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Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (universal) bundle over $G$.

On Kapranov's "Coherent sheaves on Grasmann manifold" the following result are stated:

$H^0(G, S^*) \simeq V^*$ and $H^0(G, V/S) \simeq V$

The author claims that "These facts are well known".

However after a lot of research, I could not find this statement in any reference where I looked for it.

It is reasonable that the proof has to be done "by hands", like in the case of the tautological sheaf $\mathcal O (-1)$ over $\mathbb{P}^n$.

Do you have any suggestion?

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2 Answers 2

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These are simple instances of the Bott-Borel-Weil theorem. For a complex semsimple group $G$ and a parabolic subgroup $P$ and a complex irreducible representation $W$ of $P$ consider the homogeneous vector bundle $G\times_P W\to G/P$. In this situation the BBW theorem computes the cohomology of the shaef of local holomorphic sections of this bundle as a representation of $G$. The case you need here is that the highest weight of $W$ already is $G$-dominant and integral, in which case the cohomology is concentrated in degree zero and is the $G$-irreducible representation of the same highest weight. (Observe that $S^*$ is the $P$-irreducible quotient of $V^*$, while $V/S$ is the $P$-irreducible quotient of $V$.)

The classical Borel-Weil theorem handles the case where $P=B$, the Borel subgroup of $G$, and states that the finite dimensional irredcible representations of $G$ corresponding to a dominant integral weight can be realized as the space of holomorphic sections of the homogeneous line bundle on the full flag manifold $G/B$ induced by the one-dimensional representation of $B$ defined by that weight.

A nice exposition of the BBW-theorem can be found in the book on the Penrose transform by Baston and Eastwood.

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  • $\begingroup$ I never saw Grassmanians under the point of view of representations but it seems powerful. Would you be able to give a brief introduction or a basic reference to understand this point of view? $\endgroup$
    – Raffaele C
    Jun 9, 2016 at 8:41
  • $\begingroup$ I found a nice paper "Grassmanians and representations" here: arxiv.org/pdf/math/0507482v2.pdf $\endgroup$
    – Raffaele C
    Jun 9, 2016 at 11:12
  • $\begingroup$ The basic story is that the compact homogeneous spaces of complex semisimple groups are exactly the quotients by parabolic subgroups, which are known as generalized flag manifolds. Any parabolic subgroup has a natural reductive quotient via homogeneous vector bundles representations of this so-called Levi-factor give rise to representations of the initial group. In my opinion, the first chapters of the book by Baston and Eastwood I mentioned give a nice introduction to the topic. The paper you mention contains an exposition of Bot-Borel-Weil, but not much background. $\endgroup$
    – Andreas Cap
    Jun 9, 2016 at 12:54
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    $\begingroup$ You should have $U_{k,m}\times \mathbb C^k$ in your description. To get the dual bundle, you simply replace $\mathbb C^k$ by its dual space and use the canonical action, i.e.~$(g^{-1}\cdot\lambda)(v)=\lambda(g\cdot v)$. However, the description you use is not very well suited to the description of general homogeneous bundles. The better point of view is to have $G=SL(n,\mathbb C)$ and $P\subset G$ the stabilizer of $\mathbb C^k\subset\mathbb C^n$. Then for any representation $W$ of $P$, $G\times_PW$ is the quotient of $G\times W$ by $(g,w)\sim (gh,h^{-1}\cdot w)$ for $h\in P$. $\endgroup$
    – Andreas Cap
    Jun 12, 2016 at 9:11
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    $\begingroup$ continuation: $P$ consists of block-upper-triangular matrices, so there is an obvious homomorphism $P\to S(GL(k,\mathbb C)\times GL(n-k,\mathbb C))$, Via this you can use the standard representation of $GL(k,\mathbb C)$ and its dual to induce the tautological bundle and its dual. $\endgroup$
    – Andreas Cap
    Jun 12, 2016 at 9:13
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I don't know a reference either but one can argue as follows:

Let $U\subseteq V$ be of dimension $k$ and let $P$ be its stabilizer in $GL(V)$. Then the morphism $\pi:S=GL(V)\times^PU\to V$ is proper and surjective. Moreover one checks that all of its fibers are irreducible. The normality of $V$ implies $\pi_*\mathcal O_S=\mathcal O_V$, in particular, each global function on $S$ is a pull-back from $V$. Specializing to homogeneous functions of degree $1$ one gets $H^0(G,S^*)=V^*$. The other equality is obtained from the fact that $U\mapsto U^\perp=(V/U)^*$ yields an isomorphism between $Gr_k(V)$ and $Gr_{n-k}(V^*)$ (with $n=\dim V$).

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  • $\begingroup$ In my idea, $H^0(G,S^*)$ is the group of global sections of the "dual bundle" of $S$ over the whole Grassmanian. Here it seems that you compute a global map from $S$ to the base field. What is the relation? $\endgroup$
    – Raffaele C
    Jun 7, 2016 at 15:52
  • $\begingroup$ @Ramac: I think of a global section of $S^*$ as a function on the total space of $S$ which is homogeneous of degree one on each fiber. Perhaps, I should have used a different notation for a vector bundle considered as a sheaf and a vector bundle considered as a variety. $\endgroup$
    – Friedrich Knop
    Jun 7, 2016 at 17:06

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