相对镜像重数1的分布处理
A Distributional Treatment of Relative Mirabolic Multiplicity One
摘要：
我们研究了$G=\mathbf{GL}_n(F)$（$F$一$p$ -进域）的镜像子群$P$在$G$的光滑不可约表示中的作用，该不可约表示具有相对于$H_{k} = \mathbf{GL}_k(F)\times \mathbf{GL}_{n-k}(F)$形式的子群的非零不变泛函。证明了若表示上存在非零的$H_1$不变泛函，则每个$P\cap H_1$不变泛函必等于它的标量倍，当$k>1$时，我们将同一问题约化为对称空间$G/H_k$的切空间的幂零锥上的不变分布问题.发展了一些新的分配方法，这些方法适用于非约化群的情形。
英文摘要：
We study the role of the mirabolic subgroup $P$ of $G=\mathbf{GL}_n(F)$ ($F$ a $p$-adic field) in smooth irreducible representations of $G$ that possess a non-zero invariant functional relative to a subgroup of the form $H_{k} = \mathbf{GL}_k(F)\times \mathbf{GL}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant functional exists on a representation, then every $P\cap H_1$-invariant functional must equal to a scalar multiple of it. When $k>1$, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone of the tangent space of the symmetric space $G/H_k$. Some new distributional methods, which are suitable for a setting of non-reductive groups, are developed.
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