On some explicit formulas in the theory of Weil representation
On some explicit formulas in the theory of Weil representation
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DOI：
10.2140/pjm.1993.157.335
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发表时间：
1993-02
影响因子：
0.6
通讯作者：
R. R. Rao-R.
中科院分区：
文献类型：
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作者：
R. R. Rao-R.
本文的目的是推导出一些关于Weil表示的显式公式，使我们能够对每种辛基的选择以唯一的方式定义这种射影表示。设F是特征φ2的自对偶局部紧域，X是F上的辛向量空间，V，V*是两个横截面拉格朗日子空间.然后，由Shale-SegalWeil得到的经典构造给出了辛群Sp(JSΓ)在V的Schwartz空间中的射影表示。对应于每个ξ(σe Sp(X)的算子σ)只被唯一地确定到标量倍数。本文的出发点是这些算子ξ(σ)的一个显式积分公式，它对所有的σe Sp(X)都有效。事实上(见引理3.2)对于每个σe Sp(ΛΓ)ξ(σ)φ：x-+fσ(x，x*)φ(xa+x*y)dμσJv*/kevγ
The object of this paper is to derive some explicit formulae concerning the Weil representation that allow us to define this projective representation in a unique manner for each choice of symplectic basis. Let F be a self-dual locally compact field of char φ 2 and X a symplectic vector space over F. Let V, V* be two transversal Lagrangian subspaces. Then a classical construction due to Shale-SegalWeil gives a projective representation of the symplectic group Sp(JSΓ) in the Schwartz-space of V. The operators ξ(σ) corresponding to each σ e Sp(X) are determined uniquely only up to a scalar multiple. The starting point of this paper is an explicit integral formula for these operators ξ(σ), valid for all σ e Sp(X). In fact (see Lemma 3.2) we have for each σ e Sp(ΛΓ) ξ(σ)φ :x-+ fσ(x, x*)φ(xa + x*y) dμσ JV*/kevγ