[Paper Review] Knapp-Stein type intertwining operators for symmetric pairs

Jan Möllers, Bent Ørsted+1 more|arXiv (Cornell University)|Sep 16, 2013

Advanced Algebra and Geometry26 references4 citations

TL;DR

This paper constructs Knapp–Stein type intertwining operators for symmetric pairs (G,H) of reductive groups, defining explicit integral kernels that intertwine spherical principal series representations of G and H. The key result establishes generic uniqueness of these operators—showing the space of H-intertwiners is at most one-dimensional—particularly for rank-one cases and the pair (GL(4n,ℝ), GL(2n,ℂ)).

ABSTRACT

For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$ (G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F}) imes U(n-m;\mathbb{F}))), \qquad \mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}, $$ and for the pair $$ (G,H)=(GL(4n,\mathbb{R}),GL(2n,\mathbb{C})) $$ we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners.

Motivation & Objective

• To explicitly construct intertwining operators between spherical principal series representations of G and H for symmetric pairs (G,H), generalizing classical Knapp–Stein operators.
• To address Problem 1 from Kobayashi’s program on symmetry breaking operators by providing explicit, integral kernel-based constructions.
• To classify the space of H-intertwiners between induced representations and determine conditions under which it is at most one-dimensional.
• To extend known results on invariant trilinear forms and Juhl-type operators to symmetric pairs via explicit integral operators.
• To establish meromorphic continuation and convergence of the integral kernels for an open set of parameters.

Proposed method

• Constructs intertwining operators as singular integral operators defined by explicit integral kernels involving the Harish-Chandra c-function and intertwining data.
• Imposes compatibility conditions between parabolic subgroups P of G and P_H of H to ensure the intertwining structure is preserved under restriction to H.
• Uses the theory of double flag varieties and orbit decompositions to analyze the geometry of the homogeneous space (G×H)/Δ(H) and derive convergence conditions.
• Applies representation-theoretic techniques, including rational representations of Levi subgroups and character analysis on subgroups S′ and S′′, to derive parameter conditions.
• Employs Theorem 3.15 to relate the existence of non-zero intertwiners to the vanishing of certain characters on representations of Levi subgroups.
• Derives necessary conditions on parameters ν and ν′ by analyzing the weights of irreducible components in symmetric powers of the normal space V to the orbit O.

Experimental results

Research questions

• RQ1 For which parameters ν and ν′ do the Knapp–Stein type intertwining operators between spherical principal series representations of G and H converge?
• RQ2 Under what conditions is the space of H-intertwiners between I^G(ν) and I^H(ν′) at most one-dimensional?
• RQ3 Can the proposed integral operators generically span the full space of symmetry breaking operators for specific symmetric pairs?
• RQ4 How do the integral kernels relate to H-invariant distribution vectors in tensor product representations of G×H?
• RQ5 What are the precise parameter conditions (in terms of ν and ν′) that ensure the existence of non-trivial H-intertwiners?

Key findings

• The integral kernels of the intertwining operators converge for an open set of parameters ν and ν′, and admit meromorphic continuation to the full complex parameter space.
• For the symmetric pair (G,H) = (SU(1,n;𝔽), S(U(1,m;𝔽)×U(n−m;𝔽))) with 𝔽 = ℝ, ℂ, ℍ, 𝕆, the operators generically span the space of intertwiners in the rank-one case.
• For the pair (G,H) = (GL(4n,ℝ), GL(2n,ℂ)), the space of continuous H-intertwiners I^G(ν) → I^H(−ν′)′ is at most one-dimensional when 2ν₁+ν′₁, 2ν₁−ν′₁ ∉ (n+2ℤ) and ν′₁ ∉ ℤ.
• The operators are shown to be generically unique: Hom_H(π_ν|_H, τ_ν′) = ℂ·A(α,β) with α = −(ν′+ρ_H) and β = (ν−ρ + ν′+ρ_H)/2.
• The character β on the Levi subgroup S′ is determined by |det_C A₁|^{2ν₁−ν′₁−3n} |det_C A₂|^{−ν′₁−n} |det_C A₃|^{−2ν₁−ν′₁+n}, leading to necessary integrality conditions on parameters.
• The analysis of the normal space V and its symmetric powers S^r(V_C) shows that d₁ and d₃ must be even, leading to the condition 2ν₁−ν′₁ ∈ n+2ℤ, −ν′₁ ∈ ℤ, and −2ν₁−ν′₁ ∈ n+2ℤ for non-vanishing intertwiners.