math.RT — Representation Theory
Finite Projective Source Geometry for GL2: Abel Resolvents and Central-Jet Degree Laws
Starting from the two trace functions p1 = Tr(A) and p2 = Tr(A^2) for semisimple A in GL2(C), we develop the associated projective source geometry. The coordinate y = t^2/d - 1 identifies the adjoint character, and normalized Haar measure on a specified closed subgroup H of SO(3) pushes forward to a probability measure nu_H. We describe this measure through harmonic multiplicities, moments, and its Stieltjes transform, and prove that it determines H up to conjugacy within the closed-subgroup class of SO(3). A positive Abel kernel resums the projective character tower and gives a resolvent formula whose endpoint limit recovers the mass of the identity atom. We give explicit Abel transforms for the finite rotation groups and positive-dimensional benchmark sources, and derive a finite-image Frobenius-prime Abel limit from Chebotarev. For central-jet interpolation, we prove two exact degree laws: the cyclic-versus-torus minimum degree is max(n, M+1), while the normalized tetrahedral detector against S4 and A5 has minimum degree 3 for M = 1, 2 and M+2 for M >= 3. We also compute the integral image of the tetrahedral averaging functional, obtaining 2Z for every M >= 2. All results are stated with their hypotheses. The paper does not claim a general infinite-spectrum Beyond Endoscopy trace formula or transfer of central-jet vanishing to orbital kernels.