# On the Steinberg section

Let $\chi:G\rightarrow T/W$ the Steinberg map. I assume that G is simply connected. Then $T/W=\mathbb{A}^{r}$ and Steinberg constructed a section to this map given by $\epsilon(a_{1},...,a_{r})=x_{\alpha_{1}}(a_{1})n_{1}...x_{\alpha_{r}}(a_{r})n_{r}$

where the $n_{i}\in N_{G}(T)$ represent the simple reflexion $s_{\alpha_{i}}$.

Let $\gamma\in T$ a regular semisimple element. Then his conjugacy class is isomorphic to $G/T$ and moreover $\gamma$ is conjugated to $\epsilon(\chi(a))$ by a certain element $g$.

My question is, is it possible to take $g$ is the open subset $UU^{-}/T$?

Or otherwise, is there a $U^{-}$-conjugate of the steinberg section that is in the Borel B.

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Probably the person, who can answer this question, knows what all this means. But to be on the save side, you should explain your notation and also things like "Steinberg map", which can't be found easily via Google. – Marc Palm Dec 12 '12 at 8:39
It is the notation used by Steinberg or by Ngô for the analog for the Lie algebra or even by Popov. The map is comingfrom the Chevalley theorem $k[G]^{G}=k[T]^{W}$ where G acts by conjugation. – prochet Dec 12 '12 at 18:00

EDIT: I still have the concerns expressed below about the formulation. But the answer to the final question asked is certainly no. This shows up by direct calculation in the rank one case: if $G = \mathrm{SL}_2$ and the Steinberg section consists of matrices shown in the middle of page 69 of my book, there is no matrix in $G$ which simultaneously conjugates all of these to triangular form. (In general, the obstacle apparently comes from the role in Steinberg's construction of a representative in $G$ of a Coxeter element in the Weyl group.)
Steinberg's map is best behaved for a simply connected semisimple group $G$ (over an algebraically closed field), where the orbits of the Weyl group $W$ on a maximal torus $T$ have a natural identification with the points of affine $r$-space if $r$ is the rank of $G$. This set of orbits is traditionally written $T/W$, which is not quite modern notation but is clear enough in this narrow context. Steinberg examines the fibers of his map, each of which is a union of conjugacy classes and contains a unique semisimple and a unique regular class, etc. The delicate work is to construct a reasonable cross-section of the regular classes here, in terms of simple roots and simple reflections, once the above data is fixed. There is a classical linear algebra flavor to Steinberg's set-up: for a special linear group one is setting up a bijection between regular classes and the $r$-tuples of significant coefficients of all possible characteristic polynomials.