Eric Sommers
We present a new characterization of Lusztig's canonical quotient group. We also define a duality map: to a pair consisting of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra.
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We define a partial order on the set of pairs {(O,C)}, where O is a nilpotent orbit and C is a conjugacy class in Lusztig's canonical quotient of A(O). We then show that there is a unique order-reversing duality map on this set that has certain properties analagous to those of the original Lusztig-Spaltenstein duality map. This generalizes work of E. Sommers. In this revised version, a new approach leads to a strengthened version of Theorem 2, and applications of the results are discussed in greater detail.
Let $G$ be a reductive group over a field of prime characteristic. An indecomposable tilting module for $G$ whose highest weight lies above the Steinberg weight has a character that is divisible by the Steinberg character. The resulting "Steinberg quotient" carries important information about $G$-modules, and in previous work we studied patterns in the weight multiplicities of these characters. In this paper we broaden our scope to include quantum Steinberg quotients, and show how the multiplicities in these characters relate to algebraic Steinberg quotients, Weyl characters, and evaluations o…
The Kazhdan-Lusztig parameters are important parameters in the representation theory of $p$-adic groups and affine Hecke algebras. We show that the Kazhdan-Lusztig parameters have a definite geometric structure, namely that of the extended quotient $T//W$ of a complex torus $T$ by a finite Weyl group $W$. More generally, we show that the corresponding parameters, in the principal series of a reductive $p$-adic group with connected centre, admit such a geometric structure. This confirms, in a special case, our recently formulated geometric conjecture. In the course of this study, we provide a u…
We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent group U^F where U is the unipotent radical of a rational parabolic subgroup P = L U. We show that in some particular cases the cohomology of this quotient can be expressed in terms of "smaller" parabolic Deligne-Lusztig varieties associated to the Levi subgroup L.
We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.
Lusztig's Jordan decomposition and a finite field instance of relative Langlands duality
本文为经典群选取与抛物诱导和theta对应相容的典范Lusztig对应，推广Pan的结果，并证明theta对应与有限Gan-Gross-Prasad问题的对偶，作为相对Langlands对偶的有限域实例。
We show that the Lusztig integral form is dual to the RTT integral form of the type A quantum loop algebra with respect to the new Drinfeld pairing, by utilizing the shuffle algebra realization of the former and the PBWD bases of the latter obtained in arXiv:1808.09536.
Drinfeld suggested the definition of a certain endo-functor, called the pseudo-identity functor, on the category of D-modules on an algebraic stack. We extend this definition to an arbitrary DG category, and show that if certain finiteness conditions are satisfied, this functor is the inverse of the Serre functor. We apply this to the category of (g,K)-modules, and we stipulate that the pseudo-identity functor is the analog of the Deligne-Lusztig functor. In order to support this, we show that the pseudo-identity functor for (g,K)-modules is isomorphic to the composition of cohomological and c…
We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's …
Kazhdan-Lusztig map and Langlands duality
本文利用仿射旗簇几何定义从双边胞腔到 Weyl 群共轭类的 Kazhdan-Lusztig 映射，作为经典理论的仿射类比，并通过仿射 Springer 纤维上同调中的 Springer 表示证明，为 Lusztig 猜想提供证据。
In some recent work, Lusztig outlined a generalisation of the construction of Deligne and Lusztig to reductive groups over finite rings coming from the ring of integers in a local field, modulo some power of the maximal ideal. Lusztig conjectures that all irreducible representations of these groups are contained in the cohomology of a certain family of varieties. We show that, contrary to what was expected, there exist representations that cannot be realised by the varieties given by Lusztig. Moreover, we show how the remaining representations in the case under consideration can be realised in…
Let $Bun_G(X)$ be the moduli stack of $G$-torsors on a smooth projective curve $X$ for a reductive group $G$. We prove a conjecture made by Drinfeld-Wang and Gaitsgory on the Deligne-Lusztig duality for D-modules on $Bun_G(X)$. This conjecture relates Drinfeld-Gaitsgory's pseudo-identity functors to the enhanced Eisenstein series and geometric constant term functors on $DMod(Bun_G(X))$. We also prove a "second adjointness" result for these enhanced functors.
Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials
本研究证明Mario Marietti提出的抛物型Kazhdan-Lusztig多项式组合不变性猜想等价于其在极大商上的限制，且该等价性与Barkley、Gaetz近期证明的Kazhdan-Lusztig多项式不变性猜想相关等价性处于相反极端。
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and su…
A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG category of modules over Lusztig's quantum group at a root of unity. We prove that the semi-infinite cohomology functor for positive level modules factors through the Kazhdan-Lusztig functor at positive level and the quantum group cohomology functor with respect to the positive part of Lusztig's quantum group.
We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.
Equivariant Koszul Duality, Modular Category $\mathcal{O}$, and Periodic Kazhdan--Lusztig Polynomials
本文在假设卢斯滕堡猜想成立的条件下，推导了模范畴O中延伸群维数的公式，并利用环面等变科兹尔双模理论，通过周期性卡兹德-卢斯滕堡多项式系数来表达结果。
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…
We give a generalisation of Deligne-Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne-Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne-Lusztig varieties, and show that the "unramified" varieties, including a certain natural general…
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.
$\imath$Schur duality and Kazhdan-Lusztig basis expanded
本文在 B 型 Hecke 代数的拟置换模上构造拟抛物 KL 基，建立 AIII 型 $\imath$量子群与 B 型 Hecke 代数的 $\imath$Schur 对偶，并证明该基与张量空间上的 $\imath$典范基一致，进而得到拟抛物 KL 多项式的反演公式。
In this paper we study higher Deligne--Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations coincide with certain induced representations in the generic case; this gives a solution to a problem raised by Lusztig. In particular, we determine the dimensions of these representations. As an immediate application we verify a conjecture of Letellier for $\mathrm{GL}_2$ and $\mathrm{GL}_3$.
Unipotent quotients in compactified Coxeter varieties
本文推广了Lusztig关于Coxeter簇幂么商的描述，证明有限域上连通约化群的相关结果，并应用于证明Wang的上同调不变量猜想及导出Alvis-Curtis对偶的性质。
Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality
本文研究了抛物Hecke代数的Kazhdan-Lusztig基及其在Schur-Weyl双曲面中的应用，包括不可约表示的分类和核的构造。
Let G be a connected complex reductive group and let K be a symmetric subgroup of G. We prove a formula for the Drinfeld-Gaitsgory functor for the dg-category of K-equivariant sheaves on the flag manifold of G in terms of the Matsuki duality functor. As byproducts, we obtain a description of the Serre functor and the Deligne-Lusztig duality for (g,K)-modules.
From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.
We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius-Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig characters, and the Frobenius-Schur indicators of real-valued regular and semisimple characters of finite unitary groups.
Let \H_r be the generic type A Hecke algebra defined over \ZZ[u, u^{-1}]. The Kazhdan-Lusztig bases \{C_w\}_{w \in §_r} and \{C'_w\}_{w \in §_r} of \H_r give rise to two different bases of the Specht module M_λ, λ\vdash r, of \H_r. These bases are not equivalent and we show that the transition matrix S(λ) between the two is the identity at u = 0 and u = \infty. To prove this, we first prove a similar property for the transition matrices \tilde{T}, \tilde{T}' between the Kazhdan-Lusztig bases and their projected counterparts \{\tilde{C}_w\}_{w \in §_r}, \{\tilde{C}'_w\}_{w \in §_r}, where \tild…
A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties
该研究针对可解代数群作用提出贪心开轨道准则，应用于Dynkin箭图无重表示的Aut_Q(M)作用，简化Lusztig幂零簇刚性问题，为等定向A型提供显式算法。
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…
Minimal special degenerations and duality
本文分类单李代数中相邻特殊幂零轨道间 Slodowy 切片的奇点，揭示其在 Lusztig–Spaltenstein 对偶下的互换规律，并解决 Lusztig 关于切片相交上同调的猜想。
Geometry of buildings is used to prove some homological properties of the category of smooth representations of a reductive p-adic group (Kazhdan's "pairing conjecture", Bernstein's description of homological duality in terms of Deligne-Lusztig duality). A different proof had been obtained a little earlier by Schneider and Stuhler.
Rational wavefront sets and the descent method for finite classical groups
本文通过下降法完成有限域上经典群不可约表示波前集中有理幂零轨道的显式描述，并建立单幂表示波前集与Lusztig典范商的关系及Alvis-Curtis对偶。
We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lu…
We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of $\mathfrak{gl}_m$. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particul…
Kostka numbers and Fourier duality
本文建立了 perverse 层的 Fourier 变换与一般线性群幺正表示的 Deligne-Lusztig 对偶之间的联系，并推广到任意有限 Coxeter 群。
We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.
This paper gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1), we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(S_t), equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(S_t) and a localization of the parabolic category O for gl(V) associated with the pair (V, 1). In particular, we obtain an exact contra…
We prove that the elements $A_\leq$ defined by Lusztig in a completion of the periodic module actually live in the periodic module, in the type A case. In order to prove this, we compare, using Schur duality, these elements with the Kashiwara canonical basis of an integrable module.
We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.
We classify generalized tilting modules and full exceptional sequences for the family of quasi-hereditary quotients of type A zig-zag algebras and for a related family of algebras. We also give a characterization of these quotients as quasi-hereditary algebras with simple preserving duality that are 'close' to self-injective algebras.
Schur-Weyl Duality for Toroidal Algebras of Type $A$
该研究提出并证明了A型经典2-环面李代数商代数的Schur-Weyl对偶类似物，还给出了将其推广到m>2的m-环面情形的方法，拓展了Schur-Weyl对偶的适用范围。
Quantum imaginary Schur-Weyl duality
本文建立了任意参数下无扭仿射A型箭袋Hecke代数与对称群Iwahori-Hecke代数之间的对偶，推广了Kleshchev和Muth的虚Schur-Weyl对偶，并用于计算简单模和标准模的特征。
Lusztig sheaves and integrable highest weight modules in the symmetrizable case
本文基于前人工作，研究对称可约情况下的Lusztig sheaves与可积分最高权模的构造及格罗滕迪克群的实现，揭示其与晶格结构的关系。
Local geometry of special pieces of nilpotent orbits
本文研究例外类型李代数幂零锥特殊片段的局部几何，证明了Lusztig猜想的局部版本，补全了通用奇点证明，并发现了四种非孤立奇异奇点及一个缺失的共享轨道对。
In this paper we define a two-variable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q) induced from a solvable subgroup. We construct Kazhdan-Lusztig "R-polynomials" for H and show that they may be used to define a partial order on G(b,1,n). Using a generalization of Deodhar's notion of distinguished subexpressions we give a closed formula for the R-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-…
We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.
Singularities of orbit closures in loop spaces of symmetric varieties
本文研究对称簇环路空间中Iwahori轨道闭包的奇点，推广Lusztig-Vogan工作至仿射情形，证明IC-复形的纯性与奇偶消失，并应用于仿射Lusztig-Vogan模及相对Langlands对偶，回答Lusztig的问题。
We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.
Lusztig's special pieces conjecture
研究简单代数群$G$的李代数$\mathfrak g$中特殊幂零轨道的特殊片，通过两种方法证明其是光滑$G$ - 簇被有限群作用的商，适用于经典$\mathfrak g$并给出新证，还涉及例外群及相关嵌入与定义。
Unipotent Representations of Complex Groups and Extended Sommers Duality
本文研究复约化群的幂幺表示，通过 Langlands 对偶群构造扩展 Sommers 对偶映射，将对偶李代数中的幂零轨道及其 Lusztig 商共轭类对应到原李代数中的轨道有限覆盖。
We derive and study a quantum group g(p,q) that is Kazhdan--Lusztig-dual to the W-algebra W(p,q) of the logarithmic (p,q) conformal field theory model. The algebra W(p,q) is generated by two currents $W^+(z)$ and $W^-(z)$ of dimension (2p-1)(2q-1), and the energy--momentum tensor T(z). The two currents generate a vertex-operator ideal $R$ with the property that the quotient W(p,q)/R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2 p q) of irreducible g(p,q)-representations is the same as the number of irreducible W(p,q)-representations on which $R$ acts nontriv…
In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the ($\imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Sch…
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…
Singularities of orbit closures in loop spaces of symmetric varieties II: twisted cases
本研究将轨道闭包奇点理论推广至扭曲情形，证明IC复形的逐点纯性与奇偶消失性，并应用于扭曲仿射Lusztig--Vogan模及Langlands对偶。
Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group $\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$ divides $n$. Let $W$ denote the Weyl group of $\mathbf{G}$, namely the symmetric group $\mathfrak{S}_n$. We elucidate the structure of the extended quotient $\mathbf{S}_k // W$ as an algebraic variety and of $T_k // W$ as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of $K$-theory under Langlands duality, this ca…
We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that…
In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.
We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.
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