Spherical subgroups of Kac–Moody groups and transitive actions on spherical varieties

Abstract

We define and study spherical subgroups of finite type of a Kac-Moody group. In analogy with the standard theory of spherical varieties, we introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, and we prove that it satisfies the same axioms as in the finite-dimensional case. Our main tool is a study of varieties that are spherical under the action of a connected reductive group L, and come equipped with a transitive action of a group containing L as a Levi subgroup.1 is a nilpotent Lie subalgebra of p, and we denote by U α = Exp( n>0 p nα ) the corresponding unipotent subgroup of P .We come to our crucial assumption: unless otherwise stated, in the whole paper H denotes a subgroup of P such that P/H is a spherical L-variety, i.e. it has a dense B-orbit. Invariants of spherical varietiesWe begin by recalling some notions and notations of the standard theory of spherical varieties. With any B-variety X one associates the latticewhose rank is by definition the rank of X, and the vector spaceAny discrete valuation v of C(X) can be restricted to the subset C(X) (B) . If X has an open B-orbit then this yields a well-defined element ρ(v) of N B (X), by identifying Ξ B (X) with the multiplicative group C(X) (B) modulo the constant functions. If X is normal then this applies in particular to the valuation associated with any B-stable prime divisor D of X, and in this case we denote simply by ρ(D) (or ρ X (D)) the corresponding element of N B (X).These definitions apply in particular if X is a spherical L-variety. In this case the set D(X) B \ D(X) L is called the set of colors (or L-colors) of X, and is denoted by ∆ L (X). If K is the stabilizer in L of a point in the open L-orbit of X, then X is called an embedding of its open orbit L/K. Intersecting with L/K gives a bijection between ∆ L (X) and D(L/K) B . Preliminaries on the geometry of P/HThe case where H is not contained in any proper parabolic subgroup of P will have a prominent role in the paper. We discuss in this section some consequences of this assumption.4.1. Definition. Let G be a connected reductive algebraic group, and K a subgroup. Then K is very reductive if it is not contained in any parabolic subgroup of G.Recall that a very reductive subgroup of a reductive group is reductive.The following first lemma is known, and will be useful in later proofs. We recall that, given G a reductive group and V a G-module, if V is a spherical G-variety then it is also called a spherical module. Lemma.Assume that H is not contained in any parabolic subgroup of P . Then H u ⊆ P u , and up to conjugating H in P we can assume that H has a Levi subgroup K contained in L. Under this assumption we have that (1) K is a spherical very reductive subgroup of L, and the homogeneous space P/H has a unique closed L-orbit isomorphic to L/K; 4

The trascendence of Kac-Moody algebras

“…And G. Pezzini, under the conditions of being G a complex connected reductive algebraic group, B a Borel subgroup and X a normal irreducible variety with an algebraic action of G, constructed an homogenous spherical datum for spherical subgroups H of G and showed that these objects satisfy the critical combinatorial axioms of the homogenous spherical datum for finite-dimensional spherical homogenous spaces G/H. He also proved that the (abstract) homogenous spherical datum for H is invariant under conjugation [208].…”