The Representation Type of the I-Blocks of Category O_S

Let g be a simple Lie algebra over the field C of complex numbers, with root system relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots of , which determines a standard parabolic subalgebra of g. Fix an integral weight in h*, with singular set J of simple roots. We determine when an infinitesimal block O(g, S, J) of parabolic category O_S is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Brstle-Knig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category O_S by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category O_S including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2.