论文标题
家庭中的基里洛夫模型
The Kirillov model in families
论文作者
论文摘要
让$ f $为非架构的本地领域，让$ k $为代数封闭的特征性$ \ ell $ $ $ \ el $与$ f $的剩余特征不同，而让$ a $ a $为可交换的noetherian $ w（k）$ - 代数 - $ w（k）$ the $ w（k）$表示Witt witt vict vectors。使用Rankin-Selberg功能方程并扩展了第二作者的最新结果，我们表明，如果$ v $是$ a [\ text {gl} _n（f）] $ - 惠特克类型的模块，则其惠特克空间上的mirabolic限制性图是Injective。与以前的证明相比，这与以前的证明相反，这提供了用于惠特克类型表示的基里洛夫模型的新快速证明，其中包括复杂的表示形式，该模型概括为$ \ ell $ - 模块化和家庭设置。在特殊情况下，$ a = k = \ overline {\ mathbb {f} _ {\ ell}} $ and $ v $是不可约通的，我们的结果特别回答了1989年Vignéras的问题。
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.
