Highest weight 𝔰𝔩₂-categorifications II: Structure theory

Abstract

This paper continues the study of highest weight categorical s l 2 \mathfrak {sl}_2 -actions initiated in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the refined sense. Then we prove that any highest weight s l 2 \mathfrak {sl}_2 -categorification can be filtered in such a way that the successive quotients are so-called basic highest weight s l 2 \mathfrak {sl}_2 -categorifications. For a basic highest weight categorification we determine minimal projective resolutions of standard objects. We use this, in particular, to examine the structure of tilting objects in basic categorifications and to show that the Ringel duality is given by the Rickard complex. We apply some of these structural results to categories O \mathcal {O} for cyclotomic Rational Cherednik algebras.