齐性Siegel域的自同构群

On Automorphisms of Homogeneous Siegel Domains

令Ｄ表示有界齐性Ｓｉｅｇｅｌ域，Ｇ（Ｄ）是Ｄ的自同构群，ｇ（Ｄ）是关于Ｇ（Ｄ）的李代数，则对ｇ（Ｄ），Ｓ．Ｍｕｒａｋａｍｉ得到下列直和：ｇ（Ｄ）＝ｇ－１＋ｇ－１／２十ｇ０十ｇ１／２＋ｇ１其中ｇ－１，ｇ－１／２和ｇ０是大家熟知的，本文我们给出ｇ１／２和ｇ１的构造．即在非常弱的条件下，我们证明了ｇ１／２＝｛０｝和ｇ１＝｛∑Ｐ２０Ｋ｝．同时，我们给出一些Ｓｉｅｇｅｌ域的例子，它们的自同构群可以显式给出．

Let D denote a bounded homogeneous Siegel domain, G (D) the group of automorphisms of D, and g (D) the Lie algebra associated to G (D), then for g (D), Shingo Murakami has got the direct sum (see 'Lecture Notes in Mathematics' 286):g (D) =g-1+g-1/2+g0+g1/2 +g1the g-1, g-1/2 and g0 are well known. In this paper, we describe the structure of g1/2 and g1, i. e.we prove that g1/2={0} and g1 = { P20k k} under some very weak conditions. In the meank=1time, we give some examples of Siegel domains, in which the groups of automorphisms are exactly given.