On the geometry of algebras related to the Weyl groupoid

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let $\mathfrak{g}$ be a finite dimensional classical simple Lie superalgebra over $\mathtt{k}$ or $\mathfrak{g}l(m,n)$. In the case that $\mathfrak{g}$ is a Kac-Moody algebra of finite type with set of roots $\mathrm{\Delta }$, Sergeev and Veselov introduced the Weyl groupoid $\mathfrak{W}=\mathfrak{W}(\mathrm{\Delta })$, which has significant connections with the representation theory of $\mathfrak{g}$. Let $\mathfrak{h}$, $W$ and $Z(\mathfrak{g})$ be a Cartan subalgebra of ${\mathfrak{g}}_0$, the Weyl group of ${\mathfrak{g}}_0$ and the center of $U(\mathfrak{g})$ respectively. Also let $G$ be a Lie supergroup with Lie $G=\mathfrak{g}$. There are several important commutative algebras related to $\mathfrak{W}$. Namely \begin{itemize} \item The image $I(\mathfrak{h})$ of the injective Harish-Chandra map $Z(\mathfrak{g})⟶S(\mathfrak{h}{)}^W$. \item The supercharacter $\mathbb{Z}$-algebras $J(\mathfrak{g})$ and $J(G)$ of finite dimensional representations of $\mathfrak{g}$ and $G$. \end{itemize} Let $\mathcal{A}=\mathcal{A}(\mathfrak{g})$ be denote either $I(\mathfrak{h})$ or $J(G){\otimes }_{\mathbb{Z}}\mathtt{k}$. The purpose of this paper is to investigate the algebraic geometry of $\mathcal{A}\mathrm{.}$ In many cases, the algebra $\mathcal{A}$ satisfies the Nullstellensatz. This gives a bijection between radical ideals in $\mathcal{A}$ and superalgebraic sets (zero loci of such ideals). Any superalgebraic set is uniquely a finite union of irreducible superalgebraic components. In the non-exceptional Kac-Moody case, we describe the smallest superalgebraic set containing a given (Zariski) closed set, and show that the superalgebraic sets are exactly the closed sets that are unions of groupoid orbits.