User jgordon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:09:12Z http://mathoverflow.net/feeds/user/15073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60173/cartan-subgroups-of-p-adic-groups/65614#65614 Answer by JGordon for Cartan subgroups of p-adic groups. JGordon 2011-05-21T02:22:25Z 2011-05-21T02:22:25Z <p>just stumbled upon this old thread and decided to contribute my 2 cents (though I completely agree with Moshe's answer): J.-L. Waldspurger has a very explicit description of conjugacy classes of semisimple elements in the classical groups, in his Asterisque volume "Integrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifies", I believe in Section I.7. This is, essentially, Moshe's examples 1 --3 but done in a systematic way for every classical group. One can get a parametrization of tori from this description of semisimple conjugacy classes by looking at centralizers.</p> http://mathoverflow.net/questions/65454/a-question-regarding-lie-group-actions/65502#65502 Answer by JGordon for A question regarding Lie group actions JGordon 2011-05-20T05:54:41Z 2011-05-20T05:54:41Z <p>I wanted to make this a comment, but it turned out to be too long, so have to submit it as (partial) answer... sorry if I misunderstood the question.</p> <p>Let $G$ be the Lie group. If such an example exists, it would have to be somewhat exotic -- by my previous comment, any closed subgroup of finite index has to be a union of connected components of $G$. On the other hand, if a Lie group acts on a differentiable manifold $X$ transitively, then its identity component also acts transitively (see e.g. Onischik and Vinberg, "Lie groups and algebraic groups", Chapter 1 section 3). Hence, to find such an example, one would have to look for $X$ that is not a manifold. On the other hand, if the stabilizer of a point is Lie subgroup, then $X$ would have a unique structure of a manifold with respect to which the action is differentiable. Hence, if you really want such an example, one has to look for some action that "cannot be made smooth" by putting a differentiable manifold structure on $X$, i.e., where the stabilizer of a point is not a submanifold of $G$ (I think traditionally in the "standard" theory of Lie groups such actions are not considered...)</p> http://mathoverflow.net/questions/36025/explicit-computations-using-the-haar-measure/64960#64960 Answer by JGordon for Explicit computations using the Haar measure JGordon 2011-05-14T03:31:40Z 2011-05-14T03:31:40Z <p>I just stumbled upon this old thread while searching for something else, and couldn't resist saying two things: 1. if you like p-adics, the expository article <a href="http://arxiv.org/pdf/math/0205207v2" rel="nofollow">http://arxiv.org/pdf/math/0205207v2</a> by T. C. Hales asks pretty much the same question, and gives some very interesting examples, an explanation why in general this question is very hard, and a general approach via motivic integration (a lot of progress happened in motivic integration since this article was written, but this is still a great introduction to the main ideas). 2. For a split connected reductive algebraic group over a local field, one can write down an explicit formula for the Haar measure in convenient coordinates (more precisely, one can just write down the invariant differential form that Brian Conrad mentioned): for an explicit formula, see e.g. section 2.4 in <a href="http://arxiv.org/pdf/math/0203106" rel="nofollow">http://arxiv.org/pdf/math/0203106</a> (I am sure this is a classical formula, but I have never seen a reference for it -- would be grateful if someone pointed it out). </p> http://mathoverflow.net/questions/64833/semisimplicity-of-p-adic-galois-representations/64869#64869 Answer by JGordon for semisimplicity of p-adic Galois representations JGordon 2011-05-13T04:51:30Z 2011-05-13T16:03:56Z <p>Well, actually, the "motivic Haar measure" LSpice refers to is an analogue of Haar measure that lives on ${\operatorname{GL}}_n({\mathbb C}((t)))$, not on ${\operatorname{GL}}_n({\mathbb Q}_p)$, and takes values in the Grothendieck ring of varieties, so I think it's not quite relevant here. What is closer to this discussion though is the fact that the usual Haar mesaure on ${\operatorname{GL}}_n({\mathbb Q}_p)$ if it's reasonably normalized (e.g. so that the volume of ${\operatorname{GL}}_n({\mathbb Z}_p)$<br> is $1$), actually takes values in $\mathbb Q$ on all reasonable sets that you ever want to consider. More precisely, one can define a sigma-algebra of the so-called "definable sets", and the volumes of definable sets just are in $\mathbb Q$. In this sense they are in $\mathbb Q_p$ already, so the trick is that for these sets you do not need any completion of $\mathbb Q$ in order to define their volumes, and so you do not need to worry about using the $p$-adic metric... Most sets one works with turn out to be automatically definable, so this fact may be handy in some other situation. </p> <p>Added some hours later: It was my first post on mathoverflow, and I am still not allowed to add comments to others' posts :) -- so this should be a comment to the comment by LSpice that appears in Emerton's post. Talking about measure on $\operatorname{GL}_n(\overline{{\mathbb Q}_p})$, there is a paper by E.Hrushovski and D. Kazhdan <a href="http://arxiv.org/abs/math/0510133" rel="nofollow">http://arxiv.org/abs/math/0510133</a> that talks about integration in algebraically closed valued fields (using logic). As a first approximation, as far as I understand, the values of this measure are something like equivalence classes of definable sets over the residue field (I am certainly being imprecise here). There are several papers by Yimu Yin (the ones to start with are <a href="http://arxiv.org/abs/0809.0473" rel="nofollow">http://arxiv.org/abs/0809.0473</a>, and <a href="http://arxiv.org/abs/1006.2467" rel="nofollow">http://arxiv.org/abs/1006.2467</a>) aimed at clarifying this fundamental work of Hrushovski and Kazhdan in a slightly simplified setting. Unfortunately, I do not know of any non-technical introductory paper about this. There is a short note by Moshe Kamenski <a href="http://www.nd.edu/~mkamensk/lectures/motivic.pdf" rel="nofollow">http://www.nd.edu/~mkamensk/lectures/motivic.pdf</a> -- maybe this is the best place to start. Also, I hope someone corrects me here if I made any errors in this description. </p> http://mathoverflow.net/questions/69116/discrete-series-representations-for-sl-2-over-p-adic-field/69121#69121 Comment by JGordon JGordon 2011-07-02T05:22:15Z 2011-07-02T05:22:15Z The construction described above (together with the similar construction for $J_1$) produces all of the so-called depth zero supercuspidal representations of SL(2, F). There is a nice complete treatment of this example in Joseph Rabinoff's notes: <a href="http://math.stanford.edu/~rabinoff/misc/building.pdf" rel="nofollow">math.stanford.edu/~rabinoff/misc/building.pdf</a> http://mathoverflow.net/questions/65454/a-question-regarding-lie-group-actions Comment by JGordon JGordon 2011-05-20T02:41:00Z 2011-05-20T02:41:00Z @zroslav: I think such an example cannot exist -- if a subgroup is closed and finite index, then its complement is a finite union of its cosets, hence also closed; then our subgroup is open as well. http://mathoverflow.net/questions/64968/dominant-weights/64991#64991 Comment by JGordon JGordon 2011-05-14T17:23:11Z 2011-05-14T17:23:11Z I just want to point out a very pretty paper <a href="http://arxiv.org/pdf/0908.1091v1" rel="nofollow">arxiv.org/pdf/0908.1091v1</a> -- I'm not sure it answers the question, but it seems to give a nice way of thinking of minuscule and dominant weights... http://mathoverflow.net/questions/36025/explicit-computations-using-the-haar-measure/64960#64960 Comment by JGordon JGordon 2011-05-14T05:46:00Z 2011-05-14T05:46:00Z Thank you, Thierry! I thought it was a very nice question... http://mathoverflow.net/questions/64833/semisimplicity-of-p-adic-galois-representations/64869#64869 Comment by JGordon JGordon 2011-05-13T05:32:49Z 2011-05-13T05:32:49Z Dear Anatoly, thanks for asking for references - your question appeared just as I was typing them up. Yoav Yaffe and I have another introductory note on this: <a href="http://arxiv.org/abs/0811.2160" rel="nofollow">arxiv.org/abs/0811.2160</a>; the original article by R. Cluckers and F. Loeser appeared in Inventiones, for the arxiv version see <a href="http://arxiv.org/abs/math/0410203" rel="nofollow">arxiv.org/abs/math/0410203</a> http://mathoverflow.net/questions/64833/semisimplicity-of-p-adic-galois-representations/64869#64869 Comment by JGordon JGordon 2011-05-13T05:28:32Z 2011-05-13T05:28:32Z I should have included some references: the fact I mentioned follows from the theory of motivic integration by R. Cluckers and F. Loeser; the paper <a href="http://arxiv.org/abs/math/0410223" rel="nofollow">arxiv.org/abs/math/0410223</a> is the most relevant overview.