Timeline for Conjugacy for $p$-adic matrices of finite order
Current License: CC BY-SA 4.0
8 events
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| Jan 24, 2023 at 17:41 | history | edited | LSpice | CC BY-SA 4.0 |
TeX, while this is on the front page
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| Aug 8, 2011 at 10:57 | comment | added | A Stasinski |
It is not true that two elements of $GL_n(\mathbb{Z}_p)$ are conjugate in $GL_n(\mathbb{Z}_p)$ iff they are conjugate in $GL_n(\mathbb{Q}_p)$. In particular, not every element in $GL_n(\mathbb{Z}_p)$ is $GL_n(\mathbb{Z}_p)$-conjugate to one in rational canonical form. For more on this (see arXiv:0708.1608v2 and its references).
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| Aug 8, 2011 at 10:44 | comment | added | Frieder Ladisch | When $p \nmid m$, the result is rather elementary, see the addendum to my answer. The argument (averaging) is quite common in group representation theory, but no knowledge about representations is needed to follow the argument, I think. | |
| Aug 8, 2011 at 8:43 | comment | added | Tim Dokchitser | It should probably be for order coprime to $p$; e.g. in $GL_2(Z_2)$ the reflection in the x-axis and the reflection in the line x=y (A=[[1,0],[0,-1]] and B=[[0,1],[1,0]]) are not conjugate, but they are conjugate over $Q_2$. | |
| Aug 7, 2011 at 17:54 | comment | added | Jeff Adler | Addendum: The Gelfand-Kazhdan result is special for GL(n). It certainly doesn't work for reductive p-adic groups in general. | |
| Aug 7, 2011 at 17:52 | comment | added | Jeff Adler | The result can be found in Gelfand-Kazhdan, "Representations of the group GL(n,K) where K is a local field. Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971)", published in 1975. Sorry, but I don't have my files handy, and I'm suddenly unsure if the result is for elements of order co-prime to p, or all elements. But either way, it says that two elements of a maximal compact subgroup are conjugate in the subgroup if and only if they are conjugate in GL(n,K). | |
| Aug 7, 2011 at 9:52 | comment | added | Tim Dokchitser | What is the result of Gelfand and Kazhdan that you refer to? I suppose that the corollary that you mention ("A special case of their result...") already uses the fact that the two elements have order coprime to $p$. | |
| Aug 7, 2011 at 9:30 | history | answered | Jeff Adler | CC BY-SA 3.0 |