  Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let
$E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers
$r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of
$G(U(r,s) \times U(s,r))$-Shimura variety. We construct cycles which we
conjecture to generate the Tate classes and verify our conjecture in the case
of $G(U(1,s) \times U(s,1))$. We also discuss the general conjecture regarding
special cycles on the special fibers of unitary Shimura varieties.
