Stability conditions and braid group actions on affine An quivers

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

An Application of Spherical Geometry to Hyperkähler Slices

This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$ , a reductive subgroup $H\subseteq G$ , and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$ , defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$ . This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$ , $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ -regularity of $(G,H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Satake diagrams and real structures on spherical varieties

With each antiholomorphic involution [Formula: see text] of a connected complex semisimple Lie group [Formula: see text] we associate an automorphism [Formula: see text] of its Dynkin diagram. The definition of [Formula: see text] is given in terms of the Satake diagram of [Formula: see text]. Let [Formula: see text] be a self-normalizing spherical subgroup. If [Formula: see text] then we prove the uniqueness and existence of a [Formula: see text]-equivariant real structure on [Formula: see text] and on the wonderful completion of [Formula: see text].

Spherical subgroups in simple algebraic groups

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

On Irreducible Representations of So2n+1 × So2m

In this paper, we study the restriction of irreducible representations of the group SO2n+1 × SO2m to a spherical subgroup.