Definition of Hecke operators

Question

I am confused about the definition of Hecke operators. It will be great if someone provides some references.

Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the left of $G$ and and let $\tilde\Gamma$ be the commensurator of $\Gamma$. Then we call $\Gamma\alpha\Gamma$ as the hecke operators where $\alpha\in\tilde\Gamma$.

Gelbart's "Automorphic forms on Adele groups' says: Let $K_p=G(\mathbb{Z}_p)$ be maximal compact in $G_p=G(\mathbb{Q}_p)$. Then we call $K_p\alpha K_p$as the hecke operators, where $\alpha$ = $\begin{pmatrix}p & 0\\0 & 1\end{pmatrix}$.

1) Why are above two definitions are equivalent?

2) How do we define Hecke operators in general symmetric spaces $\Gamma\backslash G/K$?

Answer 1

There is a Cartan decomposition for $PGL_2(\mathbb{Q}_p)$ meaning that double coset of the from $PGL_2(\mathbb{Q}_p) //PGL_2(\mathbb{Z}_p)$ are represented by $\alpha^k$, $k\geq0$.

So this is the reason because the definitions are equivalent.

This works well as long as you are working on $GL_2(\mathbb{Z})$ automorphic forms with preassigned behaviour on the center, for congruence subgroups. You need then a more general definition. Also it seems to me that you do not carefully distinguish between the adelic setting and classical setting concerning Lie groups only.

For the general case: The algebra of Hecke operator is $C_c^\infty( G(A_f) //K_f)$ acting by convolution on $L^2( G(\mathbb{Q}) \backslash G(A) /K_f)$ if $K_f$ is an open subgroup in $G(A_f)$ and $\Gamma = K_f \cap G(\mathbb{Q})$. ($A=$adeles, $A_f=$ finite adeles). You have on the level of $G(\mathbb{R})$-reps under certain conditions (see strong approximation) that $L^2( G(\mathbb{Q}) \backslash G(A) /K_f) \cong L^2(\Gamma \backslash G(\mathbb{R}))$.

Answer 2

In order to understand the general philosophy of Hecke algebras, I recommand to do exercice 22 of Bourbaki "Groupes et Algèbres de Lie", Chap. IV.

In this exercice a Hecke algebra is defined in great generality. Take a group $G$ endowed with a subgroup $B$ satisfying the two following equivalent conditions :

(i) $B\cap gBg^{-1}$ is of finite index in $B$ for all $g\in G$;

(ii) any double class $BgB$ is a finite union of left $B$-classes.

Then for any field of coefficients $k$, you may define the Hecke algebra $H_k (G,B)$ as a sub-vector space of the space of $k$-valued functions on $G$.

The exercice shows how to construct a set of generators and compute the structure constants.

Remark. It is funny to notice that a large part of the fondation of abstract building theory is proposed as a series of exercices in this chapter IV.

Answer 3

I think Shimura defines all operators, while Gelbart only defines generators of the algebra, hence one of them has a general $\alpha$ while the other has more specific ones -- this remark should answer your first question. For the second question, I remember Miyake's "Modular forms" has a detailed explanation which I enjoyed reading.