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arXiv 1808.09503math.RTmath.NT
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模$p$ Iwahori-Hecke ${\operatorname{Ext}}$ -代数

The modular pro-$p$ Iwahori-Hecke ${\operatorname{Ext}}$-algebra

Rachel Ollivier, Peter Schneider

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摘要：

设$\mathfrak F$是具有正剩余特征的局部紧非阿基米德域，$k$是具有特征的域.设$G$是$\mathfrak F$ -可裂的$\mathfrak{F}$上的连通约化群的$\mathfrak{F}$ -有理点群.给定$G$的一个pro-$p$ Iwahori子群$I$，考虑$G/I$上紧支集的$k$值函数空间$\mathbf X$.它自然是$G$的所有光滑$k$ -表示的范畴${\operatorname{Mod}}{(G)}$中的对象。研究了分次Ext-代数$E^*={\operatorname{Ext}}_{\operatorname{Mod}(G)}^*(\mathbf X, \mathbf X)$ .它的零度片$E^0$是通常的亲$p$ Iwahori-Hecke代数$H$。我们在$E^*$中描述了乘积，并给出了$E^*$的一个对合反自同构。当$I$是一个维数为$d$的Poincaré群时，${\operatorname{Ext}}$ -代数$E^*$在次数$i\in\{0\dots d\}$上得到支持，并建立了$E^i$与$E^{d-i}$之间的对偶定理.在同样的假设下（并假设$\mathbf G$几乎是单连通的），我们计算$E^d$作为一个$H$ -模在左边和右边。我们证明了它是平凡特征标的直和，也是超奇异模的直和。

英文摘要：

Let $\mathfrak F$ be a locally compact nonarchimedean field of positive residue characteristic $p$ and $k$ a field of characteristic $p$. Let $G$ be the group of $\mathfrak{F}$-rational points of a connected reductive group over $\mathfrak{F}$ which we suppose $\mathfrak F$-split. Given a pro-$p$ Iwahori subgroup $I$ of $G$, we consider the space $\mathbf X$ of $k$-valued functions with compact support on $G/I$. It is naturally an object in the category ${\operatorname{Mod}}{(G)}$ of all smooth $k$-representations of $G$. We study the graded Ext-algebra $E^*={\operatorname{Ext}}_{\operatorname{Mod}(G)}^*(\mathbf X, \mathbf X)$. Its degree zero piece $E^0$ is the usual pro-$p$ Iwahori-Hecke algebra $H$. We describe the product in $E^*$ and provide an involutive anti-automorphism of $E^*$. When $I$ is a Poincaré group of dimension $d$, the ${\operatorname{Ext}}$-algebra $E^*$ is supported in degrees $i\in\{0\dots d\}$ and we establish a duality theorem between $E^i$ and $E^{d-i}$. Under the same hypothesis (and assuming that $\mathbf G$ is almost simple and simply connected), we compute $E^d$ as an $H$-module on the left and on the right. We prove that it is a direct sum of the trivial character, and of supersingular modules.

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