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This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer.

I am currently reading Steinberg, Robert, An occurrence of the Robinson-Schensted correspondence, J. Algebra 113, No. 2, 523-528 (1988). ZBL0653.20039., but there are some parts that I do not understand.

Situation

I am working with the general linear group. Specifically, take $G_0=\text{GL}_{n,\mathbb{F}_q}$, and $G=G_0\times_{\mathbb{F}_q}\overline{\mathbb{F}_q}$ the base change. Let $X$ be the set of Borel subgroups of $G$. The Weyl group $W$ of $G$ is isomorphic to the symmetric group $S_3$, and we have a bijection between $W$ and the set of $G$-orbits on $X\times X$. Fix a Borel subset $B^+\in X$. For $w\in W$, define $\mathcal{O}(w)$ to be the orbit of $(B^+,\dot wB^+\dot w^{-1})$ in $X\times X$. We say that $B_1,B_2\in X$ are in relative position $w$ if $(B_1,B_2)\in\mathcal{O}(w)$, and we write $B_1\xrightarrow{w}B_2$.

My Question

I assume that two Borel subgroups $B_1$ and $B_2$ give two tableaux $T_1$ and $T_2$ respectively. The mentioned article associates to the pair $(T_1,T_2)$ a permutation $w(T_1,T_2)\in W$. What I want to say is that $B_1\xrightarrow{w}B_2$ if and only if $w=w(T_1,T_2)$. Is this true? My confusion is probably coming from the unipotent transformation $u$ mentioned in the article. How can a generic element of an irreducible component of $\mathcal F_u$ (the variety of flags fixed by $u$) be identified by a Borel subgroup $B\in X$?

Any help is appreciated.

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  • $\begingroup$ You said ``two Borel subgroups $B_1$ and $B_2$ give two tableaux $T_1$ and $T_2$ respectively''. This is not quite correct. It is a classical theorem of Spaltenstein and of Steinberg that the irreducible components of $\mathcal{F}_u$ are in $1-1$ correspondence with the tableaux associated to $u$. So, you first need to assume that $B_i$ lies in $\mathcal{F}_u$, but even though it is not quite correct to say that $B_i$ gives a tableau $T_i$: Irreducible components of $\mathcal{F}_u$ can intersect. $\endgroup$
    – user148212
    Commented May 19, 2023 at 16:42
  • $\begingroup$ I see the problem. Is there any way that I can apply Theorem 1.1 of the article to understand what it means for two Borels $B_1$ and $B_2$ to be in relative position $w$? The interpretation in terms of flags is very satisfying, so my idea was to connect Borels to tableaux and use the theorem, but you raise a very tricky point... $\endgroup$
    – EJB
    Commented May 22, 2023 at 7:46

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