# The meaning of $L_{\chi}^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$

I'm reading James Arthur's notes on the trace formula and am confused on a point on pages 65 and 66. For $$G$$ a reductive group over $$\mathbb Q$$ we are going over the decomposition of the space $$L^2(G(\mathbb Q) \backslash G(\mathbb A))$$. First, I'm a bit surprised we are not going modulo center or modulo $$A_G(\mathbb R)^{\circ}$$ and wonder if Arthur means this space to be left $$G(\mathbb Q$$)-invariant functions which are square integrable on $$G(\mathbb Q) A_G(\mathbb R)^{\circ} \backslash G(\mathbb A)$$, rather than on $$G(\mathbb Q) \backslash G(\mathbb A)$$. If so, then $$L^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$$ would then be isomorphic to the subspace of functions in $$L^2(G(\mathbb Q) \backslash G(\mathbb A))$$ which are moreover $$A_G(\mathbb R)^{\circ}$$-invariant.

Anyway, whatever is meant by $$L^2(G(\mathbb Q) \backslash G(\mathbb A))$$, Arthur says it decomposes as a direct sum of subspaces $$L^2_{\chi}(G(\mathbb Q) \backslash G(\mathbb A))$$, as $$\chi$$ runs through the cuspidal automorphic data. Here $$\chi$$ is an equivalence class of pairs $$(P,\sigma)$$, where $$P = MN$$ is a standard parabolic subgroup of $$G$$, and $$\sigma$$ is a cuspidal automorphic representation of $$M(\mathbb A)^1$$. Another pairs $$(P',\sigma')$$ is equivalent to $$(P,\sigma)$$ if there is a $$w$$ in the Weyl group such that $$wMw^{-1} = M'$$ and $$w(\sigma) \cong \sigma'$$ as representations of $$M'(\mathbb A)^1$$.

The space $$L^2_{\chi}(G(\mathbb Q) \backslash G(\mathbb A))$$ is the closed subspace of $$L^2(G(\mathbb Q) \backslash G(\mathbb A))$$ generated by pseudo-Eisenstein series

$$E \psi(g) = \sum\limits_{\delta \in P(\mathbb Q) \backslash G(\mathbb Q)}\psi(\delta g)$$ where $$\psi: N(\mathbb A)M(\mathbb Q)\backslash G(\mathbb A) \rightarrow \mathbb C$$ is a function obtained by Mellin transform

$$\psi(x) = \int\limits_{\mathfrak a_P^{\ast}} e^{\langle \lambda + \rho_P, H_P(x) \rangle} \Psi(\lambda ,x) \space d \lambda$$

of a Paley-Weiner section of an induced space $$\mathcal H_{P,\sigma}$$ (definition on page 65). So, from

$$L^2(G(\mathbb Q) \backslash G(\mathbb A)) = \bigoplus\limits_{\chi} L^2_{\chi}(G(\mathbb Q) \backslash G(\mathbb A))$$

Arthur says we should use the obvious analogous decomposition

$$L^2(G(\mathbb Q) \backslash G(\mathbb A)^1) = \bigoplus\limits_{\chi} L^2_{\chi}(G(\mathbb Q) \backslash G(\mathbb A)^1).$$

My questions are:

1 . What is meant by $$L^2_{\chi}(G(\mathbb Q) \backslash G(\mathbb A)^1)$$? Does this mean the closed subspace of $$L^2(G(\mathbb Q) \backslash G(\mathbb A)^1)$$ generated by Pseudo-Eisenstein series $$E\psi(g)|G(\mathbb A)^1$$ which are restricted to $$G(\mathbb A)^1$$?

2 . For a much more basic question, what is meant by $$L^2(G(\mathbb Q) \backslash G(\mathbb A))$$? Is it the plain meaning, of measurable $$G(\mathbb Q)$$-invariant functions on $$G(\mathbb A)$$ which are square-integrable on $$G(\mathbb Q) \backslash G(\mathbb A)$$? Or is it the space of measurable $$G(\mathbb Q)$$-invariant functions which are square-integrable on $$G(\mathbb Q) A_G(\mathbb R)^{\circ} \backslash G(\mathbb A)$$?

Arthur doesn't work mod centre or $$A_G(\mathbb R)^0$$ because he works with $$G(\mathbb A)^1$$ instead. For a nice explanation of the subtle difference between these choices, take a look at Section 6 of Knapp's Theoretical Aspects of the Trace Formula for GL(2). Essentially, it comes down to the nonuniqueness of extending characters from $$Z(\mathbb A)^1 G(F)$$ to $$Z(\mathbb A)G(F)$$, both trivial on $$G(F)$$. With this in mind, then the $$L^2$$-spaces under consideration are really as the notation suggests, and hopefully this should clarify both your questions.
• So $L^2(G(\mathbb Q) \backslash G(\mathbb A))$ is really as the notation suggests, and consists of all measureable functions on $G(\mathbb A)$ which are $G(\mathbb Q)$-invariant and satisfy $\int\limits_{G(\mathbb Q) \backslash G(\mathbb A)} |f(g)|^2 dg < \infty$? This seems strange, since $G(\mathbb Q) \backslash G(\mathbb A)$ doesn't usually have finite volume.
• I meant that one should always work with $L^2(G(\mathbb Q)\backslash G(\mathbb A)^1)$ or something like $L^2(Z(\mathbb A)G(\mathbb Q)\backslash G(\mathbb A))$ for the reasons you mentioned. Sep 9, 2022 at 16:04