Relation between Fourier coefficients and Satake parameters
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) = sum_k=0^infty fraca_nn^s = prod_p left( 1 - alpha(p)p^-s right)^-1 left( 1 - beta(p)p^-s right)^-1 left( 1 - gamma(p)p^-s right)^-1$$
Straightforwardly developing the Euler product provides expressions of the Fourier coefficients $a_n$'s in terms of the Satake parameters $alpha(p)$, $beta(p)$ and $gamma(p)$. I am not particularly aware of others standard useful relations between them. I bumped into the following one:
$$a_p^k = frac
left
$$
I guess this can be verified, but even the case $k=1$ seems obscure to me. I do not want to believe such a formula to be a (verifiable) accident. Despite it works computationally, am I missing something lying behind? How strongly is the self-contragredience assumption necessary?
Any insight is welcome, as well as other ways to embrace the relations between spectral parameters and coefficients.
nt.number-theory automorphic-forms l-functions
add a comment |Â
up vote
4
down vote
favorite
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) = sum_k=0^infty fraca_nn^s = prod_p left( 1 - alpha(p)p^-s right)^-1 left( 1 - beta(p)p^-s right)^-1 left( 1 - gamma(p)p^-s right)^-1$$
Straightforwardly developing the Euler product provides expressions of the Fourier coefficients $a_n$'s in terms of the Satake parameters $alpha(p)$, $beta(p)$ and $gamma(p)$. I am not particularly aware of others standard useful relations between them. I bumped into the following one:
$$a_p^k = frac
left
$$
I guess this can be verified, but even the case $k=1$ seems obscure to me. I do not want to believe such a formula to be a (verifiable) accident. Despite it works computationally, am I missing something lying behind? How strongly is the self-contragredience assumption necessary?
Any insight is welcome, as well as other ways to embrace the relations between spectral parameters and coefficients.
nt.number-theory automorphic-forms l-functions
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) = sum_k=0^infty fraca_nn^s = prod_p left( 1 - alpha(p)p^-s right)^-1 left( 1 - beta(p)p^-s right)^-1 left( 1 - gamma(p)p^-s right)^-1$$
Straightforwardly developing the Euler product provides expressions of the Fourier coefficients $a_n$'s in terms of the Satake parameters $alpha(p)$, $beta(p)$ and $gamma(p)$. I am not particularly aware of others standard useful relations between them. I bumped into the following one:
$$a_p^k = frac
left
$$
I guess this can be verified, but even the case $k=1$ seems obscure to me. I do not want to believe such a formula to be a (verifiable) accident. Despite it works computationally, am I missing something lying behind? How strongly is the self-contragredience assumption necessary?
Any insight is welcome, as well as other ways to embrace the relations between spectral parameters and coefficients.
nt.number-theory automorphic-forms l-functions
Let $L(s)$ be an automorphic L-function (attached to a self contragredient automorphic representation on $GL(3)$), according to the following notations for $s$ of sufficiently large real part:
$$L(s) = sum_k=0^infty fraca_nn^s = prod_p left( 1 - alpha(p)p^-s right)^-1 left( 1 - beta(p)p^-s right)^-1 left( 1 - gamma(p)p^-s right)^-1$$
Straightforwardly developing the Euler product provides expressions of the Fourier coefficients $a_n$'s in terms of the Satake parameters $alpha(p)$, $beta(p)$ and $gamma(p)$. I am not particularly aware of others standard useful relations between them. I bumped into the following one:
$$a_p^k = frac
left
$$
I guess this can be verified, but even the case $k=1$ seems obscure to me. I do not want to believe such a formula to be a (verifiable) accident. Despite it works computationally, am I missing something lying behind? How strongly is the self-contragredience assumption necessary?
Any insight is welcome, as well as other ways to embrace the relations between spectral parameters and coefficients.
nt.number-theory automorphic-forms l-functions
nt.number-theory automorphic-forms l-functions
edited 2 hours ago
asked 2 hours ago
Desiderius Severus
2,15121847
2,15121847
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.
The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.
No assumption beyond unramifiedness should be necessary.
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.
The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.
No assumption beyond unramifiedness should be necessary.
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
add a comment |Â
up vote
2
down vote
There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.
The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.
No assumption beyond unramifiedness should be necessary.
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.
The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.
No assumption beyond unramifiedness should be necessary.
There is no coincidence, this is the Weyl character formula for the representation $operatornameSym^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake isomorphism.
The general statement is: For an automorphic representation associated to a group $G$ with dual group $hatG$, the coefficient of $p^k$ in the $L$-function associated to a representation $rho$ of $hatG$ is equal to the trace of the Satake parameter (a conjugacy class on $hatG$) acting on $Sym^k rho$.
No assumption beyond unramifiedness should be necessary.
edited 52 mins ago
Desiderius Severus
2,15121847
2,15121847
answered 58 mins ago
Will Sawin
65.2k6131273
65.2k6131273
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
add a comment |Â
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
Thanks for your great and enlightening answer. Do you have any reference with some more details on why this statement is true?
â Desiderius Severus
52 mins ago
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f312863%2frelation-between-fourier-coefficients-and-satake-parameters%23new-answer', 'question_page');
);
Post as a guest
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
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
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