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what is a parabolic subgroup? what is standard parabolic subgroup?

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 The first few hits of a Google search on "parabolic subgroup" are useful. – Lee Mosher Dec 27 at 13:35
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 The notion of "standard parabolic" is motivated by a real theorem: for a Borel subgroup $B$ in a connected reductive group $G$ over an algebraically closed field $k$, there are only finitely many parabolic $k$-subgroups $P$ of $G$ containing $B$ (labelled in a natural way in terms of the root system of $G$) and every parabolic $k$-subgroup of $G$ is $G(k)$-conjugate to a unique such $P$. The parabolics containing a fixed $B$ are called the "standard" parabolic $k$-subgroups; it is a convenient set of conjugacy class representatives. (The analogue over general $k$ is similar but lies deeper.) – ayanta Dec 27 at 21:59

closed as off topic by Lee Mosher, Misha, Goldstern, GH, Simon Thomas Dec 27 at 15:25

1 Answer

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Example: Subgroups of $SL(n, \mathbf{C})$ which contain upper triangular subgroup, and closed in Zariski topology (i.e. matrices satisfying polynomial conditions on its entries) is a parabolic subgroup. (also their conjugate subgroups).

See the textbook by Humphreys Linear Algebriac Groups.

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