# Analogues of Hecke relations for Maass forms

If a (suitably normalised) holomorphic cusp newform has q-expansion $$f(z) = \sum_n \lambda_f(n) e(nz),$$

then we know the Hecke relations for $$(mn,q)=1$$, $$(\star) \qquad \lambda_f(m)\lambda_f(n) = \sum_{d | (m, n)} \lambda_f\left( \frac{mn}{d^2} \right).$$

In the case of a Maass cusp form, we can write the expansion $$u(z) = \sum_{n \geqslant 1} \rho_u(n) W_s(nz),$$

where $$s$$ is related in an explicit way to the associated eigenvalue and $$W_s$$ is a Whittaker function. Similarly, the continuous part of the spectrum made of Eisenstein series at a cusp $$a$$ admits expansion of the form $$E_a(s,z) = \phi_a y^s + \phi_a(s) y^{1-s} + \sum_{n \geqslant 1} \phi_a(n,s)W_s(nz)$$

I am interested in the following question :

Are there analogous "Hecke relations" for the coefficients of Maass forms, $$\rho_u(n)$$ and $$\phi_a(n,s)$$?

I suppose so, but I do not have any good reference for these matters.

• Yes, see Duke, Friedlander, and Iwaniec, The subconvexity problem for Artin $L$-functions (Inventiones 149, 2002), Section 6. In particular, the Hecke relations (or multiplicative commuting operators) are valid for any functions of period one. Since these operators commute with the Laplacian, Bob's your uncle and you can decompose into simultaneous eigenfunctions. Is this not also in one of Iwaniec's books? doi.org/10.1007/s002220200223 – literature-searcher Apr 27 '19 at 10:37
• See 8.5 of Iwaniec's Spectral Methods of Automorphic Forms (AMS Grad Studies in Math 53, 2002). doi.org/10.1090/gsm/053 – literature-searcher Apr 27 '19 at 10:44
• Just to be clear, in general the Eisenstein series attached to cusps are not eigenfunctions of the Hecke operators, so $\rho_u(n)$ will not satisfy the Hecke relations. This doesn't contradict what literature-searcher is saying, that a given Laplace eigenspace of Eisenstein series can be decomposed into Hecke eigenfunctions. – Matt Young Apr 27 '19 at 14:42

1. The relation $$(\star)$$ holds for every $$m$$ and $$n$$ if your restrict the sum to $$(d,q)=1$$. This is because $$f$$ is a newform (and we assume that $$f$$ has trivial nebentypus). The same is true for Maass cuspidal newforms.

2. You can get a Hecke eigenbasis of Eisenstein series if you induce the basis vectors from Dirichlet characters modulo $$q$$. See Section 2.1.1 of this article for a quick introduction. I also recommend you to study Chapter 7 of Miyake's book, which discusses these induced Eisenstein series in detail and in classical language.