Ivan Dimitrov

Weight modules of direct limit Lie algebras

The purpose of this paper is to initiate a systematic study of the irreducible weight representations of direct limits of reductive Lie algebras and, in particular, of the classical simple direct limit Lie algebras A(∞), B(∞), C(∞), andD(∞). We study arbitrary, not necessarily highest weight, irreducible weight modules and describe the supports of all such modules. The representation theory of the classical direct limit groups has been initiated in the pioneering works of G. Olshanskii [O1], [O2] and is now in an active phase (see the recent works of A. Habib [Ha], L. Natarajan [Na], K.-H. Neeb [Ne], and L. Natarajan, E. Rodriguez-Carrington, and J. A. Wolf [NRW]). Nevertheless, the structure theory of weight representations of the simple direct limit Lie algebras has until recently been still in its infancy as only highest weight modules have been discussed in written works (see the works of Yu. A. Bahturin and G. Benkart [BB], K.-H. Neeb [Ne], and T. D. Palev [P]). Our approach is based mainly on the recent paper [DMP] in which a general method for studying the support of weight representations of finite-dimensional Lie al-

On the structure of weight modules

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight 0-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of go, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.

Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups

The purpose of the present paper is twofold, to introduce the notion of a generalized flag in an infinite-dimensional vector space V (extending the notion of a flag of subspaces in a vector space) and to give a geometric realization of homogeneous spaces of the ind-groups SL(∞), SO(∞), and Sp(∞) in terms of generalized flags. Generalized flags in V are chains of subspaces which in general cannot be enumerated by integers. Given a basis E of V, we define a notion of E- commensurability for generalized flags, and prove that the set ℱl(ℱ,E) of generalized flags E-commensurable with a fixed generalized flag ℱ in V has a natural structure of an ind-variety. In the case when V is the standard representation of G = SL(∞), all homogeneous ind-spaces G/P for parabolic subgroups P containing a fixed splitting Cartan subgroup of G are of the form ℱl(ℱ,E). We also consider isotropic generalized flags. The corresponding ind-spaces are homogeneous spaces for SO(∞) and Sp(∞). As an application of the construction, we compute the Picard group of ℱl(ℱ,E) (and of its isotropic analogs) and show that ℱl(ℱ,E) is a projective ind-variety if and only if ℱ is a usual, possibly infinite flag of subspaces in V.

PARTIALLY INTEGRABLE HIGHEST WEIGHT MODULES

We prove a more general version of a result announced without proof in [DP], claiming roughly that in a partially integrable highest weight module over a Kac-Moody algebra the integrable directions from a parabolic subalgebra.

DECOMPOSING INVERSION SETS OF PERMUTATIONS AND APPLICATIONS TO FACES OF THE LITTLEWOOD-RICHARDSON CONE

If

Locally semisimple and maximal subalgebras of the finitary Lie algebras gl(∞), sl(∞), so(∞), and sp(∞)

\alpha \in S_n

Parabolic sets of roots

α∈Sn is a permutation of