Linear maps preserving invariants

Let G\subset \mathrm {GL}(V) be a complex reductive group. Let G' denote \{\varphi \in \mathrm {GL}(V)\mid p\circ \varphi =p\text { for all }p\in \mathbb {C}[V]^G\}. We show that, ``in general'', G'=G. In case G is the adjoint group of a simple Lie algebra \mathfrak {g}, we show...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-12, Vol.136 (12), p.4197-4200
Main Author: Schwarz, Gerald W.
Format: Article
Language:English
Subjects:
Adjoints
Algebra
Automorphisms
Exact sciences and technology
General mathematics
General, history and biography
Group theory
Mathematics
Scalars
Sciences and techniques of general use
Topological groups, lie groups
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Summary:Let G\subset \mathrm {GL}(V) be a complex reductive group. Let G' denote \{\varphi \in \mathrm {GL}(V)\mid p\circ \varphi =p\text { for all }p\in \mathbb {C}[V]^G\}. We show that, ``in general'', G'=G. In case G is the adjoint group of a simple Lie algebra \mathfrak {g}, we show that G' is an order 2 extension of G. We also calculate G' for all representations of \mathrm {SL}_2.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09628-7