Algebraic automorphic representation

An algebraic automorphic representation is an automorphic representation over a number field whose archimedean Langlands parameters have integral infinitesimal data. This phrase is incomplete unless its normalization is specified.

In the Buzzard–Gee terminology the two standard normalizations are $C$ -algebraic and $L$ -algebraic. They differ by the half-sum of the positive roots.

Classical convention for general linear groups

For an isobaric automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$ , Clozel's “algebraic” condition agrees with the modern $C$ -algebraic condition: its infinitesimal character at each archimedean place is that of a finite-dimensional algebraic representation. Authors often say simply “algebraic” in this setting.

A twist by a suitable half-integral power of the norm converts the $C$ -normalization to the $L$ -normalization when that twist exists. For a general reductive group, such a twist need not exist on the group itself.

Cohomology and Galois representations

A cohomological automorphic representation is $C$ -algebraic. By contrast, $L$ -algebraicity is the normalization naturally used when conjecturing a Galois representation valued directly in the usual $L$ -group. A $C$ -algebraic representation is expected to give a representation valued in the associated $C$ -group.

Usage rule

Statements such as “algebraic automorphic representations have Galois representations” must specify:

$C$ - or $L$ -normalization;
the reductive group and coefficient field;
regularity, cuspidality, polarization, or other hypotheses;
whether the claim is a theorem or a conjecture;
the target $L$ -group or $C$ -group.

References

Laurent Clozel, “Motifs et formes automorphes: applications du principe de fonctorialité,” in Automorphic Forms, Shimura Varieties, and $L$ -Functions, vol. I, 1990, pp. 77–159.
Kevin Buzzard and Toby Gee, “The conjectural connections between automorphic representations and Galois representations,” Automorphic Forms and Galois Representations, vol. 1, 2014. arXiv.