Timeline for Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
Current License: CC BY-SA 4.0
5 events
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| Jan 8 at 15:14 | comment | added | Abdelmalek Abdesselam | Yes I expect this is not a straightforward thing to do, and could e.g. need Hironaka's resolution of singularities or something of this kind. I know there is work by Aizenbud and Gurevitch on this kind of, and even more general, Schwartz spaces, in the context of the Langlands program (which you know a lot about, and definitely much more than me). | |
| Jan 7 at 20:45 | comment | added | paul garrett | @AbdelmalekAbdesselam, yes, this "geometric" characterization of Schwartz space is useful... but/and it does depend on the metrical properties of the compactification, and the fact (for $\mathbb R^n$) that there's a handy smooth one-point compactification, etc. A linear reductive Lie group (I know, sometimes the linearity is part of the defn of "reductive"...) could be viewed as a sub-thing of the analogous one-point compactification of the ambient $\mathbb R^{n^2}$, but I don't think it'd be nice at infinity. Maybe that doesn't matter? | |
| Jan 7 at 16:01 | comment | added | Abdelmalek Abdesselam | ...of $D'$ made of restrictions of distribution which live on the $n$-dimensional sphere obtained by adding the point at infinity. I was wondering if something similar works for Lie groups. Is there some natural compactification analogous to the sphere for $\mathbb{R}^n$ which would produce the correct Schwartz space in a similar manner. | |
| Jan 7 at 15:56 | comment | added | Abdelmalek Abdesselam | +1. I think this is an important question and I asked a related one here mathoverflow.net/questions/217663/… I also agree that a definition is "correct" if it is the hypothesis of lots of nice theorems. But it is also worthwhile to find an equivalent formulation of the "correct" definition which is also conceptually "natural" and does not appear contrived or the result of some clever construction. Another definition I know for $\mathbb{R}^n$, the original one by Schwartz, is roughly $(S')'$ where $S'$ is the subspace... | |
| Dec 7, 2023 at 21:45 | history | answered | paul garrett | CC BY-SA 4.0 |