AI 中文总结
该研究通过伊瓦霍-赫克代数的球型与符号幂等元，无需非分歧主系列或交错算子，推导了Casselman-Shalika公式与J.-S. Li的非分歧群不可约球型表示的泛性质判据。
AI 中文摘要
我们给出Casselman-Shalika公式与J.-S. Li的非分歧群不可约球型表示的泛性质判据的自包含赫克代数推导，无需使用非分歧主系列或交错算子。论证围绕伊瓦霍-赫克代数$\boldsymbol{\textit{H}}$的球型幂等元$\boldsymbol{e_K}$与符号幂等元$\boldsymbol{e_{\text{sgn}}}$展开：其左理想$\boldsymbol{\textit{H}}e_K=\boldsymbol{\textit{A}}e_K$与$\boldsymbol{\textit{H}}e_{\text{sgn}}=\boldsymbol{\textit{A}}e_{\text{sgn}}$在伯恩斯坦子代数$\boldsymbol{\textit{A}}$上是秩1自由模；描述$\boldsymbol{e_K}\textit{H}e_K$在$\boldsymbol{\textit{A}}e_K$内可得到Satake同构，描述$\boldsymbol{e_K}\textit{H}e_{\text{sgn}}$在$\boldsymbol{\textit{A}}e_{\text{sgn}}$内可得到Gelfand-Graev表示的$K$-不变量的秩1自由性与Casselman-Shalika公式；对称地，描述$\boldsymbol{e_{\text{sgn}}}\textit{H}e_K$在$\boldsymbol{\textit{A}}e_K$内可确定球型模的符号同型部分非零的条件，从而得到Li的泛性质判据。
英文摘要
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.
