Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V\oplus V^{\ast }{]}^G$, where $V^{\ast }$ is the dual space. We are particularly interested in the case where $V={\text{gfq}}^n$ and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in ${\text{GL}}_n(\text{gfq})$. In fact, we determine $\gfq[V \oplus V^*]^G$for$G = U_n$and$G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description.