show/hide this revision's text 2 edited body

This is kind of a complicated question, since there isn't really a single good answer.

We begin with a simple Lie group $G$ (for simplicity!). On the one hand, we hopefully have a description of the unitary representations of $G$. On the other hand, we may want to understand how spaces such as $L^2(H\backslash G)$, where $H$ may be trivial or discrete or maximal compact or etc), decompose into unitary representations of $G$ (that it will decompose is known on general (highly nontrivial) principles). At least two issues arise.

First, what does it mean for a representation to "appear" in the decomposition of $L^2(H\backslash G)$? We'd like it to mean that there exists an $f\in L^2(H\backslash G)$ such that $f$ generates the representation. This can't possibly work in general, and it already fails for $L^2(\mathbb R)$. Basically, whenever $H\backslash G$ is not compact, there will be a "continuous" part to the decomposition made up of unitary representations that can't be found as subrepresentations of $L^2(H\backslash G)$. Personally, a priori, it is surprising to me that you can integrate a bunch of stuff not in $L^2$ and wind up with something in $L^2$. But then, I think about Fourier inversion and Paley-Weiner Paley-Wiener theorems, and it's not so surprising. (In fact, if you believe your future will contain a nontrivial amount of harmonic analysis, you should try to become well-acquainted with Fourier theory.)
Now, there are functions on $H\backslash G$ that generate these representations and they usually aren't very far from being in $L^2(H\backslash G)$ (like $e^{ix}$ on $\mathbb R$ and Eisenstein series on $\Gamma\backslash\mathfrak H$), but there really is no way to force them in there. A person might wonder why a benevolent God would allow this to happen, but that is outside of my expertise.

Second, which representations will appear in $L^2(H\backslash G)$? For example, the trivial representation appears in $L^2(H\backslash G)$ if and only if $H\backslash G$ has finite volume. And complementary series representations don't seem to appear at all (usually)! (This is Selberg's Conjecture.)

On a hopefully more helpful note, with certain definitions of a Schwartz space on $H\backslash G$, you can realize these functions as tempered distributions (meaning continuous linear functionals on the Schwartz space). In fact, the space of functions with uniform moderate growth on $\Gamma\backslash \mathfrak H$ contains Eisenstein series and is contained in the dual of the Schwartz space for $\Gamma\backslash\mathfrak H$. See some of Casselman's work, here and here. In a different direction, there is Schmid and Miller's work on automorphic distributions, e.g. here.
show/hide this revision's text 1

This is kind of a complicated question, since there isn't really a single good answer.

We begin with a simple Lie group $G$ (for simplicity!). On the one hand, we hopefully have a description of the unitary representations of $G$. On the other hand, we may want to understand how spaces such as $L^2(H\backslash G)$, where $H$ may be trivial or discrete or maximal compact or etc), decompose into unitary representations of $G$ (that it will decompose is known on general (highly nontrivial) principles). At least two issues arise.

First, what does it mean for a representation to "appear" in the decomposition of $L^2(H\backslash G)$? We'd like it to mean that there exists an $f\in L^2(H\backslash G)$ such that $f$ generates the representation. This can't possibly work in general, and it already fails for $L^2(\mathbb R)$. Basically, whenever $H\backslash G$ is not compact, there will be a "continuous" part to the decomposition made up of unitary representations that can't be found as subrepresentations of $L^2(H\backslash G)$. Personally, a priori, it is surprising to me that you can integrate a bunch of stuff not in $L^2$ and wind up with something in $L^2$. But then, I think about Fourier inversion and Paley-Weiner theorems, and it's not so surprising. (In fact, if you believe your future will contain a nontrivial amount of harmonic analysis, you should try to become well-acquainted with Fourier theory.)
Now, there are functions on $H\backslash G$ that generate these representations and they usually aren't very far from being in $L^2(H\backslash G)$ (like $e^{ix}$ on $\mathbb R$ and Eisenstein series on $\Gamma\backslash\mathfrak H$), but there really is no way to force them in there. A person might wonder why a benevolent God would allow this to happen, but that is outside of my expertise.

Second, which representations will appear in $L^2(H\backslash G)$? For example, the trivial representation appears in $L^2(H\backslash G)$ if and only if $H\backslash G$ has finite volume. And complementary series representations don't seem to appear at all (usually)! (This is Selberg's Conjecture.)

On a hopefully more helpful note, with certain definitions of a Schwartz space on $H\backslash G$, you can realize these functions as tempered distributions (meaning continuous linear functionals on the Schwartz space). In fact, the space of functions with uniform moderate growth on $\Gamma\backslash \mathfrak H$ contains Eisenstein series and is contained in the dual of the Schwartz space for $\Gamma\backslash\mathfrak H$. See some of Casselman's work, here and here. In a different direction, there is Schmid and Miller's work on automorphic distributions, e.g. here.