I was reading in this wiki article about Generalized flag variety, but didn't understand the actions of $GL_n$ and $SU_n$ on the set of all complete flags.
Specially, how to prove that $SL(n,\mathbb C)/B\cong SU(n)/T$, where $B$ is the subgroup of upper triangular matrices and $T$ is the subgroup of the diagonal $n×n$-matrices?
Furthermore, when $n=3$, what is $SU(3)/T$ explicitly?
A flag in $\bf C^n$ is a family of vector spaces $E_0={0}\subset E_1\subset..\subset E_n= \bf C^n$. Both of the member of the identity identify with the set of flags : given a flag one can choose a base (resp. an orthornormal) base $B=(e_i)_{1\leq i\leq n}$ such that $E_i= Vect(e_1,...,e_i)$. Conversely such a base $B$ defines a flag, and two bases $B,B'$ give the same flag if and only the matrix of changing the base from $B$ to $B'$ is an upper triangular matrix $T$ (resp. a diagonal matrix with eigenvalues of absolute value 1). One can identify the set of bases (orthonormal bases) with $Sl(n,\bf C)$ and $U(n)$ to get the result.