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Let $G=GL(n,\mathbb{C})$ and $B$ be a Borel in $G$.

The variety $G/B$ is known to be a moduli of flags (i.e., a variety of Borels) with a bijection between a class of Borels and points of $G/B$.

The cotangent bundle of $G/B$ has a description
$ T^*(G/B)= G\times_{B} (\mathfrak{g}/\mathfrak{b})^* = G\times_B \mathfrak{n} $

$ = \{ (\mathfrak{b'}, \langle n, -\rangle) \in G/B\times \mathfrak{g}^* : n\in \mathfrak{b}' \mbox{ and } n \mbox{ is nilpotent } \}. (\star) $

  1. Besides the upper triangular matrices in $G$, what other Borels are in $G$? If this question is too hard to answer, how about just for the case $n=2$ or $3$?

  2. Concretely, say for $n=2$, can you give an example of this description $ (\star)$?

  3. How is $G$ acting on $T^*(G/B)$? I thought I saw in several papers that $ g.[h,n] = [hg^{-1},gn] \in G\times_{B}\mathfrak{n}, $ but in some other references, I thought I saw $ g.[h,n] = [hg^{-1},gng^{-1}] \in G\times_{B}\mathfrak{n}. $ Which one is correct?

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    Yes, exactly. And vice-versa.2012-07-08

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