Characters of Springer representations on elliptic conjugacy classes

From conjugacy classes in the Weyl group to semisimple conjugacy classes

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.

Euler characteristic of analogues of a Deligne-Lusztig variety for $GL_n$

In this paper we give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne-Lusztig variety if we replace Frobenius morphism with conjugation by an element for $GL_n$. The main theorem states that it only depends on the unipotent part of the Jordan decomposition of an element and the conjugacy class in the Weyl group. Also it generalizes the formula of the Euler characteristic of a Springer fiber for type A.

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry.

On C-small conjugacy classes in a reductive group

Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let C be an elliptic conjugacy class in W and let w be an element of minimal length of C. We investigate the existence of a semisimple class of G whose intersection with BwB has dimension dim(B). We show that in good characteristic such a semisimple class exists almost always.

特征p=2下的椭圆Weyl群元素与幺正等距映射

Elliptic Weyl group elements and unipotent isometries with p=2

该研究针对特征2代数闭域上的典型群，证明了与Weyl群椭圆共轭类关联的幺幂共轭类Φ(C)可通过幺幂类闭包关系刻画，补充了此前奇特征和例外群情形的相关结果。

中心同源与约化群中的共轭类

Central isogenies and conjugacy classes in reductive groups

将Steinberg关于半单元素中心化子分量群的描述推广到解释当通用覆盖非平展时幂单元素中心化子非约化的现象，并应用于L-参数模空间和通用变形环的特殊纤维中的重数计算。

Representations of the elliptic affine Hecke algebras

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-$A$. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d…

$\mathcal{W}$-代数的模性与仿射Springer纤维

Modularity for $\mathcal{W}$-algebras and affine Springer fibres

本文通过构造仿射李代数可允许表示与对偶仿射Springer纤维不动点之间的双射，将表示特征标的模性与Cherednik Verlinde代数构造相联系，并推广到$\mathcal{W}$-代数不可约表示。

通过周期Hecke模及其Langlands对偶构造椭圆类

Elliptic classes via the periodic Hecke module and its Langlands dual

本文通过周期Hecke模构造Springer解的椭圆类，引入椭圆扭曲群代数并定义其作用，证明DL算子自然组装构成周期模与Langlands对偶根系关联的有理同构。

Twisted character of a small representation of PGL(4)

We compute by a purely local method the elliptic, twisted by transpose-inverse, character χ_πof the representation π=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that χ_πis an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the tw…

PU$(n,1)$ 中的 Horn 问题

Horn's problem in PU$(n,1)$

本文研究 PU$(n,1)$ 椭圆共轭类上的乘法 Horn 问题，证明解集为凸多胞体的有限并，并完整刻画 $n=2$ 情形。

On elliptic factors in real endoscopic transfer I

This paper is concerned with the structure of packets of representations and some refinements that are helpful in endoscopic transfer for real groups. It includes results on the structure and transfer of packets of limits of discrete series representations. It also reinterprets the Adams-Johnson transfer of certain nontempered representations via spectral analogues of the Langlands-Shelstad factors, thereby providing structure and transfer compatible with the associated transfer of orbital integrals. The results come from two simple tools introduced here. The first concerns a family of splitti…

Twisted character of a small representation of GL(4)

We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with n…