  We studied the positive representations $P_\lambda$ of split real quantum
groups $U_{q\tilde{q}}(g_R)$ restricted to the Borel subalgebra
$U_{q\tilde{q}}(b_R)$. We proved that the restriction is independent of the
parameter $\lambda$. Furthermore, we prove that it can be constructed from the
GNS-representation of the multiplier Hopf algebra $U_{q\tilde{q}}^{C^*}(b_R)$
constructed earlier, which enables us to decompose their tensor product using
the theory of the "multiplicative unitary". This will be an essential
ingredient in the construction of quantum higher Teichm\"{u}ller theory from
the perspective of representation theory, generalizing earlier work by
Frenkel-Kim.
