Degenerate principal series and nilpotent invariants

We study two nilpotent invariants, namely associated cycles and wave front cycles, attached to irreducible constituents of degenerate principal series representations of $$\mathrm {Sp}(2n,{\mathbb {R}})$$ . We compute the associated cycles of those constituents with the largest Gelfand–Kirillov dimension, as well as the dimensions of the space of generalized Whittaker models associated to nilpotent orbits occurring in the wave front cycles of these constituents. Furthermore, for these constituents, we prove that the coefficients of nilpotent orbits occurring in the wave front cycles equal the dimensions of the space of generalized Whittaker models associated to the corresponding nilpotent orbits. Our main approach is based on the result of Loke and Ma (Compos Math 151(1):179–206, 2015), and, Gomez and Zhu’s (Geom Funct Anal 24(3):796–853, 2014), on the behavior of these nilpotent invariants under the local theta correspondence.