Global bases for Bosonic extensions of quantum unipotent coordinate rings

In the paper, we establish the global basis theory for the bosonic extension (Formula presented.) associated with an arbitrary symmetrizable generalized Cartan matrix. When (Formula presented.) is of simply laced finite type, (Formula presented.) is isomorphic to the quantum Grothendieck ring (Formula presented.) of the Hernandez–Leclerc category (Formula presented.) over a quantum affine algebra of untwisted type. In this case, we show that the (Formula presented.) -characters of simple modules in (Formula presented.) correspond to the normalized global basis of (Formula presented.).