A Tale of Two Idempotents: Casselman-Shalika and Spherical Genericity

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 $e_K$ and $e_{\mathrm{sgn}}$ of the Iwahori-Hecke algebra $\mathcal{H}$. Their left ideals $\mathcal{H}e_K=\mathcal{A}e_K$ and $\mathcal{H}e_{\mathrm{sgn}}=\mathcal{A}e_{\mathrm{sgn}}$ are free of rank one over the Bernstein subalgebra $\mathcal{A}$. Describing $e_K\mathcal{H}e_K$ inside $\mathcal{A}e_K$ recovers the Satake isomorphism. Describing $e_K\mathcal{H}e_{\mathrm{sgn}}$ inside $\mathcal{A}e_{\mathrm{sgn}}$ yields rank-one freeness of the $K$-invariants of the Gelfand-Graev representation and the Casselman-Shalika formula. Symmetrically, describing $e_{\mathrm{sgn}}\mathcal{H}e_K$ inside $\mathcal{A}e_K$ determines when the sign-isotypic part of a spherical module is non-zero, and hence yields Li's genericity criterion.