On Cocharacters Associated to Nilpotent Elements of Reductive Groups

Optimal Subgroups and Applications to Nilpotent Elements

Let G be a reductive group acting on an affine variety X , and let x ∈ X be a point whose G-orbit is not closed. In this paper, we study G.R. Kempf’s optimal class of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Our main result says that if K is a G-completely reducible subgroup of G which fixes x, then the optimal class ...

A classification of nilpotent orbits in infinitesimal symmetric spaces

Let G be a semisimple algebraic group defined over an algebraically closed field k whose characteristic is very good for G and not equal to 2. Suppose θ is an involution on G. We also denote the induced involution on g by θ. Let K = {g ∈ G : θ(g) = g} and let p be the −1-eigenspace of θ in g. The adjoint action of G on g induces an action of K on p and on the variety N (p), which consists of th...

Rationality Properties of Nilpotent Orbits in Good Characteristic

Let X be an F -rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F . One may associate toX certain cocharacters of Gwith favorable properties; this is an essential feature of the classification of geometric nilpotent orbits due to Bala-Carter, Pommerening, and, more recently, Premet. Suppose that the Lie algebra has a non-degenerate...

Integral Models for Shimura Varieties of Abelian Type

Introduction 967 1. Reductive groups and p-divisible groups 971 (1.1) Cocharacters and filtrations 971 (1.2) Review of S-modules 973 (1.3) Reductive groups and crystalline representations 975 (1.4) Application to p-divisible groups 978 (1.5) Deformation theory 981 2. Integral canonical models of Hodge type 986 (2.1) Shimura varieties 986 (2.2) Absolute Hodge cycles 989 (2.3) Integral models 992...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup