  We construct the positive principal series representations for $U_q(g_R)$
where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where
$r$ is the rank of $g$. We show that under the representations, the generators
of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the
generators of $U_q(g_R)$ by the transcendental relations. We define the
modified quantum group of the modular double and show that the representations
of both parts of the modular double commute with each other, and there is an
embedding into the $q$-tori polynomials.
