18 research outputs found
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
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
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 and simple current extensions
In this paper we investigate the structure of intermediate vertex algebras
associated with a maximal conformal embedding of a reductive Lie algebra in a
semisimple Lie algebra of classical type.Comment: Latex file, 45 pages. Revised versio
On the Kernel of the affine Dirac operator
Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate
invariant bilinear form, \sigma an elliptic automorphism of L leaving the form
invariant, and A a \sigma-invariant reductive subalgebra of L, such that the
restriction of the form to A is non-degenerate. Consider the associated twisted
affine Lie algebras L^, A^, and let F be the \sigma-twisted Clifford module
over A^ associated to the orthocomplement of A in L. Under suitable hypotheses
on\sigma and A, we provide a general formula for the decomposition of the
kernel of the affine Dirac operator, acting on the tensor product of an
integrable highest weight L^-module and F, into irreducible A^-submodules.
As an application, we derive the decomposition of all level 1 integrable
irreducible highest weight modules over orthogonal affine Lie algebras with
respect to the affinization of the isotropy subalgebra of an arbitrary
symmetric space.Comment: Comments: Latex file, 37 pages. This is a revised version of the
paper published in Moscow Mathematical Journal, Vol. 8, n. 4, 2008, 759--788.
The new feature in the present version is a direct argument for a key step in
the proof of Theorem 1.1, which makes the paper self-containe