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What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices of $A_1$ dynkin diagram with suitable coxeter relations?

Disclaimer

I am aware of a bunch of similar questions e.g. this mathoverflow post (Relation between Hecke Operator and Hecke Algebra). However, most of them are written in $p$-adic or even adelic language, which I am not familiar with.

Background

I understand that Hecke algebras have at least three faces:

  1. that arising classically from the space of lattices
  2. that arising from double-coset consideration
  3. that arising from the combinatorial structure of Dynkin diagrams

In particular, the first one can be obtained from the second with suitable choice of pair of groups $H \subset G$.

Questions

  1. What is a suitable choice of pair of groups? Is $G=SL(2,R)$ and $H=SL(2,Z)$ one of them? How about $G=GL(2,R)$ and $H=GL(2,Z)$?

  2. The group $G=SL(2,R)$ is roughly of type $A_1$, so I expect the Hecke operators to form a Hecke algebra of type $A_1$ in some sense (affine?). However, naive comparison denies my expectations: there are only two generators in the third picture, whereas in the first picture, we have infinitely many generators labeled by prime numbers and nonzero complex numbers. How to connect these? Is the consideration of $p$-adic groups necessary?

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  • $\begingroup$ Asking for an explanation of Hecke algebras without using p-adic language is like asking for "Hamlet" without the prince. P-adic groups are fundamental to the theory -- live with it. $\endgroup$
    – David Loeffler
    Commented Nov 11, 2019 at 12:24
  • $\begingroup$ I agree. In fact, I hope an answer to this question will lead me into the $p$-adic world. $\endgroup$
    – Student
    Commented Nov 11, 2019 at 12:27

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