论文标题
伯恩斯坦投影仪由较弱的良好cosets的弱点确定
The Bernstein projector determined by a weak associate class of good cosets
论文作者
论文摘要
让$ g $为特征零的$ p $ adic field $ f $的还原组，$ p \ gg 0 $。在[Kim04]，J.-L。金在A. Moy和G. Prasad的意义上研究了一个名为“弱关联”的等价关系，称为弱关联性。之后[kim04]，我们附上良好\ \（k \）的集合\（\ mathfrak s} \） - 在弱的积极深度未经三分的$ k $ -k $ -types $ g（f）$ g（f）$ g（f）$中的$ g（f）$ g（f）的$ g（f）$ \ \ mathfrak g（mathfrak g（f）$ siplline s $ s \ f）s \ f）中的弱点中的类型lie代数$ \ Mathfrak g（f）$ g（f）$，以及一个可接收双\（\ tilde g \）的子集$ \ tilde g _ {\ tillline { $ \ OVILLINE {\ MATHFRAK S} $。然后\（\（\ tilde g _ {\叠加{\ mathfrak s}} \）是有限的许多伯恩斯坦组件的结合，以$ g $，因此我们可以考虑Bernstein Projector $ e _ {\ edimalline {\ edimalline {\ Mathfrak s}} $确定它。我们表明，$ e _ {\ Overline {\ Mathfrak s}} $在Moy-- prasad $ g（f）$ - 域$ g（f）_r \ subset g（f）$之外消失，并重新制定了Kim的结果，因为Kim认为$ e _ {\ Mathak $ nline的限制，$ e _ $ nline {\ Mathak S}与Moy-Prasad $ g（f）$ - 域$ \ mathfrak g（f）_r \ subset \ mathfrak g（f）$的对数，同意$ \ mathfrak g（f）_r _r $，以及$ \ mathfrak g（f）_ mather的特征性函数的反傅立叶变换，nline _ {这是R. Bezrukavnikov，D。Kazhdan和Y. Varshavsky在Arxiv中给出的描述之一的变体：1504.01353对于深度为bernstein Projector。
Let $G$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$. In [Kim04], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for $G$ in the sense of A. Moy and G. Prasad. Following [Kim04], we attach to the set \(\overline{\mathfrak s}\) of good \(K\)-types in a weak associate class of positive-depth unrefined minimal $K$-types a $G(F)$-invariant open and closed subset $\mathfrak g(F)_{\overline{\mathfrak s}}$ of the Lie algebra $\mathfrak g(F)$ of $G(F)$, and a subset $\tilde G_{\overline{\mathfrak s}}$ of the admissible dual \(\tilde G\) of \(G(F)\) consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak s}$. Then \(\tilde G_{\overline{\mathfrak s}}\) is the union of finitely many Bernstein components for $G$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak s}}$ that it determines. We show that $E_{\overline{\mathfrak s}}$ vanishes outside the Moy--Prasad $G(F)$-domain $G(F)_r \subset G(F)$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak s}}$ to $G(F)_r$, pushed forward via the logarithm to the Moy--Prasad $G(F)$-domain $\mathfrak g(F)_r \subset \mathfrak g(F)$, agrees on $\mathfrak g(F)_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak g(F)_{\overline{\mathfrak s}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan and Y. Varshavsky in arXiv:1504.01353 for the depth-$r$ Bernstein projector.
