Continuous automorphic spectrum

Definition

Let $G$ be a connected reductive group over a global field $F$ . After fixing a central-character or split-center convention, the right regular representation on the automorphic Hilbert space decomposes as

L^2_{\mathrm{aut}}(G)= L^2_{\mathrm{disc}}(G)\oplus L^2_{\mathrm{cont}}(G).

The first summand is the discrete automorphic spectrum; the second is the continuous automorphic spectrum. Spectrally, it is a direct integral of representations obtained by normalized adelic parabolic induction from discrete automorphic data on proper Levi subgroups.

Eisenstein construction

Eisenstein series and their meromorphic continuation provide generalized eigenfunctions for the continuous spectrum. Their constant terms and normalized intertwining operators determine the Plancherel measure and identify redundancies among inducing data.

Poles of Eisenstein series can contribute square-integrable residues. Those belong to the residual spectrum, which is discrete rather than continuous.

Trace-formula role

The spectral side of the Arthur–Selberg trace formula includes integrals of weighted characters arising from this continuous family. It reduces to a plain sum only when the relevant automorphic quotient is compact or when a test function annihilates all proper-parabolic contributions.

References

Robert P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer, 1976.
James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings 4, 2005, §§12–14 and 21. Clay.