Parabolic projective functors in type A

TL;DR

This paper classifies projective functors in the parabolic category O of type A, showing they are determined by their action on the Grothendieck group, confirming a conjecture by Stroppel.

Contribution

It provides a complete classification of projective functors in the parabolic category O of type A, revealing their indecomposability properties and confirming a conjecture about their determination by Grothendieck group actions.

Findings

Indecomposable projective functors are either indecomposable or zero upon restriction.

Projective functors are uniquely determined by their induced linear transformations on the Grothendieck group.

The classification confirms Stroppel's conjecture in type A.

Abstract

We classify projective functors on the regular block of Rocha-Caridi's parabolic version of the BGG category $\mathcal{O}$ in type $A$. In fact, we show that, in type $A$, the restriction of an indecomposable projective functor from $\mathcal{O}$ to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category $\mathcal{O}$ 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 \cite{St}.