Let us work over $\mathbb C$. It is well known that if $G$ is a (connected) complex reductive group and $B$ a Borel subgroup, then the homogeneous space $G/B$ is a smooth projective variety. For example, $GL(n,\mathbb{C})/B$ is the full flag variety.
I want to understand quotients of the form $G/T$, where $T$ is a maximal torus. These are not projective but also important, and seem much less considered in the literature. One of the simplest concrete questions is the following:
Let $G=SL(2,\mathbb{C})$, $T=\mathbb{C}^*$ be its maximal torus consisting of diagonal matrices, and $B$ the Borel subgroup of upper-triangular matrices. Then, we have a fibration:
$$ B/T \to G/T \to G/B $$
Since $G/B\cong \mathbb{P}^1$, the projective line, and $B/T$ is naturally an affine line $\mathbb{C}$, the fibration looks like a line bundle over the Riemann sphere.
As all line bundles are classified by an integer $n\in \mathbb{Z}$, if the above reasoning is correct, how can we compute this value of $n$? Can the argument be generalized to show that other quotients of the form $G/T$ are vector bundles over $G/B$?
Any help and/or reference is appreciated.
If $G$ is a reductive algebraic group (over $\mathbb C$), with $T$ a maximal torus contained in $B$ a Borel subgroup, the fibration $p\colon G/T\to G/B$ is an affine bundle over $G/B$, but not a vector bundle -- the fibres are all affine spaces $\mathbb A^k$, but they do no have a vector space structure.
For example, if $G=\text{SL}_2(\mathbb C)$, a Borel subgroup corresponds to a line $L_1\subseteq \mathbb C^2$, where $B=\text{Stab}_{G}(L_1)$. A maximal torus in $B$ is given by a choice of a complementary line $L_2$, so that $\mathbb C^2 = L_1\oplus L_2$, and the fibre of the map $G/T\to G/B$ can be identified with the space $\{L_2 \in \mathbb P^1: L_2\neq L_1\}=\mathbb A^1$.
Another useful way to understand the quotients $G/T$ is to consider conjugacy classes: if you pick $s \in T$ regular semisimple, a condition which holds generically (in the case of $s\in\text{SL}_n(\mathbb C)$ this simply means that $s$ has $n$ distinct eigenvalues), then $T$ is the centralizer of $s$ in $G$. It follows that the surjective map $g\mapsto gsg^{-1}$ from $G$ to the conjugacy class of $s$ induces an identification of $G/T$ with the conjugacy class of $s$. Because $s$ is regular semisimple its conjugacy class is a closed subset of $\text{SL}_2(\mathbb C)$ (given by the characteristic polynomial of $s$) and hence $G/T$ is an affine variety. In particular $G/T$ map $p\colon G/T\to G/B$ cannot possess a section, since any map from a projective variety to an affine variety is constant.
One more thing that seems important to say is that in the above, everything is understood in the algebro-geometric context (the complex analytic setting is similar). In the $\mathcal C^\infty$-setting, $G/T$ is isomorphic to a smooth line bundle over $G/B$: for example, you can choose an Hermitian form on $\mathbb C^2$, and then $L_1$ has a distinguished complement $L_1^{\perp}$ giving the zero section of the bundle.
Addendum: The fact that $G/T$ is an affine variety is a special case of a general result: If $G$ is reductive and $H$ is a reductive subgroup of $G$, then $G/H$ is an affine variety (and indeed conversely if $G/H$ is affine then $H$ is reductive).