Blocks of fat category O

We generalize the category O of Bernstein, Gelfand and Gelfand to the so called fat category O, On and derive some of its properties. From a Lie theoretic point of view, contains a significant amount of indecomposable representations that do not belong to O (although it fails to add new simple ones) such as the fat Verma modules. These modules have simple top and socle and may be viewed as standard objects once a block decomposition of is obtained and each block is seen to be equivalent to a category of finite dimensional modules over a finite dimensional standardly stratified algebra. We describe the Ringel dual of these algebras (concluding that principal blocks are self dual) and we obtain the character formulae for their tilting modules. Furthermore, a double centralizer property is proved, relating each block with the corresponding fat algebra of coinvariants. As a byproduct we obtain a classification of all blocks of in terms of their representation type. In the process of determining the quiver and relations which characterize the basic algebras associated to each block of On we prove (for root systems of small rank) a formula establishing the dimension of the Ext1 spaces between simple modules. By borrowing from Soergel some results describing the behaviour of the combinatorial functor V, we are able to compute examples.