Alexander Premet提出了以下问题：经典半单限制李代数的次正则幂零表示与相应奇点的非交换变形之间的关系是什么？我们解决了$A$类型的这个问题。使用麦凯对应，我们涉及的解决方案，基地在等变$K$ -理论介绍Lusztig。
英文摘要：
Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type $A$. Using the McKay correspondence, we relate the solution to bases in equivariant $K$-theory introduced by Lusztig.
Let $\bbk$ be an algebraically closed field of prime characteristic $p$. If $p$ does not divide $n$, irreducible modules over $\frak {sl}_n$ for regular and subregular nilpotent representations have already known(see \cite{Jan2} and \cite{Jan3}). In this article, we investigate the question when $p$ divides $n$, and precisely describe simple modules of $\frak {sl}_n$ for regular and subregular nilpotent representations.
We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.
Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the $\mathfrak{sl}_{n+1}$ subregular W-algebra can be realised in terms of the $\mathfrak{sl}_{n+1}$ regular W-algebra and the half lattice vertex algebra $Π$. This generalises the realisations found for $n=1$ and $2$ in [arXiv:1711.11342, arXiv:2007.00396] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realisation to explore the representation theory of $\mathfrak{sl}_{n+1}$ subregula…
We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type $\mathfrak{sl}(1|n), \mathfrak{osp}(2|2n)$. The type A case proves a conjecture of Feigin and Semikhatov. Let $(\mathfrak{g}_1,\mathfrak{g}_2) = (\mathfrak{sl}_{n+1},\mathfrak{sl}(1|n+1))$ or $(\mathfrak{so}_{2n+1}, \mathfrak{osp}(2|2n))$ and let $r$ be the lacity of $\mathfrak{g}_1$. Let k be a complex number and $\ell$ defined by $r(k+h^\vee_1)(\ell+h^\vee_2)=1$ with $h^\vee_i$ the dual Coxeter numbers of the $\mathfrak g_i$. Our first main result is that the Heisenberg cos…
According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple …
We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra of type A of admissible level k. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl(2)-induced modules in these orbits. In particular, simple admissible sl(2)-induced modules are fully classified for the principal nilpotent orbit.
This article is a summary of the author's unpublished Ph.D thesis. Its purpose is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of type $A_r$, $D_r$, $E_6$, $E_7$ and $E_8$ to the inhomogeneous simple singularities of type $B_r$, $C_r$, $F_4$ and $G_2$. To a simple homogeneous singularity, one can associate the representation space of a particular quiver. This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity. From this we will construct and compute explicitl…
We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case …
Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学宇宙物理数学研究所（WPI））；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理解析研究所）
In this article we study quotients of deformations of simple singularities, and attempt to characterize them in terms of subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X = D_s$, $E_6$ or $E_7$, but not semiuniversal anymore. Therefore not all subdiagrams of $X$ appear as singular configurations of the fibers of the deformation. We propose a conje…
We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corr…
