Dan Barbasch, Dan Ciubotaru
Using Lusztig's geometric classification, we find the reducibility points of a standard module for the affine Hecke algebra, in the case when the inducing data is generic. This recovers the known result of Muic-Shahidi for representations of split p-adic groups with Iwahori-spherical Whittaker vectors. We also give a necessary (insufficient) condition for reducibility in the non-generic case.
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In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups.
Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius--Schur indicators for "unipotent characters" of twisted dihedral groups.
We construct the irreducible unipotent modules of the finite general linear groups using tableaux. Our construction is analogous to that of James (1976) for the symmetric groups, answering an open question as to whether such a construction exists. Our modules are defined over any field containing a nontrivial $p^\text{th}$ root of unity (where $p$ is the defining characteristic of the group). We show that our modules are isomorphic to those constructed by James (1984), although the two constructions utilize quite different approaches. Finally we look closer at the complex irreducible unipotent…
Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $λ$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by $S^λ$. It is conjectured that for $c\in \Z_{\geqslant 0}$ the number of irreducible constituents of dimension $q^c$ of the restriction $\RRes^{G}_U(S^λ)$ of $S^λ$ to $U$ is a polynomial in $q$ with integer coefficients depending only on $c$ and $λ$, not on $q$. In the special case of the partition $λ=(1^n)$ this implies a longstanding (still open) conjecture of Hi…
Special unipotent representations and the coadjoint orbit method
本文针对线性实约化群或元辛群，基于相关循环对特殊幂幺表示进行几何分类，通过变形量子化给出其哈里什 - 钱德拉模的统一构造，证明相关模有不可约循环且酉可化，突出卢茨蒂格猜想应用。
We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's o…
Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of {\em character sheaves}, a geometric version of the classical character theory of the finite group $G(F_q)$. We show that under a certain technical condition, the restriction of a character sheaf to its {\em unipotent support} (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\Z$-basis of the $\Z$-module of unipotently supported virtual char…
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…
For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G^\vee-modules, where G^\vee is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G…
Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.
Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case
本文通过字符折叠映射，将扭曲量子环代数的极小前基本模块用单态量子坐标环表示，拓展了之前关于（co）极小前基本模块的研究。
Unipotent quantum coordinate ring and cominuscule prefundamental representations
本文研究了单幂量子坐标环与共minuscule预基本表示之间的关系，证明了其特征一致性和同构性。
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As…
We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favourable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.
Drinfeld double of quantum groups, tilting modules and $\mathbb{Z}$-modular data associated to complex reflection groups
本研究推广Lusztig与Malle关于复反射群幺正特征标及$\u2124$-模数据的工作，通过研究单复李代数Borel子代数量子包络代数Drinfeld double的倾斜模范畴，构造了部分对应数据的范畴化。
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 $W$ be the Weyl group of a split semisimple group $G$. Its Hecke category $\mathsf{H}_W$ can be built from pure perverse sheaves on the double flag variety of $G$. By developing a formalism of generalized realization functors, we construct a monoidal trace from $\mathsf{H}_W$ to a category of bigraded modules over a certain graded ring: namely, the endomorphisms of the equivariant Springer sheaf over the unipotent locus of $G$. We prove that: (1) On objects attached to positive braids $β$, the output is the weight-graded, equivariant Borel-Moore homology of a generalized Steinberg scheme $…
Hodge theory, intertwining functors, and the Orbit Method for real reductive groups
本文研究实约化群幂单表示的Hodge滤过，证明其与轨道方法预测的量子化滤过一致，从而推导出酉性与K-型描述等猜想，并给出部分旗簇上混合Hodge模的新上同调消失定理。
Noncommutative tensor triangular geometry: modules, bimodules, and unipotent Hopf algebras
本文研究了单幂Hopf代数的非交换张量三角几何，探讨了稳定范畴中的厚理想、Balmer谱及Hochschild上同调支持之间的关系。
Polynomial properties of unipotent radicals of parabolic subgroups in classical groups
本文研究辛群、正交群和酉群中极大标准抛物子群的幂幺根，给出其次数为$q^e$的不可约特征标数量的显式公式。
In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Broué's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Broué's conjecture are consistent with both the mathematics and philo…
Affine highest weight structures on module categories over quiver Hecke algebras
本文证明任意类型箭图Hecke代数的有限生成分次模范畴具有多种Kleshchev分层，从而得到与Weyl群元素相关的量子幺正子群范畴为仿射最高权范畴，推广了多项早期工作，关键利用行列式模和R-矩阵技术。
A unipotent triangular relationship is established between the dual standard monomial theoretic basis and canonical basis for the negative part of the quantized universal enveloping algebra of type A.
Khovanov-Lauda-Rouquier algebras of finite Lie type come with families of standard modules, which under the Khovanov-Lauda-Rouquier categorification correspond to PBW-bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and all non-zero endomorphisms of a standard module are injective. We obtain applications to extensions between standard modules and modular representation theory of KLR algebras.
We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to the Miličić--Soergel category $\mathcal{N}$. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard …
Khovanov-Lauda-Rouquier algebras $R_θ$ of finite Lie type are affine quasihereditary with standard modules $Δ(π)$ labeled by Kostant partitions $π$ of $θ$. In type $A$, we construct explicit projective resolutions of standard modules $Δ(π)$.
We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra $\mathfrak{q}(n)^{(2)}$. We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal $\mathbb{Z}$-graded modules over certain Heisenberg Lie superalgebras contained in $\mathfrak{q}(n)^{(2)}$.
Generic modules for the category of filtered by standard modules
本文研究了标准模块过滤类别的通用模块性质，证明其tame性与有限内长度的通用模块类有关，并修正了前人研究中的错误。
We describe explicitely the structures of standard $(g,K)$-modules of $SL(3,R)$.
Let G be a quasi-split p-adic group. Under the assumption that the local coefficients $C_ψ$ defined with respect to $ψ$-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.
We introduce cell modules for the tabular algebras defined in a previous work (math.QA/0107230); these modules are analogous to the representations arising from left Kazhdan--Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type $A$ equipped with its Kazhdan--Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.
We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups $O_{m}^{\pm}(3)$ with $m\geq6$ on nonsingular points of their standard modules. These actions together with those studied in \cite{HN} are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.
We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_{2}$. We also give conjectural series for some level 2 modules of $A^{(2)}_{13}$.
Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $Δ$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (Δ, Δ)$ of standard modules, characterize the stratification property of $E$ for $\leqslant$ and $\leqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).
We introduce a sequence of $q$-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For $\mathfrak{g}=\hat{\mathfrak{sl}_{2}}$, we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category $\mathcal{O}$ of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into $q$-characters of simple modules in this category $\mathcal{O}$.
We study the principal subspaces of higher level standard $A_2^{(2)}$-modules, extending earlier work in the level one case, by Calinescu, Lepowsky, and Milas. We prove natural presentations of principal subspaces and also of certain related spaces.
Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard $\widehat{\mathfrak{sl}(n)}$-modules, $n \ge 3$. As a consequence, we obtain the multigraded dimensions of the principal subspaces $W(k_1Λ_1 + k_2 Λ_2)$ and $W(k_{n-2}Λ_{n-2} + k_{n-1} Λ_{n-1})$. This generalizes earlier work by Calinescu o…
If $K$ is a field of finite characteristic $p$, $G$ is a cyclic group of order $q=p^α$, $U$ and $W$ are indecomposable $KG$-modules with $\dim U=m$ and $\dim W=n$, and $λ(m,n,p)$ is a standard Jordan partition of $ m n$, we describe how to find a generator for each of the indecomposable components of the $KG$-module $U \otimes W$.
We establish a connection between (degenerate) nonsymmetric Macdonald polynomials and standard bases and dual standard bases of maximal parabolic modules of affine Hecke algebras. Along the way we prove a (weak) polynomiality result for coefficients of symmetric and nonsymmetric Macdonald polynomials.
The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $β= q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the…
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