A classification of irreducible prehomogeneous vector spaces and their relative invariants

Sato, Masahide; Kimura, Tatsuo

doi:10.1017/s0027763000017633

Cited by 441 publications

(543 citation statements)

References 11 publications

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“…2δi corresponds to the character det ρ(g) 2 (see [19,Prop. 8] Finally we recall the fundamental theorem of the theory of prehomogeneous vector spaces over the real number field R. For i = 1, .…”

Section: L-functions Of Prehomogeneous Vector Spaces (By Fumihiro Sato)mentioning

confidence: 99%

The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)

Cluckers¹,

Herremans²

2007

Bul. Soc. Math. France

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Section: L-functions Of Prehomogeneous Vector Spaces (By Fumihiro Sato)mentioning

confidence: 99%

The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)

Cluckers¹,

Herremans²

2007

Bul. Soc. Math. France

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“…Such an object is called reduced in the terminology of Sato and Kimura. When k is algebraically closed of characteristic zero, the set of all reduced irreducible regular prehomogeneous vector spaces has been divided into 29 types by Sato and Kimura [17]. If k is of characteristic zero and one restricts to k-split groups, then the absolute classification restricts to a relative one.…”

Section: F Loesermentioning

confidence: 99%

“…As in 4.1, we consider the quotient spaces X i / SL r i as embedded by the Plücker embedding the same affine space V , and up to renumbering the f 1,i 's and multiplying them by non-zero constants, one may assume that both the f 1,i 's and the f 2,i 's are the pullback of the same homogeneous polynomials f i of degree d i in V , cf. [17], [6]. In particular, we may write 1 = 2 = .…”

Section: The Hodge Spectrummentioning

confidence: 99%

Motivic zeta functions for prehomogeneous vector spaces and castling transformations

Loeser¹

2003

Nagoya Mathematical Journal

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Abstract. We study the behaviour of motivic zeta functions of prehomogeneous vector spaces under castling transformations. In particular we deduce how the motivic Milnor fibre and the Hodge spectrum at the origin behave under such transformations. §1. Introduction 1.1. Let us recall the classification of irreducible regular prehomogeneous vector spaces due to Sato and Kimura [18]. A prehomogeneous vector space over a field k [18] consists of the datum (G, X) of a connected linear k-algebraic group G together with a linear representation ρ of G on a finite dimensional affine space X = A m k over k having a Zariski dense G-orbit. The complement of the Zariski dense G-orbit is called the singular locus of the prehomogeneous vector space. When X is an irreducible G-module, one says the prehomogeneous vector space (G, X) is irreducible.One may find a general definition of regular prehomogeneous vector spaces in [18], which, when the group G is reductive, is equivalent to the condition that the singular locus is an hypersurface. From now on, except for Section 7, we shall always assume the group G to be reductive. When (G, X) is an irreducible regular prehomogeneous vector space, the singular locus is an irreducible hypersurface f = 0 in X.In the theory of Sato and Kimura, a fundamental role is played by castling transforms, which are defined as follows.Let G be a connected reductive k-algebraic group and ρ be a linear representation of G on A m k . Write m = r 1 + r 2 . Then set X 1 = M m,r 1 and let G × SL r 1 act on X 1 by ρ 1 (g, g 1 )x 1 = ρ(g)x 1 t g 1 .

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“…It is known that P l9 , P n are relative invariants of (G, p, V) (cf. [1]). The set {P l9 , P n } is called a complete set of irreducible relative invariants of (G, p, V).…”

Section: P( P (G)x) = X(g)p(x)mentioning

confidence: 99%

The functional equation of zeta distributions associated with prehomogeneous vector spaces

Teranishi

1985

Nagoya Mathematical Journal

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A classification of irreducible prehomogeneous vector spaces and their relative invariants

Cited by 441 publications

References 11 publications

The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)

The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)

Motivic zeta functions for prehomogeneous vector spaces and castling transformations

The functional equation of zeta distributions associated with prehomogeneous vector spaces

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