Modular Representations of Truncated current Lie algebras

In this paper we consider the structure and representation theory of truncated current algebras \(\mathfrak{g}_m = \mathfrak{g}[t]/(t^{m+1})\) associated to the Lie algebra \(\mathfrak{g}\) of a standard reductive group over a field of positive characteristic. We classify semisimple and nilpotent elements and describe their associated support varieties. Next, we prove various Morita equivalences for reduced enveloping algebras, including a reduction to nilpotent \(p\)-characters, analogous to a famous theorem of Friedlander--Parshall. We go on to give precise upper bounds for the dimensions of simple modules for all \(p\)-characters, and give lower bounds on these dimensions for homogeneous \(p\)-characters. We then develop the theory of baby Verma modules for homogeneous \(p\)-characters and, whenever the \(p\)-character has standard Levi type, we give a full classification of the simple modules. In particular we classify all simple modules with homogeneous \(p\)-characters for \(\mathfrak{g}_m\) when \(\mathfrak{g} = \mathfrak{gl}_n\). Finally, we compute the Cartan invariants for the restricted enveloping algebra \(U_0(\mathfrak{g}_m)\) and show that they can be described by precise formulae depending on decomposition numbers for \(U_0(\mathfrak{g})\).