Yurii A. Neretin
For the space $Lat_n$ of all the lattices in a $p$-adic $n$-dimensional linear space we obtain an analog of matrix beta-functions; this beta-function has a degeneration to the Tamagawa zeta-function. We propose an analog of Berezin kernels for $Lat_n$. We obtain conditions of positive definiteness of these kernels and explicit Plancherel formula.
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In this paper I study Ramanujan hypergraps and both abstract and explicit constructions is given.
For F an algebraic extension of Q2, the conjugacy classes of invertible, 2-by-2, trace-zero matrices under the action of G := SL2(F) are analyzed relative to the quadratic extension that splits the respective characteristic polynomial. The stabilizer in G of each such matrix is computed as a stabilizer of a simple, Galois invariant path in the Bruhat-Tits Tree of G.
We consider two sign characters defined on a tamely ramified maximal torus T of a twisted Levi subgroup M of a reductive p-adic group G. We show that their product extends to the stabilizer M(F)_x of any point x in the Bruhat-Tits building of T, and give a formula for this extension. This result is used in the passage between zero and positive depth in the explicit construction of supercuspidal L-packets, as well as in forthcoming work on the Harish-Chandra character formula for supercuspidal representations.
The purpose of the paper is to give a new approach to tamely-ramified descent in Bruhat-Tits theory. This descent was first studied by Guy Rousseau in his thesis.
Let L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally analytic G-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat-Tits building of G. For smooth representations, the corresponding sheaves are closely related to the sheaves constructed by S. Schneider and U. Stuhler. The functor is also compatible, in a certain sense, with the localization of g-modules on the f…
In this note we prove the classical decomposition theorems for certain subgroups of the automorphism group of the Bruhat-Tits tree associated to $\mathrm{PGL}_{2}(F)$, where $F$ is a local field. The results can be used to understand the representation theory of these subgroups. Moreover, we can also use the results to obtain similar decompositions for \textit{any} closed transitive subgroup of the automorphism group of \textit{any} Bruhat-Tits tree.
Let $G$ be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic $p$. Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let $R$ be a commutative quasi-Frobenius ring. If $H=R[I\backslash G/I]$ denotes the pro-$p$ Iwahori-Hecke algebra of $G$ over $R$ we clarify the relation between the category of $H$-modules and the category of $G$-equivariant coefficient systems on the semisimple Bruhat-Tits building of $G$. If $R$ is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler r…
The purpose of the paper is to present an alternative approach to unramified descent in Bruhat-Tits theory of reductive groups over nonarchimedean local fields. This approach appears to be conceptually simpler than the approach in the papers of Bruhat and Tits.
Coefficient systems on the A_2 Bruhat-Tits building
本文探讨了p-adic李群G表示论中的一个猜想，证明了在G的Bruhat-Tits建筑局部区域中，系数系统的链复形精确性可推导出该猜想，尤其在G=SL_3(K)时提供了有力证据。
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
In this note we make a universal construction of Bruhat-Tits group scheme on wonderful embeddings of adjoint groups in the absolute and relative settings and of adjoint Kac-Moody groups. These have natural classifying properties reflecting the orbit structure on the wonderful embeddings.
Covers of Bruhat-Tits trees
本文研究了非阿基米德局部域上元裂覆盖$\operatorname{GL}_2(F)$作用于连通局部有限树的几何结构，并证明了相关对的性质。
Cohomology of the Bruhat-Tits strata in the supersingular locus of the $\mathrm{GU}(1,n-1)$ Shimura variety at a ramified prime
本文计算了分歧素数处 GU(1,n-1) Shimura 簇超奇异轨迹中闭 Bruhat-Tits 层的 ℓ-adic 上同调，证明奇数阶为零、偶数阶为辛群表示，并发现其与交点上同调一致。
Introductory and pedagogical treatmeant of the article : P. Broussous "Distinction of the Steinberg representation", with an appendix by François Courtès, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean Morlet, Dipendra Prasad, Volker Heiermann Ed. 2017. Contains modified and simplified proofs of loc. cit. This article is written in memory of François Courtès who passed away in september 2016.
For an essentially tame supercuspidal representation $π$ of a connected reductive $p$-adic group $G$, we establish two distinct and complementary sufficient conditions for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of $G$ which is not inertially equivalent to $π$. These two results are further formulated in terms of the geometry of the Bruhat-Tits building of $G$ and its fixed points under the action of certain tori. The consequence is a set of broadly applicable tools for addressing the branching rules of $π$ and the unicity of $[G…
Orbits of a symmetric subgroup in the Bruhat-Tits building
该研究针对剩余特征为奇数的非阿基米德局部紧域上的连通约化代数群，刻画了其F-有理对合对应的对称子群在约化建筑室集合中的轨道结构。
Computing quaternionic representations via twisted forms of Bruhat-Tits trees
通过扭曲形式的布鲁哈-蒂茨树研究四元数代数的有限子群表示，计算积分形式集的大小。
We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type $A$. More precisely, consider a $p$-adic linear space $V$ and the set $Lat(V)$ of all lattices in $V$. The complex distance in $Lat(V)$ is a complete system of invariants of a pair of points of $Lat(V)$ under the action of the complete linear group. An element of a Nazarov semigroup is a lattice in the duplicated linear space $V\oplus V$. We investigate behavior of the complex distance under the action of the Nazarov semigroup on the set $Lat(V)$.
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 …
Parameterizing Conjugacy Classes of Unramified Tori via Bruhat-Tits Theory
该研究利用Bruhat-Tits理论，对非阿基米德局部域上连通约化群中几类非分歧相关子群的共轭类进行参数化，并将结果推广到有限李型群的类似对象。
We construct $p$-adic analogs of operator colligations and their characteristic functions. Consider a $p$-adic group $G=GL(α+k\infty, Q_p)$, its subgroup $L=O(k\infty,Z_p)$, and the subgroup $K=O(\infty,Z_p)$ embedded to $L$ diagonally. We show that double cosets $Γ= K\setminus G/K$ admit a structure of a semigroup, $Γ$ acts naturally in $K$-fixed vectors of unitary representations of $G$. For each double coset we assign a 'characteristic function', which sends a certain Bruhat--Tits building to another building (buildings are finite-dimensional); image of the distinguished boundary is contain…
Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We present a combinatorial description of the space of double cosets of $Hie(T_n)$ with respect to $Aut(T_n)$ and construct a 'ne…
Totally Ramified Maximal Tori and Bruhat-Tits theory
该研究利用Bruhat-Tits理论，研究非阿基米德局部域上连通约化群中$K$-迷向极大（驯顺）$k$-环面的轨道类与嵌入问题，给出稳定类、有理类及模有理共轭$k$-嵌入的显式刻画。
Bruhat-Tits buildings, representations of $p$-adic groups and Langlands correspondence
该研究将Bruhat-Tits理论相关的超尖点表示结果推广到任意$p$-进约化群表示，证明半单尖点支集的增强$L$-参数的Springer对应性质，并在特定条件下建立其与非奇异超尖点支集表示的局部Langlands对应，得出复合$L$-包的相关结论。
For any Kac-Moody group $\mathbf{G}$, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for $\mathbf{G}$ is strictly compatible with a $\mathbb{Z}$-valued length function. We conjecture in general and prove for $\mathbf{G}$ of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of "double-affine Schubert varieties."
On tamely ramified $\mathcal G$-bundles on curves
本文证明光滑射影曲线上的仿射 Bruhat-Tits 群概形可表示为驯顺覆盖上约化群概形的不变量，并建立两类扭子及其模叠之间的联系，同时发展等变扭子的局部类型理论。
We determine the maximal eigenvalue of the p-adic curvature transformations on Bruhat-Tits buildings, and we give an essentially optimal upper bound on the minimal non-zero eigenvalue of these transformations.
For a symmetrizable Kac-Moody algebra the category of admissible representations is an analogue of the category of finite dimensional representations of a semisimple Lie algebra. The monoid associated to this category and the category of restricted duals by a generalized Tannaka-Krein reconstruction contains the Kac-Moody group as open dense unit group and has similar properties as a reductive algebraic monoid. In particular there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid, [M 1]. We determine the closure relations of the Bruhat and Birkhoff cells, which…
Quantum ergodicity on the Bruhat-Tits building for $\text{PGL}(3, F)$ in the Benjamini-Schramm limit
本文在 Benjamini-Schramm 极限下证明了 $\ ext{PGL}(3, F)$ 的 Bruhat-Tits 建筑上温谱本征函数的平均等分布，并利用 Brion 公式等工具简化计算，同时给出了若干独立结果。
Normed Fiber Functors
该研究借助Tannakian形式体系，将一般线性群Bruhat-Tits建筑的非阿基米德范数描述推广到非分歧约化群情形，并给出对应仿射群概形的模论刻画。
This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine We…
We apply spectral analysis of quotients of the Bruhat-Tits buildings of type $\tilde{A}_{d-1}$ to construct isospectral non-isomorphic Cayley graphs of the finite simple groups $\operatorname{PSL}_d({\mathbb F}_q)$ for every $d \geq 5$ ($d \neq 6$) and prime power $q > 2$.
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