Parabolic modulus character

Definition

Let $F$ be a local field, let $G$ be a connected reductive $F$ -group, and let $P=MN$ be a parabolic subgroup with unipotent radical $N$ . The parabolic modulus character is the positive character

\delta_P(p)= \left|\det\!\left(\operatorname{Ad}(p) \bigm|\operatorname{Lie}N\right)\right|_F, \qquad p\in P(F).

Here $\operatorname{Lie}N$ is the Lie algebra of $N$ . The character is trivial on $N(F)$ , so it may be regarded as a character of the Levi subgroup $M(F)$ . It is also the restriction to $P(F)$ of the modular function appropriate to the quotient $G(F)/P(F)$ .

Normalization convention

For nonarchimedean $F$ , normalized parabolic induction uses $\delta_P^{1/2}\sigma$ , while the normalized Jacquet module uses $\delta_P^{-1/2}$ . These half-powers make unitary induction unitary and produce symmetric formulas for contragredients and adjunction.

Some sources call $\delta_P^{-1}$ the modulus. A formula involving $\rho_P$ , normalized induction, or a Satake parameter should therefore be checked against the author's convention.

Example

For the upper-triangular Borel $B\subset\operatorname{GL}_n(F)$ ,

\delta_B(\operatorname{diag}(a_1,\ldots,a_n)) =\prod_{i<j}|a_i/a_j|_F.

References

Armand Borel, “Automorphic $L$ -functions,” in Automorphic Forms, Representations and $L$ -Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979, §3.
I. N. Bernstein and A. V. Zelevinsky, “Induced representations of reductive $p$ -adic groups. I,” Annales scientifiques de l'École Normale Supérieure 10 (1977), 441–472. Numdam.