Catharina Stroppel (Leicester/Aarhus):

Twisting and Tilting

We consider the principal block O_{o} of the Bernstein-Gelfand category O for a semisimple complex Lie algebra 𝔤 ([BGG]). This is a highest weight category with standard objects Δ(x) indexed by the elements of the Weyl group. For each element w of the Weyl group, Arkhipov defined a right exact endofunctor of the category of 𝔤-modules which restricts to an endofunctor of O_{o}. A very nice property of these functors is that they satisfy braid relations. If w is a simple reflection then T_{w}Δ(x) has the same composition factors as Δ(sx), but the modules are not isomorphic in general. This is why we call the modules T_{w}Δ(x) “twisted” standard objects (or twisted Verma modules in [AL]) and the functors “twisting functors”. We prove that twisting functors and the functors given by tensoring with finite dimensional modules commute (moreover, they commute with Jantzen’s translation functors, [Ja]). This is a very useful property to describe twisted dual standard objects. Of course, twisting functors do not commute with the duality. It turns out that we get the right adjoint functor of T_{w} by conjugating T_{w^{−1}} with the duality. These properties are used to prove that the derived functors LT_{w} define an auto-equivalence of the bounded derived category D^{b}(O_{o}). Therefore, we get an action of the braid group via derived equivalences. We get the following applications:

• A rather easy looking reformulation for the Kazdhan-Lusztig conjecture in terms of twisted simple objects (for details see [AS]).

• A definition of “twisted tilting modules” (see [St]).

• Each tilting module has a “w-twisted standard flag” (see the special cases [Ir], [M] and in general [MS]).

References

[AL] H. H. Andersen, N. Lauritzen, Twisted Verma modules, in: Studies in Memory of Issai Schur, 1-26, v. 210, Progress in Math, Birkhäuser, Basel, 2002.

[AS] H. H. Andersen, C. Stroppel, Twisting functors on O, Preprint Series University of Aarhus (March 2003).

[BGG] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, A certain category of 𝔤modules. Funkcional. Anal. i Prilozen. 10 (1976), no. 2, 1-8.

[Ir] R. Irving:, Shuffled Verma modules and principal series modules over complex semisimple Lie algebras. J. London Math. Soc. (2) 48 (1993), no. 2, 263-277.

[Ja] J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, Berlin, 1983.

[M] V. Mazorchuk Twisted and shuffled filtrations on tilting modules, Math. Reports of the Academy of Science of the Royal Society of Canada 25, n. 1 (2003), 26-32.

[MS] V. Mazorchuk, C. Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Technical Report University of Leicester 2003/07.

[St] C. Stroppel, Homomorphisms and Extensions of principal series modules, Journal of Lie Theory 13 (2003) 193-212.