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
7931413
$endgroup$
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
7931413
$endgroup$
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
7931413
$endgroup$
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
7931413
asked Dec 20 '18 at 12:07
NotoneNotone
7931413
asked Dec 20 '18 at 12:07
NotoneNotone
7931413
asked Dec 20 '18 at 12:07
NotoneNotone
7931413
asked Dec 20 '18 at 12:07
NotoneNotone
7931413
7931413
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 |
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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