L-algebraic automorphic representation

Let $G$ be a connected reductive group over a number field $F$ , and let $\pi$ be an automorphic representation of $G(\mathbb A_F)$ . It is $L$ -algebraic if every archimedean component $\pi_v$ is $L$ -algebraic.

After choosing a maximal torus, write the restriction of the archimedean Langlands parameter to $\mathbb C^\times$ with exponent

\lambda_\sigma\in X_*(\widehat T)\otimes_\mathbb Z\mathbb C.

The condition is

\lambda_\sigma\in X_*(\widehat T).

The corresponding condition for the conjugate exponent follows from admissibility, and the property is independent of the auxiliary choices.

Meaning of the letter L

This is the normalization in which an associated $\ell$ -adic Galois representation is expected to take values directly in the $L$ -group:

\rho_{\pi,\iota}: \operatorname{Gal}(\overline F/F) \longrightarrow {}^L G(\overline{\mathbb Q}_\ell).

This assertion is a general conjecture. It includes compatibility with unramified Satake parameters and predicted Hodge–Tate cocharacters.

Difference from C-algebraicity

The $C$ -algebraic condition requires $\lambda_\sigma-\delta$ to be integral, where $\delta$ is half the sum of the positive roots. The two conditions therefore differ by the $\delta$ -shift. For $\operatorname{GL}_n$ , an appropriate norm twist relates them, but the twist need not descend to every reductive group.

Scope warning

$L$ -algebraic does not mean that the automorphic $L$ -function is algebraic, nor does it by itself imply cuspidality, regularity, or the existence of a known Galois representation. Those are separate conditions or theorems.

References

Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” Definitions 2.3.1 and 3.1.1 and Conjecture 3.2.1. arXiv.