Abstract. It is well known that the category of real Lie supergroups is equivalent to the category of the so called (real) Harish-Chandra pairs, see [3,6,7]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper, we give a proof of this result in the complexanalytic case. Furthermore, if (G, O G ) is a complex Lie supergroup and H ⊂ G is a closed Lie subgroup, i.e. it is a Lie subsupergroup of (G, O G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, O G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold.It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [15]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.

We prove that every orthogonal Gelfand-Zeitlin algebra U acts (faithfully) on its Gelfand-Zeitlin subalgebra Γ. Considering the dual module, we show that every Gelfand-Zeitlin character of Γ is realizable in a U -module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on Γ gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules which realize the given Gelfand-Zeitlin characters.

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