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Feb
27 |
comment |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Dear Dave, sorry, I made a silly mistake. Is there a reference for what you said above, namely that "$G({\Bbb Q}_p)\times{\Bbb Q}^*_p$ has only finitely many orbits on the set of vector-space lattices whenever $G$ is split, semi-simple, and simply connected, and ρ is faithful and irreducible."? This is the case I need. |
Feb
26 |
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orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Dear Dave, oh. It seems the following simple argument works. $L=A({\Bbb Q}{\Bbb Q}\times ... {\Bbb Q})$ is vector-space lattice iff $A^{-1} G({\Bbb Q}_p)A=G({\Bbb Q}_p)$, i.e. $A\in GL({\Bbb Q}^n)$ is in the normaliser of $G({\Bbb Q}_p)$. $<=$ is easy; $=>$: $ A^{-1}Stab(L)A=G({\Bbb Q}_p)\cap GL({\Bbb Q}^n)$; pass to the closures and use $Stab(L)$ is dense in $G(({\Bbb Q}_p)$: $A^{-1}G(({\Bbb Q}_p)A=cl(A^{-1}Stab(L)A)=cl(G({\Bbb Q}_p)\cap GL({\Bbb Q}^n))\subseteq G({\Bbb Q}_p)$. |
Feb
26 |
comment |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Dear Dave, thank you! It is interesting and shall take some time for me to digest; but it does solve (probably) my question so I "accepted" your answer. |
Feb
26 |
accepted | orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ |
Feb
25 |
comment |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Thanks! And what about $G=GL_n$, should it not be an exception as well? What I ask is true for $G=GL_n$... |
Feb
24 |
revised |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
added 245 characters in body |
Feb
24 |
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orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
In this case one needs to consider the action of the group of diagonal matrices with non-zero determinant... |
Feb
24 |
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orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
Thanks, I've made a few correction. Yes, I consider the action of the group of p-adic points on the subgroups L with these properties. Concerning being dense in either Zariski or p-adic topology, it is unclear what the right question is. |
Feb
24 |
revised |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
added 54 characters in body |
Feb
24 |
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orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
@Venkataramana, thank you, I corrected that. It does seem that the answer might be implied by the theory of arithmetic subgroups, but it feels the question is standard enough to be somewhere in the literature... |
Feb
24 |
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Feb
24 |
revised |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
edited body |
Feb
18 |
revised |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
added 194 characters in body |
Feb
18 |
revised |
orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$
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Feb
17 |
asked | orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ |
Jan
4 |
awarded | Necromancer |
Oct
17 |
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Jul
23 |
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Is the theory of categories decidable?
On the other hand, there is no reason for the equational fragment of the theory of categories without automorphisms be undecidable, and it is actually decidable, see "mishap.sdf.org/… |