  In the paper, we establish the global basis theory for the bosonic extension
$\widehat{\mathcal{A}}$ associated with an arbitrary generalized Cartan matrix.
When $\widehat{\mathcal{A}}$ is of simply-laced finite type, it is isomorphic
to the quantum Grothendieck ring of the Hernandez-Leclerc category over a
quantum affine algebra. In this case, we show that the $(t,q)$-characters of
simple modules in the Hernandez-Leclerc category correspond to the normalized
global basis of $\widehat{\mathcal{A}}$.
