Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

Abstract

Let 𝕊 be a Clifford module for the complexified Clifford algebra ℂℓ(ℝⁿ), 𝕊′ its dual, ρ and ρ′ be the corresponding representations of the spin group Spin(𝑛). The group G=Spin(1, 𝑛+1) is a (twofold) covering of the conformal group of ℝⁿ. For λ, μ ∈ ℂ, let πρ,λ (resp. πρ′,μ) be the spinorial representation of 𝐺 realized on a (subspace of) C∞(ℝⁿ, 𝕊) (resp. C∞(ℝⁿ, 𝕊′)). For 0 ≤ 𝑘 ≤ 𝑛 and 𝑚 ∈ ℕ, we construct a symmetry-breaking differential operator 𝘉⁽ᵐ⁾k;λ,μ from C∞(ℝⁿ×ℝⁿ, 𝕊 ⊗ 𝕊′) into C∞(ℝⁿ, Λ*ₖ(ℝⁿ)⊗ ℂ) which intertwines the representations πρ,λ⊗πρ′,μ and πτ∗ₖ,λ₊μ₊₂ₘ, where τ*ₖ is the representation of Spin(𝑛) on the space Λ*ₖ(ℝⁿ)⊗ ℂ of complex-valued alternating 𝑘-forms on ℝⁿ.