FUNCTORS ON CATEGORIES OF ATYPICAL REPRESENTATIONS

NOAM SHOMRON

Abstract. Let 𝔤 be a simple finite-dimensional Lie superalgebra (we consider specifically 𝔤 = sl(m, n), W (n), or P (n)). These present several obstacles if one tries to reproduce standard algebraic and geometric constructions in the category of atypical 𝔤-modules. Following [Se1, Se2], we should consider representations of a certain quotient Ū = U/I of U = U(𝔤), where I is the intersection of annihilators of certain generalized Verma modules.

Even for W (n), which is worse behaved than classical Lie superalgebras, this reduction method provides a way of defining translation functors, a problem which was raised in [Sh]. While it is known that the centre of U is trivial in this case, we can factor by central characters of Ū to construct functors. Analyses of representation categories, for example, such as described in [F], seem to be very interesting for Lie superalgebras. This work is part of an effort to shed new light on atypical representations, about which very little was known in the past due to lack of tools.

References

[F] Peter Fiebig, “Centers of deformed representation categories and translation functors”, to appear.

[Se1] Vera Serganova, “On representations of the Lie superalgebra P (n)”, to appear.

[Se2] Vera Serganova, “On the reduction method for atypical representations of classical Lie superalgebras”, to appear.

[Sh] Noam Shomron, “Blocks of Lie superalgebras of type W (n)”, to appear.