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Holonomic functions and prehomogeneous spaces

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Abstract

A function that is analytic on a domain of \({\mathbb {C}}^n\) 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 \({\mathcal {D}}\)-modules.

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Notes

  1. This explains the relation \(b_{h}(s) \mid b_{h_0}(s)\). The same reasoning explains why the b-function (4.1) divides that of f.

  2. We have tried using smaller subgroups and also \({\text {SL}}_2\) (on the second factor), but the computations did not terminate then.

  3. We believe that this is actually an equality, and while this would simplify the argument a bit, it is not essential.

  4. We have implemented also various Gröbner methods, but none of these computations terminated.

  5. Technically, \(({\text {det}}V)^\frac{1}{n_2}\) is only a \({\mathfrak {g}}\)-module, but we will continue calling \(V'\) a G-module. The twist by \(({\text {det}}V)^\frac{1}{n_2}\) is convenient to give G-equivariant correspondences, otherwise [18, Proposition 2.1] does not hold as stated. For example, take \(n_1=n_2=1, n=2,\) with \(G=\mathbb {C}^*\) acting on \(V=\mathbb {C}^2\) and \(W=\mathbb {C}\) by scalar multiplication.

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Acknowledgements

I am grateful to Bernd Sturmfels for inspirational conversations about Weyl closure and holonomic functions. I thank Uli Walther for providing me valuable suggestions and comments. I also thank Nero Budur for pointing out some of the relevant literature concerning V-filtrations.

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  1. David and Judi Proctor Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA

    András Cristian Lőrincz

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Lőrincz, A.C. Holonomic functions and prehomogeneous spaces. Sel. Math. New Ser. 29, 69 (2023). https://doi.org/10.1007/s00029-023-00874-7

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