Twisted Verma Modules

By H. Andersen, N. Lauritzen. 2001

47 citations recorded by the source corpus.

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6f89aceaa319e3517184dea2e1281a351060c98b

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Citations in the corpus

On functors associated to a simple root (2004, unknown): “[AL], [Ma]) that the set of twisted Verma modules are equal to the set of shuffled Verma modules.”
Projective-injective modules, Serre functors and symmetric algebras (2005, background): “For any w A W , there is a twisting functor Tw : O! O (given by tensoring with some ‘‘semi-regular bimodule’’), see [2], [45] or [3] for a precise definition.”
Twisting functors on (2003, unknown): “They can also be realized as local cohomology groups of line bundles on the corresponding ag variety (see [ AL02 ]) and are exactly the Wakimoto modules, i.e., modules with certain vanishing properties for the semi-innite cohomology (see [Vor99] and [FF90] for the ane case).”
Categorification of (induced) cell modules and the rough structure of generalized Verma modules (2007, unknown): “From [1] and [2] (see also Proposition 45), we have explicit formulas for the action of the functor T on Verma modules.”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “Recall the right exact twisting endofunctors Ti = Tsαi , from [2], [3], of O(n) attached to sαi .”
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology (2006, background): “ence of the canonical projection Endg(θ0 λM(µ)) →End(θ0 λN), it is enough to show that C surjects onto Endg(θ0 λM(µ)) naturally. However, M(µ) ∼=TxM(λ), where Tx is the twisting functor as studied in [1], [2]. Now, Tx commutes with the action of the centre (see the deﬁnition of the functors in [2]),”
Some homological properties of the category O (2007, methodology): “According to [Andersen and Lauritzen 2003], shuffled Verma modules can be equivalently described using twisting and shuffling functors, the latter being Koszul dual to each other by [Mazorchuk et al.”
On Jordan-H\"older series of some locally analytic representations (2010, methodology): “Pd−i K (Pd K,Lλ) lies in the category O p+ (i,d−i+1) alg . Remark 7.4. It is well-known that the above objects lie in O,although, mostly the case of the full ﬂag variety is considered, cf. e.g. [20], [1]. Indeed this can be seen by using the Grothendieck-Cousin complex which was used in loc.cit. to ”