由良陪集的弱伴随类确定的Bernstein投影
The Bernstein projector determined by a weak associate class of good cosets
摘要：
设$G$是特征为零的$p$-adic域$F$上的约化群，其中$p \gg 0$.在[Kim04]中，J. - L. Kim研究了在A. Moy和G.普拉萨德在[Kim04]之后，我们将正深度未加细的最小$K$-型的弱缔合类中的良好\(K\)-型的集合\(\overline{\mathfrak s}\)附加到$G(F)$的李代数$\mathfrak g(F)$的$G(F)$-不变开和闭子集$\mathfrak g(F)_{\overline{\mathfrak s}}$，以及\(G(F)\)的容许对偶\(\tilde G\)的子集$\tilde G_{\overline{\mathfrak s}}$，该子集由包含属于$\overline{\mathfrak s}$的未加细的最小$K$-型的那些表示组成。则\(\tilde G_{\overline{\mathfrak s}}\)是$G$的多个Bernstein分量的并集，因此我们可以考虑它所确定的Bernstein投影$E_{\overline{\mathfrak s}}$。我们表明，$E_{\overline{\mathfrak s}}$消失的Moy-Prasad $G(F)$-域$G(F)_r \subset G(F)$外，并重新制定的结果金说，$E_{\overline{\mathfrak s}}$的限制$G(F)_r$，推动通过对数的Moy-Prasad $G(F)$-域$\mathfrak g(F)_r \subset \mathfrak g(F)$，同意$\mathfrak g(F)_r$的特征函数的逆傅里叶变换的$\mathfrak g(F)_{\overline{\mathfrak s}}$。这是R给出的描述之一的变体。Bezrukavnikov，D. Kazhdan和Y. Varshavsky在arXiv：1504.01353的深度-$r$伯恩斯坦投影仪。
英文摘要：
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.
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