| We study the index theory of transversally elliptic pseudodifferential operators in the framework of noncommutative geometry.; For such an operator we construct a spectral triple in the sense of A. Connes and H. Moscovici (“The local index formula in noncommutative geometry” Geom. Funct. Anal., 5(2):174–243, 1995). We prove that this spectral triple satisfies the conditions which ensure the Connes-Moscovici local index formula applies.; We show that the spectral triple has discrete dimensional spectrum. A notable feature of the spectral triple is that its corresponding zeta functions have multiple poles, while in the classical elliptic cases only simple poles appear for the zeta functions. We show that the multiplicities of the poles of the zeta functions have an upper bound, which is the sum of dimensions of the base manifold and the acting compact Lie group. Moreover for our spectral triple the Connes-Moscovici local index formula involves only local transverse symbol of the operator. |
