Hey man, as much as I have specific series I want to see you continue (dude by the end of the year i'll probably be spamming your email with work on SL(2,Z) geometry applications and visualizations LOL), i am ALWAYS thrilled to see your scattered/"random" videos, and i'll take that any day over waiting for perfection. excited to see the local langlands videos ahead!! hope your research is going well m8!!!
Great series ahead, I can tell! Are you planning on talking about modular (finite fields) representations at some point? Thanks for the videos, keep it up.
Probably not this Spring as I'll be focused on those things most related to the summer workshop I'm going to. However I do have some personal interest in the subject, so possible later this summer as a means to teach myself more about it.
I am a noob and may have misunderstood something, so forgive my impertinence in the following questions: 1) How does one see that Z_p is a maximal compact subgroup of Q_p? Do you have a reference for that? 2) Why is GL(n,Z_p) a maximal compact subgroup of GL(n,Q_p), but not rather something like O(n,Q_p)? (Or does that not exist? Then I wonder how this very different flavour of maximal compact subgroup reflects on the representation theory.) 3) You say continous maps from Q_p to C are locally constant, but then you want to study continous representations of GL(n,Q_p) on complex vector spaces. How come there are enough of these representations for them to be worth studying? Naively I would expect the representations to be locally constant as well.
1) Z_p is the closed unit ball in Q_p, hence being a closed and bounded ubset of a metric space, it's compact. Note that it's also the open unit ball, and hence it's an open comapct subset. In fact, Q_p has a basis of translate of Z_p. Check out the section on p-adic numbers by Deitmar. 2) I'm blanking on the proof right now, but I suspect you take the argument in 1), extend it to Q_p^{n^2} and then intersect it with the invertible matrices which is closed in Q_p^{n_2}. You certainly can study the groups O(n, Q_p) and SO(n, Q_p), in fact it's really important and I'll probably talk about that at some point! Deitmar, mentioned in 1) probably discusses this as well. Off the top of my head, I don't have a "good" explanation as to why the theory differs like this... if you find a good conceptual explanation somewhere please let me know! I guess in some sense it's just a manifestation of the wird differences in the topologies. 3) Actually your expectation is correct! We're going to study representations which are locally constant! On the surface that might sound like it's going to be really boring, but it turns out it really is the "right" thing to look at, and there's some pretty interesting representations of this kind!
I'm excited!
Hey man, as much as I have specific series I want to see you continue (dude by the end of the year i'll probably be spamming your email with work on SL(2,Z) geometry applications and visualizations LOL), i am ALWAYS thrilled to see your scattered/"random" videos, and i'll take that any day over waiting for perfection. excited to see the local langlands videos ahead!! hope your research is going well m8!!!
Thanks! I always appreciate your support. Research is moving along, hope you're doing well too.
Great series ahead, I can tell! Are you planning on talking about modular (finite fields) representations at some point? Thanks for the videos, keep it up.
Probably not this Spring as I'll be focused on those things most related to the summer workshop I'm going to. However I do have some personal interest in the subject, so possible later this summer as a means to teach myself more about it.
I am a noob and may have misunderstood something, so forgive my impertinence in the following questions:
1) How does one see that Z_p is a maximal compact subgroup of Q_p? Do you have a reference for that?
2) Why is GL(n,Z_p) a maximal compact subgroup of GL(n,Q_p), but not rather something like O(n,Q_p)? (Or does that not exist? Then I wonder how this very different flavour of maximal compact subgroup reflects on the representation theory.)
3) You say continous maps from Q_p to C are locally constant, but then you want to study continous representations of GL(n,Q_p) on complex vector spaces. How come there are enough of these representations for them to be worth studying? Naively I would expect the representations to be locally constant as well.
1) Z_p is the closed unit ball in Q_p, hence being a closed and bounded ubset of a metric space, it's compact. Note that it's also the open unit ball, and hence it's an open comapct subset. In fact, Q_p has a basis of translate of Z_p. Check out the section on p-adic numbers by Deitmar.
2) I'm blanking on the proof right now, but I suspect you take the argument in 1), extend it to Q_p^{n^2} and then intersect it with the invertible matrices which is closed in Q_p^{n_2}. You certainly can study the groups O(n, Q_p) and SO(n, Q_p), in fact it's really important and I'll probably talk about that at some point! Deitmar, mentioned in 1) probably discusses this as well.
Off the top of my head, I don't have a "good" explanation as to why the theory differs like this... if you find a good conceptual explanation somewhere please let me know! I guess in some sense it's just a manifestation of the wird differences in the topologies.
3) Actually your expectation is correct! We're going to study representations which are locally constant! On the surface that might sound like it's going to be really boring, but it turns out it really is the "right" thing to look at, and there's some pretty interesting representations of this kind!
@@k-theory8604 Wow, thanks for the quick and extensive answer!