I).$$ Let $x \mapsto x^{\vpi_i}$ denote the corresponding characters of ${H}$. There is a unique regular function $\De_{{\vpi_i},{\vpi_i}}$ on $G$ such that $$\De_{{\vpi_i},{\vpi_i}}(g) = [g]_0^{\vpi_i}, \qquad (g\in{G_0}).$$ Moreover, $G_0 = \{ g\in G \mid \De_{{\vpi_i},{\vpi_i}}(g) \not = 0,\ i\in{I}\}$.

For $u,v\in W$ and $i\in I$, the generalized minor $\De_{u(\vpi_i),v(\vpi_i)}$ is the regular function on $G$ given by $$\De_{u(\vpi_i),v(\vpi_i)}(g) = \De_{{\vpi_i},{\vpi_i}}\left(\overline{\overline{u^{-1}}} g\overline{v}\right),
\qquad (g\in G).$$

Unipotent cells
---------------

Let ${B_-} = {N_- H}$. The group ${G}$ has a Bruhat decomposition: $${G} = \bigsqcup_{{w}\in{W}} {B_-}\overline{{w}}{B_-}.$$ For ${w}\in{W}$, define the *unipotent cell* ${N}^{w}:= {N_+}\,\cap\,({B_-}\overline{w}{B_-})$. Let $$O_{w} := \{g\in {N}({w})\mid \De_{{\vpi_i},{w}^{-1}({\vpi_i})}(g) \not = 0,\ i\in{I}\},$$ an open subset of the affine space $N(w)$.

\[Prop3.5\] We have an isomorphism $O_{{w}} \stackrel{\sim}{\to} {N}^{{w}}$. It follows that $\C[{N}^{{w}}]$ is the localization of $\C[{N}({w})] \simeq \C[{N}]^{N'({w})}$ at $\prod_{i\in {I}} \De_{{\vpi_i},{w}^{-1}({\vpi_i})}$.

Formulas for factorization parameters {#sect3.5}
-------------------------------------

Define the one-parameter subgroups of $N_+$: $$x_{{i}}({t}) := \exp({t}e_i),\qquad ({t}\in\C,\, {i}\in{I}).$$ Consider as above a reduced decomposition ${w} = {s_{i_r}}\cdots {s_{i_1}} \in W$. (Note that for reasons which will become clear in §\[sect4.3\] below, we prefer from now on to number the factors of this decomposition *from right to left*.) Then the image of the map $\mathbf{x}_{i_r,\ldots,i_1}\colon(\C^*)^r \to {N}$ given by $$\mathbf{x}_{i_r,\ldots,i_1}({t_r},\ldots,{t_1}) 