Normalized parabolic induction - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive Lie group, let $P=MAN$ be a parabolic subgroup in Langlands form, and let $\tau$ be a representation of $MA$ , extended trivially across $N$ . Normalized parabolic induction is the representation $i_P^G(\tau)$ realized with the half-modular normalization. In the right-equivariant function convention its vectors satisfy

f(gp)=\delta_P(p)^{-1/2}\tau(p)^{-1}f(g),

where $\delta_P$ is the positive reciprocal modular function, or parabolic modulus,

\delta_P(p)=\Delta_P(p)^{-1} =\left|\det\!\left(\operatorname{Ad}(p)|_{\mathfrak n}\right)\right|.

The factor $\delta_P^{-1/2}$ is what distinguishes normalized from unnormalized parabolic induction.

Compact picture and unitarity

Choose a maximal compact subgroup $K$ with $G=KP$ . Restriction to $K$ realizes the induced space using functions on $K$ with their $K\cap M$ equivariance; the Hilbert norm is then independent of the continuous induction parameter. If $\tau$ is unitary, left translation gives a unitary representation. Equivalently, the half-modular factor in the equivariant function model packages the Radon–Nikodym correction in unitary induction Knapp, Chapter VII.

Root-theoretic form

For $P=MAN$ , let $\rho_P\in\mathfrak a^*$ be half the sum of the positive restricted roots occurring in $\mathfrak n$ , counted with multiplicity. Then

\delta_P(a)^{1/2}=e^{\rho_P(\log a)} \qquad (a\in A).

Thus, with the corpus convention $\int_P f(xp)\,dx=\Delta_P(p)^{-1}\int_P f(x)\,dx$ , one has $\Delta_P(a)=e^{-2\rho_P(\log a)}$ and $\delta_P(a)=\Delta_P(a)^{-1}$ . Thus the half-modular normalization is often written as a shift by $\rho_P$ . This shift makes standard intertwining operators and induction in stages take their customary symmetric forms.

Relationship to principal series

When $P$ is minimal and $\tau=\sigma\otimes e^\lambda$ on $MA$ , normalized parabolic induction produces a principal series representation. Induction from larger parabolics gives generalized principal series and the standard modules used in the Langlands classification.

Convention warning

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on induced representations and the compact picture.
Nolan R. Wallach, Real Reductive Groups I, Pure and Applied Mathematics 132, Academic Press, 1988. Publisher record. Relevant: chapters on real reductive groups, principal series, and normalized induction.