Related papers: On the fundamental group of a spherical variety

We define an analog in characteristic $p>0$ of the proalgebraic completion of the topological fundamental group of a complex manifold.

Let G be a simple algebraic group over an algebraically closed field of characteristic zero and X be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring of X into simple G-modules.

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure,…

Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k)…

We present an algorithm to compute the torsion component $\mathrm{Pic}^\tau X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^\tau X$ as a closed subscheme of a projective…

Let k_0 be a field of characteristic 0, k its algebraic closure, G a connected reductive group defined over k. Let H\subset G be a spherical subgroup. We assume that k_0 is a large field, for example, k_0 is either the field R of real…

We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\pi$ is obtained via induction from a representation of…

In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a…

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we…

Let G be a simple algebraic group over an algebraically closed field k. We classify the spherical conjugacy classes of G.

A normal variety $X$ is called $H$-spherical for the action of the complex reductive group $H$ if it contains a dense orbit of some Borel subgroup of $H$. We resolve a conjecture of Hodges--Yong by showing that their spherical permutations…

We investigate horospherical homogeneous spaces--a class of spherical homogeneous spaces encompassing both flag varieties and algebraic tori--over fields of characteristic p>0, and establish their complete classification for p>2.

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this…

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

We investigate the space $X$ of unitary hermitian matrices over $\frp$-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic,…

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators…

The topological fundamental group $\pi_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased…

We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke…

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $\pi_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed…

We compute explicitly the group of connected components $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{R}$ of real numbers.…