Young's lattice, invariant ideals, and weak identities

For a $d$-dimensional vector space $V$ over a field of characteristic zero, we show that $GL(V)$-stable ideals of the universal enveloping algebra of the free two-step nilpotent Lie algebra $\mathfrak n_2(V)$ are in natural correspondence with filters in Young's lattice of integer partitions with at most $d$ rows. The same classification holds for $GL(V)$-stable Poisson ideals of $S(V\oplusΛ^2(V))$ with its natural Poisson bracket. As applications, we determine all weak polynomial identities of the $n$-th Weyl algebra relative to its standard generating subspace and of the universal enveloping algebra of the $(2n+1)$-dimensional Heisenberg Lie algebra relative to the Lie algebra itself.