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 $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are two equivalent 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.

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.

  • $\begingroup$ I haven't thought this through thoroughly but I think the answer is yes. The idea is that the full Hecke algebra of a restricted direct product of locally compact totally disconnected groups is the tensor product of the local Hecke algebras. $\endgroup$ – Grad student Jan 2 at 2:47
  • $\begingroup$ Restricted tensor product of the local Hecke algebras, I mean. $\endgroup$ – Grad student Jan 2 at 2:56
  • $\begingroup$ Being able to speak of $G(\mathscr O_v)$ is a matter of choosing an $\mathscr O_v$-structure on $G$, not an embedding $G \to \mathrm{GL}_n$. (Think, for example, of how you genuinely need a real structure, not just a (complex) embedding on a general linear group, on a complex group $G$ before you can speak of $G(\mathbb R)$.) $\endgroup$ – LSpice Jan 2 at 3:01
  • $\begingroup$ Also, although it's just a wording issue, it's not clear to me what makes the second definition intrinsic and the first one not. $\endgroup$ – LSpice Jan 2 at 3:03
  • 3
    $\begingroup$ Yes. Your two definitions of $C^\infty(G({\mathbb A}))$ are equivalent, and with a compact support condition, equivalent to the third. The exercise is to prove that, given a closed embedding of $k$-groups from $G$ to $GL_n$, the subspace topology on $G({\mathbb A})$ from $GL_n({\mathbb A})$ coincides with the restricted direct product topology, with respect to the open compact subgroups $G(K_v) \cap GL_n(O_v)$. $\endgroup$ – Marty Jan 2 at 5:05

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