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Let $G = SL_2(\mathbb{F}_q)$, $B$ the subgroup of all upper triangular matrices, $s = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.

What does $^sB$ mean?

I read it from page 4 of C. Bonnafé, Representations of $SL_2(\mathbb{F}_q)$, but there's no definition.

There's also a statement saying $B \cap {}^sB = T$, which is the subgroup of all diagonal matrices.

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  • $\begingroup$ This is just a guess, but could it be $sBs^{-1}$? This might be to distinguish it from $B^s = s^{-1}Bs$. $\endgroup$ – Derek Holt Aug 24 '15 at 10:11
  • $\begingroup$ $s^{-1}Bs \cap B = T$ is also true though. He's not using this kind of notation anywhere else, so I'll go with your guess. $\endgroup$ – Qian Aug 24 '15 at 10:31
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$^sB=sBs^{-1}$ is defined at the beginning of the book (page xxi "General notation"). It seems that that notation for a group action is more common in Cohomology.

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  • $\begingroup$ That's shameful of me... I should've checked the TOC first. I only have chapter 1 at hand. $\endgroup$ – Qian Aug 24 '15 at 10:45

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