Typical blocks of the category O and Whittaker modules for Takiff superalgebras

We study the simplicity of Kac induced modules over the -th Takiff superalgebras g_: = g C/ (^+1), for >0, associated with the Lie superalgebras g of type I. We formulate a general notion of typical weights and typical Jordan blocks of the category O for g_ associated with Lie superalgebras gl (m|n), osp (2|2n) and pe (n). For Lie superalgebras gl (m|n) and osp (2|2n), we establish an equivalence from an arbitrary typical Jordan block of the category O for g_ to a Jordan block of the category O for the even subalgebra of g_. This provides a solution to the problem of determining the composition multiplicities of the Verma modules over g_ with typical highest weights. We also investigate non-singular Whittaker modules over these Takiff superalgebras. In particular, we obtain a classification of non-singular simple Whittaker modules and a criterion for simplicity of non-singular standard Whittaker modules.