Lifting automorphisms of quotients of adjoint representations.

zbMATH Open1337.20051arXiv1301.6300MaRDI QIDQ2927929

Gerald W. Schwarz

Publication date: 5 November 2014

Published in: Journal of Lie Theory (Search for Journal in Brave)

Abstract: Let $mathfrak{g}_i$ be a simple complex Lie algebra, $1leqileqd$, and let $G=G_{1}imes...imes{G}_d$ be the corresponding adjoint group. Consider the $G$-module $V=oplus{r}_{i}mathfrak{g}_i$ where $r_{i}geq1$ for all $i$. We say that $V$ is emph{large} if all $r_{i}geq2$ and $r_{i}geq3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $psi$ of the quotient $Z:=V//G$ lifts to an algebraic mapping $PsicolonVoV$ which sends the fiber over $z$ to the fiber over $psi(z)$, $zinZ$. (Most cases were already handled in [Kuttler11]). We also showed that one can choose a biholomorphic lift $Psi$ such that $Psi(gv)=sigma(g)Psi(v)$, $ginG$, $vinV$, 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.

Full work available at URL: https://arxiv.org/abs/1301.6300

zbMATH Keywords

automorphismsdifferential operatorssimple complex Lie algebrasquotientsadjoint representationsactions of compact Lie groups

Mathematics Subject Classification ID

Representation theory for linear algebraic groups (20G05) Compact Lie groups of differentiable transformations (57S15) Semisimple Lie groups and their representations (22E46) Linear algebraic groups over the reals, the complexes, the quaternions (20G20) Lie algebras of linear algebraic groups (17B45)

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