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Journal of Lie Theory 24 (2014), No. 3, 625--639
Copyright Heldermann Verlag 2014

Lifting Automorphisms of Quotients of Adjoint Representations

Gerald W. Schwarz
Dept. of Mathematics, Brandeis University, Waltham, MA 02454-9110, U.S.A.


\def\g{{\frak g}} Let $\g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times\dots\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\g_i$ where $r_i\in\mathbb{N}$ for all $i$. We say that $V$ is {\it large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In ``Quotients, automorphisms and differential operators'', (2012), we showed that when $V$ is large any algebraic automorphism $\psi$ of the quotient $Z:=V/\!\!/G$ lifts to an algebraic mapping $\Psi\colon V\to V$ which sends the fiber over $z$ to the fiber over $\psi(z)$, $z\in Z$. (Most cases were already handled in J.~Kuttler, Lifting automorphisms of generalized adjoint quotients, Transformation Groups {\bf16} (2011) 1115--1135.)\ We also showed that one can choose a biholomorphic lift $\Psi$ such that $\Psi(gv)=\sigma(g)\Psi(v)$, $g\in G$, $v\in V$, where $\sigma$ 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. Part of the proof involves establishing the following result for $V$ large: Any algebraic differential operator of order $k$ on $Z$ lifts to a $G$-invariant algebraic differential operator of order $k$ on $V$. We also consider the analogues of the questions above for actions of compact Lie groups.

Keywords: Differential operators, automorphisms, quotients, adjoint representation.

MSC: 20G20, 22E46, 57S15

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