Orthogonal functions related to Lax pairs in Lie algebras

We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover, eigenfunctions for the operator $L$ for a Lax pair for $\mathfrak{sl}(d+1,\mathbb C)$ are studied in certain representations.


Introduction
The link of the Toda lattice to three-term recurrence relations via the Lax pair after the Flaschka coordinate transform is well understood, see e.g. [2], [23]. We consider a Lax pair in a specific Lie algebra, such that in irreducible * -representations the Lax operator is a Jacobi operator. A Lax pair is a pair of time-dependent matrices or operators L(t) and M(t) satisfying the Lax equationL where [ , ] is the commutator and the dot represents differentiation with respect to time. The Lax operator L is isospectral, i.e. the spectrum of L is independent of time. A famous example is the Lax pair for the Toda chain in which L is a self-adjoint Jacobi operator, L(t)e n = a n (t)e n+1 + b n (t)e n + a n−1 (t)e n−1 , where {e n } is an orthonormal basis for the Hilbert space, and M is the skew-adjoint operator given by M(t)e n = a n (t)e n+1 − a n−1 (t)e n−1 . In this case the Lax equation describes the equations of motion (after a change of variables) of a chain of interacting particles with nearest neighbour interactions. The eigenvalues of L, L being isospectral, constitute integrals of motion.
In this paper we define a Lax pair in a 2-parameter Lie algebra. In the special case of sl(2, C) we recover the Lax pair for the sl(2, C) Kostant Toda lattice, see [2, §4.6] and references given there. We give a slight generalization by allowing for a more general M(t). We discuss the corresponding solutions to the corresponding differential equations in various representations of the Lie algebra. In particular, one obtains the classical relation to the Hermite, Krawtchouk, Charlier, Meixner, Laguerre and Meixner-Pollaczek polynomials from the Askey scheme of hypergeometric functions [13] for which the Toda modification, see [11, §2.8], remains in the same class of orthogonal polynomials. This corresponds to the results established by Zhedanov [25], who investigated the situation where L, M andL act as three-term recurrence operators and close up to a Lie algebra of dimension 3 or 4. In the current paper Zhedanov's result is explained, starting from the other end. In Zhedanov's approach the condition on forming a low-dimensional Lie algebra forces a factorization of the functions as a function of time t and place n, which is immediate from representing the Lax pair from the Lie algebra element.
Moreover, we extend to non-polynomial settings by considering representations of the corresponding Lie algebras in ℓ 2 (Z) corresponding to the principal unitary series of su (1,1) and the representations of e(2), the Lie algebra of the group of motions of the plane. In this way we find solutions to the Toda lattice equations labeled by Z. There is a (non-canonical) way to associate to recurrences on ℓ 2 (Z) three-term recurrences for 2 × 2-matrix valued polynomials, see e.g. [3], [15]. However, this does not lead to explicit 2 × 2-matrix valued solutions of the non-abelian Toda lattice as introduced and studied in [4], [6] in relation to matrix valued orthogonal polynomials, see also [12] for an explicit example and the relation to the modification of the matrix weight. The general Lax pair for the Toda lattice in finite dimensions, as studied by Moser [19], can also be considered and slightly extended in the same way as an element of the Lie algebra sl(d + 1, C). This involves t-dependent finite discrete orthogonal polynomials, and these polynomials occur in describing the action of L(t) in highest weight representations. We restrict to representations for the symmetric powers of the fundamental representations, then the eigenfunctions can be described in terms of multivariable Krawtchouk polynomials following Iliev [10]. We discuss briefly the t-dependence of the corresponding eigenvectors of L(t).
In brief, in §2 we recall the 2-parameter Lie algebra as in [17] and the Lax pair. In §3 we discuss su(2) and its finite-dimensional representations, and in §4 we discuss the case of su(1, 1), where we discuss both discrete series representations and principal unitary series representations. The last leads to new solutions of the Toda equations and the generalization in terms of orthogonal functions. The corresponding orthogonal functions are the overlap coefficients between the standard basis in the representations and the tdependent eigenfunctions of the operator L. In §5 we look at the oscillator algebra as specialization, and in §6 we consider the Lie algebra for the group of plane motions leading to a solution in connection to Bessel functions. In §7 we indicate how the measures for the orthogonal functions involved have to be modified in order to give solutions of the coupled differential equations. For the Toda case related to orthogonal polynomials, this coincides with the Toda modification [11, §2.8]. Finally, in §8 we consider the case of finite dimensional representations of such a Lax pair for a higher rank Lie algebra in specific finite-dimensional representations for which all weight spaces are 1-dimensional.
A question following up on §7 is whether the modification for the weight is of general interest, cf. [11, §2.8]. A natural question following up on §8 is what happens in other finite-dimensional representations, and what happens in infinite dimensional representations corresponding to non-compact real forms of sl(d + 1, C) as is done in §4 for the case d = 1. We could also ask if it is possible to associate Racah polynomials, as the most general finite discrete orthogonal polynomials in the Askey scheme, to the construction of §8. Moreover, the relation to the interpretation as in [16] suggests that it might be possible to extend to quantum algebra setting, but this is quite open.
Dedication. This paper is dedicated to Richard A. Askey (1933-2019) who has done an incredible amount of fascinating work in the area of special functions, and who always had an open mind, in particular concerning relations with other areas. We hope this spirit is reflected in this paper. Moreover, through his efforts for mathematics education, Askey's legacy will be long-lived 2. The Lie algebra g(a, b) Let a, b ∈ C. The Lie algebra g(a, b) is the 4-dimensional complex Lie algebra with basis H, E, F, N satisfying For a, b ∈ R there are two inequivalent * -structures on g(a, b) defined by We define the following Lax pair in g(a, b).
and u ∈ C[0, ∞) be real-valued functions and let c ∈ R. The Lax pair L, M ∈ g(a, b) is given by Note that L * = L and M * = −M. Being a Lax pair means thatL = [M, L], which leads to the following differential equations.
In the following sections we consider the Lax operator L in an irreducible * -representation of g(a, b), and we determine explicit eigenfunctions and its spectrum. We restrict to the following special cases of the Lie algebra g(a, b): is the generalized oscillator algebra • g(0, 0) ∼ = e(2) ⊕ C, with e(2) the Lie algebra of the group of plane motions These are the only essential cases as g(a, b) is isomorphic as a Lie algebra to one of these cases, see [17,.
3. The Lie algebra su (2) In this section we consider the Lie algebra g(a, b) from Section 2 with (a, b) = (1, 0) and ǫ = +, i.e. the Lie algebra su(2) ⊕ C. The basis element N plays no role in this case, therefore we omit it. So we consider the Lie algebra with basis H, E, F satisfying commutation relations where (without loss of generality) we set c = 0. The differential equations for r and s from Proposition 2.2 read in this caseṡ (t) = 2u(t)r(t) and the invariant in Corollary 2.3 is given by I(r, s) = r 2 + s 2 .
Proof. Fromṡ = 2ur it follows that s is increasing. Since (r(t), s(t)) in phase space is a point on the invariant I(r, s) = I(r(0), s(0)), which describes a circle around the origin, it follows that r(t) and s(t) remain positive.
Throughout this section we assume that the conditions of Lemma 3.1 are satisfied, so that r(t) and s(t) are positive. Note that if we change the condition on r(0) to r(0) < 0, then r(t) < 0 for all t > 0.
Proof. From (3.2) and the recurrence relation of the Krawtchouk polynomials we obtain The last identity implies .
Then we find that the eigenvalue is Since s 2 + r 2 is constant, the result follows.
4. The Lie algebra su(1, 1) In this section we consider representations of g(a, b) with (a, b) = (1, 0) and ǫ = −, i.e. the Lie algebra su(1, 1) ⊕ C. We omit the basis element N again. The commutation relations are the same as in the previous section. The * -structure in this case is defined by The Lax pair (2.2) is given by where we set c = 0 again. The functions r and s satisfẏ and the invariant is given by I(r, s) = s 2 − r 2 . Throughout this section we assume that the assumptions of Lemma 4.1 are satisfied.
We consider two families of irreducible * -representations of su(1, 1). The first family is the positive discrete series representations π k , k > 0, on ℓ 2 (N). The actions of the basis elements on the standard orthonormal basis {e n | n ∈ N} are given by π k (H)e n = 2(k + n) e n π k (E)e n = (n + 1)(2k + n) e n+1 π k (F )e n = − n(2k + n − 1) e n−1 .
We use the convention e −1 = 0.
Note that both representations π + k and π λ,ε as given above define unbounded representations. The operators π(X), X ∈ su(1, 1), are densely defined operators on their representation space, where as a dense domain we take the set of finite linear combinations of the standard orthonormal basis {e n }.
Remark 4.2. The Lie algebra su(1, 1) has two more families of irreducible * -representations: the negative discrete series and the complementary series. The negative discrete series representation π − k , k > 0, can be obtained from the positive discrete series representation π k by setting π − k (X) = π k (ϑ(X)), The complementary series are defined in the same way as the principal unitary series, but the labels λ, ε satisfy ε ∈ [0, 1 2 . The results obtained in this section about the Lax operator in the positive discrete series and principal unitary series representations can easily be extended to these two families of representations.
π k (L) can be diagonalized using explicit families of orthogonal polynomials. We need to distinguish between three cases corresponding to the invariant s 2 − r 2 being positive, zero or negative. This corresponds to hyperbolic, parabolic and elliptic elements, and the eigenvalues and eigenfunctions have different behaviour per class, cf. [16]. 4.1.1. Case 1: s 2 − r 2 > 0. In this case eigenfunctions of π k (L) can be given in terms of Meixner polynomials. The orthonormal Meixner polynomials [13, Section 9.10] are defined by where β > 0 and 0 < c < 1. They satisfy the three-term recurrence relation Their orthogonality relations are given by Proof. The proof runs along the same lines as the proof of Theorem 3.2. The condition It follows from the three-term recurrence relation for Meixner polynomials that r −1 L has 4.2. Case 2: s 2 − r 2 = 0. In this case we need the orthonormal Laguerre polynomials [13, Section 9.12], which are defined by They satisfy the three-term recurrence relation and the orthogonality relations are The set {L n | n ∈ N} is an orthonormal basis for the corresponding weighted L 2 -space.
Using the three-term recurrence relation for the Laguerre polynomials we obtain the following result.
In this case we need the orthonormal Meixner-Pollaczek polynomials [13, Section 9.7] given by where λ > 0 and 0 < φ < π. The three-term recurrence relation for these polynomials is and the orthogonality relations read The set {P n | n ∈ N} is an orthonormal basis for the weighted L 2 -space.
Proof. The proof is similar as before. Using the three-term recurrence relation for the Meixner-Pollaczek polynomials it follows that the generalized eigenvalue of from which the result follows.

4.4.
The Lax operator in the principal unitary series. The action of the Lax operator L in the principal unitary series as a Jacobi operator on ℓ 2 (Z) is given by Again we distinguish between the cases where the invariant s 2 − r 2 is either positive, negative or zero.
The Meixner functions [9] are defined by for x, n ∈ Z and c ∈ (0, 1). The parameters λ and ε are the labels from the principal unitary series. The Meixner functions satisfy the three-term recurrence relation and the orthogonality relations read The set {m n | n ∈ Z} is an orthonormal basis for the weighted L 2 -space.
The differential equations for s and r are in this case given bẏ s = 2ru r = −2cu and the invariant is r 2 + 2cs. There is a family of irreducible * -representations π k,h , h > 0, k ≥ 0, on ℓ 2 (N) defined by π k,h (N)e n = −h e n π k,h (H)e n = 2(k + n) e n π k,h (E)e n = h(n + 1) e n+1 π k,h (F )e n = √ hn e n−1 .
For the diagonalization of π k,h (L) we distinguish between the cases c = 0 and c = 0.
5.1. Case 1: c = 0. In this case we need the orthonormal Charlier polynomials [13, Section 9.14], which are defined by where a > 0 and n, x ∈ N. The orthogonality relations are and {C n | n ∈ N} is an orthonormal basis for the corresponding L 2 -space. The three-term recurrence relation reads −xC n (x) = a(n + 1) C n+1 (x) − (n + a)C n (x) + √ an C n−1 (x).
Then U t is unitary and The action of L can be written in the following form, − k e n = sgn(c) hr 2 (n + 1) 4c 2 e n+1 + n + hr 2 4c 2 e n + sgn(c) hr 2 n 4c 2 e n−1 and recall that 1 2c r 2 + s is constant. The result then follows from comparing with the three-term recurrence relation for the Charlier polynomials.

5.2.
Case 2: c = 0. In this caseṙ = 0, so r is a constant function. We use the orthonormal Hermite polynomials [13, Section 9.15], which are given by They satisfy the orthogonality relations and {H n | n ∈ N} is an orthonormal basis for L 2 (R, e −x 2 dx/ √ π). The three-term recurrence relation is given by then U t is unitary and U t • π k,h (L(t)) • U * t = M(x). Proof. We have π k,h 1 r √ h (L + sh) e n = √ n + 1 e n+1 + √ n e n−1 , which corresponds to the three-term recurrence relation for the Hermite polynomials.

The Lie algebra e(2)
We consider the Lie algebra g(a, b) with a = b = 0 and ǫ = +. Similar as in the case of sl(2, C), we omit the basis element N again. The remaining Lie algebra is e (2) e(2) has a family of irreducible * -representations π k , k > 0, on ℓ 2 (Z) given by π k (H)e n = 2n e n , π k (E)e n = ke n+1 , π k (F )e n = ke n−1 .
This defines an unbounded representation. As a dense domain we use the set of finite linear combinations of the basis elements. Assume c = 0. The Lax operator π k (L(t)) is a Jacobi operator on ℓ 2 (Z) given by π k (L(t))e n = kr(t)e n+1 + 2cne n + kr(t)e n−1 .
then U t is unitary and U t • π k (L(t)) • U * t = M(2cm). Finally, let us consider the completely degenerate case c = 0. In this case r is also a constant function, so there are no differential equations to solve. We can still diagonalize the (degenerate) Lax operator, which is now independent of time.

Modification of orthogonality measures
In this section we briefly investigate the orthogonality measures from the previous sections in case the Lax operator L(t) acts as a finite or semi-infinite Jacobi matrix. In these cases the functions U t e n are t-dependent orthogonal polynomials and we see that the weight function W t of the orthogonality measure for U t e n is a modification of the weight function W 0 in the sense that  Theorem 7.1 can be checked for each case by a straightforward calculation: we express m as a function of s and r, m(t) = A 0 F (s(t), r(t)), where A 0 is a normalizing constant such that m(0) = 1. Then differentiating and using the differential equations for r and s we can expressṁ/m in terms of u and r.
Note that r = r(t) is constant in this case. Thenṡ(t) = 2ru(t) leads tȯ Remark 7.3. The result from Theorem 7.1 is also valid for the orthogonal functions from Theorems 4.6 and 4.8, i.e. for L(t) acting as a Jacobi operator on ℓ 2 (Z) in the principal unitary series for su(1, 1) in cases r 2 − s 2 = 0. However, there is no similar modification function in the other cases where L(t) acts as a Jacobi operator on ℓ 2 (Z). Furthermore, the corresponding recurrence relations for the functions on Z can be rewritten to recurrence relations for 2 × 2 matrix orthogonal polynomials, but in none of the cases the modification of the weight function is as in Theorem 7.1.

The case of sl(d + 1, C)
We generalize the situation of the Lax pair for the finite-dimensional representation of sl(2, C) to the higher rank case of sl(d + 1, C). Let E i,j be the matrix entries forming a basis for the gl(d + 1, C). We label i, j ∈ {0, 1, · · · , d}. We put H i = E i−1,i−1 − E i,i , i ∈ {1, · · · , d}, for the elements spanning the Cartan subalgebra of sl(d + 1, C).
and assume that the functions u i and r i are non-zero for all i and Note that we can write it uniformlẏ assuming the convention that s 0 (t) = s d+1 (t) = 0, which we adapt for the remainder of this section. The Toda case follows by taking u i = r i for all i, see [2], [19].
Proof. The proof essentially follows as in [2, §4.6], but since the situation is slightly more general we present the proof, see also [19, §5]. A calculation in sl(d + 1, C) gives and the last term needs to vanish, since this term does not occur in L(t) and in its derivativė L(t). Now the stated coupled differential equations correspond toL = [M, L].
Remark 8.2. Taking the representation of the Lax pair for the su(2) case in the d + 1dimensional representation as in Section 6, we get, with d = 2j, as an example Then the coupled differential equations of Proposition 8.1 are equivalent to (3.1).
Note that with the choice of Remark 8.2, the polynomials in Lemma 8.3 are Krawtchouk polynomials, see Theorem 3.2. Explicitly, where C = r 2 (t) + s 2 (t) is invariant, see Theorem 3.2 and its proof.
Proof. In the natural representation we have L(t)e n =      r 1 (t)e 1 + s 1 (t)e 0 n = 0 r n+1 (t)e n+1 + (s n+1 (t) − s n (t))e n + r n−1 (t)e n−1 1 ≤ n < d −s d (t)e d + r d (t)e d−1 n = d as a Jacobi operator. So the spectrum of L(t) is simple, and the spectrum is time independent, since (L(t), M(t)) is a Lax pair. We can generate the corresponding eigenvectors as d n=0 p n (λ; t)e n , where the recursion follows from the expression of of the Lemma. The eigenvalues are then determined by the final equation, and since Tr(L(t)) = 0 we have d i=0 λ i = 0. Let P (t) = p i (λ j ; t) d i,j=0 be the corresponding matrix of eigenvectors, so that L(t)P (t) = P (t)Λ, Λ = diag(λ 0 , λ 1 , · · · , λ d ).
Since L(t) is self-adjoint in the natural representation, we find i,j=0 is unitary. As r i and s i are real-valued, we have p n (λ s ; t) = p n (λ s ; t), so that Q(t) is a real matrix, hence orthogonal. So the dual orthogonality relations to (8.2) hold as well. We will assume moreover that r i are positive functions. The dual orthogonality relations to (8.2) hold; Note that the w r (t) are essentially time-dependent Christoffel numbers [22, §3.4]. By [19, §2], see also [5,Thm. 2], the eigenvalues and the w r (t)'s determine the operator L(t), and in case of the Toda lattice, i.e. u i (t) = r i (t), the time evolution corresponds to linear first order differential equations for the Christoffel numbers [19, §3].
Since the spectrum is time-independent, the invariants for the system of Proposition 8.1 are given by the coefficients of the characteristic polynomial of L(t) in the natural representation. Since the characteristic polynomial is obtained by switching to the threeterm recurrence for the corresponding monic polynomials, see [11, §2.2], [19, §2], this gives the same computation. For a Lax pair, Tr(L(t) k ) are invariants, and in this case the invariant for k = 1 is trivial since L(t) is traceless. In this way we have d invariants, Tr(L(t) k ), k ∈ {2, · · · , d + 1}. Lemma 8.4. With the convention that r n and s n are zero for n / ∈ {1, · · · , d} we have the invariants Proof. Write L(t) = DS + D 0 + S * D with D = diag(r 0 (t), r 1 (t), · · · , r d (t)), S : e n → e n+1 the shift operator and S * : e n → e n−1 its adjoint (with the convention e −1 = e d+1 = 0 and r 0 (t) = 0). And D 0 is the diagonal part of L(t). Then Tr(L(t) k ) = Tr((DS + D 0 + S * D) k ) and we need to collect the terms that have the same number of S and S * in the expansion. The trace property then allows to collect terms, and we get Tr(L(t) 2 ) = Tr(D 2 0 ) + 2Tr(D 2 ), Tr(L(t) 3 ) = Tr(D 3 0 ) + 3Tr(D 0 D 2 ) + 3Tr(SD 0 S * D 2 ) and this gives the result, since (SD 0 S * ) n,n = (D 0 ) n−1,n−1 .
We do not use Lemma 8.4, and we have included to indicate the analog of Corollary 2.3.
We can continue this and find e.g.
Let N ∈ N, and let C N [x] = C N [x 0 , · · · , x d ] be the space of homogeneous polynomials of degree N in d + 1-variables, then C N [x] is an irreducible representation of sl(d + 1) and is a highest weight representation corresponding to Nω 1 , ω 1 being the first fundamental weight for type A d . Then x ρ , and so we have a basis of joint eigenvectors of the Cartan subalgebra spanned by H 1 , · · · , H d and the joint eigenspace, i.e. the weight space, is 1-dimensional. It is a unitary representation for the inner product and it gives a unitary representation of SU(d + 1) as well.
From now on we assume this value for u i,j 1 ≤ i, j ≤ d. Explicit expressions for P (σ ′ , ρ ′ ) in terms of Gelfand hypergeometric series are due to Mizukawa and Tanaka [18], see [10, (1.3)]. See also Iliev [10] for an overview of special and related cases of the multivariable cases.
Proof. Observe thatx and Q(t) 0,i = w i (t) is non-zero. Now expandx ρ using (8.5) and Q(t) i,j = p i (λ j ; t) w j (t) gives the result.
By the orthogonality (8.4) of the eigenvectors of L(t) we find where we use that all entries of Q(t) are real. The second orthogonality follows by duality, and the orthogonality corresponds to [10,Cor. 5.3].
Proof. Apply Lemma 8.5 to expandx ρ in L(t)x ρ = ( d i=0 λ i ρ i )x ρ , and use the explicit expression of L(t) and the corresponding action. Compare the coefficient of x τ on both sides to obtain the result.
Remark 8.7. In the context of Remark 8.2 and (8.1) we have that the u i,j are Krawtchouk polynomials. Then the left hand side in (8.5) is related to the generating function for the Krawtchouk polynomials, see [13, (9.11.11)], i.e. the case d = 1 of (8.5). Putting p(t) w) i . Using this in the generating function, the left hand side of (8.5) gives a generating function for Krawtchouk polynomials. Comparing the powers of w k on both sides gives The left hand side is, up to a normalization, the overlap coefficient of L(t) in the sl(2, C) case for the representation of dimension Nd + 1, see §3. Indeed, the representation sl(2, C) to sl(d + 1, C) to End(C N [x]) yields a reducible representation of sl(2, C), and the vector x (0,··· ,0,N ) is a highest weight vector of sl(2, C) for the highest weight dN. Restricting to this space then gives the above connection. Since L(t) has simple spectrum, we conclude that for some constant c depending on the eigenvalue λ and t. Note that this differs from [20, Lemma 2]. For the case N = 1 we get M(t)v λr (t) = d n=0 p n−1 (λ r ; t)u n (t) − p n+1 (λ r ; t)u n+1 (t) x n with the convention that u 0 (t) = u d+1 (t) = 0, p −1 (λ r ; t) = 0. So (M(t) − c(t, λ r ))v λr (t) =v λr (t) = d n=0ṗ n (λ r ; t) x n and comparing the coefficient of x 0 , we find c(t, λ r ) = −p 1 (λ r ; t)u 1 (t). So we have obtained the following proposition.
Note that for 0 ≤ n < d we have (8.6)ṗ n (λ; t) = u n (t)p n−1 (λ; t) − u n+1 (t)p n+1 (λ; t) + u 1 (t)p 1 (λ; t)p n (λ; t) as polynomial identity. Indeed, for n = 0 this is trivially satisfied, and for 1 ≤ n < d, this is a polynomial identity of degree n due to the condition in Proposition 8.1, which holds for all λ r and hence is a polynomial identity. Note that the right hand side is a polynomial of degree n, and not of degree n + 1 since the coefficient of λ n+1 is zero because of the relation on u i and r i in Proposition 8.1.
Writing out the identity for the Krawtchouk polynomials we obtain after simplifying where the left hand side is related to the derivative. Note that the derivative of p cancels with factors u, see Theorem 3.2 and its proof and §7.