Abstract:Let $G$ be a real linear reductive group and $K$ be a maximal compact subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie algebra $\mathfrak {p}$, and $\mathfrak {n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V)$ ($i\geq 0$), where $V_{K}$ denotes the $(\mathfrak {g},K)$ module of $K$ finite vectors in $V$. This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any $k\geq 1$, $\mathfrak {n}^{k}V$ is a closed subspace of $V$ and the inclusion $V_{K}\subset V$ induces an isomorphism $V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V$. This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).

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中文翻译：

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卡塞尔曼比较定理的证明

摘要：设$G$为实线性约简群，$K$为极大紧子群。令 $P$ 是 $G$ 的最小抛物子群，具有复李代数 $\mathfrak {p}$，$\mathfrak {n}$ 是它的 nilradical。在本文中，我们表明：对于 $G$ 的任何可接受的有限生成的适度增长平滑 Fréchet 表示 $V$，包含 $V_{K}\subset V$ 诱导同构 $H_{i}(\mathfrak {n}, V_{K})\cong H_{i}(\mathfrak {n},V)$ ($i\geq 0$)，其中 $V_{K}$ 表示 $(\mathfrak {g},K)$ $V$ 中 $K$ 有限向量的模块。这被称为卡塞尔曼比较定理（参见亨利克赫克特和约瑟夫 L.泰勒 [关于卡塞尔曼比较定理的评论, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146])。因此，我们证明：对于任何 $k\geq 1$，$\mathfrak {n}^{k}V$ 是 $V$ 的封闭子空间，并且包含 $V_{K}\subset V$同构 $V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V$。这加强了 Casselman 的自动连续性定理（参见 W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] 和 Nolan R. Wallach [ Real reductive groups , Academic Press, Boston, MA, 1992]）。

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