Whittaker model - Knowlpedia

Definition

Let $G$ be a quasi-split reductive group over a local field, let $B=TU$ be an $F$ -rational Borel subgroup, and let $\psi:U(F)\to\mathbb C^\times$ be a nondegenerate unitary character of its unipotent radical. A Whittaker functional on a representation $(\pi,V)$ of $G(F)$ is a linear map $\lambda:V\to\mathbb C$ satisfying

\lambda(\pi(u)v)=\psi(u)\lambda(v) \qquad(u\in U(F)).

If a nonzero such functional exists, $\pi$ is $\psi$ -generic. The associated Whittaker model is the realization by functions

W_v(g)=\lambda(\pi(g)v), \qquad W_v(ug)=\psi(u)W_v(g).

Nondegeneracy

Nondegeneracy means that $\psi$ is nontrivial on every simple-root quotient of $U$ determined by $B$ . The $G(F)$ -conjugacy class of the pair $(B,\psi)$ is a Whittaker datum.

Uniqueness

For irreducible admissible representations in the standard quasi-split local setting, the Whittaker functional is unique up to scalar when it exists. This multiplicity-one result makes the Whittaker model a canonical realization after normalizations are chosen.

Role in local Langlands

A Whittaker datum normalizes the internal parametrization of an $L$ -packet. For a tempered packet of a quasi-split $p$ -adic group, the generic-packet conjecture predicts a unique member generic for the chosen datum.

References

François Rodier, “Whittaker models for admissible representations of reductive $p$ -adic split groups,” in Harmonic Analysis on Homogeneous Spaces, Proceedings of Symposia in Pure Mathematics 26, 1973.
Tasho Kaletha, “Representations of reductive groups over local fields,” §§2.2–2.3, 2022. arXiv.