What does it mean for a complex valued function on $G(mathbb A)$ to be smooth (or smooth of compact support)?
$begingroup$
Let $G$ be a linear algebraic group over a number field $k$. Let $mathbb A$ denote the adeles of $k$, $mathbb A_f$ the finite adeles, and $k_{infty} = prodlimits_{v mid infty} k_v$. Here are a couple of possible definitions for a function $phi: G(mathbb A) rightarrow mathbb C$ to be "smooth."
$phi$ is smooth if, writing $G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, $phi$ is smooth (infinitely differentiable) in the first variable and smooth (locally constant) in the second variable. I saw this definition here.
$phi$ is smooth if for every $g in G(mathbb A)$, there exists an open neighborhood $V$ of $g$ and a smooth (infinitely differentiable) function $phi_g: G(k_{infty}) rightarrow mathbb C$ such that for all $h = (h_{infty}, h_f) in V subset G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, we have $phi(h) = phi_g(h_{infty})$. I saw this definition in Daniel Bump's book, Automorphic Forms and Representations, page 299.
My first question is: are these two notions of smooth the same? It looks like the second definition implies the first one. But for the first definition to imply the second, we would have to say something about when smooth functions on open sets in $G(k_{infty})$ can be extended to ones on the whole $G(k_{infty})$.
Let me take the first definition of smooth as the correct one. Choose an embedding $G rightarrow operatorname{GL}_n$, so we can talk about $G(mathcal O_v) := G(k_v) cap operatorname{GL}_n(mathcal O_v)$. Denote the space of smooth functions on $G(mathbb A)$ by $C^{infty}(G(mathbb A))$. Let $C_c^{infty}(G(mathbb A))$ be the space of $phi in C^{infty}(G(mathbb A))$ which vanish outside of a compact set in $G(mathbb A)$. There is another definition of $C_c^{infty}(G(mathbb A))$ which I have seen before (see my previous question here). Namely:
$C_c^{infty}(G(mathbb A))$ is defined to be the space of finite linear combinations of "factorizable" functions $prod_v f_v$ for $f_v: G(k_v) rightarrow mathbb C$ smooth (infinitely differentiable or locally constant, depending on whether $v$ is infinite or finite), where $f_v = operatorname{Char} G(mathcal O_v)$ for almost all places $v$.
My second question is: are these two definitions of $C_c^{infty}(G(mathbb A))$ the same? The first one defines it as a subspace of $C^{infty}(G(mathbb A))$, and the second defines it intrinsically as combinations of factorizable functions.
number-theory reductive-groups adeles langlands-program
asked Jan 2 at 1:35
D_SD_S
13.7k61552
$endgroup$
add a comment |
$begingroup$
Let $G$ be a linear algebraic group over a number field $k$. Let $mathbb A$ denote the adeles of $k$, $mathbb A_f$ the finite adeles, and $k_{infty} = prodlimits_{v mid infty} k_v$. Here are a couple of possible definitions for a function $phi: G(mathbb A) rightarrow mathbb C$ to be "smooth."
$phi$ is smooth if, writing $G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, $phi$ is smooth (infinitely differentiable) in the first variable and smooth (locally constant) in the second variable. I saw this definition here.
$phi$ is smooth if for every $g in G(mathbb A)$, there exists an open neighborhood $V$ of $g$ and a smooth (infinitely differentiable) function $phi_g: G(k_{infty}) rightarrow mathbb C$ such that for all $h = (h_{infty}, h_f) in V subset G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, we have $phi(h) = phi_g(h_{infty})$. I saw this definition in Daniel Bump's book, Automorphic Forms and Representations, page 299.
My first question is: are these two notions of smooth the same? It looks like the second definition implies the first one. But for the first definition to imply the second, we would have to say something about when smooth functions on open sets in $G(k_{infty})$ can be extended to ones on the whole $G(k_{infty})$.
Let me take the first definition of smooth as the correct one. Choose an embedding $G rightarrow operatorname{GL}_n$, so we can talk about $G(mathcal O_v) := G(k_v) cap operatorname{GL}_n(mathcal O_v)$. Denote the space of smooth functions on $G(mathbb A)$ by $C^{infty}(G(mathbb A))$. Let $C_c^{infty}(G(mathbb A))$ be the space of $phi in C^{infty}(G(mathbb A))$ which vanish outside of a compact set in $G(mathbb A)$. There is another definition of $C_c^{infty}(G(mathbb A))$ which I have seen before (see my previous question here). Namely:
$C_c^{infty}(G(mathbb A))$ is defined to be the space of finite linear combinations of "factorizable" functions $prod_v f_v$ for $f_v: G(k_v) rightarrow mathbb C$ smooth (infinitely differentiable or locally constant, depending on whether $v$ is infinite or finite), where $f_v = operatorname{Char} G(mathcal O_v)$ for almost all places $v$.
My second question is: are these two definitions of $C_c^{infty}(G(mathbb A))$ the same? The first one defines it as a subspace of $C^{infty}(G(mathbb A))$, and the second defines it intrinsically as combinations of factorizable functions.
number-theory reductive-groups adeles langlands-program
asked Jan 2 at 1:35
D_SD_S
13.7k61552
$endgroup$
add a comment |
$begingroup$
Let $G$ be a linear algebraic group over a number field $k$. Let $mathbb A$ denote the adeles of $k$, $mathbb A_f$ the finite adeles, and $k_{infty} = prodlimits_{v mid infty} k_v$. Here are a couple of possible definitions for a function $phi: G(mathbb A) rightarrow mathbb C$ to be "smooth."
$phi$ is smooth if, writing $G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, $phi$ is smooth (infinitely differentiable) in the first variable and smooth (locally constant) in the second variable. I saw this definition here.
$phi$ is smooth if for every $g in G(mathbb A)$, there exists an open neighborhood $V$ of $g$ and a smooth (infinitely differentiable) function $phi_g: G(k_{infty}) rightarrow mathbb C$ such that for all $h = (h_{infty}, h_f) in V subset G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, we have $phi(h) = phi_g(h_{infty})$. I saw this definition in Daniel Bump's book, Automorphic Forms and Representations, page 299.
My first question is: are these two notions of smooth the same? It looks like the second definition implies the first one. But for the first definition to imply the second, we would have to say something about when smooth functions on open sets in $G(k_{infty})$ can be extended to ones on the whole $G(k_{infty})$.
Let me take the first definition of smooth as the correct one. Choose an embedding $G rightarrow operatorname{GL}_n$, so we can talk about $G(mathcal O_v) := G(k_v) cap operatorname{GL}_n(mathcal O_v)$. Denote the space of smooth functions on $G(mathbb A)$ by $C^{infty}(G(mathbb A))$. Let $C_c^{infty}(G(mathbb A))$ be the space of $phi in C^{infty}(G(mathbb A))$ which vanish outside of a compact set in $G(mathbb A)$. There is another definition of $C_c^{infty}(G(mathbb A))$ which I have seen before (see my previous question here). Namely:
$C_c^{infty}(G(mathbb A))$ is defined to be the space of finite linear combinations of "factorizable" functions $prod_v f_v$ for $f_v: G(k_v) rightarrow mathbb C$ smooth (infinitely differentiable or locally constant, depending on whether $v$ is infinite or finite), where $f_v = operatorname{Char} G(mathcal O_v)$ for almost all places $v$.
My second question is: are these two definitions of $C_c^{infty}(G(mathbb A))$ the same? The first one defines it as a subspace of $C^{infty}(G(mathbb A))$, and the second defines it intrinsically as combinations of factorizable functions.
number-theory reductive-groups adeles langlands-program
asked Jan 2 at 1:35
D_SD_S
13.7k61552
$endgroup$
Let $G$ be a linear algebraic group over a number field $k$. Let $mathbb A$ denote the adeles of $k$, $mathbb A_f$ the finite adeles, and $k_{infty} = prodlimits_{v mid infty} k_v$. Here are a couple of possible definitions for a function $phi: G(mathbb A) rightarrow mathbb C$ to be "smooth."
$phi$ is smooth if, writing $G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, $phi$ is smooth (infinitely differentiable) in the first variable and smooth (locally constant) in the second variable. I saw this definition here.
$phi$ is smooth if for every $g in G(mathbb A)$, there exists an open neighborhood $V$ of $g$ and a smooth (infinitely differentiable) function $phi_g: G(k_{infty}) rightarrow mathbb C$ such that for all $h = (h_{infty}, h_f) in V subset G(mathbb A) = G(k_{infty}) times G(mathbb A_f)$, we have $phi(h) = phi_g(h_{infty})$. I saw this definition in Daniel Bump's book, Automorphic Forms and Representations, page 299.
My first question is: are these two notions of smooth the same? It looks like the second definition implies the first one. But for the first definition to imply the second, we would have to say something about when smooth functions on open sets in $G(k_{infty})$ can be extended to ones on the whole $G(k_{infty})$.
Let me take the first definition of smooth as the correct one. Choose an embedding $G rightarrow operatorname{GL}_n$, so we can talk about $G(mathcal O_v) := G(k_v) cap operatorname{GL}_n(mathcal O_v)$. Denote the space of smooth functions on $G(mathbb A)$ by $C^{infty}(G(mathbb A))$. Let $C_c^{infty}(G(mathbb A))$ be the space of $phi in C^{infty}(G(mathbb A))$ which vanish outside of a compact set in $G(mathbb A)$. There is another definition of $C_c^{infty}(G(mathbb A))$ which I have seen before (see my previous question here). Namely:
$C_c^{infty}(G(mathbb A))$ is defined to be the space of finite linear combinations of "factorizable" functions $prod_v f_v$ for $f_v: G(k_v) rightarrow mathbb C$ smooth (infinitely differentiable or locally constant, depending on whether $v$ is infinite or finite), where $f_v = operatorname{Char} G(mathcal O_v)$ for almost all places $v$.
My second question is: are these two definitions of $C_c^{infty}(G(mathbb A))$ the same? The first one defines it as a subspace of $C^{infty}(G(mathbb A))$, and the second defines it intrinsically as combinations of factorizable functions.
number-theory reductive-groups adeles langlands-program
number-theory reductive-groups adeles langlands-program
asked Jan 2 at 1:35
D_SD_S
13.7k61552
asked Jan 2 at 1:35
D_SD_S
13.7k61552
asked Jan 2 at 1:35
D_SD_S
13.7k61552
asked Jan 2 at 1:35
D_SD_S
13.7k61552
asked Jan 2 at 1:35
D_SD_S
13.7k61552
13.7k61552
add a comment |
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%2f3059060%2fwhat-does-it-mean-for-a-complex-valued-function-on-g-mathbb-a-to-be-smooth%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%2f3059060%2fwhat-does-it-mean-for-a-complex-valued-function-on-g-mathbb-a-to-be-smooth%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
