120 research outputs found
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
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
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
Compatible Discrete Series
Several very interesting results connecting the theory of abelian ideals of
Borel subalgebras, some ideas of D. Peterson relating the previous theory to
the combinatorics of affine Weyl groups, and the theory of discrete series are
stated in a recent paper (\cite{Ko2}) by B. Kostant. In this paper we provide
proofs for most of Kostant's results extending them to -nilpotent ideals
and develop one direction of Kostant's investigation, the compatible discrete
series.Comment: AmsTex file, 27 Pages; minor corrections; to appear in Pacific
Journal of Mathematic
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
Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting
We extend classical results of Kostant and al. on multiplets of
representations of finite-dimensional Lie algebras and on the cubic Dirac
operator to the setting of affine Lie algebras and twisted affine cubic Dirac
operator. We prove in this setting an analogue of Vogan's conjecture on
infinitesimal characters of Harish-Chandra modules in terms of Dirac
cohomology. For our calculations we use the machinery of Lie conformal and
vertex algebras.Comment: Latex file, 89 pages. Several misprints corrected. To appear in
Advances in Mathematic
Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebras
We give uniform formulas for the branching rules of level 1 modules over
orthogonal affine Lie algebras for all conformal pairs associated to symmetric
spaces. We also provide a combinatorial intepretation of these formulas in
terms of certain abelian subalgebras of simple Lie algebras.Comment: Latex, 56 pages, revised version: minor corrections, Subsection 6.2
added. To appear in Advances in Mathematic
Conformal embeddings in affine vertex superalgebras
This paper is a natural continuation of our previous work on conformal
embeddings of vertex algebras [6], [7], [8]. Here we consider conformal
embeddings in simple affine vertex superalgebra where
is a basic
classical simple Lie superalgebras. Let
be the subalgebra of generated by . We
first classify all levels for which the embedding in is conformal. Next we prove that, for a
large family of such conformal levels, is a completely
reducible --module and obtain
decomposition rules. Proofs are based on fusion rules arguments and on the
representation theory of certain affine vertex algebras. The most interesting
case is the decomposition of as a finite, non
simple current extension of . This
decomposition uses our previous work [10] on the representation theory of
.Comment: Latex file, 45 pages, to appear in Advances in Mathematic
- …