Ivan Penkov

Generic Irreducible Representations of Finite-Dimensional Lie Superalgebras

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

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-

Representations of classical Lie superalgebras of type I

Abstract An explicit description of a generic irreducible module (possibly infinite dimensional and not necessarily diagonalizable over some Cartan subsuperalgebra) over a finite dimensional classical Lie superalgebra G of type I is given in terms of its irreducible G 0-submodules. For a finite dimensional module the result reduces to a character formula, proved earlier by J. Bernstein and D. Leites for the Lie superalgebras gl(1 + me), gl(m + e), osp(2 + ne). In the finite dimensional case a character formula for another class, of so-called relatively typical, irreducible modules is also proved and illustrated by explicit examples.

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.

Characters of Finite-Dimensional Irreducible q(n) -Modules

The solution of the Kac character problem for the‘queer’ series of Lie superalgebras q(n) is announced. An explicit algorithm which computes the character of an arbitrary finite-dimensional irreducible q(n)-module is presented. As an illustration, the ‘correction terms’ to the generic character formula of Penkov (Monatsh. Math.118 (1994), 419) are written down for all finite-dimensional irreducible representations of q(n), for n≤4, with nongeneric character.

Generic representations of classical Lie superalgebras and their localization

We complete the study of arbitrary generic irreducible modules over a classical complex Lie superalgebraG initiated in [14] (whereG was assumed to be of type I) by presenting a full description of the underlyingG0-module of any suchG-module. This enables us in particular to extend Beilinson-Bernsteins localization theorem to a certain full subcategory of the category ofG-modules with fixed central character and also to describe the image of the enveloping algebraU(G) in the global sections of a generic twisted ring of differential operators on any flag superspace. As an application we construct an infinite family of full subcategories of the category ofG-modules with fixed generic atypical central character, each of which is equivalent to the category ofG0-modules with fixed regular central character.

CARTAN SUBALGEBRAS OF ROOT-REDUCTIVE LIE ALGEBRAS

Abstract Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras sl ∞ , so ∞ , and sp ∞ . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl ∞ were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of gl ∞ , Canad. Math. Bull. 46 (2003) 597–616]. In the present paper we refine and extend the results of [K.-H. Neeb, I. Penkov, Cartan subalgebras of gl ∞ , Canad. Math. Bull. 46 (2003) 597–616] to the case of a general root-reductive Lie algebra g . We prove that the Cartan subalgebras of g are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras sl ∞ , so ∞ , and sp ∞ . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras gl ∞ , sl ∞ , so ∞ , and sp ∞ with respect to the group of automorphisms of the natural representation which preserve the Lie algebra.

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.

Tensor Representations of Classical Locally Finite Lie Algebras

The structure of tensor representations of the classical finite-dimensional Lie algebras was described by H. Weyl.In this paper we extend Weyl’s results to the classical infinite-dimensional locally finite Lie algebras \({\mathfrak{g}{{\mathfrak{l}}}_{\infty}},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},\,\,{\mathfrak{s}{{\mathfrak{{p}}}_{\infty}}}\,\,\,{\rm and}\,\,\, {\mathfrak{s}{{\mathfrak{{o}}}_{\infty}}},\) and study important new features specific to the infinite-dimensional setting. Let \(\mathfrak{g}\) be one of the above locally finite Lie algebras and let v be the natural representation of \(\mathfrak{g}.\) The tensor representations of \(\mathfrak{g}.\) have the form V ⊗p ⊗ V ⊗q * for the cases \({\mathfrak{g}}=\,\,{\mathfrak{g}}{{\mathfrak{{l}}}}_{\infty},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},\) where V * is the restricted dual of V. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan’Hollder constituents, socle filtrations, and indecomposable direct summands.

Elements of supergeometry

This paper is devoted to an exposition of the structure theory of supermanifolds and bundles on them, a description of Serre duality on supermanifolds, investigation of inverse sheaves, definition of characteristic classes and proof of the Grothendieck-Riemann-Roch theorem for supermanifolds.