Yang's System of Particles and Hecke Algebras

Abstract

The graded Hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions. This operator algebra arises from the study of the root system analogue of Yang's system of n particles on the real line with delta function potential. It turns out that the spectral problem for this generalization of Yang's system is related to the problem of finding the spherical tempered representations of the graded Hecke algebra. This observation turns out to be very useful for both these problems. Application of our technique to affine Hecke algebras yields a simple formula for the formal degree of the generic Iwahori spherical discrete series representations.