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Given a linear action of a group $G$ on a $K$-vector space $V$, we considerthe invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We areparticularly interested in the case where $V =\gfq^n$ and $G$ is the group$U_n$ of all upper unipotent matrices or the group $B_n$ of all uppertriangular matrices in $\GL_n(\gfq)$. In fact, we determine $\gfq[V \oplusV^*]^G$ for $G = U_n$ and $G =B_n$. The result is a complete intersection forall values of $n$ and $q$. We present explicit lists of generating invariantsand their relations. This makes an addition to the rather short list of "doublyparametrized" series of group actions whose invariant rings are known to have auniform description.

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