the root system of $\g$, by $R:= \oplus_{i\in I} \Z\a_i$ the root lattice, by $R^+:= \oplus_{i\in I} \N\a_i$ its positive cone, and by $\De^+ := \De\cap R^+$ the subset of positive roots. We have $\De = \De^+\sqcup (-\De^+)$. The Weyl group $W$ acts on $\De$, and we define the subset of real roots as the $W$-orbit of $\{\a_i \mid i\in I\}$. For $w\in W$, put $$\De_w := \{\a\in\De^+ \mid w(\a) \in \De^-\}.$$ This is a finite set of positive real roots, with cardinality $\ell(w)$. Finally, set $$\n({w}) := \bigoplus_{\a\in\De_{w}} \g_\a \subset \n_+,$$ a nilpotent subalgebra of $\g$ of dimension $\ell(w)$.

\[example1\] Take $$C = 
\begin{pmatrix}
2 & -2 \\
-2 & 2 
\end{pmatrix}.$$ Then $\g=\slchap_2$ is an affine Lie algebra of Dynkin type $\widetilde{A}_1$. We have $$W = \langle s_1, s_2 \mid s_1^2 = s_2^2 = 1 \rangle.$$ Let $w = s_2s_1s_2s_1$. Then $$\De_w = \{{\a_1},\, 2{\a_1} + {\a_2} ,\, 3{\a_1} + 2{\a_2} ,\, 4{\a_1} + 3{\a_2}\},$$ and $$\n(w) = \mathrm{Span}\langle e_1,\ [e_1,[e_2,e_1]], 
[e_1,[e_2,[e_1,[e_2,e_1]]]],\ [e_1,[e_2,[e_1,[e_2,[e_1,[e_2,e_1]]]]]]\rangle.$$

Kac-Moody groups
----------------

The enveloping algebra $U(\n)$ of the Lie algebra $\n$ is a cocommutative Hopf algebra, with an $R^+$-grading given by $\deg(e_i) = \a_i$. Let $$U(\n)_{\mathrm{gr}}^* := \bigoplus_{d\in R^+} U(\n)_d^*$$ be its graded dual. 