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— We discuss different aspects of the c-function of HarishChandra with focus on its connection with the horospherical transform. I have known Vadim Schechtman for many years, but it was only a few years ago that I found out that we share two strong interests: surprises which it is possible yet to mine in works of great mathematicians and product-formulas for some special functions on Lie groups. Vadim has collected a very impressive exhibition of such formulas (see unpublished notes on his webpage). It starts with the product-formula for the c-function which Karpelevich and I found more than 50 years ago [19] in the beginning of my mathematical life. Vadim’s anniversary for me is a pleasant reason to talk about this old subject. Starting at least with Gauss (who continuously produced new proofs of the Fundamental Theorem of Algebra), old mathematicians like to return to subjects of their first mathematical love. Perhaps it has a similar nature to criminals, returning to the scene of the crime? What about the c-function, for the past several years, there were quite a few popular generalizations of the product-formula on arbitrary fields, but in these notes we will stay inside old fashioned real or complex considerations. (1) Departm. of Math., Hill Center, Rutgers University, 110 Frelinghysen Road, Piscataway, NJ 08854 gindikinath.rutgers.edu

Harish-Chandra’s c-function; 50 years later