431 research outputs found
ad-Nilpotent ideals of a Borel subalgebra II
We provide an explicit bijection between the ad-nilpotent ideals of a Borel
subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q}
under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter
number of g). From this result we deduce in a uniform way a counting formula
for the ad-nilpotent ideals.Comment: AMS-TeX file, 9 pages; revised version. To appear in Journal of
Algebr
On some modules of covariants for a reflection group
Let be a simple Lie algebra with Cartan subalgebra and Weyl group . We build up a graded map of
-modules, where
is the space of -harmonics. In this way we prove an enhanced
form of a conjecture of Reeder for the adjoint representation.
New version with different title. Various improvements. New section 7.Comment: 18 Page
Spherical nilpotent orbits and abelian subalgebras in isotropy representations
Let be a simply connected semisimple algebraic group with Lie algebra
, let be the symmetric subgroup defined by an
algebraic involution and let be
the isotropy representation of . Given an abelian subalgebra
of contained in and stable under the action of
some Borel subgroup , we classify the -orbits in
and we characterize the sphericity of . Our main
tool is the combinatorics of -minuscule elements in the affine Weyl
group of and that of strongly orthogonal roots in Hermitian
symmetric spaces.Comment: Latex file, 29 pages, minor revision, to appear in Journal of the
London Mathematical Societ
Abelian subalgebras in Z_2-graded Lie algebras and affine Weyl groups
Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed
Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable
subalgebras of g_1.Comment: 21 pages, amstex file. Minor corrections. Introduction slightly
expanded. To appear in IMR
The adjoint representation inside the exterior algebra of a simple Lie algebra
For a simple complex Lie algebra we study the space of
invariants , (which describes the isotypic component of type
in ) as a module over the algebra of
invariants . As main result
we prove that is a free module, of rank twice the rank of ,
over the exterior algebra generated by all primitive invariants in , with the exception of the one of highest degree.Comment: Final version. More misprints corrected. To appear in Advances in
Mathematic
Nilpotent orbits of height 2 and involutions in the affine Weyl group
Let G be an almost simple group over an algebraically closed field k of
characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of
G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent
elements in g whose height is at most 2. We provide a parametrization of the
B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we
provide a complete description of the inclusion order among the B-orbit
closures in terms of the Bruhat order on certain involutions in the affine Weyl
group of g.Comment: v2: 28 pages, 1 table. Minor revision. To appear in Indag. Mat
The -orbit of , Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z
Let an affine Weyl group act as a group of affine transformations on
a real vector space V. We analyze the -orbit of a regular element in V
and deduce applications to Kostant's formula for powers of the Euler product
and to the representations of as permutations of the integers.Comment: Latex, 27 pages, minor corrections, to appear in Journal of Pure and
Applied Algebr
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