A Classification of Certain Finite Double Coset Collections in the Classical Groups

Let G be a classical algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension criterion which involves root systems. 1 Statement of results Let G be a classical algebraic group defined over an algebraically closed field, let X be a maximal rank reductive subgroup, and let P be a parabolic subgroup. The property of finiteness for X\G/P is preserved under taking isogenies, quotients by the center of G, connected components and conjugates (see Lemma 2.2 for a precise statement). Thus, if desired, we can specify only the Lie type of G. Similarly, we can specify only the conjugacy class of X and P ; thus we usually give the Lie type of X and describe P by crossing off nodes from the Dynkin diagram for G. For the purpose of classifying finiteness, it suffices to consider only those X which are defined over Z. A subgroup X is spherical if X\G/B is finite for some (or, equivalently, for each) Borel subgroup B. For each classical group we list in Table 1 those maximal rank reductive spherical subgroups which are defined over Z. We first describe the notation which is used for the list, and for the rest of the paper, and then describe how the list is obtained. We write X = AnAmT1 if X is a group of Lie type An +Am which has a 1-dimensional central torus, ∗AMS subject codes: 14L, 20G15.