Superizations of Cahen-Wallach symmetric spaces and spin representations of the Heisenberg algebra

Let M_0=G_0/H be a (n+1)-dimensional Cahen-Wallach Lorentzian symmetric space associated with a symmetric decomposition g_0=h+m and let S(M_0) be the spin bundle defined by the spin representation \rho:V\to GL(S) of the stabilizer H. This article studies the superizations of M_0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to \Lambda(S^*(M_0)). Here, G is a Lie supergroup which is the superization of the Lie group G_0 associated with a certain extension of the Lie algebra g_0 to a Lie superalgebra g=g_{\bar 0}+g_{\bar 1}=g_0+S, via the Kostant construction. The construction of the superization g consists of two steps: extending the spin representation \rho:h\to gl(S) to a representation \rho:g_{0}\to gl(S) and constructing appropriate \rho(g_0)-equivariant bilinear maps on S. Since the Heisenberg algebra heis is a codimension one ideal of the Cahen-Wallach Lie algebra g_0, first we describe spin representations of heis and then determine their extensions to g_0. There are two large classes of spin representations of heis and g_0: the zero charge and the non-zero charge ones. The description strongly depends on the dimension (n+ 1) mod 8. Some general results about superizations g=g_{\bar 0}+g_{\bar 1} are stated and examples are constructed.