A Tale of Two Idempotents: Casselman-Shalika and Spherical Genericity
Abstract
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.
Disclosure
“g the paper, the human author used AI tools at two levels. At the model level, Claude Opus 4.8, Claude Fable 5, and GPT 5.6-Sol were used. At the agent level, a custom agent built by the author for mathematical research was used along with Claude Code and Codex. These tools assisted with mathematical development, exposition, ci- tation checking, and proofreading. The human author verified all mathematical content and takes full responsibility for it. 2 Preliminaries 2.1 The”
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