On Arkhipov?s and Enright?s functors

Abstract

We give a description of Arkhipov's and (Joseph's and Deodhar-Mathieu's versions of) Enright's endofunctors on the category O, associated with a fixed triangular decomposition of a complex finite-dimensional semi-simple Lie algebra, in terms of (co)approximation functors with respect to suitably chosen injective (resp. projective) modules. We establish some new connections between these functors, for example we show that Arkhipov's and Joseph's functors are adjoint to each other. We also give several proofs of braid relations for Arkhipov's and Enright's functors.

A new approach to Kostant’s problem

“…As the module M 1 is the largest submodule of M 1 that belongs to Ꮿ 1 , we get M 1 /M 1 ∈ Ꮿ 2 , which proves (iii). For statement (iv) we refer to [Khomenko and Mazorchuk 2005 Theorem 5. Let w ∈ W be an involution and R be the right cell of W , containing w. Then the following conditions are equivalent:…”

“…In other words, the kernels of nat θ w I and θ w (nat I ) coincide. Now the statement of the lemma follows from [Khomenko and Mazorchuk 2005, Lemma 1], applied to the situation F = Aθ w , G = θ w A and H = θ w .…”

A Twisted Approach to Kostant's Problem

“…In [18] it was shown that Arkhipov's functors are adjoint to Joseph's completion functors (see [15]), which suggests a close connection to Kostant's problem. We base our arguments mostly on the results of [3] and also use some results from [17,18,22].…”

“…In Section 3 we show how one can apply the twisting functors to obtain the classical results related to Kostant's problem. In principle if one takes into account the relation between the twisting functors and Joseph's completion functors, obtained in [18], our approach here is rather similar to the original approach. However, here it is formulated in a shorter way.…”

A geometric model for Hochschild homology of Soergel bimodules

“…The braid group B G of G is a finitely presented group, with generators s for each simple reflection s 2 W , which is defined by the presentation: There are several natural weak actions of the braid group on category O by families of functors (see, for example, Andersen and Stroppel [2] and Khomenko and Mazorchuk [9]), which have an avatar on the bimodule side of the picture in the form of a complex of bimodules attached to each braid group element. The description of these bimodule complexes can be found in various sources, for example Khovanov [11], or for general Coxeter groups in Rouquier [13].…”