Central character of a representation

Definition

Let $G$ be a group with center $Z(G)$ , and let $(\pi,V)$ be a representation. A central character of $\pi$ is a character $\omega_\pi:Z(G)\to k^\times$ such that

\pi(z)=\omega_\pi(z)\operatorname{id}_V \qquad(z\in Z(G)).

For an irreducible representation over an algebraically closed field, the existence and uniqueness of $\omega_\pi$ follow from Schur's lemma whenever the endomorphism version of that lemma applies. For topological groups and continuous or smooth representations, the central character is required to have the corresponding continuity or smoothness.

Distinctions

This is different from the infinitesimal character, which records the action of the center of a universal enveloping algebra. It is also different from a character of $G$ itself: only the center must act by the specified scalar.

Langlands compatibility

For $G=\operatorname{GL}_n(F)$ , the determinant of a local Langlands parameter corresponds under local class field theory to the central character of the associated representation. Analogous statements for a general reductive group use the map from its $L$ -group to the $L$ -group of its center.

References

Jean-Pierre Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42, Springer, 1977, §2.2.
Tasho Kaletha, “Representations of reductive groups over local fields,” 2022, §2.1. arXiv.