# Hecke algebra $\mathcal H(\operatorname{GL}_2(\mathbb Q_p)/\operatorname{GL}_2(\mathbb Z_p))$ and Hecke operators

I was reading James Cogdell's notes here on automorphic representations and came to the following claim about the spherical Hecke algebra $$\mathcal H(\operatorname{GL}_2(\mathbb Q_p), \operatorname{GL}_2(\mathbb Z_p))$$.

He remarks that this Hecke algebra identifies with the algebra spanned by the classical Hecke operators $$T_{p^r}$$.

I have read about Hecke operators in Diamond and Shurman's book on modular forms, but have never seen a connection with the $$p$$-adic Hecke algebra. What is the connection between the classical Hecke operators $$T_{p^r}$$ and the spherical Hecke algebra?

• You can always look at Bump's book for GL_2 things like this. Or Deligne's "Formes Modulaires" article in the Antwerp volume II (Modular Forms of One Variable II, 1973). But for a more general picture, I learned these things from Gross's paper "On the Satake Isomorphism". See math.harvard.edu/~gross/preprints/sat.pdf, especially Sections 1,2,3,5,8. Sep 23, 2018 at 21:14
• Another possibility to learn these things: Diamond-Im, Modular forms and modular curves, section 11 and the references therein. Sep 23, 2018 at 21:17

Let us consider the case when the centre acts trivially, for simplicity. Let us call $$H_p$$ to be the polynomial algebra generated by the classical Hecke operators $$\{T_{p^r}\mid r\ge 0\}$$. Using the Hecke multiplicativity relations one can say that $$H_p$$ is a polynomial algebra generated by $$T_p$$. Let $$K_p:=\mathrm{PGL}_2(\mathbb{Z}_p)$$ and $$G_p:=\mathrm{PGL}_2(\mathbb{Q}_p)$$ Consider the map $$\varphi: \mathcal{H}(K_p\backslash G_p/K_p)\to H_p,$$ $$S_p:=\frac{1}{\mathrm{Vol}(K_p\begin{pmatrix}p&\\ &1\end{pmatrix}K_p)}\mathrm{Char}_{K_p\begin{pmatrix}p&\\ &1\end{pmatrix}K_p}\mapsto T_p.$$ By a theorem of Satake $$S_p$$ generates $$\mathcal{H}$$ polynomially, and $$\varphi$$ is an algebra isomorphism. Using elementary divisor theorem it is not very difficult to find preimage of $$T_{p^r}$$ under $$\varphi$$, as well.
To have more geometric understanding you may think $$G_p/K_p$$ as a $$p+1$$-regular tree. One can find a coset representatives of $$K_p\begin{pmatrix}p&\\ &1\end{pmatrix}K_p\bigg/K_p$$ to be $$\{\begin{pmatrix}1&b\\ &p\end{pmatrix}\mid b\in \mathbb{Z}/p\mathbb{Z},\begin{pmatrix}p&\\ &1\end{pmatrix}\}.$$ These cosets correspond to the $$p+1$$ branches of the identity in the tree, and then $$S_p$$ can be thought as an averaging operator over these branches. If you recall the definition of the classical Hecke operator $$T_p$$ on the upper half plane, say at a point $$z$$, you will see that that also averages over the points $$\{\begin{pmatrix}1&b\\ &p\end{pmatrix}.z\mid b=0,\dots,p-1,\begin{pmatrix}p&\\ &1\end{pmatrix}.z\}.$$ Hope this describes the connection between two descriptions of the $$p$$-adic Hecke algebra.