Automorphic representation

At finite places one works with smooth admissible representations. At the archimedean places one ordinarily records the underlying $(\mathfrak g_\infty,K_\infty)$-Harish–Chandra module. Consequently, the displayed adelic representation is shorthand for a compatible archimedean Harish-Chandra module and a smooth representation of $G(\mathbb A_F^\infty)$.

An automorphic representation need not occur as a closed irreducible subrepresentation of an $L^2$-space. Constituents of Eisenstein series can appear naturally as subquotients. By contrast, a representation in the discrete spectrum occurs unitarily, with a multiplicity, in the automorphic $L^2$-space.

An irreducible admissible automorphic representation factors as a restricted tensor product

Almost every finite-place component $\pi_v$ is unramified. These local components carry Satake parameters or local Langlands parameters and are the inputs to Euler products.

Two automorphic representations are nearly equivalent if their local components are isomorphic at all but finitely many places. Strong multiplicity one makes near equivalence very rigid for $\operatorname{GL}_n$, but it need not identify representations for a general reductive group. Global packets and multiplicity formulas account for this distinction.

An automorphic form is a vector-valued analytic object; an automorphic representation is an irreducible representation generated or detected by such vectors. The two terms should not be used interchangeably.

1. A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” in Automorphic Forms, Representations and $L$-Functions, Proc. Sympos. Pure Math. 33, part 1, 1979, pp. 189–207.
2. James Arthur, “An introduction to the trace formula,” in Harmonic Analysis, the Trace Formula, and Shimura Varieties, 2005. Clay Mathematics Proceedings.