Sascha Orlik
Let F be a local non-archimedean field and let G be the group of F-valued points of a reductive algebraic group over F. In this paper we compute the Ext-groups of generalized Steinberg representations in the category of smooth G-representations with coefficients in a certain self-injective ring.
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We describe the equivariant K-groups of a family of generalized Steinberg varieties that interpolates between the Steinberg variety of a reductive, complex algebraic group and its nilpotent cone in terms of the extended affine Hecke algebra and double cosets in the extended affine Weyl group. As an application, we use this description to define Kazhdan-Lusztig "bar" involutions and Kazhdan-Lusztig bases for these equivariant K-groups.
Extensions of locally analytic generalized Steinberg representations
本文针对Qp有限扩张L上的GLn(L)，研究局部Qp解析广义抛物Steinberg表示，计算其Ext-群，揭示其携带的Breuil简单L-不变量与p进局部Langlands对应自守侧的关联。
In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our construction including the construction of a sort of Gelfand Zetlin basis for a generic irreducible representation of quantum GL(n) at odd roots of unity.
We generalize results of P. Schneider and U. Stuhler for GL_l+1 to a reductive algebraic group G defined and split over a non-archimedean local field K. Following their lines, we prove that the generalized Steinberg representations of G with coefficients in any abelian group are cyclic. When G is semi-simple of adjoint type, we give an expression of these representations, whenever it is possible and in particular for those that are of maximal degree, in terms of the parahoric subgroups of G.
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …
Let $F$ be a nonarchimedean local field, let $E$ be a Galois quadratic extension of $F$ and let $G$ be a quasisplit group defined over $F$; a conjecture by Dipendra Prasad states that the Steinberg representation of $G(E)$ is then $χ$-distinguished for a given unique character $χ$ of $G(F)$. In the first two papers of the series, Broussous and the author have proved that result when $G$ is $F$-split and $E/F$ is unramified; this paper deals with the tamely ramified case, still with $G$ $F$-split.
Iwahori Spherical Whittaker Functions for Steinberg Representations
研究Steinberg表示中Iwahori固定向量的性质，确定其在IwahoriHecke代数下的作用，并推导与之相关的Whittaker函数，扩展了GL_n(F)上的先前结果。
Twisted Yangians and Steinberg varieties of type C
研究型C广义steinberg变种的equivariant同调，并证明扭转杨代数AIII_{2n}^{(τ)}到它的代数同态映射
We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.
Let $F$ be a non archimedean local field of characteristic not $2$. Let $D$ be a division algebra of dimension $d^2$ over its center $F$, and $E$ a quadratic extension of $F$. If $m$ is a positive integer, to a character $χ$ of $E^*$, one can attach the Steinberg representation $St(χ)$ of $G=GL(m,D\otimes_F E)$. Let $H$ be the group $GL(m,D)$, we prove that $St(χ)$ is $H$-distinguished if and only if $χ_{|F^*}$ is the quadratic character $η_{E/F}^{md-1}$, where $η_{E/F}$ is the character of $F^*$ with kernel the norms of $E^*$. We also get multiplicity one for the space of invariant linear for…
Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and…
Koszul duality for generalised Steinberg representations of $p$-adic groups
本文证明了 p-进半单群的广义 Steinberg 表示扩张代数模范畴与 Langlands 参数簇上等变反常层子范畴之间的 Koszul 对偶等价。
On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications
本文研究了局部解析广义Steinberg表示之间的Ext群及其应用，推广了Schraen的结果，并定义了更高L-不变量。
We generalise the concept of a Steinberg cross-section to non-connected Kac-Moody group. As in the connected case, which was treated by G. Brüchert, a quotient map w.r.t the conjugacy action exists only on a certain submonoid of the Kac-Moody group. Non-connected Kac-Moody groups appear naturally as semidirect product of \C^* with a central extension of loop groups LG, where the underlying simple group G is no longer simply connected and might even be non-connected. In contrast to the connected case, the understanding of central extensions of non-connected loop groups is a rather complicated i…
Let $G$ be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and anti-invariants of the cohomology of the Springer fibres of the cone of nilpotent elements, $\mathcal N$, of $\operatorname{Lie}(G)$ for the Steinberg variety $Z$ of triples. Using a general specialization argument we show that for a parabolic subgroup $W_P \times W_Q$ of $W \times W$ the space of $W_P \times W_Q$-invariants and the space of $W_P \times W_Q$-anti-invariants of $H_{4n}(Z)$ are isomorphic to the top Borel-Moore homology gro…
Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field
本文通过确定Iwahori Hecke代数特征标在特定元素上的取值，完整计算了p-进域上$GL(n)$的广义Steinberg表示的Whittaker函数，并证明该表示限制到$SL_n(F)$上仍不可约，推广了Baruch-Purkait关于$GL_2(F)$的结果。
We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an…
On generalized Steinberg theory for type AIII
针对AIII型对称对的双旗簇$K$对角作用轨道，本文给出轨道的部分置换对参数化等描述，并通过部分置换组合算法显式刻画对称化与异值Steinberg映射，推广了$p=q=r$特殊情形下的已有结果。
A generalization of Steinberg theory and an exotic moment map
本研究将Steinberg关于约化群的理论推广到对称对情形，构造了广义Steinberg映射与异动量映射，聚焦特定一般线性群对称对，给出对应几何背景与组合算法，拓展了Robinson-Schensted对应。
This is the second paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. It contains a first formula for generalized exponents of small weights which extends the Shapiro-Steinberg formula for classical exponents. The formula is made possible by a computation of Fourier coefficients of the degenerate Cherednik kernel. Unlike the usual partition function coefficients, the answer reflects only the combinatorics of minimal expressions as a sum of roots.
Good position braid representatives and transversal slices of unipotent orbits
本文利用Weyl群关联辫幺半群中的好元素，为代数闭域上连通约化群的所有幂幺轨道构造横截面，并以此重新表述仿射Springer纤维的维数公式。
Symmetry breaking for $\operatorname{PGL}(2)$ over non-archimedean local fields
本文研究非阿基米德局部域二次扩张中 PGL(2) 主序列表示间的对称破缺算子，通过全纯分布核确定支集与映射性质并完成分类，还证明了 Steinberg 表示的直和项嵌入。
Let $S$ be a reduced $E$-Fountain semigroup. If $S$ satisfies the congruence condition, there is a natural construction of a category $\mathcal{C}$ associated with $S$. We define a $\Bbbk$-module homomorphism $φ:\Bbbk S\to\Bbbk\mathcal{C}$ (where $\Bbbk$ is any unital commutative ring). With some assumptions, we prove that $φ$ is an isomorphism of $\Bbbk$-algebras if and only if some weak form of the right ample identity holds in $S$. This gives a unified generalization for a result of the author on right restriction $E$-Ehresmann semigroups and a result of Margolis and Steinberg on the Catala…
On Higher Multiplicity upon Restriction from $\mathrm{GL}(n)$ to $\mathrm{GL}(n-1)$
针对非阿基米德局部域上从$\text{GL}(n)$到$\text{GL}(n-1)$的表示限制问题，证明了主系列表示与本质平方可积表示的同调性质及例外情况，给出非一般表示商为Steinberg的列表并否定广义Steinberg的情况。
Let $p>2$ be a prime number. Let $G:=GL_2(Q_p)$ and $π$, $τ$ smooth irreducible representations of $G$ on $\bar{F}_p$-vector spaces with a central character. We show if $π$ is supersingular then $Ext^1_G(τ,π)\neq 0$ implies $τ\cong π$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ext^1_G(τ,π)$, when $π$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ext^1_G(τ,π)$ for all irreducible representations of $G$.
Gluing parabolic character sheaves
该研究引入并研究伴随型约化群对应Vinberg半群非退化轨迹上幂幺特征层范畴，证明其偏对象可由抛物特征层粘合，应用中证明He的猜想并否定Lusztig关于Steinberg特征层中间扩张的猜想。
Geometric construction of Schur algebras
本文通过广义Steinberg簇的Borel-Moore同调与等变K群构造了任意型的广义Schur代数，并以此建立了局部几何Langlands对应的类似物、仿射AIII型拟分裂$\imath$量子群的实现及几何Howe对偶。
Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished c…
Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We establish two generalizations of this result. On the one hand to the extension of the Hecke algebra associated to the normaliser of a reflection subgroup and on the other hand to the extension of the Hecke algebra by a lattice.
Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\oper…
In this paper we investigate some methods on calculating the spaces of generalized semi-invariant distributions on p-adic spaces. Using homological methods, we give a criterion of automatic extension of (generalized) semi-invariant distributions. Based on the meromorphic continuations of Igusa zeta integrals, we give another criteria with purely algebraic geometric conditions, on the extension of generalized semi-invariant distributions.
A geometric model for blocks of Frobenius kernels
本文基于 Steinberg 张量积公式的几何对应，定义并研究了 Langlands 对偶群的仿射 Grassmannian 上的反常层范畴，为代数群 Frobenius 核的表示块提供几何模型，并证明其具有射影、内射对象且为最高权范畴。
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its opposite algebra. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions.
In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of semisimple $\check{G}$-local systems. To see this, we first rephrase the question in terms of Deligne-Lusztig duality and then study the Deligne-Lusztig functor DL_G^\spec acting on the spectral Langlands DG category IndCoh_N(LS_G). We prove that DL_G^\spec is the projection IndCoh_N(LS_G) \to QCoh(LS_G), followed by the action of a coherent D-module St_G which w…
C. Jantzen has defined a correspondence which attaches to an irreducible representation of a classical $p$-adic group, a finite set of irreducible representations of classical $p$-adic groups supported in a single or in two cuspidal lines (the case of the single cuspidal lines is interesting for the unitarizability). It would be important to know if this correspondence preserves the unitarizability (in both directions). The main aim of this paper is to complete the proof started in the previous paper of the fact that if we have an irreducible unitarizable representation $π$ of a classical $p$-…
Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments I: reduction to combinatorics
本文为系列研究首篇，引入最高导数多段概念，将局部非阿基米德域上$\text{GL}_n(F)$不可约表示的Bernstein-Zelevinsky导数单商构造问题约化为组合问题，并证明了双导数相关结果。
We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.
Generalized Grassmann invariant-redrawn
本文是旧作《广义Grassmann不变量》的重绘版，保留正文仅修改图，新增用箭图与$A_n$型根系解释新图的附录，提出两种补充Morse图缺失根的思路，相关改动以蓝色标注。
Central isogenies and conjugacy classes in reductive groups
将Steinberg关于半单元素中心化子分量群的描述推广到解释当通用覆盖非平展时幂单元素中心化子非约化的现象，并应用于L-参数模空间和通用变形环的特殊纤维中的重数计算。
The Geometrical Lemma for Smooth Representations in Natural Characteristic
本文在p进约化群光滑模p表示的导出范畴中建立了经典几何引理，并应用其计算抛物诱导表示间的高阶扩张群以及导出抛物诱导左伴随的上同调函子。
In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.
Separable extensions in tensor-triangular geometry and generalized Quillen stratification
本文分析张量三角范畴可分扩张诱导的谱映射，确定其像与纤维基数，并建立直至幂零的弱下降，从而将 Quillen 分层定理推广至等变导出范畴。
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