b-Function of a Prehomogeneous Vector Space with No Regular Component
b-Function of a Prehomogeneous Vector Space with No Regular Component
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无正则分量的预齐次向量空间的 b 函数
DOI：
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发表时间：
2005
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通讯作者：
K. Sugiyama
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文献类型：
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作者：
K. Sugiyama
本文的目的是确定正则2-简单预齐次向量空间（SL5 × GL9， Λ2⊗Λ1 + 1⊗Λ * 1）的b函数。在利用微局部微积分确定不可约的预齐次向量空间的b函数（cf. M. Sato, Kashiwara, Kimura and Oshima [13], Kimura[5]）之后，同样的方法被应用于一些可约的预齐次向量空间（cf. Kasai[3]）。由于轨道结构的复杂性，将微局部微积分应用于可约预齐次向量空间需要更多的复杂计算。最近，Ukai[16]证明了可约正则预齐次向量空间的b函数可以通过结合结构定理、泛函方程、局部b函数和一元b函数的已知结果的相当初等信息来确定。利用这种方法，Ukai计算了由异常简单李代数的幂零元引起的预齐次向量空间的b函数。受Ukai工作的启发，在与Wakatsuki[17]的联合项目中，我们确定了(I)型的简化正则2-简单预齐次向量空间的b函数，这些向量空间由Kimura， Kasai， Inuzuka和Yasukura分类（完整列表参见[8，pp. 395-398]），除了以下三种情况：(a) （GL1 × SL5 × SL8， Λ2⊗Λ1 + 1⊗Λ∗1）。(b) （gl1 × sl5 × sl9， Λ2⊗Λ1 + 1⊗Λ * 1）。(C) (GL1×Spin10×SL15, (half-spin众议员)⊗Λ1 + 1⊗Λ∗1)。计算这些空间的b函数的困难来自于它们没有规则分量的事实。这里，我们所说的约化预齐次向量空间（G, ρ， V）的正则分量，是指由ρ: G→GL(V)的适当子表示σ: G→GL(E)所定义的正则预齐次向量空间（G, σ，E）。当不存在规则分量时，单纯的局部化应用不能提供足够的信息来确定b函数，并且以前已知的单变量b函数的结果不适用。
The purpose of the present paper is to determine the b-function of the regular 2-simple prehomogeneous vector space (SL5 × GL9, Λ2 ⊗ Λ1 + 1 ⊗ Λ∗1). After the b-functions of irreducible prehomogeneous vector spaces were determined by using the microlocal calculus (cf. M. Sato, Kashiwara, Kimura and Oshima [13], Kimura [5]), the same method was applied to some reducible prehomogeneous vector spaces (cf. Kasai [3]). It then has been recognized that more involved calculations are necessary to apply the microlocal calculus to reducible prehomogeneous vector spaces because of the complexity of orbit structures. Recently, Ukai [16] showed that b-functions of reducible regular prehomogeneous vector spaces can be determined by combining rather elementary information from structure theorems, functional equations, local b-functions, and the known results on b-functions of one variable. By using this method, Ukai calculated the b-functions of prehomogeneous vector spaces arising from nilpotent elements of exceptional simple Lie algebras. Inspired by Ukai’s work, in a joint project with Wakatsuki [17], we have determined the b-functions of the reduced regular 2-simple prehomogeneous vector spaces of type (I), which are classified by Kimura, Kasai, Inuzuka and Yasukura (see [8, pp. 395–398] for the complete list), except for the following three cases: (A) (GL1 × SL5 × SL8, Λ2 ⊗ Λ1 + 1 ⊗ Λ∗1). (B) (GL1 × SL5 × SL9, Λ2 ⊗ Λ1 + 1 ⊗ Λ∗1). (C) (GL1 × Spin10 × SL15, (half-spin rep.) ⊗ Λ1 + 1 ⊗ Λ∗1). The difficulty in calculating the b-functions of these spaces arises from the fact that they have no regular components. Here, by a regular component of a reductive prehomogeneous vector space (G, ρ, V ), we mean a regular prehomogeneous vector space (G, σ,E) defined by some proper subrepresentation σ : G → GL(E) of ρ : G → GL(V ). When there are no regular components, naive application of localization does not give us enough information to determine b-functions, and previously known results on b-functions of one variable are not applicable.
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发表时间：
2007
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作者：
Tomoyoshi Ibukiyama;Hidenori Katsurada;Yumiko Hironaka;木村 達雄;Tatsuo Kimura;佐藤 文広;佐藤 文広;伊吹山 知義;Tomoyoshi Ibukiyama;佐藤 文広;佐藤 文広;Fumihiro Sato;広中 由美子;伊吹山 知義;Tomoyoshi Ibukiyama;広中 由美子;広中 由美子;Yumiko Hironaka;木村 達雄;Tatsuo Kimura
通讯作者：
Tatsuo Kimura
I型非正则2-简预齐次向量空间及其相对不变量
DOI：
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发表时间：
2002
期刊：
J.Algebra 251
影响因子：
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作者：
T.Kogiso;G.Miyabe;M.Kobayashi;T.Kimura
通讯作者：
T.Kimura
DOI：
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发表时间：
2003
期刊：
T.M.M.(American Mathematical Society) vol.215
影响因子：
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作者：
T.Kogiso;G.Miyabe;M.Kobayashi;T.Kimura;T.Kimura
通讯作者：
T.Kimura