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arxiv: 2604.15943 · v1 · submitted 2026-04-17 · 🧮 math.GR · math.RT

Recognition: unknown

Masures associated with split Kac--Moody groups over valued fields

Auguste Hebert (IECL, UL)

Pith reviewed 2026-05-10 07:22 UTC · model grok-4.3

classification 🧮 math.GR math.RT
keywords Kac-Moody groupsmasuresBruhat-Tits buildingsvalued fieldssplit groupsgeometric group theoryaxioms for buildingsroot systems
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0 comments X

The pith

A masure is constructed for any split Kac-Moody group over a valued field and shown to obey the authors' simplified axiomatic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a geometric space called a masure for split Kac-Moody groups defined over valued fields. This space generalizes the Bruhat-Tits building that works for finite-dimensional reductive groups. The authors apply a simplified version of Rousseau's axiomatic and verify that their construction meets every axiom in that list. Readers care because the result supplies an explicit geometric model where the group acts with controlled stabilizers and apartments, opening concrete calculations for these infinite-dimensional groups.

Core claim

We construct the masure associated with a split Kac-Moody group over a valued field, and we prove that it satisfies our axiomatic. The construction follows the pattern used for Bruhat-Tits buildings but adapts the building blocks to the Kac-Moody root system and the valuation on the ground field, producing a space equipped with apartments, facets, and a group action that respects the simplified axioms.

What carries the argument

The masure itself, a generalization of a Bruhat-Tits building consisting of apartments and facets on which the Kac-Moody group acts with prescribed stabilizers.

If this is right

  • The group admits a well-defined action on the masure with stabilizers that can be described in terms of parahoric subgroups.
  • Any representation or cohomology theory previously studied on Bruhat-Tits buildings extends in principle to these masures.
  • The simplified axiomatic becomes a practical checklist for verifying future constructions of similar spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction technique may extend to non-split Kac-Moody groups once the axioms are checked in that setting.
  • Low-rank examples such as affine Kac-Moody groups over local fields should allow direct computation of the masure to test its metric properties.
  • The masure may serve as a model space for studying twin buildings or other related geometric objects in the same algebraic setting.

Load-bearing premise

The simplified axiomatic captures enough of the original Rousseau axioms that any object satisfying the simplified version still behaves as a usable geometric model for the group.

What would settle it

An explicit split Kac-Moody group over a valued field whose constructed space violates one of the simplified axioms, such as the existence of a unique apartment containing any two given points.

Figures

Figures reproduced from arXiv: 2604.15943 by Auguste Hebert (IECL, UL).

Figure 0.0
Figure 0.0. Figure 0.0: 1: The tree of SL2(F) for F = Q2 or F = F2((t)). The dotted “line” and the red ”line” are two apartments [PITH_FULL_IMAGE:figures/full_fig_p009_0_0.png] view at source ↗
Figure 0.0
Figure 0.0. Figure 0.0: 3: Two 3D printed models of parts of the building of [PITH_FULL_IMAGE:figures/full_fig_p010_0_0.png] view at source ↗
Figure 0.0
Figure 0.0. Figure 0.0: 4: Set of walls containing 0 of the standard apartment of affine SL2. Not all the walls are represented since there are infinitely many. If we draw a copy of Z in the line δ = 1, then the set of walls is the set of lines containing 0 and an element of Z. (MA3) was introduced in [Rou11] (in a slightly different version). It replaces the fact that two faces are always contained in a common apartment in a B… view at source ↗
Figure 0.0
Figure 0.0. Figure 0.0: 5: Set of walls of the standard apartment of affine [PITH_FULL_IMAGE:figures/full_fig_p012_0_0.png] view at source ↗
Figure 1.8
Figure 1.8. Figure 1.8: 1: The vectorial Bruhat decomposition (or axiom (A4)) for the tree: [PITH_FULL_IMAGE:figures/full_fig_p026_1_8.png] view at source ↗
Figure 1.8
Figure 1.8. Figure 1.8: 2: The affine Bruhat decomposition (or axiom (A3’)) for the tree: [PITH_FULL_IMAGE:figures/full_fig_p027_1_8.png] view at source ↗
Figure 1.8
Figure 1.8. Figure 1.8: 3: No apartment can contain 0, u.ϵ and ϵ (note that apartments branch almost everywhere but are not represented). 1.9.1 Axioms (A2) to (A4) An apartment of type (A, Wf) is a set A equipped with a non-empty set Isom(A, A) of bijections from A to A such that if f0 ∈ Isom(A, A), then Isom(A, A) = {f0 ◦ w | w ∈ Wf}. In an apartment of type (A, Wf), we can define any notion defined in A and preserved by Wf. F… view at source ↗
Figure 1.9
Figure 1.9. Figure 1.9: 1: The exchange condition for the tree. The apartment [PITH_FULL_IMAGE:figures/full_fig_p029_1_9.png] view at source ↗
Figure 1.9
Figure 1.9. Figure 1.9: 2: The standard apartment is the red one. The distance is distorted on the picture: [PITH_FULL_IMAGE:figures/full_fig_p031_1_9.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: 1: Vectorial apartment of sl3. We have ∆ = ±{α1, α2, α1 + α2} and ∆∨ = ±{α ∨ 1 , α∨ 2 , α∨ 1 + α ∨ 2 }. The fundamental chamber C v f is a Weyl chamber and Wv acts sim￾ply transitively on the Weyl chambers. 2.2 Kac–Moody algebras In this section, we define Kac–Moody algebras by generators and relations. We first Kac– Moody data, which are the data to which Kac–Moody algebras and groups are associated. 37… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: 1: Vectorial apartment of SLg2 with thirteen walls 2.4.2 Vectorial apartment in the indefinite case We now describe the vectorial apartment in the indefinite case. Let S be a Kac–Moody data associated to an indefinite Kac–Moody matrix. The following proposition gathers well-known facts on imaginary roots. Proposition 2.4.8. ([Héb18, Proposition 2.3.9]) Suppose that A is an indefinite Kac–Moody matrix. Th… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: 2: Vectorial apartment of SLg2 with forty-one walls We now study the case where A is of size 2. We set I = {1, 2}. We write A = A(a, b) =  2 −a −b 2  with a, b ∈ Z>0. We have ab ≥ 5 by Proposition 2.3.5. In particular, A is invertible and the minimal free realization A of A over R has dimension 2. The vectorial faces of A are {0}, the vectorial panels and the vectorial chambers. Except {0}, these faces… view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: 3: Vectorial apartment of A(1,5) [PITH_FULL_IMAGE:figures/full_fig_p049_2_4.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: 1: Exemple of a prenilpotent pair {α, β} 1. Let Ψ ⊂ Φ. Then Ψ is prenilpotent if and only if T ∩ T α∈Ψ α −1 (R+) and −T ∩ T α∈Ψ α −1 (R+) has nonempty interior. 2. Let α, β ∈ Φ be such that α ̸= −β. Then at least one pair of {{α, β}, {α, −β}} is prenilpotent. Proposition 3.1.4. (see [Héb18, Proposition 3.3.2]) Suppose that A is of finite type. Then a pair {α, β} ⊂ Φ is prenilpotent if and only if α ̸= −β… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: 2: Exemple of a non prenilpotent pair {α, β} Definition 3.1.7. We define U +, U − and U 0 as the Z-subalgebras of UC(g) generated respec￾tively by {e (s) i | i ∈ I, s ∈ N}, {f (s) i | i ∈ I, s ∈ N} and { [PITH_FULL_IMAGE:figures/full_fig_p053_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: 1: Affine apartment of SL3 when Λ = Z. All the sets represented are chimneys: q1 and q2 have the same direction, which is a vectorial chamber, r1, r2 and r3 have the same direction, which is a ray, the direction of r is a ray and x, F and C have direction {0}. The sectors q1 and q2 define the same germ at infinity, which is up and left of the picture, r1,r2 and r3 define the same germ, which is a “strip … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: 2: Affine apartment of affine SL2 when Λ = Z. All the filters represented are chimneys. The set F v + is a vectorial panel, C+ is an alcove and the hatched strip represents the germ at infinity of the chimney r = cl(C+, f+) (actually it is the “strip at infinity” in the direction of the hatched strip). The filter F+ is a local panel and C− is an alcove. The sets q+ and q− are sectors and f+ is a sector-p… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: 1: Sundial configuration. We use the notation of Lemma [PITH_FULL_IMAGE:figures/full_fig_p080_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: 1: Local sundial configuration when the masure is associated with affine [PITH_FULL_IMAGE:figures/full_fig_p081_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: 1: In the figure, Qa is a sector-germ of an apartment A. The sector-germ Qb is adjacent to Qa and not contained in A. Let D1 (resp. D2) be the half-apartments delimited by Ma,b and on the left (resp. right) of Ma,b. The apartments Ai contains Qb and Di , for i ∈ {1, 2}. The wall Ma,b separates Qa and Qb in A1. Lemma 4.1.8, there exists an apartment isomorphism fj : B(j) → A fixing A ∩ B(j) . For i ∈ J1, … view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: 2: We keep the same notation as in Figure [PITH_FULL_IMAGE:figures/full_fig_p084_4_5.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: 3: We keep the same notation as in Figure [PITH_FULL_IMAGE:figures/full_fig_p085_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: 1: Folding of a the retraction of a segment-germ [PITH_FULL_IMAGE:figures/full_fig_p086_4_6.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: 2: Proof of Lemma 4.6.12. The point x = s(0) belongs to P2 ∩ P3 ∩ P4 ∩ P5. We consider the translate sϵ of s by a vector ϵ⃗u, for ϵ ∈ R>0 and some well-chosen vector ⃗u ∈ A. We choose ϵ and ⃗u so that for all t ∈ R such that sϵ(t) is well-defined and sϵ(t) belongs to at most two Pi . We can then apply Lemma 4.6.11. Proof. The proof is illustrated by [PITH_FULL_IMAGE:figures/full_fig_p089_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: 1: We drop the indices Q+∞ and Q−∞ in the figure. Let δx be the green dotted ray and δy be the purple dotted ray. The point x is contained in the apartment A− x =]Q−∞, 0]∪δx (resp. A+ x = [0, Q+∞[∪δx), which contains Q−∞ (resp. Q+∞). Then x +Q−∞ ν − (resp. x +Q+∞ ν + 1 ) is the translate of x by 1 in the direction Q−∞ (resp. Q+∞). The apartment A+ y = δy ∪ [1, Q+∞[ (resp. A− y = δy∪]Q−∞, 1]) contains y a… view at source ↗
read the original abstract

Masures are generalizations of Bruhat-Tits buildings adapted to the study of Kac--Moody groups over valued fields. They were introduced by Gaussent and Rousseau in 2007. Rousseau defined an axiomatic for these object and we simplified it. In this paper, which is mainly expository, we construct the masure associated with a split Kac--Moody group over a valued field, and we prove that it satisfies our axiomatic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript, which is primarily expository, constructs the masure associated to a split Kac-Moody group over a valued field and verifies that the resulting object satisfies a simplified axiomatic for masures introduced by the authors (a simplification of Rousseau's original axiomatic). The central claim is that this explicit construction meets the listed axioms, providing a concrete realization in the split case.

Significance. If the verification holds, the paper supplies a self-contained reference for the masure construction in the split setting, which may aid researchers working with Kac-Moody groups over valued fields by clarifying the geometric structure through an accessible axiomatic framework. The explicit construction and direct verification against the authors' axioms constitute a concrete contribution to the literature.

minor comments (2)
  1. The abstract states that the authors simplified Rousseau's axiomatic but does not indicate in which section the equivalence (or sufficiency) of the simplified version for the purposes of the construction is established; adding a brief pointer would improve readability.
  2. Notation for the valued field and the root datum is introduced without a dedicated preliminary subsection; a short table or list of standing assumptions would help readers track the data throughout the verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for recognizing its expository value as a self-contained reference, and for recommending minor revision. The report does not raise any specific major comments or point to particular issues with the construction or the verification of the axioms. We have therefore prepared a revised version addressing any minor editorial or presentational improvements we identified during our own review.

Circularity Check

0 steps flagged

No significant circularity: explicit construction verified against simplified axiomatic

full rationale

The paper is expository and presents a direct construction of the masure for a split Kac-Moody group over a valued field, followed by a verification that this object satisfies the authors' simplified axiomatic (itself derived from Rousseau's 2007 definition). No equations reduce the result to a fitted parameter, self-definition, or prior self-citation chain; the central claim is an explicit build-and-check against listed axioms, with the simplification framed as a deliberate expository choice rather than a load-bearing premise that forces the outcome. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work relies on the prior definition of split Kac-Moody groups and valued fields from the literature.

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Reference graph

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