  We define for real $q$ a unital $*$-algebra
$U_q(\mathfrak{sl}(2,\mathbb{R}))$ quantizing the universal enveloping
$*$-algebra of $\mathfrak{sl}(2,\mathbb{R})$. The $*$-algebra
$U_q(\mathfrak{sl}(2,\mathbb{R}))$ is realized as a $*$-subalgebra of the
Drinfeld double of $U_q(\mathfrak{su}(2))$ and its dual Hopf $*$-algebra
$\mathcal{O}_q(SU(2))$, generated by the equatorial Podle\'s sphere coideal
$*$-subalgebra $\mathcal{O}_q(K\backslash SU(2))$ of $\mathcal{O}_q(SU(2))$ and
its associated orthogonal coideal $*$-subalgebra $U_q(\mathfrak{k}) \subseteq
U_q(\mathfrak{su}(2))$. We then classify all the irreducible
$*$-representations of $U_q(\mathfrak{sl}(2,\mathbb{R}))$.
