# Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem to find one. Thanks a bunch in advance!

Edit: I understand $G=\mathrm{Sl}_n(\Bbbk)$ as a connected algebraic group and define a parabolic subgroup $P\subseteq G$ to be one that contains a maximal connected solvable subgroup. I know how this is equivalent to $G/P$ being complete (or projective).

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Is this not the definition of parabolic subgroups of $SL_{n}(\mathbb{k})$ or are you using the definition of a parabolic subgroup being the normalizer of a radical subgroup? –  David Ward Aug 21 '12 at 12:22
I would use the definition that the group contains a maximal connected solvable (i.e. Borel) subgroup. –  Jesko Hüttenhain Aug 21 '12 at 13:30

## 1 Answer

Books on "buildings" will discuss such a definition/theorem, in the example of $SL_n$, for example, my book that is also on-line, "Buildings and Classical Groups" (see http://www.math.umn.edu/~garrett/m/buildings/). In fact, this idea is what Jacques Tits was trying to abstract for exceptional groups by his more general notion of "building".

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That's sweet - could you tell me the chapter/section, or even better, the page number? –  Jesko Hüttenhain Aug 21 '12 at 13:25
Chapter 9 of my book is exactly about "the spherical building for GL(n) (or SL(n))". Starts on page 131. –  paul garrett Aug 21 '12 at 14:56
Thanks this looks good. I like that you also have the symplectic group worked out explicitly. –  Jack Schmidt Aug 22 '12 at 17:15