Tatsuo Kimura

A classification of 2-simple prehomogeneous vector spaces of type I

Let p: G + GL( V) be a rational representation of a connected linear algebraic group G on a finite-dimensional vector space V, all defined over an algebraically closed field K of characteristic zero. If V has a Zariskidense G-orbit, we call a triplet (G, p, V) a prehomogeneous oector space (abbrev. P.V.). When p is irreducible, such P.V.s have been classified in [ 11. Since then, it has turned out gradually that the complete classification of reductive P.V.s (i.e., P.V.s with reductive groups G) is an extremely laborious task. Therefore it is natural to classify some restricted class of P.V.s (e.g., [2]) to get some insight into the general situation. A P.V. (G, p, V) is called a 2-simple P. V. when ( 1) G = GL( 1)’ x G, x G, with simple algebraic groups G, and Gz, (2) p is the composition of a rational representation p’ of G, x G, of the form p’ = p1 0 p’, + . . . + PkOL&+(a,+ ... +a,)@l+l@(z,+ ... +t,) with k+s+t=l, where pi, (TV (resp. pi, T,) are nontrivial irreducible representations of G, (resp. G,), and the scalar multiplications GZ,( 1)’ on each irreducible component V, for i = 1, . . . . f, where V= I/, @ . . . @ V,. We say that a 2-simple P.V. (G,p, V) is of type Zif k>l and at least one of (GL(l)xG,xG,, pi@ pi) (i = 1, . . . . k) is a nontrivial P.V. (see Definition 5, p. 43 in [ 11). On the other hand, if k>l and all (GL(l)xG,xG,, p,Op() (i=l,...,k) are trivial P.V.s, it is called a 2-simple P.V. of type II. In [3], all 2-simple P.V.s of type II has been already classified. In this paper, we shall classify all 2simple P.V.s of type I. Thus, together with [3], we complete a classification of all 2-simple P.V.s. For example, the fact that all irreducible P.V.s are castling-equivalent to 2-simple P.V.s (or to (,X(m) x &Y(m) x G,!,(2), /1 1 @ /i I @ n 1 ) with m = 2,3) (see [ 1 ] ) indicates the importance of 2-simple P.V.s. For simplicity, we write (G, p’, V) or (G, p’) instead of (G, p, V).

A Classification Theory of Prehomogeneous Vector Spaces

This chapter presents a classification theory of pre-homogeneous vector spaces. It presents an assumption where G is a connected linear algebraic group, ρ a rational representation of G on a finite-dimensional vector space V. When V has a Zariski-dense G -orbit Y , one says that a triplet ( G, ρ, V) is a pre-homogenous vector space (P.V.). A point of Y is called a generic point. However, the converse is not true in general. As this condition is very strong, one can classify all such P.V. under the assumption that G is reductive without assuming the irreducibility of p. A P.V. with a finitely many orbits is called a finite P. V. It discusses indecomposable commutative Frobenius algebras and δ -functions.

On M. Sato's Classification of Some Reductive Prehomogeneous Vector Spaces

Under some condition, M. Sato classified reductive prehomogeneous vector spaces of the form (G0 × G,Λ1 ⊗ ρ, V (n) ⊗ V ). In this paper, under another condition, we classify the prehomogeneous vector spaces of the same form. We consider everything over the complex number field C. 2010 Mathematics Subject Classification: Primary 11S90; Secondary 20G20.

Relative invariants of some 2-simple prehomogeneous vector spaces

In this paper, we shall construct explicitly irreducible relative invariants of two 2-simple prehomogeneous vector spaces. Together with a preprint by the same authors, this completes the list of all relative invariants of regular 2-simple prehomogeneous vector spaces of type I.

Fundamental theorem of prehomogeneous vector spaces of characteristic p

For a local field of characteristic 0, the functional equations of zeta distributions of prehomogeneous vector spaces have been obtained by M. Sato, Shintani, Igusa, F. Sato and Gyoja. In this paper, we shall consider the case of local fields of characteristic p > 0.

Complex Powers on p-adic Fields and a Resolution of Singularities

Publisher Summary This chapter discusses the complex powers on p-adic fields and a resolution of singularities. The chapter also discusses the integration formula corresponding to the blowing-up at the origin, and the resolution of singularities modulo π. Various theorems are proven in the chapter, and several propositions are discussed.

On prehomogeneity of a rank variety

If a linear algebraic group G acts on M(m,n), then it also acts on a rank variety M (r)(m,n) = {X ∈ M(m,n)| rankX = r}. In this paper, we give the necessary and sufficient condition that this variety has a Zariski-dense G-orbit. We consider everything over the complex number field C. Introduction Let G be a linear algebraic group. Let ρ : G → GL(m) and σ : G → GL(n) be its rational representations over C. Then G acts on M(m,n) by ρ ⊗ σ, i.e., X → ρ(g)Xσ(g) (X ∈ M(m,n), g ∈ G). By this action, G also acts on a rank variety M (m,n) = {X ∈ M(m,n)| rankX = r}. On the other hand, if a rational representation τ : H → GL(V ) of a linear algebraic group H on a finitedimensional vector space V has a Zariski-dense H-orbit, we call a triplet (H, τ, V ) a prehomogeneous vector space (abbrev. PV). A point of the Zariski-dense H-orbit is called a generic point, and the isotropy subgroup at a generic point is called a generic isotropy subgroup. For the basic facts of PVs, see [K]. In this paper, we shall prove the following theorem. Theorem 0.1. The following assertions are equivalent. (1) M (m,n) has a Zariski-dense G-orbit by the action ρ⊗ σ. (2) (G×GL(r), ρ⊗ Λ1 + σ ⊗ Λ1, M(m, r)⊕M(n, r)) is a PV. Here the action of (2) is given by (X,Y ) → (ρ(g)XA, σ(g)Y A−1) for (X,Y ) ∈ M(m, r)⊕M(n, r)) and (g,A) ∈ G×GL(r). Note that we have m ≥ r and n ≥ r in (2). 1. Proof of theorem The following lemma is the key for our proof. Lemma 1.1 (M. Sato). Assume that an algebraic group G acts on both of two irreducible algebraic varieties W and W ′. Let φ : W → W ′ be a morphism satisfying (I) φ(gw) = gφ(w) (g ∈ G,w ∈ W ), (II) φ(W ) = W ′. Received by the editors May 26, 2011. 2010 Mathematics Subject Classification. Primary 11S90; Secondary 15A03.