In all finite Coxeter types but I 2 (12), I 2 (18) and I 2 (30), we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I 2 (2k), where k ≥ 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations.
In this paper we investigate Donkin's (p, r)-Filtration Conjecture, and present two proofs of the "if" direction of the statement when p ≥ 2h − 2. One proof involves the investigation of when the tensor product between the Steinberg module and a simple module has a good filtration. One of our main results shows that this holds under suitable requirements on the highest weight of the simple module. The second proof involves recasting Donkin's Conjecture in terms of the identifications of projective indecomposable G r -modules with certain tilting G-modules, and establishing necessary cohomological criteria for the (p, r)-filtration conjecture to hold.
Abstract. We classify projective functors on the regular block of RochaCaridi's parabolic version of the BGG category O in type A. In fact, we show that, in type A, the restriction of an indecomposable projective functor from O to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category O in type A are completely determined, up to isomorphism, by the linear transformations they induce on the level of the Grothendieck group, which was conjectured by Stroppel in [St3].
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