# Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$

Let $$G$$ be an adjoint semisimple group over $$\mathbb Q$$ with parabolic subgroup $$P = MN$$ in good position relative to a compact subgroup $$U= \prod\limits_v K_v$$ of $$G(\mathbb A)$$. Let $$L$$ be the space of square integrable functions on $$G(\mathbb Q) \backslash G(\mathbb A)$$ which are invariant on the right by $$U$$. In Euler Products, Langlands defines a cusp form in $$L$$ to be an element $$\phi$$ satisfying

$$\int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} \phi(ng)dn = 0\tag{1}$$

for almost all $$g \in G(\mathbb A)$$. However, it is not clear to me why the left hand side converges at all. All we know is that

$$\int\limits_{G(\mathbb Q) \backslash G(\mathbb A)} |\phi(g)|^2 dg < \infty$$ Using the Iwasawa decomposition, we can write $$G(\mathbb A) = N(\mathbb A)M(\mathbb A)K$$, so that, at least formally,

$$\int\limits_{G(\mathbb A)} \phi(g)dg = \int\limits_{M(\mathbb A)} \int\limits_{N(\mathbb A)} \int\limits_K \phi(nmk) \delta_P(m) dk dm dn = \operatorname{vol}(K)\int\limits_{M(\mathbb A)} \int\limits_{N(\mathbb A)}\phi(nm) \delta_P(m)dndm$$

One can probably finagle from here something like:

$$\int\limits_{G(\mathbb Q) \backslash G(\mathbb A)}|\phi(g)|^2 dg = \operatorname{vol}(K)\int\limits_{M(\mathbb Q)\backslash M(\mathbb A)} \int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} |\phi(nm)|^2 \delta_P(m)dn dm$$

from which we should have

$$\int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} |\phi(ng)|^2 dn < \infty$$ for almost all $$g \in G$$. However, this says nothing about the convergence of $$n \mapsto \phi(ng)$$, only of $$n \mapsto |\phi(ng)|^2$$.

• Isn't $\mathrm{N}(\mathbb{Q}) \backslash \mathrm{N}(\mathbb{A})$ compact? – Peter Humphries Aug 14 at 19:59
• Yes, but $\phi$ is not necessarily continuous there – D_S Aug 14 at 21:17
• Hmmm, I guess usually one only considers smooth functions. – Peter Humphries Aug 14 at 21:18
• Maybe Langlands should say the cusp forms are the closure of the space generated by such smooth functions – D_S Aug 14 at 21:19
• It's alright to use arbitrary $L^2$ functions here. Like you wrote, the $L^2$ norm over $N({\mathbb Q})\backslash N({\mathbb A})$ is finite for almost all $g$, then, as the quotient is compact, the constant function is $L^2$, and therefore the inner product of $|\phi|$ and the constant function exists for almost all $g$. – Zero yesterday