$X_n$ over $\C$, and $N_+$ is a maximal unipotent subgroup $N_+ = N_{X_n}(\C)$. Thus, if $X_n = A_n$, ${G} = \SL(n+1,\C)$, and $N_+$ is the subgroup of unipotent upper triangular matrices.

\(b) If $\g$ is an affine Lie algebra of affine Dynkin type $\widetilde{X}_n$, then $G$ is a central extension of $G_{X_n}(\C[z,z^{-1}])$ by $\C^*$. Moreover, $${N_+} \simeq \{ g \in {G}_{X_n}(\C[z]) \mid g|_{z=0} \in {N}_{X_n}(\C) \}.$$ Thus, continuing Example \[example1\], if $\g$ is of type $\widetilde{A}_1$, $$N_+ = \left\{
\begin{pmatrix}
a & b \\
c & d 
\end{pmatrix}
\in \SL(2,\C[z])\mid
 a(0)=d(0)=1,\ c(0)=0 
\right\}.$$

\(c) If $\g$ is of indefinite type, no “concrete” realization of ${G}$ is known.

Generalized minors {#sect3.3}
------------------

We have $\mathrm{Norm}_{{G}}({H})/{H} \cong {W}$. For $i\in{I}$, put $$\overline{s}_i := \exp(f_i)\exp(-e_i)\exp(f_i),
\quad
\overline{\overline{s}}_i := \exp(-f_i)\exp(e_i)\exp(-f_i).$$ To ${w} = {s_{i_1}}\cdots {s_{i_r}} \in W$ with $\ell({w})=r$, we attach two representatives in $\mathrm{Norm}_{{G}}({H})$: $$\overline{{w}}=\overline{s}_{i_1}\cdots \overline{s}_{i_r},
\qquad
\overline{\overline{w}}=\overline{\overline{s}}_{i_1}\cdots \overline{\overline{s}}_{i_r}.$$ Let ${G_0} = {N_-} {H} {N_+}$ be the Zariski open subset of $G$ consisting of elements $g$ having a Birkhoff decomposition. For $g\in G_0$, this unique factorization is written $$g=[g]_-[g]_0[g]_+,\qquad
([g]_-\in{N_-},\ [g]_0\in{H},\  [g]_+\in{N_+}).$$ Let $\{\varpi_i \mid i\in I\} \subset \h^*$ be a fixed choice of fundamental weights, that is, $$\varpi_i(h_j) = \delta_{ij}, \qquad (i,j\in 