Timeline for Geometrically quantizing real Grassmannians

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7 events

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Jan 15, 2018 at 0:06	comment	added	John Baez		Okay, that's a good hint. Now that I think about it, these quadrics should be the Grassmannians corresponding to the "first dot" of the $\mathfrak{so}(n)$ Dynkin diagram, so geometrically quantizing them for $p = 1$ should give simply $\mathbb{C}^n$. For larger $p$, we should get the $p$th symmetric power $S^p(\mathbb{C}^n)$. (Here I'm using the convention where $p = 1$ corresponds to the "anticanonical" bundle, the one that has nontrivial sections.) At least this should be the story except for some low dimensions $n$ where "first dot" becomes ambiguous.
Jan 14, 2018 at 21:23	comment	added	Ben McKay		The space of global sections of those bundles is given by Borel-Bott-Weil; I can't remember how that story goes. Maybe if no one else answers, I will look it up.
Jan 14, 2018 at 20:48	comment	added	John Baez		Excellent! So, I'm just curious about the geometric quantization. In simple terms: say we take the $p$th power of the canonical line bundle on $\mathbb{CP}^{n-1}$ and restrict to the quadric hypersurface $Q \subset \mathbb{CP}^{n-1}$ you described. What's the space of holomorphic sections on the resulting line bundle over $Q$? This should be well-known. Here $p \in \mathbb{Z}$, and we expect to get a 0-dimensional space when $p$ has one sign and something interesting when it has the other sign.
Jan 14, 2018 at 11:38	history	edited	Ben McKay	CC BY-SA 3.0	added quadric
Jan 14, 2018 at 10:19	comment	added	Ben McKay		Careful: this only works for the oriented Grassmannian. The Grassmannian of unoriented 2-planes in $\mathbb{R}^n$ is not Kaeher, as it has vanishing second Betti number.
Jan 14, 2018 at 10:11	history	edited	Ben McKay	CC BY-SA 3.0	found Nijenhuis tensor
Jan 14, 2018 at 10:01	history	answered	Ben McKay	CC BY-SA 3.0