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We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of $SL(n,\mathbb{C})$

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This is a standard textbook question coming out of the 1950s work of Borel and Chevalley. See the textbooks on linear algebraic groups, or the more recent book by Fulton-Harris (etc.). Beyond the linear algebra description in this special case, the interesting point is that the flag variety is projective (in the general setting of semisimple or reductive groups over any algebraically closed field). In the complex setting it's a compact manifold. – Jim Humphreys Oct 8 '13 at 16:28
Jim Humphreys, thanks – baba ab dad Oct 8 '13 at 18:28
Fulton and Harris discuss this variety and some generalizations in their chapter on homogeneous spaces in their book on Representation Theory. – Ben McKay Oct 8 '13 at 19:20
up vote 6 down vote accepted

The variety $\mathrm{SL}(n,\mathbb{C})/B$ is the variety of complete flags in $\mathbb{C}^n$. That is, a point in the variety can be identified with a chain

$$\{0\}=V_0\subset V_1\subset\dotsc\subset V_{n-1}\subset V_n=\mathbb{C}^n$$

such that $\dim{V_i}=i$. This reduces to $\mathbb{P}^1$ in the case $n=2$, as the only non-proscribed datum at each point is the line $V_1\subset\mathbb{C}^2$.

To prove this, generalizing the $n=2$ result, one checks that the "standard" Borel subgroup stabilizes the "standard" flag

$$\{0\}\subset\langle e_1\rangle\subset\langle e_1,e_2\rangle\subset\cdots\subset\mathbb{C}^n$$

where the $e_i$ are the standard basis vectors.

To expand a bit more, and give an answer that judging from the comments you might like better, let $P$ be a maximal parabolic subgroup of $\mathrm{SL}(n,\mathbb{C})$, so $P$ is proper, contains a Borel, and is maximal with respect to this property. Then $\mathrm{SL}(n,\mathbb{C})/P$ is the Grassmannian of $k$-planes in $\mathbb{C}^n$, which more naturally generalizes $\mathbb{P}^1$ (the Grassmannian of $1$-planes in $\mathbb{C}^2$). What's happening in the $n=2$ case is that $\mathrm{SL}(2,\mathbb{C})$ is so small that the only parabolic subgroup (up to conjugacy) is Borel.

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Matt Pressland, but what about $n>2$. This is my question – baba ab dad Oct 8 '13 at 15:43
@HassanJolany I expanded my answer a bit - hopefully it is now clearer that I am addressing the $n>2$ case. – Matthew Pressland Oct 8 '13 at 15:51
This answer is correct, so if it isn't the nice variety you are looking for, no one can do anything about your disappointment. For $n=3$, this is the variety of pointed lines in the projective plane, which is the same thing as the variety of complete flags in $\mathbb{C}^2$. – Ben McKay Oct 8 '13 at 19:19
@HassanJolany If you want an explicit description in coordinates, see the following MathOverflow question:… – S. Carnahan Oct 9 '13 at 1:32
I understood it to mean that you can get a projective space, but you have to quotient by the right maximal parabolic instead of by a Borel. (I don't know enough to know whether there's any sense in which this parabolic is $U(n-1)$ though). – Matthew Pressland Oct 10 '13 at 18:13

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