# Tautological and normal bundles over flag manifolds and jet bundles

Hello! Recently, doing my research on jet bundles, I was led to consider the following construction.

Let $V$ be a real vector space of dimension $n$. Consider the flag manifold $G(V,k,l)$ and the two Grassmann manifolds $G(V,k)$ and $G(V,l)$. It seems to me that there are two natural projections $\eta:G(V,k,l)\to G(V,k)$ and $\pi:G(V,k,l)\to G(V,l)$. If I'm not mistaken, $\eta^{-1}(K)=G(K,l)$ and $\pi^{-1}(L)=G((V/L)^*,k-l)$. Now each Grassmann manifold is accompanied by its own tautological bundle, and I can use the above projections to lift the tautological bundles over $G(V,k,l)$. Namely, let $\Theta$ (resp., $\theta$) the tautological bundle over $G(V,k)$ (resp., over $G(V,l)$), and consider their pullbacks $\overline{\Theta}:=\eta^\ast(\Theta)$ and $\overline{\theta}:=\pi^\ast(\theta)$. By definition, the $\overline{\Theta}$-fiber over a point $(K,L)\in G(V,k,l)$ is $K$, while the $\overline{\theta}$-fiber over the same point is $L$. Hence, I can obtain a "normal tautological bundle" $\nu:=\overline{\Theta}/\overline{\theta}$ of rank $k-l$.

QUESTION A

• Is the bundle $\nu$ I've just described a well-known construction? If yes, where I can find it?
• In the case of a complete flag manifolds, we would have an $n$--tuple of line bundles. Does it have any special interpretation?

Now let $\pi:E\to M$ a smooth vector bundle of smooth manifolds, of rank $n$. It seems natural to associate to $\pi$ a smooth bundle $G(\pi,k,l)$ by replacing each fiber with its flag (or, in particular, Grassmann) manifold. In its turn, $G(\pi,k,l)$ is the base of vector bundles of tautological type (constructed, fiber by fiber, just as above). I'm aware of a case when this construction is of some use: $G(\tau_M,k)$ is the 1-st jet bundle $J^1(M,k)$ over the $k$--dimensional manifold $M$ ($\tau_M$ being the tangent bundle); the corresponding tautological $k$-dimensional bundle over $J^1(M,k)$ is by some authors called the $R$-distribution, and it plays a key role in defining higher order jet spaces and the Cartan distribution over them.

QUESTION B

• Is the construction of "flag bundles" out of vector ones a well-known one? If yes, are their properties (e.g., behavior w.r.t. bundle morphisms) examined somewhere?
• Does exist an analogous construction of jet spaces, carried out by using the "flag bundles" of the tangent bundle, instead of the Grassmann bundles? If yes, does the "normal tautological bundle" (describe above) play any relevant role in such a theory?

## 1 Answer

The short answer is "yes." I'm not sure about the jet bundle stuff (not really my area), but everything else you've written is extremely well-known stuff in Lie theory. I've used them dozens of times in my own work. The quotient bundles you're calling "normal tautological bundle" i would just call "tautological bundle." In fact, it's well known enough that it's hard for me to give a reference, though you can try these notes of Brion or chapter 10 of Fulton's Young Tableaux (thought these are both for complex manifolds, not real ones; perhaps someone who knows the real theory better can help me with a reference). The line bundles on the flag variety indeed play an important role: they generate the Picard group of the variety and their Chern classes generate the cohomology, with all the relations in cohomology manifest from the fact that these filter a trivial bundle. They are also a special class of the line bundles that come up in discussions of Borel-Weil attached to weights of SL(n); these are ones attached to the weights which appear in the vector representation.

The notion of "flag bundles" you mention is also standard: you just quotient the frame bundle by upper-triangular matrices. This often comes up in discussion of the splitting principle.

• @Ben: thanks for the highlights on the Picard group and the splitting principle! Thanks also for the references - though I'm still looking for the construction in the real case. I knew it's a trivial fact, but this make it harder to find a reference. Still waiting for some hints about my questions on jet bundles... – Giovanni Moreno Apr 19 '12 at 8:12