Complex sentence: The semigroup acts by contraction operators on the holomorphic discete series and the positive part of their polar decomposition is given by exponentials of imaginary multiples of infinitesimal operators in the maximal cone.
Split into simpler sentences: The semigroup acts by contraction operators on the holomorphic discete series. The positive part of their polar decomposition is given by exponentials of imaginary multiples of infinitesimal operators corresponding to elements in the maximal cone.