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Twisted Verma modules. (English) Zbl 1079.17002
Joseph, Anthony (ed.) et al., Studies in memory of Issai Schur. Basel: Birkhäuser (ISBN 0-8176-4208-0/hbk). Prog. Math. 210, 1-26 (2003).
In [R. S. Irving, “Shuffled Verma modules and principal series modules over complex semisimple Lie algebras”, J. Lond. Math. Soc., II. Ser. 48, No. 2, 263–277 (1993; Zbl 0801.17007)] translation functors in the BGG-category \(\mathcal{O}\) for a semi-simple finite-dimensional complex Lie algebra were applied to construct principal series modules. The present paper is inspired by the inductive procedure in this construction. The authors define an axiomatic setup for what they call a family of twisted Verma modules. They show that the axioms describe a unique (up to isomorphism) family of modules, indexed by pairs of elements in the Weyl group. This family has all properties, expected after Irving’s work. The authors give three different constructions for this family. The first one is the one by Irving. The second one uses local cohomology with supports in Schubert cells. The last one uses Arkhipov’s twisting functors on the category \(\mathcal{O}\) (this is where the name twisted Verma modules come from). Finally, the authors show that twisted Verma modules possess natural Jantzen type filtrations with corresponding sum formulae.
For the entire collection see [Zbl 1005.00049].

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras