Realisation of Lusztig cones

On crystal operators in Lusztig's parametrizations and string cone defining inequalities

Let $\mathbf{w}_0$ be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and Kashiwara's (string) parametrizations of the canonical basis associated with $\mathbf{w}_0$. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities def…

Perverse Sheaves on the Nilpotent Cone and Lusztig's Generalized Springer Correspondence

In this note, we consider perverse sheaves on the nilpotent cone. We prove orthogonality relations for the equivariant category of sheaves on the nilpotent cone in a method similar to Lusztig's for character sheaves. We also consider cleanness for cuspidal perverse sheaves and the generalized Lusztig--Shoji algorithm.

The structure of the nilpotent cone, the Kazhdan-Lusztig map and algebraic group analogues of the Slodowy slices

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.

Kazhdan-Lusztig基与优化

Kazhdan-Lusztig Basis and Optimization

本文通过连续二次优化研究Hecke代数在q=1处的canonical基，证明了Kazhdan-Lusztig基在特定形的分拆中张成最大锥，并验证了优化方法在小秩下的有效性。

Computing the Lusztig--Vogan Bijection

Let $G$ be a connected complex reductive algebraic group with Lie algebra $\mathfrak{g}$. The Lusztig--Vogan bijection relates two bases for the bounded derived category of $G$-equivariant coherent sheaves on the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$. One basis is indexed by $Λ^+$, the set of dominant weights of $G$, and the other by $Ω$, the set of pairs $(\mathcal{O}, \mathcal{E})$ consisting of a nilpotent orbit $\mathcal{O} \subset \mathcal{N}$ and an irreducible $G$-equivariant vector bundle $\mathcal{E} \rightarrow \mathcal{O}$. The existence of the Lusztig--Vogan bijection $γ\c…

Degree cones and monomial bases of Lie algebras and quantum groups

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n…