nt.number theory - Relation between Hecke Operator and Hecke Algebra

The fact that Hecke operators (double coset stuff coming from $SL_2(mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL_2(mathbf{Q}_p)$) are related has nothing really to do with the Satake isomorphism. The crucial observation is that instead of thinking of modular forms as functions on the upper half plane, you can think of them as functions on $GL_2(mathbf{R})$ which transform in a certain way under a subgroup of $GL_2(mathbf{Z})$, and then as functions on $GL_2(mathbf{A})$ ($mathbf{A}$ the adeles) which are left invariant under $GL_2(mathbf{Q})$ and right invariant under some compact open subgroup of $GL_2(widehat{mathbf{Z}})$.

Now there's just some general algebra yoga which says that if $H$ is a subgroup of $G$ and $f$ is a function on $G/H$, and $gin G$ such that the $HgH$ is a finite union of cosets $g_iH$, then you can define a Hecke operator $T=[HgH]$ acting on the functions on $G/H$, by $Tf(g)=sum_i f(gg_i)$; the lemma is that this is still $H$-invariant.

Next you do the tedious but entirely elementary check that if you consider modular forms not as functions on the upper half plane but as functions on $GL_2(mathbf{A})$, then the classical Hecke operators have interpretations as operators $T=[HgH]$ as above, with $T_p$ corresponding to the function supported at $p$ and with $g=(p,0;0,1)$. Because the action is "all going on locally" you may as well compute the double coset space locally, that is, if $H=H^pH_p$ with $H_p$ a compact open subgroup of $GL_2(mathbf{Q}_p)$, then you can do all your coset decompositions and actions locally at $p$.

Now finally you have your link, because you can think of $T$ as being the characteristic function of the double coset space $HgH$ which is precisely the sort of Hecke operator in your Hecke algebra of locally constant functions. Furthermore the sum $f(gg_i)$ is just an explicit way of writing convolution, so everything is consistent.

I don't know a book that explains how to get from the classical to the adelic point of view in a nice low-level way, but I am sure there will be some out there by now. Oh---maybe Bump?

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