LIMITS OF COMPLEMENTARY SERIES, SPECIAL REPRESENTATIONS AND COHOMOLOGY OF CAT(-1) GROUPS.

The paper explicitly states and proves the equivalence between (1) rough conditional negative type of the distance on the convex core, (2) positivity of all Knapp–Stein operators Is on L2(∂X,μo) for s>1/2, and (3) positivity of the infinitesimal operator I1 on L2(∂X,μo)⊕C, via Proposition 3.5 (Is→I1 near s=1), Proposition 4.4 (a distance decomposition Q11(c(x,y))=d(x,y)+r with orbitwise limiting behavior), and Corollary 4.5 using [4, Thm. 1] to connect (1) and (2) . The model’s solution proves the same equivalence but by a different route: it uses Schoenberg’s theorem to relate positivity of Is to the conditional negative definiteness of the boundary kernel C(ξ,η)=−log do(ξ,η), and a GNS-type construction to obtain a c.n.d. kernel on X, then a coarse distance formula to conclude rough negative type. This alternate proof is broadly correct in strategy, but it requires technical qualifications (handling the diagonal divergence of −log do, justifying approximation of Dirac masses in L2, fixing a sign in the change-of-basepoint formula) and a more explicit proof of the coarse distance estimate. The paper’s argument avoids these pitfalls with a cocycle-based identity that directly yields the desired decomposition.