Über die Glattheit von Quotientenabbildungen
Let G be a semisimple group acting linearly on a complex vector space V.
Let h(t) be the generating function of the graded algebra C[V]^G of
G-invariants. Then it was known that h(t) satisfies a functional equation
h(1/t)=±t^q h(t) where q is some integer. The principal result is the
proof of a conjecture of V.L.Popov: q<=dim V.
As a corollary, one obtains a pretty good necessary criterion for C[V]^G
to be a polynomial ring: dim V <= 2dim G, provided V^G=0.
The principal technique is to calculate the canonical module of C[V]^G
using the following observation: the categorical quotient morphism
V-->V//G is smooth in codimension one.
Appeared in: Manuscripta Mathematica 56
(1986) 419-427
Available files:
Remark: In a later paper with Peter
Littelmann, all representations with q<dim V are classified when G
is simple or V is irreducible. There is also another
follow-up paper, in which the methods are
extended to arbitrary reductive groups.
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Last updated: August 25, 2006