
Journal of Lie Theory 24 (2014), No. 3, 625639 Copyright Heldermann Verlag 2014 Lifting Automorphisms of Quotients of Adjoint Representations Gerald W. Schwarz Dept. of Mathematics, Brandeis University, Waltham, MA 024549110, U.S.A. schwarz@brandeis.edu [Abstractpdf] \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'', http://arxiv.org/abs/1201.6369 (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) 11151135.)\ 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 [ Fulltextpdf (351 KB)] for subscribers only. 