Knapp–Stein intertwining operator - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive group, $P=MAN$ a parabolic subgroup, and $I_P(\sigma,\lambda)$ a representation formed by normalized parabolic induction. For a Weyl-group element $w$ carrying the inducing data to $(w\sigma,w\lambda)$ , a Knapp–Stein intertwining operator

R(w,\sigma,\lambda)\colon I_P(\sigma,\lambda) \longrightarrow I_{wPw^{-1}}(w\sigma,w\lambda)

is the meromorphic continuation in $\lambda$ of the standard integral over the appropriate unipotent subgroup, multiplied by scalar normalizing factors. It is a representation intertwiner wherever regular, and the normalization is chosen to satisfy Weyl-group composition and unitarity identities.

Integral and continuation

In a chamber where the unipotent integral converges absolutely, the operator is defined directly on smooth induced functions. Analytic continuation then extends it meromorphically to the full complex parameter space. Poles, zeros, and noninvertibility detect reducibility phenomena in principal series. The original construction and continuation are established in Knapp–Stein, pp. 489–578.

Normalization and consequences

After normalization, one obtains identities of the form

R(w_1w_2,\lambda)=R(w_1,w_2\lambda)R(w_2,\lambda)

when lengths and parameters permit the corresponding factorization, together with $R(w,\lambda)^*=R(w^{-1},-\overline{\lambda})$ . On the unitary axis the regular normalized operators are unitary. Their eigenspaces can control reducible constituents, while positivity of an associated Hermitian form produces complementary series.

Conventions and scope

References

Anthony W. Knapp and Elias M. Stein, “Intertwining Operators for Semisimple Groups,” Annals of Mathematics 93 (1971), 489–578. DOI record. Relevant: construction, meromorphic continuation, normalization, and functional equations.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on principal-series intertwining operators.