p-adic 场上连通可约群的Gelfand-Graev表示的Iwahori分量
Iwahori component of the Gelfand--Graev representation for reductive groups
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中文总结 AI 辅助
研究p-adic域上连通可约群的Gelfand-Graev表示的Iwahori分量，证明其与Hecke代数和符号表示的同构关系，扩展了Chan-Savin定理的应用范围。
中文摘要 AI 辅助
设G为一个p-adic域F上的连通可约群，U为最小抛物子群的单位元 radical，ψ为U(F)上的深度零非退化特征标，I为G(F)的一个Iwahori子群。我们证明，作为Iwahori-Hecke代数H的模，Gelfand-Graev表示ind_U^Gψ中I-不变向量的空间同构于H⊗_{H_{W_0}} sgn。这里sgn是与相对Weyl群相关的有限Hecke子代数H_{W_0}的符号表示。此结果扩展了Chan-Savin在分裂群上的定理到所有连通可约群。
英文摘要
Let $G$ be a connected reductive group over a $p$-adic field $F$, $U$ the unipotent radical of a minimal parabolic subgroup, $ψ$ a depth-zero non-degenerate character of $U(F)$, and $I$ an Iwahori subgroup of $G(F)$. We show that, as a module over the Iwahori-Hecke algebra ${H}$, the space of $I$-fixed vectors in the Gelfand-Graev representation $\mathrm{ind}_U^Gψ$ is isomorphic to ${H} \otimes_{{H}_{W_0}} \mathrm{sgn}$. Here $\mathrm{sgn}$ is the sign representation of the finite Hecke subalgebra ${H}_{W_0}$ attached to the relative Weyl group. This extends the theorem of Chan-Savin from split groups to all connected reductive groups.
