Timeline for on a decomposition lemma in adelic groups
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 28, 2012 at 17:45 | comment | added | prochet | Dear xuhan. For groups, It doesn't work, but actually I need a version of this problem for Lie algebras and your argument works for two points. So, when I look for n points, can we find suitable Borel so that it works? | |
| Nov 24, 2012 at 22:29 | comment | added | user29283 | Dear prochet: You're right, I was mistaken -- the product of opposite Borels is an open cell and so doesn't work. Probably it never works. And the fact that one is using a constant point is a sign that this may not be a fruitful thing to consider. (You never explained the motivation for your question, which might be helpful for suggesting deeper tools to use, like strong approximation, etc.) | |
| Nov 24, 2012 at 12:38 | comment | added | prochet | In fact, I'm not sure to understand why, in the case where $B_{x}$ and $B_{y}$ are oppposite, $g(x)\in B_{x}B_{y}$? | |
| Nov 24, 2012 at 6:37 | comment | added | S. Carnahan♦ | Perhaps you should rename your ground field, since you use $k$ later. | |
| Nov 24, 2012 at 3:40 | comment | added | prochet | Correction to the precedent comment in (i) And for n points is it still true? Say, is there a choice of Iwahoris $I_{x_{1}}$,..., $I_{x_{n}}$ such that : $\forall$ $(h_{2},....,h_{n})\in G(X−x_{2})\times...\times G(X−x_{n})$ there exists $(l_{2},...,l_{n})$ such that: $\\$ (i) $\forall i, l_{i}\in G(X-\{x_{1},...,x_{n}\}\cap I_{x_{i}}$ $\\$ (ii)$ l_{2}h_{2}=....=l_{n}h_{n}$ $\\$ (iii) $l_{i}h_{i}\in I_{x_{1}}$ ? | |
| Nov 24, 2012 at 3:32 | comment | added | prochet | Thanks. And for n points is it still true? Say, is there a choice of Iwahoris $I_{x_{1}}$,..., $I_{x_{n}}$ such that : $\forall (h_{2},....,h_{n})\in G(X-x_{2})\times...\times G(X-x_{n})$ there exists $(l_{2},...,l_{n})$ such that: (i) $\forall i$, $l_{i}\in G(X-x)\cap I_{x_{i}}$ (ii) l_{2}h_{2}=....=l_{n}h_{n} (iii)l_{i}h_{i}\in I_{x_{1}} ? | |
| Nov 24, 2012 at 3:05 | comment | added | user29283 | It depends on how you're choosing the Iwahori subgroups $I_x$ and $I_y$. (You say "the" Iwahori, but that's as bad as "the" Borel.) Since $g$ is regular at $x \in X - y$, all elements of $G(k(X))$ lying in $I_xg^{-1}$ (viewed inside $G({\rm{Frac}}(O^{\wedge}_x))$) are regular at $x$. Hence, the hypothetical $k$ is regular at $x$, so it is everywhere regular and hence constant (over the ground field denoted as $k$...). So $k$ lies in whatever Borel $B_y$ of $G$ "corresponds" to $I_y$ and you're asking if $g(x) \in B_y B_x$. If $B_y$ and $B_x$ are opposite Borels, you win; if equal, you lose... | |
| Nov 24, 2012 at 2:45 | history | asked | prochet | CC BY-SA 3.0 |