  Given a linear action of a group $G$ on a $K$-vector space $V$, we consider
the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are
particularly 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 upper
triangular matrices in $\GL_n(\gfq)$. 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.
