A stupid question about Automorphic forms

Question

Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions:

(a) invariance with respect to left $\Gamma -$ translations.

(b) Right $K -$ finiteness.

(c) Annihilated with respect to a finite co-dimension ideal of the center $Z(\mathcal{U}(\mathfrak{g}_{\mathbb{C}}))$ ( of the complexified universal enveloping algebra).

(d) Growth conditions.

Now as I understand, it is clear why condition (a) is relevant. Condition (b) comes from the idea that representation of $G$ associated to $f$ is admissible.

My understanding is that condition (d) ensures that we can "extend" the automorphic form to the cusps of the symmetric space.

I would like to know: what exactly is the intuition behind condition (c)? (I hope it is not just to provide a framework which includes classical modular forms over upper half-plane and the Maass forms.)

Also any comments about "my understanding" of conditions (a), (b) and (d) are appreciated!

Answer 1

These conditions have technical/subtle interactions. It probably suffices to think about automorphic forms on a Lie group, and not think of the interaction with finite places.

For example, on each K-isotype, the Casimir element is an elliptic operator, but it is not elliptic without specifying the behavior under K. From elliptic regularity, such eigenfunctions are real-analytic.

With K-finite and z-finite, moderate growth implies that all derivatives are of moderate growth, etc. (Borel's little book talks about this for SL(2,R), and his Park City notes recapitulate this. The general argument is similar to SL(2,R), anyway.)

K-finite, z-finite, moderate-growth cuspforms are (up to adjustment of central character) provably of rapid decay, so certainly L^2. This and the previous assertion are part of the "theory of the constant term".

Dropping the moderate growth condition is useful occasionally, as in the "weak Maass form" business.

In a more elementary vein, a choice to require annihilation by a finite-codimension ideal in z is akin to looking at functions on the real line annihilated by a product (D^2-lambda_1)...(D^2-lambda_n), namely, polynomial multiples of exponentials. Certainly not every function on the line is annihilated by such, but the spectral theory (Fourier transform) decomposes a general function into a superposition of certain of these special functions. (Here the analogue of K is {1}.)

Answer 2

I am no expert in the general theory, but let me share some thoughts. In some sense (c) accounts for the fact that $K$ does not have enough open subgroups (unlike at nonarchimedan places), so that the usual convolution definition of Hecke operators is too crude. Instead, one needs to work with the convolution algebra of all distributions on $G$ with support in $K$, i.e. with $U(\mathfrak{g})\otimes_{U(\mathfrak{k})}A_K$, where $\mathfrak{k}$ is the Lie algebra of $K$ and $A_K$ is the convolution algebra of finite measures on $K$. For compatibility with (b) one needs to restrict to elements of $U(\mathfrak{g})$ which commute with elements of $\mathfrak{k}$, and a simple way to achieve this is to restrict to $Z(U(\mathfrak{g}))$. In this way one can obtain a unified treatment at all places for which the adelic framework is most suitable. For example, a Hecke cusp form on $\mathrm{GL}_n$ can be characterized by $n$ parameters at each place: these parameters define the local $L$-functions whose product is the global $L$-function.

For a special case see Section 2.3 in Goldfeld: Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$. For a brief discussion of the general theory see Borel-Jacquet: Automorphic forms and automorphic representations, Proc. Symp. Pure Math. 33 (1979), 189-202.