Summary: TWISTING FUNCTORS ON O
HENNING HAAHR ANDERSEN AND CATHARINA STROPPEL
Abstract. This paper studies twisting functors on the main block of the
Bernstein­Gelfand­Gelfand category O and describes what happens to (dual)
Verma modules. We consider properties of the right adjoint functors and show
that they induce an autoequivalence of derived categories. This allows us
to give a very precise description of twisted simple objects. We explain how
these results give a reformulation of the Kazhdan­Lusztig conjectures ([KL79])
in terms of twisting functors.
Introduction
In the following we study the structure of certain modules for a semisimple
complex Lie algebra arising from twisting functors and explain connections to mul­
tiplicity formulas for composition factors.
We fix a semisimple complex Lie algebra g and choose a Borel and a Cartan subal­
gebra inside g. We consider the corresponding BGG­category O of g­modules with
certain finiteness conditions ([BGG76]). For any element w of the corresponding
Weyl group we define following [Ark] an endofunctor Tw of O which is given by
tensoring with a certain g­bimodule Sw (a semiinfinite analog of the universal en­
veloping algebra of g). Such functors can be defined in a very general setup. In
[Soe98], for example, they were used to get character formulas for tilting modules

Source: Andersen, Henning Haahr - Department of Mathematics, Aarhus Universitet

Collections: Mathematics