Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
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2014
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| Цитувати: | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859846960182198272 |
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| author | Maarten van Pruijssen Román, P. |
| author_facet | Maarten van Pruijssen Román, P. |
| citation_txt | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 113, 28 pages
Matrix Valued Classical Pairs
Related to Compact Gelfand Pairs of Rank One
Maarten VAN PRUIJSSEN † and Pablo ROMÁN ‡
† Universität Paderborn, Institut für Mathematik,
Warburger Str. 100, 33098 Paderborn, Germany
E-mail: vanpruijssen@math.upb.de
URL: http://www.mvanpruijssen.nl
‡ CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria,
Córdoba, Argentina
E-mail: roman@famaf.unc.edu.ar
URL: http://www.famaf.unc.edu.ar/~roman
Received April 30, 2014, in final form December 12, 2014; Published online December 20, 2014
http://dx.doi.org/10.3842/SIGMA.2014.113
Abstract. We present a method to obtain infinitely many examples of pairs (W,D) consis-
ting of a matrix weight W in one variable and a symmetric second-order differential opera-
tor D. The method is based on a uniform construction of matrix valued polynomials starting
from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation.
The heart of the construction is the existence of a suitable base change Ψ0. We analyze
the base change and derive several properties. The most important one is that Ψ0 satisfies
a first-order differential equation which enables us to compute the radial part of the Casimir
operator of the group G as soon as we have an explicit expression for Ψ0. The weight W
is also determined by Ψ0. We provide an algorithm to calculate Ψ0 explicitly. For the
pair (USp(2n),USp(2n− 2)×USp(2)) we have implemented the algorithm in GAP so that
individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs
(G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit
expressions for most of these cases.
Key words: matrix valued classical pairs; multiplicity free branching
2010 Mathematics Subject Classification: 22E46; 33C47
1 Introduction
Matrix valued orthogonal polynomials (MVOPs) in one variable are generalizations of scalar
valued orthogonal polynomials and they already show up in the 1940s [24, 25]. Since then,
MVOPs have been studied in their own right and they have been applied and studied in different
fields such as scattering theory, spectral analysis and representation theory [2, 6, 13, 14, 15]. In
this paper we are concerned with obtaining families of MVOPs whose members are simultaneous
eigenfunctions of a symmetric second-order differential operator.
Fix N ≥ 1 and an interval I ⊂ R. We write M = End(CN ) and the Hermitian adjoint of
A ∈M is denoted by A∗. The space M[x] is an M-bimodule. A matrix weight is a function W :
I →M with finite moments and W (x) = W (x)∗ and W (x) > 0 almost everywhere. The pairing
M[x]×M[x]→M : (P,Q) 7→ 〈P,Q〉W :=
∫
I
P (x)∗W (x)Q(x)dx
is an M-valued inner product that makes M[x] into a right pre-Hilbert M-module. A family
of MVOPs (with respect to this pairing) is a family (Pn : n ∈ N) with Pn ∈ M[x] satis-
fying (1) deg(Pn) = n, (2) the leading coefficient of Pn is invertible, for all n ∈ N and
mailto:vanpruijssen@math.upb.de
http://www.mvanpruijssen.nl
mailto:roman@famaf.unc.edu.ar
http://www.famaf.unc.edu.ar/~roman
http://dx.doi.org/10.3842/SIGMA.2014.113
2 M. van Pruijssen and P. Román
(3) 〈Pn, Pm〉W = Mnδm,n for all n,m ∈ N. Existence of such a family is guaranteed by ap-
plication of the Gram–Schmidt process on (1, x, x2, . . .). Moreover, up to right multiplication
by GL(CN ), a family of MVOPs is uniquely determined by the weight W .
In [9] the question was raised if there exists a matrix weight together with a differential
operator of degree two that has the corresponding family of orthogonal polynomials as a family
of simultaneous eigenfunctions. If N = 1 then the answer is well known, we get the classical
orthogonal polynomials [3]. In fact, the algebra of differential operators that have the classical
polynomials as simultaneous eigenfunctions is a polynomial algebra generated by a second-order
differential operator, see e.g. [28].
In general, the algebra of differential operators that act on the M-valued polynomials is
End(M)[x, ∂x]. We identify End(M) = M ⊗ M such that a simple tensor A ⊗ B acts on an
element C ∈ M via (A ⊗ B)C = ACB∗. A polynomial P ∈ M[x] is an eigenfunction of
a differential operator D ∈ End(M)[x, ∂x] if there exists an element Λ ∈M such that DP = PΛ.
This is justified by the fact that we consider M[x] as right pre-Hilbert M-module.
A pair (W,D) consisting of a matrix weight and an element D ∈ End(M)[x, ∂x] of order two
that is symmetric with respect to 〈·, ·〉W is called a matrix valued classical pair (MVCP). Any
family of MVOPs is automatically a family of simultaneous eigenfunctions.
Consider the two subalgebras (M⊗ C)[x, ∂x] and (C⊗M)[x, ∂x] of End(M)[x, ∂x]. If (W,D)
is a classical pair with D ∈ (C ⊗M)[x, ∂x] then the weight can be diagonalized by a constant
matrix [9]. After publication of [9] examples of MVCPs (W,D) with D ∈ (M ⊗ C)[x, ∂x] came
about, arising from analysis on compact homogeneous spaces in a series of papers starting in [16]
and ending in [30].
A uniform construction of MVCPs arising from the representation theory of compact Lie
groups is presented in [19, 33] and was inspired by [21, 22, 23, 38]. The input datum is a compact
Gelfand pair (G,K) of rank one and a certain face F of the Weyl chamber of K. For each µ ∈ F ,
the output is an orthogonal family of Mµ = End(CNµ) valued functions (Ψµ
d : d ∈ N) on the
circle S1 and a commutative algebra of differential operators D(µ) for which the functions Ψµ
d
are simultaneous eigenfunctions. Here, Nµ is a natural number that depends on the weight
µ ∈ F . Moreover the functions Ψµ
d are determined by this property and a normalization. We
call Ψµ
d the full spherical function of type µ and degree d. The datum (G,K,F ) for which this
construction applies is called a multiplicity free system and they are classified in [19].
We obtain families of MVOPs by multiplying the functions Ψµ
d from the left with the inverse
of Ψµ
0 . The matrix weight Wµ is expressed in terms of Ψµ
0 , some data from the irreducible K-
representation and a scalar Jacobi weight that is associated to the symmetric space G/K. Con-
jugating the elements of D(µ) with Ψµ
0 yields a commutative algebra Dµ of differential operators
for which the MVOPs are simultaneous eigenfunctions. In fact, the MVOPs are determined by
this property and a normalization. The commutative algebra Dµ is contained in (M⊗C)[x, ∂x].
The exact relation between Wµ and Dµ is not yet understood on the level of the polynomials,
i.e. it is not clear what exactly characterizes Dµ. From this point of view it is not clear what
the right notion of a matrix valued classical pair should be. In the spirit of classical orthogonal
polynomials the MVOPs should be determined as eigenfunctions of a commutative algebra of
differential operators. The algebras D(µ) and Dµ are isomorphic by definition and the first
algebra is studied in [7, Chapter 9] and [27]. It would be interesting to determine its generators
in our situation, where the branching is multiplicity free and the rank is one.
Since we do not have a precise description of the algebra D(µ), we content to stick to the
original definition of a MVCP and we provide a method to find infinitely many examples of
them. To this end we exploit the existence of a special differential operator Ω ∈ D(µ), the (image
of the) second-order Casimir operator Ω on G. After conjugation with Ψµ
0 we find a second-order
differential operator Dµ ∈ Dµ that is symmetric with respect to the pairing and thus has the
MVOPs as simultaneous eigenfunctions.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 3
We show that our method is effective by applying it to low dimensional examples. First we
classify the data for which we obtain MVCPs of size 2 × 2 from [19]. The short list that we
obtain contains old and new items. Among the new ones are the symplectic (symmetric) pairs
(USp(2n),USp(2n− 2)×USp(2)) and the spherical (but non-symmetric) pair (G2, SU(3)). For
almost all these examples we provide the corresponding MVCPs below.
This paper is organized as follows: In Section 2 we review the construction of families of
MVOPs based on the representation theory of compact Gelfand pairs (G,K). We restrict the
functions Ψµ
d to the Cartan circle A ⊂ G and we identify A = S1. The coordinate on S1 is the
fundamental zonal spherical function φ, normalized by two constants c, d so that x = cφ+ d ∈
[0, 1]. With this change of variables we denote Ψ̃µ
d(x) = Ψµ
d(φ(a)), which is a function on [0, 1].
The MVOPs are basically a reflection on the three term recurrence relations of the func-
tions Ψµ
d and Ψ̃µ
d , introduced in this section. More precisely let Q̃d be defined by
Q̃µd(x) =
(
Ψ̃µ
0 (x)
)−1
Ψ̃µ
d(x).
Then it follows from the three term recurrence relation for Ψ̃µ
d that Q̃µd is a polynomial of degree d
with non singular leading coefficient.
Section 3 is the heart of this paper. Here we discuss how the algebra of differential opera-
tors Dµ comes about. By means of the bispectral property, we prove that there exist constant
matrices R̃ and S̃ such that Ψ̃µ
0 satisfies the first-order differential equation
x(1− x)∂xΨ̃µ
0 (x) = Ψ̃µ
0 (x)
(
S̃ + xR̃
)
. (1.1)
Remark 1.1. If we let ∂x act on both sides (1.1), we obtain an instance of Tirao’s matrix valued
differential equation [36]. However, the techniques in [36] do not apply directly because S̃ might
have eigenvalues in −N0. This is indeed the case for all the examples considered in this paper.
Remark 1.2. Observe that (1.1), can be seen as a differential operator acting on the right on Ψ̃µ
0 .
On the other hand Ψ̃µ
0 is also an eigenfunction of the radial part of the Casimir operator Ω of G
acting on the left.
In Corollary 3.6, we exploit again the bispectral property of the functions Ψ̃µ
d to deduce
that the image (radial part) D̃µ ∈ Dµ of the Casimir operator of G can be expressed in terms
of R̃, S̃ and an additional constant matrix coming from (G,K). More precisely we show that
the polynomials Q̃µd satisfy D̃µQ̃µd = Q̃µdΛd/(rp
2), where
D̃µ = x(x− 1)∂2
x +
[
λ1m
rp2(M −m)
− 2S̃ + x
(
λ1
rp2
− 2R̃
)]
∂x +
Λ0
rp2
,
where M , m are the maximum and minimum of φ|S1 , p the period of φ and r a scaling factor.
This data is provided in Table 2 for the various cases. The diagonal matrix Λ0 is the eigenvalue
of Ψ̃µ
0 as an eigenfunction of the Casimir operator Ω and it can be calculated for each pair
(G,K). It follows that the explicit knowledge of the function Ψ̃µ
0 implies explicit knowledge of
the corresponding pair (Wµ, D̃µ).
In Section 4 we discuss an algorithm to obtain an explicit expression for Ψ̃µ
0 . This algorithm
can be implemented in GAP [12, 34] for each specific pair (G,K). Once we have a formula for
Ψµ
0 we can calculate the corresponding MVCP by differentiation and matrix multiplication. We
also propose a method of finding families of MVCPs.
(1) Take a family (Gn,Kn, µn)n∈N for which the construction applies. For instance, take a con-
stant family of Gelfand pairs and consider (G,K, nµ), where µ ∈ F , or take a canonical
element of a face F , for instance the first fundamental weight ω, and consider the family
(Gn,Kn, ω)n∈N, where we let the Gelfand pairs (Gn,Kn)n∈N vary with n.
4 M. van Pruijssen and P. Román
(2) Calculate the first so many functions Ψ̃µ
0 of the family until a pattern shows up. This
provides an ansatz for a family of MVCPs.
(3) Show that every pair is indeed a MVCP. This is not difficult, one needs to check three
equations [10, Theorem 3.1]. It turns out that in many cases the group parameter, which
is a priori discrete, may vary continuously within a certain range.
In Section 5 we discuss the implementation in GAP [12] for the Gelfand pair (USp(2n),
USp(2n−2)×USp(2)) and the appropriate faces F . We discuss a branching rule that is necessary
to implement the algorithm. The branching laws for the symplectic groups are more difficult
than those for the special unitary and orthogonal groups. Indeed, the multiplicities are not only
determined by interlacing condition, but also by an alternating sum of partition functions. At
this point it is important to select the right irreducible K-types. Selecting an irreducible K-
representation of highest weight µ ∈ F , where (G,K,F ) is a multiplicity free system, guarantees
that the branching rules simplify and that the involved algebras are commutative. In fact, the
whole construction of MVOPs would not work for more general irreducible K-representations.
There are two families of MVCPs of size 2× 2 related to (USp(2n),USp(2n− 2)× USp(2)).
We calculate the first family by hand. For the other family we calculated the corresponding
family of MVCPs using our method. We apply the machinery once more to give an example of
size 3× 3 that is associated to this Gelfand pair.
In Section 6 we classify all possible triples (G,K, µ) that give rise to 2× 2 MVCPs according
to the uniform construction described in [19]. Subsequently we determine the corresponding
functions Ψ̃µ
0 .
To indicate that our method is effective we display most of the MVCPs of size 2× 2 below.
Some of these MVCPs were already known (Cases a1, b, d, see [30, 32, 37]) but they were
obtained by different means. The other MVOPs (Cases a2, c1, c2, g1, g2, f) are new as far as
we know. The matrix weights are of the form
Wµ(x) = (1− x)αxβΨ̃µ
0 (x)∗TµΨ̃µ
0 (x).
We provide the expressions for Ψ̃µ
0 (x) and Tµ instead of working out this multiplication, because
the expressions become rather lengthy. The parameters n, i, m are a priori all integers for which
we give the bounds in each case, see Remark 1.3.
Case a. The pair (G,K) = (SU(n+1),U(n)). We have α = n−1, β = 0, n ≥ 2, 1 ≤ i ≤ n−1
and m ∈ Z. We have two families of MVCPs associated to this example, depending on the sign
of m, but in either case
D̃µ = x(x− 1)∂2
x +
[
−1− 2S̃ + x(n+ 1− 2R̃)
]
∂x +
Λ0
2
.
Case a1. For m ≥ 0:
Ψ̃µ
0 (x) = x
m
2
√x √
x
1
(m+ 1)− x(m+ n− i+ 1)
i− n
, Tµ =
(
i 0
0 n− i
)
,
Λ0 =
(
0 0
0 2(m+ n− i+ 1)
)
, R̃ =
−m+ 1
2
1
2
0 −m+ 2
2
,
S̃ =
−
(m+ 1)(i−m− n)−m− 1− i+ n
2(m+ n+ 1− i)
m+ 1
2(m+ n+ 1− i)
i− n
2(i− n−m− 1)
(m+ 1)(i−m− n)
2(i−m− n− 1)
.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 5
Case a2. For m < 0:
Ψ̃µ
0 (x) = x−
(m+1)
2
m+ (i−m)x
i
1
x
1
2 x
1
2
, Tµ =
(
i 0
0 n− i
)
, Λ0 =
(
0 0
0 m− i
)
,
R̃ =
m− 1
2
0
1
2
m
2
, S̃ =
−
m(i−m− 1)
2(i−m)
i
2(i−m)
− m
2(i−m)
−mi+ i−m2
2(i−m)
.
Case b. The pair (G,K) = (SO(2n + 1), SO(2n)). Let α = β = n − 1, 1 ≤ i ≤ n − 2. The
corresponding MVCP is given by
D̃µ = x(x− 1)∂2
x +
(
− 2− 2S̃ + 2x(n− R̃)
)
∂x + Λ0,
Ψ̃µ
0 (x) =
(
2x− 1 1
1 2x− 1
)
, Tµ =
(
i 0
0 n− i
)
,
Λ0 =
(
0 0
0 2(n− i)
)
, R̃ =
(
−1 0
0 −1
)
, S̃ =
1
2
1
2
1
2
1
2
.
Case c. The pair (G,K) = (USp(2n),USp(2n − 2) × USp(2)). We have α = 2n − 3, β = 1
with n ≥ 3. We have two families of MVCPs associated to this example but in either case
D̃µ = x(x− 1)∂2
x +
(
−n− 2S̃ + 2x(2n− R̃)
)
∂x + Λ0.
Case c1:
Ψ̃µ
0 (x) =
√x √
x
1
x(n− 1)− 1
n− 2
, Tµ =
(
2 0
0 2n− 4
)
,
Λ0 =
(
0 0
0 4(n− 1)
)
, R̃ =
−1
2
1
2
0 −1
, S̃ =
1
2(n− 1)
1
2(n− 1)
(n− 2)
2(n− 1)
(n− 2)
2(n− 1)
.
Case c2:
Ψ̃µ
0 (x) =
x+ 1
2
(n+ 1)x− 2
n− 1
√
x
√
x((n+ 3)x+ n− 5)
2(n− 1)
, Tµ =
2
n+ 1
0
0
2
(n− 2)
,
Λ0
(
0 0
0 2n+ 6
)
, R̃ =
−1
(n+ 1)
(n− 1)
0 −3
2
, S̃ =
4
n+ 3
2n− 10
(n− 1)(n+ 3)
n− 1
n+ 3
n− 5
2(n+ 3)
.
For Case c2, we do not provide a formal proof that the family of MVCPs that are produced
in this way, are the ones associated to the Lie theoretical datum. For this we have to study the
various representations, which is quite involved.
Case g. The pair (G,K) = (G2, SU(3)). There is a single 2 × 2 MVCP associated to this
pair. We have α = β = 2 and
D̃µ = x(x− 1)∂2
x +
(
−3− 2S̃ + x(6− R̃)
)
∂x +
Λ0
2
,
6 M. van Pruijssen and P. Román
where
Ψµ(x) =
(
x x
√
x 3x
3
2 − 2
√
x
)
, Tµ =
(
1 0
0 2
)
,
Λ0 =
(
0 0
0 6
)
, R̃ =
−1
1
2
0 −3
2
, S̃ =
5
6
1
3
1
6
2
3
.
We omit Case d, (SO(2n), SO(2n − 1)), as it is similar to Case b. We make a few remarks
concerning these examples.
Remark 1.3. The parameters n, m, i in the various examples may vary in R rather than
in N, within certain bounds. The bounds are determined by the question whether the matrix
weight is positive. To see that the pairs (W,D) remain MVCPs, one has to check the symmetry
relations (4.2). These expressions are meromorphic in the parameters, so they remain valid.
Remark 1.4. In each case the determinant of the weight is a product of powers of x and (1−x)
times a constant. On the one hand this is quite remarkable, for the weight matrices do not seem
to have much structure. However, it turns out that all the weights that we construct have this
property. This follows from Corollary 3.4, which also settles an earlier Conjecture 1.5.3 of [33].
Remark 1.5. The matrix weights W may have symmetries, i.e. they may be conjugated by
a constant matrix into a diagonal matrix weight. We check whether this occurs by looking at the
commutant of Wµ. It turns out that we only have non trivial commutant in Cases b1 and b2
for specific parameters.
2 Lie theoretical background
Let (G,K) be a pair of compact connected Lie groups from Table 1 and let g, k denote their
Lie algebras. The quotient G/K is a two-point-homogeneous space (cf. [39]) which implies
that K acts transitively on the unit sphere in TeKG/K. We denote TeKG/K = p and fix a one
dimensional abelian subspace a ⊂ p. The one dimensional subspace a ⊂ g is the Lie algebra of
a subtorus A ⊂ G and it follows that we have a decomposition G = KAK.
Let M = ZK(A) denote the centralizer of A in K with Lie algebra m ⊂ k. Let TM ⊂ M be
a maximal torus and let TK ⊂ K be a maximal torus that extends TM . Then M ∩ TK = TM
and ATM is a maximal torus of G. The Lie algebras of TK and TM are denoted by tK and tM .
The complexifications of the Lie algebras are denoted by gc, . . ..
The (restricted) roots of the pairs (gc, ac⊕ tM,c), (gc, ac), (mc, tM,c) and (kc, tK,c) are denoted
by RG, R(G,A), RM and RK respectively. We fix systems of positive roots R+
G, R+
(G,A), R
+
M
and R+
K such that the natural projections RG → R(G,A) and RG → RM respect positivity.
The lattices of integral weights of G, K and M are denoted by PG, PK and PM , the cones of
positive integral weights by P+
G , P+
K and P+
M . The theorem of the highest weight implies that the
equivalence classes of the irreducible representations are parametrized by the cones of positive
integral weights. Given λ ∈ P+
G we denote by πGλ : G → GL(V G
λ ) an irreducible representation
of highest weight λ. The restriction πGλ |K decomposes into a finite sum of irreducible K-
representations and for µ ∈ P+
K we denote the multiplicity by mG,K
λ (µ) = [πGλ |K : πKµ ].
Definition 2.1.
• Let µ ∈ P+
K . A triple (G,K, µ) is called a multiplicity free triple if mG,K
λ (µ) ≤ 1 for all
λ ∈ P+
G .
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 7
• Let F ⊂ P+
K be a face, i.e. the N-span of some fundamental weights of K. A triple (G,K,F )
is called a multiplicity free system if (G,K, µ) is a multiplicity free triple for all µ ∈ F .
The notion of a multiplicity free system can be considered for any compact Lie group G with
closed subgroup K. In [19, 33] it is shown that for (G,K,F ) to be a multiplicity free triple, the
pair (G,K) is necessarily a Gelfand pair. Furthermore, the multiplicity free systems (G,K,F )
with (G,K) a Gelfand pair of rank one are classified by the rows of Table 1. The multiplicity
free systems with (G,K) a compact symmetric pair have been classified in [18].
Table 1. Compact multiplicity free systems of rank one. In the third column we have given the highest
weight λsph ∈ P+
G of the fundamental zonal spherical representation in the notation for root systems of
Knapp [20, Appendix C], except for the case (G,K) = (SO4(C),SO3(C)), where G is not simple and
λsph = $1 +$2 ∈ P+
G = N$1 +N$2. The groups M are isogenous to U(n− 2), SO(2n− 2), SO(2n− 1),
USp(2n− 4)×USp(2), Spin(7), SU(3) and SU(2) respectively.
G K λsph faces F
SU(n+ 1) n ≥ 1 U(n) $1 +$n any
SO(2n) n ≥ 2 SO(2n− 1) $1 any
SO(2n+ 1) n ≥ 2 SO(2n) $1 any
USp(2n) n ≥ 3 USp(2n− 2)×USp(2) $2 dimF ≤ 2
F4 Spin(9) $1 dimF ≤ 1 or
F = Nω1 + Nω2
Spin(7) G2 $3 dimF ≤ 1
G2 SU(3) $1 dimF ≤ 1
Let (G,K,F ) be a multiplicity free system from Table 1, let µ ∈ F and define P+
G (µ) = {λ ∈
P+
G : mG,K
λ (µ) = 1}. Let P+
M (µ) = {ν ∈ P+
M : mK,M
µ (ν) = 1}. The structure of the set P+
G (µ) is
important for the construction of the families of matrix valued orthogonal polynomials that we
associate to (G,K, µ). In the special case µ = 0 we have P+
G (0) = Nλsph, where λsph is called
the fundamental spherical weight. We have indicated the weights λsph in Table 1.
The complexified Lie algebra gc has a decomposition gc = kc ⊕ ac ⊕ n+, where n+ is the sum
of the root spaces of the positive restricted roots R+
(G,A). If (G,K) is symmetric this is just the
Iwasawa decomposition. For the two non-symmetric cases see, e.g., [33].
For λ ∈ P+
G (µ) the action of M on (V G
λ )n
+
= {v ∈ V G
λ : n+v = 0} is irreducible of highest
weight λ|tM . Moreover, λ|tM ∈ P+
M (µ), see e.g. [19]. It follows that the natural projection
q : PG → PM induces a map P+
G (µ) → P+
M (µ) which turns out to be surjective, see [19]. On
the other hand, if λ ∈ P+
G (µ) then λ + λsph ∈ P+
G (µ), which follows from an application of the
Borel–Weil theorem. Define the degree d : P+
G (µ)→ N by
d(λ+ λsph) = d(λ) + 1, min{d(P+
G (µ) ∩ (λ+ Zλsph))} = 0.
Let B(µ) = {λ ∈ P+
G (µ) : d(λ) = 0}. The set P+
G (µ) is called the µ-well and B(µ) the bottom
of the µ-well.
Theorem 2.2. We have P+
G (µ) = Nλsph +B(µ). There is an isomorphism
λ : N× P+
M (µ)→ P+
G (µ)
such that λ(d, ν)|tM = ν and d(λ(d, ν)) = d. If λ, λ′ ∈ P+
G (µ) and [πGλ ⊗ πGλsph : πGλ′ ] ≥ 1 then
d(λ)− 1 ≤ d(λ′) ≤ d(λ) + 1.
The proof of Theorem 2.2 is based on a case by case inspection, see [19, 33]. Let �µ be
the partial ordering on P+
G (µ) induced from the partial ordering on Nλsph +B(µ) given by the
lexicographic ordering (≤,�), where ≤ comes first.
8 M. van Pruijssen and P. Román
Corollary 2.3. If λ, λ′ ∈ P+
G (µ) and
[
πGλ ⊗ πGλsph : πGλ′
]
≥ 1, then λ − λsph �µ λ′ �µ λ + λsph
(whenever λ− λsph ∈ P+
G ).
Indeed, the highest weights of the irreducible representations that occur in the tensor prod-
uct decompositions are of the form λ′ = λ + λ′′, where λ′′ is a weight of λsph (see, e.g., [20,
Proposition 9.72]). The lowest weight of the fundamental spherical representation is w0(λsph),
which equals −λsph by inspection. Here, w0 denotes the longest Weyl group element.
Fix a multiplicity free system (G,K,F ) from Table 1 and fix µ ∈ F . Let πKµ : K → GL(V K
µ )
be an irreducible representation of highest weight µ. Let R(G) denote the (convolution) algebra
of matrix coefficients of G and define the (K ×K)-action on R(G)⊗ End(V K
µ ) by
(k1, k2)(m⊗ Y )(g) = m
(
k−1
1 gk2
)
⊗ πKµ (k1)Y πKµ (k2)−1.
The space Eµ := (R(G)⊗End(V K
µ ))K×K is called the space of µ-spherical functions. Note that
Φ ∈ Eµ satisfies
Φ(k1gk2) = πKµ (k1)Φ(g)πKµ (k2) ∀ k1, k2 ∈ K, g ∈ G.
Furthermore, note that E0 is a polynomial algebra and that Eµ is a free, finitely generated
E0-module. In fact, Eµ ∼= E0 ⊗ C|B(µ)| as E0-modules.
Let λ ∈ P+
G (µ) and let πGλ : G → GL(V G
λ ) denote the corresponding representation. Then
V G
λ = V K
µ ⊕ (V K
µ )⊥ and we denote by b : V K
µ → V G
λ a unitary K-equivariant embedding and by
b∗ : V G
λ → V K
µ its Hermitian adjoint.
Definition 2.4. The elementary spherical function of type µ associated to λ ∈ P+
G (µ) is
defined by
Φµ
λ : G→ End
(
V K
µ
)
: g 7→ b∗ ◦ πλ(g) ◦ b.
It is clear that the elementary spherical functions have the desired transformation behaviour.
We equip the space Eµ with a sesqui-linear form that is linear in the second variable,
〈Φ1,Φ2〉µ,G =
∫
G
tr (Φ1(g)∗Φ2(g)) dg
with dg the normalized Haar measure. As a consequence of Schur orthogonality and the Peter–
Weyl theorem we have the following result.
Theorem 2.5.
• The pairing 〈·, ·〉µ,G : Eµ × Eµ → C is a Hermitian inner product and 〈Φµ
λ,Φ
µ
λ′〉µ,G =
cλδλ,λ′.
• {Φµ
λ : λ ∈ P+
G (µ)} is an orthogonal basis of Eµ.
Denote φ = Φ0
λsph
, the fundamental zonal spherical function and write
φd = Φ0
dλsph
. (2.1)
Then E0 = C[φ], i.e. E0 is a polynomial ring generated by φ. Note that φ(k1gk2) = φ(g) for all
k1, k2 ∈ K and all g ∈ G. This implies that φΦλ can be expressed as a linear combination of
elementary spherical functions. It follows from Corollary 2.3 that
φΦλ =
∑
λ−λsph�µλ′�µλ+λsph
cµλ,λ′Φλ′ . (2.2)
The Borel–Weil theorem implies that cµλ,λ+λsph
6= 0 and we can express the elementary spherical
function Φλ as a E0-linear combination of the functions Φλ(0,ν), with ν ∈ P+
M (µ).
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 9
Definition 2.6.
• For λ ∈ P+
G (µ) define Qλ(φ) = (qµλ,ν(φ) : ν ∈ B(µ)) in C|B(µ)|[φ] by
Φλ =
∑
ν∈B(µ)
qµλ,ν(φ)Φν .
• For d ∈ N define Qd(φ) ∈ End(C|B(µ)|)[φ] as the matrix valued polynomial having the
Qλ(d,ν)(φ) as columns (ν ∈ B(µ)).
Theorem 2.7. The matrix valued polynomial Qd is of degree d and has invertible leading coef-
ficient.
Indeed, from (2.2) we deduce that for each d ∈ N there are Ad, Bd, Cd in End(C|B(µ)|) such
that
φQd(φ) = Qd+1(φ)Ad +Qd(φ)Bd +Qd−1(φ)Cd. (2.3)
Corollary 2.3 implies that the matrices Ad are upper triangular and the non vanishing of cµλ,λ+λsph
that the diagonals are non-zero.
Define V µ : G → End(C|B(µ)|) by V µ(g)ν,ν′ = tr(Φλ(0,ν)(g)∗Φλ(0,ν′)(g)). We see that V µ
is K-biinvariant, hence it is of the form V µ = W̃µ(φ) ∈ End(C|B(µ)|)[φ].
The pairing 〈Q,Q′〉 =
∫
GQ(φ(g))∗W̃µ(φ)Q′(φ(g))dg is a matrix valued inner product, see
e.g. [19]. Note that all the functions in the integrand are polynomials in φ, and φ is K-biinvariant.
In view of G = KAK and the integral formulas for this decomposition, we have
〈Q,Q′〉 =
∫ 1
0
Q(x)∗W̃µ(x)Q′(x)(1− x)αxβdx, (2.4)
where x = cφ+d (with constants c, d depending on the pair (G,K)) is a normalization such that x
attains all values in [0, 1]. The factor (1−x)αxβ is the ordinary Jacobi weight that is associated to
the Riemann symmetric space G/K on the interval [0, 1]. We denote Wµ(x) = (1−x)αxβW̃µ(x)
Now we come to a different discription of the family (Qd : d ∈ N), one that allows us to
transfer differentiability properties of the elementary spherical functions to similar properties of
the matrix valued polynomials.
The spherical functions Φ ∈ Eµ are determined by their restriction to A and Φ(a) ∈
EndM (Vµ) because A and M commute. Since mK,M
µ (ν) ≤ 1, EndM (Vµ) consists of diagonal
matrices. More precisely, with respect to a basis of the M -subrepresentations of Vµ, the ma-
trix Φ(a) is block diagonal, and every block is a constant times the identity matrix of size
dim ν. Sending such a matrix to a vector containing these constant provides an isomorphism
u : EndM (Vµ) ∼= C|B(µ)|.
Definition 2.8. The function Ψµ
λ : A→ C|B(µ)| is defined by Ψµ
λ(a) = u(Φµ
λ(a)). The function
Ψµ
d : A → End(C|B(µ)|) is the matrix valued function whose columns are the vector valued
functions Ψµ
λ(d,ν), with ν ∈ P+
M (µ), and where λ : N×P+
M (µ)→ P+
G (µ) is defined in Theorem 2.2.
It follows that Ψµ
d(a) = Ψµ
0 (a)Qd(φ(a)) and W̃µ(φ(a)) = Ψµ
0 (a)∗TµΨµ
0 (a), with Tµ the di-
agonal matrix whose entries are dim ν. Moreover, we know precisely which matrix coefficients
occur in the entries of the functions Ψµ
d .
Theorem 2.9. The entries of Ψµ
d are indexed by the set P+
M (µ). Hence, the entry (Ψµ
d)ν1,ν2 is
equal to the matrix coefficient mλ
v,v, where λ = λ(d, ν2) and v ∈ VM
ν1 ⊂ V K
µ ⊂ V G
λ is any vector
of length one.
10 M. van Pruijssen and P. Román
The proof is immediate from the definition of Ψµ
d . Note that the construction of the func-
tions Ψµ
d can also be performed for the complexified pair (GC,KC). We obtain a function on
AC ∼= C× that takes values in End(C|B(µ)|) and we denote it with the same symbol, Ψµ
d : C× →
End(CN ). For later reference, we state the following result concerning the entries of Ψµ
d , of
which the proof is straightforward.
Proposition 2.10. The entries of the function Ψµ
λ : AC → C|B(µ)| are Laurent polynomials
in C[z]. The maximal degree is less than or equal to |λ(HA)|, where HA is defined by A =
a/2πiHAZ. In fact, the maximal degree occurs precisely in the entry labeled with λ|tM .
The functions Ψµ
d are analytic, as the entries are matrix coefficients. Moreover, they satisfy
the three term recurrence relation
φ(z)Ψµ
d(z) = Ψµ
d+1(z)Aµd + Ψµ
d(z)Bµ
d + Ψµ
d−1(z)Cµd for all z ∈ AC, (2.5)
where the matrices Ad, Bd, Cd are the same as in (2.3). Define ∆µ(Ψµ
d) = Ψµ
d+1Ad + Ψµ
dBd +
Ψµ
d−1Cd. The operator ∆µ is a second-order difference operator acting on the variable d that
has Ψµ
d as eigenfunction and with eigenvalue φ.
3 Differential properties
In this section we discuss the second-order differential operator that we obtain from the quadratic
Casimir operator Ω of the group G. Moreover, using the fact that Ψµ
0 is an eigenfunction of Ω
whose eigenvalue is a diagonal matrix and that the functions Ψµ
d satisfy a three term recurrence
relation, we deduce that Ψµ
0 satisfies a first-order differential equation. From the singularities
of this equation we deduce that Ψµ
0 (a) is invertible whenever a ∈ A is a regular point for φ|A.
In this section we work with spherical functions on the complexified Lie groups GC and AC.
Let U(gc) be the universal enveloping algebra of gc and let U(gc)
K denote the algebra of
Ad(K)-invariant elements. Let I(µ) ⊂ U(kc) be the kernel of the representation U(kc) →
End(Vµ) and define
D(µ) := U(gc)
K
/ (
U(gc)
K ∩ U(gc)I(µ)
)
.
The irreducible representations of D(µ) correspond to irreducible representations of gc whose
restriction to kc has a subrepresentation of highest weight µ, see e.g. [7, Théorème 9.1.12].
The differential operators D ∈ D(µ) leave the space of µ-spherical functions invariant. As
the µ-spherical functions are determined by their values on A, in view of G = KAK and the
transformation behavior of the µ-spherical functions, every D ∈ D(µ) defines a differential
operator R(µ,D) satisfying R(µ,D)(Φµ|A) = (DΦµ)|A for all Φ ∈ Eµ. Since the spherical
functions are analytic, we obtain, after identifying AC = C×, a map R(µ) : D(µ) → C(z) ⊗
End(EndM (V K
µ ))[∂z], and the map R(µ) is an algebra homomorphism [5]. The differential
operator R(µ,D) is called the radial part of D. We denote the image of R(µ) by DR(µ).
Theorem 3.1. For every element D ∈ DR(µ) and every d ∈ N there is an element Λd(D) ∈
End(CN ) such that
DΨµ
d = Ψµ
dΛd(D).
Moreover, the map Λd : D 7→ Λd(D) is a representation. The family of representations {Λd}d∈N
separates the points of the algebra D(µ).
Proof. The first part is proved in [19, 33]. The matrix Λd(D) is diagonal and its entries are
Λd(D)ν,ν = πGλ(d,ν)(D). For the second statement, suppose the converse. Then there are two
differential operators D, D′ that have the same eigenvalues and it follows that D −D′ acts as
zero on the elementary spherical functions. This implies that D −D′ = 0. �
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 11
It follows that for any D ∈ DR(µ) and ∆µ (defined in (2.5)), the triple (∆µ, D, {Ψµ
d : d ∈ N})
has a bispectral property, i.e. the operators ∆µ and D have the members of the family {Ψµ
d :
d ∈ N} as simultaneous eigenfunctions (albeit in different variables). For more on the bispectral
property, see e.g. [8, 17]. In the case µ = 0 we have |B(µ)| = 1 and we denote the eigenvalues
by lower case letters, Dφ = λ(D)φ.
The Casimir operator on G is given as follows. Let {Xi : i = 1, . . . ,dim gc} be a basis of gc
and let {X̃i : i = 1, . . . ,dim gc} be a dual basis with respect to the Killing form κ on gc. Then
Ω =
∑
i,j
κ(Xi, Xj)X̃iX̃j . On the gc-representation Vλ the Casimir operator Ω acts with the scalar
κ(λ, λ) + 2κ(λ, ρG), where ρG is half the sum of the positive roots of G. The image of Ω in D(µ)
is denoted by Ω or by Ω(µ) if we want to indicate on which function space we let it act. Then
R(µ,Ω) = r
(
(z∂z)
2 + c(z)z∂z + Fµ(z)
)
, (3.1)
where c(z) is a meromorphic function on AC, Fµ(z) is a meromorphic function with matrix
coefficients and r is a constant that depends on the pair (G,K). For the particular case µ = 0 we
have F0(z) = 0. For the symmetric pairs these statements follow from [40, Proposition 9.1.2.11].
For the two non-symmetric pairs see [19] or [33, Paragraph 3.4.28].
Lemma 3.2. The function φ satisfies (z∂zφ)2 = p2(φ −M)(φ −m), where M and m are the
maximum and the minimum of φ restricted to the circle S1 ⊂ AC and p = #(K ∩A).
Proof. The fundamental spherical functions are of the form φ(z) = a(zp + z−p)/2 + b (see
e.g. [33, Table 3.3]). On the other hand, φ is an eigenfunction of Ω(0) and the statement follows
from a calculation. �
In what follows we only consider the operator R(µ,Ω), the radial part of the image of the
second-order Casimir operator in D(µ). The eigenvalues are denoted by Λd for general µ ∈ F
and λd for µ = 0, i.e. we have
Ω(0)φd = λdφd, (3.2)
Ω(µ)Ψµ
d = Ψµ
dΛd, (3.3)
where φd is given by (2.1).
Theorem 3.3. The function Ψµ
0 satisfies the first-order differential equation
2rz2∂zφ(z)(∂zΨ
µ
0 )(z) = Ψµ
0 (z)(Rφ(z) + S), (3.4)
on C×, where R = A−1
0 Λ1A0 − Λ0 − λ1 and S = Λ0B0 −B0A
−1
0 Λ1A0.
Proof. We need the identities (2.5), (3.2) and (3.3) for d = 1, 2 to prove this result. Let the
operator r(z∂z)
2 act on both sides of the equality
φ(z)Ψµ
0 (z) = Ψµ
1 (z)A0 + Ψµ
0 (z)B0
and work out the differentiation. The derivatives of order >1 can be written in terms of φ, Ψµ
0
and Ψµ
1 and their first-order derivatives. We get
r(z∂z)
2(φ(z)Ψµ
0 (z)) = 2rz2φ′(z)(Ψµ
0 )′(z)
+ φ(z)
[
Ψµ
0 (z)Λ0 − Fµ(z)Ψµ
0 (z)− c(z)z(Ψµ
0 )′(z)
]
+
[
λ1φ(z)− c(z)zφ′(z)
]
Ψµ
0 (z),
and
r(z∂z)
2(φ(z)Ψµ
0 (z)) = r(z∂z)
2
(
Ψµ
1 (z)A0 + Ψµ
0 (z)B0
)
12 M. van Pruijssen and P. Román
=
(
Ψµ
1 (z)Λ1 − Fµ(z)Ψµ
1 (z)− c(z)z(Ψµ
1 )′(z)
)
A0
+
(
Ψµ
0 (z)Λ0 − Fµ(z)Ψµ
0 (z)− c(z)z(Ψµ
0 )′(z)
)
B0.
Equating and using the three term relation and its derivative, we find
2rz2φ′(z)(Ψµ
0 )′(z) = Ψµ
1 (z)Λ1A0 + Ψµ
0 (z)Λ0B0 − φ(z)Ψµ
0 (z)(Λ0 + λ1).
Using the three term recurrence relation once more we get
2rz2φ′(z)(Ψµ
0 )′(z)
=
(
φ(z)Ψµ
0 (z)−Ψµ
0 (z)B0
)
A−1
0 Λ1A0 + Ψµ
0 (z)Λ0B0 − φ(z)Ψµ
0 (z)(Λ0 + λ1)
= Ψµ
0 (z)
[(
A−1
0 Λ1A0 − Λ0 − λ1
)
φ(z) + Λ0B0 −B0A
−1
0 Λ1A0
]
.
Plugging in R and S yields the desired equation. �
Note that the matrix R measures the fact whether the matrices An of the recurrence relations
are diagonal or not. In the symplectic case these recurrence matrices are not diagonal in general,
see Sections 5.2 and 5.3. For the orthogonal groups (SO(m+ 1),SO(m)) the recurrence matri-
ces An are diagonal. In this case the matrix R is diagonal (even scalar in many examples). For
m = 3 this follows from [22, Theorem 3.1] or [31, Theorem 9.4]. In general this follows from
the description of P+
G (µ) and the decomposition of the tensor product of a general irreducible
representation and the fundamental spherical representation, see [33, Chapter 2.4].
Let φAC denote the restriction of φ to AC. Let AC,reg be the set of points where φAC is an
immersion, i.e., where dφAC is injective. Denote Areg = A ∩ AC,reg. Finally let AC,µ−reg denote
the set of points, where det Ψµ
0 (z) 6= 0. Since the weight function Wµ is polynomial in φ and
moreover, the highest degree polynomial occurs precisely once in each column and once in each
row, the determinant of Wµ is a polynomial in φ of positive degree. It follows that AC,µ−reg ⊂ AC
is a dense open subset. See also [33].
Corollary 3.4. If z ∈ AC,reg then Ψµ
0 (z) is an invertible matrix.
Proof. The linear system (3.4) has singularities precisely in AC\AC,reg. Hence, locally in AC,reg,
there exists a fundamental solution matrix which is holomorphic and invertible. By Theorem 3.3
the function Ψµ
0 has the same properties on the set AC,µ−reg, whose intersection with AC,reg is
dense in AC. The claim follows. �
Corollary 3.4 settles a conjecture on the determinant of the weight matrices, see [33, 1.5.3]
and [22, Theorem 2.3]. Namely, the determinant of V µ(φ) is a polynomial in φ that is non-
zero outside the critical points of φAC . In view of Lemma 3.2 det(V µ(φ)) is a multiple times
(φ−M)nM (φ−m)nm , for some nM , nm ∈ N.
We can now conjugate the image Ω(µ) ∈ D(µ) of the quadratic Casimir operator Ω with
the function Ψµ
0 to obtain a differential operator of order two for the family of matrix valued
orthogonal polynomials. Since Ω(µ) has real eigenvalues the resulting operator is symmetric
with respect to the pairing (2.4). Indeed, the matrices 〈Qd, Qd〉 are diagonal matrices.
Theorem 3.5. The polynomials Qd, d ∈ N, satisfy DµQd = QdΛd, where
Dµ = (Ψµ
0 )−1ΩΨµ
0 = r(z∂z)
2 +
[
rc(z) + (Rφ(z) + S)/(zφ′(z))
]
(z∂z) + Λ0.
Proof. It is a straightforward computation that
Ω(µ)
(
Ψµ
0 (z)Q(z)
)
= rΨµ
0 (z)(z∂z)
2Q(z) + r
[
2z∂zΨ
µ
0 (z) + c(z)Ψµ
0 (z)
]
(z∂zQ(z))
+ r
[
(∂zΨ
µ
0 )2 + c(z)(z∂z)Ψ
µ
0 (z) + Fµ(z)Ψµ
0 (z)
]
Q(z). (3.5)
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 13
It follows from (3.1) and Lemma 3.2 that the coefficient of Q(z) in (3.5) is exactly Λ0. On
the other hand, the coefficient of (z∂zQ(z)) in (3.5) is obtained from Theorem 3.3. Now the
theorem follows by multiplying with (Ψµ
0 )−1 from the left on both sides of (3.5). �
Corollary 3.6. Let Ψ̃µ
0 be the function on the interval [0, 1], defined by Ψ̃µ
0 ((φ(z) −m)/(M −
m)) = Ψµ
0 (z), where z ∈ {eit : 0 ≤ t < π/p}. Then Ψ̃µ
0 satisfies the (right)-first-order differential
equation
x(1− x)∂xΨ̃µ
0 (x) = Ψ̃µ
0 (x)
(
S̃ + xR̃
)
, S̃ = − S +mR
2rp2(M −m)
, R̃ = − R
2rp2
. (3.6)
Let Q̃d = Qd ◦ ((φ −m)/(M −m)). Then D̃µQ̃d = Q̃d(Λd/(rp
2)), where D̃µ is the differential
operator
D̃µ = x(x− 1)∂2
x +
[
λ1m
rp2(M −m)
− 2S̃ + x
(
λ1
rp2
− 2R̃
)]
∂x +
Λ0
rp2
. (3.7)
Here M , m are the maximum and minimum of φ|S1, see Lemma 3.2.
Proof. The proof is a straightforward consequence of Lemma 3.2, Theorem 3.5 and (3.2). �
Remark 3.7. We observe that the function (φ−m)/(M −m) is a bijection from {eit : 0 ≤ t <
π/p} onto the interval [0, 1] (see Table 2), so that Ψ̃µ
0 in Corollary 3.6 is well defined.
In Section 4.3 we provide all the data r, p. The scaling r is determined by comparing the radial
part of Ω(0) to the hypergeometric differential operator for the Jacobi polynomials on G/K.
Basically we only need to know (α, β).
Remark 3.8. The operator D̃µ in (3.7) is a matrix valued hypergeometric equation [36]. In the
scalar case, the polynomial eigenfunctions of (3.7) can be written in terms of hypergeometric
series. In the matrix valued setting, in order to give a simple expression of Q̃d as matrix valued
hypergeometric series it is necessary to perform a deeper analysis. See Corollary 5.6 for the case
of the symplectic group.
Remark 3.9. Theorem 3.5 allows us to calculate the conjugation of the Casimir operator with
the function Ψµ
0 for individual cases, without calculating the radial part of the Casimir operator.
The latter is in general very technical so Theorem 3.5 makes it much easier to generate explicit
examples of differential operators. For an explicit expression of the differential operator Dµ
we only need to know the eigenvalue λ1 and an explicit expression of the function Ψµ
0 . Indeed,
using Lemma 3.3 we find expressions for R and S, using computer algebra, and these, together
with λ1 gives an expression for Dµ by (3.7).
We conclude that a numerical expression of the functions Ψµ
0 allows us, using computer
algebra, to generate examples of a matrix valued classical pairs (Wµ, Dµ).
Remark 3.10. The operator D̃µ in Corollary 3.6 is given explicitly by the following expressions
(use Table 4.3).
• For SU(n+ 1): D̃µ = x(x− 1)∂2
x +
[
−1− 2S̃ + x(n+ 1− 2R̃)
]
∂x + Λ0
2 .
• For SO(2n): D̃µ = x(x− 1)∂2
x +
[
−2n−1
2 − 2S̃ + x(2n− 1− 2R̃)
]
∂x + Λ0.
• For SO(2n+ 1): D̃µ = x(x− 1)∂2
x +
[
−n− 2S̃ + x(2n− 2R̃)
]
∂x + Λ0.
• For USp(2n): D̃µ = x(x− 1)∂2
x +
[
−2− 2S̃ + x(2n− 2R̃)
]
∂x + Λ0
2 .
• For F4: D̃µ = x(x− 1)∂2
x +
[
−6− 2S̃ + x(12− 2R̃)
]
∂x + Λ0.
• For Spin(7): D̃µ = x(x− 1)∂2
x +
[
−7
2 − 2S̃ + x(7− 2R̃)
]
∂x + 4
3Λ0.
• For G2: D̃µ = x(x− 1)∂2
x +
[
−3− 2S̃ + x(6− 2R̃)
]
∂x + Λ0
2 .
14 M. van Pruijssen and P. Román
4 A method to calculate MVCPs
In this section we describe an algorithm to calculate the functions Ψµ
d from Definition 2.8.
We have implemented this algorithm in the computer package GAP for the symmetric pair
(USp(2n),USp(2n − 2) × USp(2)) (see Section 5 for this pair and [34] for the GAP-files). From
the description of the algorithm it is clear how to implement it for other multiplicity free triples
in Table 1. In the Section 4.1 we discuss the general algorithm to calculate Ψµ
n. In Section 4.2
discuss the implementation of the algorithm in GAP. Finally we provide the necessary data from
the compact Gelfand pairs to calculate the MVCPs.
4.1 The algorithm
The algorithm to calculate the functions Ψµ
d has the following input: (1) a multiplicity free
system (G,K,F ) of rank one from Table 1, (2) an element µ ∈ F and (3) an integer d. The
output is an expression for the function Ψµ
d : S1 → End(CNµ), where Nµ is the number of
elements in the bottom B(µ) of the µ-well.
The Algorithm
1. Determine P+
M (µ) = {ν1, . . . , νNµ}.
2. Determine the bottom B(µ) = {b(µ, ν1), . . . , b(µ, νNµ)} of the µ-well.
3. Determine Bd(µ) := {λj = b(µ, νj) + dλsph : j = 1, . . . , Nµ}.
4. for all λj ∈ Bd(µ) do
5. Determine a K-equivariant embedding γ : Vµ → Vλj .
6. for all νi ∈ P+
M (µ) do
7. Determine a vector vλj ,µ,νi ∈ γ(Vµ) of highest weight νi
of length one.
8. Determine the matrix coefficient eit 7→ 〈atvλj ,µ,νi , vλj ,µ,νi〉.
9. Put all the entries in the j-th column
10. Build Ψµ
d with all the columns j = 1, . . . , Nµ.
Step 1. Determine P+
M (µ). The set P+
M (µ) parametrizes the elementary spherical functions
of a fixed degree d (Theorem 2.2) and also the entries of the elementary spherical functions
restricted to A (Theorem 2.9). Thus, the number of elements N in P+
M (µ) determines the size
of the matrices Ψµ
d(a), a ∈ A.
Steps 2 and 3. Determine Bd(µ). This set determines the spectrum P+
G (µ) and in turn
parametrizes the elementary spherical functions.
Step 5. Determine a K-equivariant embedding γ : V K
µ → V G
λj
. To this end we calculate the
weight spaces V G
λj
(µ′), where µ′ is a weight in PG that projects onto µ ∈ P+
K , i.e. with q(µ′) = µ.
If rankG = rankK then µ′ = µ. The highest weight vector v0 ∈ V K
µ is annihilated by all the
simple root vectors of K and the image γ(v0) is the unique vector with this property. Hence we
calculate⋂
µ′:q(µ′)=µ
V G
λj
(µ′) ∩
⋂
β∈ΠK
ker(Eβ), (4.1)
which is a one-dimensional subspace of Vλj because mG,K
λj
(µ) = 1. Fix a non-zero element v′0
in (4.1). Put γ(v0) = v′0 and define γ : V K
µ → V G
λj
by stipulating that γ is K-equivariant. Letting
the root vectors in kC of the negative simple roots act on v′0 gives a basis of γ(V K
µ ) ⊂ V G
λj
.
Step 7. Determine a vector vλj ,µ,νi in γ(V K
µ ) of highest M -weight νi. This is similar to the
first part of Step 5.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 15
Step 8. Determine the matrix coefficient eit 7→ 〈atvλj ,µ,νi , vλj ,µ,νi〉. By Proposition 2.10 we
know that any matrix coefficient of πGλj restricted to A is a Laurent polynomial whose terms
are of degree k with |k| ≤ |λ(HA)|. On the other hand we have at = exp(iHAt), which implies
〈atvλj ,µ,νi , vλj ,µ,νi〉Vλ =
∞∑
p=0
(it)p
p!
〈Hp
Avλj ,µ,νi , vλj ,µ,νi〉V Gλj
.
It suffices to calculate 〈Hk
Avλj ,µ,νi , vλj ,µ,νi〉V Gλ for k ≤ 2|λ(HA)|. Indeed, the Laurent polyno-
mial mλ
v,w|A is determined by the the first 2|λ(HA)| terms of its power series. To get an invariant
inner product on the representation spaces we have implemented the Shapovalov form, see
e.g. [35].
Steps 9 and 10. The matrix coefficients we obtain in Step 8 are put in a column vector of
length Nµ and these Nµ column vectors are in turn put as columns in an (Nµ ×Nµ)-matrix.
4.2 Implementation in GAP
In Step 1 we need to determine the P+
M (µ). Descriptions of P+
M (µ) are given by classical branch-
ing rules for the lines 1, 2 and 3, see [33]. For line 4 see Section 5, for line 5 see [1] and for
the two remaining lines see [19]. Upon choosing suitable tori, the branching rules amount to
interlacing conditions of strings of integers which are easily implemented in GAP.
To implement the bottom B(µ) for Steps 2 and 3 we need a precise description. For the first
three lines of Table 1 see [4, 33] and for the other lines see [19]. Once Bd(µ) is known for d = 0,
it is known for all d. The bottom depends (piecewise) affine linearly on P+
M (µ) and is thus easily
implemented in GAP.
Having the highest weights of the involved irreducible G, K and M -representations we can
do all the Lie algebra calculations in the appropriate G-representation space V G
λ using GAP. This
settles Steps 5 and 7.
The linear system that we need to solve in Step 8 to determine the coefficients of the Laurent
polynomials mλ
vνi ,vνi
|A is easily implemented. The actual output of the algorithm is a number
of N2 strings of real numbers, each string representing a Laurent polynomial mλ
vνi ,vνi
|A.
A simple loop provides the implementation of Steps 9 and 10.
Remark 4.1. The representation spaces that are used soon become very large, which makes
the implementation only suitable for relatively small K-types µ. In this case, small means
that the calculation is performed in a reasonable time. It depends on, among other things, the
dimensions of the representations spaces V G
λ , λ ∈ B(µ). It would be interesting to make these
things more precise, once the algorithm is improved. For instance, of the spaces V G
λ , λ ∈ B(µ),
we only need a few vectors, whereas in our implementation we first need the whole space V G
λ
to calculate these few vectors. Unfortunately, we do not see how we can realize these kind of
improvements now.
4.3 Obtaining the MVCPs
Once we have constructed the function Ψµ
0 whose entries are polynomials in cos(t) we follow the
next steps
(a) We make the change of variables x = (cos(pt) + 1)/2 so that Ψ̃µ
0 (x) = Ψµ
0 (t).
(b) We calculate x(1−x)
(
Ψ̃µ
0 (x)
)−1
∂xΨ̃µ
0 (x) which gives the matrices R̃, S̃ according to Corol-
lary 3.6.
16 M. van Pruijssen and P. Román
Table 2. Data needed to calculate MVCPs.
G K λ1 φ(at) (α, β) r
SU(n+ 1) U(n) 2n+ 2 (n+1) cos2(t)−1
n (n− 1, 0) 2p−2
SO(2n) SO(2n− 1) 2n− 1 cos(t) (n− 3
2 , n−
3
2) p−2
SO(2n+ 1) SO(2n) 2n cos(t) (n− 1, n− 1) p−2
USp(2n) USp(2n− 2)×USp(2) 4n n cos2(t)−1
n−1 (2n− 3, 1) 2p−2
F4 Spin(9) 12 cos(2t) (7, 3) p−2
Spin(7) G2 21/4 cos(3t) (5
2 ,
5
2) 3
4p
−2
G2 SU(3) 12 cos(2t) (2, 2) 2p−2
(c) The differential operator D̃µ is now given by Remark 3.10. The missing data m, M , p is
contained in the fundamental spherical function φ (see the proof of Lemma 3.2), which, as
well as λ1, we provide in Table 2. The eigenvalue Λ0 has to be calculated in the individual
cases using for example Weyl’s dimension formula. We do this in the Sections 5 and 6 for
the (2× 2)-examples.
(d) To obtain an expression for the weight Wµ(x) = W̃µ(x)w(x) we need the scalar weight
w(x) = (1 − x)αxβ and the matrix Tµ that has the dimensions of the irreducible M -
representations ν ∈ P+
M (µ) on the diagonal, see (2.4) and the discussion following Defini-
tion 2.8. The parameters (α, β) of the scalar weights are provided in Table 2. The weight
is given by
Wµ(x) = (1− x)αxβ Ψµ
0 (x)∗TµΨµ
0 (x), x = (φ−m)/(M −m),
Hence, knowledge of (Ψµ
0 , φ, T
µ, λ1,Λ0) allows us to find explicit expressions of the pair
(Wµ, D̃µ).
(e) A formal proof that (Wµ, D̃µ) is a MVCP can be done showing that D̃µ is symmetric
with respect to Wµ. This boils down to prove that the following symmetry equations, [10,
Theorem 3.1], hold true
A∗1W = −WA1 + 2(x(x− 1)W )′, A∗0W = WA0 − (WA1)′ + (x(x− 1)W )′′, (4.2)
where
A1(x) =
λ1m
rp2(M −m)
− 2S̃ + x
(
λ1
rp2
− 2R̃
)
, A0 =
Λ0
rp2
.
We observe that the boundary conditions in [10, Theorem 3.1] are always satisfied in our
case.
5 The symplectic case
Let n ≥ 3 and G = USp(2n), K = USp(2n − 2) × USp(2). The branching rules for G to K
are due to Lepowsky [26]. Let K1 = USp(2)× USp(2n− 4)× USp(2) ⊂ K and M = USp(2)×
USp(2n − 4) ⊂ K1, where the embedding K1 ⊂ K is the canonical one and where M ⊂ K1
is given by (x, y) 7→ (x, y, x). The branching rules for K to M are due to Baldoni-Silva [1].
If we choose the K-type in a 2-dimensional face then the branching rules become considerably
simple. We employ the standard choices for roots and weights of the symplectic group [20,
Appendix C]. We have rankG = rankK = rankK1 = n and rankM = n− 1. The weight lattices
of G, K, K1 are equal to P = Zn. Let {εi : i = 1, . . . , n} denote the standard basis of P . The
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 17
fundamental weights of G are $i =
i∑
j=1
εj for i = 1, . . . , n. The fundamental weights of K are
ωi =
i∑
j=1
εj for i = 1, . . . , n − 1 and ωn = εn. The fundamental weights of K1 are ξ1 = ε1,
ξi =
i∑
j=2
εj for i = 2, . . . , n− 1 and ξn = εn. The fundamental weights of M are identified with
η1 = 1
2(ε1 + εn) and ηi =
i∑
j=2
εj for i = 2, . . . , n− 1. The fundamental weights generate the semi
group of integral dominant weights P+
K , P+
K1
and P+
M which in turn parametrize the equivalence
classes of irreducible representations of K, K1 and M .
5.1 Branching rule from K to M
The branching rule fromK toM can be calculated in two steps: first fromK toK1, then fromK1
to M . A crucial ingredient of the branching rule is the partition function pΞ : Zn−1 → N, where
Ξ = {εi ± ε1 : i = 2, . . . , n− 1} and
pΞ(z) = #
(nξ)ξ ∈ N|Ξ| :
∑
ξ∈Ξ
nξξ = z
. (5.1)
Proposition 5.1. Let µ = xωi + yωj with i < j and x, y ∈ N. Write µ =
n∑
k=1
bkεk and let
ν =
n∑
k=1
ckεk ∈ P+
K1
. Define C1 = b1 − max(b2, c2), Ck = min(bk, ck) − max(bk+1, ck+1) for
k = 2, . . . , n− 2 and Cn−1 = min(bn−1, cn−1) and Cn = 0. We have mK,K1
µ (ν) = 1 if and only if
(1) cn = bn,
(2) Ci + Cj − c1 is even,
(3) Ci + Cj ≥ c1 ≥ |Ci − Cj |,
(4) Ck ≥ 0 for k = 1, . . . , n− 2,
Proof. The statement is Lepowsky’s branching rule for USp(2n− 2) to USp(2)×USp(2n− 4)
with the additional information that we have control over the multiplicity. The support of the
function mK,K1
µ : P+
K1
→ N is contained in the set that is determined by condition (1) and (4) and
the additional condition that
n−1∑
k=1
Ck is even, see [26, Theorem 2]. In this case the multiplicity
is given by
mK,K1
µ (ν) = pΞ((C1 − c1)ε1 + C2ε2 + · · ·+ Cn−1εn−1)
− pΞ((C1 + c1 + 2)ε1 + C2ε2 + · · ·+ Cn−1εn−1), (5.2)
where pΞ is the partition function (5.1). See [20, Theorem 9.50] for a proof of an equivalent
statement (the roots are permuted because the embedding is different). If mK,K1
µ (ν) = 1 then ν
is in the support of mK,K1
µ which implies that Ck ≥ 0 for k = 1, . . . , n− 2. The condition on µ
implies that Ck = 0 unless k = i or k = j, hence (2) is satisfied. Condition (1) is trivially
satisfied. It remains to check (3), which we do below. Conversely, if (1)–(4) are satisfied, then ν
is in the support of mK1,K
µ , because Ck = 0 unless k = i or k = j. It remains to show that (3)
implies mK,K1
µ (µ) = 1, which we check by calculating (5.2). We distinguish two cases: i = 1 and
18 M. van Pruijssen and P. Román
i > 1. Suppose that i = 1. Then Ck = 0 unless k = 1 or k = j. Without loss of generality we
may assume that j = 2 (if j = n then we take C2 = 0) and Ξ = {ε2 ± ε1}. Then
mK,K1
µ (ν) = pΞ((C1 − c1)ε1 + C2ε2)− pΞ((C1 + c1 + 2)ε1 + C2ε2).
Both terms are zero or one, because |Ξ| = 2. The first term is one if and only if C1−C2 ≤ c1 ≤
C1 +C2 and C1 +C2− c1 even. The second term is minus one if and only if c1 ≤ C2−C1−2. As
c1 ≥ 0 in the first place, we see that mK,K1
µ (ν) = 1 if and only if (1)–(4) are satisfied. Suppose
that i > 1. Without loss of generality we assume that i = 2 and j = 3 and Ξ = {ε1± ε2, ε1± ε3}.
Then
mK,K1
µ (ν) = pΞ(−c1ε1 + C2ε2 + C3ε3)− pΞ((c1 + 2)ε1 + C2ε2 + C3ε3).
We have −c1ε1 +C2ε2 +C3ε3 = B2(ε2− ε1)+(C2−B2)(ε2 + ε1)+B3(ε3− ε1)+(C3−B3)(ε3 + ε1)
from which it follows that
pΞ(−c1ε1 + C2ε2 + C3ε3)
= #
{
(B2, B3) ∈ N2 : B2 ≤ C2, B3 ≤ C3, B2 +B3 =
1
2
(C2 + C3 + c1)
}
. (5.3)
A similar calculation shows
pΞ((c1 + 2)ε1 + C2ε2 + C3ε3)
= #
{
(B2, B3) ∈ N2 : B2 ≤ C2, B3 ≤ C3, B2 +B3 =
1
2
(C2 + C3 − c1 − 2)
}
. (5.4)
The quantities (5.3), (5.4) can only be nonzero if C2 + C3 − c1 is even. In this case (5.3) is the
number of integral points of the intersection of the line ` = {(B2, B3) : B2+B3 = 1
2(C2+C3+c1)}
with the rectangle {(B2, B3) : 0 ≤ B2 ≤ C2, 0 ≤ B3 ≤ C3}. The quantity (5.4) is equal to the
number of points of the intersection of the same rectangle with the line `+ (1, 0). It follows that
mK,K1
µ (ν) = 1 if and only if 2 max(C2, C3) ≤ 1
2(C2 + C3 + c1) ≤ C2 + C3 which is equivalent to
|C2 − C3| ≤ c1 ≤ C2 + C3. �
The branching from K1 to M is equivalent to the branching USp(2)×USp(2) to the diagonal
USp(2). Given a K1 type (c′1ε1 + c2ε2 + · · · + cn−1εn−1 + c′nεn), the M -types that occur upon
restricting are of the form
(c1ε1 + c2ε2 + · · ·+ cn−1εn−1 + c1εn), c1 =
|c′1 − c′n|
2
,
|c′1 − c′n|
2
+ 1, . . . ,
c′1 + c′n
2
.
Corollary 5.2. The matrix valued orthogonal polynomials of size 2×2 are obtained from µ = ω1
or µ = ωn−1.
Proof. We have to do a case by case investigation. Let µ = xωi + yωj with x, y non-negative
integers, y positive and i ≤ j. Consider the cases
1) i = j,
2) 1 ≤ i < n, j = n,
3) 1 ≤ i < j < n.
Case 1. If i = n we have only one M -type. Indeed, this situation boiles down to restricting
holomorphic or anti-holomorphic representation of SL2(C) to SU(2). If i = 1 then we have
µ = (x+ y)ω1 and
P+
K1
(µ) = {sε1 + (x+ y − s)ε2 : s = 0, . . . , y + x},
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 19
where P+
K1
(µ) is the support of mK,K1
µ . If i = n− 1 then µ = (x+ y)ωn−1 and
P+
K1
(µ) = {sε1 + (x+ y)(ε2 + · · ·+ εn−2) + sεn−1 : s = 0, . . . , y + x}.
If 1 < i < n − 1 then µ = (x + y)ωi then (ci, ci+1) ∈ {(0, 0), (x + y, 0), (x + y, x + y)} lead to
different representations of K1.
Case 2. We have µ = xωi + yωn with xy 6= 0. If 1 < i < n− 1 then we find three K1 types
that occur upon restriction. If i = 1 then we can choose c2 = x and c2 = 0. The first choice
implies c1 = 0, the second c1 = x. We find two K1-types, one of them decomposes in at least
two M -types upon restriction.
Case 3. If x, y are both non zero, then we have at least three K1-types. Indeed, if either x
or y is zero, then we are in Case 1. Assume xy 6= 0. Then µ, µ− ε1 − εi, µ− ε1 − εj are three
K1 types by Proposition 5.1.
We conclude that µ = ω1 and µ = ωn−1 are the only K-types that give matrix valued
orthogonal polynomials of size 2× 2. �
5.2 The algorithm for a (2 × 2)-case
We consider (G,K) = (USp(2n),USp(2n−2)×USp(2)) and the irreducible USp(2n−2)×USp(2)-
representations with highest weight µ in a two dimensional face. The complexified Lie algebra of
USp(2n) is denoted by sp2n(C). We calculate the spherical function φ and Ψµ
0 . The realizations
of the fundamental representations that are involved in this calculation are described in [11].
We employ the same notation as in [11]. In particular, the root vectors of εi − εj and 2εk are
denoted by Xi,j and Uk respectively.
The spherical representation λsph = $2 = ε1 + ε2 is realized in the kernel of
∧2 V → C :
v ∧ w 7→ Q(v, w). The kernel is of dimension
(
2n
2
)
− 1 = n(2n − 1) − 1. The weight zero space∧2 V (0) is spanned by {ek ∧ en+k : k = 1, . . . , n}. The weight zero space V G
$2
(0) is spanned by
the vectors {ek ∧ en+k− ek+1 ∧ en+k+1 : k = 1, . . . , n− 1}. The vector in V G
$2
that is fixed under
USp(2n − 2) × USp(2) is of weight zero because the groups G and K are both of semisimple
rank n. We look for a vector that is killed by sp2n−2(C) ⊕ sp2(C) and it is easily seen that
e1 ∧ en+1 + . . .+ en−1 ∧ e2n−1 − (n− 1)en ∧ e2n is killed by this subalgebra.
The torus A has as its Lie algebra a = R · (X1,n −Xn,1). Put
at =
cos t 0 sin t 0 0 0
0 In−2 0 0 0 0
− sin t 0 cos t 0 0 0
0 0 0 cos t 0 sin t
0 0 0 0 In−2 0
0 0 0 − sin t 0 cos t
.
Then A = {at|t ∈ [0, 2π)} and 〈v0, atv0〉 = n2 cos2(t)−n and it follows that φ(at) = (n cos2(t)−
1)/(n− 1).
Now we calculate the function Ψω1
0 by means of the algorithm. Upon identifying V G
$1
∼= C2n
we have vεi = ei and v−εi = en+i.
Step 1. We have Vω1 |M = C2 ⊕ C2n−4, with C2 ∼= Cvε1 ⊕ Cv−ε1 and C2n−4 the standard
representation of USp(2n− 4). Hence P+
M (ω1) = {η1, η2}.
Step 2. The degree is zero, so we consider B(ω1) = {$1, $3}
Steps 4–10. We calculate Ψω1
$1
(at). The weight vectors of V G
$1
are {v±εi , 1 ≤ i ≤ n}. The
vector space V K
ω1
is spanned by the weight vectors v±εi , i = 1, . . . , n − 1 and the embedding
V K
ω1
→ V G
$1
is clear. We choose v$1,ω1,η1 = e1 and v$1,ω1,η2 = e2. It follows that Ψω1
$1
(at) =
(cos(t), 1)T .
20 M. van Pruijssen and P. Román
Steps 4–10: we calculate Ψω1
$3
(at). Let V = V G
$1
as above. We realize V G
$3
in the kernel of
the map ϕ3 :
∧3 V → V given by
vk1 ∧ vk2 ∧ vk3 7→
∑
i<j
Q(vi, vj)(−1)i+j−1 · · · ∧ v̂i ∧ · · · ∧ v̂j ∧ · · · .
In
∧3 V , the weight vectors of weight ±εi ± εj ± εk are v±εi ∧ v±εj ∧ v±εk . There are 8
(
n
3
)
of
them. The weight vectors of weight ±εi are vεj ∧ v−εj ∧ v±εi . There are 2n(n− 1) of them. The
kernel of this map is of dimension
(
2n
3
)
− 2n = 8
(
n
3
)
+ 2(n− 2)n, the multiplicities of the short
roots are n− 2 and there are 2n of them. It follows that the restricted map
∧3 V (±εi)→ V has
an (n− 2)-dimensional kernel. Since
ϕ3(vεj ∧ v−εj ∧ v±εi) = (−1)n−1Q(vεj , v−εj )v±εi ,
we have
ker
(
ϕ3
∣∣∧3 V (εi)
)
=
∑
j 6=i
ajvεj ∧ v−εj ∧ vεi
∣∣∣∣∣∑
j 6=i
aj = 0
.
In order to calculate the embedding V K
ω1
into V G
$3
we need to calculate⋂
α∈R+
K
ker
(
eα
∣∣
ker(ϕ3|∧3 V (ε1)
)
)
.
It is sufficient to calculate the kernels of the root vectors for the simple roots. Note that X1,2
already acts as zero. For the others we have
Xk,k+1
n∑
j=2
ajvεj ∧ v−εj ∧ vε1
= (ak − ak+1)vεk ∧ v−εk+1
∧ vε1
for k = 2, . . . , n− 2 and
Uk
n∑
j=2
ajvεj ∧ v−εj ∧ vε1
= 0
for k = n− 1, n. Hence ak = ak+1 for k = 2, . . . , n− 2 and
⋂
α∈R+
K
ker
(
eα
∣∣
ker(ϕ3|∧3 V (ε1)
)
)
= C
n−1∑
j=2
vεj ∧ v−εj ∧ vε1 − (n− 2)(vεn ∧ v−εn ∧ vε1)
.
It follows that V K
ω1
→ V G
$3
is determined by
vε1 7→ w :=
n−1∑
j=2
vεj ∧ v−εj ∧ vε1 − (n− 2)(vεn ∧ v−εn ∧ vε1).
The highest weight vector of the M -type η2 in V K
ω1
is X2,1vε1 which maps to
X2,1w = X2,1
n−1∑
j=2
vεj ∧ v−εj ∧ vε1 − (n− 2)(vεn ∧ v−εn ∧ vε1)
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 21
= vε1 ∧ v−ε1 ∧ vε2 +
n−1∑
j=3
vεj ∧ v−εj ∧ vε2 − (n− 2)(vεn ∧ v−εn ∧ vε2) ∈ V$3 .
Hence (atw,w) = (n−2) cos(t)+(n−2)2 cos(t) and (atX2,1w,X2,1w) = (n−1)2 cos2(t)−(n−1),
so that Ψω1
$3
(at) = (cos(t), ((n− 1) cos2(t)− 1)/(n− 2))T . We find
Ψω1
0 (at) =
(
cos(t) cos(t)
1 ((n− 1) cos2(t)− 1)/(n− 2)
)
.
We resume this discussion in the following statements.
Proposition 5.3. In the variable x = cos2(t), we have
Ψ̃ω1
0 (x) =
√x √
x
1
x(n− 1)− 1
n− 2
.
Theorem 5.4. The multiplicity free triple (USp(2n),USp(2n − 2) × USp(2), ω1) gives rise to
the MVCPs (Wω1
n , Dω1
n ) given by
Wω1
n (x) = x(1− x)2n−3
2x+ 2n− 4 2xn− 2
2xn− 2 2
((n− 1)2x2 − nx+ 1)
n− 2
,
D̃ω1 = x(x− 1)∂2
x +
(
− 2− 2S̃ + 2x(n− R̃)
)
∂x + Λ0/2,
with
R̃ =
−1
2
1
2
0 −1
, S̃ =
1
2(n− 1)
(
1 1
(n− 2) (n− 2)
)
, Λ0 =
(
0 0
0 4(n− 1)
)
.
Proof. We follow the steps in Section 4.3. From the explicit expression of Ψ̃ω1
0 (x) in Proposi-
tion 5.3, we compute x(1− x)(Ψ̃µ
0 (x))−1∂xΨ̃µ
0 (x) which gives the matrices R̃, S̃. We determine
the expression for the eigenvalue Λ0 by calculating how the Casimir operator acts on the rep-
resentation spaces at hand. Remark 3.10 and Table 2 give the expression of the differential
operator D̃ω1 . Observe that Λ0 is normalized so that the (1, 1)-entry is zero by adding a multi-
ple of the identity matrix. �
Remark 5.5. Let us consider a differential operator of the form
z(1− z)F ′′(z) + (C − zU)F ′(z)− V F (z) = 0, z ∈ C, (5.5)
where C, U and V are N × N matrices and F : C → CN is a (column-)vector-valued function
which is twice differentiable. It is shown by Tirao [36] that if the eigenvalues of C are not in −N,
then the matrix-valued hypergeometric function 2H1 defined as the power series
2H1
(
U, V
C
; z
)
=
∞∑
i=0
zi
i!
[C,U, V ]i,
[C,U, V ]0 = 1, [C,U, V ]i+1 = (C + i)−1
(
i2 + i(U − 1) + V
)
[C,U, V ]i
converges for |z| < 1 in Md(C). Moreover, for F0 ∈ CN the (column-)vector-valued function
F (z) = 2H1
(
U, V
C
; z
)
F0 (5.6)
is a solution to (5.5) which is analytic for |z| < 1, and any analytic (on |z| < 1) solution to (5.5)
is of this form.
22 M. van Pruijssen and P. Román
In the following Corollary we write the monic MVOPs with respect to Wω1 in terms of the
matrix-valued hypergeometric functions 2H1.
Corollary 5.6. The unique sequence of monic MVOP {Pd}d≥0 with respect to W$1 is given by
(
Pd(x)M−1
)
i,j
=
(
2H1
(
2(n− R̃),Λ0/2− λd(i)
2S̃ + 2
;x
)
Fd(i)
)
j
, M =
1
d
d+ 2n− 2
0 1
,
where λd(1) = d(d+ 2n), λd(2) = d(d+ 2n+ 1) + 2n− 2 and
Fd(i) = d!
[
2S̃ + 2, 2(n− R̃),Λ0/2− λd(i)
]−1
d
ei.
Here ei is the standard basis vector.
Proof. Since the eigenvalues of 2 + 2S̃ are not in −N, we can apply Remark 5.5. By looking at
the leading coefficient, it is easy to see that the eigenvalue of D̃ω1 for Pd is given by the upper
triangular matrix Λ̃d = d(d − 1) + d2(n − R̃) + Λ0/2. It is readily seen that the polynomial
Pd(x)M−1, d ∈ N0, is an eigenfunction of D̃ω1 with eigenvalue
Λ̃Md = M Λ̃dM
−1 =
(
λd(1) 0
0 λd(2)
)
=
(
d(d+ 2n) 0
0 d(d+ 2n+ 1) + 2n− 2
)
.
Therefore the i-th column of PdM
−1, i = 1, 2, is a polynomial solution of the equation
x(1− x)∂2
x +
(
2 + 2S̃ − 2x(n− R̃)
)
∂x − (Λ0/2− λd(i)) = 0. (5.7)
It can be easily verified that λd(i) = λd′(i
′) if and only if d = d′ and i = i′, see for instance [29,
Lemma 2.2]. This implies that there is a unique (up to a scalar multiple) polynomial solution
of (5.7), see also [22, Theorem 4.5] and the discussion above. The explicit expression for the
columns of Pd(x)M−1 follows from (5.6). �
5.3 The second family of 2 × 2 MVOP
In this subsection we use our implementation in GAP [34] to obtain a one-parameter sequence of
MVOPs. We conjecture that it is the one associated to the spherical functions of type µn = $n−1
for the pair (G,K) = (USp(2n),USp(2n − 2) × USp(2)). As we were not able to calculate Ψµ
0
by hand, we computed Ψ̃µn
0 for small values of n in GAP and made an ansatz for its general
expression.
Conjecture 5.7. For all n ≥ 3, we have
Ψ̃µn
0 (x) =
x+ 1
2
(n+ 1)x− 2
n− 1
√
x
√
x((n+ 3)x+ n− 5)
2(n− 1)
.
The matrix Ψ̃µn
0 is the building block of the weight matrix Wµn . If this function is indeed
the desired function Ψµ
0 that comes from the representation theory, then the construction to
make a MVCP out of it should work. To collect the matrices that appear as coefficients in the
differential operator, we compute explicitly the first-order differential equation.
Lemma 5.8. The matrix Ψ̃µn
0 satisfies (3.6) with
S̃ =
4
n+ 3
2n− 10
(n− 1)(n+ 3)
n− 1
n+ 3
n− 5
2(n+ 3)
, R̃ =
−1
(n+ 1)
(n− 1)
0 −3
2
.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 23
Proof. The lemma follows by computing explicitly
x(1− x)
(
Ψ̃µn
0 (x)
)−1 d
dx
Ψ̃µn
0 (x).
Then S̃ is minus the coefficient of degree 0 in x and R̃ is the coefficient of degree 1 in x. �
Theorem 5.9. Let Wµn and D̃µn be defined by
Wµn(x) = (1− x)2n−3xΨ̃µ
0 (x)∗TµnΨ̃µ
0 (x), Tµn =
2
n+ 1
0
0
(2n− 2)
(n− 1)(n− 2)
.
D̃µn = x(1− x)∂2
x +
[
2 + 2S̃ − 2x(n− R̃)
]
∂x − Λ0/2, Λ0 =
(
0 0
0 2n+ 6
)
,
with R̃ and S̃ given in Lemma 5.8. Then (Wµn , D̃µn) is a MVCP.
Proof. The proof is analogous to that of Theorem (5.4). The matrix Tµ, or rather a multiple
of it, is computed by means of Weyl’s dimension formula. Finally we use the explicit expressions
of (Wµn , D̃µn) to verify the symmetry equations (4.2). �
Remark 5.10. The MVCP that we obtained in Theorem 5.9 is only by conjecture the MVCP as-
sociated to the indicated multiplicity free triple. The statement is somewhat misleading because
we persisted in using the same notation for the weight and the differential operator. However,
the fact that the construction works for the conjectured expression of Ψµ
0 , convinces us that it is
the right one. Indeed, computer experiments show that it is unlikely that the wrong expression
of Ψµ
0 , yield a MVCP after all.
Remark 5.11. Observe that, as in the previous example, the eigenvalues of S̃ are 0 and 1/2.
This implies that the eigenvalues of 2 + 2S̃ are 2 and 3 so that the polynomial eigenfunctions of
the differential operator in Theorem 5.9 can be written in terms of matrix valued hypergeometric
functions [36].
5.4 A family of 3 × 3 MVOP
In this subsection we fix the Gelfand pair (G,K) = (USp(6),USp(4) × USp(2)) and we study
the one-parameter family of K-types µj = $1 + j$3. For small values of j ∈ N we can use our
implementation in GAP to compute Ψ̃
µj
0 . We use this information to make the following ansatz
of Ψ̃
µj
0 for all j: For all j ∈ N, we have
Ψ̃
µj
0 =
x
(j−1)
2 ((j + 1)x− 1)
j
x
(j−1)
2
x
(j−1)
2 (1 + 4x)
5
x
(j−1)
2 x
(j−1)
2 (−j + x(j + 1)) x
(j−1)
2
(−1 + x(j + 5))
5
x
j
2 x
j
2 x
j
2
(−(2j + 5) + 2x(j + 5))
5
.
24 M. van Pruijssen and P. Román
As in the previous example, we obtain the matrices R̃ and S̃ in the differential equation (3.7)
for Ψ̃
µj
0
S̃ =
−(j3 + 7j2 + 7j − 5)
2(j + 1)(j + 5)
− j(j + 4)
(j + 1)(j + 5)
− j(3j + 10)
5(j + 1)(j + 5)
− 3
2(j + 2)
−(2j2 + 1)
4(j + 1)
(2j − 5)
20(j + 1)
− 5
2(2j + 10)
− 5
2(2j + 10)
−(2j2 + 10j + 5)
4(j + 5)
,
R̃ =
−(j + 1)
2
0
2j
5(j + 1)
0 −(j + 1)
2
(j + 5)
10(j + 1)
0 0 −(j + 2)
2
Now we proceed as in Section 4.3 to construct the MVCP associated to µj .
Theorem 5.12. Let Wµn and Dµn be defined by
Wµn = (1− x)3xΨ̃
µj
0 (x)∗Tµj Ψ̃
µj
0 (x),
D̃µ = x(1− x)∂2
x +
[
2 + 2S̃ − 2x(n− R̃)
]
∂x −
Λ0
2
, (5.8)
where
Tµj =
i 0 0
0 i+ 2 0
0 0 2i+ 2
, Λ0 =
0 0 0
0 i+ 1 0
0 0 i+ 5
.
Then (Wµn , D̃µn) is a MVCP.
Remark 5.13. The eigenvalues of S̃ are (i+1)/2, i/2, (i−1)/2 so that 2+ S̃ has always positive
eigenvalues. This ensures that the hypergeometric operator (5.8) fits into the theory developed
in [36]. However, further analysis is required to find an expression of the orthogonal polynomial
with respect to Wµn as matrix hypergeometric functions.
6 Classif ication of all (2 × 2)-cases
In this section we give the other MVCPs of size 2 × 2 from Table 1. First we classify all the
possible 2 × 2 cases, then we indicate the general approach of how we calculate the matrix
coefficients and then we give explicit expressions for the MVCPs. The fundamental weights
of G are denoted by $i, those of K by ωi and those of M by ηi. We employ the standard
definitions and conventions for weights according to [20]. The embeddings K → G are those as
in [19].
Theorem 6.1. Let (G,K,F ) be a multiplicity free system of Table 1 and let µ ∈ F . Then
(Wµ, Dµ) is of size 2× 2 if and only if (G,K, µ) is one of the following:
a) (G,K) = (SU(n+ 1),U(n)) and µ = $i +m$n with i ∈ {1, . . . , n− 1} and m ∈ Z;
b) (G,K) = (SO(2n+ 1), SO(2n)) and µ = ωi with i ∈ {1, . . . , n− 2};
d) (G,K) = (SO(2n),SO(2n− 1)) and µ = ωi with i ∈ {1, . . . , n− 1};
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 25
c) (G,K) = (USp(2n),USp(2n− 2)×USp(2)) and µ ∈ {ω1, ωn−1};
g) (G,K) = (G2,SU(3)) and µ = ω1 or µ = ω2;
f) (G,K) = (F4, Spin(9)) and µ = ω1.
Proof. We have to prove that the indicated irreducible K-representations decompose into two
irreducible M -representations upon restriction to M . In all cases apart from Case f this follows
from the branching rules K ↓M described in [4] (Cases a, b and d), Section 5 (Case c) and [19]
(Case g). To exclude all the other cases for the pairs (G,K) in Cases a, b and d we refer to
the same branching rules. These are all given by interlacing conditions and the statement is
easily checked. Roughly speaking, the K-weight is given by a string of ordered integers and
as soon as we make more than one jump, or a jump larger than one, we will find more than
two M -weights that interlace. Case c is dealt with in Section 5. For Case g we have to check
the branching SU(3) ↓ SU(2) which can also be done in stages, SU(3) ↓ U(2) ↓ SU(2). On the
first step, branching Case a applies.
For Case f we use the branching rules Spin(9) ↓ Spin(7) as described in [1, 19] to see that
|P+
M (aω1 + bω2)| =
a∑
c=0
b∑
d=0
d∑
e=0
1, |P+
M (aω3)| =
a∑
b=0
b∑
c=0
a−b∑
d=0
1 and |P+
M (aω4)| = 2a+ 2 if a > 0 (and
one if a = 0). It follows that µ = ω1 yields the only (2× 2)-case for the pair (F4,Spin(9)).
To exclude the pair (Spin(7),G2), we use the branching rule for G2 ↓ SU(3). The SU(3)-types
that occur are parametrized by the integral points of a triangle with integral vertices, i.e. there
are no 2× 2 cases. �
In order to calculate the MVCPs we have to determine the function Ψ̃µ
0 and the fundamental
spherical function φ. The latter is already used in the proof of Lemma 3.2. To calculate Ψ̃µ
0 we
introduce the following notation. Let B(µ) = {λ, λ′} and let P+
M (µ) = {ν, ν ′}, where λ = λ(0, ν)
and λ′ = λ(0, ν ′) in the notation of Theorem 2.2. Let VM
ν , V K
µ , V G
λ denote the representations
spaces and consider the following commutative diagram
V G
λ
VM
ν ⊕ VM
ν′
i //
f
::
f ′ $$
V K
µ
γ
OO
γ′
��
V G
λ′
(6.1)
The maps f , f ′ are M -equivariant, i is an M -isomorphism and γ, γ′ are the embeddings
of the K-representation spaces and are thus K-equivariant. Let wν , wν′ denote the highest
weight vectors of the M -representations. We need to calculate the four vectors x1,1 = f(wν),
x2,1 = f(wν′), x1,2 = f ′(wν) and x2,2 = f ′(wν′) and subsequently the inner products 〈xi,j , atxi,j〉.
Together with information from Table 2 we calculate the data (Ψ̃µ
0 , φ, T
µ, λ1,Λ0). Then we follow
the steps in Section 4.3 to obtain explicit expressions for the corresponding pair (Wµ, D̃µ).
In the remainder of this section we provide expressions for almost all the functions Ψ̃µ
0 of size
2 × 2 that can be obtained from Table 1 using our method (note that Case c is dealt with in
Section 5). See Remark 1.3 for further comments on the parameters.
Case a. (G,K) = (SU(n+ 1),U(n)). In this case we need to distinguish between two cases.
The first case has been already investigated in a series of papers ending with [30].
Case a1. We have a two-parameter family of of classical pairs corresponding to the K-types
µ = $i + m$n, where i ∈ {1, . . . , n − 1} and m ∈ Z≥0. In this case the bottom of the µ-well
26 M. van Pruijssen and P. Román
is given by B(µ) = {µ, µ + εi+1 − εn+1}. The function Ψ̃µ
0 can be obtained by applying our
algorithm or from [30, Section 5.2]
Ψ̃µ
0 (x) = x
m
2
√x √
x
1
(m+ 1)− x(m+ n− i+ 1)
i− n
.
Case a2. This case was not considered in the literature before as far as we know. We take
the K-types µ = $i +m$n but now we let m ∈ Z<0. Note that although all ingredients needed
for this case are already given in [30], the weight matrix constructed there does not have finite
moments for negative values of m, thus it is excluded [30, Section 6]. Our case is essentially
a conjugation of the case considered in [30] and does not have any problem of integrability. The
function Ψµ
0 is given by
Ψ̃µ
0 (x) = x−
(m+1)
2
(m+ (i−m)x))
i
1
x
1
2 x
1
2
.
Cases b and d. We only treat (G,K) = (SO(2n+1),SO(2n)) and µ = ωi with i ∈ {1, . . . , n−2}.
This case is essentially the same as the one in Section 5 but simpler. We use the notation
of [11, Chapter 19]. The K representation is of highest weight ωi for indicated i is realized in∧iC2n and this space decomposes as
∧i−1 C2n−1 ⊕
∧iC2n−1 as M -representation, where M ∼=
SO(2n − 1). It follows that P+
M (ωi) = {ηi−1, ηi}. On the other hand, B(ωi) = {$i, $i+1} and
λ(0, ηi) = $i+1. Everything is now explicit and the maps f , f ′ of the commutative diagram (6.1)
are easily calculated on highest weight vectors. We have A = {at = exp(tHA) : t ∈ R}, where
HA = X1,n −Xn,1. We find
Ψωi
0 (at) =
(
cos(t) 1
1 cos(t)
)
.
Case g1. (G,K) = (G2,SU(3)) and µ = ω1. Let µ = ω1. We have M ∼= SU(2). Following [19]
we find P+
M (µ) = {0, η1} and B(µ) = {$1, $2}, with λ(0, 0) = $1 and λ(0, η1) = $2. Fol-
lowing [33, Paragraph 2.2.7] we see that the Lie algebra of A is generated by E$1 − E−$1 . It
follows, as in all cases, that the calculations of the restricted matrix coefficients in X$1 and X$2
are really SU(2)-calculations. From the weight diagrams (see, e.g., [11, Chapter 22]) we read:
Ψωi
0 (at) =
(
m1
1,1(b2t) m1
1,1(b2t)
m
1/2
1/2,1/2(b2t) m
3/2
1/2,1/2(b2t)
)
=
(
cos2(t) cos2(t)
cos(t) cos3(t)− 2 cos(t) sin2(t)
)
,
where the m`
m,n denote the matrix coefficients of SU(2) and where b2t is the rotation over an
angle t, see [23].
The Cases g2 and f are left: Case g2 is similar to Case g1, and Case f is too complicated to
do by hand. The involved F4 representations are of dimension 52 and 1274. We will handle it
as soon as our GAP implementation is improved to deal with general cases.
Acknowledgements
The research for this paper was partly conducted when the first author visited the University of
Córdoba in August and September 2012. We would like to thank the Mathematics Departments
of the Universities of Nijmegen and Córdoba for their generous supports that made this visit
possible. Finally we would like to thank to the anonymous referees, whose comments and
suggestions have helped us to improve the paper.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One 27
References
[1] Baldoni Silva M.W., Branching theorems for semisimple Lie groups of real rank one, Rend. Sem. Mat. Univ.
Padova 61 (1979), 229–250.
[2] Berezans’kĭı J.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Mono-
graphs, Vol. 17, Amer. Math. Soc., Providence, R.I., 1968.
[3] Bochner S., Über Sturm–Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730–736.
[4] Camporesi R., A generalization of the Cartan–Helgason theorem for Riemannian symmetric spaces of rank
one, Pacific J. Math. 222 (2005), 1–27.
[5] Casselman W., Miličić D., Asymptotic behavior of matrix coefficients of admissible representations, Duke
Math. J. 49 (1982), 869–930.
[6] Damanik D., Pushnitski A., Simon B., The analytic theory of matrix orthogonal polynomials, Surv. Approx.
Theory 4 (2008), 1–85, arXiv:0711.2703.
[7] Dixmier J., Algèbres envellopantes, Éditions Jacques Gabay, Paris, 1996.
[8] Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys.
103 (1986), 177–240.
[9] Durán A.J., Matrix inner product having a matrix symmetric second order differential operator, Rocky
Mountain J. Math. 27 (1997), 585–600.
[10] Durán A.J., Grünbaum F.A., Orthogonal matrix polynomials satisfying second-order differential equations,
Int. Math. Res. Not. 2004 (2004), no. 10, 461–484.
[11] Fulton W., Harris J., Representation theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag,
New York, 1991.
[12] GAP – Groups, Algorithms, and Programming, Ver. 4.6.4, 2013, available at http://www.gap-system.org.
[13] Geronimo J.S., Scattering theory and matrix orthogonal polynomials on the real line, Circuits Systems
Signal Process. 1 (1982), 471–495.
[14] Groenevelt W., Ismail M.E.H., Koelink E., Spectral decomposition and matrix-valued orthogonal polyno-
mials, Adv. Math. 244 (2013), 91–105, arXiv:1206.4785.
[15] Groenevelt W., Koelink E., A hypergeometric function transform and matrix-valued orthogonal polynomials,
Constr. Approx. 38 (2013), 277–309, arXiv:1210.3958.
[16] Grünbaum F.A., Pacharoni I., Tirao J., Matrix valued spherical functions associated to the complex pro-
jective plane, J. Funct. Anal. 188 (2002), 350–441.
[17] Grünbaum F.A., Tirao J., The algebra of differential operators associated to a weight matrix, Integral
Equations Operator Theory 58 (2007), 449–475.
[18] He X., Ochiai H., Nishiyama K., Oshima Y., On orbits in double flag varieties for symmetric pairs, Trans-
form. Groups 18 (2013), 1091–1136, arXiv:1208.2084.
[19] Heckman G., van Pruijssen M., Matrix valued orthogonal polynomials for Gelfand pairs of rank one,
arXiv:1310.5134.
[20] Knapp A.W., Lie groups beyond an introduction, Progress in Mathematics, Vol. 140, 2nd ed., Birkhäuser
Boston, Inc., Boston, MA, 2002.
[21] Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to (SU(2) ×
SU(2), diag), Int. Math. Res. Not. 2012 (2012), 5673–5730, arXiv:1012.2719.
[22] Koelink E., van Pruijssen M., Román P., Matrix-valued orthogonal polynomials related to (SU(2) ×
SU(2), diag), II, Publ. Res. Inst. Math. Sci. 49 (2013), 271–312, arXiv:1203.0041.
[23] Koornwinder T.H., Matrix elements of irreducible representations of SU(2) × SU(2) and vector-valued or-
thogonal polynomials, SIAM J. Math. Anal. 16 (1985), 602–613.
[24] Krein M.G., Infinite J-matrices and a matrix-moment problem, Doklady Akad. Nauk SSSR 69 (1949),
125–128.
[25] Krein M.G., Fundamental aspects of the representation theory of Hermitian operators with deficiency index
(m,m), Amer. Math. Soc. Translations Ser. 2 97 (1971), 75–143.
[26] Lepowsky J., Multiplicity formulas for certain semisimple Lie groups, Bull. Amer. Math. Soc. 77 (1971),
601–605.
http://dx.doi.org/10.1007/BF01180560
http://dx.doi.org/10.2140/pjm.2005.222.1
http://dx.doi.org/10.1215/S0012-7094-82-04943-2
http://dx.doi.org/10.1215/S0012-7094-82-04943-2
http://arxiv.org/abs/0711.2703
http://dx.doi.org/10.1007/BF01206937
http://dx.doi.org/10.1216/rmjm/1181071926
http://dx.doi.org/10.1216/rmjm/1181071926
http://dx.doi.org/10.1155/S1073792804132583
http://dx.doi.org/10.1007/978-1-4612-0979-9
http://www.gap-system.org
http://dx.doi.org/10.1007/BF01599024
http://dx.doi.org/10.1007/BF01599024
http://dx.doi.org/10.1016/j.aim.2013.04.025
http://arxiv.org/abs/1206.4785
http://dx.doi.org/10.1007/s00365-013-9207-1
http://arxiv.org/abs/1210.3958
http://dx.doi.org/10.1006/jfan.2001.3840
http://dx.doi.org/10.1007/s00020-007-1517-x
http://dx.doi.org/10.1007/s00020-007-1517-x
http://dx.doi.org/10.1007/s00031-013-9243-8
http://dx.doi.org/10.1007/s00031-013-9243-8
http://arxiv.org/abs/1208.2084
http://arxiv.org/abs/1310.5134
http://dx.doi.org/10.1093/imrn/rnr236
http://arxiv.org/abs/1012.2719
http://dx.doi.org/10.4171/PRIMS/106
http://arxiv.org/abs/1203.0041
http://dx.doi.org/10.1137/0516044
http://dx.doi.org/10.1090/S0002-9904-1971-12767-2
28 M. van Pruijssen and P. Román
[27] Lepowsky J., Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176
(1973), 1–44.
[28] Miranian L., On classical orthogonal polynomials and differential operators, J. Phys. A: Math. Gen. 38
(2005), 6379–6383.
[29] Pacharoni I., Román P., A sequence of matrix valued orthogonal polynomials associated to spherical func-
tions, Constr. Approx. 28 (2008), 127–147, math.RT/0702494.
[30] Pacharoni I., Tirao J., One-step spherical functions of the pair (SU(n+ 1),U(n)), in Lie Groups: Structure,
Actions, and Representations, Progr. Math., Vol. 306, Birkhäuser/Springer, New York, 2013, 309–354,
arXiv:1209.4500.
[31] Pacharoni I., Tirao J., Zurrián I., Spherical functions associated to the three dimensional sphere, Ann. Mat.
Pura Appl. 146 (2014), 1727–1778, arXiv:1203.4275.
[32] Pacharoni I., Zurrián I., Matrix ultraspherical polynomials: the 2 × 2 fundamental cases, arXiv:1309.6902.
[33] van Pruijssen M., Matrix valued orthogonal polynomials related to compact Gel’fand pairs of rank one, Ph.D.
Thesis, Radboud University Nijmegen, 2012, available at http://repository.ubn.ru.nl/dspace31xmlui/
handle/2066/100840.
[34] van Pruijssen M., Román P., GAP codes SP.g and SP2.g, available at http://www.mvanpruijssen.nl.
[35] Shapovalov N.N., A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie
algebra, Funct. Anal. Appl. 6 (1972), 307–312.
[36] Tirao J.A., The matrix-valued hypergeometric equation, Proc. Natl. Acad. Sci. USA 100 (2003), 8138–8141.
[37] Tirao J.A., Zurrián I.N., Spherical functions of fundamental K-types associated with the n-dimensional
sphere, SIGMA 10 (2014), 071, 41 pages, arXiv:1312.0909.
[38] Vretare L., Elementary spherical functions on symmetric spaces, Math. Scand. 39 (1976), 343–358.
[39] Wang H.C., Two-point homogeneous spaces, Ann. of Math. 55 (1952), 177–191.
[40] Warner G., Harmonic analysis on semi-simple Lie groups. II, Die Grundlehren der mathematischen Wis-
senschaften, Vol. 189, Springer-Verlag, New York – Heidelberg, 1972.
http://dx.doi.org/10.1090/S0002-9947-1973-0346093-X
http://dx.doi.org/10.1088/0305-4470/38/28/010
http://dx.doi.org/10.1007/s00365-007-0673-1
http://arxiv.org/abs/math.RT/0702494
http://dx.doi.org/10.1007/978-1-4614-7193-6_14
http://arxiv.org/abs/1209.4500
http://dx.doi.org/10.1007/s10231-013-0354-6
http://dx.doi.org/10.1007/s10231-013-0354-6
http://arxiv.org/abs/1203.4275
http://arxiv.org/abs/1309.6902
http://repository.ubn.ru.nl/dspace31xmlui/handle/2066/100840
http://repository.ubn.ru.nl/dspace31xmlui/handle/2066/100840
http://www.mvanpruijssen.nl
http://dx.doi.org/10.1073/pnas.1337650100
http://dx.doi.org/10.1073/pnas.1337650100
http://dx.doi.org/10.3842/SIGMA.2014.071
http://arxiv.org/abs/1312.0909
http://dx.doi.org/10.2307/1969427
1 Introduction
2 Lie theoretical background
3 Differential properties
4 A method to calculate MVCPs
4.1 The algorithm
4.2 Implementation in GAP
4.3 Obtaining the MVCPs
5 The symplectic case
5.1 Branching rule from K to M
5.2 The algorithm for a (22)-case
5.3 The second family of 22 MVOP
5.4 A family of 33 MVOP
6 Classification of all (22)-cases
References
|
| id | nasplib_isofts_kiev_ua-123456789-146404 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:39:33Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Maarten van Pruijssen Román, P. 2019-02-09T11:22:55Z 2019-02-09T11:22:55Z 2014 Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E46; 33C47 DOI: http://dx.doi.org/10.3842/SIGMA.2014.113 https://nasplib.isofts.kiev.ua/handle/123456789/146404 We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases. The research for this paper was partly conducted when the first author visited the University of Cordoba in August and September 2012. We would like to thank the Mathematics Departments of the Universities of Nijmegen and Cordoba for their generous supports that made this visit possible. Finally we would like to thank to the anonymous referees, whose comments and suggestions have helped us to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Article published earlier |
| spellingShingle | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One Maarten van Pruijssen Román, P. |
| title | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_fullStr | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_full_unstemmed | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_short | Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One |
| title_sort | matrix valued classical pairs related to compact gelfand pairs of rank one |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146404 |
| work_keys_str_mv | AT maartenvanpruijssen matrixvaluedclassicalpairsrelatedtocompactgelfandpairsofrankone AT romanp matrixvaluedclassicalpairsrelatedtocompactgelfandpairsofrankone |