Equations of hypergeometric type and Newton polyhedra. (English. Russian original) Zbl 0667.33010
Sov. Math., Dokl. 37, No. 3, 678-682 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 3, 529-534 (1988).
The authors consider the holonomic system of differential equations of hypergeometric type
\[
[\prod_{a_ j>0}(\partial /\partial v_ j)^{a_ j}-\prod_{a_ j<0}(\partial /\partial v_ j)^{-a_ j}]F=0,
\]
\[ [(\sum^{N}_{j=0}b_{ij}v_ j(\partial /\partial v_ j))-\beta_ i]F=0, \] where \((a_ 1,...,a_ N)\in Z^ N,\) \(b_ j=(b_{1j},...,b_{nj})\in Z^ n\) and \(a_ i,\) \(b_{jk}\) satisfy some additional conditions. The polyhedron P (called Newton polyhedron) corresponds to this system of equations, which is the convex hull of the vectors \(b_ j,\) \(j=1,...,N,\) and zero. A theorem is formulated which says that the number of linearly independent solutions of this system at a generic point is equal to the volume of P. The authors construct so called higher-dimensional \(\Gamma\)-series and show when they constitute a basis in the space of solutions of the system of equations, given above. Let D be the sheaf of rings of differential operators on \(C^ N\), and let J be the left ideal generated by the equations of the system. Let \(M=D/J\) be the sheaf of left D-modules corresponding to the system, and let SS(M) be the characteristic variety of M (i.e. the set of common zeros of the highest symbols of the operators belonging to J). The authors give descriptions of the variety SS(M) and of the characteristic cycle. Examples related to compact Hermitian symmetric spaces are considered.
\[ [(\sum^{N}_{j=0}b_{ij}v_ j(\partial /\partial v_ j))-\beta_ i]F=0, \] where \((a_ 1,...,a_ N)\in Z^ N,\) \(b_ j=(b_{1j},...,b_{nj})\in Z^ n\) and \(a_ i,\) \(b_{jk}\) satisfy some additional conditions. The polyhedron P (called Newton polyhedron) corresponds to this system of equations, which is the convex hull of the vectors \(b_ j,\) \(j=1,...,N,\) and zero. A theorem is formulated which says that the number of linearly independent solutions of this system at a generic point is equal to the volume of P. The authors construct so called higher-dimensional \(\Gamma\)-series and show when they constitute a basis in the space of solutions of the system of equations, given above. Let D be the sheaf of rings of differential operators on \(C^ N\), and let J be the left ideal generated by the equations of the system. Let \(M=D/J\) be the sheaf of left D-modules corresponding to the system, and let SS(M) be the characteristic variety of M (i.e. the set of common zeros of the highest symbols of the operators belonging to J). The authors give descriptions of the variety SS(M) and of the characteristic cycle. Examples related to compact Hermitian symmetric spaces are considered.
Reviewer: A.Klimyk
MSC:
33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
33C05 | Classical hypergeometric functions, \({}_2F_1\) |
32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
53C35 | Differential geometry of symmetric spaces |
35L25 | Higher-order hyperbolic equations |
58J99 | Partial differential equations on manifolds; differential operators |
32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |