Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1F1 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.


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 [7].Around the origin, they have the series expansion (1.1) p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; x) 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 the connection with toric geometry established in [10,11] and the interplay with mirror symmetry, see also the article [34] in this volume for more details and further references.
A natural generalization are hypergeometric functions of a matrix argument X as introduced by Herz in [14, Section 2] using the Laplace transform.Herz was building on work of Bochner [2].Ever since, they have been a recurrent topic in the theory of special functions.In [4, Section 5], Constantine expressed these functions as a series of zonal polynomials, thereby establishing a link with the representation theory of GL n .This series expansion bears a striking likeness to (1.1) and is usually written as (1.2) p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; X) where the λ are partitions of n and the (a i ) 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 F 1 (a; c; X) 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 x 1 , . . ., x m counted with multiplicities.So we may equally well assume that X = diag(x 1 , . . ., x m ) is a diagonal matrix.In [31], Muirhead showed that the linear partial differential operators . ., m, annihilate 1 F 1 (a; c; X) wherever they are defined.We denote by P k the differential operator obtained from g k by clearing denominators and consider the left ideal I m (P 1 , . . ., P m ) in the Weyl algebra D m , see Section 4. We refer to I m as the Muirhead ideal or the Muirhead system of differential equations and denote by W(I m ) its Weyl closure.Our main result is: Theorem 5.1.The singular locus of I m agrees with the singular locus of W(I m ).It is the hyperplane arrangement This leads to a lower bound for the characteristic variety of I m , by which we essentially mean the characteristic variety of the D m -module D m /I m .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 schemetheoretic structures into account.For details, see Definition 2.1 and the remarks thereafter.
Here, (•) means that the corresponding entry gets deleted.Note that the varieties on the right hand side of (1.5) are conormal varieties for the natural symplectic structure on T * A m , see Section 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(I m ), we introduce the following notation.Let J 0 |J 1 . . .J k denote a partition of [m] = {1, . . ., m}, such that only J 0 may possibly be empty.We denote by Z J 0 |J 1 ...J k the linear subspace given by the vanishing of all x i for i ∈ J 0 and all x i − x j for i, j ∈ J and ∈ [k].For a smooth subvariety Y ⊆ A m , we denote by N * Y ⊆ T * A m the conormal variety to Y. Then our conjecture can be phrased as follows: The (reduced) characteristic variety of W(I m ) is the following arrangement of m-dimensional linear spaces: In particular, it has B m+1 many irreducible components, where B n denotes the n-th Bell number.
By an explicit analysis of the differential operators in I m , we also obtain an upper bound for Char(I m ).For a partition J 0 |J 1 . . .J k , we define certain subspaces 3) for the precise definition.Proposition 6.3.The (reduced) characteristic variety of I m is contained in the arrangement of the linear spaces C J 0 |J 1 ...J k : It is the upper and lower bound together with explicit computations in the computer algebra system Singular for small values of m, see Section 6.3, that led us to formulate Conjecture 6.2.We believe that it may contribute to a better understanding of the hypergeometric function 1 F 1 .As I m turns out to be nonholonomic in general, it seems that one should rather work with its Weyl closure W(I m ), for which, in general, generators are not known.Clearly, one has Char(W(I m )) ⊆ Char(I m ).Therefore, Proposition 6.3 in particular also gives an upper bound for Char(W(I m )).

Applications and related work.
Hypergeometric functions of a matrix argument possess a rich structure and are highly fascinating objects.Not surprisingly, there is by now a long list of interesting applications in various areas such as number theory, numerical mathematics, random matrix theory, representation theory, statistics, and others; the following short list does not claim to be exhaustive.
The relation to representation theory and statistics is classical.For the link to representation theory, we refer to [1] and references therein.The connection with multivariate statistics was already present in [14] through the connection to the Wishart distribution, see [14,Section 8].
Unlike in the one-variable case, hypergeometric functions of a matrix argument have been studied from the point of view of holonomic systems only recently.The first instance we know of appeared in arithmetic [19].Motivated by the study of Siegel modular forms and the computation of special values of L-functions, the authors of [19] study solutions of certain systems of differential equations.They are equivalent to Muirhead's system, see e.g.their Proposition 7.4 and Theorem 7.5.
Holonomicity is shown explicitly in [19,Theorem 9.1].Apart from number theory, hypergeometric functions of a matrix argument and holonomic systems also made an appearance in random matrix theory [6].
A large impetus came from numerical analysis with the advent of the holonomic gradient descent and the holonomic gradient method developed in [33].These methods allowed to numerically evaluate and minimize several functions that are of importance in multivariate statistics.In [33] and [27], these methods are applied to the Fisher-Bingham distribution.In [12], the holonomic gradient method is used to approximate the cumulative distribution function of the largest root of a Wishart matrix.Motivated by this method, several teams, mainly in Japan, have studied Muirhead's systems from the D-module point of view such as [12,13,38].This is the starting point for our contribution.We examine the D-module theoretic properties of Muirhead's ideal for the hypergeometric function 1 F 1 of a matrix argument from a completely and consistently algebraic point of view.

Outline.
This article is organized as follows.In Section 2, we recall some basic facts about the Weyl algebra and D m -ideals.We recall the notion of holonomic functions and give a characterization that is well suited for testing holonomicity.In Section 3, we discuss hypergeometric functions of a matrix argument.In Section 4, we define the Muirhead ideal I m and collect what is known about holonomicity of I m and its Weyl closure.Section 5 contains our main results.We investigate the Muirhead ideal of operators annihilating 1 F 1 and determine its singular locus.This section also contains some results about holomorphic and formal solutions of the Muirhead system.The characteristic variety of this ideal and its Weyl closure is investigated in Section 6. Conjecture 6.2 suggests that the characteristic variety of the Weyl closure can be described in a combinatorial way, using partitions of sets.We also discuss some basic computations in low dimensions.

The Weyl algebra
In this section, we recall basic facts about the Weyl algebra, the characteristic variety, and the definition of holonomic functions.We mainly follow the presentation and notation given in [35,37].

Ideals and characteristic varieties
We start by introducing some notation and terminology.Throughout this article, N denotes the natural numbers including 0. For m ∈ N >0 , we denote by the m-th Weyl algebra and by the ring of differential operators with rational functions as coefficients.In this article, we refer to R m as m-th rational Weyl algebra.For a commutative ring A, we will abbreviate A[x] = A[x 1 , . . ., x m ] the polynomial ring and A(x) = A(x 1 , . . ., x m ) the field of rational functions.We will also use ξ as a set of variables so that e.g.
For a vector w = (u, v) ∈ R 2m with u + v ≥ 0 componentwise, we define a partial order on the monomials where the indices refer to the coordinates of the vectors.We refer to w as a weight vector and to deg w as the w-degree.With the notation e = (1, . . ., 1) ∈ N m and w = (0, e) we recover the order of a partial differential operator as the leading exponent for this w-degree.
Given an operator P ∈ D m and a weight vector w ∈ R 2m , we define its initial form in w (P) to be the sum of all terms of maximal w-degree.Note that one has to write P in the basis x α ∂ β in order to compute the w-degree, i.e., one has to bring all differentials to the right.
The initial form in w (P) can be viewed as the class of P of the associated graded algebra gr w (D m ) to the filtration of D m induced by w.The relation To highlight this commutator relation notationally, one writes ξ i instead of ∂ i in gr (u,v) (D m ) for all indices i with u i + v i = 0.In particular, For a D m -ideal I, the initial ideal with respect to w is the left ideal  Remark 2.2.
(1) Note that (0) and D m are not holonomic.Therefore, if I is a holonomic ideal, it is a non-zero, proper D m -ideal.(2) Recall that as a consequence of an important theorem of Sato-Kawai-Kashiwara [36], we have dim Z ≥ m for all irreducible components Z of Char(I), see also the discussion in Section 2.

Conormality of the characteristic variety
We remark that A 2m = Spec C[x 1 , . . ., x m , ξ 1 , . . ., ξ m ] should actually be considered as the cotangent bundle T * A m where the ξ i are the coordinates in the fiber of the canonical morphism T * A m − → A m and the x i are the coordinates in the base.Being a cotangent bundle, T * A m carries a natural (algebraic) symplectic form σ which can explicitly be described in coordinates as The symplectic structure gives rise to the notion of a Lagrangian subvariety, that is, a subvariety Z ⊆ T * A m such that at every smooth point z ∈ Z reg , the tangent space T z Z ⊆ T z (T * A m ) = T * A m is isotropic (i.e., σ vanishes identically on this subspace) and maximal with this property.Note that a Lagrangian subvariety automatically has dimension m.Examples for Lagrangian subvarieties in T * A m are conormal varieties.Given a subvariety X ⊆ A m , the associated conormal variety N * X is defined as the Zariski closure of the conormal bundle N * X reg /A m ⊆ T * A m .This is always a Lagrangian subvariety.We will make use of the following (special case of) important results due to Sato-Kawai-Kashiwara [36,Theorem 5.3.2],see also Gabber's article [9, Theorem I] for an algebraic proof.
Theorem 2.3.Let I be a D m -ideal.Then Char(I) ⊆ T * A m is coisotropic.If I is holonomic, every irreducible component Z of the characteristic variety Char(I) is a conormal variety.In particular, Z is Lagrangian.
To be more precise, the references above show that Z is Lagrangian.By definition, the characteristic variety is stable under the C * -action given by scalar multiplication in the fibers of T * A m − → A m , and therefore it is conormal by [23,Lemma (3.2)], see also [15,Theorem E.3.6].

Holonomic functions
In this section, we recall the definition of a holonomic function and give a characterization of this notion which turns out to be very useful in practice.
The definition generalizes in an obvious way to arbitrary subsets N ⊆ M. If M is a space of functions (e.g.holomorphic, multivalued holomorphic, smooth etc.) and f ∈ M is holonomic, then we refer to f as a holonomic function.The definition of a holonomic function first appeared in the article [43] of Zeilberger.
Definition 2.5.The Weyl closure of a D m -ideal I is the D m -ideal We clearly have In general, it is a challenging task to compute the Weyl closure of a D m -ideal, see [41] for the one-dimensional case and [42] in general.The following property is in particular shared by spaces of functions.
This class of D m -modules allows to deduce further properties of annihilating D m -ideals.Proof.Write a given P ∈ W(Ann D m (N)) as P = i q i P i where q i ∈ R m and Definition 2.8.For a D m -ideal I, its singular locus is the set Sing(I) where π denotes the projection T * A m − → A m and the union is over all irreducible components Z of Char(I) distinct from the zero section {ξ 1 = • • • = ξ m = 0} as sets.Moreover, we denote by The second equality is a standard fact, we refer to [35,Section 1.4].If I is a holonomic D m -ideal, rank(I) gives the dimension of the space of holomorphic solutions to I in a simply connected domain outside the singular locus of I by the theorem of Cauchy-Kowalevski-Kashiwara Theorem [24, p. 44], see also [35,Theorem 1.4.19].The following result clarifies the relationship between the holonomic rank and holonomicity.The following characterization of holonomicity is useful.Proposition 2.10.Let M be a torsion-free D m -module and f ∈ M. Then the following statements are equivalent.
(2) For all k = 1, . . ., m, there exists a natural number m(k) ∈ N and a nonzero differential operator The annihilator of f has finite holonomic rank.

Hypergeometric functions of a matrix argument
In this section, we are going to introduce the hypergeometric functions of a matrix argument in the sense of Herz [14], see Definition 3.2.We will follow Constantine's approach [4] via zonal polynomials.

Zonal polynomials
Zonal polynomials are important in multivariate analysis with applications in multivariate statistics.Their theory has been developed by James in [21,22] and subsequent works, see the introduction of Chapter 12 of Farrell's monograph [8] for a more complete list.The definition given by James in [22] relies on representation theoretic work of É. Cartan [3] and James also credits Hua [16,17], see [18] for an English translation.As a general reference, the reader may consult the monographs of Farrell [8,Chapter 12], Takemura [40], and Muirhead [32].The presentation here follows [32,Chapter 7].
Let m be a fixed positive integer.Throughout, we only consider partitions of the form λ = (λ 1 , . . ., λ m ) of an integer Definition 3.1.For all partitions λ = (λ 1 , . . ., λ m ) of d, the zonal polynomials C λ ∈ C[x 1 , . . ., x m ] are defined to be the unique symmetric homogeneous polynomials of degree d satisfying the following three properties.
(1) The leading monomial with respect to the lexicographic order ≺ lex with The uniqueness and existence of course have to be proven, we refer to [32, Section 7.2], where also the eigenvalues α λ are determined to be Zonal polynomials can be explicitly calculated by a recursive formula for the coefficients in a basis of monomial symmetric functions.From this it follows that zonal polynomials have in fact rational coefficients.The space of symmetric polynomials has a basis given by symmetrizations of monomials.We can enumerate this basis by ordered partitions; the partition of a given basis element is its leading exponent in the lexicographic order.For a partition λ = (λ 1 , . . ., λ m ) we put: where S m .λdenotes the orbit of the m-th symmetric group S m .We write the zonal polynomials with respect to this basis: Zonal polynomials can now be computed explicitly thanks to the following recursive formula: where the sum runs over all (not necessarily ordered) partitions κ = (κ 1 , . . ., κ m ) such that there exist i < j with κ k = µ k for all k i, j and κ i = µ i + t, κ j = µ j − t for some t ∈ {1, . . ., µ j } and such that µ < κ ≤ λ after reordering κ.

Hypergeometric functions of a matrix argument
Let X ∈ C m×m be a square matrix and λ = (λ 1 , . . ., λ m ) a partition.One defines the zonal polynomial C λ (X) as where x 1 , . . ., x m are the eigenvalues of X counted with multiplicities.Note that C λ (X) is well-defined because C λ is a symmetric polynomial.
The parameters a 1 , . . ., a p and c 1 , . . ., c q in this definition are allowed to attain all complex values such that all the denominators (c i ) λ do not vanish.Explicitly, Remark 3.3.If X = diag(x 1 , 0, . . ., 0), it follows straight forward from Definition 3.1 of zonal polynomials that p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; X) is the classical hypergeometric function p F q (a 1 , . . ., a p ; c 1 , . . ., c q ; x 1 ) in one variable.Therefore, Definition 3.2 is indeed an appropriate generalization of hypergeometric functions in one variable.
The convergence behavior of the hypergeometric function of a matrix argument is analogous to the one-variable case, basically with the same proof.For p ≤ q, this series converges for all X.For p = q + 1, this series converges for X < 1, where • denotes the maximum of the absolute values of the eigenvalues of X.If p > q + 1, the series diverges for all X 0. 4. Annihilating ideals of 1 F 1 Let 1 F 1 be the hypergeometric function of a matrix argument as introduced in Definition 3.2.In this section, we systematically study a certain ideal that annihilates 1 F 1 .This function depends on two complex parameters a, c satisfying (3.2).As discussed in the last section, the value of this function on a symmetric matrix X ∈ C m×m is the same as the value on the unique semisimple element in the GL m (C) (conjugacy) orbit closure of X.We may thus restrict our attention to the case that X is diagonal.Then this hypergeometric function satisfies the following differential equations.
In fact, we will point out in Proposition 5.6 that in this theorem, the condition of symmetry in x 1 , . . ., x m can be dropped as it is implied by the other conditions.By using the identity the operators from (4.1) can be written as Clearing the denominators in (4.1), we obtain Definition 4.2.We denote by I m the D m -ideal generated by P 1 , . . ., P m and call it the Muirhead ideal.
Our goal is to systematically study the ideal I m .In this direction, Hashiguchi-Numata-Takayama-Takemura obtained the following result in [12].Proof.This immediately follows from Theorem 4.3 and Lemma 2.9.
At the end of Section 5 in [12], it is conjectured that I m is holonomic.Via direct computation they show that I 2 is holonomic in Appendix A of the paper.One can still verify holonomicity of I 3 for generic parameters a, c through a computation in Singular.It turns out, however, that the above conjecture does not hold.We are thankful to N. Takayama for pointing out that the D 4 -ideal I 4 was shown to be non-holonomic in the Master's thesis [25].We give an easy alternative argument for this in Example 6.6.

Analytic solutions to the Muirhead ideal
In this section, we determine the singular locus of the Muirhead ideal I m and of its Weyl closure: Then the singular locus of I m agrees with the singular locus of W(I m ).It is the hyperplane arrangement The inclusion Sing(I m ) ⊆ A is readily seen from in (0,e) (P i ) = x i j i To prove the reverse containment, we investigate analytic solutions to the Muirhead system locally around points in the components of the arrangement A .Our main technical tool is the following observation resembling [35, Theorem 2.5.5]: where Sol C x (•) denotes the solution space in the formal power series ring C x .
1 Our notation for the initial of a formal power series differs from the one used, among others, in [35,37].Ours is more coherent with the definition of initial forms of linear differential operators.
If P = in (−u,u) (P) and all monomials appearing in the expanded expression P • f + in (−u,u) (P) • f are of higher u-degree than those of in (−u,u) (P) • in −u ( f ).Hence, in (−u,u) (P) annihilates in −u ( f ).This shows that for every D m -ideal I, we have Let F be a basis of the solution space Sol C x (I).Replacing F by a suitable linear combination of its elements, we can assure that the initial forms in −u ( f ) for f ∈ F are linearly independent.Then (5.2) implies In the following two lemmata, we apply Lemma 5.2 to the Muirhead system and bound the spaces of analytic solutions locally around general points in A .Note that up to S m -symmetry, there are two types of components in A , namely {x ∈ C m | x 1 = 0} and {x ∈ C m | x 1 = x 2 }.Lemma 5.3 considers points that lie in exactly one component of A of the first type, while Lemma 5.4 is concerned with the second type.
Lemma 5.3.Let p ∈ C m be a point with distinct coordinates, one of which is zero.Then the space of formal power series solutions to I m centered at p is of dimension at most 2 m−1 .
Proof.Since I m is invariant under the action of the symmetric group S m , we may assume that the point p = (p 1 , . . ., p m ) has the unique zero coordinate p 1 = 0. Studying formal power series solutions to I m around p is equivalent to substituting x i by x i + p i in each of the generators P 1 , . . ., P m and to studying the solutions in C x of the resulting operators.Let us define u (3, 2, . . ., 2) ∈ R m .Examining the expression for P 1 , . . ., P m , we observe that for all i ≥ 2, where θ i x i ∂ i and P i | x →x+p denotes the operator obtained from P i by replacing x with x + p.Note that an operator P(θ 1 , . . ., θ m ) ∈ C[θ 1 , . . ., θ m ] ⊆ D m acts on the one-dimensional vector spaces C • x α for α ∈ N m with eigenvalue P(α).In particular, the solution space in C x of the operators (5.3) is spanned by the 2 m−1 monomials x α with α 1 = 0 and α i ∈ {0, 1} for all i ≥ 2. Here, we have used that m+1 2 − c N >0 by the assumption on c, which guarantees that θ 1 + c − m+1 2 has no non-constant solutions in C x 1 .In particular, from Lemma 5.2, we conclude Lemma 5.4.Let p = (p 1 , . . ., p m ) ∈ (C * ) m with #{p 1 , . . ., p m } = m − 1.Then the space of formal power series solutions to I m centered at p is of dimension at most 2 m−2 • 3.
Proof.We proceed similar to the proof of Lemma 5.3.By symmetry of I m , we may assume that p 1 = p 2 , while all other pairs of coordinates of p are distinct.Denote e = (1, . . ., 1) ∈ N m .Then in (−e,e) , where α 3 , . . ., α m ∈ {0, 1} and where f varies over a basis of The latter is a 3-dimensional vector space spanned by {1, }.This can be easily verified as follows.After the change of variables this system becomes From summing these two equations, we observe that a solution f ∈ C y 1 , y 2 needs to be annihilated by the operator ∂ y 1 ∂ y 2 .Therefore, we can write any solution as With this, we have argued that the solution space of in (−e,e) I m | x →x+p is at most 3 • 2 m−2 -dimensional.Together with Lemma 5.2, this proves the claim.
Proof of Theorem 5.1.First, we observe that in (0,e) (P i ) = x i j i where π : T * A m − → A m denotes the natural projection.By definition of the singular locus, this proves the containment For the reverse inclusion, consider a point p ∈ C m contained in exactly one irreducible component of A .By Lemma 5.3 and Lemma 5.4, the space of formal power series solutions to I m (or, equivalently, to W(I m )) around p is of dimension strictly smaller than 2 m = rank(I m ) = rank(W(I m )).In particular, p needs to be a singular point of I m and of W(I m ), as otherwise the Cauchy-Kowalevski-Kashiwara Theorem implies the existence of 2 m linearly independent analytic solutions around p. In particular, the singular loci of I m and of W(I m ) must contain those points.Since singular loci are closed, we conclude that they contain the entire arrangement A .This description of the singular locus gives rise to the following lower bound on the characteristic variety.In Section 6, we will also discuss an upper bound and a conjectural description of the characteristic variety.
Corollary 5.5.The characteristic variety of W(I m ) contains the zero section and the conormal bundles of the irreducible components of A , i.e., Proof.As already noted in the introduction after (1.5), the linear spaces on the right hand side of the claimed inclusion are conormal varieties.By Theorem 2.3, the conormal varieties to the irreducible components of Sing(W(I m )) are contained in Char(W(I m )).Moreover, the zero section V(ξ 1 , . . ., ξ m ) is always contained in the characteristic variety.Theorem 5.1 concludes the proof.
Above, we have studied bounds on solutions to the Muirhead system locally around points in C m contained in exactly one component of A , while the Cauchy-Kowalevski-Kashiwara Theorem describes the behavior around points in C m \ A .A more detailed study around special points p ∈ A where several components of A intersect may be of interest.
We finish this section by looking at the most degenerate case: p = 0. Recall from Theorem 4.1 that 1 F 1 is the unique analytic solution to I m around 0 that is symmetric and normalized to attain the value 1 at the origin.In fact, the restricting factor assuring uniqueness here is not the symmetry, but the analyticity around 0. Namely, using the techniques presented before, we arrive at the following refinement of Theorem 4.1: is the unique formal power series solution to I m around 0 with 1 F 1 (a; c)(0) = 1.In particular, 1 F 1 (a; c) is the unique convergent power series solution to I m around 0 with 1 F 1 (a; c)(0) = 1.

Proof. Consider any weight vector
From the definition of P 1 , . . ., P m , we see that for all i ∈ {1, . . ., m}: where θ i x i ∂ i .In particular, the Weyl closure of in (−u,u) (I) contains the operators x diagonalizes with respect to the basis of C x given by the monomials.In particular, Sol C x (Q 1 , . . ., Q m ) is a subspace of Sol C x (in (−u,u) (I)) spanned by monomials.Therefore, by Lemma 5.2, it suffices to show that the only monomial annihilated by Q 1 , . . ., Q m is 1.
Let α ∈ N m be such that x α is annihilated by Q 1 , . . ., Q m .Assume for contradiction that α 0 and let i ∈ {1, . . ., m} be maximal such that α i 0. Then for all positive integers .This contradicts the assumption α i 0. We conclude that and therefore dim Sol C x (I m ) ≤ 1.The last claim is now immediate.

Characteristic variety of the Muirhead ideal
In this section, we give a conjectural description of the (reduced) characteristic variety of the Weyl closure of the Muirhead ideal I m , see Conjecture 6.2.The conjecture based on our computations and further evidence is provided by the partial results obtained in Corollary 5.5 and Proposition 6.3.The description of Char(W(I m )) is combinatorial in nature and would imply that the number of irreducible components is given by the (m + 1)-st Bell number B m+1 .

Conjectural structure of the characteristic variety
Let us first explain some notations.Notation 6.1.We denote [m] = {1, . . ., m}.We consider partitions of this set [m] = J 0 J 1 . . .J k , where J 0 is allowed to be empty, the J i with i 0 are nonempty, and we consider the J 1 , . . ., J k as unordered.Taking into account that J 0 plays a distinguished role, we denote such a partition by In particular, Char(W(I m )) has B m+1 many irreducible components.
As I 4 is not holonomic, it does not seem reasonable to make predictions about Char(I m ).The better object to study is its Weyl closure, which is challenging to compute.The appearance of the Bell numbers in the conjecture is explained by the following observation: We have a bijection of sets Ordered partitions {1, . . ., m} = J 0 J 1 . . .
defined by J 0 Ji \ {0} for 0 ∈ Ji , where on the right hand side of (6.2), the symmetric group S k acts on J 1 . . .J k .It is important to note that J 0 is allowed to be empty, and J 0 is the only set among the J i and J j with this property.

Bounds for the characteristic variety
Next, we give an upper bound for the reduced characteristic variety Char(I m ) red and hence a fortiori an upper bound for Char(W(I m )) red .By upper bound, we mean a variety containing the given variety.Note that we already proved a lower bound for Char(W(I m )) in Corollary 5.5.
For a partition J 0 | J 1 . . .J k of [m], we defined the linear subspace C J 0 |J 1 ...J k of A 2m in (6.1).We denote by C J 0 |J 1 ...J k the linear space In particular, this also gives an upper bound for Char(W(I m )) red .
Proof.The characteristic variety of I m is defined by the vanishing of the symbols in (0,e) (P) ∈ C[x][ξ] of all operators P ∈ I m .Hence, describing explicit symbols in in (0,e) (I m ) bounds Char(I m ) from above.We observe that in (0,e) (P i ) Moreover, for i j, consider the following operators in I m : This expression can be seen as the S -pair of the operators P i and P j for graded term orders on R m .A straightforward computation by hand reveals that in (0,e) (S i j ) = − 1 2 x i x j k i, j Since these operators lie in the Muirhead ideal, we have Char(I m ) ⊆ V in (0,e) (P i ), in (0,e) (S i j ) | i j Z, so it suffices to see that Z is set-theoretically contained in the union of all C J 0 |J 1 ...J k .We prove this by the comparing their fibers over Note that this partition is uniquely determined by the point z up to permuting J 1 , . . ., J k .Let F denote the fiber of Z over the point z.We claim that F is set-theoretically contained in the fiber of C J 0 |J 1 ...J k over z.
To prove this claim, it suffices to see that for all singletons J = {n} and twoelement sets J = {i, j} in our partition, where 1 ≤ , ≤ k, the polynomials ξ 2 n and (ξ i + ξ j ) 3 vanish on F. But for those n, i, j, the polynomial (6.5) in (0,e) (P n is a non-zero multiple of ξ 2 n by (6.4), since J is a singleton, and is a non-zero multiple of (ξ i + ξ j ) 3 .Here, we have used that z i = z j by construction of the partition J 0 |J 1 . . .J k .Both (6.5) and (6.6) vanish on F by the definition of Z, and hence ξ n and ξ i + ξ j vanish on the set F red , disregarding the scheme structure.This shows that concluding the proof.

Examples
The computational difficulty of questions concerning the characteristic variety Char(I m ), the Weyl closure W(I m ), its characteristic variety, irreducible components, and more increases rapidly with the number of variables m.For m = 2, 3, we succeed with straightforward computations in Singular to obtain the characteristic variety and its decomposition into irreducible components.For m = 2, also the Weyl closure W(I m ) is computable, but already for m = 3 this is no longer feasible.For m = 4, none of the computer calculations terminate.We provide more precise information in the following examples.Example 6.4.We consider the case m = 2.We perform our computations for generic a, c, i.e., in Q(a, c)[x 1 , . . ., x m ] ∂ 1 , . . ., ∂ m with indeterminates a, c.Computations in Singular show that the characteristic variety Char(I 2 ) set-theoretically decomposes into the following five irreducible components (6.7) Already for m = 2, the ideal I m and its Weyl closure W(I m ) differ.The operator is clearly in W(I 2 ) \ I 2 .In fact, W(I 2 ) = I 2 + (P).Moreover, Char(I 2 ) red = Char(W(I 2 )) red but the multiplicities of the irreducible components are different.
In the order of appearance in (6.7), the irreducible components have multiplicities 4, 2, 2, 4, 3 in I 2 and 3, 2, 2, 4, 1 in W(I 2 ).The decomposition (6.7) will also turn out to be a byproduct of our more general result presented in Proposition 6.3.Example 6.5.Next we consider the case m = 3. Computations for generic a, c in Singular show that Char(I 3 ) decomposes into the 15 = B 4 irreducible components as predicted by Conjecture 6.2.
If we compare this to our upper bound for the characteristic variety Char(W(I 3 )) from Proposition 6.3, we see that the only difference between the components in (6.1) and ( 6 in the upper bound.However, the Weyl closure is holonomic by Lemma 2.9 and thus the components of its characteristic variety are the conormals to their projections to A 3 by Theorem 2.3.Such a projection is a closed subvariety of the diagonal V(x 1 − x 2 , x 2 − x 3 ) ⊆ A 3 , hence either equal to it or equal to a point.The corresponding conormal varieties are V(x 1 − x 2 , x 2 − x 3 , ξ 1 + ξ 2 + ξ 3 ) and the cotangent spaces to the points p λ (λ, λ, λ) for some λ ∈ C. It turns out that the components V(x 1 − x 2 , x 2 − x 3 , ξ 1 + ξ 2 + ξ 3 ) and V(x 1 , x 2 , x 3 ) of Char(W(I 3 )) are the only ones contained in B. In other words, the cotangent spaces to p λ are not contained in the characteristic variety unless λ = 0.It does not seem to be very pleasant to verify this last claim by hand.The operator P of lowest order we found in D 3 whose symbol in (0,e) (P) does not vanish on p λ with λ 0 has order 4 and one needs coefficients of order 6 to show that P ∈ I 3 .
It is striking that the components of Char(W(I 3 )) contained in B are exactly those conormal bundles contained in B that are bihomogeneous in the x i and the ξ j .According to Conjecture 6.2, all components should have this property but for the time being we do not see how to deduce bihomogeneity in general, see also Problem 6.8.Example 6.6.Computations in Singular for fixed a, c over a finite field suggest that Char(I 4 ) decomposes into 51 = B 5 − 1 irreducible components.One of them, K V(x 1 − x 2 , x 1 − x 3 , x 1 − x 4 ), is 5-dimensional.The analogous computations over Q(a, c) do not terminate.We can nevertheless verify its existence via the following trick.Instead of I 4 , we consider the ideal J 4 I 4 + (x 1 − x 2 ).Then we clearly have: Char(I 4 ) ⊇ Char(I 4 ) ∩ V(x 1 − x 2 ) ⊇ Char(J 4 ).
The computation of Char(J 4 ) is much simpler and immediately terminates.It turns out that K ⊆ Char(J 4 ).Therefore, Char(I 4 ) contains the 5-dimensional component K and we conclude that I 4 is not holonomic.This would of course be an immediate consequence of a proof of Conjecture 6.2.It should however be easier to tackle Problem 6.8 directly.One strategy could be to write down a flat one-parameter family of ideals {J t } t∈A 1 , such that J 1 = I m and J 0 has an action by C * × C * and then to see how to relate the characteristic varieties in a flat family.
One way to realize such a one-parameter family concretely is to apply a suitable C * -action to I m and take the limit as the parameter t of C * goes to zero.If e.g.we decree the x i to have weight zero and the ξ i have weight one, the commutator relation of the Weyl algebra is preserved and for each t we obtain an ideal J t as claimed.The flat limit is stable under the C * -action and can be found by applying the action to a Gröbner basis.Note that the action on J 0 induces an action of C * × C * on Char(J 0 ), as the latter always has a C * -action given by scalar multiplication on the fibers of T * A m − → A m .
There are also other instances of annihilating ideals related by one-parameter families.It is classically known that the hypergeometric functions 0 F 1 and 1 F 1 are related to one another through a scaling and limit process.More precisely, 1 F 1 (a; c; 1 a X) − → 0 F 1 (c; X) as a − → ∞, see [32,Section 7.5].Also, the hypergeometric function 0 F 1 is known to be annihilated by the operators where k = 1, . . ., m.One directly checks that the g k from (4.1) scale accordingly to give the system (6.8),see [32,Theorem 7.5.6].
Problem 6.9.Can the scaling relation between 0 F 1 and 1 F 1 be used to deduce a relation between the characteristic varieties of I m and the corresponding ideal generated by the operators (6.8)?
We would like to mention that 0 F 1 naturally appears when investigating the normalizing constant of the Fisher distribution on SO(3), as described in [39].

Outlook
We think that Conjecture 6.2 deserves further study and that it will be helpful to get a better understanding of the hypergeometric function 1 F 1 of a matrix argument.The goal of the present article was to put forward this very clear and intriguing conjecture and to provide some evidence for it.The context in which we studied the function 1 F 1 was rather conceptual, but our methods were mainly ad hoc.We believe that, eventually, the problem should be addressed using more advanced methods from D-module theory.For this, one should look for a more intrinsic description of the Muirhead ideal-or rather its Weyl closure.In particular, it would be interesting to understand if there is some generalization of GKZ systems and a relation to the hypergeometric function of a matrix argument similar to the one-variable case.We hope to be able to tackle these problems in the future.

(2. 1 )
in w (I) ( in w (P) | P ∈ I ) ⊆ gr w (D m ).A D m -module is a left D m -module.Mod(D m ) denotes the category of D m -modules.Likewise for R m -ideals and R m -modules, respectively.Next we recall the important notions of a characteristic variety and of holonomicity.

Definition 2 . 1 .
The characteristic variety of a D m -ideal I is the subscheme of A 2m determined by the ideal in (0,e) (I) ⊆ C[x 1 , . . ., x m ][ξ 1 , . . ., ξ m ] and is denoted by Char(I).The D m -ideal I is called holonomic if in (0,e) (I) has dimension m.

2 . ( 3 )
It is worthwhile to remark that the scheme structure of the characteristic variety is not uniquely determined by the D m -module D m /I.Intrinsic invariants of D m /I are the set Char(I) red and the multiplicity of its irreducible components, see e.g.[15, Section 2.2].The point is that-unlike in the commutative world-I cannot be recovered as the annihilator of the D m -module D m /I, and so there can be I J ⊆ D m with D m /I D m /J .

Lemma 2 . 7 .
Let M ∈ Mod(D m ) be torsion-free and N a subset of M. Then Ann D m (N) is Weyl closed.

Lemma 2 . 9 (
[35],Theorem 1.4.15).Let I be a D m -ideal.If I has finite holonomic rank, then its Weyl closure W(I) is a holonomic D m -ideal.

4. 1 .
Setup and known results about the annihilator Theorem 4.1.[32, Theorem 7.5.6]Let m ∈ N >0 .The function 1 F 1 (a, c; X) of a diagonal matrix argument X = diag(x 1 , . . ., x m ) is the unique solution F of the system of the m linear partial differential equations given by the operators

Theorem 4 . 3 ([ 12 ,Corollary 4 . 4 .
Theorem 2]).For the graded lexicographic term order on R m , a Gröbner basis of R m I m is given by{g k = ∂ 2 k + l.o.t.| k = 1, . . ., m}.An immediate consequence is: The holonomic rank of I m is given by rank(I m ) = 2 m .In particular, the Weyl closure W(I m ) of I m and the function 1 F 1 of a diagonal matrix are holonomic.

6. 4 .Problem 6 . 7 .Problem 6 . 8 .
Open problems concerning the characteristic variety As the examples above indicated, there are a lot of open problems which we would like to put forward.Compute the Weyl closure W(I m ) of I m for any m.A first step would be to explicitly write down differential operators in W(I m ) \ I m .Show that Char(W(I m )) (and possibly Char(I m )) are invariant under the action of C * × C * on T * A m = A m × A m given by scalar multiplication on the factors.
with equality if and only if |J | ≤ 2 for = 1, . . ., k.Further evidence for Conjecture 6.2 is given by the following result.Proposition 6.3.The (reduced) characteristic variety of I m is contained in the arrangement of the linear spaces C J 0 |J 1 ...J k :