Contemporary Mathematics
Volume 442, 2007
A class of gradings of simple Lie algebras
Karin Baur and Nolan Wallach
ABSTRACT. In this paper we give a classification of parabolic subalgebras of
simple Lie algebras over IC that satisfy two properties. The first property is
Lynch's sufficient condition for the vanishing of certain Lie algebra cohomol-
ogy spaces for generalized Whittaker modules associated with the parabolic
subalgebra and the second is that the moment map of the cotangent bundle
of the corresponding generalized flag variety be birational onto its image. We
will call this condition the moment map condition.
1.
Introduction
The purpose of this paper is to give a classification of parabolic subalgebras of
simple Lie algebras over C that satisfy two properties. The first property is Lynch's
sufficient condition for the vanishing of certain Lie algebra cohomology spaces for
generalized Whittaker modules associated with the parabolic subalgebra and the
second is that the moment map of the cotangent bundle of the corresponding
generalized flag variety be birational onto its image. We will call this condition
the moment map condition. Associated to each parabolic subalgebra of a simple
Lie algebra
g
is a Z-grading and to each Z-grading of
g
corresponds a parabolic
subalgebra. The first condition is that the parabolic subalgebra has a Richardson
element in the first graded part (where the grading is the one associated to the
parabolic subalgebra).
If G
is a semi-simple Lie group over
JR.
and if
P
is a parabolic subgroup of
G such that the intersection of the complexification of its Lie algebra intersected
with each simple factor of the complexification of Lie( G) satifies the two conditions
then one can prove holomorphic continuation of Jacquet integrals and a variant
of a multiplicity one theorem for degenerate principal series associated with P
([14], [15]). The full classification corresponding to the first condition was the
subject of our joint paper
[3]
where we classified the so-called "nice" parabolic
subalgebras of simple Lie algebras over C. Thus the point of this paper is to
list the elements of the list in
[3]
that satisfy the second condition. The second
condition is just the assertion that the stabilizer of a Richardson element in the
nilradical of the parabolic subgroup corresponding to the parabolic subalgebra of
1991 Mathematics Subject Classification. Primary 17B45, Secondary 17B10.
Key words and phrases. Parabolic subgroups, moment map.
The first author was supported by the Freie Akademische Stiftung and by DARPA contract
#AFRL F49620-02-C-0010.
@2007 American Mathematical Society
3
http://dx.doi.org/10.1090/conm/442/08517
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