New realization of cyclotomic $q$-Schur algebras I

We introduce a Lie algebra $\mathfrak{g}_{\mathbf{Q}}(\mathbf{m})$ and an associative algebra $\mathcal{U}_{q,\mathbf{Q}}(\mathbf{m})$ associated with the Cartan data of $\mathfrak{gl}_m$ which is separated into $r$ parts with respect to $\mathbf{m}=(m_1, \dots, m_r)$ such that $m_1+ \dots + m_r =m$. We show that the Lie algebra $\mathfrak{g}_{\mathbf{Q}} (\mathbf{m})$ is a filtered deformation of the current Lie algebra of $\mathfrak{gl}_m$, and we can regard the algebra $\mathcal{U}_{q, \mathbf{Q}}(\mathbf{m})$ as a "$q$-analogue" of $U(\mathfrak{g}_{\mathbf{Q}}(\mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $\mathcal{U}_{q, \mathbf{Q}}(\mathbf{m})$ under a certain mild condition. We also study the representation theory for $\mathfrak{g}_{\mathbf{Q}}(\mathbf{m})$ and $\mathcal{U}_{q,\mathbf{Q}}(\mathbf{m})$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.