scholarly journals On the microlocal structure of regular simple prehomogeneous vector space $(GL(1)^{2}\times SL(7), \Lambda_{3}＋\Lambda^{*}_{1})$

2000 ◽  
Vol 24 (1) ◽  
pp. 209-219
Author(s):  
Shin-ichi Kasai
Keyword(s):  
Vector Space ◽  
Prehomogeneous Vector Space

GLOBAL ZETA FUNCTIONS OF FREUDENTHAL QUARTICS

2002 ◽  
Vol 13 (08) ◽  
pp. 797-820
Author(s):  
HIROSHI SAITO
Keyword(s):  
Vector Space ◽  
Zeta Function ◽  
Explicit Formula ◽  
Explicit Form ◽  
Zeta Functions ◽  
The Other ◽  
Vector Spaces ◽  
Prehomogeneous Vector Space ◽  
Prehomogeneous Vector Spaces

We give two applications of an explicit formula for global zeta functions of prehomogeneous vector spaces in Math. Ann.315 (1999), 587–615. One is concerned with an explicit form of global zeta functions associated with Freudenthal quartics, and the other the comparison of the zeta function of a unsaturated prehomogeneous vector space with that of the saturated one obtained from it.

scholarly journals A classification of irreducible prehomogeneous vector spaces and their relative invariants

1977 ◽  
Vol 65 ◽  
pp. 1-155 ◽  
Author(s):  
M. Sato ◽  
T. Kimura
Keyword(s):  
Vector Space ◽  
Linear Algebraic Group ◽  
Vector Spaces ◽  
Rational Representation ◽  
Finite Dimensional Vector Space ◽  
Prehomogeneous Vector Space ◽  
Finite Dimensional ◽  
Prehomogeneous Vector Spaces ◽  
Relative Invariants

LetGbe a connected linear algebraic group, andpa rational representation ofGon a finite-dimensional vector spaceV, all defined over the complex number fieldC.We call such a triplet (G, p, V) aprehomogeneous vector spaceifVhas a Zariski-denseG-orbit. The main purpose of this paper is to classify all prehomogeneous vector spaces whenpis irreducible, and to investigate their relative invariants and the regularity.

scholarly journals Zeta functions of prehomogeneous affine spaces

1993 ◽  
Vol 132 ◽  
pp. 91-114
Author(s):  
Atsushi Murase ◽  
Takashi Sugano
Keyword(s):  
Vector Space ◽  
Transformation Group ◽  
Zeta Functions ◽  
Preceding Paper ◽  
Linear Algebraic Group ◽  
Algebraic Subset ◽  
Affine Spaces ◽  
Finite Dimensional Vector Space ◽  
Prehomogeneous Vector Space ◽  
Finite Dimensional

Let ρ be an algebraic homomorphism of a linear algebraic group G into the affine transformation group Aff(V) of a finite dimensional vector space V. We say that a triplet (G, V, ρ) is a prehomogeneous affine space, if there exists a proper algebraic subset S of V such that V — S is a single ρ(G)-orbit. In particular, (G, V, ρ) is a usual prehomogeneous vector space (PV, briefly) in the case where ρ(G) ⊂ GL(V) (cf. [5], [7]). In the preceding paper [2], we defined zeta functions associated with certain prehomogeneous affine spaces and proved their analytic continuation and functional equations.

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