Bruhat-Tits Building Research Articles

Distance formulas in Bruhat–Tits building of SLd(ℚp)

We study the distance on the Bruhat–Tits building of the group [Formula: see text] (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance [Formula: see text] of two vertices [Formula: see text] and [Formula: see text] (without having to specify their common apartment). Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of [Formula: see text] and give a formula for the number of edges shared by two given apartments.

Stable lattices in p-adic representations III. Indecomposable reduction and holonomy

Let G be a group, V a finite dimensional vector space over a complete discrete valuation field K, and ρ:G⟶GLK(V) an irreducible representation with bounded image (in the metric topology). First, we show that there are many stable lattices with indecomposable reduction, in the sense that their simplicial convex hull in the Bruhat-Tits building of GLK(V) is the set S(ρ) of stable norms.Then we investigate the reductions more closely, in case K has finite residue field and the simplicial set S(ρ) has dimension 2. When the reduction type (defined below) is (2,1), we show that there exists a stable lattice with indecomposable reduction that has at most 2 composition series (uniserial or biserial). When the reduction type is (3), we show that the number of v∈S(ρ)0 with semisimple reduction is bounded above by a constant times |S(ρ)0|.To prove these last two statements, we prove and apply an index theorem (of Gauss-Bonnet type).

Drinfeld discriminant function and Fourier expansion of harmonic cochains

Let $$F_{\infty }={{\mathbb {F}}_q}\left( \!\left( {1/T}\right) \!\right) $$ be the completion of $${\mathbb {F}}_q(T)$$ at 1/T. We develop a theory of Fourier expansions for harmonic cochains on the edges of the Bruhat–Tits building of $${{\,\textrm{PGL}\,}}_r(F_{\infty })$$ , $$r\ge 2$$ , generalizing an earlier construction of Gekeler for $$r=2$$ . We then apply this theory to study modular units on the Drinfeld symmetric space $$\Omega ^r$$ over $$F_{\infty }$$ , and the cuspidal divisor groups of Satake compactifications of certain Drinfeld modular varieties. In particular, we obtain a higher dimensional analogue of a result of Ogg for classical modular curves $$X_0(p)$$ of prime level.

An Intrinsic Characterization of Bruhat–Tits Buildings Inside Analytic Groups

Given a reductive group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat–Tits building into the analytic space associated with the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

A note on the rational homological dimension of lattices in positive characteristic

AbstractWe show via $\ell^2$ -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building.

Growth of mod-2 homology in higher-rank locally symmetric spaces

Let X be a higher-rank symmetric space or a Bruhat–Tits building of dimension at least 2 such that the isometry group of X has property (T). We prove that for every torsion-free lattice Γ⊂IsomX, any homology class in H1(Γ∖X,F2) admits a representative cycle of total length oX(Vol(Γ∖X)). As an application, we show that dimF2H1(Γ∖X,F2)=oX(Vol(Γ∖X)).

Abelianization of the Cartwright-Steger lattice

The Cartwright-Steger lattice is a group whose Cayley graph can be identified with the Bruhat-Tits building of PGLd over a local field of positive characteristic. We give a lower bound on the abelianization of this lattice, and report that the bound is tight in all computationally accessible cases.

Typical representations via fixed point sets in Bruhat–Tits buildings

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

Chabauty limits of diagonal Cartan subgroups of [formula omitted

Let C be the subgroup of all diagonal matrices in SL(n,Qp). In the first part of this paper we study and give a classification of the Chabauty limits of SL(n,Qp)-conjugates of C using the action of SL(n,Qp) on its associated Bruhat–Tits building. Along the way we construct an explicit homeomorphism between the Chabauty compactification in sl(n,Qp) of SL(n,Qp)-conjugates of the p-adic Lie algebra of C and the Chabauty compactification of SL(n,Qp)-conjugates of C. In the second part of the paper we compute all of the Chabauty limits for n≤4 (up to conjugacy). In contrast, for n≥7 we prove there are infinitely many SL(n,Qp)-nonconjugate Chabauty limits.

Presqu’un immeuble pour le groupe des automorphismes modérés

Inspired by the Bruhat-Tits building of SL$_n$($\mathbb Q_p$), we construct a complete metric space X with an action of the tame automorphism group of the affine space Tame($K^n$). The points in X are certain monomial valuations, and X admits a natural structure of Euclidean CW-complex of dimension n-1. When n = 3, and for K of characteristic zero, we prove that X has non-positive curvature and is simply connected, hence is a CAT(0) space. As an application we obtain the linearizability of finite subgroups in Tame($K^3$).

Mustafin Models of Projective Varieties and Vector Bundles

Abstract Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

Equivalence of categories between coefficient systems and systems of idempotents

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of Rep R ⁡ ( G ) \operatorname {Rep}_R(G) , the category of smooth representations of a p p -adic group G G with coefficients in R R . In particular, they were used to construct level 0 decompositions when R = Z ¯ ℓ R=\overline {\mathbb {Z}}_{\ell } , ℓ ≠ p \ell \neq p , by Dat for G L n GL_{n} and the author for a more general group. Wang proved in the case of G L n GL_{n} that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of G L n GL_{n} and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.

On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2

We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree 2 over a ramified odd prime p if the level at p is given by a special maximal compact open subgroup. More precisely, we show that such a locus is purely 2-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport–Zink space for the quaternionic unitary similitude group of degree 2, with a special parahoric level. We prove that such a scheme is purely 2-dimensional, and every irreducible component is isomorphic to the Fermat surface. We also determine its connected components, irreducible components and their intersection behaviors by means of the Bruhat–Tits building of \({{\,\mathrm{PGSp}\,}}_4({\mathbb {Q}}_p)\). In addition, we compute the intersection multiplicity of the GGP cycles associated to an embedding of the considering Rapoport–Zink space into the Rapoport–Zink space for the unramified \({{\,\mathrm{GU}\,}}_{2,2}\) with hyperspecial level for the minuscule case.

Smoothness of Schubert varieties in twisted affine Grassmannians

We give a complete list of smooth and rationally smooth normalized Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. The particular case of the quasi-minuscule Schubert variety in the quasi-split but non-split form of ${\rm Spin}_8$ ("ramified triality") provides an input needed in the article by He-Pappas-Rapoport classifying Shimura varieties with good or semi-stable reduction.

Rigid connections on P1 via the Bruhat–Tits building

We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection of minimal slope associated to the Coxeter torus $\mathcal{C}$. In the present paper, we consider connections on $\mathbb{G}_m$ which have an irregular homogeneous $\mathcal{C}$-toral singularity at zero of slope $i/h$, where $h$ is the Coxeter number and $i$ is a positive integer coprime to $h$, and a regular singularity at infinity with unipotent monodromy. Our main result is the characterisation of all such connections which are rigid.

On continued fraction expansions of quadratic irrationals in positive characteristic

Let R=\mathbb F_q[Y] be the ring of polynomials over a finite field \mathbb F_q , let \widehat K=\mathbb F_q((Y^{-1})) be the field of formal Laurent series over \mathbb F_q , let f\in\widehat K be a quadratic irrational over \mathbb F_q(Y) and let P\in R be an irreducible polynomial. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as P^nf as n \to +\infty , proving, in sharp contrast with the case of quadratic irrationals in \mathbb R over \mathbb Q considered in [1], that they have one such degree very large with respect to the other ones. We use arguments of [2] giving a relationship with the discrete geodesic flow on the Bruhat–Tits building of (\mathrm {PGL}_2,\widehat K) and, with A the diagonal subgroup of \mathrm {PGL}_2(\widehat K) , the escape of mass phenomena of [7] for A -invariant probability measures on the compact A -orbits along Hecke rays in the moduli space \mathrm {PGL}_2(R)\backslash \mathrm{PGL}_2(\widehat K) .

On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space

In this article we study the special fiber of the Rapoport–Zink space attached to a quaternionic unitary group. The special fiber is described using the so called Bruhat–Tits stratification and is intimately related to the Bruhat–Tits building of a split symplectic group. As an application we describe the supersingular locus of the related Shimura variety.

Nilpotent Orbits of Orthogonal Groups over p-adic Fields, and the DeBacker Parametrization

For local non-archimedean fields k of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups GLn(D), SLn(D) (where D is a central division algebra over k) or classical groups, give a new characterisation of the “building set” (defined by DeBacker) of an $\mathfrak {sl}_{2}(k)$ -triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker’s parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building.

Congruences of parahoric group schemes

Let $F$ be a non-archimedean local field and let $T$ be a torus over $F$. With $\cT^{NR}$ denoting the N\'eron-Raynaud model of $T$, a result of Chai and Yu asserts that the model $\cT^{NR} \times_{\fO_F} \fO_F/\fp_F^m$ is canonically determined by $(\Tr_l(F), \Lambda)$ for $l>>m$, where $\Tr_l(F) = (\fO_F/\fp_F^l, \fp_F/\fp_F^{l+1}, \epsilon)$ with $\epsilon$ denoting the natural projection of $\fp_F/\fp_F^{l+1}$ on $\fp_F/\fp_F^l$, and $\Lambda:=X_*(T)$. In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat-Tits building of a connected reductive group over $F$.

On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow

Let X X be a locally finite irreducible affine building of dimension ≥ 2 \geq 2 , and let Γ ≤ Aut ⁡ ( X ) \Gamma \leq \operatorname {Aut}(X) be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is Γ \Gamma linear? More generally, when does Γ \Gamma admit a finite-dimensional representation with infinite image over a commutative unital ring? If X X is the Bruhat–Tits building of a simple algebraic group over a local field and if Γ \Gamma is an arithmetic lattice, then Γ \Gamma is clearly linear. We prove that if X X is of type A ~ 2 \widetilde {A}_2 , then the converse holds. In particular, cocompact lattices in exotic A ~ 2 \widetilde {A}_2 -buildings are nonlinear. As an application, we obtain the first infinite family of lattices in exotic A ~ 2 \widetilde {A}_2 -buildings of arbitrarily large thickness, providing a partial answer to a question of W. Kantor from 1986. We also show that if X X is Bruhat–Tits of arbitrary type, then the linearity of Γ \Gamma implies that Γ \Gamma is virtually contained in the linear part of the automorphism group of X X ; in particular, Γ \Gamma is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic Γ \Gamma -space attached to the the building X X , which we call the singular Cartan flow.