Bruhat–Tits Stratification in Arithmetic Geometry

• Bruhat–Tits stratification is a systematic decomposition of p-adic spaces linked to reductive groups into locally closed pieces based on combinatorial invariants like facets and vertex lattices.
• It integrates analytic compactifications via Berkovich geometry with the combinatorial structure of buildings, facilitating precise geometric and arithmetic analyses.
• Applications include modeling Shimura varieties, moduli of shtukas, and Deligne–Lusztig varieties, supporting advanced cohomological and representation-theoretic studies.

Bruhat–Tits stratification refers to a systematic decomposition of certain spaces attached to $p$-adic reductive groups—such as buildings, Rapoport–Zink spaces, special fibers of Shimura varieties, and moduli of shtukas—into locally closed, often smooth, pieces (strata) indexed by combinatorial or group-theoretic invariants. The indexing data are determined by the incidence structure of vertices, facets, or more general faces in the Bruhat–Tits building of the underlying group or related root data. These stratifications play a central role in arithmetic geometry, representation theory, and the paper of $p$-adic period domains, relating the geometry of moduli spaces to the combinatorics and arithmetic of group theory.

1. Basic Structures and Origins

The Bruhat–Tits building $\mathcal{B}(G,k)$ of a reductive group $G$ over a non-Archimedean local field $k$ is constructed by gluing Euclidean apartments (affine spaces modeled on $$\Sigma_{\mathrm{vect}} = X_*(T) \otimes_{\mathbb{Z}} \mathbb{R}$$, where $T$ is a maximal split torus in $G$) via the action of the affine Weyl group, yielding a polysimplicial, piecewise-linear structure (Remy et al., 2011). The stratification arises from the combinatorial cell structure: facets (vertices, edges, higher faces) correspond naturally to parahoric and more generally parahoric-type subgroups. The underlying idea is that the geometry and topology of $\mathcal{B}(G,k)$, when realized in moduli problems or analytic spaces, governs the decomposition into strata, with each stratum reflecting algebraic or representation-theoretic data.

Closures of these strata often correspond to Deligne–Lusztig varieties or their generalizations; in moduli-theoretic or arithmetic applications, they describe the interaction between the group-theoretic incidence geometry and the reduction structure of moduli spaces (Muller, 2021, Muller, 1 Oct 2025, Muller, 19 Jul 2024).

2. Stratification in Compactifications and Analytic Realizations

A pivotal development is the embedding of Bruhat–Tits buildings into analytic spaces via Berkovich geometry, thus enabling compactification and analytic stratification (Remy et al., 2011, Chanfi, 2022).

• The process begins with a $G(k)$-equivariant map $\vartheta:\mathcal{B}(G,k)\to G^{\mathrm{an}}$ that sends points (often “virtually special” points) to seminorms or Shilov boundary points in Berkovich analytic spaces. Composing this map with projections onto analytic flag varieties or generalized flag spaces (arising from pseudo-parabolic subgroups) yields compactifications in which the image of $\mathcal{B}(G,k)$ is stratified by boundary pieces.
• Each stratum in the compactification is $G(k)$-equivariantly homeomorphic to the building of a quotient group—specifically, the maximal quasi-reductive quotient of a pseudo-parabolic subgroup $Q$, i.e., $B(Q/R_{\mathrm{us},k}(Q),k)$, where $R_{\mathrm{us},k}(Q)$ is the $k$-split unipotent radical. The stratification is polyhedral, determined by cones from Weyl fans adapted to the root system (Chanfi, 2022).
• Two principal methods of constructing compactifications—flag mapping and polyhedral fans—interplay to realize both the analytic and combinatorial aspects of the stratification (Remy et al., 2011).

This analytic realization and its stratification reflect deep relations with Satake compactifications, harmonic analysis, and geometrization of reduction theory.

3. Stratification in Moduli and Arithmetic Applications

In arithmetic geometry, particularly for Rapoport–Zink spaces and Shimura varieties at parahoric or deeper level, Bruhat–Tits stratification decomposes the special fiber into locally closed, often smooth, subschemes indexed by vertex lattices, chains of lattices, or facets in the building of a related $p$-adic group (Wang, 2019, Wang, 2019, Muller, 2021, Muller, 1 Oct 2025, He et al., 10 Feb 2025, Zachos et al., 16 Oct 2025, Muller, 19 Jul 2024).

• Each stratum corresponds to a specific combinatorial invariant (vertex lattice, chain, or Bruhat–Tits index), with explicit moduli-theoretic incarnations given—for example, chains of lattices subject to certain index or duality conditions, or points in moduli spaces whose associated Dieudonné (or window) lattices fit prescribed positions with respect to the building.
• Isomorphisms between Bruhat–Tits strata and (often generalized) Deligne–Lusztig varieties are established, enabling the use of representation theory of finite groups of Lie type to analyze their geometry and cohomology (Muller, 2021, Muller, 19 Jul 2024, He et al., 10 Feb 2025, Muller, 1 Oct 2025, Zachos et al., 16 Oct 2025). For instance, each closed Bruhat–Tits stratum in the basic locus of a $GU(1,n-1)$ Rapoport–Zink space may be identified with the closure of a fine Deligne–Lusztig variety for a product of unitary and general linear groups.
• Incidence relations among strata are governed by the Bruhat–Tits building; partial orders on the combinatorial indices reflect inclusion patterns of the corresponding vertex lattices or facets, and determine how closures of strata intersect (Muller, 1 Oct 2025, He et al., 10 Feb 2025, Zachos et al., 16 Oct 2025).
• The geometry of the strata is well-behaved: in many instances, the closed Bruhat–Tits strata are smooth, irreducible, normal and Cohen–Macaulay (Muller, 1 Oct 2025, He et al., 10 Feb 2025, Zachos et al., 16 Oct 2025). Even when isolated singularities occur (e.g., in the closures of certain strata of the supersingular locus), intersection cohomology coincides with the ordinary cohomology, and the spectral sequence associated to their stratification degenerates for weight reasons (Muller, 19 Jul 2024).

4. Cohomological and Arithmetic Frameworks

Bruhat–Tits stratification is instrumental in analyzing cohomological and arithmetic properties of group schemes and moduli spaces. Recent works employ advanced cohomological methods to relate local–global phenomena, descent, and exact calculations of obstructions via stratification data (Zidani, 23 Jan 2025, Zidani, 22 Sep 2025).

• When a Bruhat–Tits group scheme over a Dedekind or Henselian ring is chosen as a model for a reductive group, patching techniques show that the first étale cohomology injects into that of the generic fiber. Schematic formulas such as

$$H^1_{\mathrm{\acute{e}t}}(R, G) \cong \ker\left( G(R)\backslash G(K)/G(K)^+ \to \cdots\right)$$

appear, connecting double coset spaces (related to facets of the building) to the kernel of cohomological maps (Zidani, 23 Jan 2025).

• The exact sequence

$$1 \to H^c \to H \stackrel{\widetilde{\xi}}{\longrightarrow} \Xi_H \to H^1(\Gamma, H^c) \to \cdots$$

features critically in determining obstructions to triviality of torsors or descent for group schemes modeled on the stratification (Zidani, 22 Sep 2025). Kernels such as

$$\ker\left( H^1(\Gamma, G_{\widetilde{F}}) \to H^1(\Gamma, G) \right) \cong (\mathrm{Orb}(\widetilde{F})_H)^{\Gamma}/H$$

locate the arithmetic obstruction in the combinatorics of the types and orbits of facets, refining classical cohomological approaches and resolving local–global triviality problems in new contexts.

• Explicit computations show that for quasi-split adjoint groups, the order of the obstruction kernel is a 2-power, non-trivial only for certain types—specifically, types ${}^2 D_n$ or ${}^2 A_{4n+3}$—providing a precise, combinatorially formulated local–global principle (Zidani, 22 Sep 2025).

5. Stratifications in Affine Flag Varieties and Affine Deligne–Lusztig Varieties

In the context of affine flag varieties and affine Deligne–Lusztig varieties (ADLVs), the Bruhat–Tits stratification is defined by partitioning ADLVs according to the "Bruhat–Tits type" of corresponding points, often in terms of their position relative to the building or in terms of relative position invariants (Goertz, 2018).

• In Coxeter-type cases, Bruhat–Tits stratification coincides with the "J-stratification," i.e., the stratification defined via relative position data with respect to the $\sigma$-centralizer group $J_b$ of a basic element $b$. Each stratum is locally closed and isomorphic (in many cases) to a Deligne–Lusztig variety for a finite group of Lie type, and the closure of any stratum is a union of strata (Goertz, 2018).
• The identification of these strata with combinatorial data from the building allows analysis of cohomological and representation-theoretic invariants, with direct application to the paper of moduli of $p$-divisible groups and the arithmetic of Shimura varieties.

6. Model-theoretic and Functorial Aspects

Bruhat–Tits stratification is preserved and clarified by advances in group scheme theory and geometric representation. For quasi-reductive groups, extensions of Bruhat–Tits theory prove that the integrality and stratification properties familiar for reductive groups persist, allowing for smooth parahoric models, affine Grassmannian quotients, and functoriality in scalar extension (Lourenço, 2020, Chanfi, 2022).

• The functoriality of the building under field extension (in the sense of Rousseau) ensures that the stratification is canonical and well-adapted to arithmetic and moduli-theoretic base change. Stratifications of integral models of shtukas with deep Bruhat–Tits level structures are controlled by the closure and inclusions of bounded subsets within the apartment structure, yielding models with robust properness and Newton stratification properties (Bieker, 2022).
• In group-scheme compactifications—especially in universal wonderful compactifications—the Bruhat–Tits stratification is encoded at a structural level via orbits or strata where the local group scheme recovers the parahoric or parabolic model corresponding to the associated facet (Balaji et al., 2021).

7. Topological Stratification and Real Bruhat–Tits Cells

In real or topological contexts (e.g., real flag varieties, $GL_{n+1}$, $SO_{n+1}$, $Spin_{n+1}$), Bruhat–Tits stratification arises via intersections of Bruhat (or Schubert) cells. Stratification of such intersections, using combinatorial data such as “ancestries” attached to reduced words, naturally partitions the space into contractible strata that assemble into a CW complex whose topology is often determined by the combinatorics of the compactification (Alves et al., 2020, Alves et al., 2021). These topological stratifications provide insight into the homotopy type of the stratified spaces, with explicit calculations showing (for instance) that all connected components are contractible for $n \leq 4$ and more intricate topology (e.g., components homotopically equivalent to $S^1$) arises for larger $n$.

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In summary, the Bruhat–Tits stratification provides a foundational paradigm for decomposing non-Archimedean and moduli-theoretic spaces arising from reductive or quasi-reductive groups into stratified, geometrically tractable pieces. These strata are canonically indexed by the combinatorics of the Bruhat–Tits building (facets, apartments, vertex lattices, etc.), are often realized as or related to Deligne–Lusztig varieties, and their incidence properties reflect the underlying group-theoretic and arithmetic structure. Recent research elucidates the deep relationship between this stratification and cohomological, topological, and arithmetic phenomena, as well as the compactification and model-theoretic structure of moduli spaces and group schemes (Remy et al., 2011, Salle et al., 2015, Prasad, 2016, Goertz, 2018, Wang, 2019, Lourenço, 2020, Muller, 2021, Chanfi, 2022, Muller, 19 Jul 2024, Zidani, 23 Jan 2025, He et al., 10 Feb 2025, Zidani, 22 Sep 2025, Muller, 1 Oct 2025, Zachos et al., 16 Oct 2025).