Geometric Coxeter Type in Affine Deligne–Lusztig Theory

• Geometric Coxeter Type is a condition on Coxeter data that imposes rigid geometric structure on affine Deligne–Lusztig varieties, reflection representations, and incidence geometries.
• It organizes Deligne–Lusztig reductions via uniform reduction trees, where strong multiplicity one and minimal Coxeter type ensure each component fibers into classical DL varieties, Gₘ, and A¹ factors.
• The framework extends earlier Coxeter-type notions by unifying fully Hodge–Newton decomposable cases and positive Coxeter types into a product geometry that precisely captures dimension and orbit structure.

Searching arXiv for papers on “geometric Coxeter type” and closely related notions to ground the article in current literature. Searching arXiv for “geometric Coxeter type” exact phrase. “Geometric Coxeter type” denotes several closely related but technically distinct Coxeter-theoretic notions that arise when geometric structure is imposed on Coxeter data. In recent arithmetic geometry, the term refers most specifically to a class of elements in an Iwahori–Weyl group introduced to control the geometry of affine Deligne–Lusztig varieties, where these varieties decompose into products of classical Deligne–Lusztig varieties with affine spaces and pointed affine spaces (Nie et al., 24 Jul 2025). In representation theory, closely related constructions generalize the classical geometric representation of a Coxeter group by allowing homological twists encoded by characters of graph homology (Hu, 2023, Hu, 2021). In incidence geometry, “geometries of Coxeter type” designates thick, residually connected incidence geometries whose rank-two residues are generalized polygons dictated by a Coxeter matrix (Struyve, 2015). The common theme is that Coxeter combinatorics becomes rigid enough to determine substantial geometric structure, whether in affine flag varieties, reflection representations, or incidence geometries.

1. Affine Deligne–Lusztig interpretation

In the setting of affine Deligne–Lusztig theory, let $F$ be either a finite extension of $\Q_p$ or a local Laurent series field $\F_q\lb t\rb$, let $L$ be the completion of its maximal unramified extension, let $\sigma$ be Frobenius, and fix a $\sigma$-stable Iwahori subgroup $I\subset G(L)$. The affine flag variety is $LG/I$, and the Iwahori–Weyl group is

$$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$

For $w\in\widetilde W$ and $\Q_p$0, the affine Deligne–Lusztig variety is

$\Q_p$1

This is the ambient context in which geometric Coxeter type was introduced (Nie et al., 24 Jul 2025).

A key input is Deligne–Lusztig reduction: if $\Q_p$2 is attached to an affine simple root $\Q_p$3, then $\Q_p$4 is related to either $\Q_p$5 or to pieces fibering over $\Q_p$6 and $\Q_p$7 by $\Q_p$8- and $\Q_p$9-bundles up to universal homeomorphism. Iterating these reductions organizes the geometry of $\F_q\lb t\rb$0 by means of a finite rooted reduction tree whose leaves are minimal-length elements (Nie et al., 24 Jul 2025).

This reduction-theoretic control is the immediate background for geometric Coxeter type. The notion is designed so that every branch in the reduction tree behaves uniformly and ends in a particularly tractable minimal element. This suggests that geometric Coxeter type is not primarily a combinatorial label, but a geometric regularity condition on the entire Deligne–Lusztig reduction process.

2. Definition via reduction trees

A reduction tree $\F_q\lb t\rb$1 for $\F_q\lb t\rb$2 is obtained by iterating Deligne–Lusztig reduction until one reaches minimal-length elements. A path from the root $\F_q\lb t\rb$3 to a leaf $\F_q\lb t\rb$4 is a reduction path; its edges are of type I or II according to whether the reduction produces a $\F_q\lb t\rb$5-bundle piece or an $\F_q\lb t\rb$6-bundle piece. If $\F_q\lb t\rb$7 is such a path, $\F_q\lb t\rb$8 and $\F_q\lb t\rb$9 denote the numbers of type I and type II steps (Nie et al., 24 Jul 2025).

The intermediate notion is strong multiplicity one. One says that $L$0 satisfies strong multiplicity one if among all reduction paths in $L$1 exactly one ends in the unique minimal-length element representing $L$2; this is independent of the choice of $L$3 (Nie et al., 24 Jul 2025).

A minimal-length element $L$4 is of minimal Coxeter type if there exist a finite parabolic subgroup $L$5, a $L$6-straight element $L$7 with $L$8, and a $L$9-Coxeter element $\sigma$0, such that $\sigma$1 is $\sigma$2-conjugate by cyclic shift to $\sigma$3 (Nie et al., 24 Jul 2025).

An arbitrary $\sigma$4 is then of geometric Coxeter type if for any reduction tree $\sigma$5:

1. strong multiplicity one holds for each $\sigma$6, and
2. every leaf $\sigma$7 of $\sigma$8 is a minimal Coxeter type element (Nie et al., 24 Jul 2025).

This definition strictly contains previously studied classes. In particular, any element of the form $\sigma$9 with $\sigma$0 dominant and $\sigma$1 a partial $\sigma$2-Coxeter element is of geometric Coxeter type, and any positive Coxeter type element is of geometric Coxeter type (Nie et al., 24 Jul 2025). The paper also gives examples in types $\sigma$3, $\sigma$4, and twisted $\sigma$5 that are neither of the form $\sigma$6 nor positive Coxeter.

3. Geometric structure theorem

The main structural consequence is that affine Deligne–Lusztig varieties attached to geometric Coxeter type elements have a product geometry controlled by the unique reduction path corresponding to a given $\sigma$7. If $\sigma$8 is of geometric Coxeter type and $\sigma$9, then all top-dimensional irreducible components of $I\subset G(L)$0 lie in a single $I\subset G(L)$1-orbit, and each such component is universally homeomorphic to

$I\subset G(L)$2

where $I\subset G(L)$3 is the unique reduction path ending in $I\subset G(L)$4 (Nie et al., 24 Jul 2025).

The underlying mechanism is a lifting criterion. If strong multiplicity one holds and the terminal minimal element $I\subset G(L)$5 admits a section $I\subset G(L)$6, then $I\subset G(L)$7 admits a section and is universally homeomorphic to the trivial bundle

$I\subset G(L)$8

The proof uses three upgrade steps showing that lifts propagate through the two kinds of Deligne–Lusztig reduction pieces and then proceeds inductively on the reduction tree (Nie et al., 24 Jul 2025).

A stronger combinatorial description is available for the exponents $I\subset G(L)$9 and $LG/I$0. For geometric Coxeter type $LG/I$1 and any $LG/I$2, every reduction path to $LG/I$3 has

$LG/I$4

and

$LG/I$5

(Nie et al., 24 Jul 2025). This gives a uniform description of the geometry purely in terms of Newton data and twisted reflection length.

A useful comparison point is positive Coxeter type. For positive Coxeter type elements, affine Deligne–Lusztig varieties already admit an explicitly described geometry in which irreducible components are iterated fibrations with $LG/I$6- or $LG/I$7-fibers over a classical Deligne–Lusztig variety of finite Coxeter type (Schremmer et al., 2023). Geometric Coxeter type extends this mechanism from a previously restricted class to a strictly larger one (Nie et al., 24 Jul 2025).

4. Newton stratification, purity, and interval structure

For $LG/I$8, write

$LG/I$9

For geometric Coxeter type elements, the Newton-theoretic structure is unusually rigid. There are unique minimal and maximal classes $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$0 and $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$1, and $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$2 is saturated between them: $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$3 Moreover, for all $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$4,

$$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$5

The closure of a Newton stratum in the affine Schubert cell $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$6 is a union of lower strata, in the manner of purity statements for classical Newton stratifications (Nie et al., 24 Jul 2025).

The characterization of minimal Coxeter type itself is tied to a sharp length–Newton inequality: $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$7 with equality if and only if $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$8 is minimal Coxeter type (Nie et al., 24 Jul 2025). This makes minimal Coxeter type the extremal case in which combinatorial length is completely accounted for by Newton slope data, twisted reflection length, and defect.

These results place geometric Coxeter type in a broader program concerning tractable affine Deligne–Lusztig varieties. The paper explicitly states that the framework uniformly explains known cases including fully Hodge–Newton decomposable situations, positive Coxeter type, certain EL/PEL Rapoport–Zink spaces, and Chan–Ivanov semi-infinite Deligne–Lusztig varieties (Nie et al., 24 Jul 2025). A plausible implication is that geometric Coxeter type functions as a unifying organizing principle for several previously disparate “simple geometry” phenomena in mixed-characteristic and equal-characteristic local models.

5. Relation to earlier Coxeter-type conditions in arithmetic geometry

Before geometric Coxeter type was introduced, a central notion was Coxeter type for generalized affine Deligne–Lusztig varieties at parahoric level. In that setting, a datum $$\widetilde W=N_G(T)(L)/T(L)\cong X_*(T)\rtimes W_0.$$9 is of Coxeter type if $w\in\widetilde W$0 is fully Hodge–Newton decomposable and every EKOR stratum is indexed by a $w\in\widetilde W$1-twisted Coxeter element (Görtz et al., 2020). One of the main characterizations is a dimension criterion: $w\in\widetilde W$2 under the stated non-centrality hypothesis (Görtz et al., 2020).

That earlier theory classifies quadruples $w\in\widetilde W$3 of Coxeter type and connects them to Bruhat–Tits stratifications by classical Deligne–Lusztig varieties in the basic locus of Shimura varieties and Rapoport–Zink spaces (Görtz et al., 2020). Geometric Coxeter type in the sense of elements $w\in\widetilde W$4 is different in formulation: it is defined by reduction-tree behavior rather than by full Hodge–Newton decomposability plus twisted Coxeter indexing. Nevertheless, both theories single out situations in which affine Deligne–Lusztig geometry collapses to a controlled mixture of Coxeter-type finite-dimensional pieces and elementary affine factors.

Positive Coxeter type provides the intermediate stage. An element $w\in\widetilde W$5 is of positive Coxeter type if there exists $w\in\widetilde W$6 such that

$w\in\widetilde W$7

is a partial $w\in\widetilde W$8-Coxeter element (Schremmer et al., 2023). For such elements, $w\in\widetilde W$9 is an interval, dimensions admit a closed formula, and irreducible components are controlled by $\Q_p$00 and by classical Deligne–Lusztig varieties of finite Coxeter type (Schremmer et al., 2023). Geometric Coxeter type strictly extends this class (Nie et al., 24 Jul 2025). This suggests a three-stage progression:

Notion	Defining mechanism	Geometric outcome
Coxeter type datum	EKOR strata indexed by twisted Coxeter elements	Bruhat–Tits stratification by classical DL varieties
Positive Coxeter type element	Conjugacy to partial $\Q_p$01-Coxeter via $\Q_p$02	Iterated $\Q_p$03-fibrations over Coxeter DL varieties
Geometric Coxeter type element	Strong multiplicity one plus minimal Coxeter type leaves	Product geometry along unique reduction paths

The terminology can therefore be misleading if treated as a single invariant notion across the literature. The arithmetic-geometric usage is highly specific to affine Deligne–Lusztig theory and should not be conflated with the representation-theoretic or incidence-geometric usages described below.

6. Representation-theoretic generalizations of geometric type

A different line of work studies generalized geometric representations of Coxeter groups. For a Coxeter system $\Q_p$04, the classical geometric representation uses a bilinear form with matrix entries

$\Q_p$05

and generators act by reflections (Hu, 2023). Hu generalizes this by allowing integers $\Q_p$06 with

$\Q_p$07

for finite edges and nonzero scalars $\Q_p$08, giving a complex reflection representation

$\Q_p$09

(Hu, 2023).

The classification is governed by graph homology. From the chosen $\Q_p$10 one forms an associated graph $\Q_p$11; generalized geometric representations are classified, up to isomorphism, by pairs consisting of the $\Q_p$12-matrix and a character

$\Q_p$13

(Hu, 2023). When all $\Q_p$14 and $\Q_p$15, one recovers the classical geometric representation. Finite irreducible Coxeter graphs $\Q_p$16 are trees, so $\Q_p$17, hence only the classical geometric representation occurs. By contrast, $\Q_p$18 has a cycle, so nontrivial twists exist (Hu, 2023).

A related 2021 treatment classifies a class of complex representations of Coxeter groups via characters of the integral homology of certain graphs and links them to the second-highest two-sided cell in the sense of Kazhdan–Lusztig (Hu, 2021). In the simply laced case with at most one circuit, trees again force the classical geometric representation, while affine $\Q_p$19 yields a one-parameter family of homologically twisted variants (Hu, 2021).

These papers do not use “geometric Coxeter type” in the arithmetic-geometric sense. Rather, they generalize the classical geometric representation itself. The conceptual connection is that both usages promote Coxeter combinatorics to geometry by supplementing the Coxeter matrix with extra data—reduction-tree geometry in one case, graph-homology characters in the other.

7. Geometries of Coxeter type in incidence and projective geometry

In incidence geometry, a geometry over a type set $\Q_p$20 with Coxeter matrix $\Q_p$21 is of Coxeter type $\Q_p$22 if it is thick, residually connected, and every rank-two residue is a generalized $\Q_p$23-gon when $\Q_p$24, or an infinite tree when $\Q_p$25 (Struyve, 2015). Equivalently, for each pair of types $\Q_p$26, minimal circuits have length $\Q_p$27 and no shorter circuits occur (Struyve, 2015).

Struyve establishes two free constructions. The first applies to connected Coxeter diagrams containing no subdiagram of type $\Q_p$28, by iteratively adjoining vertices, alternating paths, and connections of disconnected residues while preserving flatness and partial polygonal constraints (Struyve, 2015). The second treats diagrams of type $\Q_p$29 and $\Q_p$30 by starting from a projective $\Q_p$31-building over an infinite countable skew field and extending it with new vertices of types $\Q_p$32 and $\Q_p$33 (Struyve, 2015).

This usage is again distinct from geometric Coxeter type elements in affine Deligne–Lusztig theory. Here the phrase “geometry of Coxeter type” refers to incidence structures determined by Coxeter rank-two residues. Yet the methodological similarity is striking: local Coxeter data controls global geometry via inductive extension principles.

A further geometric direction appears in projective and Hilbert geometry. Marquis studies Coxeter groups acting on properly convex domains built from the Tits–Vinberg reflection construction, obtaining criteria for finite covolume, convex-cocompactness, geometric finiteness, Zariski closure, and strict convexity of invariant domains (Marquis, 2014). Although this work does not use the specific phrase “geometric Coxeter type,” it exemplifies another sense in which Coxeter data becomes geometric: a Coxeter polytope and its facet reflections produce properly convex projective actions with Hilbert-geometric structure (Marquis, 2014).

Taken together, these strands show that “geometric Coxeter type” is not a single universal definition but a family resemblance across several subfields. In the current literature, the most specific and technically developed use is the 2025 notion for elements of an Iwahori–Weyl group controlling affine Deligne–Lusztig varieties (Nie et al., 24 Jul 2025). Its importance lies in converting a priori complicated moduli-theoretic spaces into objects assembled from classical Deligne–Lusztig varieties, $\Q_p$34, and $\Q_p$35, while preserving strong control over Newton stratification, dimensions, and component orbits.