Mini-Workshop: Spherical Varieties and Automorphic Representations

Abstract

Let G be a reductive algebraic group over an algebraicailly closed field k and fix a Borel subgroup B ⊂ G. A subgroup H ⊂ G is called spherical if B acts with finitely many orbits on G/H, or equivalently if H acts with finitely many orbits on the falg variety G/B. We denote by B(G/H) the set of the B-orbits on G/H, the talk surveyed some of the main results concerning this set. The set B(G/H) comes naturally endowed with the Bruhat order, namely the partial order 6 induced by the inclusion of orbit closures. For instance, if H = B and if T is a maximal torus contained in it, then there is a bijection between B(G/H) and the Weyl group W = N/T (where N denotes the normalizer of T in G), and the partial order 6 coincides with the classical Bruhat order. When H is a symmetric subgroup of G (namely the set of points fixed by an algebraic involution of G), the partially ordered set B(G/H) was studied by R. W. Richardson and T. A. Springer in [6]. Let H ⊂ G be a spherical subgroup. Fix a maximal torus T ⊂ B, let W be the Weyl group of T and let R ⊃ S resp. be the attached sets of roots and of simple roots. The Richardson-Springer monoid is the monoid W ∗ generated by the simple reflection sα with the relations s 2 α = sα for all α ∈ S and the braid relations. As a set, W ∗ is the Weyl group W of G but with a different multiplication. An action of W ∗ on B(G/H) was defined by Richardson and Springer in [6] as follows: if w ∈W ∗ and O ∈ B(G/H), then w ∗O is the unique open B-orbit contained in the B-stable subset BwO. The weak order is the partial order on B(G/H) induced by the action of W ∗: if O,O′ ∈ B(G/H), then O O′ if and only if O′ = w ∗ O for some w ∈ W ∗. The Bruhat order is compatible with the W ∗-action and with the dimension function, namely the following properties hold for all α ∈ S and for all O,O′ ∈ B(G/H): i) O 6 sα ∗ O, ii) If O 6 O′, then sα ∗ O 6 sα ∗ O′, iii) If O 6 O′ and if dim(O) = dim(O′), then O = O′.