Algebraic analysis of the hypergeometric function $$\,{_1F_{\!\!\;1}}\,$$ 1 F 1 of a matrix argument

Beitr Algebra Geom https://doi.org/10.1007/s13366-020-00546-z ORIGINAL PAPER Algebraic analysis of the hypergeometric function 1 F1 of a matrix argument Paul Görlach1 · Christian Lehn1 · Anna-Laura Sattelberger2 Received: 21 October 2020 / Accepted: 21 October 2020 © The Author(s) 2020 Abstract In this article, we investigate Muirhead’s classical system of differential operators for the hypergeometric function 1 F1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system. Keywords Algebraic analysis · Hypergeometric function · Characteristic variety · Singular locus · Holonomic function Mathematics Subject Classification Primary 34M15 · 33C70; Secondary 34M35 · 13P10 · 14Q15 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 The Weyl algebra . . . . . . . . . . . . . . . . . 3 Hypergeometric functions of a matrix argument 4 Annihilating ideals of 1 F1 . . . . . . . . . . . . 5 Analytic solutions to the Muirhead ideal . . . . 6 Characteristic variety of the Muirhead ideal . . . Appendix A: Singular locus for special parameters References . . . . . . . . . . . . . . . . . . . . . . B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Lehn Paul Görlach Anna-Laura Sattelberger 1 Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Straße 39, 09126 Chemnitz, Germany 2 Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22, 04103 Leipzig, Germany 123 Beitr Algebra Geom 1 Introduction Hypergeometric functions are probably the most famous special functions in mathematics and their study dates back to Euler, Pfaff, and Gauß, earlier contributions to the development of the theory are due to Wallis, Newton, and Stirling, we refer to Dutka (1984). Around the origin, they have the series expansion p Fq (a1 , . . . , a p ; c1 , . . . , cq )(x) := ∞  (a1 )n . . . (a p )n x n , (c1 )n . . . (cq )n n! (1.1) n=0 where p, q are non-negative integers with q + 1 ≥ p and (a)n = a . . . (a + n − 1) denotes the Pochhammer symbol. Hypergeometric functions are ubiquitous in mathematics and physics: they are intimately related to the theory of differential equations and show up at prominent places in physics such as the hydrogen atom. In recent years, there has been renewed interest in the subject coming from the connection with toric geometry established in Gelfand et al. (1989, 1990) and the interplay with mirror symmetry, see also the article (Reichelt et al. 2020) in this volume for more details and further references. A natural generalization are hypergeometric functions of a matrix argument X as introduced by Herz (1955, Section 2) using the Laplace transform. Herz was building on work of Bochner (1952). Ever since, they have been a recurrent topic in the theory of special functions. In Constantine (1963, Section 5), Constantine expressed these functions as a series of zonal polynomials, thereby establishing a link with the representation theory of GLn . This series expansion bears a striking likeness to (1.1) and is usually written as p Fq (a1 , . . . , a p ; c1 , . . . , cq )(X ) := ∞   (a1 )λ . . . (a p )λ Cλ (X ) , (c1 )λ . . . (cq )λ n! (1.2) n=0 λ n where the λ are partitions of n and the (ai )λ , (c j )λ are certain generalized Pochhammer symbols, see Definition 3.2. In this article, we examine the differential equations the hypergeometric function 1 F1 (a; c) of a matrix argument X satisfies from the point of view of algebraic analysis. If X is an (m × m)-matrix, the function (1.2) only depends on the eigenvalues x1 , . . . , xm counted with multiplicities. So we may equally well assume that X = diag(x1 , . . . , xm ) is a diagonal matrix. Muirhead (1970) showed that the linear partial differential operators 123 Beitr Algebra Geom ⎞ ⎛  x 1 gk := xk ∂k2 + (c − xk )∂k + ⎝ (∂k − ∂ )⎠ − a, 2 xk − x (1.3) =k k = 1, . . . , m, annihilate 1 F1 (a; c) wherever they are defined. We denote by Pk the differential operator obtained from gk by clearing denominators and consider the left ideal Im := (P1 , . . . , Pm ) in the Weyl algebra Dm , see Sect. 4. We refer to Im as the Muirhead ideal or the Muirhead system of differential equations and denote by W (Im ) its Weyl closure. Our main result is: Theorem 5.1 The singular locus of Im agrees with the singular locus of W (Im ). It is the hyperplane arrangement ⎧ ⎫ m ⎨ ⎬ A := x ∈ Cm xk (xk − x ) = 0 . (1.4) ⎩ ⎭ =k k=1 This leads to a lower bound for the characteristic variety of Im , by which we essentially mean the characteristic variety of the Dm -module Dm /Im . We would like to point out that the terminology used in this article is a slight modification and refinement of the usual definition in the theory of D-modules, taking scheme-theoretic structures into account. For details, see Definition 2.1 and the remarks thereafter. Corollary 5.7 The characteristic variety of W (Im ) contains the zero section and the conormal bundles of the irreducible components of A , i.e., Char(W (Im )) ⊇ V (ξ1 , . . . , ξm ) ∪ ∪   V (xi , ξ1 , . . . , ξi , . . . , ξm ) i V (xi − x j , ξi + ξ j , ξ1 , . . . , ξi , . . . , ξj , . . . , ξm ). (1.5) i= j  means that the corresponding entry gets deleted. Note that the varieties on Here, (·) the right hand side of (1.5) are conormal varieties for the natural symplectic structure on T ∗ Am , see Sect. 2.2. More precisely, they are the conormal varieties to the irreducible components of the divisor A of singularities of the Muirhead system. To formulate our conjecture about the structure of the characteristic variety of W (Im ), we introduce the following notation. Let J0 |J1 . . . Jk denote a partition of [m] = {1, . . . , m}, such that only J0 may possibly be empty. We denote by Z J0 |J1 ...Jk the linear subspace given by the vanishing of all xi for i ∈ J0 and all xi − x j for i, j ∈ J and  ∈ [k]. For a smooth subvariety Y ⊆ Am , we denote by N ∗ Y ⊆ T ∗ Am the conormal variety to Y . Then our conjecture can be phrased as follows: Conjecture 6.2 Let C J0 |J1 ...Jk := N ∗ Z J0 |J1 ...Jk . The (reduced) characteristic variety of W (Im ) is the following arrangement of m-dimensional linear spaces: Char(W (Im ))red =  C J0 |J1 ...Jk . [m] = J0 ··· Jk 123 Beitr Algebra Geom In particular, it has Bm+1 many irreducible components, where Bn denotes t (...truncated)