Double Coset Decompositions of Reductive Lie Groups Arising from Two Involutions
Double Coset Decompositions of Reductive Lie Groups Arising from Two Involutions
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两次对合产生的还原李群的双陪集分解
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发表时间：
1997
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通讯作者：
T. Matsuki
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作者：
T. Matsuki
(G)0和（G）0 L G。这里G = {g ∈ G| ρ（g）= g}，F_0表示F的连通分支，其中包含李群F的单位元e.在[8]中，我们仅用“代数”方法给出了几种典型三元组（G，H，L）的双陪集分解HG/L的标准表示。作为最基本的例子，我们研究了任意域IF的情形（G，H，L）=（GL（n，IF），GL（p，IF）×GL（n-p，IF），GL（r，IF）× GL（n-r，IF））。在[8]中，我们还研究了域IR、C和IH上向量空间上二次型的其他例子。本文假设G是一个约化李群，我们将描述任意（G，H，L）的双陪集分解HG/L的结构。首先我们研究紧的情形（第3节），然后研究非紧的情形（第4节）.在第三节中，我们假设G的半单部分Gs是紧的。(Gs是G的解析子群，其中gs = [g，g]。当G = Gs时，
(G)0 ⊂ H ⊂ G and (G )0 ⊂ L ⊂ G . Here G = {g ∈ G | ρ(g) = g} for an automorphism ρ of G and F0 denote the connected component of F containing the identity e for a Lie group F . In [8], we gave standard representatives of double coset decompositions HG/L for some typical triples (G,H,L) by only using “algebraic” method. As the most elementary example, we studied the case (G,H,L) = (GL(n, IF), GL(p, IF)×GL(n−p, IF), GL(r, IF)× GL(n − r, IF)) for an arbitrary field IF. We also studied in [8] other examples related to quadratic forms on vector spaces over the fields IR,C and IH. In this paper, we assume that G is a reductive Lie group and we will describe the structure of the double coset decomposition HG/L for an arbitrary (G,H,L). First we study the compact case (Section 3) and next the noncompact case (Section 4). In Section 3, we assume that the semisimple part Gs of G is compact. (Gs is the analytic subgroup of G for gs = [g, g].) When G = Gs and