Contemporary Mathematics
Volume 322, 2003
Minuscule Representations, Invariant Polynomials, and
Spectral Covers
Robert Friedman and John W. Morgan
Introduction
Let
G
be a simple and simply connected complex linear algebraic group, with
Lie algebra
g.
Let p: G
-t
Aut V be an irreducible finite-dimensional representation
of G, and let p*: g
-t
End V be the induced representation of
g.
A goal of this
paper is to study
p*
and
p,
and in particular to give normal forms for the action
of
p*(X)
and
p(g)
for regular elements
X
of g or regular elements
g
E
G.
Of
course, in the case of
p*,
when
X
is semisimple, the action of
p*(X)
on
V
can
be diagonalized, and its eigenvalues are given by evaluating the weights of
p
with
respect to a Cart an subalgebra
I)
of g on an element in
I)
conjugate to
X. If
on the
other hand
X
is a principal nilpotent element of
g,
then
X
can be completed to
an st2-triple (X,
h0
,
X_), where
h0
is regular and semisimple. In this case, if
I)
is
the Cartan subalgebra containing h0
,
then the eigenvalues for p*(ho) together with
their multiplicities, which are given by evaluating the weights of
p
with respect to
I)
on ho, completely determine Vas an s[2-module and hence determine the action
of
p*(X)
on
V.
A minor modification of these ideas will then describe the action
of
p*(X)
for every regular element
X.
In this paper, we attempt to give a different algebraic model for the action of
p*(X),
where
X
is regular, and to glue these different models together over the set
of all regular elements. Related methods also handle the case of
p(g),
where
g
is
a regular element of G. While we are only successful in case pis minuscule, and
partially successful in case
p
is quasiminuscule, it seems likely that the techniques
of this paper can be extended to give information about an arbitrary
p.
We believe
that the techniques and results of this paper are also of independent interest. For
example, they have been applied elsewhere to study sections of the adjoint quotient
morphism [7].
2000 Mathematics Subject Classification. Primary: 14F05, 17B10; Secondary: 14H52,
20G05.
Key words and phmses. Minuscule representation, spectral cover, Weierstrass cubic, regular
element, Kostant section.
The first author was partially supported by NSF grant DMS-99-70437. The second author
was partially supported by NSF grant DMS-97-04507.
©
2003 American Mathematical Society
http://dx.doi.org/10.1090/conm/322/05677
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