家庭中的基里洛夫模型
The Kirillov model in families
摘要：
设$F$是非阿基米德局部域，$k$是特征$\ell$不同于$F$的剩余特征的代数闭域，$A$是交换Noether $W(k)$ -代数，其中$W(k)$表示Witt向量.利用Rankin-Selberg函数方程，推广了第二作者最近的结果，证明了若$V$是Whittaker型的$A[\text{GL}_n(F)]$ -模，则其Whittaker空间上的奇异限制映射是内射的.这给出了一个新的快速证明Kirillov模型的存在性表示的Whittaker类型，包括复杂的表示，推广到$\ell$ -模块和家庭设置，在以前的证明相比。在$A=k=\overline{\mathbb{F}_{\ell}}$和$V$是不可约泛型的特殊情况下，我们的结果特别回答了1989年Vignéras的一个问题。
英文摘要：
Let $F$ be a non-archimedean local field, let $k$ be an algebraically closed field of characteristic $\ell$ different from the residual characteristic of $F$, and let $A$ be a commutative Noetherian $W(k)$-algebra, where $W(k)$ denotes the Witt vectors. Using the Rankin-Selberg functional equations and extending recent results of the second author, we show that if $V$ is an $A[\text{GL}_n(F)]$-module of Whittaker type, then the mirabolic restriction map on its Whittaker space is injective. This gives a new quick proof of the existence of Kirillov models for representations of Whittaker type, including complex representations, which generalizes to the $\ell$-modular and families setting, in contrast with the previous proofs. In the special case where $A=k=\overline{\mathbb{F}_{\ell}}$ and $V$ is irreducible generic, our result in particular answers a question of Vignéras from 1989.
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