# What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?

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.

• 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. – Grad student Jan 2 at 2:47
• Restricted tensor product of the local Hecke algebras, I mean. – Grad student Jan 2 at 2:56
• 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 $G(\mathbb R)$.) – LSpice Jan 2 at 3:01
• 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. – LSpice Jan 2 at 3:03
• 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 $G(K_v) \cap GL_n(O_v)$. – Marty Jan 2 at 5:05