AI 中文总结
研究将奥伯特和伯恩斯坦对偶性扩展到非连通约化\(p\)-adic群，证明相关函子性质，通过示例应用得到斯坦伯格表示定义、计算其特征并讨论扭曲内窥镜性质。
AI 中文摘要
我们将奥伯特定义的光滑有限长度复表示范畴上的对偶性，以及伯恩斯坦定义的上同调类似对偶性，扩展到任意非连通约化\(p\)-adic群，并证明了这些函子的各种性质，如唯一性、不可约性的保持、与抛物诱导和限制的兼容性以及特征公式。作为一个示例应用，我们得到了非连通约化\(p\)-adic群的斯坦伯格表示的定义，计算了其特征，并讨论了其扭曲内窥镜性质。
英文摘要
We extend to arbitrary disconnected reductive $p$-adic groups the duality on the category of smooth finite-length complex representations defined by Aubert, as well as its cohomological analog defined by Bernstein, and prove various properties of these functors, such as uniqueness, preservation of irreducibility, compatibility with parabolic induction and restriction, and a character formula. As a sample application, we obtain a definition of the Steinberg representation for a disconnected reductive $p$-adic group, compute its character, and discuss its twisted endoscopic properties.
