Gelfand-Fuchs cohomology of invariant formal vector fields

格尔福恩-基里洛夫维数与模 $p$ 余数上同调对于 $\mathrm{GL}_2$ 的内形式

Gelfand--Kirillov dimension and mod $p$ cohomology for inner forms of $\mathrm{GL}_2$

本文计算了内形式 $D^\times$ 的模 $p$ 上同调中海德克特征空间的格尔福恩-基里洛夫维数，并在 $D$ 在 $p$ 处为除环时给出了简化证明。

Traces, Schubert calculus, and Hochschild cohomology of category $\mathcal{O}$

We discuss how the Hochschild cohomology of a dg category can be computed as the trace of its Serre functor. Applying this approach to the principal block of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$, we obtain its Hochschild cohomology as the compactly supported cohomology of an associated space. Equivalently, writing $\mathcal{O}$ as modules over the endomorphism algebra $A$ of a minimal projective generator, this is the Hochschild cohomology of $A$. In particular our computation gives the Euler characteristic of the Hochschild cohomology of $\mathcal{O}$ in type A.

Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resol…

The Binary Invariant Differential Operators on Weighted Densities on the superspace $\mathbb{R}^{1|n}$ and Cohomology

Over the $(1,n)$-dimensional real superspace, $n>1$, we classify $\mathcal{K}(n)$-invariant binary differential operators acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of %the Lie superalgebra $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities--a superisation of a result by Feigin and Fuchs. We explicitly give 1-cocycles spanning these cohomology spaces.

Singular Virasoro vectors and Lie algebra cohomology

We present an explicit formula for a new family of Virasoro singular vectors. As a corrolary we get formulas for differentials of Feigin-Fuchs-Rocha-Carridi-Wallach resolution of the the positive nilpotent part of Virasoro (or Witt) algebra $L_1$.

GL2 的 Gelfand-Kirillov 维数与模 p 上同调

Gelfand-Kirillov dimension and mod p cohomology for GL2

本文研究全实域上四元数代数情形下 GL2 的模 p Galois 表示，证明相应 Hecke 特征空间中许多可容许光滑表示具有 Gelfand-Kirillov 维数 [F_v:Q]。

Permutation representations on Schubert varieties

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation represent…

Homological algebra for osp(1/2n)

We discuss several topics of homological algebra for the Lie superalgebra osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical results although the cohomology is not given by the kernel of the Kostant quabla operator. Based on this cohomology we can derive strong Bernstein-Gelfand-Gelfand resolutions for finite dimensional osp(1|2n)-modules. Then we state the Bott-Borel-Weil theorem which follows immediately from the Bott-Kostant cohomology by using the Peter-Weyl theorem for osp(1|2n). Finally we calculate the projective dimension of irreducible and Verma modules in th…

Bernstein-Gelfand-Gelfand resolutions for linear superalgebras

In this paper we construct resolutions of finite dimensional irreducible gl(m|n)-modules in terms of generalized Verma modules. The resolutions are determined by the Kostant cohomology groups and extend the strong (Lepowsky-)Bernstein-Gelfand-Gelfand resolutions to the setting of Lie superalgebras. It is known that such resolutions for finite dimensional representations of Lie superalgebras do not exist in general. Thus far they have only been discovered for gl(m|n) in case the parabolic subalgebra has reductive part equal to gl(m)+gl(n) and for tensor modules. In the current paper we prove th…

Zhelobenko Invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford Algebra Conjecture

Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of the simple 2m+1-dimensional module of the principal s-triple obtained from the Langlands dual. Bazlov settled this conjecture positively in type A. The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of the Harish-Chandra map for the enveloping algebra U(g) composed with …

On the Stability and Gelfand Property of Symmetric Pairs

A symmetric pair of reductive groups $(G,H,θ)$ is called stable, if every closed double coset of $H$ in $G$ is preserved by the anti-involution $g\mapsto θ(g^{-1})$. In this paper, we develop a method to verify the stability of symmetric pairs over local fields of characteristic 0 (Archimedean and $p$-adic), using non-abelian group cohomology. Combining our method with results of Aizenbud and Gourevitch, we classify the Gelfand pairs among the pairs \begin{align*} &(SL_n(F), (GL_k(F) \times GL_{n - k}(F)) \cap SL_n(F)), (U(B_1 \oplus B_2),U(B_1) \times U(B_2)),\\ &(GL_n(F),O(B)), (GL_n(F),U(B)…

正特征中受限 Lie-Rinehart 代数的形变与上同调

Deformations and Cohomology of restricted Lie-Rinehart algebras in positive characteristic

本文研究正特征下受限 Lie-Rinehart 代数的形式形变，证明 p>2 时由 Evans-Fuchs 受限上同调控制，并为 p=2 构造新的上同调复形，同时分析了受限 Heisenberg 代数的结构与上同调。

Generalised Gelfand-Graev Representations in Small Characteristics

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ of the finite field of prime order $p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism with $G = \mathbf{G}^F$ the corresponding $\mathbb{F}_q$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig, which decomposes Kawanaka's Generalised Gelfand-Graev Representations (GGGRs) in terms of characteristic functions of intersection c…

Cohomology of the Lie Superalgebra of Contact Vector Fields on $\mathbb{R}^{1|1} $ and Deformations of the Superspace of Symbols

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that an…

正特征中受限李代数及其态射的上同调与形变

Cohomology and deformations of restricted Lie algebras and their morphisms in positive characteristic

本文研究正特征下受限李代数及其态射的形变理论，分别针对 p≥3 和 p=2 引入或完善相应的受限上同调与形变上同调，并给出受限 Heisenberg 李代数的例子。

On the mod $p$ cohomology for $\mathrm{GL}_2$: the non-semisimple case

Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to \mathrm{GL}_2(\bar{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic. Under some mild assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(F_v)$ occurring in the corresponding Hecke eigenspaces of the mod $p$ cohomology of Shimura varieties associated to $D$ have…

Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we…

紧支撑微分同胚群的广义正能表示

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

受广义相对论启发，本文研究了光滑流形上紧支撑微分同胚群的广义正能射影酉表示，证明了在流形连通且维数大于1时该表示在单位分量上必然平凡，并确定了紧支撑向量场李代数的连续第二上同调。

有限李型群的幂单尖顶表示的Tate上同调

Tate cohomology of unipotent cuspidal representations of finite groups of Lie type

本文针对有限李型群的幂单尖顶表示，利用广义Gelfand-Graev表示与Harish-Chandra诱导构造的投射生成元的重数1性质，证明了特定Tate上同调群的不可约性与消失性。

Algebra of formal vector fields on the line and Buchstaber's conjecture

Let L_1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L_1) is isomorphic to the tensor product of the Landweber-Novikov algebra S in complex cobordism theory by reals. The cohomology H*(L_1) has trivial multiplication. Buchstaber conjectured that H*(L_1) is generated with respect to non-trivial Massey products by H^1(L_1). Feigin, Fuchs and Retakh found representation of H*(L_1) by trivial Massey products. In the present article we prove tha…

Theory of Finite Pseudoalgebras

Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of conformal algebras. They are defined as Lie algebras in a certain ``pseudotensor'' category instead of the category of vector spaces. A pseudotensor category (as introduced by Lambek, and by Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a way to compose them. This allows for the definition of Lie algebras, represent…

有限刚性对数共形场论中相关子的构造

The Construction of Correlators in Finite Rigid Logarithmic Conformal Field Theory

本文针对由非半单模化范畴描述的有限刚性对数共形场论，利用模化微观世界原理与因子化同调构造并分类开闭相关子，证明配分函数系数非负及局部算子导出代数的 BV 结构等价性。

Special representations of nilpotent Lie groups and the associated Poisson representations of current groups

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extension…

A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence

We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\widehat{\mathfrak g}$ of level $-2h^\vee-k$, where $h^\vee$ is the dual Coxeter number of the underlying semisimp…

关于四元数代数在 $\mathbb{Q}_p$ 上的一些模 $p$ 表示

On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$

本文证明了在满足 Taylor-Wiles 假设的模 Galois 表示下，来自 Shimura 簇模 p 上同调的四元数代数表示具有 Gelfand-Kirillov 维数 1，并由此推出 Scholze 函子二阶在超奇异表示上消失，同时给出了可约情形下像的结构定理。