Regular Prehomogeneous Vector Spaces Research Articles

Regular prehomogeneous vector spaces for valued Dynkin quivers

We introduce regular prehomogeneous vector spaces associated with an arbitrary valued Dynkin graph $(\Gamma, \boldsymbol{v})$ having a fixed oriented modulation $(𝔐, \Omega)$ over the ground field $K$. Here $K$ is of characteristic zero, but it may not be algebraically closed. We will construct a fundamental theory of such prehomogeneous vector spaces. Each generic point of a regular prehomogeneous vector space corresponds to a hom-orthogonal partial tilting $\Lambda$-module, where $\Lambda$ is the tensor $K$-algebra of $(𝔐, \Omega)$. We count the number of isomorphism classes of hom-orthogonal partial tilting $\Lambda$-modules of type $\mathbf{B}_n$, $\mathbf{C}_n$, $\mathbf{F}_4$ and $\mathbf{G}_2$. As a consequence of our theorem, we estimate lower and upper bounds for the number of basic relative invariants of regular prehomogeneous vector spaces for any valued Dynkin quiver.

Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

The aim of this paper is to study relations between regular reductive prehomogeneous vector spaces (PVs) with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_{2}$-triplet of a graded Lie algebra.

Hom-Orthogonal Partial Tilting Modules for Dynkin Quivers

We count the number of isomorphism classes of hom-orthogonal partial tilting modules over path algebras of Dynkin quiver of type 𝔸 n , 𝔻 n , 𝔼 n . This number is independent on the choice of an orientation of the arrows, and the number for 𝔸 n or 𝔻 n -type can be expressed as a special value of a hypergeometric function. As a consequence of our theorem, we obtain a minimum value of the number of basic relative invariants of corresponding regular prehomogeneous vector spaces.

Decomposition of reductive regular Prehomogeneous Vector Spaces

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a connected reductive algebraic group over C. If $V= \oplus_{i=0}^{n}V_{i}$ is a decomposition of V into irreducible representations, then, in general, the PV's $(G,V_{i})$ are no longer regular. In this paper we introduce the notion of quasi-irreducible PV (abbreviated to Q-irreducible), and show first that for completely Q-reducible PV's, the Q-isotopic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's of parabolic type.

Algebras of invariant differential operators on a class of multiplicity free spaces

Let G be a connected reductive algebraic group and let G ′ = [ G , G ] be its derived subgroup. Let ( G , V ) be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra D ( V ) G ′ of G ′ -invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to U ( sl 2 ) . Our result generalizes the case of the Weil representation, where the associative algebra generated by Q ( x ) and Q ( ∂ ) ( Q being a non-degenerate quadratic form on V) is a quotient of U ( sl 2 ) . Other structure results are obtained when ( G , V ) is a regular prehomogeneous vector space of commutative parabolic type. To cite this article: H. Rubenthaler, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Construction of irreducible relative invariant of the prehomogeneous vector space [formula omitted

We explicitly construct the irreducible relative invariant of the prehomogeneous vector space ( SL 5× GL 4,Λ 2( C 5)⊗ C 4) . This prehomogeneous vector space has been known as the “most difficult” case in irreducible regular prehomogeneous vector spaces.

Generalized character sums associated to regular prehomogeneous vector spaces

In this paper we consider the analogue of the Sato's functional equation for the prehomogeneous vector spaces over finite fields. The corresponding character sums depend on a relative invariant on such a space and an irreducible representation of the group of components of the stabilizer of a generic point. The proof is based on the Picard-Lefschetz formula in $l$-adique cohomology.

Racines orthogonales et orbites d'algèbres de Lie semi-simple graduées

We define a particular class of regular prehomogeneous vector spaces of parabolic type in relation with orthogonal roots, on a field of characteristic 0. We give the structure and the orbits of simple elements associated to the nonzero nilpotent elements of the prehomogeneous vector spaces in terms of these orthogonal roots.

On the Convergence of the Adelic Zeta Functions Associated to Irreducible Regular Prehomogeneous Vector Spaces

Introduction. Let (G, X) be an irreducible regular prehomogeneous vector space defined over an algebraic number field k. D denotes the derived group of G. X is an affine n-space Affn. f(x) E k[xl, ... ,xn] is the relative invariant. f(gx) = v(g)f(x) for every g E G, where vi E Hom (G,Gm). Y = X -f-1(0). The subscript A denotes the adelization relative to k. The subscript k denotes the taking of k-rational points. Let AUG be a Haar measure on GA. Consider the following integral

Highest weight modules associated with classical irreducible regular prehomogeneous vector spaces of commutative parabolic type

Highest weight modules associated with classical irreducible regular prehomogeneous vector spaces of commutative parabolic type

On the Microlocal Structure of the Regular Prehomogeneous Vector Space Associated with $\textit{SL}(5) \times \textit{GL}(4)$

Let p = A2 (x) AI be an irreducible representation of G = SX(5, C) x GL(4, C) on V=AC® C(= F(10) (x) F(4)). Then we have a Zariski-dense G-orbit, namely, the triplet (G, p, V) is a prehomogeneous vector space (abbrev. P.V.). All irreducible P.V.'s are completely classified (See [SK]), and the fc-functions of irreducible regular P.V.'s are already calculated (See [SKKO], [Ki], [KM], [KO], [O 2]) by using microlocal analysis (See [SKKO]) except the case of our P.V. (G, p, V) which has the most complicated microlocal structure among all the reduced irreducible regular P.V.'s (See [O 1]). The table of G-orbits in V was first given in [O 1]. However, one orbit 2 corresponding to a generic point of (SL(5) x GL(2), A2 ® A±,AC 5 (x) C) is missed in [O 1] (Remark 2.3). Later, Prof. N.Kawanaka gave the orbital decomposition of (G, p, V) by using a classification of nilpotent orbits of the exceptional Lie algebra E8 (See [Ka 1], [Ka 2]). In this paper, we shall give the detailed explanation of the original method used in [O 1] for the orbital decomposition of (G, p, V) (Proposition 2.1) and determine all good holonomic varieties (Theorem 3.4). These results are fundamental to investigate the microlocal structure of our P.V. from which we can get some information of the b-function (See [O 1]). We shall roughly explain our method. Let G' = SL(5, C) x GL(3, C), p' 2 = A2®Al, V =AC 5 (x) C. Then (G', p', V) is an irreducible P.V. and its orbital decomposition has been completed in [Ki]. By an injection GL(3, C)

Universal transitivity of certain classes of reductive prehomogeneous vector spaces

Let k be a field of characteristic zero. Let G be a connected &-split linear algebraic group, p: G^GL(X) with X=Affra a &-homomorphism. If there exists a Zariski-dense p(G)-orbit Y, we say that (G, p, X) is a prehomogeneous vector space (abbrev. P. V.). When each irreducible component is castling equivalent to a non-trivial reduced irreducible prehomogeneous vector space or each irreducible component is a regular prehomogeneous vector space, we have completed a classification of reductive prehomogeneous vector spaces over a complex number field C (see [4], [5]). Put G = p(G). Let / be the number of G(£)-orbits in Y(k), i. e., /= | G(k)\Y(k)\. We say that Y is a universally transitive open orbit if l=\G(k)\Y(k)\=l for all k satisfying H\k, Aut(SL2))=£0, i.e., there exists a nonsplit quaternion kalgebra. This condition is satisfied by every local field k other than C. Actually our classification depends on the transitivity of G(k) on Y(k) for just one k satisfying H\k, Aut(SL2))^0 (see Remarks 2.13 and 3.5). In [1] and [2], allirreducible regular prehomogeneous vector spaces with universally transitive open orbits are classified. In [10], we have classified simple or 2-simple prehomogeneous vector spaces with universally transitive open orbits. In this paper, we shall classify reductive prehomogeneous vector spaces with universally transitive open orbits when each irreducible component is castling equivalent to a non-trivial reduced irreducible prehomogeneous vector space or each irreducible component is a regular prehomogeneous vector space. This paper consists of the following three sections. §1. Preliminaries. §2, Reductive P.V.'s with universally transitive open orbits: the case I. §3. Reductive P.V.'s with universally transitive open orbits: the case II. The results are given in Theorems 2.11, 3.4 and Corollary 2.12.

The dimension of the space of relatively invariant hyperfunctions on regular prehomogeneous vector spaces

The dimension of the space of relatively invariant hyperfunctions on regular prehomogeneous vector spaces

Formes réelles des espaces préhomogènes irréductibles de type parabolique

M. Sato et T. Shintani’s theory associate to each real form of an irreducible and regular prehomogeneous vector spaces (where the group is reductive) a zeta function which has a remarkable functional equation. In this paper, we classify the infinitesimal real forms of the irreducible prehomogeneous vector spaces of parabolic type. The classification is obtained by means of weighted Stakaze diagrams.

La surjectivité de l'application moyenne pour les espaces préhomogènes

Let ƒ be an homogeneous polynomial on R n . First an analog of the Borel theorem is proved for the distributions which appear at the poles of the distribution |ƒ| s (s ∈ C ). If ƒ is the relative invariant of an irreducible regular prehomogeneous vector space, the preceding result is used to characterize the functions which are obtained by integration on the fibers of ƒ.

Theb-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces

The purpose of this paper is to investigate the micro-local structure and to calculate, by constructing the holonomy diagrams, theb-functions (See [2]) of irreducible regular prehomogeneous vector spaces (See [1]).

On the microlocal structure of a regular prehomogeneous vector space associated with Spin$\left( {10} \right) \times GL\left( 3 \right)$

On the microlocal structure of a regular prehomogeneous vector space associated with Spin$\left( {10} \right) \times GL\left( 3 \right)$

On the microlocal structure of a regular prehomogeneous vector space associated with $\mathrm{GL}\left( 8 \right)$

On the microlocal structure of a regular prehomogeneous vector space associated with $\mathrm{GL}\left( 8 \right)$

Microlocal Calculus and Fourier Transforms

§ 1. In this note we shall explain the general scheme for calculation of the Fourier transforms of relatively invariant hyperfunctions on regular prehomogeneous vector spaces by Microlocal Calculus. The details of the theory will appear in [1], [3] and [6]. The principal purpose of this note is to state Theorem 6. A prehomogeneous vector space over the complex number field C is by definition a triple (G, p, V) of an affine algebraic group G, a finite dimensional vector space V and a linear representation p of G on V, all defined over C, such that there exists a point xEiV, (which is called a generic point of V), whose G-orbit p(G) -.ris dense in V. A polynomial f(x) on V is called a relative invariant if f(p(g} -x) — %((?) 'f(x) for every g^G, with % denoting a rational character of G. It is a homogeneous polynomial and is uniquely determined by % up to a constant factor. Further, (G, p, V) is called regular if it possesses a relative invariant f(x) whose logarithmic Hessian det (92log/(.r) /dx^x^) does not vanish indentically. For example (G, p, V) is known to be regular if G and the isotropy group at a generic point of V are both reductive. And we call it irreducible if p is an irreducible representation. An irreducible regular prehomogeneous vector space has just one irreducible relative invariant up to a constant multiple. Let g be the Lie algebra of G, and x\-^>A-x be the derived repre