Majorization and Spherical Functions
aa r X i v : . [ m a t h . R T ] J un MAJORIZATION AND SPHERICAL FUNCTIONS
COLIN MCSWIGGEN AND JONATHAN NOVAK
Abstract.
Majorization is a partial order on real vectors which plays animportant role in a variety of subjects, ranging from algebra and combinatoricsto probability and statistics. A basic goal in the study of majorization is toconstruct functions which characterize this order. In this paper, we introducea generalized notion of majorization associated to an arbitrary root system Φ , and show that it admits a natural characterization in terms of the values ofspherical functions on any Riemannian symmetric space with restricted rootsystem Φ . Introduction
Given vectors λ, µ ∈ W N , where W N = { ( λ , . . . , λ N ) ∈ R N : λ ≥ · · · ≥ λ N } is the fundamental type A Weyl chamber, we declare λ (cid:23) µ if and only if j X i =1 λ i ≥ j X i =1 µ j , ≤ j ≤ N, with equality holding for j = N. This defines a partial order on W N known as majorization. This partial order was introduced by Hardy, Littlewood, and P´olyain 1934, and has since come to play an important role in a variety of subjects,ranging from algebra and combinatorics to probability and statistics; see [2] for acomprehensive treatment.Majorization admits an intuitive geometric characterization: we have λ (cid:23) µ ifand only if P λ ⊇ P µ , where P λ , P µ ⊂ R N are the convex sets whose extreme pointsare obtained by permuting the coordinates of λ and µ. This paper is about a nu-merical characterization of majorization which carries over to a much more generalcontext. To state this, we begin by considering a second geometric object associ-ated to each λ ∈ W N , namely the compact symplectic manifold O λ of Hermitianmatrices X whose eigenvalues are the coordinates of λ . According to the classicalSchur–Horn theorem [2], the function that sends a Hermitian matrix to its diagonalentries is a surjection O λ → P λ . Since O λ is a homogeneous space for the conju-gation action of the unitary group U( N ) , it carries a unique conjugation-invariantprobability measure often referred to as the orbital measure on O λ . The pushfor-ward of this measure under the Schur–Horn map gives a probability measure m λ on P λ known as the Duistermaat–Heckman measure [8]. Let L λ denote the Laplace Division of Applied Mathematics, Brown UniversityDepartment of Mathematics, UC San Diego
E-mail addresses : colin [email protected], [email protected] . transform of the Duistermaat–Heckman measure, i.e. the function L λ : C N → C defined by L λ ( a , . . . , a N ) = Z P λ e P Ni =1 a i x i dm λ ( x ) . Thus, the restriction of L λ to real arguments is the moment generating function of m λ , while restricting to imaginary arguments gives the characteristic function. Theorem 1.1.
For any λ, µ ∈ W N , we have λ (cid:23) µ if and only if L λ ≥ L µ pointwiseon R N . Theorem 1.1 generalizes an ex-conjecture of Cuttler, Greene, and Skandera [6]which characterizes the majorization order on Young diagrams via nonnegative spe-cializations of Schur polynomials (the original conjecture was proved by Sra [33],and refined by Khare and Tao [20]). The purpose of this paper is to prove a generalprinciple that yields a large family of majorization-characterizing inequalities in avery general context; in particular, Theorem 1.1 emerges as a special case of thisconstruction. Briefly put, we consider an extended notion of majorization associ-ated to the Weyl group of a given root system Φ, and show that it is characterizedby a pointwise inequality for spherical functions on any Riemannian symmetricspace with restricted root system Φ.The paper is organized as follows.In Section 2, we introduce the notion of Weyl group majorization, and proveour main result in the special case of spherical functions on the Lie algebra g of acompact group G (Theorem 2.2). We treat this case separately because sphericalfunctions reduce to Harish-Chandra orbital integrals in this setting, leading to avery natural and direct generalization of Theorem 1.1. Moreover, the proofs inthe Lie algebra case do not require a discussion of symmetric spaces, and are thussomewhat more elementary.In Section 3, we treat the general case of spherical functions on a Riemanniansymmetric space of non-compact type, and prove our main result (Theorem 3.6).We then use this theorem to deduce majorization-characterizing inequalities forspherical functions on symmetric spaces of Euclidean type (Proposition 3.7) andcompact type (Proposition 3.8). Furthermore, we discuss how the result for thecompact case implies inequalities for various families of orthogonal polynomials,such as Schur polynomials.In Section 4, we develop an even more general framework based on Heckman–Opdam hypergeometric functions [15]. We state a conjectural characterization ofmajorization in this context, and prove one direction of this conjecture. Moreover,we show that the full conjecture holds in rank one, where it reduces to an inequalityfor the classical Gauss hypergeometric function.2. Lie Algebras
Let G be a connected, compact Lie group with Lie algebra g . Let T ⊂ G be amaximal torus, t = Lie( T ) ⊂ g the corresponding Cartan subalgebra, and W theWeyl group generated by reflections in the root hyperplanes in t . Definition 2.1.
Let λ, µ be two vectors in t , or in any other vector space on which W acts by reflections. We say that λ W - majorizes µ , written λ (cid:23) µ , if µ lies in theconvex hull of the Weyl orbit of λ . △ AJORIZATION AND SPHERICAL FUNCTIONS 3
The relation (cid:23) defines a preorder on t and a partial order on each Weyl cham-ber. When G = U( N ) , so that W ∼ = S N and t ∼ = R N , W -majorization coincideswith the usual notion of majorization for vectors, as described in the Introduction.Generalized majorization orders associated to group actions have been studied sincethe 1960’s; see [2, 9, 10, 25].In this section, we prove that W -majorization may be characterized by comparingthe pointwise behavior of Laplace transforms of invariant probability measures oncoadjoint orbits of G . Concretely, choose an invariant inner product h· , ·i identifying g ∼ = g ∗ , and for λ ∈ g let O λ = { Ad g λ | g ∈ G } denote its (co)adjoint orbit. Define(1) L λ ( x ) = Z O λ e h y,x i dy = Z G e h Ad g λ,x i dg, λ, x ∈ t C , where dy is the unique invariant probability measure on O λ , dg is the Haar proba-bility measure on G , and t C ∼ = t ⊕ i t is a Cartan subalgebra of the complexified Liealgebra g C .The functions L λ ( x ) are ubiquitous objects that arise in many different areasof mathematics and physics. They were originally studied by Harish-Chandra inthe context of harmonic analysis on Lie algebras [12], and they play an importantrole in the orbit method in representation theory [21]. When G = U( N ), thetransform L λ ( x ) is known as the Harish-Chandra–Itzykson–Zuber (HCIZ) integral;it has been widely studied in theoretical physics and random matrix theory sincethe 1980’s [19]. More recently, the HCIZ integral has become an important objectin combinatorics [11, 28] and probability [3, 27]. For further background on thesefunctions and their diverse applications, we refer the reader to [23, 24].The main result of this section is the following characterization theorem. Theorem 2.2.
For any λ, µ ∈ t , the following are equivalent: (i) λ (cid:23) µ , (ii) L λ ( x ) ≥ L µ ( x ) for all x ∈ t .Proof. We first show (ii) implies (i), by proving the contrapositive. The discrimi-nant of g is the polynomial ∆ g ( x ) = Q α ∈ Φ + h α, x i , where Φ + is the set of positiveroots. Let x ∈ t with ∆ g ( x ) = 0. We assume for now that ∆ g ( λ ) , ∆ g ( µ ) = 0 aswell; later we will remove this assumption. The Laplace transform (1) admits anexact expression, due to Harish-Chandra [12]:(2) L λ ( x ) = ∆ g ( ρ )∆ g ( λ )∆ g ( x ) X w ∈ W ǫ ( w ) e h w ( λ ) ,x i , where ρ = P α ∈ Φ + α and ǫ ( w ) is the sign of w ∈ W . Taking t > L µ ( tx ) − L λ ( tx ) = ∆ g ( ρ )∆ g ( tx ) X w ∈ W ǫ ( w ) (cid:18) e t h w ( µ ) ,x i ∆ g ( µ ) − e t h w ( λ ) ,x i ∆ g ( λ ) (cid:19) . The W -majorization preorder is not the same as the height partial order on t . If λ, µ ∈ t , wesay that λ is higher than µ if λ − µ can be written as a linear combination of positive roots withnonnegative coefficients. The resulting order coincides with W -majorization when λ and µ areboth dominant, but the height partial order is not W -invariant. Here we take the roots to be real-valued linear functionals in t ∗ , which we identify with t viathe inner product. As a result, our roots differ by a factor of i from those typically used in thesetting of complex semisimple Lie algebras. MAJORIZATION AND SPHERICAL FUNCTIONS
This expression is manifestly W -invariant in λ , µ and x , so we may assume withoutloss of generality that all three are dominant. Then as t → ∞ we have:(4) L µ ( tx ) − L λ ( tx ) = ∆ g ( ρ )∆ g ( tx ) (cid:18) e t h µ,x i ∆ g ( µ ) − e t h λ,x i ∆ g ( λ ) (cid:19) + (lower-order terms) . Now suppose λ µ . Then µ lies outside the convex hull of the W -orbit of λ , soby the hyperplane separation theorem there is some x ∈ t and C > h µ, x i > C while h w ( λ ) , x i < C for all w ∈ W . By making a small perturbationto x if necessary, we can ensure that ∆ g ( x ) = 0. Take x in (4) to be the dominantrepresentative of the Weyl orbit of x . Then we still have h µ, x i > C , h λ, x i < C ,and from (4) we find:lim t →∞ h L µ ( tx ) − L λ ( tx ) i ≥ lim t →∞ ∆ g ( ρ )∆ g ( tx ) (cid:18) e t h µ,x i ∆ g ( µ ) − e tC ∆ g ( λ ) (cid:19) = ∞ , which implies L µ ( tx ) > L λ ( tx ) for some t >
0, so that (ii) cannot hold.Now we remove the assumption that ∆ g ( λ ) , ∆ g ( µ ) = 0. In this case the expres-sion (3) may be singular, so we instead take a limit:(5) L µ ( tx ) − L λ ( tx ) = lim η → ∆ g ( ρ )∆ g ( tx ) X w ∈ W ǫ ( w ) (cid:18) e t h w ( µ + ηρ ) ,x i ∆ g ( µ + ηρ ) − e t h w ( λ + ηρ ) ,x i ∆ g ( λ + ηρ ) (cid:19) . To evaluate this limit, we apply l’Hˆopital’s rule as many times as needed, treatingthe λ and µ terms separately. After j applications to the λ terms and k applicationsto the µ terms for some j, k ≥
0, in place of (4) we find:(6) L µ ( tx ) − L λ ( tx ) = ∆ g ( ρ )∆ g ( tx ) t k h ρ, x i k e t h µ,x i ∂ kρ ∆ g ( µ ) − t j h ρ, x i j e t h λ,x i ∂ jρ ∆ g ( λ ) ! + (lower-order terms) , where ∂ kρ ∆ g ( µ ) = d k dη k ∆ g ( µ + ηρ ) (cid:12)(cid:12) η =0 . The remainder of the argument then goesthrough as before, and we conclude that (ii) implies (i).The other direction of the proof, (i) implies (ii), amounts to showing that for all x ∈ t , the function λ L λ ( x ) is W -convex. This function is clearly W -invariant,and by [10, Theorem 1], a W -invariant, convex function is W -convex. It thereforeremains only to show that L λ ( x ) is convex in λ , and for this it is sufficient to showmidpoint convexity. For u, v ∈ t we have L ( u + v ) ( x ) = Z G e h Ad g ( u + v ) / ,x i dg = Z G p e h Ad g u,x i e h Ad g v,x i dg ≤ Z G (cid:0) e h Ad g u,x i + e h Ad g v,x i (cid:1) dg = 12 L u ( x ) + 12 L v ( x ) , where in the final line we have applied the inequality of arithmetic and geometricmeans. This proves the theorem. (cid:3) Remark 2.3.
It is easily verified from the definition (1) that L λ ( x ) = L x ( λ ), socondition (ii) in Theorem 2.2 could equivalently be written: L x ( λ ) ≥ L x ( µ ) for all x ∈ t . △ AJORIZATION AND SPHERICAL FUNCTIONS 5 Symmetric spaces
This section contains our main results, which are majorization inequalities forspherical functions on Riemannian symmetric spaces. After introducing some back-ground on symmetric spaces and spherical functions, we state and prove separateinequalities for each of the three types of irreducible symmetric space. In each casethe theorem takes the form of an inequality between the pointwise values of anytwo spherical functions, which reflects the W -majorization order on the space ofvectors that index the spherical functions. As we explain below, these results im-ply Theorem 2.2 and discretizations thereof, such as the Schur function inequalitystudied in [6, 33].3.1. Background on symmetric spaces and spherical functions.
We intro-duce only the minimum background on symmetric spaces and spherical functionsthat is required to state and prove the theorems. We refer the reader to [29, Ap-pendices B and C] for a concise introduction to these topics, and to [17, 18] fordetailed references. The definitions below mostly follow [18, ch. 4]
Definition 3.1.
Let G be a connected Lie group, and K ⊂ G a compact subgroup.We say that ( G, K ) is a symmetric pair if K is the fixed-point set of an involutiveautomorphism σ : G → G . For our purposes, a Riemannian symmetric space is aquotient X = G/K , where (
G, K ) is a symmetric pair. When G is non-compactand semisimple with finite center, and K is a maximal compact subgroup, we saythat X is of non-compact type . △ Below, the term “symmetric space” always means a Riemannian symmetric spaceas defined above.
Definition 3.2.
Let X = G/K be a Riemannian symmetric space, and write[ g ] ∈ X for the image of g ∈ G under the quotient map G → G/K . Let D ( X ) bethe algebra of differential operators on X that are invariant under all translations[ x ] [ gx ], g ∈ G . A complex-valued function φ ∈ C ∞ ( X ) is called a sphericalfunction if all of the following hold:(i) φ ([id]) = 1,(ii) φ ([ kx ]) = φ ([ x ]) for all k ∈ K ,(iii) Dφ = γ D φ for each D ∈ D ( X ), where γ D is some complex eigenvalue. △ Spherical functions play a central role in the theory of harmonic analysis onsymmetric spaces, and many important families of special functions can be realizedas spherical functions on some symmetric space.
Example 3.3.
Let G be a compact connected Lie group, and K ⊂ G × G thediagonal subgroup. Then ( G × G, K ) is a symmetric pair, and we can identify( G × G ) /K ∼ = G via ( g , g ) K g g − . The spherical functions on G are preciselythe functions of the form φ λ ( g ) = χ λ ( g )dim V λ , where V λ is the irreducible representation of G with highest weight λ , and χ λ is itscharacter. △ MAJORIZATION AND SPHERICAL FUNCTIONS
Example 3.4.
Let G again be a compact connected Lie group. If we regard itsLie algebra g as an abelian Lie group, we can form the semidirect product G ⋉ g with multiplication ( g , x ) · ( g , x ) = ( g g , Ad g x + x ). Then ( G ⋉ g , G ) is asymmetric pair, and we can identify ( G ⋉ g ) /G ∼ = g via ( g, x ) Ad g x . Thus g is asymmetric space, and the spherical functions on g reduce to the Laplace transformsstudied in Section 2, L λ ( x ) = Z G e h Ad g λ,x i dg, x ∈ g , λ ∈ t C , where t C is the complexification of a Cartan subalgebra t ⊂ g . △ If X is a symmetric space then its universal cover ˜ X is also symmetric, and thespherical functions on X may be identified with spherical functions on ˜ X that areconstant on the fibers of the covering map. In this sense the spherical functionson ˜ X subsume those on X , so that in what follows we may assume without lossof generality that X is simply connected. We say that X is irreducible if it can-not be written as a nontrivial product of symmetric spaces. A simply connected,irreducible symmetric space is always:(1) of non-compact type; or,(2) a Euclidean space; or,(3) compact.These three types correspond respectively to the cases in which X is negativelycurved, flat, or positively curved. There is a a well-known correspondence betweenthe three types, which we now describe. If X − = G/K is a symmetric space ofnon-compact type, σ : G → G is the associated involution fixing K , and g = Lie( G ),then we have the Cartan decomposition g = k + p , where k = Lie( K ) is the fixed-point set of dσ . From these data we can construct both a Euclidean symmetricspace and a compact symmetric space. First define g + = k + i p ⊂ g C , which is theLie algebra of the compact real form G + of G . The symmetric space X + = G + /K is obviously compact. Next define the algebra g , which is equal to g as a vectorspace but is endowed with a different Lie bracket [ · , · ] defined by[ x, y ] = ( , x, y ∈ p , [ x, y ] , otherwise.Then the group G = exp( g ) ∼ = K ⋉ p acts on p by affine transformations, ( k, p ) · x =Ad k x + p , and X = G /K ∼ = p is a Euclidean symmetric space.Thus we have constructed a triple of symmetric spaces ( X − , X , X + ) that belongrespectively to the three types listed above. Moreover, every simply connected,irreducible symmetric space occurs in such a triple. In the following subsections,we study spherical functions on the spaces X − , X , and X + .3.2. Symmetric spaces of non-compact type.
When X − = G/K is a symmet-ric space of non-compact type, the spherical functions admit a convenient integralrepresentation, due to Harish-Chandra [13, 14]. The version that we use here isproved in [18, ch. 4, Theorem 4.3]. Let G = N AK , g = n + a + k be the Iwasawadecompositions of G and g . For g ∈ G , let a ( g ) be the unique element of a suchthat g ∈ N e a ( g ) K . The Killing form h· , ·i on g restricts to an inner product on a .For α ∈ a , define g α = { x ∈ g | [ h, x ] = h α, h i x for all h ∈ a } . AJORIZATION AND SPHERICAL FUNCTIONS 7
The restricted root system
Φ of X − consists of all nonzero α ∈ a for which g α is nontrivial. Fix a choice Φ + of positive roots, and let W be the Weyl groupgenerated by reflections in the root hyperplanes. For α ∈ Φ, define m α = dim g α ,and set ρ = P α ∈ Φ + m α α . Write dk for the normalized Haar measure on K . Theorem 3.5 (Harish-Chandra) . The spherical functions on X − are exactly thefunctions of the form (7) φ − λ ([ g ]) = Z K e h iλ + ρ,a ( kg ) i dk, g ∈ G, as λ ranges over a C . Moreover, two such functions φ λ and φ µ are identical if andonly if µ = w ( λ ) for some w ∈ W . The following theorem is the main result of this paper.
Theorem 3.6.
Let X − = G/K be a Riemannian symmetric space of non-compacttype. For any λ, µ ∈ a , the following are equivalent: (i) λ (cid:23) µ , (ii) φ − iλ ( x ) ≥ φ − iµ ( x ) for all x ∈ X .Proof. The argument generalizes the proof of Theorem 2.2. We first show that (ii)implies (i) by proving the contrapositive, and then that (i) implies (ii) using theintegral representation (7) for the spherical functions.Suppose λ µ . Since the map λ
7→ − λ is an isometry of a , we have λ (cid:23) µ ifand only if − λ (cid:23) − µ . Accordingly, to prove that (ii) implies (i), it suffices to showthat φ −− iµ ( x ) > φ −− iλ ( x ) for some x ∈ X . By hyperplane separation, we can obtain y ∈ a and C > h µ, y i > C and h w ( λ ) , y i < C for all w ∈ W . Clearlyboth of these inequalities still hold if we replace y with the dominant representativeof its Weyl orbit, and by Theorem 3.5 we have φ − iλ = φ − iw ( λ ) for w ∈ W . Thereforewithout loss of generality we may take all three of λ , µ and y to be dominant.With these assumptions, we will study the asymptotic behavior of the sphericalfunctions φ −− iλ and φ −− iµ at infinity. This topic is well understood; see e.g. [7]. Inparticular we have the following sharp estimate as t → + ∞ , which is also a specialcase of (22) below:(8) φ −− iλ ([ e ty ]) ≍ e t h λ − ρ, y i Y α ∈ Φ + h α,λ i =0 (1 + 4 t h α, y i ) . This estimate implies that φ −− iµ ([ e ty ]) − φ −− iλ ([ e ty ]) > e − t h ρ,y i (cid:18) C e t h µ,y i − C e tC Y α ∈ Φ + h α,λ i =0 (1 + 4 t h α, y i ) (cid:19) for some constants C , C >
0. For t sufficiently large, the quantity on the right-hand side above is positive, proving that φ −− iµ ( x ) > φ −− iλ ( x ) for some x ∈ X , asdesired.We next prove that (i) implies (ii). It suffices to show that the function f x ( λ ) = φ iλ ( x ) is W -convex. As in the proof of Theorem 2.2, we use the result of [10,Theorem 1], which states that a W -invariant, convex function is W -convex. ByTheorem 3.5, f x is W -invariant, so we need only prove that f x is convex, for whichit suffices to check midpoint convexity. Write x = [ g ] for some g ∈ G . Using the MAJORIZATION AND SPHERICAL FUNCTIONS integral representation (7) and the inequality of arithmetic and geometric means,we find: f x (cid:18)
12 ( λ + µ ) (cid:19) = Z K e h ρ − ( λ + µ ) / , a ( kg ) i dk = Z K e h ρ,a ( kg ) i p e −h λ,a ( kg ) i e −h µ,a ( kg ) i dk ≥ Z K e h ρ,a ( kg ) i ( e −h λ,a ( kg ) i + e −h µ,a ( kg ) i ) dk = 12 (cid:0) f x ( λ ) + f x ( µ ) (cid:1) , which shows that f x is convex, completing the proof. (cid:3) Euclidean symmetric spaces.
The spherical functions on the Euclideansymmetric space X ∼ = p are precisely the functions(9) φ λ ( x ) = lim ε → φ − λ/ε ([ e εx ]) = Z K e i h λ, Ad k x i dk, x ∈ p , as λ ranges over a C ; see [18, ch. 4, Proposition 4.8]. Taking the limit (9) in theproof of Theorem 3.6, we obtain the following. Proposition 3.7.
Let X be a Euclidean symmetric space. For any λ, µ ∈ a , thefollowing are equivalent: (i) λ (cid:23) µ , (ii) φ iλ ( x ) ≥ φ iµ ( x ) for all x ∈ X . In particular, Theorem 2.2 is a special case of Proposition 3.7, corresponding tothe Euclidean symmetric space described in Example 3.4.3.4.
Compact symmetric spaces.
We now consider the compact symmetricspace X + = G + /K . Let V λ be the irreducible G + -representation with highestweight λ , and χ λ its character. If V λ contains a nontrivial K -fixed vector, we saythat V λ is a spherical representation and λ is a spherical highest weight. By [18,ch. 4, Theorem 4.2], the spherical functions on X + are precisely the functions(10) φ + λ ([ g ]) = Z K χ λ ( g − k ) dk, g ∈ G + , where χ λ is the character of an irreducible spherical representation of G + .Here we depart in two ways from the conventions used above in Section 2. First,we now use the notation h· , ·i to indicate the Killing form, which restricts to a negative -definite form on g + rather than an inner product. Second, we now regardthe roots and weights of G + as imaginary -valued linear functionals on a Cartansubalgebra t ⊂ g + with i a ⊂ t . We then use the Killing form to identify the weightsand roots with elements of i t .With these conventions, the spherical highest weights of G + correspond to certainlattice points in a ⊂ i t ; see [18, ch. 5 § G + is simply connected and semisimple then the spherical highest weightsare exactly those λ ∈ a satisfying(11) h λ, α ih α, α i ∈ Z ≥ for all α ∈ Φ + , AJORIZATION AND SPHERICAL FUNCTIONS 9 where Φ + are the positive restricted roots of X − . Given λ, µ ∈ a , we write λ (cid:23) µ toindicate that λ W -majorizes µ , where W is the Weyl group generated by reflectionsin the restricted roots.The function φ + λ can be analytically continued to the complexification G C , so thatwe may evaluate φ + λ ([ e x ]) for any x ∈ t C . We then have the following majorizationinequality. Proposition 3.8.
Let λ, µ ∈ a be two spherical highest weights of G + . The follow-ing are equivalent: (i) λ (cid:23) µ , (ii) φ + λ ([ e ix ]) ≥ φ + µ ([ e ix ]) for all x ∈ t .Proof. Consider the spherical function φ −− i ( λ − ρ ) on X − , regarded as a function onthe non-compact group G . When λ is a spherical highest weight, this function alsoadmits an analytic continuation to G C , which coincides with φ + λ ; see [35, § t = t ∩ k + i a and [ e y + x ] = [ e x ] ∈ X + for y ∈ t ∩ k , we can take x ∈ i a , so that e ix ∈ G . The desired result is then immediate from Theorem 3.6. (cid:3) Application to Schur polynomials.
Let us explain how Proposition 3.8implies the results of [6, 33] relating majorization and Schur polynomials. Let Y N ⊂ W N be the set of points in the type A Weyl chamber whose coordinatesare nonnegative integers. Associated to each λ ∈ Y N is the corresponding Schurpolynomial(12) s λ ( x , . . . , x N ) = det[ x N − i + λ i j ] Ni,j =1 det[ x N − ij ] Ni,j =1 . Each λ ∈ Y N corresponds to the highest weight of an irreducible polynomial repre-sentation of U( N ), and we can identify R N with a Cartan subalgebra in u ( N ) suchthat(13) s λ ( e iy , . . . , e iy N ) = χ λ ( e y ) , y = ( y , . . . , y N ) ∈ R N , and s λ (1 , . . . ,
1) = dim V λ . If we then regard the group U( N ) as a compact sym-metric space as in Example 3.3, we find φ + λ ([ e iy ]) = s λ ( e y , . . . , e y N ) s λ (1 , . . . , . Writing x i = e y i , Proposition 3.8 then yields Conjecture 7.4 in [6] under the stricterassumption that all x , . . . , x N >
0. Since Schur polynomials are continuous, wecan relax this to x , . . . , x N ≥
0, which proves Conjecture 7.4 in [6] in full. Wenote that a characterization of weak majorization in terms of Schur polynomialswas recently obtained in [20].Many families of orthogonal polynomials can be realized as spherical functionson a compact symmetric space [35]. In all such cases, Proposition 3.8 gives aninequality for the orthogonal polynomials that is analogous to the Schur functioncase. Hypergeometric functions
The Heckman–Opdam hypergeometric functions are a family of special functionsassociated to root systems, which generalize the classical Gauss hypergeometricfunction to higher dimensions. They are eigenfunctions of the hyperbolic quan-tum Calogero–Sutherland Hamiltonian and were introduced in the paper [15] inorder to prove the complete integrability of quantum Calogero–Sutherland models.Many special functions of interest can be expressed via limits or specializations ofHeckman–Opdam hypergeometric functions, including the spherical functions on allRiemannian symmetric spaces of non-compact type. Also in [15], Heckman and Op-dam defined the multivariable Jacobi polynomials, now known as Heckman–Opdampolynomials. These are closely related to hypergeometric functions and generalizenumerous widely studied families of orthogonal polynomials, such as Schur andJack polynomials.In this final section, we conjecture that Heckman–Opdam hypergeometric func-tions satisfy a fundamental monotonicity property with respect to W -majorization.If true, this conjecture would unify and generalize all of the majorization resultsdiscussed in this paper. We prove one of the two directions of implication thatcomprise the conjecture, and we show that the full conjecture holds in rank one.Just as Heckman–Opdam hypergeometric functions generalize the spherical func-tions φ − λ on different symmetric spaces of non-compact type, the Heckman–Opdampolynomials generalize of the functions φ + λ , up to some differences in normaliza-tion. Similarly, another related class of functions, the generalized Bessel functions,interpolate between the functions φ λ on different Euclidean symmetric spaces. Ac-cordingly, if the conjecture is true, then we should expect that analogous resultshold for both generalized Bessel functions and Heckman–Opdam polynomials. Thisintuition appears to be correct: we show below that the conjecture would immedi-ately imply an analogue for Heckman–Opdam polynomials of the Schur polynomialinequality proved in [6, 33].To define the Heckman–Opdam hypergeometric functions, we first must fix somepreliminary data. Here we largely follow the conventions of Anker [1] and Heckmanand Schlichtkrul [16]. Let V ∼ = R r be a Euclidean space, Φ ⊂ V a crystallographicroot system spanning V , and W the Weyl group acting on V by reflections in theroot hyperplanes. The Heckman–Opdam hypergeometric function F k,λ depends ona point λ in the complexification V C , as well as on a multiplicity parameter , which isa function k : Φ → C such that k w · α = k α for w ∈ W . Unless stated otherwise, weassume in what follows that λ ∈ V and that all k α are nonnegative real numbers.We now define F k,λ in terms of solutions to certain differential-difference equa-tions. Fix a choice of positive roots Φ + . For α ∈ Φ + let s α be the reflection throughthe hyperplane { x ∈ V | h α, x i = 0 } , and define ρ ( k ) = P α ∈ Φ + k α α. Definition 4.1.
For y ∈ V , the Cherednik operator D k,y is the differential-differenceoperator(14) D k,y = ∂ y + X α ∈ Φ + h y, α i k α − e − α (1 − s α ) − h y, ρ ( k ) i . △ The Cherednik operators were originally defined and studied in [4, 5]. For detailsof their properties, we refer the reader to these papers as well as to [1, §
4] and [30,
AJORIZATION AND SPHERICAL FUNCTIONS 11 § k α are nonnegative, for any λ ∈ V C there is a unique smooth function G k,λ on V satisfying the system ofdifferential-difference equations(15) D k,y G k,λ = h y, λ i G k,λ for all y ∈ V and normalized so that G λ (0) = 1. Definition 4.2.
For k ≥ λ ∈ V C , the Heckman–Opdam hypergeometricfunction F k,λ is defined as(16) F k,λ = 1 | W | X w ∈ W G k,λ ( w ( x )) . △ The functions F k,λ unify and interpolate between many widely studied specialfunctions, as illustrated in the following examples. Example 4.3.
In the 1-dimensional case where V ∼ = R , the root system Φ can beeither A or BC . For BC there are two Weyl orbits {± } , {± } ⊂ R , and theHeckman–Opdam hypergeometric function reduces to the Gauss hypergeometricfunction:(17) F k,λ ( x ) = F (cid:16) k k + λ, k k − λ ; k + k + 12 ; − sinh x (cid:17) . The Heckman–Opdam hypergeometric function for A corresponds to the specialcase k = 0. △ Example 4.4.
Suppose Ψ is the restricted root system of a symmetric space X = G/K of noncompact type, and m α = dim g α for each α ∈ Ψ. Take V = a , Φ = { α | α ∈ Ψ } and k α = m α . Then(18) φ − λ ([ e x ]) = F k,i λ (2 x ) , x ∈ a , λ ∈ a C . See [1, § △ Example 4.5.
The generalized Bessel function J k,λ on V can be obtained as the rational limit of F k,λ :(19) J k,λ ( x ) = lim ε → F k,λ/ε ( εx ) . From the previous example and the relation (9), it is clear that J k,λ generalizesthe spherical functions on Euclidean symmetric spaces in the same way that F k,λ generalizes the spherical functions on symmetric spaces of non-compact type. See[1, § § △ For any multiplicity parameter k , we define the function(20) δ k ( x ) = Y α ∈ Φ + ( e h α,x i / − e −h α,x i / ) k α . Example 4.6.
When Φ is reduced, we can identify V with a Cartan subalgebra t of a compact semisimple Lie algebra g with root system Φ. We write k = ~ k α = 1 for all α ∈ Φ. Then(21) F ~ ,λ ( x ) = ∆ g ( x ) δ ~ ( x ) L λ ( x ) , x ∈ t . △ We conjecture the following monotonicity property for Heckman–Opdam hyper-geometric functions.
Conjecture 4.7.
Let λ, µ ∈ V . The following are equivalent: (i) λ (cid:23) µ , (ii) F k,λ ( x ) ≥ F k,µ ( x ) for all x ∈ V . Here we show one half of the conjecture, namely that (ii) implies (i), usingSchapira’s sharp asymptotics for F k,λ [32]. We then give an elementary proofof the conjecture in rank one, where it amounts to an inequality for the Gausshypergeometric function. Proposition 4.8.
Let λ, µ ∈ V . If F k,λ ( x ) ≥ F k,µ ( x ) for all x ∈ V , then λ (cid:23) µ .Proof. As in the proof of Theorem 3.6, we show the contrapositive. Suppose λ µ ,and again use hyperplane separation to obtain a y ∈ V and C > h µ, y i > C and h w ( λ ) , y i < C for all w ∈ W . Without loss of generality, we take λ , µ and y to be dominant. We have the following sharp asymptotic estimate, dueto Schapira [32, Theorem 3.1 and Remark 3.1]:(22) F k,λ ( ty ) ≍ e t h λ − ρ ( k ) , y i Y α ∈ Φ + h α,λ i =0 (1 + t h α, y i )as t → + ∞ . We thus find that F k,µ ( ty ) − F k,λ ( ty ) > e − t h ρ ( k ) , y i (cid:18) C e t h µ,y i − C e tC Y α ∈ Φ + h α,µ i =0 (1 + t h α, y i ) (cid:19) for some constants C , C >
0. For t sufficiently large, the quantity on the right-hand side above is positive, which implies that F k,λ ( x ) < F k,µ ( x ) for some x ∈ V ,completing the proof. (cid:3) Remark 4.9.
As in the proof of Theorem 3.6, to complete the proof of Conjecture4.7 it suffices to check that the function λ F k,λ ( x ) is midpoint-convex. However,integral representations analogous to (7) for general Heckman–Opdam hypergeo-metric functions are not known, so we cannot directly apply the same technique.Although there are known integral expressions in certain cases where the multiplic-ity parameter does not correspond to a symmetric space (see e.g. [31, 34]), theseare more complicated than (7) and have so far resisted a similar analysis. A morepromising approach to a general proof of Conjecture 4.7 might be to use hyperge-ometric differential equations, as illustrated in the following proposition. △ Proposition 4.10.
When dim V = 1 , Conjecture 4.7 is true.Proof. In light of Proposition 4.8, we need only show that (i) implies (ii) in Con-jecture 4.7. Following the discussion in Example 4.3, it is sufficient to considerthe case Φ = BC . It is a classical result that the Gauss hypergeometric function F ( z ) = F ( a, b ; c ; z ) satisfies Euler’s hypergeometric equation: z (1 − z ) d Fdz + [ c − ( a + b + 1) z ] dFdz − ab F = 0 . Comparing to (17), we find:(23) F ′′ k,λ ( x ) + (cid:16) k coth x k coth x (cid:17) F ′ k,λ ( x ) + h(cid:16) k k (cid:17) − λ i F k,λ ( x ) = 0 , AJORIZATION AND SPHERICAL FUNCTIONS 13 for λ, x ∈ R . The function F k,λ is determined by (23) and by the initial conditions(24) F ′ k,λ (0) = 0 , F k,λ (0) = 1 , which follow respectively from the fact that F k,λ is W -invariant (i.e. even) and fromthe normalization of the function G k,λ in the definition (16).Suppose λ (cid:23) µ , which in dimension one just means that | λ | ≥ | µ | . Since theequation (23) depends only on λ and not on the sign of λ , we find F k, − λ = F k,λ ,so we can in fact take | λ | > | µ | . We will show that F k,λ ( x ) ≥ F k,µ ( x ) for all x ∈ R ,with equality only at x = 0.Since F k,λ is even, it suffices to consider x ≥
0. From (23) and the initialconditions (24), we have: F k,λ (0) = F k,µ (0) = 1 ,F ′ k,λ (0) = F ′ k,µ (0) = 0 ,F ′′ k,λ (0) = λ − (cid:16) k k (cid:17) > µ − (cid:16) k k (cid:17) = F ′′ k,µ (0) . Therefore there is some ε > x ∈ (0 , ε ), F k,λ ( x ) > F k,µ ( x ) , F ′ k,λ ( x ) > F ′ k,µ ( x ) , F ′′ k,λ ( x ) > F ′′ k,µ ( x ) . Consider the set E = { x > | F k,λ ( x ) = F k,µ ( x ) } . If E is empty, then F k,λ ( x ) >F k,µ ( x ) for all x >
0, and there is nothing to prove. Assume for the sake ofcontradiction that E is not empty, and let x = inf E. Clearly x > ε. Since F k,λ and F k,µ are continuous, we have F k,λ ( x ) = F k,µ ( x ), and(25) F k,λ ( x ) > F k,µ ( x ) for all x ∈ (0 , x ) . However, since F k,λ ( x ) = 1 + Z x F ′ k,λ ( τ ) dτ, F k,µ ( x ) = 1 + Z x F ′ k,µ ( τ ) dτ, and F ′ k,λ > F ′ k,µ on (0 , ε ), in order to have F k,λ ( x ) = F k,µ ( x ) there must be some x ∈ ( ε, x ) such that F ′ k,λ ( x ) < F ′ k,µ ( x ). By the intermediate value theoremand by (25), there must then be some x ∈ ( ε, x ) such that the following hold: F ′ k,λ ( x ) = F ′ k,µ ( x ), F ′′ k,λ ( x ) < F ′′ k,µ ( x ), and F k,λ ( x ) > F k,µ ( x ). But by directinspection of the equation (23) for F k,λ and the corresponding equation for F k,µ ,it is impossible for all three of these conditions to hold at a single point, yielding acontradiction and completing the proof. (cid:3) We next turn our attention from hypergeometric functions to the closely relatedfamily of Heckman–Opdam polynomials, which we now define. For α ∈ Φ, write α ∨ = 2 α h α, α i . The fundamental weights ω , . . . , ω r are defined by h ω i , α ∨ j i = δ ij , where α , . . . , α r are the simple roots. They span the weight lattice P ⊂ V . The dominant integralweights are the lattice points P + ⊂ P that lie in the dominant Weyl chamber.The Heckman–Opdam polynomials P k,λ depend on a nonnegative multiplicityparameter k and a dominant integral weight λ ∈ P + . They are elements of R [ P ],the group algebra of the weight lattice, and are therefore polynomials in an abstract-algebraic sense. However, it is typical to identify R [ P ] with the algebra spannedby the functions e h λ,x i , λ ∈ P , so that as functions on V the Heckman–Opdampolynomials are actually exponential polynomials. We write an element f ∈ R [ P ] as f = P λ ∈ P f λ e λ , where only finitely many f λ are nonzero, and set ¯ f = X λ ∈ P f − λ e λ . Define a bilinear form ( · , · ) k on R [ P ] by( f, g ) k = ( f ¯ gδ k ¯ δ k ) , which extracts the constant term (i.e. the coefficient of e = 1) in f ¯ gδ k ¯ δ k , where δ k is the function defined in (20). This bilinear form is symmetric and positivedefinite, and therefore defines an inner product on R [ P ].For λ ∈ P + , let M λ = | W · λ || W | X w ∈ W e w ( λ ) be the monomial W -invariant (exponential) polynomial, and define low( λ ) as theset of µ ∈ P + such that λ − µ can be written as a linear combination of positiveroots with non-negative integer coefficients. Definition 4.11.
For λ ∈ P + , the Heckman–Opdam polynomial P k,λ is defined by(26) P k,λ = X µ ∈ low( λ ) c λµ M µ , c λλ = 1 , and by the orthogonality relations(27) ( P k,λ , M µ ) k = 0 , µ ∈ low( λ ) , µ = λ. △ As λ ranges over P + with k fixed, the P k,λ form an orthogonal R -basis of the W -invariant elements R [ P ] W . Example 4.12.
When all k α are 0, P ,λ = M λ . Let g be a compact semisimple Lie algebra with root system Φ, and identify V with a Cartan subalgebra t ⊂ g . Let G be the connected, simply connected Liegroup with Lie( G ) = g . When k = ~ P ~ ,λ ( ix ) = χ λ ( e x ) is the character of theirreducible G -representation with highest weight λ .When Φ = A N − , if we identify M λ with the monomial symmetric polynomial m λ = | S N · λ | N ! X σ ∈ S N N Y i =1 x λ σ ( i ) i , then Heckman–Opdam polynomials are Jack polynomials. In particular, for k = ~ △ Up to a normalizing factor, the Heckman–Opdam polynomials turn out to bespecializations of the Heckman–Opdam hypergeometric function. In particular, for λ ∈ V , define(28) ˜ c ( λ, k ) = Y α ∈ Φ + Γ( h λ, α ∨ i + k α )Γ( h λ, α ∨ i + k α + k α ) , AJORIZATION AND SPHERICAL FUNCTIONS 15 where k α = 0 if α Φ. Observe that if Φ is the root system of a compact Liealgebra g , we have ˜ c ( λ,~
1) = ∆ g ( λ ) − . Set c ( λ, k ) = ˜ c ( λ, k )˜ c ( ρ ( k ) , k ) . We then have the following relation between Heckman–Opdam polynomials andhypergeometric functions [16, eq. 4.4.10]:(29) F k,λ + ρ ( k ) ( x ) = c ( λ + ρ ( k ) , k ) P k,λ ( x ) , x ∈ V. This relation generalizes the relation between the spherical functions φ −− i ( λ − ρ ) and φ + λ discussed in Section 3.4. It immediately yields the following proposition. Proposition 4.13.
Let λ, µ ∈ P + . If Conjecture 4.7 holds, then the following areequivalent: (i) λ (cid:23) µ , (ii) ˜ c ( λ + ρ ( k ) , k ) P k,λ ( x ) ≥ ˜ c ( µ + ρ ( k ) , k ) P k,µ ( x ) for all x ∈ V . In other words, Conjecture 4.7 would imply a generalization of Proposition 3.8to the case of Heckman–Opdam polynomials.It is well known that Heckman–Opdam polynomials can be realized as a limit ofMacdonald polynomials [22]. It is interesting to speculate about whether Conjec-ture 4.7, if it holds, is itself a manifestation of an even more general monotonicityproperty of Macdonald polynomials with respect to W -majorization. Acknowledgements
Colin McSwiggen would like to thank Patrick McSwiggen for helpful discussionsduring the preparation of this manuscript. The work of Colin McSwiggen is partiallysupported by the National Science Foundation under Grant No. DMS 1714187.The work of Jonathan Novak is partially supported by a Lattimer Fellowship, aswell as by the Natural Science Foundation under Grant No. DMS 1812288.
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