Spherical Functions of Fundamental $K$-Types Associated with the $n$-Dimensional Sphere

In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=({\mathrm{SO}}(n+1),{\mathrm{SO}}(n))$ in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same $K$-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$ supported on $[0,1]$. Moreover, we show that $W$ has a second order symmetric hypergeometric operator $D$.


Introduction
The theory of spherical functions dates back to the classical papers ofÉ. Cartan and H. Weyl; they showed that spherical harmonics arise in a natural way from the study of functions on the n-dimensional sphere S n = SO(n + 1)/SO(n). The first general results in this direction were obtained in 1950 by Gel'fand, who considered zonal spherical functions of a Riemannian symmetric space G/K. In this case we have a decomposition G = KAK. When the Abelian subgroup A is one dimensional, the restrictions of zonal spherical functions to A can be identified with hypergeometric functions, providing a deep and fruitful connection between group representation theory and special functions. In particular when G is compact this gives a one to one correspondence between all zonal spherical functions of the symmetric pair (G, K) and a sequence of orthogonal polynomials.
In light of this remarkable background it is reasonable to look for an extension of the above results, by considering matrix-valued irreducible spherical functions on G of a general K-type. This was accomplished for the first time in the case of the complex projective plane P 2 (C) = SU(3)/U(2) in [5]. This seminal work gave rise to a series of papers including [6,7,8,10,14,15,16,17,18,19], where one considers matrix valued spherical functions associated to a compact symmetric pair (G, K) of rank one, arriving at sequences of matrix valued orthogonal polynomials of one real variable satisfying an explicit three-term recursion relation, which are also eigenfunctions of a second order matrix differential operator (bispectral property).
The very explicit results contained in this paper are obtained for certain K-types, namely the fundamental K-types. Also, the detailed construction of sequences of matrix orthogonal polynomials out of these irreducible spherical functions, following the general pattern established in [5], gives new examples of classical sequences of matrix-valued orthogonal polynomials of size 2 and 3. For the general notions concerning matrix-valued orthogonal polynomials see [9].
Interesting generalizations of these sequences are given in [20], where the coefficients of the three term recursion relation satisfied by them is exhibited.
The present paper is an outgrowth of the results of [25,Chapter 5] and we are currently working on the extension of these results for the spherical functions of any K-type associated with the n-dimensional sphere. Using [23], one can obtain the corresponding results for the spherical functions of any K-type associated with n-dimensional real projective space. The starting point is to describe the irreducible spherical functions associated with the pair (G, K) = (SO(n+1), SO(n)) in terms of eigenfunctions of a matrix linear differential operator of order two. The output of this description is that the irreducible spherical functions of the same fundamental K-type are encoded in a sequence of matrix valued orthogonal polynomials.
Briefly the main results of this paper are the following. After some preliminaries, in Section 3 we study the eigenfunctions of an operator ∆ on G, which is closely related to the Casimir operator. Every spherical function Φ has to be eigenfunction of this operator ∆; considering the KAK-decomposition SO(n + 1) = SO(n)SO(2)SO(n) and choosing an appropriate coordinate y on an open subset of A, we translate the condition ∆Φ = λΦ, λ ∈ C, into a matrix valued differential equation DH = λH on the open interval (0, 1), where H is the restriction of Φ to SO (2). The property of the spherical functions Φ(xgy) = π(x)Φ(g)π(y), g ∈ G, x, y ∈ K, tell us that Φ is determined by its K-type and the function H.
In Section 4 we first explicitly describe all the irreducible spherical functions of the symmetric pair (G, K) = (SO(n + 1), SO(n)) with M -irreducible K-types, with M = SO(n − 1), the centralizer of the subgroup A in K; we give these expressions in terms of the hypergeometric function 2 F 1 .
In Section 5 the operator D is studied in detail when the K-types correspond to fundamental representations. Certain K-fundamental types are M irreducible, and therefore they were already considered en Section 4; besides, when n is odd there is a particular fundamental K-type which has three M -submodules, this case is studied in the last section of this work. For the rest of the cases we considered separately when n is even and when n is odd. Although, in both cases we worked with the concrete realizations of the fundamental representations considering the exterior powers of the standard representation of SO(n): Λ 1 C n , Λ 2 C n , . . . , Λ −1 C n , with n = 2 or n = 2 + 1.
Also we prove that every spherical function gives a vector polynomial eigenfunction P of D.
Therefore we obtain the following explicit expression of P in terms of the matrix hypergeometric function for any irreducible spherical function P (y) = w j=0 y j j! [C; U ; V + λ] j P (0), see Theorem 7.6. In Section 8 for each pair (n, p) we construct a sequence of matrix orthogonal polynomials {P w } w≥0 of size 2 with respect to the weight function W , which are eigenfunctions of the symmetric differential operator D. Namely, where λ(w, δ) = −w(w + n + 1) − p if δ = 0, −w(w + n + 1) − n + p if δ = 1.
Finally, in Section 9 we develop the same techniques in order to obtain analogous results for irreducible spherical functions of the particular K-fundamental type Λ (C n ) for which we have three M -submodules instead of only two. This only occurs when n is of the form 2 + 1.
It is worth to notice that, unlike the other cases, the 3 × 3 matrix-weight built here does reduce to a smaller size.

Spherical functions
Let G be a locally compact unimodular group and let K be a compact subgroup of G. LetK denote the set of all equivalence classes of complex finite dimensional irreducible representations of K; for each δ ∈K, let ξ δ denote the character of δ, d(δ) the degree of δ, i.e. the dimension of any representation in the class δ, and χ δ = d(δ)ξ δ . We shall choose once and for all the Haar measure dk on K normalized by K dk = 1.
We shall denote by V a finite dimensional vector space over the field C of complex numbers and by of all linear transformations of V into V . Whenever we refer to a topology on such a vector space we shall be talking about the unique Hausdorff linear topology on it.
Definition 2.1. A spherical function Φ on G of type δ ∈K is a continuous function on G with values in End(V ) such that i) Φ(e) = I (I is the identity transformation); The reader can find a number of general results in [21] and [4]. For our purpose it is appropriate to recall the following facts.
ii) k → Φ(k) is a representation of K such that any irreducible subrepresentation belongs to δ.
Concerning the definition, let us point out that the spherical function Φ determines its type univocally (Proposition 2.2) and let us say that the number of times that δ occurs in the representation k → Φ(k) is called the height of Φ.
A spherical function Φ : If G is a connected Lie group, it is not difficult to prove that any spherical function Φ : , and moreover that it is analytic. Let D(G) denote the algebra of all left invariant differential operators on G and let D(G) K denote the subalgebra of all operators in D(G) which are invariant under all right translations by elements in K.
In the following proposition (V, π) will be a finite dimensional representation of K such that any irreducible subrepresentation belongs to the same class δ ∈K. ii) Φ(k 1 gk 2 ) = π(k 1 )Φ(g)π(k 2 ), for all k 1 , k 2 ∈ K, g ∈ G, and Φ(e) = I; Moreover, we have that the eigenvalues [DΦ](e), D ∈ D(G) K , characterize the spherical functions Φ as stated in the following proposition. Let us observe that if Φ : G −→ End(V ) is a spherical function, then Φ : D → [DΦ](e) maps D(G) K into End K (V ) (End K (V ) denotes the space of all linear maps of V into V which commutes with π(k) for all k ∈ K) defining a finite dimensional representation of the associative algebra D(G) K . Moreover, the spherical function is irreducible if and only if the representation Φ : D(G) K −→ End K (V ) is irreducible. We quote the following result from [19]. ii) every irreducible spherical function of (G, K) is of height one.
In this paper the pair (G, K) is (SO(n + 1), SO(n)). Then, it is known that D(G) K is an Abelian algebra; moreover, D(G) K is isomorphic to D(G) G ⊗ D(K) K (see in [13,Theorem 10.1] or [1]), where D(G) G (resp. D(K) K ) denotes the subalgebra of all operators in D(G) (resp. D(K)) which are invariant under all right translations by elements in G (resp. K).
An immediate consequence of this is that all irreducible spherical functions of our pair (G, K) are of height one.
Spherical functions of type δ (see in [21,Section 3]) arise in a natural way upon considering representations of G. If g → U (g) is a continuous representation of G, say on a finite dimensional vector space E, then If the representation g → U (g) is irreducible then the associated spherical function Φ is also irreducible. Conversely, any irreducible spherical function on a compact group G arises in this way from a finite dimensional irreducible representation of G.

Root space structure of so(n, C)
Let E ik denote the square matrix with a 1 in the ik-entry and zeros elsewhere; and let us consider the matrices Then, the set {I ki } i<k is a basis of the Lie algebra so(n). These matrices satisfy the following commutation relations [I ki , I rs ] = δ ks I ri + δ ri I sk + δ is I kr + δ rk I is .
If we assume that k > i, r > s then we have is right invariant under SO(n), i.e.
Proof . To prove that Q n is right invariant under SO(n) it is enough to prove thatİ p,p−1 (Q n ) = 0 for all 2 ≤ p ≤ n. We havė This proves the proposition.

The operator Q 2
Let us assume that n = 2 . We look at a root space decomposition of so(n) in terms of the basis elements I ki , 1 ≤ i < k ≤ n. The linear span is a Cartan subalgebra of so(n, C). To find the root vectors it is convenient to visualize the elements of so(n, C) as × matrices of 2 × 2 blocks. Thus h is the subspace of all diagonal matrices of 2 × 2 skew-symmetric blocks. The subspaces of all matrices A with a block A jk of size two, 1 ≤ j < k ≤ , in the place (j, k) and −A t jk in the place (k, j) with zeros in all other places, are ad(h)-stable. Let if and only if for every A jk we have Up to a scalar, the nontrivial solutions of these linear equations are the following: Let j ∈ h * be defined by j (H) = x j for 1 ≤ j ≤ . Then for 1 ≤ j < k ≤ , the following matrices are root vectors of so(2 , C): Thus, if we choose the following set of positive roots then the Dynkin diagram of so(2 , C) is D : By looking at the 2 × 2 blocks A jk of the different roots, namely it is easy to obtain the following inverse relations From this it follows that Therefore Now using the expressions in (2.2) we get Thus Q 2 becomes

The operator Q 2 +1
Now we look at a root space decomposition of so(n) in terms of the basis elements I ki , 1 ≤ i < k ≤ n when n = 2 + 1.
The linear span h = I 21 , I 43 , . . . , I 2 ,2 −1 C is a Cartan subalgebra of so(n, C). To find the root vectors it is convenient to visualize the elements of so(n, C) as × matrices of 2 × 2 blocks occupying the left upper corner of the square matrices of size 2 + 1, with the last column (respectively row) made up of columns (respectively rows) of size two and a zero in the place (2 + 1, 2 + 1). The subspaces of all matrices A with a block A jk , 1 ≤ j < k ≤ , in the place (j, k), with the block −A t jk in the place (k, j) and with zeros in all other places, are ad(h)-stable. Also the subspaces of all matrices B with a column B j of size two, 1 ≤ j ≤ , in the place (j, + 1), with the row −B t j in the place ( + 1, j) and with zeros in all other places, are ad(h)-stable.
On the other hand [H, B] = λB if and only if Up to a scalar this linear equation has two linearly independent solutions: Let ∈ h * be defined by (H) = x j for 1 ≤ j ≤ . Then for 1 ≤ j < k ≤ and 1 ≤ r ≤ , the following matrices are root vectors of so(2 + 1, C): Thus, if we choose the following set of positive roots then the Dynkin diagram of so(2 + 1, C) is B : By looking at the 2 × 1 columns of the different roots, namely it is easy to obtain the following inverse relations . From this it follows that I 2 n,2r−1 + I 2 n,2r = 1 2 (X r X − r + X − r X r ) = −iI 2r,2r−1 + X − r X r , since [X r , X − r ] = −2iI 2r,2r−1 . Therefore we have that Then

Gel'fand-Tsetlin basis
For any n we identify the group SO(n) with a subgroup of SO(n + 1) in the following way: given k ∈ SO(n) we have k k 0 0 1 ∈ SO(n + 1).
Let T m be an irreducible unitary representation of SO(n) with highest weight m and let V m be the space of this representation. Highest weights m of these representations are given by the -tuples of integers m = m n = (m 1n , . . . , m n ) for which and m jn are all integers.
The restriction of the representation T m of the group SO(2 + 1) to the subgroup SO(2 ) decomposes into the direct sum of all representations T m , m = m n−1 = (m 1,n−1 , . . . , m ,n−1 ) for which the betweenness conditions are satisfied. For the restriction of the representations T m of SO(2 ) to the subgroup SO(2 − 1) the corresponding betweenness conditions are All multiplicities in the decompositions are equal to one (see [24, p. 362]).
If we continue this procedure of restriction of irreducible representations successively to the subgroups SO(n − 2) > SO(n − 3) > · · · > SO(2), then we finally get one dimensional representations of the group SO(2). If we take a unit vector in each one of these one dimensional representations we get an orthonormal basis of the representation space V m . Such a basis is called a Gel'fand-Tsetlin basis. The elements of a Gel'fand-Tsetlin basis {v(µ)} of the representation T m of SO(n) are labelled by the Gel'fand-Tsetlin patterns µ = (m n , m n−1 , . . . , m 3 , m 2 ), where the betweenness conditions are depicted in the following diagrams.
If n = 2 + 1 The chain of subgroups SO(n − 1) > SO(n − 2) > · · · > SO(2) defines the orthonormal basis {v(µ)} uniquely up to multiplication of the basis elements by complex numbers of absolute value one.

An explicit expression forπ(Q n )
Since Q n ∈ D(SO(n)) SO(n) , givenπ ∈ŜO(n) it follows thatπ(Q n ) commutes with π(k) for all k ∈ SO(n). Hence, by Schur's Lemmaπ(Q n ) = λI. From expressions (2.3) and (2.4) we can give the explicit value of λ in terms of the highest weight of π, by computingπ(Q n ) on a highest weight vector.
The dif ferential operator ∆ We shall look closely at the left invariant differential operator ∆ of SO(n + 1) defined by in order to study its eigenfunctions and eigenvalues. Later we will use all this to understand the irreducible spherical functions of fundamental K-types associated with the pair (G, K) = (SO(n + 1), SO(n)).
Proposition 3.1. Let G = SO(n + 1) and K = SO(n). Let us consider the following left invariant differential operator of G Then ∆ is also right invariant under K.
Proof . This is a direct consequence of the identity Q n+1 = Q n + ∆ and Proposition 2.6.
Let us define the one-parameter subgroup A of G as the set of all elements of the form where I n−1 denotes the identity matrix of size n − 1, and let M = SO(n − 1) be the centralizer of A in K. Now we want to get the expression of [∆Φ](a(s)) for any smooth function Φ on G with values in End(V π ) such that Φ(kgk ) = π(k)Φ(g)π(k ) for all g ∈ G and all k, k ∈ K.
We have Hence, we will use the decomposition G = KAK to write a(s) exp tI n+1,j = k(s, t)a(s, t)h(s, t), with k(s, t), h(s, t) ∈ K and a(s, t) ∈ A. Let us take on A \ {a(π)} the coordinate function x(a(s)) = s, with −π < s < π, and let .
From now on we will assume that −π < s, t, s + t < π.
If j = n we have a(s) exp tI n+1,n = a(s)a(t) = a(s + t). Thus we may take Since x(a(s + t)) = s + t, we obtain For 1 ≤ j ≤ n − 1, when s / ∈ Zπ, we may take Then, for 0 < s < π, we have x(a(s, t)) = arccos(cos s cos t) and ∂ ∂t x(a(s, t)) = cos s sin t √ 1 − cos 2 s cos 2 t .
From here we get We observe that k(s, 0) = h(s, 0) = e and that a(s, 0) = a(s). Then We also have We will need the following proposition, whose proof appears in the Appendix and its idea is taken from [5].
Moreover in each case, for 1 ≤ j ≤ n − 1 and 0 < s < π, we have Now we obtain the following corollaries.
Notice that the expression in Corollary 3.4 generalizes the very well known situation when the K-type is the trivial one, as we state in the following corollary (cf. [11, p. 403, equation (10)]). Corollary 3.5. Let Φ be an irreducible spherical function on G of the trivial K-type. Then, for F (s) = Φ(a(s)) we have for some λ ∈ C and 0 < s < π.
we obtain

In terms of this new variable Corollary 3.4 becomes
Corollary 3.6. Let Φ be an irreducible spherical function on G of type π ∈K. Then, if H(y) = Φ(a(s)) with y = (1 + cos s)/2, we have for some λ ∈ C and 0 < y < 1.
Remark 3.7. Let us notice that, for any y ∈ (0, 1), H(y) is a scalar linear transformation when restricted to any M -submodule, see Proposition 2.2. Therefore, if m is the number of Msubmodules contained in (V, π), we consider the vector valued function H : (0, 1) → C m whose entries are given by those scalar values that H(y) takes on every M -submodule.
If the End(V )-valued function H satisfies the differential equation given in Corollary 3.6, then the vector valued function H satisfies where E, N 1 and N 2 are matrices of size m × m.
Even more, since (I n,j ) 2 is scalar valued when restricted to any M -submodule. Hence, N 1 = N 2 and the equation above is equivalent to where N is a diagonal matrix of size m × m. To obtain an explicit expression of E for any K-type is a very serious matter; in the following sections we shall find explicitly the expressions of E and N , for certain K-types. Since V m is irreducible as M -module it follows that m 1n = · · · = m ,n = 0. The converse is also true, therefore V m is M -irreducible if and only if it is the trivial representation.
Let now consider the case K = SO(n), M = SO(n − 1), with n = 2 and let m n = (m 1n , . . . , m n ) be a K-type such that V m is irreducible as M -module. The highest weights m n−1 of the M -submodules of V m are those that satisfies the following intertwining relations The converse is also true, therefore V m is M -irreducible if and only if m n = dα or m n = dβ for any d ∈ N 0 .
If Φ is an irreducible spherical function on SO(n + 1) of type π, whose highest weight is m n = dα or m n = dβ, then from Corollary 3.6 we get that the associated function H satisfies To compute Let us first consider m n = dα. If v ∈ V mn is a highest weight vector, theṅ see (2.5) and (2.6). Therefore Let us now consider m n = dβ.
Hence, if Φ is an irreducible spherical function on SO(n + 1), n = 2 , of type m n = (d, . . . , d, ±d) ∈ C , then the associated scalar value function H = h satisfies Let us now compute the eigenvalue λ corresponding to the spherical function of type π ∈ SO(2 ), of highest weight m n = dα, associated with the irreducible representation τ ∈ SO(2 +1), of highest weight m n+1 = (w, d, . . . , d) ∈ C . If v ∈ V m n+1 is a highest weight vector, then from (2.6) we havė If v ∈ V mn is a highest weight vector, then from (2.5) we havė To solve (4.1) we write h = y α f . Then we get Thus the indicial equation is α(α − 1) + α − d(d + − 1) = 0 and α = d is one of its solutions. If we take h = y d f , then we obtain A fundamental system of solutions of this equation near y = 0 is given by the following functions Since h = y d f is bounded near y = 0 it follows that where the constant u is determined by the condition h(1) = 1.
Remark 4.1. Let h w = h w (y), w ≥ d, be the function h above. Then h w is a polynomial of degree w. Moreover observe that the function y d used to hypergeometrize (4.1) is precisely h d .
Let us now compute the eigenvalue λ corresponding to the spherical function of type m n = dβ associated with an irreducible representation τ of SO(n + 1) of highest weight If v ∈ V mn is a highest weight vector, thenπ(Q n )v = −d (d + − 1)v as above, because Q n v does not depend on the sign of the last coordinate of m n . Since ∆ = Q n+1 − Q n we also have Therefore we have proved the following result.
where the constant u is determined by the condition h w (1) = 1.

The operator ∆ for fundamental K-types
We are interested in finding a more explicit expression of the differential equation given in Corollary 3.6: for certain representations π ∈ŜO(n), including those that are fundamental. The obvious place to start to look for irreducible representations of SO(n) is among the exterior powers of the standard representation of SO(n). It is known that Λ p (C 2 ) are irreducible SO(2 )-modules for p = 1, . . . , −1, and that Λ (C 2 ) splits into the direct sum of two irreducible submodules. While in the odd case Λ p (C 2 +1 ) are irreducible SO(2 +1)-modules for p = 1, . . . , . See Theorems 19.2 and 19.14 in [3].
Moreover, Λ p (C n ) and Λ n−p (C n ) are isomorphic SO(n)-modules. In fact, if {e 1 , . . . , e n } is the canonical basis of C n , then the linear map ξ : where u 1 < · · · < u p and v 1 < · · · < v n−p are complementary ordered set of indices, is an SO(n)-isomorphism.
All these statements can be established directly upon observing that the elements I ki = E ki − E ik with 1 ≤ i < k ≤ n form a basis of the Lie algebra so(n), and that I ki e k = e i , I ki e i = −e k and I ki e j = 0 if j = k, i.

The even case: K = SO(2 )
First we will study the case n = 2 , with > 2. The fundamental weights of so(2 , C) are Here we will consider the fundamental K-modules We will show that the highest weight of Λ p (C n ) is 1 + · · · + p for 1 ≤ p ≤ − 1. Observe that λ −1 and λ are not analytically integral and therefore they will not be considered, although we will also consider the K-module with highest weight λ −1 + λ = 1 + · · · + −1 . Notice that we have already considered the cases 2λ −1 and 2λ in Section 4, which are M -irreducible. We will also show that the fundamental K-modules are direct sum of two irreducible M -submodules.
In order to obtain the explicit expression of E in (3.2) for a given irreducible representation π of K = SO(n), of highest weight ε 1 + · · · + ε p , we are interested to compute (I nj )P 0π (I nj )(e 1 ∧ · · · ∧ e p−1 ∧ e n ).
Therefore, we obtain a more explicit version of Corollary 3.6 using (3.2) and Remark 3.8.

The odd case: K = SO(2 + 1)
We now study the case n = 2 + 1, with ≥ 1. The fundamental weights of so(2 + 1, C) are Here we will consider the fundamental K-modules We will show that the highest weight of Λ p (C n ) is 1 + · · · + p for 1 ≤ p ≤ . Also we will establish that Λ p (C n ) splits into the direct sum of two M -submodules for 1 ≤ p ≤ − 1, while Λ (C n ) splits into the sum of three M -submodules; for this reason it will be treated separately in Section 8. Observe that λ is not analytically integral and therefore it will not be considered, although we will consider the K-module with highest weight 2λ .
Thus the vector valued function F 1 (s) given by the irreducible spherical function Φ 1 (a(s)) is F 1 (s) = 1 cos s .
Definition 6.1. We shall consider the 2 × 2 matrix-valued function Ψ = Ψ(y), for 0 < y < 1, whose columns are given by the functions H 0 (y) = F 0 (s) and H 1 (y) = F 1 (s), with cos s = 2y −1: Since the functions H 0 (y) and H 1 (y) are associated with irreducible spherical functions, they satisfy the differential equation given in Corollary 5.1; moreover, the respective eigenvalues are λ = −p and λ = p − n. Therefore, we have Furthermore, it is easy to check that the function Ψ(y) also satisfy the equation above even when n is odd. Theorem 6.2. The function Ψ can be used to obtain a hypergeometric differential equation from the one given in Corollaries 5.1 and 5.2. Precisely, if H is a vector-valued solution of the differential equation in Corollaries 5.1 or 5.2, with eigenvalue λ, then P = Ψ −1 H is a solution of DP = λP , where D is the hypergeometric differential operator given by Proof . By hypothesis we have that Then, writing H = ΨP , we have Therefore This completes the proof of the theorem.

∆-eigenvalues of spherical functions
As we said, when n = 2 the irreducible spherical functions of the pair (SO(n+1), SO(n)), of type m n = (1, . . . , 1, 0 . . . , 0) ∈ C with p ones, 1 ≤ p ≤ − 1 are those associated with the irreducible representations τ of G of highest weights of the form m n+1 = (w + 1, 1, . . . , 1, δ, 0, . . . , 0) ∈ C with p − 1 ones, such that the following pattern holds Let Φ w,δ be the corresponding spherical function. Then ∆Φ w,δ = λΦ w,δ , where the eigenvalue λ = λ n (w, δ) can be computed from the expression ∆ = Q n+1 − Q n . If v ∈ V m n+1 is a highest weight vector from (2.6) we havė If v ∈ V m 2 is a highest weight vector, then from (2.5) we havė Since ∆ = Q n+1 − Q n it follows that Analogously, we obtain that the eigenvalues of the spherical functions Φ w,δ of the pair (SO(2 + 2), SO(2 + 1)) are of the form here δ is 0 or 1 when we are in the cases 1 ≤ p < but δ could also be −1 in the particular case p = .
Therefore, we have that the eigenvalues of the spherical functions Φ w,δ of the pair (SO(n + 1), SO(n)) are of the form (6.2)

Polynomial eigenfunctions of the hypergeometric operator D
Let D be the differential operator on the real line introduced in Theorem 6.2: where n is of the form 2 or 2 + 1 for ∈ N and 1 ≤ p < . We will study the C 2 -vector valued polynomial eigenfunctions of D.
The equation DP = λP is an instance of a matrix hypergeometric differential equation studied in [22]. Since the eigenvalues of C, n/2 and n/2 + 2, are not in −N 0 the function P is determined by P 0 = P (0). For |y| < 1 it is given by where the symbol [C; U ; V + λ] j is inductively defined by for all j ≥ 0. Therefore, we have that there exists a polynomial solution if and only if the coefficient [C; U ; V + λ] j+1 is a singular matrix for some j ∈ Z. Since the matrix C + j is invertible for all j ∈ N 0 , we have that there is a polynomial solution of degree j for DP = λP if and only if there exists P 0 ∈ C 2 such that [C; U ; V + λ] j P 0 = 0 and (j(U + j − 1) + V + λ)[C; U ; V + λ] j P 0 = 0. Now we easily observe that the only possible values for λ such that j(U + j − 1) + V + λ has non trivial kernel are those given in (6.2). Then, if λ = −w(w + n + 1) − p, it is easy to check that the first and only j for which j(U + j − 1) + V + λ is singular is j = w, and its kernel (of dimension 1) is the subspace generated by ( 1 0 ). Analogously, if λ = −w(w + n + 1) − n + p, it is easy to check that the first and only j for which j(U + j − 1) + V + λ is singular is j = w, and its kernel (of dimension 1) is the subspace generated by ( 0 1 ) respectively. Therefore we have the following result. Theorem 6.3. For a given ∈ N take n = 2 or 2 + 1 and 1 ≤ p ≤ − 1, then the polynomial eigenfunctions of DP = y(1 − y)P + (C − yU )P − V P, with C = (n/2 + 1) 1 1 (n/2 + 1) , U = (n + 2)I, V = p 0 0 n − p have eigenvalues −w(w + n + 1) − p or −w(w + n + 1) − n + p, with w ∈ N 0 ; in both cases the degree of the polynomial is w with leading coefficient a multiple of ( 1 0 ) or ( 0 1 ), respectively.

The inner product
Given a finite dimensional irreducible representation π of K in the vector space V π let (C(G) ⊗ End(V π )) K×K be the space of all continuous functions Φ : G −→ End(V π ) such that Φ(k 1 gk 2 ) = π(k 1 )Φ(g)π(k 2 ) for all g ∈ G, k 1 , k 2 ∈ K. Let us equip V π with an inner product such that π(k) becomes unitary for all k ∈ K. Then we introduce an inner product in the vector space (C(G) ⊗ End(V π )) K×K by defining where dg denote the Haar measure on G normalized by G dg = 1, and where Φ 2 (g) * denotes the adjoint of Φ 2 (g) with respect to the inner product in V π . By using Schur's orthogonality relations for the unitary irreducible representations of G, it follows that if Φ 1 and Φ 2 are non equivalent irreducible spherical functions, then they are orthogonal with respect to the inner product ·, · , i.e.
Recall that, given an irreducible spherical function Φ of type π of the pair (G, K), the function Φ(a(s)) is scalar valued when restricted to any SO(n − 1)-module (see (3.1) for a(s)). We shall denote by m the number of SO(n − 1)-submodules of π, and by d 1 , d 2 , . . . , d m the respective dimensions of each one of those submodules.
with ω * = π if n is even and ω * = 2 if n is odd. where I n−1 denotes the identity matrix of size n − 1.

Proof . Let
Now [12,Theorem 5.10,p. 190] establishes that for every f ∈ C(G/K) and a suitable constant c * where dg K and dk M are respectively the invariant measures on G/K and K/M normalized by G/K dg K = K/M dk M = 1 and the function δ * : A −→ R is defined by δ * (a(s)) = ν∈Σ + | sin isν(I n+1,n )|, with Σ + the set of those positive roots whose restrictions to a, the Lie algebra of A, are not zero. In our case we have δ * (a(s)) = | sin n−1 s|.
To find the value of c * we consider the function f ≡ 1, having then 1 = 2c * π 0 sin n−1 sds.
If we put y = 1 2 (cos s + 1) for 0 < s < π we have and the proposition follows.
Proof . If we apply a left invariant vector field X ∈ g, to the function on G given by g → tr(Φ 1 (g)Φ 2 (g) * ), and then we integrate over G we obtain Therefore XΦ 1 , Φ 2 = − Φ 1 , XΦ 2 . Now let τ : g C −→ g C be the conjugation of g C with respect to the real linear form g. Then −τ extends to a unique antilinear involutive * operator on D(G) such that (D 1 D 2 ) * = D * 2 D * 1 for all D 1 , D 2 ∈ D(G). This follows easily from the fact that the universal enveloping algebra over C of g is canonically isomorphic to D(G). Then it follows that DΦ 1 , Finally, it is easy to verify that ∆ * = ∆.

Spherical functions as polynomial solutions of DP = λP
Let us consider D, the differential operator on (0, 1) introduced in Corollaries 5.1 and 5.2: Recall that the operator D that appears in (6.3) extends the differential operator D = Ψ DΨ −1 to the whole real line, where is the matrix function given in (6.1) and used in Theorem 6.2. We want to focus our attention on the following vector spaces of C 2 -valued analytic functions on (0, 1): From Theorem 6.2 we know that the correspondence P → ΨP is an injective linear map from W λ into S λ . Now we want to prove that this map is bijective.
Proof . A vector valued function P ∈ W λ is an eigenfunction of the hypergeometric operator D. Since it is analytic at y = 1 it is determined by P (1), therefore dim(W λ ) = 2.
On the other hand, if H ∈ S λ then there is a function F (s) analytic at s = 0, such that it extends the function H( cos s+1 2 ) defined on (0, π). Then, F satisfies the following differential equation Let a j ∈ C 2 and α j , β j , γ j ∈ C, for j ≥ 0, be the Taylor coefficients of F , sin, sin 2 and cos at s = 0: Therefore, from (7.2) we have Hence, since β 2 = α 1 = γ 0 = 1, we have that is a linear combination with matrix coefficients of {a 0 , a 1 , . . . , a j−1 }; it is clear that for j = 1 and j > 2 the matrix above is non singular, therefore {a 0 , a 2 } determine completely the sequence {a j } j≥0 . Also it is clear that when j = 0 or 2, that matrix has nullity 1. Therefore we can conclude that dim(S λ ) = 2. The theorem follows.
Proof . We know that the function H is analytic in (0, 1), and from Corollary 5.1 we know that it is an eigenfunction of the operator D (see (7.1)). Also we know that the function H( 1+cos s 2 ) is analytic at s = 0, since Φ(a(s)) it is. Therefore from Theorem 7.3 the function P = Ψ −1 H is an analytic eigenfunction of D on the closed interval [0, 1].
If we introduce the following matrix-weight function V = V (y) supported on the interval [0, 1] with ω * = π if n is even and ω * = 2 if n is odd, then from Proposition 7.1 we have It follows from Propositions 7.1 and 7.2 that D is a symmetric operator with respect to the inner product defined among continuous vector-valued functions on [0, 1] by Then, since D = Ψ −1 DΨ, we have that D is a symmetric operator with respect to the inner product defined among continuous vector-valued functions on [0, 1] by Actually, we have that (W, D) is a classical pair in the sense of [7], see also [2]. As the weight W has finite moments there exists a sequence {Q r } r≥0 of 2 × 2 matrix-valued orthonormal polynomials, such that DQ r = Q r Λ r where Λ r is a real diagonal matrix (for precise definitions and general facts on matrix-valued orthogonal polynomials see [5] and [2]). Let {e 1 , e 2 } be the canonical basis of C 2 . Then Therefore, for r ≥ 0, j = 1, 2, {Q r e j } is a family of C 2 -valued orthonormal polynomials such that where Λ r = diag(λ 1 r , λ 2 r ). Now we write our function P = Ψ −1 H as P = r,j a r,j Q r e j , where a r,j = P, Q r e j W . Since P is analytic on [0, 1] the sum converges not only in the L 2 -norm but also in the topology based on uniform convergence of sequences of functions and their successive derivatives. Therefore, Then a r,j = 0 if λ j r = λ. Since dim W λ = 2 it follows that P is a polynomial.
Remark 7.5. It is easy to see from (5.1) and (5.2) that the dimensions of the M -submodules of the fundamental representation of K with highest weight of the form (1, . . . , 1, 0 . . . , 0), with p ones, are given by therefore the weight W is given by with ω * = π if n is even and ω * = 2 if n is odd. Then, W is a scalar multiple of p(2y − 1) 2 + n − p n(2y − 1) n(2y − 1) (n − p)(2y − 1) 2 + p .
Even more, since 0 < p < and n = 2 , 2 + 1 it follows that p = n − p. Then it can be proved that the weight W does not reduce to a smaller size, i.e., there is not any invertible matrix M such that M * W (y)M is diagonal for all y ∈ [0, 1].
Proof . It only remains to prove that P w,δ (0) can be computed.
Let us consider the case δ = 0. We know from (6.2) and Theorem 6.3 that there is some c ∈ C such that [C; U ; V + λ] w P w,0 (0) = c 1 0 .
Since [C; U ; V + λ] w is invertible, this c is univocally determined by the condition Φ(e) = I, which implies Similarly, we can prove the same for P w,1 (0).
Therefore, our construction encodes all equivalent classes of irreducible spherical functions of a fundamental K-type of highest weight λ p , 0 < p < , in the orthogonal set of C 2 -valued polynomials {P w,0 , P w,1 }. The degree of P w,0 and P w,1 is w, and the leading coefficient is a multiple of ( 1 0 ) or ( 0 1 ), respectively.
Theorem 8.1. The matrix-valued polynomial functions P w , w ≥ 0, form a sequence of orthogonal polynomials with respect to W , which are eigenfunctions of the symmetric differential operator D in (6.3). Moreover, Proof . From Theorem 6.2 we have that the δ-th column of P w is an eigenfunction of the operator D with eigenvalue λ(w, δ), see (6.2) and (6.3). Therefore we have From Theorem 7.6 we know that each column of P w is a polynomial function of degree w and, even more, that P w is a polynomial whose leading coefficient is a nonsingular diagonal matrix.
Given w and w , non negative integers, by using Remark 7.7 we have which proves the orthogonality. Even more, it also shows us that P w , P w W is a diagonal matrix. Also, making a few simple computations we have that for every w, w ∈ N 0 , since Λ w is real and diagonal. This concludes the proof of the theorem. 9 The SO(2 + 1)-type with highest weight 2λ In this section K = SO(2 + 1). We will focus on the particular case when the K-type is given by an irreducible representation π with highest weight 2λ = (1, 1, . . . , 1). We will first see that such K-module is the direct sum of three M -submodules, and we will find similar results to those obtained for the fundamental K-types λ 1 , . . . , λ −1 that are direct sum of two M -submodules. Let us consider the irreducible K-module Λ (V ), with V = C n , n = 2 + 1. The vector v = (e 1 − ie 2 ) ∧ (e 3 − ie 4 ) ∧ · · · ∧ (e 2 −1 − ie 2 ) is the unique, up to a scalar, dominant vector and its weight is 2λ = (1, 1, . . . , 1).
Analogously we obtain Therefore, we obtain a more explicit version of Corollary 3.6 using (3.2) and Remark 3.8. Confront Corollary 5.2.
We obtain a similar result to Theorem 6.3, with an analogous proof: have eigenvalues −w(w + n + 1) − or −w(w + n + 1) − − 1, with w ∈ N 0 . In both cases the degree of the polynomial is w and the leading coefficient can be any multiple of Let us consider D, the differential operator on (0, 1) introduced in Corollary 9.1: Recall that the operator D that appears in Theorem 9.3 extends the differential operator D = Ψ DΨ −1 to the whole real line.
Then, we have an analogous result to Theorem 7.3, whose proof is quite similar and therefore we will omit it.
Theorem 9.4. The linear map P → ΨP is an isomorphism from W λ onto S λ . Now, we can easily make a proof similar to that one of Theorem 7.4 in order to obtain next theorem.
Proof . It only remains to prove that P w,δ (0) can be computed.
Let us consider the case δ = 0. We know from (6.2) and Theorem 9.3 that there is some c ∈ C such that [C; U ; V + λ] w P w,0 (0) = c since [C; U ; V + λ] w is invertible, this condition tells us that P w,δ (0) belongs to a plane which contains the origin and does not depend on δ.
Besides, the condition Φ w,δ (e) = I, for δ = ±1, tells us From [24, p. 364, equation (8)] we can easily compute d ds Φ w,δ (a(s)) at s = 0, which is obtained by looking at the action ofτ (I n+1,n ) and considering the corresponding projection, see (2.1); having then This last condition establishes that P w,1 (0) and P w,−1 (0) are in two different and parallel planes, and the line mentioned above does not belong to any of them since each plane has to intersect it. Therefore the values of P w,1 (0) and P w,−1 (0) are univocally determined.

Matrix-valued orthogonal polynomials of size 3
In this subsection, given n of the form 2 + 1 with ∈ N, we shall construct a sequence of matrix-valued polynomials {P w } w≥0 directly related to irreducible spherical functions of type π ∈ŜO(n) of highest weight m π = (1, . . . , 1) ∈ C .
We also get Now we will first consider the case A(s, t) = k(s, t). A direct computation yields to in particular ∂k ∂t t=0 = 1 sin s I n,j . Differentiating once more with respect to t and evaluating at t = 0 we obtain ∂ 2 k ∂t 2 t=0 = − 1 sin 2 s (E jj + E n,n ). Then we get in particular ∂h ∂t t=0 = − cos s sin s I n,j . Differentiating once more with respect to t and evaluating at t = 0 we obtain ∂ 2 h ∂t 2 t=0 = − cos 2 s sin 2 s (E jj + E n,n ). Then we get = − cos 2 s sin 2 s (E jj + E n,n ) − cos 2 s sin 2 s I 2 n,j = 0.