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 ${\mathfrak{g}\mathfrak{l}}_m$ which is separated into $r$ parts with respect to $\mathbf{m}=(m_1,\dots ,m_r)$ such that $m_1+\cdots +m_r=m$. We show that the Lie algebra ${\mathfrak{g}}_{\mathbf{Q}}(\mathbf{m})$ is a filtered deformation of the current Lie algebra of ${\mathfrak{g}\mathfrak{l}}_m$, 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.