Right Regular Representation and Automorphic Forms

Question

Let $R(g)$ denote the right regular representation of $SL_{2}(\mathbb{R})$ in $L^{2}(\Gamma\backslash SL_{2}(\mathbb{R}))$ with $\Gamma$ a congruence subgroup. Why is decomposing $R(g)$ equivalent to knowing how to construct automorphic forms on $SL_{2}(\mathbb{R})$?

Answer 1

In one direction, an irreducible subrepresentation $\pi$ of $L^2(\Gamma\backslash SL_2(\mathbb R))$ is either an odd or even principal series, holomorphic discrete series, or anti-holomorphic discrete series, since we know the irreducible unitary repns of $G=SL_2(\mathbb R)$. The holomorphic discrete series repns have a distinguished vector, namely, the lowest $K$-type (annihilated by suitable "lowering" operator). This vector is a holomorphic modular form. Similarly for anti-holomorphic discrete series. For even principal series, there is a distinguished vector, the $K$-invariant one, which gives a waveform. Almost-the-same for odd principal series. The Casimir operator is a scalar on irreducibles, so waveforms are eigenfunctions for the invariant Laplacian on $G/K$ (which Casimir descends-to on right $K$-invariant functions).

On the other hand, an $L^2$ function $f$ generates a unitary repn under right translation. If $f$ is annihilated by raising (or lowering) operators, then this repn is holomorphic (or anti-holo). If $f$ is a waveform, including being an eigenfunction for the invariant Laplacian, then the repn it generates is a principal series.

If the right regular repn on the adele group is decomposed, instead, then one also discusses the repns of the p-adic groups, and Hecke operators on automorphic forms. The choice of distinguished vector in irreducibles of p-adic groups is about newforms, treated classically by Atkin-Lehner and representation-theoretically by Casselman.

For larger groups, such as $GL(n)$, useful classifications of irreducible unitaries are more complicated, and perhaps not perfectly known. Conjectural parametrizations (local Langlands conjectures) are known for GL(n) by Harris-Taylor/Henniart. The archimedean cases have been substantially understood since the 1960s and work of Langlands, Schmid, Zuckerman, Vogan, and others, but comparison of different constructions seems still problemmatical. "Square-free level" repns of p-adic groups are those with Iwahori-fixed vectors, and the spherical ones among these are well understood (see Casselman's 1980 Compositio paper). The general case of repns with Iwahori-fixed vectors is more delicate. General constructions of supercuspidal repns for GL(n), for example, is from Kutzko and his collaborators. There are over-arching conjectures of Langlands and Arthur about global parametrizations of these repns... There is obviously much more to be said... (Inevitably, a short paragraph like this omits names of many important contributors.)