Applications of representation theory

This thesis presents two applications of representation theory of locally compact groups. The first is concerned with random walks, the second with Mackey's Intertwining Number Theorem. Firstly, we consider the random walk on a collection of chambers bounded by hyperplanes in a given subspace E of Rn+1 . Initially, a particular transition probability is used in the first part of this analysis, and the identification of the collection of chambers with a reflection group provides necessary tools for obtaining a criterion for the recurrence of that walk. Next, the techniques of representation theory are used to deal with the generalization of the random walk when transition probability is considered to be a general probability measure on the group concerned. Secondly, Mackey's Intertwining Number Theorem for one dimensional representations of open and closed subgroups of a given locally compact group G is generalized. A similar result to Mackey's is obtained in the case where the representations are finite dimensional. The recent developments in the theory of Aqp, spaces (in which such spaces are recognized as preduals of spaces of intertwining operators of induced representations) are being simplified under the condition that the subgroups are open and closed. These results, together with the fact that the space of intertwining operators between two representations can be identified with the dual of the G-tensor product of the corresponding representation spaces (endowed with the greatest cross-norm) are used to carry out the analysis.