Orthogonality within the Families of C-, S-, and E-Functions of Any Compact Semisimple Lie Group

The paper is about methods of discrete Fourier analysis in the context of Weyl group symmetry. Three families of class functions are defined on the maximal torus of each compact simply connected semisimple Lie group $G$. Such functions can always be restricted without loss of information to a fundamental region $\check F$ of the affine Weyl group. The members of each family satisfy basic orthogonality relations when integrated over $\check F$ (continuous orthogonality). It is demonstrated that the functions also satisfy discrete orthogonality relations when summed up over a finite grid in $\check F$ (discrete orthogonality), arising as the set of points in $\check F$ representing the conjugacy classes of elements of a finite Abelian subgroup of the maximal torus $\mathbb T$. The characters of the centre $Z$ of the Lie group allow one to split functions $f$ on $\check F$ into a sum $f=f_1+...+f_c$, where $c$ is the order of $Z$, and where the component functions $f_k$ decompose into the series of $C$-, or $S$-, or $E$-functions from one congruence class only.


Introduction
The purpose of this paper is to prove continuous and discrete orthogonality relations for three families of special functions that arise in connection with each compact simple simply connected Lie group G of rank n. These functions are defined on R n (more precisely on the Lie algebra of a maximal torus of G). They are periodic and have invariance properties with respect to the affine Weyl group W aff of G. Following [1] we call them C-, S-, and E-functions to underline the fact that they are generalizations of the cosine, sine, and the exponential functions.
Group transforms, similar to familiar Fourier and cosine transforms, motivate this paper. The C-, S-, and E-functions serve as fundamental functions into which functions with suitable invariance properties can be expanded. Their orthogonality properties make the expansions easy to perform.
It is discrete transforms that primarily interest us here because they have a number of practically useful properties [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In particular, continuous extension of the discrete transforms smoothly interpolate digital data in any dimension and for any lattice symmetry afforded by the structure of the given Lie group G. Many examples show that relative to the amount of available data, these transforms provide much smoother interpolation than the conventional Fourier transform.
At first encounter with such a theory, one might suspect that there are two major limitations to its applicability: first that it involves symmetry that probably does not exist in most of problems, and second that it involves data grids which reflect this symmetry and that these may be difficult to find and laborious to create. In fact neither of these is the case.
First regarding the symmetry, suppose that some compact domain D in R n is given and that we are to analyze certain function f defined on D. After enclosing D inside the fundamental regionF of the affine Weyl group W aff , we may extend f immediately to a W aff -invariant functionf on all of R n . Thusf is completely periodic and also carries full W -symmetry. Analysis off yields information about f by restriction toF . Second, with regards to grids, the finite subgroups of the torus T of G provide us with infinitely many different W aff -invariant grids. There is an elegant and rapidly computable method of labeling the points of such grids that lie in the fundamental regionF [3]. An additional feature of the connection with the Lie groups is that we may use the central elements to effect basic splitting of functions, similar to the familiar decomposition of a function into its even and odd parts.
The purpose of this paper is to record for future reference the exact form of orthogonality relations and central splittings of the various classes of the functions (C-, S-, E-) in the context of discrete subgroups of T of G. The special functions considered here are new in their intended use though the C-and S-functions are well known in Lie theory. More precisely, C-functions, which have also been called Weyl orbit sums or Weyl orbit functions, are constituents of the irreducible characters of G. The discrete version of the C-transform has been described in general in our [3]. Its various mathematical applications are found in [4,5,6,7,8,9,10,11,12,13,14,15,16]. For other applications, see [15,16,17,18,19,20]. Existence of the continuous version of the transform is implied there as well. A review of properties of C-functions is in [13].
The S-functions appear in the Weyl character formula [21], where each character is expressed as the ratio of two particular S-functions. In this paper the functions are put to a different use. A detailed description of the four special cases of S-functions depending on two variables (semisimple Lie groups of rank two), is found in [11]. A general review of properties of Sfunctions is in [14]. The E-functions are not found in the literature except [1]. Their definition is rather natural, once the even subgroup of the Weyl group is invoked.
The names C-, S-, and E-functions make allusion to the fact that, for rank n = 1, the functions become cosine, sine, and the exponential functions. The underlying compact Lie groups are SU (2) for the C-and S-functions, and either SU (2) or U (1) for the E-functions, because W e = 1 in the SU (2) case. From a general perspective, our functions are special cases of functions symmetrized by summing group dependent terms over all elements of a finite group. In our cases the finite groups are the Weyl group W for C and S, and its even subgroup W e for the E-functions.
In Section 2 we recall pertinent facts from the Lie theory. In Section 3 orthogonality properties of common exponential function are invoked, when the function is integrated over n-dimensional torus, or when it is summed up over a suitable finite subgroup of the torus. In the subsequent Sections 4-6, the families of C-, S-, and E-functions are dealt with respectively. The three families are defined for any compact semisimple Lie group. The rank of the group is equal to the number of independent variables. In each case the definition is followed by the proof of the continuous and discrete orthogonality relations. In Section 7, it is shown how the general problem of decomposition of a function into series of C-, or S-functions can be split into c subproblems, where c is the order of the centre of the Lie group. Central splitting is illustrated on the examples of the groups SU (3), Sp(4), and SU (2) × SU (2). Concluding remarks are in the last section.

Preliminaries
The uniformity of the theory of semisimple Lie groups furnishes us with many standard concepts for any compact simply connected simple Lie group G of rank n. We assume that the reader is familiar with the basic theory of such groups but we outline some of it here in order to establish the notation. For more details see [6] and [4].
We denote by G a compact simply connected simple Lie group of rank n; g is the Lie algebra of G and g C := C ⊗ R g is its complexification. Let T be a maximal torus of G, dim R T = n. Let t ′ be the Lie algebra of T. Then t : The kernelQ of exp 2πi(·) is a lattice of rank n in t called the co-root lattice.
Let t * be the dual space of t and let be the natural pairing of t * and t.
For any finite dimensional (smooth complex) representation where Ω(V ) ⊂ t * is a subset of the dual space t * of t. Then on each weight space V λ , π V (exp 2πix) is simply a multiplication by e 2πi λ,x , for all x ∈ t. The set Ω(V ) is the set of weights of π V . SinceQ = ker (exp 2πi(·)) one has e 2πi λ,x = 1 for all λ ∈ Ω(V ) and for all x ∈Q.
Thus λ ∈ Ω(V ) implies that λ is in the Z-dual lattice P toQ. This is a rank n lattice in t * called the weight lattice. Every λ ∈ P occurs in Ω(V ) for some representation π V . For the adjoint representation, the eigenspace decomposition is i.e. {0} ∪ ∆ = Ω(adjoint rep.) and ∆ is called the root system of G relative to T. The Z-span Q of ∆ is an n-dimensional lattice in t * , called the root lattice. Its Z-dual in t is the co-weight latticeP . Let us summarize relations between the lattices: We have the index of connection: be the transpose of the Killing form of g restricted to t. Then we can introduce the following four bases: • {α 1 , . . . , α n }, basis of Q formed by a set of simple roots, • {ω 1 , . . . ,ω n }, basis ofP dual to the basis {α 1 , . . . , α n }, • {α 1 , . . . ,α n }, basis ofQ ⊂ t of simple co-roots. They are defined by • {ω 1 , . . . , ω n }, basis of P , the fundamental weights dual to {α 1 , . . . ,α n }.
For each M ∈ Z >0 , we introduce two finite subgroups of T. For x ∈ t, such that Then 1 MQ /Q is isomorphic to the group of all elements of T whose order divides M . This is our first point group.
Then we have the group 1 MP /Q, of all elements with the property This is the second finite group. Thus For each of these two groups we have natural dual, in the sense of duality in Abelian groups. Namely, we have the map a or equivalently, which is a dual pairing. Likewise is a dual pairing.
Of course the critical object in this theory is the Weyl group W which describes the inherent symmetry of the situation. It acts on T, t, and t * .
All the objects (2)-(4) are W -invariant. In particular the finite groups 1 MQ /Q, 1 MP /Q, P/M Q, P/M P play a fundamental role in the introduction of discrete methods into computational problems involving class functions on G or t.
We let P + denote the set of dominant weights, and P ++ the set of strictly dominant weights. In the sequel Γ will denote some W -invariant finite subgroup of T. Mostly we have in mind the groups 1 MQ /Q and 1 MP /Q, but this is not assumed unless explicitly stated so. Usually we will denote the elements of Γ by elements of t via the exponential map.
On t, W can be combined with the translation group defined byQ to give the affine Weyl group W aff = W ⋉Q.
A convenient fundamental regionF for the affine Weyl group of a simple compact Lie group G of rank n is the simplex which is specified by the n + 1 vertices, where q 1 , . . . , q n are the coefficients of the highest root of G relative to the basis of simple roots. By definitionF is closed. In case G is semisimple but not simple, its fundamental region is the cartesian product of fundamental regions of its simple components.
In the case of W e aff , the affine W e , its fundamental region is the union of (any) two adjacent copies ofF of the corresponding affine Weyl group W aff .

Basic orthogonality relations
Let us fix an n-dimensional torus T of a compact simple Lie group G. For λ, µ ∈ P one has the well known basic continuous orthogonality relation of the elementary functions: assuming T 1dt = 1.
For instance consider (7). For any λ ∈ P (in particular for λ − µ in (7)), The proof of (6) is even simpler. Let λ, µ ∈ P . We say that they satisfy the separation condition on Γ, if one has

C-functions
Since T = t/Q is a compact Abelian group, its Fourier analysis is expressed in terms of objects coming from the lattice dual toQ, namely P . Thus Fourier expansions of T are of the form wherex ∈ t. These are literally functions on t but are effectively functions on T since e 2πi λ,x+Q = e 2πi λ,x for all λ ∈ P.

Definition of C-functions
Orbit sums, equivalently C-functions, C λ (t) are the finite sums We can assume that λ is dominant since C λ (x) depends on the Weyl group orbit W λ, not on λ.
As is well-known, the characters of finite dimensional representations are linear combinations of C-functions (9). Coefficients of such a linear combination are the multiplicities of the dominant weights in the weight system of the representation. In general, it is a laborious problem to calculate the multiplicities. For simple Lie groups of ranks n < 9 a limited help is offered by the tabulation [22].

Continuous orthogonality of C-functions
For λ, µ dominant, we have the continuous orthogonality relations: Here overline denotes complex conjugation, and |W |, |W λ| respectively stand for the size of the Weyl group and for the size of the orbit containing λ.

Discrete orthogonality of C-functions
Let Γ be a W -invariant finite subgroup of T. Leť which we view as a set of elements of t. Let λ, µ ∈ P and assume that they satisfy the separation condition (8) on Γ. We wish to prove the orthogonality relation To see this, note that We have from (7) and (8) that for λ ′ ∈ W λ and µ ′ ∈ W µ, from which (11) follows.

Definition of S-functions
where Here l(w) is the length of w as word in the elementary reflections of W .
The W -skew-invariance of S λ also implies that S λ vanishes on the boundaries of affine chambers of t.

Continuous orthogonality of S-functions
Let λ, µ be strictly dominant integral weights. Then

Discrete orthogonality of S-functions
Let Γ be a finite W -invariant subgroup of T, and let λ, µ ∈ P ++ . Assume the separation condition (8).
We prove that To see this we note first that ifǔ j is on the boundary ofF , then there is an reflection r such that rǔ j =ǔ j and At the same time, and Putting together (16) and (18), we obtain (15).

Definition of E-functions
Let W e ⊂ W be the even subgroup of W , and W e aff the even subgroup of W aff , Let r be any fixed simple affine reflection, and seť Then F e is a fundamental region for W e aff , the affine even Weyl group. Let P + e := P + ∪ rP + .
For λ ∈ P , Since E λ (x) depends on W e λ, not on λ, we can suppose λ ∈ P + e .
which also shows that E rλ depends only on λ, not on the choice of the reflection r. Let Γ be as above. Put Γ ∩F e = {ǔ e 1 ,ǔ e 2 , . . . ,ǔ e m }.

Discrete orthogonality of E-functions
Assume λ, µ ∈ P + e , and assume the separation condition (8) for the group W e . Since E λ , E µ are W e -invariant, 7 Central splitting of the functions Each λ ∈ P determines an irreducible character on Z, so, and j is called the congruence class of λ. It depends only on λ mod Q. Hence it is constant on the W -and W e -orbits of λ.
The following arguments can be made for C-, S-, and E-functions. Using just the C-functions, we have where j is the congruence class of λ. Thus for any class k we have From this, if f is a linear combination of C-or S-functions, we can determine its splitting into the sum of congruence class components (central splitting) as follows, where

Example SU (3)
The centre Z has 3 elements, {ž 0 ,ž 1 ,ž 2 } (cyclic group of 3 elements). Its characters are χ 0 :ž → 1 (trivial character), According to (26) a function f onF is decomposed into the sum of three functions, where By substitution of (29) into (30), followed by use of (25), one verifies directly the basic property of the central splitting (29) of f (x): Each of the components f 0 , f 1 , f 2 of f decomposes into a linear combination of Cor Sfunctions from one congruence class only, respectively class 0, 1, and 2.
For specific application of (30) using a function f (x) given onF , it is useful to make one more step, namely to express each component function as a sum of three terms, each term being the given function evaluated at particular points inF . Forx ∈F , the pointsx +ž 1 andx +ž 2 need not be inF and need to be brought there by suitable transformations from the affine Weyl group of A 2 .

Example Sp(4)
The fundamental region is the isosceles triangleF with vertices Whenx = 0, we havex +ω 2 outside ofF . An appropriate affine reflection bringsx +ω 2 intoF . A pointx = x 1ω1 + x 2ω2 = (x 1 , x 2 ) is inF , if 0 ≤ 2x 1 + x 2 ≤ 1. Thenx +ω 2 is reflected intoF as follows: Therefore the component functions can be written with their arguments given explicitly in thě ω-basis: The functions (35) have definite symmetry properties with respect to the line connecting the points 1 2ω 1 and 1 2ω 2 on the boundary ofF . In case one has f (x) expanded in terms of C-functions (S-functions), f 0 (x 1 , x 2 ) is symmetric (antisymmetric), while f 1 (x 1 , x 2 ) is antisymmetric (symmetric) with respect to that line.
Equivalently, in case f (x) is expanded in terms of C-functions, then f 0 (x 1 , x 2 ) is expanded in terms of C-functions of congruence class 0, while f 1 (x 1 , x 2 ) is expanded in terms of Cfunctions of congruence class 1. Similarly, in case the expansion of f (x) is in terms of Sfunctions, the previous conclusions about congruence classes pertinent to f 0 and f 1 expansions are interchanged.
Recall that the functions C λ and also S λ , where λ = aω 1 + bω 2 ∈ P , are said to be of congruence class 0 or 1, according to the value of a mod 2.