Timeline for Finding the action of the symplectic group on the Siegel-half plane
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2011 at 18:19 | comment | added | paul garrett | @Gunnar: the very last page or two of that little essay "does" the Harish-Chandra realization for Sp(n,R) on the last page-or-two. For general simple G with G/K admitting an invariant hermitian structure, the same thing works, suitably abstracted. I think Helgason's "Symmetric spaces..." treats the general case. For classical groups, the computations of course succeed without necessarily couching things in Lie-theoretic or geometric terms. | |
| Oct 16, 2011 at 16:54 | vote | accept | Gunnar Þór Magnússon | ||
| Oct 16, 2011 at 16:53 | comment | added | Gunnar Þór Magnússon | @Paul: I just found your paper "The classical groups and domains". It looks like my second question is treated in the part on self-adjoint cones; I think the group action on $U$ simply corresponds to the one of $GL_n(\mathbb C)$ restricted to the symplectic group. I had a strong case of not seeing the forest for the trees here - in my head the Siegel half-plane and the space of $(1,1)$-forms are intimitely related, since they both pop up while studying complex tori, while for this question that doesn't matter at all. | |
| Oct 16, 2011 at 15:58 | answer | added | David E Speyer | timeline score: 7 | |
| Oct 16, 2011 at 15:51 | comment | added | paul garrett | The HarishChandra-Borel realizations of non-compact type hermitian symmetric spaces inside their compact duals may be of interest. | |
| Oct 16, 2011 at 14:04 | answer | added | Donu Arapura | timeline score: 4 | |
| Oct 16, 2011 at 12:01 | comment | added | Gunnar Þór Magnússon | Well, yes I was wondering about a conceptual interpretation. The action is exactly the same, $M \cdot Z = (AZ + B)(CZ + D)^{-1}$, where $(Z - {}^t \overline Z)/2i$ is positive-definite. It gives a transitive action of $Sp_{2n}$ on the space $U$ of $(1,1)$-forms, and I wonder if there is any way of seeing this action pop up naturally, instead of ad-hoc like this. | |
| Oct 16, 2011 at 11:25 | comment | added | David Loeffler | I don't quite understand your revised question. You seem to be simultaneously saying that you know how to define an action of $Sp_n$ on $U$, and asking how to define such an action. Are you asking for a conceptual interpretation of an action on $U$ that you know how to define in coordinates? | |
| Oct 16, 2011 at 8:59 | history | edited | Gunnar Þór Magnússon | CC BY-SA 3.0 |
added 804 characters in body
|
| Oct 16, 2011 at 8:38 | answer | added | David Loeffler | timeline score: 7 | |
| Oct 16, 2011 at 8:33 | history | edited | Gunnar Þór Magnússon | CC BY-SA 3.0 |
Deleted a remark that needs more work before it makes sense.
|
| Oct 16, 2011 at 8:11 | history | edited | Gunnar Þór Magnússon | CC BY-SA 3.0 |
added 31 characters in body
|
| Oct 16, 2011 at 8:01 | history | asked | Gunnar Þór Magnússon | CC BY-SA 3.0 |