In this note I wish to announce certain preliminary results concerning a class of special functions arising in the theory of representations of groups defined over local fields. Before the preparation of this manuscript, the author obtained a copy of the recent paper  of I. M. Gelfand and D. A. Kajdan. Several of the results announced here are proved in their paper. I hope to indicate below what I have obtained independently and also what new results I have obtained since the paper of Gelfand and Kajdan became available. I wish to thank A. W. Knapp for the proof of a central lemma stated below.