In the first part of this paper the projective dimension of the structural modules in the BGG category $\mathcal{ O}$ 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 $\mathbf{ a}$-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 $\mathbb{ Z}^2$-graded and that it has two $\mathbb{ Z}$-graded Koszul subalgebras. The dimension of the space $\mathrm{ Ext}^1$ 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.
Categories: math.RT