Remark 3

If $\mathfrak{g}$ is semi-simple, the permissible lattices in $\mathfrak{h}$ are the subgroups $\mathscr{H}$ of $\mathfrak{h}$ such that $Q(R^{\vee})\subset \mathscr{H}\subset P(R^{\vee})$.

In the remainder of this number, we assume fixed a split reductive Lie algebra $(\mathfrak{g},\mathfrak{h})$, a basis B of $R = R(\mathfrak{g},\mathfrak{h})$ and, for each $\alpha \in B$, a pair $(x_{\alpha}, y_{\alpha})$ with

If we denote by $\mathfrak{n}_+$ (resp. $\mathfrak{n}_-$) the subalgebra of $\mathfrak{g}$ generated by the $x_{\alpha}$ (resp. the $y_{\alpha}$), we know (§3, no. 3, Prop. 9 (iii)) that

(where $U(\mathfrak{g}), . .$. are the enveloping algebras of $\mathfrak{g}, . . .$).

Denote by $\mathscr{U}_+$ the $\mathbf{Z}$-subalgebra of $U(\mathfrak{n}_+)$ generated by the $x^{(n)}_{\alpha}$ for $\alpha \in B$ and $n\in \mathbf{N}$. Let W be the Weyl group of $R, R_+$ the set of positive roots relative to B.