C-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 $C$ -algebraic if every archimedean component $\pi_v$ is $C$ -algebraic.

Concretely, after choosing a maximal torus and positive roots, write the restriction of the archimedean Langlands parameter to $\mathbb C^\times$ using a cocharacter exponent $\lambda_\sigma\in X_*(\widehat T)\otimes_\mathbb Z\mathbb C$ . If $\delta$ is the half-sum of the positive roots, then the condition is

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

Although the formula uses choices, the integrality condition does not.

Meaning of the letter C

Cohomological automorphic representations are $C$ -algebraic. Equivalently, the infinitesimal character is integral in the Harish-Chandra normalization appropriate to the infinitesimal character of a finite-dimensional algebraic representation.

For isobaric representations of $\operatorname{GL}_n$ , this agrees with Clozel's convention for an algebraic automorphic representation.

Difference from L-algebraicity

The $L$ -algebraic condition is $\lambda_\sigma\in X_*(\widehat T)$ , without the $\delta$ -shift. Thus the two conditions differ by half the sum of the positive roots. They coincide when $\delta$ lies in the relevant integral lattice, but can be disjoint for some groups.

Expected arithmetic object

The most direct Galois-valued conjecture for $C$ -algebraic representations uses the $C$ -group, the $L$ -group of a canonical central extension of $G$ . It is generally incorrect to assert without further qualification that a $C$ -algebraic representation gives a Galois representation into the ordinary $L$ -group of $G$ .

References

Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” Definitions 2.3.3 and 3.1.2. arXiv.
Laurent Clozel, “Motifs et formes automorphes: applications du principe de fonctorialité,” 1990.