Normalized parabolic induction for a p-adic group

Definition

Let $G=\mathbf G(F)$ for a connected reductive group over a nonarchimedean local field, let $P=MN$ be a parabolic subgroup with Levi factor $M$ and unipotent radical $N$ , and let $\sigma$ be a smooth representation of $M$ , inflated across $N$ . The normalized parabolic induction is

i_P^G(\sigma)=\operatorname{Ind}_P^G \bigl(\delta_P^{1/2}\otimes\sigma\bigr),

where $\delta_P$ is the parabolic modulus character. The half-modulus factor makes induction carry unitary representations to unitary representations.

Function model

In the right-equivariant convention, the representation consists of locally constant functions $f:G\to V_\sigma$ , with the usual support condition modulo $P$ , satisfying

f(gmn)=\delta_P(m)^{-1/2}\sigma(m)^{-1}f(g).

The group $G$ acts by left translation.

Role in the Langlands classification

The p-adic Langlands classification says that every irreducible admissible representation is the unique irreducible quotient of $i_P^G(\tau\otimes\chi)$ for suitable tempered $\tau$ on a Levi subgroup and a character $\chi$ in a positive chamber. Supercuspidal representations provide the primitive inducing data further down the classification.

Convention warning

Some authors build the factor $\delta_P^{1/2}$ into the symbol $\operatorname{Ind}$ , while others call that functor unnormalized induction. Left-versus-right equivariance also reverses the displayed power. The invariant content is the half-density correction.

References

Joseph Bernstein and Andrei Zelevinsky, “Induced representations of reductive $p$ -adic groups I,” Annales scientifiques de l’École Normale Supérieure 10 (1977), 441–472. Numdam.
Tasho Kaletha, “Representations of reductive groups over local fields,” §1, 2022. arXiv.