### Modular Reduction of Nilpotent Orbits

## Abstract

Suppose πΊ_{π} is a connected reductive algebraic π-group where π is an algebraically closed field. If π_{π} is a πΊ_{π}-module then, using geometric invariant theory, Kempf has defined the nullcone π©(π_{π}) of π_{π}. For the Lie algebra π€_{π} = Lie(πΊ_{π}), viewed as a πΊ_{π}-module via the adjoint action, we have π©(π€_{π}) is precisely the set of nilpotent elements.

We may assume that our group πΊ_{π} = πΊ Γ_{β€} π is obtained by base-change from a suitable β€-form πΊ. Suppose π is π€ = Lie(G) or its dual π€* = Hom(π€, β€) which are both modules for πΊ, that are free of finite rank as β€-modules. Then π β¨_{β€} π, as a module for πΊ_{π}, is π€_{π} or π€_{π}* respectively.

It is known that each πΊ_{β} -orbit πͺ β π©(π_{β}) contains a representative ΞΎ β π in the β€-form. Reducing ΞΎ one gets an element ΞΎ_{π} β π_{π} for any algebraically closed π. In this talk, we will explain two ways in which we might want ΞΎ to have βgood reductionβ and how one can find elements with these properties. We will also discuss the relationship to Lusztigβs special orbits.

This is on-going joint work with Adam Thomas (Warwick).