Quantum ${\mathfrak{g}\mathfrak{l}}_{\mathrm{\infty }}$, infinite $q$-Schur algebras and their representations

In this paper, we investigate the structure and representations of the quantum group $\mathbf{U}(\mathrm{\infty })={\mathbf{U}}_{\upsilon }({\mathfrak{g}\mathfrak{l}}_{\mathrm{\infty }})$. We will present a realization for $\mathbf{U}(\mathrm{\infty })$, following Beilinson–Lusztig–MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism ${\zeta }_r$ from $\mathbf{U}(\mathrm{\infty })$ to the infinite $q$-Schur algebra $\mathcal{S}(\mathrm{\infty },r)$ is not surjective for any $r\ge 1$. We will give a BLM type realization for the image $\mathbf{U}(\mathrm{\infty },r):={\zeta }_r(\mathbf{U}(\mathrm{\infty }))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra ${\widehat{\mathcal{K}}}^{†}(\mathrm{\infty })$ so that ${\zeta }_r$ can be extended to an algebra epimorphism $\tilde\zeta_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will investigate the representation theory of $\mathbf{U}(\mathrm{\infty })$, especially the polynomial representations of $\mathbf{U}(\mathrm{\infty })$.