Timeline for Which groups can have $GSp(4)$ as local component?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 15, 2017 at 8:05 | comment | added | Desiderius Severus | @paulgarrett Ok, many thanks for those clarifications! | |
| Sep 15, 2017 at 8:05 | vote | accept | Desiderius Severus | ||
| Sep 7, 2017 at 13:07 | comment | added | paul garrett | To construct a(n algebraic) group isomorphic to $Sp_n$ over a finite extension of a local field $k$, let $B$ be the unique quaternion division algebra over $k$, and $S$ a quaternion-hermitian form of signature $p,q$ on a $B-(p+q)$ -dimensional vector space over $B$. This is certainly not literally the split symplectic group over $k$, but when we extend scalars to split $B$, it becomes such the split symplectic group over that extension. | |
| Sep 6, 2017 at 15:35 | comment | added | Desiderius Severus | @paulgarrett I believe I have a problem with the definition of the algebraic group you use. If it is the group of matrices $A$ such that $A^\star J A = \lambda J$ for $J$ the usual symplectic matrix, why it is not again the symplectic group at any places? (Maybe that should correspond to the "boring extension of scalars" of $GSp$ you talked about. Otherwise, what is the definition via non-degenerate quaternion-symmetric forms with quaternion algebras over the ground field? | |
| Aug 28, 2017 at 15:13 | vote | accept | Desiderius Severus | ||
| Sep 6, 2017 at 15:32 | |||||
| Aug 28, 2017 at 13:02 | comment | added | paul garrett | @DesideriusSeverus, a quaternion division algebra splits almost everywhere locally, so the local group is the split $Sp(4)$ almost everywhere locally. At the (finitely-many) ramified places: at real ones, one has a "signature", while at finite ones there is a maximal possible size of isotropic subspace (as with $p$-adic quadratic forms). | |
| Aug 28, 2017 at 12:50 | comment | added | Venkataramana | @Desiderius Severus, You had asked about $Sp_4$, whereas I was looking at $SO(f)$ (the isometry group of the quadratic form with determinant one). These two groups are not that different: $Sp_4$ is the two sheeted simply connected cover of $SO(f)$. | |
| Aug 28, 2017 at 10:23 | comment | added | Desiderius Severus | @paulgarrett You answer seems to be exactly the general answer I am seeking. However why is it clear that the isometry group is the symplectic group almost everywhere? Is it the case exactly at the split places? Do you know what happen in the ramified places? I am really not at ease with this kind of computations... | |
| Aug 28, 2017 at 10:04 | comment | added | Desiderius Severus | @Venkataramana Thanks for the answer, however when you say "the group", you are talking about the group of isometry defined for the quadratic form you consider? | |
| Aug 10, 2017 at 4:41 | comment | added | Venkataramana | This answer gives a more general result; if one is to confine oneself to $GSp_4=GSpin(5)$, matters are a bit easier to describe: take any non-degenrate quadratic form $f$ in 5 variables over $\mathbb Q$; then for almost all primes $p$, the group becomes the (quasi-split=split) group $GSpin(5)=GSp_4$. Over any (number) field, this (namely $GSpin (f)$) is the only way to get (inner) forms of $GSpin (5)$ . | |
| Aug 10, 2017 at 0:59 | comment | added | paul garrett | ... and see A. Weil's "Involutions and the classical groups" (or similar title) from about 1964 for proof that there are no other classical groups than what we can imagine by devices similar to my answer... I heard an exposition of this from G. Shimura on some day in the mid 1970s. | |
| Aug 10, 2017 at 0:30 | history | edited | paul garrett | CC BY-SA 3.0 |
added 444 characters in body
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| Aug 9, 2017 at 23:41 | history | answered | paul garrett | CC BY-SA 3.0 |