论文标题
$（{\ Mathfrak g}，k）$ - $ {\ mathrm o}的模块（p，q）$
Annihilator of $({\mathfrak g},K)$-modules of ${\mathrm O}(p,q)$
论文作者
论文摘要
令$ {\ mathfrak g} $表示复杂的lie代数为$ g = {\ mathrm o}（p，q）$和$ k $ a最大紧凑型亚组为$ g $。在上一篇论文中，我们构建了$（{\ Mathfrak g}，k）$ - 与$ {\ Mathfrak sl} _2 $ m+1 $的有限维表示相关的模块，我们用$ m^{+}（+}（m）$和$ m^{ - } { - }（ - }（m）$。本文的目的是证明$ m^{\ pm}（m）$的歼灭者是约瑟夫的理想，并且仅当$ m = 0 $。我们将看到，根据$ {\ Mathfrak G} $的Casimir元素给出的Square $ S^{2}（{\ Mathfrak G}）$的对称元素，$ k $的复杂的lie代数在主要结果的证明中起着关键作用。
Let ${\mathfrak g}$ denote the complexified Lie algebra of $G={\mathrm O}(p,q)$ and $K$ a maximal compact subgroup of $G$. In the previous paper, we constructed $({\mathfrak g},K)$-modules associated to the finite-dimensional representation of ${\mathfrak sl}_2$ of dimension $m+1$, which we denote by $M^{+}(m)$ and $M^{-}(m)$. The aim of this paper is to show that the annihilator of $M^{\pm}(m)$ is the Joseph ideal if and only if $m=0$. We shall see that an element of the symmetric of square $S^{2}({\mathfrak g})$ that is given in terms of the Casimir elements of ${\mathfrak g}$ and the complexified Lie algebra of $K$ plays a critical role in the proof of the main result.
