Residual automorphic spectrum

The residual automorphic spectrum of a connected reductive group $G$ over a global field $F$ is the orthogonal complement of the cuspidal spectrum inside the discrete automorphic spectrum:

L^2_{\mathrm{res}} = L^2_{\mathrm{disc}}\ominus L^2_{\mathrm{cusp}}.

An irreducible constituent of this space is a residual automorphic representation.

Eisenstein-series origin

Start with a cuspidal automorphic representation $\sigma$ of a Levi subgroup $M$ . Normalized parabolic induction at the finite places produces a meromorphic family of representations and Eisenstein series. Residues at suitable poles can be square-integrable even though their constant terms do not vanish. Their irreducible constituents generate the residual spectrum.

This construction is recursive in the Levi subgroups and is controlled by the poles of global intertwining operators and automorphic $L$ -functions.

Simple example

For $G=\operatorname{GL}_2$ , the constant functions on an appropriate finite-volume automorphic quotient give a noncuspidal discrete representation. They arise through a residue of an Eisenstein series. This illustrates why “discrete” and “cuspidal” are not synonyms.

Terminology warning

The residual automorphic spectrum is not the residual spectrum of a single operator in functional analysis. It is a named summand of the global automorphic $L^2$ -decomposition.

Relation to classification

Residual representations often correspond to non-tempered global parameters with a nontrivial Arthur $\operatorname{SL}_2$ -factor. Arthur packets and multiplicity formulas organize this phenomenon for groups where the classification is known.

References

Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, 1976. DOI.
C. Mœglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995. DOI.