THE COINVARIANT ALGEBRA AND REPRESENTATION TYPES OF BLOCKS OF CATEGORY O

Abstract

Let [Gfr ] be a finite-dimensional semisimple Lie algebra over the complex numbers. Let A be the finite-dimensional algebra of a (regular or singular) block of the BGG-category [Oscr ] . By results of Soergel, A has a combinatorial description in terms of a subalgebra C0 of the coinvariant algebra C. König and Mazorchuk have constructed an embedding from C0-mod into the category [Fscr ](Δ) of A-modules having a Verma flag. This is the main tool for the classification of [Fscr ] (Δ) into finite, tame and wild representation types presented here. As a consequence a classification of A-mod into finite, tame and wild representation types is obtained, thus re-proving a recent result of Futorny, Nakano and Pollack.