About definition of the global Hecke algebra and the automorphic form

Question

Introduction

I am trying to understand various definitions (which I hope them equivalent) of automorphic forms.

My main problem is to describe with the global Hecke algebra rather than the group and the universal enveloping algebra.

My main purpose is to understand the contents of [GJ] or [JL].

Let $G$ be a connected reductive group over a global field $F$.

the Global Hecke Algebra

Firstly, I introduce the definition of the global Hecke algebra. There is no problem to define the global Hecke algebra as the restricted tensor product of local Hecke algebras, and there is no problem to define the local Hecke algebra for a non-Archimedean place. The problem is how to define the Archimedean parts. Let $v$ be an Archimedean place and $K_v$ is a maximal compact subgroup of $G_v$

$\mathcal{H}_v(G_v,K_v)$ is defined to be the convolution algebra of distributions of $G_v$ supported on $K_v$ ([Ge] Def.3.6), and with additionally the $K_v$-finiteness condition. (one can show left finiteness $\Leftrightarrow$ right finiteness) ([Bu] p311, "by Flath")

I notice that to be connected with the universal enveloping algebra one need the following structure theorem:

$\mathcal{H}(G_v,K_v)\simeq \mathcal{H}(K_v)\otimes_{U(\mathfrak{k}_\mathbb{C})}U(\mathfrak{g}_\mathbb{C})$. ([Ge] p28, 5.4; [Bu] p311; [KV] p71, Cor.1.71)

Remark: I think the $K_v$-finiteness condition is necessary. My question is: is this definition which we use most?

the Automorphic Forms

Secondly, (assume the definition of the global Hecke algebra is correct) I am now give some definitions of automorphic forms on $G(\mathbb{A})$ from different books.

Let $\phi: G(\mathbb{A}) \to \mathbb{C}$ be a complex valued function on $G(\mathbb{A})$, and $\omega: F^\times\backslash\mathbb{A}^\times\to U(1)$ a fixed unitary idelic character. We introduce the following conditions:

(LI): (left invirance)$\phi(\gamma g)=\phi(g)$ for all $\gamma \in G(F)$ and $g \in G(\mathbb{A})$.

(CC): (central unitary character) $\phi(ag)=\omega(a)\phi(g)$ for all $a \in Z(G(\mathbb{A}))$ and $g \in G(\mathbb{A})$.

(SM): (smoothness) $\phi$ is smooth.

(GC): (growth condition) If $F$ is a number field then $\phi$ is of moderate growth/ slowly increasing.

(KF): ($K$-finiteness) $\phi$ is right $K-$ finite.

Above are the common conditions of different versions.

(ZF): ($\mathcal{Z}-$finiteness) $\phi$ is $\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})-$ finite.

(AD): (admissibility) The $\mathcal{H}-$space $V$ which is the $\mathcal{H}-$ spanning of $\phi$ is admissible, i.e., for any elementary idempotent element $\xi$ in $\mathcal{H}$, $\xi V$ is finite dimensional.

(ND): (non-degeneration) There is an elementary idempotent element $\xi$ in $\mathcal{H}$, $\xi\phi=\phi$.

(IZ): (ideal of $\mathcal{Z}$) This is a (finite-codimensional [?]) ideal of $\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$ that annihilates $\phi$.

Then I give three version of additional conditions for automorphic forms:

Version 1: (ZF). ([GF] Def.4.7.6; [Bu] p299)

Version 2: (AD). ([GJ] p145-146; [JL] Def.10.2)

Version 3: (ND) + (IZ). ([Yo] Def.4.1; [Ge] Def.6.12)

Remark: I think the finite-codimensional condition is needed in (IZ). My main interest is to show Version 2 is equivalent to Version 1, and to show Version 2 implies (ND) (since it is a very useful condition). However I am really not familiar with the Archimedean case and also I am a little worried when seeing such many different but expectedly equivalent definitions. So I wish help for the explicit problem and also advise on how to deal with it.

Reference

[GJ] Zeta functions of simple algebras, Godement, Jacquet.

[JL] Automorphic Forms on $GL(2)$ ,Jacquet, Langlands.

[GH] Automorphic Representations and L-Functions for the General Linear Group, Volume 1 , Goldfeld, Hundley.

[Ge] An Introduction to Automorphic Representations, (course notes), Getz.

[Bu] Automorphic Forms and Representations, Bump.

[KV] Cohomological Induction and Unitary Representations, Knapp, Vogan.

[Yo] 保型形式論-現代整数論講義, Yoshida.

Answer 1

A bit of information, though (as @Kimball comments) you are asking a very broad question... :

As you seem to be aware, the equivalence between smooth repns of p-adic groups and repns of the (full) Hecke algebra is not hard to prove. Ok. In particular, irreducibles of one are irreducibles of the other.

In contrast, formulating the archimedean analogue is not so easy to do in a manner that captures what we want, either for the local repn theory, or for its applications to automorphic forms. One difference is that there are differential operators in the archimedean case, while in the p-adic case there's nothing "transverse to support". At the same time, sure, test functions, or $K$-finite test functions, on the group do act, and really give all the information we truly need. But this needs a group action, and in the archimedean case the group action wrecks $K$-finiteness. For other reasons as well, as Harish-Chandra discovered, we sometimes prefer $\mathfrak g,K$ modules to $G$-modules. The differentiation action of the universal enveloping algebra is the same as "convolution" action of distributions supported at the identity. We also want to be able to talk about the $K$ action, and the smallest associative algebra containing both is indeed the algebra of $K$-finite distributions supported on $K$. Some little theorems in analysis are probably necessary to make this absolutely clear.

Perhaps annoyingly, for a person who has invested lots of energy in understanding such things, it seems that we rarely refer to this for any purpose, in practice. That is, that attempt at some sort of unifying formalism seems to have more an aesthetic goal than a utilitarian one. To talk about "automorphic repns" usually is a very small distance away from assuming that $\pi$ is an irreducible ... afc repn". So the various equivalent ways of talking about larger repns do not necessarily arise.

Yes, there is the issue of recovering the $G$-repns from $\mathfrak g,K$-repns, called "globalization". There is a range of "globalizations", discussed (independently) by Casselman and Wallach. This is a pretty tricky business, in my opinion.

The number-theoretically significant issue of expressing a fairly arbitrary (generalized?) function on $G_k\backslash G_{\mathbb A}$, or similar, is much more complicated than those set-up, aesthetic issues. E.g., in the simplest possible case, Fourier series on a circle, pointwise convergence is already subtle, not to mention Fourier series of distributions, Sobolev inequalities, and so on.

To recapitulate: I have the impression that those definitions are not too much in peoples' minds while doing actual work in automorphic forms. Either people say "let $\pi$ be an irreducible afc repn", or assume that spectral decompositions of "Poincare series" really do work as well as they hope. :) (In any case, until we see an interesting outcome, there's not so much interest in worry about filling in technical details. After all, for automorphic forms, in contrast to Fourier series on the circle, giving reasonable sup-norm estimates on eigenfunctions for the Laplacian is highly non-trivial already! A rough foundational situation! :)s