# Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators

I am reading Langlands' pape Euler Products and have a few questions. Let $$G$$ be a split adjoint semisimple group over $$\mathbb Q$$. If $$p$$ is a place of $$\mathbb Q$$, finite or infinite, let $$G_{\mathbb Z_p}$$ be a maximal compact subgroup of $$G_{\mathbb Q_p}$$. Let $$K = \prod\limits_p G_{\mathbb Z_p} \subset G_{\mathbb A}$$, and let $$L$$ be the Hilbert space of square integrable functions on $$G_{\mathbb Q} \backslash G_{\mathbb A}$$ which are right invariant under $$K$$.

Let $$L_0 \subset L$$ be the subspace of cusp forms (whose definition as given by Langlands does not quite make sense to me). Langlands writes:

• Is $$H_p$$ always commutative?

• If $$p$$ is finite, there is an injection of the "spherical Hecke algebra" $$C_c^{\infty}(G_{\mathbb Q_p}, G_{\mathbb Z_p})$$, the space of locally constant, compactly supported functions on $$G_{\mathbb Q_p}$$ which are left and right invariant under $$G_{\mathbb Z_p}$$, into $$H_p$$, where a function $$f \in C_c^{\infty}(G_{\mathbb Q_p}, G_{\mathbb Z_p})$$ is associated with the measure $$\mu_f \in H_p$$ on $$G_{\mathbb Q_p}$$ defined by $$\mu_f(E) = \int\limits_{E} f(x) d\mu_{\textrm{Haar}}(x)$$Is $$f \mapsto \mu_f$$ an isomorphism of the spherical Hecke algebra onto $$H_p$$?

• Why in the $$p$$-adic case are all measures in $$H_p$$ absolutely continuous with respect to Haar measure?

• Why is there a countable orthonormal basis of $$L_0$$ consisting of eigenfunctions for all operators $$\lambda(\mu)$$ over all $$\mu \in H_p$$ and all $$p$$? Is this some sort of version of spectral theorem?

• Commutativity of $H_{p}$ can be proved in the same way as the commutativity of spherical Hecke algebra. Namely one uses Gelfand's trick (wrt transpose involution) and Cartan decomposition $G=KAK$. – GTA Aug 15 '19 at 18:04

These things are not trivial at all, but by the time Langlands was writing "Euler products" they were known, and quite familiar to many people at Princeton and Yale, even if not so many other places.

$$H_p$$ is certainly mostly commutative. As commented by @GTA, at least for classical groups the Gelfand criterion is easy to verify (from a Cartan decomposition, both $$p$$-adic and archimedean). I do not know whether there is an intrinsic proof that treats, for example, Galois twists of exceptional groups.

About the continuity of that class of measures with respect to Haar measure: the left-and-right (or even one-sided) $$G_{\mathbb Z}$$-invariance, together with compact support, implies (by some abstract uniqueness-of-invariant-distributions result) that the functional is a finite linear combination of (Haar) integrals over cosets of $$G_{\mathbb Z}$$. So, yes, barring some technicalities, your map is an isomorphism of Hecke algebras.

(At archimedean places, similar things can be said, but since there are derivatives, one cannot purely echo what is true for $$p$$-adic places.)

The existence of an orthonormal basis of spherical-Hecke eigenvectors for the space of cuspforms is highly non-trivial. True, for holomorphic cuspforms, the finite-dimensionality makes things elementary at a given level, for good primes. The general argument (with or without level one, etc.) does indeed use the spectral theorem for families of compact operators closed under adjoints. The difficult piece is proving that the integral operators attached to test functions are compact on cuspforms.

The case of compact $$\Gamma\backslash G$$ is often treated in introductory sources, since the compactness of integral operators follows from their being Hilbert-Schmidt, and proving the latter is essentially elementary.

The compactness of integral operators on cuspforms on non-compact quotients is a much bigger production. One sort of argument was sketched in Lax-Phillips "Scattering theory for automorphic forms" (Princeton orange series), although a scrupulous reader will notice many analytical details needing to be filled-in. They treated $$SL_2(\mathbb Z)$$, proving that spaces of pseudo-cuspforms decompose discretely for (a certain self-adjoint extension of a restriction of) the invariant Laplacian. The ideas admit generalization.

Another argument for compactness of suitable integral operators on cuspforms was given by R. Godement in the Boulder Conference from 1965/6, and is visible in the AMS Proc. Symp. Pure Math IX. It, too, has pretty steep functional analysis prerequisites, which may be not obvious to a casual number-theorist reader.

Both types of argument are illustrated in several examples in my recent books, which, I hasten to point out, are legally available in a PDF http://www.math.umn.edu/~garrett/m/v/current_version.pdf at my website, published by Cambridge Univ. Press ("Modern Analysis of Automorphic Forms, by Example"). Although the analytical backdrop may not be of great interest to all number theorists, I hope that at least the demonstrable existence of (perhaps tedious or uninteresting) proofs will be comforting. :) Seriously, many of these details plagued me for decades...

(Also, Gelfand's criterion is treated in my books, and several other of these apocryphal things...)

• I've never really thought about exceptional groups, is there an example where $H_{p}$ is not commutative? – GTA Aug 16 '19 at 19:16
• @GTA, I've not heard of an example where there's no good (commutative) choice of $H_p$ is not commutative, but that doesn't mean much. I do not know how "hyperspecial" max'l compacts pan out for exceptional groups, for example. – paul garrett Aug 16 '19 at 19:18