An analogue of Kreĭn's theorem for semisimple Lie groups (Q418442)

For \(K\)-positive definite functions on a real rank \(n\) connected, noncompact, semisimple Lie group with finite centre, the author derives an integral representation of them. Moreover, the author considers \(\tau\)-positive definite functions, and gives an example in which the set of \(\tau\)-positive definite functions is same as the set of positive definite functions. Then the author characterizes the \(\lambda\)'s for which the \(\tau\)-spherical function \(\phi^{\tau}_{\sigma,\lambda}\) is a positive definite function for the group \(G=\text{Spin}_e(n,1)\) and the complex spin representation \(\tau\).