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In this paper we discuss the highest weight $frak k_r$-finite representations of the pair $(frak g_r,frak k_r)$ consisting of $frak g_r$, a real form of a complex basic Lie superalgebra of classical type $frak g$ (${frak g} eq A(n,n)$), and the maximal compact subalgebra $frak k_r$ of $frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.
Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
In this paper, we classify all indecomposable Harish-Chandra modules of the intermediate series over the twisted Heisenberg-Virasoro algebra. Meanwhile, some bosonic modules are also studied.
In this paper, we give geometric realizations of Lusztigs symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztigs symmetries.
Let $V$ be a highest weight module over a Kac-Moody algebra $mathfrak{g}$, and let conv $V$ denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv $V$, i.e. we completely classify the faces and their inclusions