Principal series for $operatorname{GL}_2$, question about an exact sequence
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I have a question about Proposition 7.2 of these notes by Gordon Savin.
Here $G = operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$, and $delta$ is the modulus character $delta begin{pmatrix} a_1 \ & a_2 end{pmatrix} = frac{|a_1|}{|a_2|}$.
I follow all the steps in the proof, except for the very last sentence. Maybe I would understand things if I knew what the map
$$0 rightarrow delta^{frac{1}{2}} chi^w rightarrow Ind_B^G(chi)_N$$
was in the first place.
representation-theory p-adic-number-theory reductive-groups
asked Dec 2 at 4:21
D_S
13.1k51551
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I have a question about Proposition 7.2 of these notes by Gordon Savin.
Here $G = operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$, and $delta$ is the modulus character $delta begin{pmatrix} a_1 \ & a_2 end{pmatrix} = frac{|a_1|}{|a_2|}$.
I follow all the steps in the proof, except for the very last sentence. Maybe I would understand things if I knew what the map
$$0 rightarrow delta^{frac{1}{2}} chi^w rightarrow Ind_B^G(chi)_N$$
was in the first place.
representation-theory p-adic-number-theory reductive-groups
asked Dec 2 at 4:21
D_S
13.1k51551
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a question about Proposition 7.2 of these notes by Gordon Savin.
Here $G = operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$, and $delta$ is the modulus character $delta begin{pmatrix} a_1 \ & a_2 end{pmatrix} = frac{|a_1|}{|a_2|}$.
I follow all the steps in the proof, except for the very last sentence. Maybe I would understand things if I knew what the map
$$0 rightarrow delta^{frac{1}{2}} chi^w rightarrow Ind_B^G(chi)_N$$
was in the first place.
representation-theory p-adic-number-theory reductive-groups
asked Dec 2 at 4:21
D_S
13.1k51551
I have a question about Proposition 7.2 of these notes by Gordon Savin.
Here $G = operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = begin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix}$, and $delta$ is the modulus character $delta begin{pmatrix} a_1 \ & a_2 end{pmatrix} = frac{|a_1|}{|a_2|}$.
I follow all the steps in the proof, except for the very last sentence. Maybe I would understand things if I knew what the map
$$0 rightarrow delta^{frac{1}{2}} chi^w rightarrow Ind_B^G(chi)_N$$
was in the first place.
representation-theory p-adic-number-theory reductive-groups
representation-theory p-adic-number-theory reductive-groups
asked Dec 2 at 4:21
D_S
13.1k51551
asked Dec 2 at 4:21
D_S
13.1k51551
asked Dec 2 at 4:21
D_S
13.1k51551
asked Dec 2 at 4:21
D_S
13.1k51551
asked Dec 2 at 4:21
D_S
13.1k51551
13.1k51551
add a comment |
add a comment |
1 Answer
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$DeclareMathOperator{Ind}{Ind}$Oh wait, I think I get it. The kernel of $alpha_w: Ind_B^G(chi)_w rightarrow mathbb C$ is exactly $Ind_B^G(chi)_w(N)$, so we get an isomorphism of the Jacquet module $Ind_B^G(chi)_{w,N}$ with $mathbb C$. This is how we get an embedding
$$mathbb C rightarrow Ind_B^G(chi)_{w,N} rightarrow Ind_B^G(chi)_N$$
answered Dec 2 at 4:26
D_S
13.1k51551
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$DeclareMathOperator{Ind}{Ind}$Oh wait, I think I get it. The kernel of $alpha_w: Ind_B^G(chi)_w rightarrow mathbb C$ is exactly $Ind_B^G(chi)_w(N)$, so we get an isomorphism of the Jacquet module $Ind_B^G(chi)_{w,N}$ with $mathbb C$. This is how we get an embedding
$$mathbb C rightarrow Ind_B^G(chi)_{w,N} rightarrow Ind_B^G(chi)_N$$
answered Dec 2 at 4:26
D_S
13.1k51551
add a comment |
up vote
0
down vote
$DeclareMathOperator{Ind}{Ind}$Oh wait, I think I get it. The kernel of $alpha_w: Ind_B^G(chi)_w rightarrow mathbb C$ is exactly $Ind_B^G(chi)_w(N)$, so we get an isomorphism of the Jacquet module $Ind_B^G(chi)_{w,N}$ with $mathbb C$. This is how we get an embedding
$$mathbb C rightarrow Ind_B^G(chi)_{w,N} rightarrow Ind_B^G(chi)_N$$
answered Dec 2 at 4:26
D_S
13.1k51551
add a comment |
up vote
0
down vote
up vote
0
down vote
$DeclareMathOperator{Ind}{Ind}$Oh wait, I think I get it. The kernel of $alpha_w: Ind_B^G(chi)_w rightarrow mathbb C$ is exactly $Ind_B^G(chi)_w(N)$, so we get an isomorphism of the Jacquet module $Ind_B^G(chi)_{w,N}$ with $mathbb C$. This is how we get an embedding
$$mathbb C rightarrow Ind_B^G(chi)_{w,N} rightarrow Ind_B^G(chi)_N$$
answered Dec 2 at 4:26
D_S
13.1k51551
$DeclareMathOperator{Ind}{Ind}$Oh wait, I think I get it. The kernel of $alpha_w: Ind_B^G(chi)_w rightarrow mathbb C$ is exactly $Ind_B^G(chi)_w(N)$, so we get an isomorphism of the Jacquet module $Ind_B^G(chi)_{w,N}$ with $mathbb C$. This is how we get an embedding
$$mathbb C rightarrow Ind_B^G(chi)_{w,N} rightarrow Ind_B^G(chi)_N$$
answered Dec 2 at 4:26
D_S
13.1k51551
answered Dec 2 at 4:26
D_S
13.1k51551
answered Dec 2 at 4:26
D_S
13.1k51551
answered Dec 2 at 4:26
D_S
13.1k51551
13.1k51551
add a comment |
add a comment |
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