In the first part of this
paper the projective dimension of the structural modules in the BGG category 𝒪 is
studied. This dimension is computed for simple, standard and costandard modules.
For tilting and injective modules an explicit conjecture relating the result
to Lusztig’s a-function is formulated (and proved for type A). The second
part deals with the extension algebra of Verma modules. It is shown that
this algebra is in a natural way ℤ2-graded and that it has two ℤ-graded
Koszul subalgebras. The dimension of the space Ext1 into the projective
Verma module is determined. In the last part several new classes of Koszul
modules and modules, represented by linear complexes of tilting modules, are
constructed.
