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As is well known, $CP^{N-1}$ is the base space of the principal bundle $SU(N)$ with fibre $U(N-1)$. Moreover, $S^{2N-1}$ is the base space of the principal bundle $SU(N)$ with fibre $SU(N-1)$. Finally, $CP^{N-1}$ is the base space of the principal bundle $S^{2N-1}$ with fibre $U(1)$. What is interesting is that all the line bundles of $CP^{N-1}$ arise as $U(1)$-associated bundles to $S^{2N-1}$.

My question is, does this generalise to the setting of generalised flag manifolds $M=G/P$, where $P$ is a parabolic manifold and $G$ a semi-simple algebraic group? That is, if we take $P$ as generalising $U(N-1)$, $G$ as generalising $SU(N)$, and $M$ as generalising $CP^{N-1}$, what are the generalisations $X$ and $H$ of $S^{2N-1}$ and $U(1)$ so that all the line bundles of $M$ arise as $H$-associated bundles to $X$, or do such objects even exist?

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2 Answers 2

I would like to add a further analogy and an application for the case of the complex complete flag manifold $G/B$. In this case $ X = G = SU(N)$ is isomorphic (as a homogeneous space) to its principal homogeneous space: the Stiefel manifold $V_{n-1}(\mathbb{R}^n)$, as given in the following wikipedia page.

In both cases of the sphere and the Stiefel manifold, they can be given as (intersection of) quadrics. in $\mathbb{C}^n$ (the stiefel manifold can be identified with the space of full rank $n\times (n-1)$ matrices satisfying $\bar{M}^t M = 1$). Of course, the case of the standard Hopf fibration (X = S^3, M = S^2) is mutual to both cases.

This construction can be applied to perform integrations over the $SU(n)$ invariant measures of the complex projective spaces and complete flag manifolds: Integration over spheres and Stiefel manifolds is relatively straightforward because they can be performed on quadric constraint surfaces in $\mathbb{C}^n$. The integrals can be converted to (a series of) Gaussian integrals by means of Fourier transforms.

Functions on the complex projective spaces and complete flag manifolds can be extended to functions on the total manifolds, the sphere and the Stiefel manifold by defining them to be constants along the torus fibers.

In the case of the complex projective spaces, the sphere is identified with the space of unit vectors in $\mathbb{C}^n$ and the complex projective space with the projectors onto these vectors. Thus any function on the sphere which depends solely on the projectors of the unit vector is an extension of a function on the complex projective space and its integral on the sphere is proportional to the integral of the original function on the complex projective space.

In the case of the complete flag manifold, the Stiefel manifold may be considered as the space of orthonormal frames in $C^n$, (the column vectors of $M$) and the flag manifold as the space of projectors onto these vectors. Again, functions on the Stiefel manifold depending solely on the projectors of the frame vectors are extensions of functions on the flag manifolds.

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From one point of view, X=G and H=P works (though of course this does not recover the special case X=sphere and H=circle that motivated you). This is (closely related to) the Borel-Weil-Bott theorem: fixing a maximal torus $T \subseteq P$ and assuming $G$ simply connected, line bundles on $G/P$ are in bijection with certain homomorphisms $\lambda:T \rightarrow \mathbb{C}^\times$ (if $P=B$ then it's just all hom's). Or did you have more requirements in mind that make this answer not very useful?

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