Abstract
We introduce a reduction method for studying representations of classical Lie superalgebras with atypical central character. We show that the atypical quotient of universal enveloping algebra has a non-trivial Jacobson radical. The factor by this radical has a new center, which is calculated for sl(1| n) and psl(2|2). Using this center we obtain new character formulae, generalization of Borel–Weil–Bott and Beilinson–Bernstein localization theorems.
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