Holonomic functions and prehomogeneous spaces

A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the s...

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Vydáno v:Selecta mathematica (Basel, Switzerland) Ročník 29; číslo 5
Hlavní autor: Lőrincz, András Cristian
Médium: Journal Article
Jazyk:angličtina
Vydáno: Cham Springer International Publishing 01.11.2023
Springer Nature B.V
Témata:
Algebra
Analytic functions
Decomposition
Differential equations
Hyperspaces
Lie groups
Linear algebra
Mathematical analysis
Mathematics
Mathematics and Statistics
Modules
Polynomials
Sheaves
Vector spaces
13A50
11S90
Primary 14F10
16S32
14L30
32S40
ISSN:1022-1824, 1420-9020
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Shrnutí:A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein–Sato polynomials. When the space is endowed with the action of a linear algebraic group G , we study the class of G -finite analytic functions, i.e. functions that under the action of the Lie algebra of G generate a finite dimensional rational G -module. These are automatically algebraic functions on a variety with a dense orbit. When G is reductive, we give several representation-theoretic techniques toward the determination of Bernstein–Sato polynomials of G -finite functions. We classify the G -finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein–Sato polynomials for distinguished G -finite functions. The results can be used to construct explicitly equivariant D -modules.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-023-00874-7