I. Pacharoni
This paper is partially supported by CONICET, FONCyT, SecytUNC and the ICTP.
Abstract. In the scalar case, it is well known that the zonal spherical functions of any compact Riemannian symmetric space of rank one can be expressed in terms of the Jacobi polynomials. The main purpose of this paper is to revisit the matrix valued spherical functions associated to the complex projective plane to exhibit the interplay among these functions, the matrix hypergeometric functions and the matrix orthogonal polynomials. We also obtain very explicit expressions for the entries of the spherical functions in the case of 2 × 2 matrices and exhibit a natural sequence of matrix orthogonal polynomials, beyond the group parameters.
The well known Legendre polynomials are a special case of spherical harmonics: the homogeneous harmonic polynomials of , considered as functions on the unit sphere . Let be ordinary polar coordinates in : , and In terms of these coordinates the Riemannian structure of is given by the symmetric differential form and the Laplace operator is
Legendre and Laplace found that the Legendre polynomials satisfy the following addition formula

where the 's are the associated Legendre polynomials.
By integrating (1) we get

Moreover the Legendre polynomials can be determined as solutions to (2). This integral equation can now be expressed in terms of the function on defined by . In fact (2) is equivalent to

where denotes the compact subgroup of of all elements which fix the north pole , and denotes the normalized Haar measure of .
In fact, let denote the subgroup of all elements of which fix the point . Then . Thus to prove (3) it is enough to consider rotations and around the axis through the angles and , respectively. Then if denotes the rotation of angle around the axis we have
]]> Thus andThe functional equation (3) has been generalized to many different settings. One is the following. Let be a locally compact unimodular group and let be a compact subgroup. A nontrivial complex valued continuous function on is a zonal spherical function if (3) holds for all . Note that then for all and all , and that where is the identity element of .
The example above arises when and . The other compact connected rank one symmetric spaces have zonal spherical functions which are orthogonal polynomials in an appropriate variable. These polynomials are special cases of Jacobi polynomials and they can be given explicitly as hypergeometric functions.
The complex projective plane is another rank one symmetric space. In this case the zonal spherical functions are .
A very fruitful generalization of the functional equation (3) is the following (see [T1] and [GV]. Let be a locally compact unimodular group and let be a compact subgroup of . Let denote the set of all equivalence classes of complex finite dimensional irreducible representations of ; for each , let and denote, respectively, the character and the dimension of any representation in the class , and set . We shall denote by a finite dimensional complex vector space and by the space of all linear transformations of into .
A spherical function on of type is a continuous function such that , (= identity transformation) and
When is the class of the trivial representation of and , the corresponding spherical functions are precisely the zonal spherical functions. From the definition it follows that is a representation of , equivalent to the direct sum of representations in the class , and that for all and all . The number is the height of . The height and the type are uniquely determined by the spherical function.
2. Matrix valued spherical functions associated to
]]> In [GPT1] the authors consider the problem of determining all irreducible spherical functions associated to the complex projective plane . This space can be realized as the homogeneous space , and . In this case all irreducible spherical functions are of height one. Let be any irreducible representation of in the class . Then an irreducible spherical function can be characterized as a function such that is analytic,
 , for all , , and ,
 , for all and .
Here denotes the algebra of all left and right invariant differential operators on . In our case it is known that the algebra is a polynomial algebra in two algebraically independent generators and , explicitly given in [GPT1].
The set can be identified with in the following way: If then
For any we denote by the left upper block of , and we consider the open dense subset . Then is left and right invariant under . For any we introduce the following function defined on :

where is supposed to be a spherical function of type . Then satisfies
 ,
 , for all ,
 , for all .
The canonical projection defined by where maps the open dense subset onto the affine space of those points in whose last homogeneous coordinate is not zero. Then property ii) says that may be considered as a function on , and moreover from iii) it follows that is determined by its restriction to the cross section of the orbits in , which are the spheres of radius centered at the origin. That is is determined by the function on the interval . Let be the closed subgroup of of all diagonal matrices of the form , . Then fixes all points . Therefore iii) also implies that for all . Since any as an module is multiplicity free, it follows that there exists a basis of such that is simultaneously represented by a diagonal matrix for all . Thus, if , we can identify with a vector
Making the change of variables these operators become
]]> If we denote by the matrix with entry equal to 1 and 0 elsewhere, then the coefficient matrices areThe following result, which characterizes the spherical functions associated to the complex projective plane is taken from Theorem 3.8 of [RT], see also [GPT1].
Theorem 2.1. The irreducible spherical functions of of type , correspond precisely to the simultaneous valued polynomial eigenfunctions of the differential operators and , introduced in (5) and (6), such that for all with polynomial and .
We also obtain, from [GPT1] or [PT1], that there is a bijective correspondence between the equivalence classes of all irreducible spherical functions of type and the set of pairs of integers

Under this correspondence the function associated to the spherical function satisfies and where

2.1. Hypergeometric operators. A key result to characterize the spherical functions of of any type is the fact that the differential operator is conjugated, by a matrix polynomial function , to a hypergeometric operator . From [RT], (or [PT2], for a more general situation) we obtain that the function , where

This fact allows us to describe the eigenfunctions of the differential operator in term of the matrix valued hypergeometric functions, introduced in [T2]: Let be a dimensional complex vector space, and let and . The hypergeometric equation is
where stands for a function of with values in .
More generally we can consider the equation

In the scalar case the differential operator (11) is always of the form (10). Nevertheless in a noncommutative setting the equations and may have no solutions , .
If the eigenvalues of are not in we define the function

where the symbol is defined inductively by and

2.2. Spherical functions as matrix hypergeometric functions. The irreducible spherical functions of of type are in a one to one correspondence with certain simultaneous polynomial eigenfunctions of the differential operators and (see Theorem 2.1).
A delicate fact establish in [RT] is that the functions are also polynomials functions which are eigenfunctions of the differential operators and .
In the variable , these operators have the form


where the coefficient matrices are
 (15) 
To describe all simultaneous polynomial eigenfunctions of the differential operators and we start by considering the eigenfunctions of of eigenvalues , with (see (8)). We let
]]>Remark. It is not difficult to prove that that if and only if , for some .
Therefore if then it is of the form
Since the initial value determines , we have that the linear map defined by is a surjective isomorphism. Since and commute, the differential operators and also commute. Moreover, since has polynomial coefficients whose degrees are less or equal to the corresponding orders of differentiation, restricts to a linear operator of . Thus we have the following commutative diagram

where is the matrix given by

The eigenvalues of are given by (see Theorem 10.3 in [GPT1])

Then the characterization of the irreducible spherical functions is summarize in the following theorem, taking from [RT].
Theorem 2.2. The function associated to a spherical function of type and parameters is of the form , where

and is the unique eigenvector of normalized by . The expressions of the matrices are given in (15) and the eigenvalues and are given in (18).
2.3. Orthogonality. Let be the space of all continuous functions such that for all , . Let us equip with an inner product such that becomes unitary for all . We have the following inner product in :

where denotes the adjoint of with respect to the inner product in . Then we have the following inner product on the corresponding functions 's associated to the spherical functions

Since the Casimir operator is symmetric with respect to the inner product for matrix valued functions on given in (20), it follows that the differential operators and are symmetric with respect to the weight function , that is they satisfy
Now it is easy to verify that the differential operators and are symmetric with respect to the weight function

To illustrate the above result we will display the cases (the scalar case) and , where the size of our matrices will be .
3.1. The case .. In this case the functions are scalar functions. If the parameter is 0 then we have the zonal spherical functions.
The operator is proportional to , () and

are linearly independent solutions. By Theorem 2.1 we have that should be a polynomial function such that . Moreover if the function have to satisfies with a polynomial function. Therefore we get: For and
By using the Pfaff's identity we get
Proposition 3.1. The spherical functions associated to the complex projective plane of type are
3.2. The case . In this case the operators and are
In [GPT1], Section 11.1 we exhibit the complete list of spherical function of type . We have two families of such functions, corresponding with the choice of the parameter or . For the parameter is in the range , and if .
First family. For we have , . The (vector valued) function is given by, up to the normalizing constant such that .
with .
Second family. For we have and . The functions is
]]>with .
By taking the Taylor expansion at these functions takes the following unified expression. We recall that corresponds to the identity of the group .
Theorem 3.2. The complete list of spherical functions associated to of type are given by
 For we have , and
The parameter is an integer that satisfies and .
 For , we have , and
where . ]]> The parameter is an integer that satisfies and .
In this case the function , where and is

In the variable , the conjugated operators and are

The matrix (see (17)), is
The explicit expression of the entries of these functions 's are given in the following theorem.
Theorem 3.3. The functions associated to the spherical functions of the pair of type are given by
 For we have , and
The parameter is an integer that satisfies and .
 For , we have , and
where .
The parameter is an integer that satisfies and .

From

(26) we only have to prove the expression for the second entry of the function . For the first family, from Theorem 3.2 we get
For the second family we obtain, with
This concludes the proof of the theorem. □
3.3. Matrix valued orthogonal polynomials coming from spherical functions. In the scalar case, it is well known that the zonal spherical functions of the sphere are given, in spherical coordinates, in terms of Gegenbauer polynomials. Therefore, it is not surprising that in the matrix valued setting the same phenomenon occurs: the matrix spherical functions are closely related to matrix orthogonal polynomials.
For a given nonnegative integers and we define the matrix polynomial as the matrix function whose row is the polynomial , associated to the spherical functions of type , given in the previous section. In other words
The columns of are eigenfunctions of the differential operators and given in (25), thus we have that satisfies
3.4. Extension of the group parameters. These results have a direct and fruitful generalization by replacing the complex projective plane by the dimensional complex projective space , which can be realized as the homogeneous space , where and .
In this case, the finite dimensional irreducible representations of , are parameterized by the tuples of integers such that . By considering the irreducible spherical functions of type and proceeding as we explained for the complex projective plane, one obtains a situation that generalizes the one of . Then by extending the parameters , we have the following results.
Theorem 3.4. Let and let us define
Then is a sequence of orthogonal polynomials with respect the weight matrix ]]>
Let be the following second order differential operator
In [PT1] for or in general in [P08], we obtain a multiplication formula for spherical functions by tensoring certain irreducible representations of and decomposing them into irreducible representations. From this formula we derive a three term recursion relation for the "packages" of spherical functions. Restricting this to the variable (the variable that parameterizes a section of the orbits in ), we obtain a three term recursion relation for the packages of functions associated to the spherical functions. In this case we obtain the following
Theorem 3.5. The sequence satisfies the following three term recursion relation

with
Remark 3.6. The three term recursion relation can be seen as a difference operator in the variable , given by a semiinfinite matrix . The vector matrix is an eigenfunction of because it satisfies .
We observe that the semiinfinte matrix have the interesting property that the sum of all the matrix elements in any row is equal to one. Moreover all the entries of are nonnegative real numbers. This have important applications in the modeling of some stochastic phenomena.
[GV] Gangolli R. and Varadarajan V. S. Harmonic analysis of spherical functions on real reductive groups, SpringerVerlag, Berlin, New York, 1988. Series title: Ergebnisse der Mathematik und ihrer Grenzgebiete, 101. [ Links ]
[GPT1] F. A. Grünbaum, I. Pacharoni and J. Tirao, Matrix valued spherical functions associated to the complex projective plane, J. Funct. Anal. 188 (2002), 350441. [ Links ]
[P08] Pacharoni I.Three term recursion relation for spherical functions. Preprint, 2008. [ Links ]
[PT1] Pacharoni I. and Tirao J. A. Three term recursion relation for spherical functions associated to the complex projective plane. Math Phys. Anal. Geom. 7 (2004), 193221. [ Links ]
[PT2] Pacharoni I. and Tirao J. A. Matrix valued orthogonal polynomials arising from the complex projective space. Constr. Approxim. 25, No. 2 (2007) 177192. [ Links ]
[PR] Pacharoni, I. Román, P. A sequence of matrix valued orthogonal polynomials associated to spherical functions Constr. Approxim. 28, No. 2 (2008) 127147. [ Links ]
[RT] P. Román and J. A. Tirao. Spherical functions, the complex hyperbolic plane and the hypergeometric operator. Intern. J. Math. 17, No. 10, (2006), 11511173. [ Links ]
[T1] J. Tirao. Spherical Functions. Rev. de la Unión Matem. Argentina, 28 (1977), 7598. [ Links ]
[T2] J. Tirao, The matrixvalued hypergeometric equation. Proc. Natl. Acad. Sci. U.S.A., 100 No. 14 (2003), 81388141. [ Links ]
I. Pacharoni
CIEMFaMAF,
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
pacharon@mate.uncor.edu
Recibido: 18 de mayo de 2008 ]]> Aceptado: 11 de agosto de 2008
]]>