Micro-local analysis of prehomogeneous vector spaces

Sato, Masahide; Kashiwara, Masaki; Kimura, Tatsuo; Oshima, Toshio

doi:10.1007/bf01391666

Cited by 81 publications

(65 citation statements)

References 7 publications

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“…In Section 3, we study b-functions of prehomogeneous determinants and prove our main result in Theorem 3.5 which states that the roots of the b-function of any reductive prehomogeneous determinant are symmetric about −1, which implies, in particular, that condition (b) of Theorem 1.3 holds true. In Section 4, we compute the b-function for all linear free divisors in dimension up to 4, and, using the microlocal calculus from [SKKŌ81], for some prehomogeneous determinants arising from quiver theory. Based on this list of examples we expect that the symmetry property holds true for any linear free divisor.…”

Section: §1 Logarithmic Comparison Theorem and B-functionsmentioning

confidence: 99%

On the Symmetry of $b$-Functions of Linear Free Divisors

Granger¹,

Schulze²

2010

Publ. Res. Inst. Math. Sci.

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We introduce the concept of a prehomogeneous determinant as a possibly nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the b-function are symmetric about −1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry persists.Combined with Kashiwara's theorem on the roots of b-functions, our symmetry result shows that −1 is the only integer root of the b-function. This gives a positive answer to a problem posed by Castro-Jiménez and Ucha-Enríquez in the above cases.We study the condition of strong Euler homogeneity in terms of the action of the stabilizers on the normal spaces.As an application of our results, we show that the logarithmic comparison theorem holds for reductive linear Koszul free divisors exactly when they are strongly Euler homogeneous.

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Section: §1 Logarithmic Comparison Theorem and B-functionsmentioning

confidence: 99%

On the Symmetry of $b$-Functions of Linear Free Divisors

Granger¹,

Schulze²

2010

Publ. Res. Inst. Math. Sci.

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“…For a long time it was also unclear how to compute b f (s) for given f . In [53] many interesting examples of Bernstein-Sato polynomials are worked out by hand, while in [1,6,28,41] algorithms were given that compute b f (s) under certain conditions on f . The general algorithm we are going to explain was given by T. Oaku.…”

Section: D-modulesmentioning

confidence: 99%

D-modules and Cohomology of Varieties

Walther

2002

Computations in Algebraic Geometry With Macaulay 2

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In this chapter we introduce the reader to some ideas from the world of differential operators. We show how to use these concepts in conjunction with Macaulay 2 to obtain new information about polynomials and their algebraic varieties.Gröbner bases over polynomial rings have been used for many years in computational algebra, and the other chapters in this book bear witness to this fact. In the mid-eighties some important steps were made in the theory of Gröbner bases in non-commutative rings, notably in rings of differential operators. This chapter is about some of the applications of this theory to problems in commutative algebra and algebraic geometry.Our interest in rings of differential operators and D-modules stems from the fact that some very interesting objects in algebraic geometry and commutative algebra have a finite module structure over an appropriate ring of differential operators. The prime example is the ring of regular functions on the complement of an affine hypersurface. A more general object is theČech complex associated to a set of polynomials, and its cohomology, the local cohomology modules of the variety defined by the vanishing of the polynomials. More advanced topics are restriction functors and de Rham cohomology.With these goals in mind, we shall study applications of Gröbner bases theory in the simplest ring of differential operators, the Weyl algebra, and develop algorithms that compute various invariants associated to a polynomial f . These include the Bernstein-Sato polynomial b f (s), the set of differential operators J(f s ) which annihilate the germ of the function f s (where s is a new variable), and the ring of regular functions on the complement of the variety of f .For a family f 1 , . . . , f r of polynomials we study the associatedČech complex as a complex in the category of modules over the Weyl algebra. The algorithms are illustrated with examples. We also give an indication what other invariants associated to polynomials or varieties are known to be computable at this point and list some open problems in the area.

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“…Hoionomy Diagrams 5.0. In this section, we give a modification of the techniques developped in [24], suitably for the calculation of fo-functions of semi-invariants. The main difficulty of the modification lies in finding codimension one intersections of irreducible components of W 0 (/' 5 ) (the characteristic variety of ^(/)) and in showing the local irreducibility of components.…”

Section: Commutative Parabolic Cases (1)mentioning

confidence: 99%

Highest Weight Modules and $b$-Functions of Semi-invariants

Gyoja

1994

Publ. Res. Inst. Math. Sci.

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Micro-local analysis of prehomogeneous vector spaces

Cited by 81 publications

References 7 publications

On the Symmetry of $b$-Functions of Linear Free Divisors

On the Symmetry of $b$-Functions of Linear Free Divisors

D-modules and Cohomology of Varieties

Highest Weight Modules and $b$-Functions of Semi-invariants

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