Weil group and conjugacy classes of cocharacters
$begingroup$
We have the following setup: $G$ a compact reductive group over $mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$
In Variétiés de Shimura Lemma 1.2.4 p.258 Deligne concludes the following isomorphism:
$$ Hom(U^1,G)/G(mathbb{R})cong Hom(mathbb{G}_m, G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
from these isomorphisms:
$$ Hom(U^1,T)/Wcong Hom(U^1,G)/G(R)\
Hom(mathbb{G}_{m},T_mathbb{C})/Wcong Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
- What is the "Weil group of T"? I am aware of the definition of a Weil group of a local field, maybe he just means the Weil group of the local field over which $T$ is defined?
- I do not know how to conclude.
I know any two maximal tori are conjugate and that the image of a torus will be contained in a torus, which will give me e.g. a map $Hom(mathbb{G}_{m},T_mathbb{C})to Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$, which is surjective, but from there I am stuck. Is he maybe referring to the Weyl group of $T$?
Given the two latter isomorphisms, how do I conclude the first one?
Is there an "obvious" isomorphism $Hom(U^1,T)/Wto Hom(mathbb{G}_{m},T_mathbb{C})/W$ given by complexification?
I would also welcome a reference which possibly elaborates on such isomorphisms between tori and reductive groups.
algebraic-geometry reference-request group-schemes reductive-groups
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
$endgroup$
add a comment |
$begingroup$
We have the following setup: $G$ a compact reductive group over $mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$
In Variétiés de Shimura Lemma 1.2.4 p.258 Deligne concludes the following isomorphism:
$$ Hom(U^1,G)/G(mathbb{R})cong Hom(mathbb{G}_m, G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
from these isomorphisms:
$$ Hom(U^1,T)/Wcong Hom(U^1,G)/G(R)\
Hom(mathbb{G}_{m},T_mathbb{C})/Wcong Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
- What is the "Weil group of T"? I am aware of the definition of a Weil group of a local field, maybe he just means the Weil group of the local field over which $T$ is defined?
- I do not know how to conclude.
I know any two maximal tori are conjugate and that the image of a torus will be contained in a torus, which will give me e.g. a map $Hom(mathbb{G}_{m},T_mathbb{C})to Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$, which is surjective, but from there I am stuck. Is he maybe referring to the Weyl group of $T$?
Given the two latter isomorphisms, how do I conclude the first one?
Is there an "obvious" isomorphism $Hom(U^1,T)/Wto Hom(mathbb{G}_{m},T_mathbb{C})/W$ given by complexification?
I would also welcome a reference which possibly elaborates on such isomorphisms between tori and reductive groups.
algebraic-geometry reference-request group-schemes reductive-groups
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
$endgroup$
2
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36
add a comment |
$begingroup$
We have the following setup: $G$ a compact reductive group over $mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$
In Variétiés de Shimura Lemma 1.2.4 p.258 Deligne concludes the following isomorphism:
$$ Hom(U^1,G)/G(mathbb{R})cong Hom(mathbb{G}_m, G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
from these isomorphisms:
$$ Hom(U^1,T)/Wcong Hom(U^1,G)/G(R)\
Hom(mathbb{G}_{m},T_mathbb{C})/Wcong Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
- What is the "Weil group of T"? I am aware of the definition of a Weil group of a local field, maybe he just means the Weil group of the local field over which $T$ is defined?
- I do not know how to conclude.
I know any two maximal tori are conjugate and that the image of a torus will be contained in a torus, which will give me e.g. a map $Hom(mathbb{G}_{m},T_mathbb{C})to Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$, which is surjective, but from there I am stuck. Is he maybe referring to the Weyl group of $T$?
Given the two latter isomorphisms, how do I conclude the first one?
Is there an "obvious" isomorphism $Hom(U^1,T)/Wto Hom(mathbb{G}_{m},T_mathbb{C})/W$ given by complexification?
I would also welcome a reference which possibly elaborates on such isomorphisms between tori and reductive groups.
algebraic-geometry reference-request group-schemes reductive-groups
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
$endgroup$
We have the following setup: $G$ a compact reductive group over $mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$
In Variétiés de Shimura Lemma 1.2.4 p.258 Deligne concludes the following isomorphism:
$$ Hom(U^1,G)/G(mathbb{R})cong Hom(mathbb{G}_m, G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
from these isomorphisms:
$$ Hom(U^1,T)/Wcong Hom(U^1,G)/G(R)\
Hom(mathbb{G}_{m},T_mathbb{C})/Wcong Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$$
- What is the "Weil group of T"? I am aware of the definition of a Weil group of a local field, maybe he just means the Weil group of the local field over which $T$ is defined?
- I do not know how to conclude.
I know any two maximal tori are conjugate and that the image of a torus will be contained in a torus, which will give me e.g. a map $Hom(mathbb{G}_{m},T_mathbb{C})to Hom(mathbb{G}_m,G_mathbb{C})/G_mathbb{C}(mathbb{C})$, which is surjective, but from there I am stuck. Is he maybe referring to the Weyl group of $T$?
Given the two latter isomorphisms, how do I conclude the first one?
Is there an "obvious" isomorphism $Hom(U^1,T)/Wto Hom(mathbb{G}_{m},T_mathbb{C})/W$ given by complexification?
I would also welcome a reference which possibly elaborates on such isomorphisms between tori and reductive groups.
algebraic-geometry reference-request group-schemes reductive-groups
algebraic-geometry reference-request group-schemes reductive-groups
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
asked Dec 20 '18 at 12:07
NotoneNotone
8181413
8181413
2
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36
add a comment |
2
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36
2
2
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047469%2fweil-group-and-conjugacy-classes-of-cocharacters%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047469%2fweil-group-and-conjugacy-classes-of-cocharacters%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
$begingroup$
He is definitely referring to the Weyl group of $T$!
$endgroup$
– Jef L
Dec 20 '18 at 15:37
$begingroup$
Yes, this seems very sensible. I am still missing out on the conclusion so far though
$endgroup$
– Notone
Dec 22 '18 at 20:36