Whittaker datum - Knowlpedia

Definition

Let $G$ be a quasi-split connected reductive group over a local field $F$ . A Whittaker datum for $G$ is a $G(F)$ -conjugacy class of pairs $\mathfrak w=(B,\psi)$ , where $B=TU$ is an $F$ -rational Borel subgroup with unipotent radical $U$ and

\psi:U(F)\longrightarrow\mathbb C^\times

is a nondegenerate unitary character. Nondegeneracy means that the induced character on every simple-root quotient of $U$ is nontrivial.

Representations generic for a datum

A representation is $\mathfrak w$ -generic if it has a Whittaker model for a representative $(B,\psi)$ . This property is independent of the representative within the conjugacy class.

Normalizing packet parametrizations

A Whittaker datum chooses an origin for the internal parametrization of a tempered $L$ -packet on a quasi-split $p$ -adic group: the conjecturally unique $\mathfrak w$ -generic member corresponds to the trivial representation of the appropriate component group. Changing the datum twists that parametrization by a character; it does not change the underlying packet.

Existence and multiplicity

A Whittaker datum exists precisely in the quasi-split setting. It need not be unique up to $G(F)$ -conjugacy, even though the underlying rational Borel subgroups are conjugate in the appropriate sense.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §§2.2 and 2.3.1, 2022. arXiv.
Robert Kottwitz and Diana Shelstad, Foundations of Twisted Endoscopy, Astérisque 255, 1999. Numdam.