Pdf on jan 1, 1990, fumihiro sato and others published the maass zeta function attached to positive definite quadratic formstheory of prehomogeneous vector spaces find, read and cite all the. Orbital exponential sums for prehomogeneous vector spaces by takashi taniguchi and frank thorne dedicated to professor tomohide terasoma on his sixtieth birthday. Sato when he constructed the theory of prehomogeneous vector spaces in 60s. Prehomogeneous vector spaces let g be a connected reductive group defined over the complex number field and let. V be a prehomogeneous vector space, let obe any gf qinvariant subset of v f q, and let be the characteristic function of o.
Character sums asociated to prehomogeneous vector spaces. This book explains the basic concepts of prehomogeneous vector spaces, the fundamental theorem, the zeta functions associated with prehomogeneous vector spaces, and a classification theory of irreducible prehomogeneous vector spaces. We prove a handful of these lifting laws, and show how they can be used to help solve certain problems in arithmetic. This is an interesting example of the cuspidal prehomogeneous vector spaces.
Let g,vbe a prehomogeneous vector space, let o be any gf. The author was an early and important developer of the theory and continues. Advanced studies in pure mathematics 14, 1988 representations of lie groups, kyoto, hiroshima, 1986 pp. Our treatment of prehomogeneous vector spaces really begins in the third chap. Some notes on prehomogeneous vector spaces associated. Pdf the maass zeta function attached to positive definite. Assume that g is connected and reductive, and that v has an open gorbit.
Prehomogeneous vector spaces and ergodic theory ii, trans. Let be a connected reductive complex algebraic group. The theory of prehomogeneous vector spaces was founded by m. A collection of vectors v is a real vector space if the fol. A classification theory of prehomogeneous vector spaces.
On a certain class of cuspidal prehomogeneous vector spaces. Orbital exponential sums for prehomogeneous vector spaces takashi taniguchi and frank thorne july 26, 2016 abstract let g. An introduction to the theory of local zeta functions. Let g,vbe a prehomogeneous vector space, let o be any gfqinvariant subset of vf q,andlet. We denote by its derived subgroup, which is a connected semisimple group. Prehomogeneous vector spaces were introduced by mikio sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The zeta functions attached to prehomogeneous vector spaces.
Prehomogeneous vector spaces and ergodic theory iii. Radial components, prehomogeneous vector spaces, and rational. Also it means a necessary and sufficient condition for a triplet g. Such vectors belong to the foundation vector space rn of all vector spaces. Jan 22, 2016 the purpose of this paper is to introduce afunctions and bfunctions of prehomogeneous vector spaces in the original way of m. Vector spaces handwritten notes these are lecture notes of prof. Let fn igdenote a set of representatives for the corresponding orbits of min n. On residues of zeta functions associated with prehomogeneous. Differential calculi on some quantum prehomogeneous vector spaces. Prehomogeneous vector spaces and field extensions iii. Microlocal analysis of prehomogeneous vector spaces. Prehomogeneous spaces and projective geometry seminario. Prehomogeneous vector spaces are also of use in representation theory, algebraic geometry and invariant theory. In this paper we develop a method for explicitly and e ciently.
Theory of prehomogeneous vector spaces algebraic partthe english translation of satos lecture from shintanis note translated from the japanese by masakazu muro. We call such a triplet g, p, v a prehomogeneous vector space if v has a zariskidense gorbit. Tatsuo kimura, institute of mathematics, university of tsukuba, tsukuba, japan. On a certain class of cuspidal prehomogeneous vector spaces and its basic relative invariants takeyoshi kogiso. On zeta functions associated with prehomogeneous vector spaces. Addition is by components and the action of k is by multiplication on all components. The purpose of this chapter is explain the elementary theory of such vector spaces, including linear independence and notion of the dimension. Thus, it was natural to consider the email address. On zeta functions associated with prehomogeneous vector. Get a printable copy pdf file of the complete article 259k, or click on a page image below to browse page by page. Regular prehomogeneous vector spaces for valued dynkin quivers kamiyoshi, tomohiro, kurosawa, yoshiteru, nagase, hiroshi, and nagura, makoto, tsukuba journal of mathematics, 2019. You should be reasonably familiar with these spaces and other nite dimensional vector spaces. An affine algebraic group g over an algebraically closed field k of characteristic 0 is said to act prehomogeneously on an affine variety w over k if g has a unique open orbit og in w.
We call a triple g, p, v a prehomogeneous vector space if there exists a proper algebraic subset s of v such that v s. It is one of the important topics in the theory of invariant holonomic systems and hyperfunctions on prehomogeneous vector spaces. Full text get a printable copy pdf file of the complete article 259k, or click on a page image below to browse page by page. In section 1, we give a certain class of cuspidal prehomogeneous vector spaces which was observed in kas, theorem 3. An irreducible representation of a connected reductive group g,v over kis called a regular prehomogeneous vector space if. Well go over set theory, the axioms for vector spaces, and examples of axioms using vector spaces.
The answer is that there is a solution if and only if b is a linear combination of the columns column vectors of a. The work presented here depends upon the theory of prehomogeneous vector spaces. Introduction to prehomogeneous vector spaces book, 2003. Ifn is the opposite of n, choose m i 2msuch that w1 0 n idm in 0 i n i, n0 i2n, n e i 2n. The main purpose of this paper is to classify all prehomogeneous vector spaces when p is irreducible, and to investigate their relative invariants and the regularity. We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. Then by the general theory of a irreducible prehomogeneous vector space 15, the hessian determinant deth f of the relative invariant fis also a relative invariant of the same representation, so we have 9c2c such that deth f cfm by the uniqueness of the relative invariant.
A structure theorem for prehomogeneous vector spaces. Internal direct sum, external direct sum, vector space homomorphism and related theorems. The purpose of this paper is to introduce afunctions and bfunctions of prehomogeneous vector spaces in the original way of m. In this note, we give a structure theorem for all prehomogeneous vector spaces defined over the complex number field c. Sato and shintani published the paper sash2 in 1974 on zeta functions associated with prehomoge neous vector spaces, but a very narrow class of. Introduction to prehomogeneous vector spaces tatsuo kimura. On the quasiirreducibility and complete quasireducibility of some reductive prehomogeneous vector spaces hamada, michio, tsukuba journal of mathematics, 2011. Hence we have 9c2c such that deth f cfm by the uniqueness of the relative invariant. Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. The eld k which is either r or c is a vector space over itself. Theory of prehomogeneous vector spaces algebraic partthe english translation of satos lecture from shintanis note translated from the. Lie algebras associated with a standard quadruplet and prehomogeneous vector spaces sasano, nagatoshi, tsukuba journal of mathematics, 2015.
In mathematics, a prehomogeneous vector space pvs is a finitedimensional vector space v. Cuspidal prehomogeneous vector spaces for reductive lie groups. Elements of the set v are called vectors, while those of fare called scalars. Sato when he constructed the theory of prehomogeneous vector spaces.
Then, based on the general theory of prehomogeneous vector spaces 15, the hessian deth f of the relative invariant f is also a relative invariant of the same representation. The vector spaces kn consisting of ordered ntuples of elements of k. To be precise, let g be a quasisplit connected reductive group over a non. A classification of irreducible prehomogeneous vector spaces. The irreducible pvs were classified by sato and tatsuo kimura in 1977, up to a transformation known as castling. Chapter 8 vector spaces in quantum mechanics we have seen in the previous chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states. Translations of mathematical monographs, issn 00659282. If the eld f is either r or c which are the only cases we will be interested in, we call v a real vector space or a complex vector space, respectively. Request pdf geometryofnumbers methods over global fields i. Indeed, the canonical generator of the defining ideal of the closure of the onecodimensional orbit is nothing but the irreducible relative invariant. Muhammad khalid of university of sargodha, sargodha written by atiq ur rehman. We call g0,g1 a prehomogeneous vector space of parabolic type. A vector space v is a collection of objects with a vector.
Prehomogeneous vector spaces we develop geometryofnumbers methods to count orbits in prehomogeneous vector spaces. Theory of prehomogeneous vector spaces, noted by t. Introduction to prehomogeneous vector spaces about this title. Pdf differential calculi on some quantum prehomogeneous. The aim here is to use the example of the sterngerlach experiment to develop this idea further, and to show that the. Full text full text is available as a scanned copy of the original print version. Theory of prehomogeneous vector spaces algebraic partthe english translation of satos lecture from shintanis note translated from. Journal of number theory nt2182 journal of number theory 67, 115 7 1997. Let g be a complex linear algebraic group and g glv a finite dimensional rational representation. The idea of a prehomogeneous vector space was introduced by mikio sato. Yukie, prehomogeneous vector spaces and ergodic theory i, duke math.
Orbital exponential sums for prehomogeneous vector spaces. Let f in cv be a nonzero relative invariant with character. A linear algebra and vector space theory we will present the basic elements of vector space theory needed for the development of material in the text. Title prehomogeneous vector spaces over finite fields. An introduction to the theory of local zeta functions junichi igusa. In this course you will be expected to learn several things about vector spaces of course. In this paper we discuss lifting laws which, roughly, are ways of lifting elements of the open orbit of one prehomogeneous vector space to elements of the minimal nonzero orbit of another prehomogeneous vector space. Theory and practice observation answers the question given a matrix a, for what righthand side vector, b, does ax b have a solution. Representation theory of reductive lie groups and algebras. Choose a nonzero relative invariant fv in cvv, with character. We will then consider vector bundles on homogeneous spaces, and use them to construct interesting birational transformations, including important types of ops.
Our primary application is the determination, when the characteristic is not 2, of the density of discriminants of. Theory of prehomogeneous vector spaces algebraic part. In mathematics, a prehomogeneous vector space pvs is a finitedimensional vector space v together with a subgroup g of the general linear group glv such that g has an open dense orbit in v. The space of square matrices, to which shintani applied his method, is a prehomogeneous vector space of commutative parabolic type arising from special linear lie algebra sln. However, pvs can also be studied from the point of view of lie theory. A classification of irreducible prehomogeneous vector. Funtional analysis lecture notes for 18 mit mathematics. Title prehomogeneous vector spaces over finite fields theory of prehomogeneous vector spaces authors gyoja, akihiko citation.
On a certain class of cuspidal prehomogeneous vector. We hope that this will lead us to a better understanding of possible connections of our work with invariant theory. To justify this viewpoint, we prove the convergence of adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and rederive a large portion of godementjacquet theory. Quantum deformations of prehomogeneous vector spaces of commutative parabolic type associated to classical simple lie algebras are intensively studied in. Tatsuo kimura this is the first introductory book on the theory of prehomogeneous vector spaces, introduced in the 1970s by mikio sato. In this lesson, well discuss the definition and provide some common examples of vector spaces. Also important for time domain state space control theory and stresses in materials using tensors. For a more in depth treatment, the reader is encouraged to turn to halmos 1958 or eaton 1983. Our strategy is based on the theory of pre homogeneous vector spaces, especially castling transfor. A survey of the theory of prehomogeneous vector spaces 1v let g be a connected, complex, linear algebraic group and p a rational linear representation of g on an ndimensional vector space v.
In general, if x is a homogeneous space, and h o is the stabilizer of some marked point o in x a choice of origin, the points of x correspond to the left cosets gh o, and the marked point o corresponds to the coset of the identity. Cholesky decomposition of a hyper inverse wishart matrix a. Representation theory of reductive lie groups and algebras in honor of hisayosi matumoto on the occasion of his 60th birthday date 27 wed29 fri, march, 2019 location room 002 27th, room 123 28th, 29th, changed, graduate school of mathematical sciences, the university of tokyo banquet 28 thu, at lever son verre komaba. Prehomogeneous vector spaces and field extensions springerlink. In section 2, we construct two basic relative invariants of the cuspidal prehomogeneous vector space gl4. On the zeta functions of prehomogeneous vector spaces for a. Secondary 11f70, 11s90 keywords zeta integrals, schwartz spaces, prehomogeneous vector spaces contents 1 introduction 1 2 harmonic analysis on symmetric spaces 5 3 prehomogeneous vector spaces 7 4 holonomic dmodules. On the zeta functions of prehomogeneous vector spaces for. In this paper, we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees, and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois. An example is gl1 acting on a onedimensional space. On a classification of 3simple prehomogeneous vector spaces with two irreducible components kurosawa, yoshiteru, tsukuba journal of mathematics, 2012. Letk be an infinite field of characteristic not equal to 2, 3, 5. It is a finitedimensional vector space v with a group action of an algebraic group g, such that there is an orbit of g that is open for the zariski topology and so, dense.
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