The Heat Operator of a Transversally Elliptic Operator

Let $G$ be a connected compact Lie group. We study the heat operator of a $G$-transversally elliptic operator. After we review the spectral properties of a $G$-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 $f_\alpha(t)$, which essentially determines the convergence of the character.