Seminar Algebra and Geometry: Gerry Schwarz (Brandeis University)

Lifting automorphisms of adjoint representations

Let $\mathfrak g$ be a simple complex Lie algebra and let G be the corresponding adjoint group. Consider the G-module V which is the direct sum of r copies of $\mathfrak g$.  We say that V is \emph{large\/} if r≥2 and r≥3 if G has rank 1. We showed that when V is large any algebraic automorphism ψ of the quotient Z:=V//G lifts to an algebraic  mapping Ψ:V→V which sends the fiber over z to the fiber over ψ(z), z ∈ Z. (Most cases were already handled by work of Kuttler)
We also showed that one can choose a biholomorphic lift Ψ such that Ψ(gv)=σ(g)Ψ(v), g ∈ G, v ∈ V, where σ is an automorphism of G. This leaves open  the following questions: Can one lift holomorphic automorphisms of Z? Which automorphisms lift if  V is not large? We answer  the first question in the affirmative and also answer the second question.

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