On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.Comment: Latex, 75 pages. Minor corrections. Final version, to appear in the Japanese Journal of Mathematic

On the classification of non-equal rank affine conformal embeddings and applications

We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra Vk (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 → C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 → C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs