Automorphic form and Hecke eigenvalues

Let $F$ be a number field and $G$ a connected reductive $F$ -group. An automorphic form is a suitably smooth, finite, moderate-growth function on

G(F)\backslash G(\mathbb A_F),

usually with a prescribed central character. At an unramified finite place $v$ , a vector fixed by a hyperspecial subgroup $K_v$ is a spherical Hecke eigenvector if there is a character

\chi_v:\mathcal H(G(F_v),K_v)\longrightarrow\mathbb C

such that $T\phi=\chi_v(T)\phi$ for every spherical Hecke operator $T$ .

Modern representation-theoretic formulation

The right translates of an eigenform can generate an automorphic representation

\pi\simeq\bigotimes_v'\pi_v.

This is the restricted tensor product of the local components. For almost every finite $v$ , $\pi_v$ is unramified and $\pi_v^{K_v}$ is one-dimensional. The spherical Hecke algebra acts on that line by $\chi_v$ . The normalized Satake isomorphism converts $\chi_v$ into the Satake parameter of $\pi_v$ .

Three objects to distinguish

The automorphic form $\phi$ is a vector in a function space.
The automorphic representation $\pi$ is an irreducible global representation generated or detected by such vectors.
The Satake parameter is an unramified local semisimple conjugacy class.

A single form can be a simultaneous eigenvector at many places, while the restricted tensor product records the entire automorphic representation.

Relation to the letter

The letter moves directly from unramified Hecke eigencharacters to semisimple dual-group classes. That is the seed of the modern local parameter language, but it precedes $L$ -packets, refined local Langlands, and the global automorphic-spectrum formalism.

References

A. Borel and H. Jacquet, “Automorphic forms and automorphic representations,” Proc. Sympos. Pure Math. 33, part 1, 1979.
Ichirō Satake, “Theory of spherical functions on reductive algebraic groups over $p$ -adic fields,” Publications Mathématiques de l'IHÉS 18 (1963), 5–69. Numdam.