# New Realization of Cyclotomic $q$-Schur Algebras

### Kentaro Wada

Shinshu University, Matsumoto, Japan

## Abstract

We introduce a Lie algebra $g_{Q}(m)$ and an associative algebra $U_{q,Q}(m)$ associated with the Cartan data of $gl_{m}$ which is separated into $r$ parts with respect to $m=(m_{1},…,m_{r})$ such that $m_{1}+⋯+m_{r}=m$. We show that the Lie algebra $g_{Q}(m)$ is a filtered deformation of the current Lie algebra of $gl_{m}$, and we can regard the algebra $U_{q,Q}(m)$ as a \lq\lq $q$-analogue" of $U(g_{Q}(m))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $U_{q,Q}(m)$ under a certain mild condition. We also study the representation theory for $g_{Q}(m)$ and $U_{q,Q}(m)$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.

## Cite this article

Kentaro Wada, New Realization of Cyclotomic $q$-Schur Algebras. Publ. Res. Inst. Math. Sci. 52 (2016), no. 4, pp. 497–555

DOI 10.4171/PRIMS/188