Amnon Yekutieli
We present an averaging process for sections of a torsor under a unipotent group. This process allows one to integrate local sections of such a torsor into a global simplicial section. The results of this paper have applications to deformation quantization of algebraic varieties.
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Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U'}$ is a cyclic $U/U'$-module. Second, we study "symmetries" of Poincare series for $U'$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polyno…
Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect groups conjecture for …
The paper is devoted to the problem of finding free generators in the fields of invariants for actions of unipotent groups on affine varieties. We consider the case when the unipotent group is the unipotent radical in an arbitrary parabolic subgroup in the reductive group of classical type GL(n), SL(n), O(n) or Sp(2n). In the explicit form, we present a system of free generators in the field of invariants for the action of the unipotent radical on the reductive group by conjugation.
We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types B or C, in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broue's abelian defect group conjecture for unipoten…
Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters
本文研究下幺幂群坐标环与带量子参数多项式环的联系，构造对称群作用并定义差分算子，进而研究 Schubert 多项式的量子化。
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup of G. Let W be a subgroup of G such that Ad(W) is contained in the Zariski closure (in the group of automorphisms of the Lie algebra of G) of the subgroup generated by the unipotent elements of Ad(W). Then any finite ergodic invariant measure for the action of W on G/Gamma is a h…
In this paper we introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an algebraic variety X consists of a rational morphism γ:X-->T and a compatible family e_i:G_m\times X-->X, i\in I of rational actions of the multiplicative group G_m satisfying certain braid-like relations. Such a structure induces a rational action of W on X. Quite surprisingly, many interesting ration…
Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group $W$ (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from $W$. Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with $W$ (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond …
Let P be a maximal parabolic of a classical group over a field F. Then the Levi subgroup M is isomorphic to the product of a classical group and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W is less than or equal to dim X and that dim W is even. When F is a local field, we obtain a Weyl-type integration formula for N.
Classification of coadjoint orbits for the maximal unipotent subgroup in the simple group of type F_4
本文通过支持集方法对F_4类型简单群中最大单位子群的共轭轨道进行分类。
Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under conjugation action) \bar{Q_l}-complexes on G. We can associate to G and e a modular category M_{G,e}. In this article, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class C_p^{\pm}.
We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table.
We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups.
Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p \ge 0$. We give a case-free proof of Lusztig's conjectures [Unipotent elements in small characteristic, {\em Transform. Groups} 10 (2005), 449--487] on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of $G$ which can be viewed as an extension of the Dynkin--Kostant theory, but is valid without restriction on $p$. We also obtain analogous results for the adjoint action of $G$ on its Lie algebra $\gl$ and the coadjoint action of $G$ on $\gl^*$.
Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construct…
We study locally transitive actions of the commutative unipotent group $\Gan$ on a nondegenerate quadric in the projective space $¶^{n+1}$. It is shown that for each $n$ such an action is unique up to isomorphism.
We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their stabilizer in $G$. Using the recent result by Brunat--Dudas--Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix, whenever they are related via Bonnafé--Dat--Rouquier's equivalence to a unipotent block. Th…
Decomposition numbers for unipotent blocks with small $\mathfrak{sl}_2$-weight in finite classical groups
本文证明A型抛物Kazhdan-Lusztig多项式可计算有限典型群幂零特征在酉素数情形下的分解数，借助仿射李代数范畴作用与Fock空间晶体，给出d-小Harish-Chandra系列分解数的闭公式。
Character sheaves on unipotent groups in positive characteristic: foundations
研究正特征下幂幺群上特征层理论，证明其主要定理，即容许对与特征层的\(L\)-包的对应关系，还通过附录探讨了相关范畴理论模式，如幺半范畴对偶、拓扑场论方法定义结构及证明导出范畴定义等价等。
If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite group arising in this way is called an algebra group. One can also consider G as a unipotent algebraic group over k. We study representations of G from the point of view of ``geometric character theory'' for algebraic groups over finite fields (cf. G. Lusztig, ``Character sheaves and generalizations'', math.RT/0309134). The main result of this paper is a cons…
We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types Bn, Cn, Dn. We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the field of invariants.
Hodge theory, intertwining functors, and the Orbit Method for real reductive groups
本文研究实约化群幂单表示的Hodge滤过，证明其与轨道方法预测的量子化滤过一致，从而推导出酉性与K-型描述等猜想，并给出部分旗簇上混合Hodge模的新上同调消失定理。
We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $χ: N(\!(t)\!) \to \mathbb{G}_a$ and a category $\mathcal{C}$ acted upon by $N(\!(t)\!) $, we define the category $\mathcal{C}^{N(\!(t)\!), χ}$ of $(N(\!(t)\!), χ)$-invariant objects, along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), χ}$. These are the Whittaker catego…
For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…
Categorical actions and derived equivalences for finite odd-dimensional orthogonal groups
本文证明了 Broué 的阿贝尔亏群猜想对有限奇数维正交群成立，通过约化定理、构造 Kac-Moody 代数的范畴作用及研究孤立 RoCK 块，最终建立了块及其 Brauer 对应间的导出等价。
A good knowledge of the Jordan canonical form (JCF) for a tensor product of `Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $π$ introduced by Norman in 1995. These permutations $π= π(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $π(r,s,p)$ to be trivial, …
Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent bundle. A long time ago, Gelfand and Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hami…
We study a natural action of the Heisenberg group of integer unipotent matrices of the third order on distribution space of a two-dimensional local field for a flag on a two-dimensional scheme.
Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics
本文研究约化李群在有限维向量空间上的线性作用，以自然的群论或几何条件解决由数论齐次动力学问题转化而来的线性动力学问题。
Let $G$ be a finite group isomorphic to $SL_n(q)$ or $SU_n(q)$ for some prime power $q$. In this paper, we give an explicit description of the action of automorphisms of $G$ on the set of its irreducible complex characters. This is done by showing that irreducible constituents of restrictions of irreducible characters of $GL_n(q)$ (resp. $GU_n(q)$) to $SL_n(q)$ (resp. $SU_n(q)$) can be distinguished by the rational classes of their unipotent support which are equivariant under the action of automorphisms. Meanwhile, we give a criterion to explicitly determine whether an irreducible character i…
Let $U(G)$ be a maximal unipotent subgroup of one of classical groups $G=GL(V),O(V),Sp(V)$. Let $W$ be a direct sum of copies of $V$ and its dual $V*$. For the natural action $U(G):W$, we describe a minimal system of homogeneous generators for the algebra of $U(G)$-invariant regular functions on $W$. For $G=GL(V)$, we also describe the syzygies among these generators in some particular cases.
In the first part, we construct a new isomorphism between the endomorphism algebra of an induced cuspidal character sheaf and the group algebra of the relative Weyl group involved. We show it differs from the isomorphism of Lusztig by a linear character, and we relate this linear character to some stabilizers. Some consequences for characteristic functions of character sheaves are obtained. In the second part, we compute explicitly this linear character whenever the cuspidal local system is supported by the regular unipotent class and, as an application of these methods, we obtain a refinement…
Invariant algebraic D-modules over affine algebraic groups
该研究探讨仿射代数群上的不变代数D-模，明确线性代数群中这类模与李代数表示的对应关系，证明幂单群中模同构的GIT商纤维等价条件，并完成代数环面与一般线性群Borel子群上不变D-模的分类。
An analogue of Whittaker reduction for group-valued moment maps
本文为半单复Poisson-Lie群的Poisson作用构造了Whittaker约化的类似，该约化沿幂零轨道横截面进行，并在Dirac几何框架下描述其辛叶。
A Lie group analog for the Monster Lie algebra
本文围绕魔群李代数$\frak m$展开，通过引入新的可和性概念，构造了相关李群类似物$G(\frak m)$及其完备幂幺群$\Uhp$等，还构造了指数映射和伴随表示类似物，证明了魔群$\mathbb{M}$对相关结构的作用及兼容性。
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