Spectral Correspondences for Rank One Locally Symmetric Spaces - The Case of Exceptional Parameters

The paper’s SU(n,1) spectral correspondence (Theorem 6.5) gives the same characterization as the candidate solution: at exceptional parameters µ=−(ρ+2ℓα), the socle is the irreducible, unitarizable subrepresentation with K-types Y_{p,q} (p,q≥ℓ+1), and the minimal K-type Poisson transform is an isomorphism onto Γ-invariant sections valued in Y_{ℓ+1,ℓ+1} satisfying the first-order system (i)–(vi). The paper proves surjectivity via an explicit Fourier construction, while the model argues via general intertwining/gradient identities and representation theory. The statements, operators, scalars, and K-type decompositions align closely with the paper’s Lemma 3.13 and Theorem 6.5, including the non–M-spherical constituents V1,V2 and the vanishing conditions Djψ=0. Hence both are correct with different proof styles.