2026/07/30 by Yi Luo
Mathematics · #math.RT
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We give a self-contained Hecke-algebraic derivation of the Casselman-Shalika formula and J.-S. Li's genericity criterion for irreducible spherical representations of unramified groups, without using the unramified principal series or intertwining operators. The argument centers on the spherical and sign idempotents eK and esgn of the Iwahori-Hecke algebra H. Their left ideals HeK=AeK and Hesgn=Aesgn are free of rank one over the Bernstein subalgebra A. Describing eKHeK inside AeK recovers the Satake isomorphism. Describing eKHesgn inside Aesgn yields rank-one freeness of the K-invariants of the Gelfand-Graev representation and the Casselman-Shalika formula. Symmetrically, describing esgnHeK inside AeK determines when the sign-isotypic part of a spherical module is non-zero, and hence yields Li's genericity criterion.