For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$ , after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .

中文翻译：

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HECKE 类别、字符组和表示的内窥镜检查

对于还原组 $G$ 在有限域上，我们证明了在环面作用下具有固定单向性的混合 Hecke 范畴的中性块单曲面等价于相应内窥镜组的混合 Hecke 范畴 $H$ 与琐碎的单调。我们还将这种等价性扩展到所有块。我们给出两个应用程序。一个是角色之间的关系 $G$ 内窥镜组具有固定的半单参数和单能字符滑轮 $H$ ，在传递到渐近版本之后。另一种是表示之间的类似关系 $G(\mathbb{F}_{q})$ 具有固定的半简单参数和单能表示 $H(\mathbb{F}_{q})$ .