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

Let $\mathscr{R}_α^Λ$ 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 $Tr(\mathscr{R}_α^Λ)$ is nonzero only if $0\leq d\leq d_{Λ,α}$. 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 $Tr(\mathscr{R}_α^Λ), Tr(\mathscr{R}_α^Λ)_0, Tr(\mathscr{R}_α^Λ)_{d_{Λ,α}}$ to arbitrary ground field $K$. Moreover, we show that the dimension of the degree $0$ component $Tr(\mathscr{R}_α^Λ)_0$ is always equal to $\dim V(Λ)_{Λ-α}$, where $V(Λ)$ is the integrable highest weight $U(\mathfrak{g})$-module with highest weight $Λ$, and we obtain a basis for $Tr(\mathscr{R}_α^Λ)_0$.