Regular algebraic cuspidal automorphic representation

Let $F$ be a number field. A regular algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb A_F)$ , often abbreviated RACAR, is a cuspidal automorphic representation whose archimedean infinitesimal character is integral and regular: it agrees with that of an irreducible algebraic representation having pairwise distinct shifted weights at every archimedean embedding.

Unless another normalization is stated, “algebraic” here is Clozel's condition, equivalently the $C$ -algebraic normalization.

Regularity

For each embedding $\tau:F\hookrightarrow\mathbb C$ , the archimedean parameter determines $n$ weight exponents. Regularity means that these exponents, after the chosen standard shift, are pairwise distinct. It is stronger than algebraicity and rules out singular infinitesimal character.

Authors package the same data as a dominant highest weight $\lambda_{\tau,1}\geq\cdots\geq\lambda_{\tau,n}$ , often with strictness appearing only after adding the $\rho$ -shift. Formulas should therefore state whether they use motivic, cohomological, $C$ -algebraic, or $L$ -algebraic normalization.

Importance

RACAR representations are a principal theorem-level setting for the automorphic–Galois correspondence. Under hypotheses that vary with the base field and polarization, they have associated compatible systems of $n$ -dimensional $\ell$ -adic Galois representations, whose local representations above $\ell$ have prescribed Hodge–Tate weights, with extensive local–global compatibility.

Variants

“Regular algebraic essentially self-dual cuspidal” and “regular algebraic polarizable cuspidal” add self-duality conditions needed by many construction and automorphy-lifting theorems. These adjectives are genuine extra hypotheses, not part of RACAR itself.

References

Laurent Clozel, “Motifs et formes automorphes: applications du principe de fonctorialité,” 1990.
Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, “Potential automorphy and change of weight,” Annals of Mathematics 179 (2014), 501–609. arXiv.