The Heat Operator of a Transversally Elliptic Operator

Let G be a connected compact Lie group.We study the heat operator of a transversally elliptic operator.After we review the spectral properties of a transversally elliptic operator, we define the character, that is a distribution on G generalizing the trace of the heat operator to the G-equivariant case.The main theorem of this paper gives the estimate of f α (t), which essentially determines the convergence of the character.