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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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  • $\begingroup$ 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. $\endgroup$
    – Marc Palm
    Commented Dec 12, 2012 at 8:39
  • $\begingroup$ 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. $\endgroup$
    – prochet
    Commented Dec 12, 2012 at 18:00

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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.)

This is an extended comment, since the question seems to have some serious content even though the formulation is inadequate. It also lacks any indication of motivation. There is some standard background here, including the belatedly quoted Chevalley theorem and the fundamental paper by Steinberg on regular elements in semisimple algebraic groups. The paper is available via Numdam here.

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.

There are some further developments by Popov, Friedman-Morgan, along with generalizations to Kac-Moody groups. (There is some exposition of Steinberg's work in Chapter 4 of my AMS book on conjugacy classes.) But it's not clear to me from the sketchy indications what the precise question here is, or what consequences it might have. As far as I can understand what is being asked, I suspect that the answer will be negative. Anyway, some experimentation with rank 2 matrix groups would certainly help to clarify what is possible. Has that been tried?

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