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 $\Gamma\backslash G/K$?



Definition of Hecke operators
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 $L^2( G(\mathbb{Q}) \backslash G(A) /K\_f) \cong L^2(\Gamma \backslash G(\mathbb{R}))$.


