On the maximal and minimal degree components of the cocenter of the cyclotomic KLR algebras

Abstract

Let ${\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v){\}}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ has characteristic $0$, the degree $d$ component of the cocenter $Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›)$ is nonzero only if $0\le d\le d_{\hat{I}›,\hat{I}\pm }$. In this paper we show that this holds true for arbitrary ground field $K$, arbitrary $\mathfrak{g}$ and arbitrary polynomials $\{Q_{ij}(u,v){\}}_{i,j\in I}$. We generalize our earlier results on the $K$-linear generators of $Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›),Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›{)}_0,Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›{)}_{d_{\hat{I}›,\hat{I}\pm }}$ to arbitrary ground field $K$. Moreover, we show that the dimension of the degree $0$ component $Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›{)}_0$ is always equal to $\dim V(\hat{I}›{)}_{\hat{I}›-\hat{I}\pm }$, where $V(\hat{I}›)$ is the integrable highest weight $U(\mathfrak{g})$-module with highest weight $\hat{I}›$, and we obtain a basis for $Tr({\mathcal{R}}_{\hat{I}}{\pm }^{\hat{I}}›{)}_0$.