Representations of graded Hecke algebras associated to noncrystallographic root systems.

Abstract

Lusztig's presentation of the graded Hecke algebra in terms of generators and relations allows for the definition of algebras associated to the dihedral root systems $I\sb2(n)$ and the icosahedral root system $H\sb3,$ which are noncrystallographic. The representation theory of these algebras is investigated, the expected number of tempered representations is accounted for in all cases, and a proof is given for $I\sb2(2m + 1)$ that there are no others. This involves the adaptation of several techniques from the setting of p-adic groups, such as the Langlands classification, principal series and intertwining operators, and the decomposition of Jacquet modules. In addition, one of the representations found proves to be especially interesting because it is of a substantially different type than is conjectured for those which arise in the p-adic group case.