ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE ( ANTI ) SYMMETRIC TRIGONOMETRIC FUNCTIONS

The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of A1 and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of A2 and a specific version of two variable Jacobi polynomials. The connection with recently studied G2-polynomials is established. Formulas for connection between the four types of orbit functions of Bn or Cn and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.


Introduction
Special functions associated with the root systems of simple Lie algebras, e.g.Weyl-orbit functions, play an important role in several domains of mathematics and theoretical physics, in particular in representation theory, harmonic analysis, numerical integration and conformal field theory.The purpose of this paper is to link Weyl-orbit functions with various types of orthogonal polynomials, namely Chebyshev and Jacobi polynomials and their multivariate generalizations, and thus to motivate further development of the remarkable properties of these polynomials in connection with orbit functions.
The collection of Weyl-orbit functions includes four different families of functions called C-, S-, S s -and S l -functions [6,15,16,25].They are induced from the sign homomorphisms of the Weyl groups of geometric symmetries related to the underlying Lie algebras.The symmetric C-functions and antisymmetric Sfunctions also appear in the representation theory of simple Lie algebras [32,34]; the S-functions appear in the Weyl character formula and every character of irreducible representations of simple Lie algebra can be written as a linear combination of C-functions.Unlike C-and S-functions, S s -and S l -functions exist only in the case of simple Lie algebras with two different lengths of roots.
A review of several pertinent properties of the Weylorbit functions is contained in [8,15,16,25].These functions possess symmetries with respect to the affine Weyl group -an infinite extension of the Weyl group by translations in dual root lattice.Therefore, we consider C-, S-, S s -and S l -functions only on specific subsets of the fundamental domain F of the affine Weyl group.Within each family, the functions are continuously orthogonal when integrated over F and form a Hilbert basis of squared integrable functions on F [25,27].They also satisfy discrete orthogonality relations which is of major importance for the processing of multidimensional digital data [8,10,27].Using discrete Fourier-like transforms arising from discrete orthogonality, digital data are interpolated in any dimension and for any lattice symmetry afforded by the underlying simple Lie algebra.Several special cases of simple Lie algebras of rank two are studied in [28][29][30].
The properties of orbit functions also lead to numerical integration formulas for functions of several variables.They approximate a weighted integral of any function of several variables by a linear combination of function values at points called nodes.In general, such formulas are required to be exact for all polynomial functions up to a certain degree [3].Furthermore, the C-functions and S-functions of simple Lie algebra A 1 coincide, up to a constant, with the common cosine and sine functions respectively.They, are therefore, related to the extensively studied Chebyshev polynomials and, consequently, to the integration formulas, quadratures, for the functions of one variable [5,31].In [24], it is shown that there are analogous formulas for numerical integration, for multivariate functions, that depend on the Weyl group of the simple Lie algebra A n and the corresponding C-and S-functions.The resulting rules for functions of several variables are known as cubature formulas.The idea of [24] is extended to any simple Lie algebra in [9,25,26].Optimal cubature formulas in the sense of the nodal points that are required are known only for S-and S s -functions.
Besides the Chebyshev polynomials, the Weyl-orbit functions are related to other orthogonal polynomials.For example, orbit functions of A 2 and C 2 coincide with two-variable analogues of Jacobi polynomials [21].It can also be shown that the C-, S-, S s -and S l -functions arising in connection with simple Lie algebras B n and C n become, up to a constant, (anti)symmetric multivariate cosine functions and (anti)symmetric multivariate sine functions [17].Note that these generalizations lead to multivariate analogues of Chebyshev polynomials and are used to derive optimal cubature formulas.Therefore, this fact indicates that it might be possible to obtain such formulas for all families of orbit functions.As for Chebyshev polynomials, it is of interest to study the accuracy of approximations and interpolations using cubature formulas.
This paper starts with a brief introduction to Weyl groups in Section 2. Then there is a review of the relations of the Weyl-orbit functions with other special functions which are associated with the Weyl groups.In Section 3.1, the connection of the C-and S-functions of one variable with Chebyshev polynomials is recalled.In Sections 3.2 and 3.4, we show that each family of Weyl-orbit functions corresponding to A 2 and C 2 can be viewed as a two-variable analogue of Jacobi polynomials [21].In Section 3.4, we also provide the exact connection with generalizations of trigonometric functions [34].

Weyl groups of simple Lie algebras
In this section, we summarize the properties of the Weyl groups that are neeeded for definition of orbit functions.There are four series of simple Lie algebras Note that we use the standard normalization for the lengths of roots, namely α i , α i = 2 if α i is a long simple root.
In addition to the basis of R n consisting of the simple roots α i , it is convenient for our purposes to introduce the basis of fundamental weights ω j given by ω j , α This allows us to express the weight lattice P defined by as Z-linear combinations of ω j .The subset of dominant weights P + is standardly given as We consider the usual partial ordering on P given by µ λ if and only if λ − µ is a sum of simple roots with non-negative integer coefficients.
To each simple root α i there is a corresponding reflection r i with respect to the hyperplane orthogonal to α i , The finite group W generated by such reflections r i , i = 1, . . ., n is called the Weyl group.For the properties of the Weyl groups see e.g.[12,14].
In the case of simple Lie algebras with two different lengths of the roots, we need to distinguish between short and long simple roots.Therefore, we denote by ∆ s the set of simple roots containing only short simple roots and we denote by ∆ l the set of long simple roots.We also define the following vectors:

Weyl-orbit functions
Each type of Weyl-orbit function arises from the sign homomorphism of Weyl groups σ : W → {±1}.There exist only two different sign homomorphisms on W connected to simple Lie algebras with one length of the roots: identity, denoted by 1, and the determinant [8,25].They are given by their values on the generators r i of W as In the case of simple Lie algebras with two different lengths of roots, i.e.B n , C n , F 4 and G 2 , there are two additional sign homomorphisms denoted by σ s and σ l and given as Labelled by the parameter a ∈ R n , the Weyl-orbit function of the variable b ∈ R n corresponding to sign homomorphism σ is introduced via the formula Each sign homomorphism 1, det, σ s , σ l determines one family of complex valued Weyl-orbit functions, called C-, S-,S s -and S l -functions respectively, and denoted by Several remarkable properties of Weyl-orbit functions, such as continuous and discrete orthogonality are usually achieved by restricting a to some subsets of the weight lattice P (see for example [8,10,15,16,27]).Note also that symmetric C-functions and antisymmetric S-functions appear in the theory of irreducible representations of simple Lie algebras [32,34].
It is also convenient to use an alternative definition of the Weyl-orbit functions via sums over the Weyl group orbits [15,16,25].These orbit sums C a , S a+ , S s a+ s , S l a+ l differ from Φ a , ϕ a+ , ϕ s a+ s , ϕ l a+ l only by a constant with |Stab W c| denoting the number of elements of W which leave c invariant.

Case A 1
The symmetric C-functions and antisymmetric Sfunctions of A 1 are, up to a constant, the common cosine and sine functions [15,16], It is well known that such functions appear in the definition of the extensively studied Chebyshev polynomials [5,31].Several types of Chebyshev polynomials are widely used in mathematical analysis, in particular, as efficient tools for numerical integration and approximations.The Chebyshev polynomials of the first, second, third and fourth kind are denoted by T m (x), U m (x), V m (x) and W m (x), respectively.If x = cos(θ), then for any m ∈ Z ≥0 it holds that Therefore, for specific choices of parameter a 1 and 2πb 1 = θ, we can view the Weyl-orbit functions of A 1 as these Chebyshev polynomials.

Case G 2
Since, for G 2 its two simple roots are of different lengths, all four families of C-, S-, S s and S lfunctions are obtained [15,16,25].The symmetric and antisymmetric orbit functions are given by the following formulas for a = a 1 ω 1 +a 2 ω 2 and b The hybrid cases can be expressed as These functions have been studied in [22], under the notation CC k , SS k , SC k and CS k with no. / Weyl Orbit Functions and (Anti)symmetric Trigonometric Functions where the variable H . Indeed, performing the following change of variables and parameters, we obtain the following relations.
In [22], the functions C a , S a+ /S , S s a+ s /S s s and S l a+ l /S l l are expressed as two-variable polynomials in variables and it is shown that they are orthogonal within each family with respect to a weighted integral on the region (see Fig. 2) containing points (x, y) satisfying with the weight function w α,β (x, y) equal to for S l -functions.

Cases B n and C n
It is shown in this section that the C-, S-, S sand S l -functions arising from B n and C n are related to the symmetric and antisymmetric multivariate generalizations of trigonometric functions [17].
The symmetric cosine functions cos + λ (x) and the antisymmetric cosine functions cos − λ (x) of the variable x = (x 1 , . . ., x n ) ∈ R n are labelled by the parameter λ = (λ 1 , . . ., λ n ) ∈ R n and are given by the following explicit formulas, where S n denotes the symmetric group consisting of all permutations of numbers 1, . . ., n, and sgn(σ) is the signature of σ.The symmetric sine functions sin + λ (x) and the antisymmetric sine functions sin − λ (x) are defined similarly, Firstly, consider the Lie algebra B n and an orthonormal basis {e 1 , . . ., e n } of R n such that α i = e i − e i+1 for i = 1, . . ., n − 1 and α n = e n .
If we determine any a ∈ R n by its coordinates with respect to the basis {e 1 , . . ., e n }, then it holds for the generators r i , i = 1, . . ., n − 1 and r n of the Weyl group W (B n ) of B n that r i (a 1 , . . ., a i , a i+1 , . . ., a n ) = (a 1 , . . ., a i+1 , a i , . . ., a n ), Therefore, W (B n ) consists of all permutations of the coordinates a i with possible sign alternations of some of them, and we actually have that W (B n ) is isomorphic to (Z/2Z) n S n [12].This implies that Since det is a homomorphism on W (B n ), we also obtain Similar connections are valid for S s -functions and S l -functions, Since Lie algebras B n and n are dual to each other, we can deduce that the symmetric and antisymmetric generalizations are also connected to the Weyl-orbit functions of C n .In order to obtain explicit relations, one can proceed by analogy with case B n and introduce an orthogonal basis {f 1 , . . ., f n } such that for i = 1, . . ., n − 1 We denote by ãi the coordinates of any point a ∈ R n with to the basis {f 1 , . . ., f n }, i.Thus, proceeding as before, we derive the following.Note that the S s -functions are related to cos − a and the S l -functions are related to sin + a in the case of C n , whereas the S s -functions correspond to sin + a and the S l -functions correspond to cos − a if we consider the simple Lie algebra B n .This follows from the fact that the short (long) roots of C n are dual to the long (short) roots of B n .

Concluding remarks
(1.) Symmetric and antisymmetric cosine functions can be used to construct multivariate orthogonal polynomials analogous to the Chebyshev polynomials of the first and third kind.The method of construction is based on decomposition of the products of these functions and is fully described in [7].
To build polynomials analogous to the Chebyshev polynomials of the second and fourth kind, it seems that the symmetric and antisymmetric generalizations of sine functions have to be analysed.This hypothesis is supported by the decomposition of the products of two-dimensional sine functions which can be found in [11].
( .)Another approach to generalization of the multivariate polynomials related to the Weyl-orbit functions stems from the shifted orthogonality of the orbit functions developed in [2].This generalization encompasses shifts of the points of the sets over which the functions are discretely orthogonal, and also shifts of the labeling weights.As a special case it contains for A 1 all four kinds of Chebyshev polynomials.The existence of analogous polynomials obtained through this approach and their relations to already known generalizations deserves further study. (3.)Besides the methods of polynomial interpolation and numerical integration, the Chebyshev polynomials are connected to other efficient methods in numerical analysis such as numerical solutions of differential equations, solutions of difference equations, fast transforms and spectral methods.The existence and the form of these methods, connected in a multivariate setting to Weyl-orbit functions, are open problems.

A 2 Since
, for A 2 the two simple roots are of the same length, there are only two corresponding families of Weyl-orbit functions, C-and S-functions.For a = a 1 ω 1 +a 2 ω 2 and b = b 1 α ∨ 1 +b 2 α ∨ 2 the explicit formulas of C-functions and S-functions are given by C a (b) = 1 |Stab W a| e 2πi(a1b1+a2b2)

Figure 1 .
Figure 1.The region of orthogonality bounded by the three-cusped deltoid.

Figure 2 .
Figure 2. The region of orthogonality for the case G2.

Figure 3 .
Figure 3.The region of orthogonality bounded by two lines and parabola.