  The purpose of this paper is to give explicit constructions of unitary
$t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the
explicit constructions were so far known only for very special cases. Here
explicit construction means that the entries of the unitary matrices are given
by the values of elementary functions at the root of some given polynomials. We
will discuss what are the best such unitary $4$-designs in $U(4)$ obtained by
these methods.
  Indeed we give an inductive construction of designs on compact groups by
using Gelfand pairs $(G,K)$. Note that $(U(n),U(m) \times U(n-m))$ is a Gelfand
pair. By using the zonal spherical functions for $(G,K)$, we can construct
designs on $G$ from designs on $K$.
  We remark that our proofs use the representation theory of compact groups
crucially. We also remark that this method can be applied to the orthogonal
groups $O(d)$, and thus provides another explicit construction of spherical
$t$-designs on the $d$ dimensional sphere $S^{d-1}$ by the induction on $d$.
