A bideterminant basis for a reductive monoid

Classification of Reductive Monoid Spaces Over an Arbitrary Field

In this semi-expository paper we review the notion of a spherical space. In particular we present some recent results of Wedhorn on the classification of spherical spaces over arbitrary fields. As an application, we introduce and classify reductive monoid spaces over an arbitrary field.

一般基上的约化幺半群

Reductive monoids over general base

本文发展了一般基概形上单位群为分裂约化群的仿射代数幺半群理论，给出了分类定理，并应用于轨道闭包、Steinberg型定理和Vinberg幺半群的有限型积分模型。

约化幺半群的分类

Classification of reductive monoids

该论文通过下降方法对任意基概形上的约化幺半群进行分类，构造了相关结构层并应用于判据、扩张定理及 Vinberg 幺半群的构造。

On the Resolution of Reductive Monoids and Multiplicativity of $γ$-factors

In this article, we give a proof of multiplicativity for $γ$-factors, an equality of parabolically induced and inducing factors, in the context of the Braverman-Kazhdan/Ngo program, under the assumption of commutativity of the corresponding Fourier transforms and a certain generalized Harish-Chandra transform. We also discuss the resolution of singularities and their rationality for reductive monoids, which are among the basic objects in the program.

An analogue of a reductive algebraic monoid, whose unit group is a Kac-Moody group

By a generalized Tannaka-Krein reconstruction we associate to the admissible representations of the category O of a Kac-Moody algebra, and its category of admissible duals a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by Kac and Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat a…

Representations of the Renner Monoid

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalizes the Munn-Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn-Solomon formula for calculating the character of the Renner monoids, in terms of the characters of…

Białynicki-Birula decomposition for reductive groups in positive characteristic

We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.

Generic Hecke algebra for Renner monoids

We associate with every Renner monoid $R$ a \emph{generic Hecke algebra} $\H(R)$ over $\mathbb{Z}[q]$ which is a deformation of the monoid $\mathbb{Z}$-algebra of $R$. If $M$ is a finite reductive monoid with Borel subgroup $B$ and associated Renner monoid $R$, then we obtain the associated Iwahori-Hecke algebra $\H(M,B)$ by specialising $q$ in $\H(R)$ and tensoring by $\mathbb{C}$ over $\mathbb{Z}$, as in the classical case of finite algebraic groups. This answers positively to a long-standing question of L. Solomon.

An algebraic geometric model of an action of the face monoid associated to a Kac-Moody group on its building

The face monoid described in [M1] acts on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a Kac-Moody group. We found in [M5] two natural extensions of the action of the Kac-Moody group on its building to actions of the face monoid on the building. Now we give an algebraic geometric model of one of these actions of the face monoid. The building is obtained as a part of the spectrum of homogeneous prime ideals of the Cartan algebra of the Kac-Moody group. We describe the full spe…

p-进Iwahori Hecke代数与对偶Vinberg单形

Pro-p Iwahori Hecke algebras and the dual Vinberg monoid

本文研究了p-进Iwahori Hecke代数与对偶Vinberg单形的关系，并在GL(n)情况下推广了SpecZ由半单表示参数化的结果。

On extended weight monoids of spherical homogeneous spaces

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H \subset G$, the extended weight monoid $\widehat Λ^+_G(G/H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line bundles on $G/H$. Assuming that $G$ is semisimple and simply connected and $H$ is specified by a regular embedding in a parabolic subgroup $P \subset G$, in this paper we obtain a description of $\widehat Λ^+_G(G/H)$ via the set of simple spherical roots of $G/H$ together with certain combinatorial data explicitly computed from the pair $(P,H)$. As an application…

关于仿射 Hecke 代数的两个模几何实现

On two modular geometric realizations of an affine Hecke algebra

本文通过构造三类范畴的幺半等价，将正特征域上仿射 Hecke 代数与 Harish-Chandra 双模、Steinberg 簇等变凝聚层及仿射旗簇构造层联系起来，并据此证明了 Finkelberg-Mirković 关于主块几何实现的猜想。

Crystals, quiver varieties and coboundary categories for Kac-Moody algebras

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commuto…