论文标题
一类P-ADIC对称空间的本地Zeta功能
Local Zeta Functions for a class of p-adic symmetric spaces
论文作者
论文摘要
这是即将发表的论文的第一部分的扩展版本，我们将研究最小球形系列的局部ZETA函数，用于对称空间，这是在P-Adic领域上作为抛物线前的抛物线前固定前空间的开放轨道。 Nicole BOPP和第二作者（[7]）已经考虑了地面场为$ \ Mathbb {R} $的情况。如果$ f $是一个特征性$ 0 $的P-ADIC领域，我们认为$ f $上的还原性LIE代数$ \ wideTilde {\ Mathfrak {\ Mathfrak {g}} $上的$ f $，它具有短$ \ Mathbb {Z} $ - 分级 - 级别 - $ \ \ \ \ \ \ \ \ \ \ tidetilde {\ Mathfrak {\ Mathfrak { \ Mathfrak {g} _ { - 1} \ oplus \ mathfrak {g} _ {0} \ oplus \ mathfrak {g} _1 $。我们还假设表示$（\ mathfrak {g} _0，\ mathfrak {g} _1）$是绝对不可修复的。在所谓的规律性条件下，我们研究了$ g_ {0} $的轨道，in $ \ mathfrak {g} _ {1} $，其中$ g_ {0} $是$ f $定义的代数组，其lie algebra as lie algebra as $ \ mathfrak {g} g} _ _ {0} $。我们还调查了$ p $ -Orbits，其中$ p $是$ g $的最小$σ$σ$ split抛物线子组（$σ$是在任何开放$ g_ {0} $ - $ \ sathfrak {g} _1 $中定义对称空间结构的相关性。
This is an extended version of the first part of a forthcoming paper where we will study the local Zeta functions of the minimal spherical series for the symmetric spaces arising as open orbits of the parabolic prehomogeneous spaces of commutative type over a p-adic field. The case where the ground field is $\mathbb{R}$ has already been considered by Nicole Bopp and the second author ([7]). If $F$ is a p-adic field of characteristic $0$, we consider a reductive Lie algebra $\widetilde{\mathfrak{g}}$ over $F$ which is endowed with a short $\mathbb{Z}$-grading: $\widetilde{\mathfrak{g}} = \mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus \mathfrak{g}_1$. We also suppose that the representation $(\mathfrak{g}_0, \mathfrak{g}_1)$ is absolutely irreducible. Under a so-called regularity condition we study the orbits of $G_{0}$ in $\mathfrak{g}_{1}$, where $G_{0}$ is an algebraic group defined over $F$, whose Lie algebra is $\mathfrak{g}_{0}$. We also investigate the $P$-orbits, where $P$ is a minimal $σ$-split parabolic subgroup of $G$ ($σ$ being the involution which defines a structure of symmetric space on any open $G_{0}$-orbit in $\mathfrak{g}_1$).
