AI 中文总结
本文引入p进域上分裂环面的Brylinski-Deligne覆盖的伪球表示概念，枚举并计算其维数，证明覆盖环面不可约真表示限制到极大紧子群重数自由，还描述了相关Hecke代数，为G的BD覆盖的Borel-Casselman定理提供环面输入。
AI 中文摘要
我们引入了p进域上分裂环面的任意Brylinski-Deligne（BD）覆盖的伪球表示概念，特别地，不假设该覆盖是驯顺的。我们对伪球表示进行了枚举并计算了其维数。当该环面是连通分裂约化群G的极大分裂环面时，在BD数据的温和假设下，我们枚举了一类特殊的伪球表示。我们期望这些特殊表示能为G的BD覆盖的Borel-Casselman定理提供环面输入，且我们已在另一项工作中对某些非驯顺二重覆盖验证了该期望。在研究伪球表示的过程中，我们还证明了覆盖环面的不可约真表示限制到其极大紧子群时是重数自由的，由此可知该极大紧子群的不可约真表示是Bushnell-Kutzko意义下的型。最后，我们描述了与这些型相关的Hecke代数。
英文摘要
We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a $p$-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group $G$, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of $G$. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.
