et al., 17 Mar 2025). The algebraization problem—determining which smooth representations of $p$-adic groups arise geometrically—remains open for general groups and ramification.

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Table: Summary of Key Deep Level Deligne–Lusztig Constructions

Construction	Level Structure	Parameter Data
Classical Deligne–Lusztig	Finite field ($r=1$)	$(T, \theta)$
Deep Level DL Varieties	Parahoric/level $r>1$	$(T, r, \theta)$
Semi-infinite DL Varieties	Infinite-level, full loop group	$(T, \theta)$
Deep Level Coxeter Type	Coxeter element, large $r$	$(T, \mathrm{Coxeter}~w)$
Drinfeld Stratification	Strata by Levi subgroups	$(L, \theta)$

The development of deep level Deligne–Lusztig varieties provides a uniform geometric framework for the construction, realization, and decomposition of supercuspidal representations of $p$-adic groups, connecting the geometry of affine flag manifolds, induction theory, and the representation-theoretic correspondences of the Langlands program (Takamatsu, 2023, Ivanov et al., 11 Jan 2026, Ivanov et al., 2024, Ivanov et al., 17 Mar 2025, Hamacher, 2020).