The Hom-spaces between projective functors

Abstract:

The category of projective functors on a block of the category $\mathcal O(\mathfrak g)$ of Bernstein, Gelfand and Gelfand, over a complex semisimple Lie algebra $\mathfrak g$, embeds to a corresponding block of the category $\mathcal O(\mathfrak g \times \mathfrak g)$. In this paper we give a nice description of the object $V$ in $\mathcal O(\mathfrak g \times \mathfrak g)$ corresponding to the identity functor; we show that $V$ is isomorphic to the module of invariants, under the diagonal action of the center $\mathcal Z$ of the universal enveloping algebra of $\mathfrak g$, in the so-called anti-dominant projective. As an application we use Soergel’s theory about modules over the coinvariant algebra $C$, of the Weyl group, to describe the space of homomorphisms of two projective functors $T$ and $T’$. We show that there exists a natural $C$-bimodule structure on $\operatorname {Hom}_{\{\operatorname {Functors}\}}(T, T’)$ such that this space becomes free as a left (and right) $C$-module and that evaluation induces a canonical isomorphism $k \otimes _C \operatorname {Hom}_{\{\operatorname {Functors}\}} (T, T’) \cong \operatorname {Hom}_{\mathcal O(\mathfrak g)}(T(M_e), T’(M_e))$, where $M_e$ denotes the dominant Verma module in the block and $k$ is the complex numbers.