Algebraic construction of contragradient quasi-Verma modules in positive characteristic

On Arkhipov?s and Enright?s functors

“…s α s β s α = w be a braid relation in the Weyl group. By [Ar2,Section 2.3], the functor A α is exactly the functor α • S α . If we denote by w the twist with respect to the automorphism of g, which corresponds to w (this one certainly does not depend on the expression for w), we get that the left hand side of the braid relation for Arkhipov's functor reads w • · · · • (S s β s α (β) ⊗ U(g) −) • (S s β (α) ⊗ U(g) −) • (S β ⊗ U(g) −).…”

“…The last isomorphism was first established by Soergel in [So1], using a special endofunctor on O, which was inspired by the work [Ar1] of Arkhipov. Later on, in [Ar2], Arkhipov proposed a construction, which associates an analogous functor to every simple root of g. Basicly, every Arkhipov's functor is tensoring with a bimodule. Reading [Ar2] one gets a very strong impression that Arkhipov's functors must satisfy the braid relation, especially as the statement of [Ar2,Lemma 2.1.10] says that two braid tensor products of Arkhipov's bimodules are isomorphic as left modules.…”

“…Later on, in [Ar2], Arkhipov proposed a construction, which associates an analogous functor to every simple root of g. Basicly, every Arkhipov's functor is tensoring with a bimodule. Reading [Ar2] one gets a very strong impression that Arkhipov's functors must satisfy the braid relation, especially as the statement of [Ar2,Lemma 2.1.10] says that two braid tensor products of Arkhipov's bimodules are isomorphic as left modules. However, we did not manage to derive the braid relations for functors (or tensor products of correspond-…”

Representations of quantum groups defined over commutative rings II

“…with respect to the multiplicative set {F n | n ∈ N}. Note that modules of the form A sT −1 π above appear naturally in other mathematical work (see [6] and [7]) besides our own (see [4]). We choose a homomorphism Ψ : M F → M and set…”

Categories with projective functors

“…Soergel's functor V is the main ingredient in a combinatorial description of the blocks of category O. The functor introduced by Arkhipov [4] was used by Soergel in [25] to determine the characters of tilting modules. One of the results in [3] is another reformulation of the Kazhdan --Lusztig conjecture by means of Arkhipov's functors.…”