Timeline for Linear symmetric spaces are spaces with ''orthogonal complements''?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2018 at 21:55 | answer | added | JHM | timeline score: 1 | |
| Apr 2, 2012 at 18:08 | vote | accept | JHM | ||
| Mar 30, 2012 at 13:22 | answer | added | Misha | timeline score: 1 | |
| Mar 29, 2012 at 17:53 | comment | added | JHM | Have edited the question--have deleted any reference to an equivariant homeomorphism because I'm not really sure how it fits in here. I'll have to think about this last part. | |
| Mar 29, 2012 at 17:52 | history | edited | JHM | CC BY-SA 3.0 |
deleted a group-theoretic description of linearity--i'll thnk about it, and edit accordingly. Am responding to R. Bryants comments.
|
| Mar 29, 2012 at 17:37 | comment | added | JHM | Even the question "how do we compactify a disk" I don't see as so clear. More unclear, is of course,"how do we compactify a disk modulo $SL_n(\mathbb{Z})$". So my question above, on interpreting the linearity of $S_n$ as the existence of canonical rational complements, is really a twisted way of trying to learn more about these various compactifications. From what I've been learning, flags of rational subspaces are almost the boundary components of the (Borel-Serre or Satake) compactifications. On these points I am over my head--but not too over. I am trying to work these questions out. | |
| Mar 29, 2012 at 17:30 | comment | added | JHM | @Bryant: As originally phrased, my description of $S_n$ as linear space iswrong--i shall correct it. From MacPherson/McConnell it seems that ''linearity'' describes how the symmetric space $S_n/$ compactifies. Roughly (and this is what i'd like to refine for myself) a symmetric space is linear if it compactifies like the open ball. But this is a very amateur description. The business of these various compactifications (Satake, Borel-Serre, toroidal, ...) I find quite difficult to understand. | |
| Mar 29, 2012 at 17:13 | comment | added | JHM | I cannot provide a reference for a proper definition of ''linearity'' because I only know the term as described in MacPherson & McConnell's Inventiones paper "Explicit reduction theory for Siegel modular threefolds". | |
| Mar 29, 2012 at 15:28 | comment | added | Robert Bryant | @jmart: I'm a little puzzled by your omission of the unimodularity condition. The space $S_n$ is actually embedded as a hypersurface in the convex cone of positive definite quadratic forms on $\mathbb{R}^n$, a hypersurface that is asymptotic to the boundary of the convex cone. Are you asking for noncompact Riemannian symmetric spaces $M = G/K$ for which there is a $G$-module $V$ such that there is an open, proper convex cone $C\subset V$ that is $G$-invariant and such that $K$ is the stabilizer of a point inside $C$ and $\dim V = 1+\dim M$? | |
| Mar 29, 2012 at 7:04 | comment | added | Thomas Richard | I haven't heard the terminology "linear symmetric spaces" before, could you give a reference where the term is defined, I'm just curious about it. | |
| Mar 28, 2012 at 19:11 | history | edited | JHM | CC BY-SA 3.0 |
added 5 characters in body
|
| Mar 28, 2012 at 18:43 | history | edited | JHM | CC BY-SA 3.0 |
elaborated.
|
| Mar 28, 2012 at 18:25 | history | asked | JHM | CC BY-SA 3.0 |