调和分析中的正交辛超群
The orthosymplectic supergroup in harmonic analysis
摘要：
正交辛超群OSp（m| 2n）被引入作为平坦黎曼超空间R^{m| 2n}稳定原点。它也对应于超球面S^{m-1}的等距超群|2n}。R^{m}上的Laplace算子和范数平方|2n}是正交辛不变的，因此我们得到了Howe对偶对（osp（m| 2n），sl（2））。这个Howe对偶解决了在以前的论文中考虑的对偶对（SO（m）xSp（2n），sl（2））的问题。特别地，我们刻画了平坦黎曼超空间上的不变函数，并证明了超球面上的积分由其正交辛不变性唯一定义。超球流形也引入了严格的数学方法。最后研究了Osp（m）的表示|2n）的球面调和函数。这对应于m的超对称张量空间的分解|sl（2）xosp（m）作用下的2n维超向量空间|2n）。作为一个副结果，我们获得了关于不可约osp（m）的信息|2n）-表示L_{(k,0,...,0)} ^{m| 2n}。特别地，我们发现关于OSP（m-1）的分支规则|2n）。
英文摘要：
The orthosymplectic supergroup OSp(m|2n) is introduced as the supergroup of isometries of flat Riemannian superspace R^{m|2n} which stabilize the origin. It also corresponds to the supergroup of isometries of the supersphere S^{m-1|2n}. The Laplace operator and norm squared on R^{m|2n}, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m|2n),sl(2)). This Howe dual pair solves the problems of the dual pair (SO(m)xSp(2n),sl(2)), considered in previous papers. In particular we characterize the invariant functions on flat Riemannian superspace and show that the integration over the supersphere is uniquely defined by its orthosymplectic invariance. The supersphere manifold is also introduced in a mathematically rigorous way. Finally we study the representations of osp(m|2n) on spherical harmonics. This corresponds to the decomposition of the supersymmetric tensor space of the m|2n-dimensional super vectorspace under the action of sl(2)xosp(m|2n). As a side result we obtain information about the irreducible osp(m|2n)-representations L_{(k,0,...,0)}^{m|2n}. In particular we find branching rules with respect to osp(m-1|2n).
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