If $G$ is a complex semisimple Lie group and $B$ is a Borel subgroup, then I am interested in a particular family of "nice" parametrizations of the the homogeneous space $G/B$. I call these "weight-preserving diffeomorphisms" $WP(G,B)$ for the pair. (This group can be defined for any pair $(G,P)$, where $P$ is a parabolic subgroup, but for simplicity, I assume here that $P=B$ is a Borel subgroup.)
Before defining $WP(G,B)$, I need some more background and notation. Suppose $\mathfrak{g}$ is the Lie algebra of $G$. Choose a Cartan subalgebra $\mathfrak{h}$ and a system $\Delta^+$ of positive roots, let $\mathfrak{n}$ be the corresponding nilpotent Lie subalgebra generated by the root vectors $X_\alpha$, $\alpha\in\Delta^+$ and let $\mathfrak{b}$ be the complementary parabolic subalgebra. Let $H$, $N$, and $B$ be the corresponding subgroups of $G$. (It is well-known that $B$ is a Borel subgroup of $G$.)
Given $(G,B)$ as above, one may simply parametrize $G/B$ using the exponential map. Thus, take $X\in\mathfrak{n}$ and define $f(X)\in G/B$ as the coset $\exp(X)B$. This provides a map from $\mathfrak{n}\rightarrow G/B$ which restricts to a diffeomorphism between $\mathfrak{n}$ and the big Bruhat cell in the Bruhat decomposition of $G/B$. However, I feel that the use of the exponential map to be somewhat awkward and arbitrary. Indeed, there is a continuous family of maps $\mathfrak{n}\rightarrow G/B$ which have this similarly nice property. The transitions between these maps which are just as good as the exponential map are what I call the "weight-preserving diffeomorphisms".
And now, finally, here is a definition of $WP(G,B)$. Choose $X\in\mathfrak{n}$. Then one may write $X$ as a linear combination $$X=\sum{z_\alpha X_\alpha}$$ where the sum is over $\Delta^+$. The coefficients ${z_\alpha}$ provide a coordinatization of $\mathfrak{n}$. Define a "weight-preserving diffeomorphism" as a change of variables $z_\alpha\mapsto z_\alpha'$ such that for each $\alpha\in\Delta^+$, one has $z_\alpha'=z_\alpha+f$, where $f$ is a polynomial in ${z_\alpha}$ with weight $\alpha$ and every term of $f$ has degree greater than 1.
Here is an example. Let $G=B_2=O(5)$. Choose $\alpha_1=e_1-e_2$, $\alpha_2=e_1$, $\alpha_3=e_1+e_2$, and $\alpha_4=e_2$. Notice $\alpha_2=\alpha_1+\alpha_4$ and $\alpha_3=\alpha_2+\alpha_4=\alpha_1+2\alpha_4$. Then every weight-preserving diffeomorphism for $G$ has the form
$$z_1' = z_1,\
z_2' = z_2+c_1z_1z_4,\
z_3' = z_3+c_2z_2z_4+c_3z_1z_4^2,\
z_4' = z_4, $$
for some choice of $c_1,c_2,c_3$. One may check in this case that $WP(G,B)$ is isomorphic to the three-dimensional Heisenberg group.
One may check that the dimension of $WP(G,B)$ when $G=A_r$ is roughly $2^r$, but I don't know a formula when $G=B_r$, $C_r$, or $D_r$. It is not hard to see that $WP(G,B)$ is always unipotent. There seem to be some very interesting combinatorial things happening with these groups. (Try computing $WP(G,B)$ when $G=B_3$.) If one extends this definition for pairs $(G,P)$ where $P$ is any parabolic subgroup, then one can check that $WP(G,P_1)$ is a subgroup of $WP(G,P_2)$ whenever $P_2$ is a subgroup of $P_1$. However I don't know much about them beyond these facts. (In fact, one may generalize these groups in a big way by enlarging the types of sets $\Delta^+$ one might use in the definition. Indeed, the definition dependsonly on the structure of $\Delta^+$ as a ``positive system of roots''.)
My questions: What is known about these groups? I am particularly interested in better understanding their structure and connections to other families of Lie groups. (I have tagged this with "combinatorial-geometry" because there is a nice connection to fiber polytopes when $G=A_r$.) A reference would be very much appreciated.
