Spherical Functions on Spheres of Rank Two

SPHERICAL FUNCTIONS ON SPHERES OF RANK TWO

MAARTEN VAN PRUIJSSEN

Abstract. Under a suitable multiplicity freeness assumption we show that the spherical functions of the pair (SU(n + 1), SU(n)) can be described by families of vector-valued polynomials in two variables. The matrix weight that establishes the orthogonality is supported on the closed unit disc and depends on n. We calculate the matrix weight in a couple of examples and we show that in these examples the weight depends actually on n ∈ (1, ∞).

Contents 1. Introduction 1 2. Invariant functions 3 3. Multiplicity Free Induction 8 4. Vector-valued orthogonal polynomials 11 5. Examples 12 References 15

1. Introduction Consider the Gelfand pair (U, L) = (SU(n + 1), SU(n)) with n ≥ 2 and an irreducible representation πL : L → GL(V ) with the following property: for every irreducible U- U 0 U L 0 representation π : U → GL(V ) the restriction π |L contains π at most once, i.e. dim HomL(V,V ) ≤ 1. We are interested in the space of functions F : U → End(V ) such that

L L (1.1) F (`1u`2) = π (`1)F (u)π (`2) for all (`1, u, `2) ∈ L × U × L.

For example, if πL is the trivial representation, then the space of algebraic functions on U satisfying (1.1) can be given the structure of a polynomial algebra generated by two elements. The building blocks of this space are the zonal spherical functions, i.e. certain matrix coeﬃcients of irreducible U-representations that contain an L-invariant vector. These functions form a vector space basis which is moreover orthogonal with respect to integration over U against the Haar measure (Schur orthogonality). Because the double cosets L\U/L can be parametrized by the closed unit disc {(x, y) ∈ R2|x2 + y2 ≤ 1} one obtains families

Date: June 6, 2017. 1 of orthogonal polynomials in two variables. The orthogonality measure is given by (1 − x2 − y2)n−1dxdy.

1.1. Results. In this note we show that under an additional assumption on the irreducible representation πL, the space of functions F : U → End(V ) that satisfy (1.1) can be described as a space of vector-valued orthogonal polynomials in two variables. The additional condition amounts to requiring the existence of a face F of the Weyl chamber of L such that the highest weight of πL is in F and the multiplicity freeness condition holds for all irreducible representations of L whose highest weight is contained in F. Every face F of the Weyl chamber of L of codimension one gives an example for this particular pair (U, L). The vector-valued polynomials that describe the spherical functions are orthogonal with respect to a matrix weight that is supported on the unit disc. We explain how to calculate the matrix weight in general. Because explicit calculations become rather involved we have used GAP [5] to obtain a couple of formulas. These formulas indicate that the matrix weight actually depends continuously on n for these examples.

1.2. Related work. If we take for πL the trivial representation we obtain families of scalar- valued polynomials that have been investigated for example in [2]. The dependence on the parameter n has also been investigated. For these questions one brings differential operators into the game. The orthogonal polynomials that one obtains are related to Heckman-Opdam theory [20]. The pair (U, L) together with a face F of the Weyl chamber of L for which the multiplicity freeness condition holds is called a multiplicity free system. These systems have been classi- fied, see e.g. [15]. In the same paper the existence of families of multivariable vector-valued polynomials has been shown. Those families of polynomials, as well as the ones in this paper, have the property that they are determined as simultaneous eigenfunctions of a commutative algebra of differential operators. This is worked out systematically for multiplicity free systems related to compact symmetric pairs in [11]. The theory of multivariable vector-valued polynomials related to matrix coefficients on compact Gelfand pairs is a generalization of the theory of zonal spherical functions on compact symmetric spaces [6, Ch.5] and of vector-valued orthogonal polynomials related to compact Gelfand pairs of rank one [7].

1.3. Future work. The existence of the vector-valued polynomials is based on an ad hoc U L analysis of the spectrum of the induced representation indL π . The spectra for the other multiplicity free systems are currently being investigated by means of the theory of spherical varieties [13]. Once the existence (in the generality of the classification [15]) is established, many questions remain. For example, do shift operators exist and are the families of polynomials associated to different multiplicity free systems related by shift operators? The 2 answers are affirmative for particular classes of examples in the one-variable matrix valued cases [10, 19] and in the multivariable scalar-valued cases (Heckman-Opdam theory).

1.4. Organization. In Section 2 we introduce the zonal spherical functions on U and we show that it is suﬃcient to study their restrictions to a two dimensional torus A ⊂ U for which LAL = U. In Section 3 we use the theory of spherical varieties to obtain the decomposition of U L indL π under the appropriate multiplicity freeness assumption. We show that the spherical functions can be described by multivariable vector-valued orthogonal polynomials. In Section 4 we construct these polynomials and the matrix weight with respect to which they are orthogonal. In Section 5 we calculate a couple of examples using the GAP-code [16].

2. Invariant functions Before we look at the invariant functions on U we establish some facts about the coset space L\U/L.

n+1 2.1. Spheres of rank two. Consider C with the standard basis (e1, . . . , en+1) equipped Pn+1 Pn+1 with the standard Hermitian inner product (v, w) = i=1 viwi, where v = i=1 viei and Pn+1 n+1 w = i=1 wiei. Let U = SU(n + 1) ⊂ GL(C ) denote the group of linear transformations with determinant one that leave the standard Hermitian form invariant, i.e. u ∈ U if and only if det(u) = 1 and for all v, w ∈ Cn+1 the equality (uv, uw) = (v, w) holds. n+1 n+1 Let L ⊂ U be the stabilizer of en+1 for the natural action U × C → C :(u, v) 7→ uv. ∼ 2n+1 Then L is isomorphic to SU(n). The image Uen+1 is isomorphic to U/L = S = {v ∈ Cn+1 :(v, v) = 1}. Deﬁne the elements     ei(n−1)s 0 0 cos(t) 0 − sin(t)  −2is    a1(s) =  0 e In−1 0  , a2(t) =  0 In−1 0  0 0 ei(n−1)s sin(t) 0 cos(t) in U. Furthermore, deﬁne ( ) s ∈ [0, π) if n − 1 even

(2.1) A1 = a1(s) s ∈ [0, 2π) if n − 1 odd

(2.2) A2 = {a2(t)|t ∈ [0, 2π)}.

which are one-dimensional tori contained in U. Note that the two dimensional torus A =

A1 · A2 is isomorphic to (A1 × A2)/(A1 ∩ A2). Since A1 ∩ A2 = {1} if n − 1 is even and

A1 ∩ A2 = {1, a2(π)} if n − 1 is odd, we have

A = {a1(s)a2(t)|(s, t) ∈ [0, π) × [0, 2π)}.

Lemma 2.1. The map L × A × L → U :(`, a, `0) 7→ `a`0 is surjective. 3 2n+1 Proof. First we show that the image of A under the orbit map U → S : u 7→ uen+1 2n+1 0 parametrizes the L-orbits. Indeed, let v = v1e1 + ··· + vn+1en+1 ∈ S and write v1 = p 2 2 0 |v1| + ... + |vn| . Use the L-action to rotate v to v1e1 + vn+1en+1. There is a unique i(n−1)s 0 s ∈ [0, 2π/(n−1)) such that vn+1 = e |vn+1|. Use L once more to rotate v1e1 +vn+1en+1 i(n−1)s 0 0 to e (−|v1|e1 + |vn+1|en+1). There is a unique t ∈ [0, π/2] such that |v1| = sin(t) and |vn+1| = cos(t). Hence we can use L to rotate v to a1(s)a2(t)en+1. It follows that L × A → U/L is surjective from which we deduce that L × A × L → U is surjective. In view of Lemma 2.1 we call U/L a sphere of rank two. Note that the map A → AL/L is generically 2(n − 1)-to-one if n − 1 is odd and n − 1-to-one if n − 1 is even. We shall make this more precise in the next subsection

2.2. Zonal Spherical Functions. We identify irreducible representations of compact Lie groups with their highest weights according to the Theorem of the Highest Weight, see e.g. [8, + Thm.5.110]. The set of dominant integral weights of U is denoted PU . We have + PU = N0$1 + ... + N0$n, where the choice and the notation of the fundamental weights is taken from [8]. This means that $i is a character of the torus of diagonal elements TU ⊂ U given by the formula

$i(diag(t1, . . . , tn+1)) = t1 ··· ti. + The elements in PU are the weights that can occur as the weight of a highest weight + vector, so PU classiﬁes all irreducible representations modulo equivalence. Given an element + U U λ ∈ PU we denote the corresponding representation by πλ : U → GL(Vλ ). We equip every U representation space Vλ with a Hermitian inner product h·, ·iλ for which the U-representation U is unitary. Note that π0 denotes the trivial U-representation. We employ the same notation for the subgroup L, although we denote the fundamental weights by ω1, . . . , ωn−1. Furthermore, we denote TL = TU ∩ L. Note that $i|TL = ωi for i = 1, . . . , n − 1 and $n|TL = 1.

+ + U L + Deﬁnition 2.2. Let P(U,L)(0) = {λ ∈ PU :[πλ |L : π0 ] = 1}. The set P(U,L)(0) is called the 0-well.

The 0-well is a semigroup that is generated by $1, $n, see e.g. [12, Table 1]. Consider the left-right-regular representation of U × U auf L2(U, du), where du is the normalized Haar measure. The space of invariants of this action restricted to L × L is given by

2 L×L Md U L×L L (U, du) = + End(Vλ ) , λ∈P(U,L)(0) U L×L where the hat indicates the Hilbert completion of the direct sum. The spaces End(Vλ ) U are all one-dimensional since (U, L) is a Gelfand pair. Let v ∈ Vλ be an L-invariant vec- U tor of length one, i.e. ||v|| = 1 and πλ (`)v = v for all ` ∈ L. The matrix coeﬃcient 4 U u 7→ hv, πλ (u)viλ is a zonal spherical function which we denote by φλ : U → C. Fur- R U 0 0 thermore, Schur-orthogonality [8, Cor.4.10] implies that U φλ(u)φλ (u)du = δλ,λ / dim(Vλ ). + 2 L×L This shows that {φλ|λ ∈ P(U,L)(0)} is an orthogonal basis of L (U, du) .

× λ1 λ2 Proposition 2.3. Let λ = λ1$1+λ2$n. Then there exists c ∈ C with φλ = cφ$1 φ$n +l.o.t., 0 0 λ1 λ2 0 0 where a monomial φ$1 φ$n is called of lower order if λ1 ≤ λ1 or λ2 ≤ λ2. Suppose that λ > 1. The tensor product V U ⊗ V U decomposes into irreducible Proof. 1 λ−$1 $1 U 0 + + U-representations Vλ0 with λ ∈ λ − Qsph, where Qsph = N0(α1 + ··· + αn) = N0($1 + $n). Here we use the theory of spherical roots, see e.g. [4, §4]. The spherical roots of the pair (SL(n + 1, C), SL(n, C)) are calculated in [3, §3(5)]. In terms of the Borel-Weil re- alizations of the representation spaces as sections of equivariant bundles, the Cartan projection V U ⊗ V U → V U is given by the multiplication of algebraic functions. Hence λ−$1 $1 λ φ φ = Pmin(λ1,λn) c φ with c 6= 0. Moreover, Pmin(λ1,λn) |c |2 = 1. λ−$1 $1 k=0 ek λ−k($1+$n) e0 k=1 ek U U The analogous statement holds for Vλ−$n ⊗ V$n if λ2 > 0 and the proof is completed by induction on λ1 + λ2.

+ Deﬁnition 2.4. Let λ ∈ P(G,L)(0). Then qλ ∈ C[z1, z2] denotes the unique polynomial for which φλ = qλ(φ$1 , φ$n ).

2 Let φ : A → C : a 7→ (φ$1 (a), φ$n (a)). One readily veriﬁes i(n−1)s −i(n−1)s φ$1 (a1(s)a2(t)) = e cos(t), φ$n (a1(s)a2(t)) = e cos(t),

2 which shows that φ : A → C maps into the real subspace spanned by g1 = e1 + e2 and 2 g2 = e1 − e2 which is just the underlying real subspace of the diagonal in C . The image φ(A) is equal to the closed unit disc

2 2 (2.3) D = {xg1 + iyg2| x + y ≤ 1}.

2n+1 2 2n+1 Let h : S → C : v 7→ hv, e1i, d : C → C : z 7→ z(e1 +e2) and let γ : A → S : a 7→ ae1. Then the diagram φ A / 2 CO γ d S2n+1 / h C commutes. We mentioned at the end of the previous subsection that A → AL/L is generically 2(n − 1)-to-one if n − 1 is odd and n − 1-to-one if n − 1 is even. This is due to the fact that L ∩ A has n − 1 and (n − 1)/2 elements depending on n − 1 odd or even. Besides, there is also an element in L that normalizes A. Indeed, the element we = diag(i, In−1, −i) conjugates a2(t) to a2(−t) and leaves A1 invariant. Conjugation with w.ae 1(3π/(4(n − 1))) ∈ L yields the same result. This conjugation generates the a group with two elements that we denote 5 by W . The actions of A ∩ L and W commute, hence AL/L ∼= A/((A ∩ L) × W ). Since the functions φ$1 and φ$2 restricted to A are invariant for the action of (A ∩ L) × W , they provide coordinates for AL/L, identifying AL/L with the closed disc. We describe the fibers of φ. • Let n − 1 be odd. The fibers of φ contain 2(n − 1) elements when 0 < x2 + y2 < 1, 2 2 −1 n − 1 elements when x + y = 1 and φ ({0}) = A1. • Let n − 1 be even. The fibers of φ contain n − 1 elements when 0 < x2 + y2 < 1, 2 2 −1 (n − 1)/2 elements when x + y = 1 and φ ({0}) = A1. It remains to simplify the integral of L-biinvariant functions over U to an integral over D. We invoke the discussion [2, §3.1] from which we deduce the equality Z n Z (2.4) f(φ(u))du = f(z, z)(1 − |z|2)n−1dλ(z, z), f ∈ C(D) U π D where the disc D from (2.3) is identified with {z ∈ C : |z| ≤ 1} via d : C → C2 and where dλ is the normalized Lebesgue-measure.

Remark 2.5. Let K ⊂ U be the image of the homomorphism U(n) → SU(n + 1) : k 7→ diag(k, det k−1). Then (U, K) is a compact Riemannian symmetric pair for which U =

KA2K. Harmonic analysis for such pairs greatly beneﬁts from this decomposition and the rest of the well developed structure theory like the existence of restricted root systems. An

important subgroup of K in this structure theory is M = ZK (A2) and we observe that 2n+1 M = A1L∗, where L∗ is the stabilizer of the L-action on S for the point e1. Note that

L∗ is isomorhic to SU(n − 1). The analysis of zonal spherical functions for a Riemannian symmetric pair like (U, K)

restricted to a torus like A1 is closely related to the analysis of hypergeometric functions associated to root systems.

For compact Gelfand pairs there is not such a rich structure theory available. However, in this particular example, which is close to a Riemannian symmetric pair, we can also parametrize the double cosets by a torus. This is the motivation for stating the following result. Let da denote the normalized Haar measure on A.

L×L R R Proposition 2.6. Let f ∈ C(U) . Then U f(u)du = c A f(a)|δn(a)|da, where

2 n−1 δn(a1(s)a2(t)) = | sin (t)| | sin(2t)|

−1 R and c = A |δn(a)|da.

Proof. Let a = Lie(A) and consider a → D : H 7→ φ(exp(H)). With respect to the

coordinates (s, t) on a for which exp(s, t) = a1(s)a2(t) the Jacobian is given by a multiple of (s, t) 7→ sin(2t). In view of (2.4) this implies the claim. 6 L 2.3. Spherical functions. Let πµ be an irreducible representation that induces multiplicity + + L U free to U, i.e. µ ∈ PL is such that for all λ ∈ PU we have dim HomL(Vµ ,Vλ ) ≤ 1. In this case we call (U, L, µ) a multiplicity free triple. + Let F be a face of the Weyl chamber of L such that for all elements µ ∈ PL ∩ F the L representation πµ induces multiplicity free to U. In that case we call the triple (U, L, F ) a multiplicity free system. Since (U, L) is a Gelfand pair, (U, L, {0}) is a multiplicity free system. In Section 3 L we investigate the induction of πµ to U in greater detail and we shall see that there are more examples of multiplicity free systems other than only (U, L, {0}). In this section we investigate the spherical functions up to the point where we need this induction. Throughout + the rest of this section we ﬁx µ ∈ PL such that (U, L, µ) is a multiplicity free triple.

+ + U L Deﬁnition 2.7. The set P(U,L)(µ) = {λ ∈ PU |[πλ |L : πµ ] = 1} is called the µ-well.

L L L −1 Consider the L × L-action on End(Vµ ) via ((`1, `2), x) 7→ πµ (`) ◦ x ◦ πµ (`2) . The inner ∗ L ∗ product (x, y) 7→ tr(x y) on End(Vµ ), where x denotes the Hermitian adjoint with respect L to the invariant Hermitian inner product on Vµ , is L × L-invariant. Together with the left- right-regular representation of L × L in L2(U, du) we obtain the L × L-representation in the tensor product 2 L L (U, du) ⊗ End(Vµ ), which is unitary for the Hermitian inner product given by Z ∗ (2.5) hΦ, Ψiµ = tr (Φ(u) Ψ(u)) du. U The space of L × L-invariants decomposes as the Hilbert completion

2 L L×L Md L U U L (2.6) (L (U, du) ⊗ End(Vµ )) = + HomL(Vµ ,Vλ ) ⊗ HomL(Vλ ,Vµ ). λ∈P(U,L)(µ)

L U U L The spaces HomL(Vµ ,Vλ ) ⊗ HomL(Vλ ,Vµ ) are one dimensional by assumption. Let j : L U U L Vµ → Vλ be an isometric L-invariant linear map and let p : Vλ → Vµ denote its adjoint. Then µ L U Φλ : U → End(Vµ ): u 7→ p ◦ πλ ◦ j µ L U is called a spherical function of type µ associated to λ. In fact, Φλ ∈ HomL(Vµ ,Vλ ) ⊗ U L HomL(Vλ ,Vµ ) in the decomposition (2.6). Schur orthogonality implies

L 2 µ µ dim(Vµ ) 0 (2.7) hΦλ, Φλ0 iµ = U δλ,λ , dim(Vλ ) µ + see e.g. [14, Prop.3.3.25]. According to the decomposition (2.6) the set {Φλ|λ ∈ P(U,L)(µ)} 2 L L×L is an orthogonal basis of (L (U, du) ⊗ End(Vµ )) . 7 2 L L×L Finally we note that in view of the transformation behavior of Ψ ∈ (L (U, du)⊗End(Vµ )) , given by L L Ψ(`1u`2) = πµ (`1)Ψ(u)πµ (`2) for all (`1, `2, u) ∈ L × L × U, ∼ it is enough to know the restrictions Ψ|A. Let L∗ = SU(n − 1) be the isotropy group 2n+1 2n+1 of e1 ∈ S for the action of L on S . Then L∗ commutes with A. We see that L 2n+1 Ψ(A) ⊂ EndL∗ (Vµ ). Let v ∈ S \{en+1}. Then the the stabilizer Lv is conjugated to L∗. This implies that the complexiﬁcation of L∗ is equal to the generic stabilizer of the action of the complexiﬁcation L of L on the annihilator l⊥ = {f ∈ u∗ | ∀X ∈ l : f(X) = 0} of l C C C C C in u∗ . The restriction πL| decomposes multiplicity free, see e.g. [15, Prop.2.4]. It follows C µ L∗ that the elements Ψ(a), a ∈ A can be diagonalized simultaneously. For later reference we record the following remark.

Remark 2.8. The integrand in (2.5) is L × L-invariant. This implies that the inner product (2.5) can be given as an integration of A, Z ∗ hΦ, Ψiµ = c Ψ(a) Φ(a)|δn(a)|da, A where we have used Proposition 2.6.

3. Multiplicity Free Induction + Pn−1 Let i ∈ {1, . . . , n − 1} and let µ ∈ PL be of the form µ = j=1 µjωj with µi = 0 and Pn write λ = j=1 λj$j. We want to describe the irreducible U-representations that contain L πµ upon restriction to L. Instead of the classical branching rules adapted to our situation we apply the theory of spherical varieties, in particular the extended weight semigroup (see e.g. [1]). To explain this object we consider the complexifications G = SL(n + 1, C) and H = SL(n, C) of the compact groups U and L respectively. The roots and choices for positivity are essentially the same for U and G and for L and H. Using Weyl’s unitary trick + + + + we employ the identifications PU = PG and PL = PH . In the same spirit we define and + + + identify P(G,H)(µ) = P(U,L)(µ) for all µ ∈ PH .

Let Pαi ⊂ H be the standard parabolic subgroup of H associated to the positive root αi.

This means that the semisimple part of Pαi is isomorphic to SL(2, C) with positive root αi. × × Pαi The character µ : TL → C can be extended to a character µ : Pαi → C . Let H × C H denote the associated line bundle over H/Pαi . The representation πµ can be realized in the Pαi space of sections of the vector bundle H × C over H/Pαi .

It is known (see [15, Thm.2.1]) that Pαi ⊂ G is a spherical subgroup, i.e. G contains

a Borel subgroup that admits an open orbit in the quotient G/Pαi . This is equivalent to

the fact that the spaces of sections of all G-equivariant line bundles over G/Pαi decompose multiplicity free into direct sums of irreducible G-representations, see [21, Thm.25.1]. Let ∗ X (Pαi ) denote the group of characters of Pαi . 8 + ∗ Deﬁnition 3.1. The extended weight semigroup Λ(b G, Pαi ) ⊂ PU × X (Pαi ) consists of the G pairs (λ, χ) such that Vλ contains a line on which Pαi acts with character −χ.

It turns out that Λ(b G, Pαi ) is a ﬁnitely generated free semigroup, see [1, Thm.2]. Using + ∗ induction in stages and Frobenius reciprocity one shows: λ ∈ P(G,H)(µ) if and only if (λ , µ) ∈ ∗ G Λ(b G, Pαi ), where λ is the highest weight of the representation dual to Vλ .

Lemma 3.2. Let i ∈ {1, . . . , n − 1}. The extended weight semigroup Λ(b G, Pαi ) is freely generated by the 2n-2 elements

∗ (3.1) ($j , ωj) with 1 ≤ j ≤ n and j 6= i, ∗ (3.2) ($j , ωj−1) with 1 ≤ j ≤ n and j 6= i + 1.

Proof. Let GL(n, C) ⊂ G denote the image of the map x 7→ diag(x, det x−1). The standard Borel subgroup BGL(n,C) ⊂ GL(n, C) consisting of upper-triangular matrices remains spherical in G. The extended weight semigroup Λ(b G, BGL(C,n)) has been calculated in [17, L.2.2]. Its generators are

∗ ∗ (3.3) ($j , $j), ($j , $j−1 − $n), j = 1, . . . , n.

An element of Λ(b G, Pαi ) can be viewed as an element of Λ(b G, BGL(C,n)) by extending the second component to the center of GL(n, C) trivially. As such it can be written as a linear combination of the elements (3.3) where the coeﬃcients are in N0. Certainly the coeﬃcients ∗ ∗ of ($i , $i) and ($i+1, $i − $n) are zero. This shows that the elements (3.1),(3.2) generate

Λ(b G, Pαi ). Since these elements are also linearly independent, the result follows. + Pn−1 Let i ∈ {1, . . . , n − 1} and let µ ∈ PL be of the form µ = j=1 µj$j with µi = 0 and Pn ∗ n n write λ = j=1 λj$j. Then (λ , µ) ∈ Λ(b G, Pαi ) if and only if there are (r, s) ∈ N0 × N0 with

• rj + sj = λj for j = 1, . . . , n,

• rj + sj+1 = µj for j = 1, . . . , n − 1. n n Note that ri = 0, si+1 = 0 automatically. We say that the pair (r, s) ∈ N0 × N0 determines the pair (λ∗, µ) and vice versa. + ∗ n n If λ ∈ P(G,H)(µ) so that (λ , µ) is determined by the pair (r, s) ∈ N0 ×N0 , then λ−rn$n − + s1$1 ∈ P(G,H)(µ). This inspires the following deﬁnition, + B(µ) := {λ ∈ P(G,H)(µ)|rn = s1 = 0}, called the bottom of the µ-well. Moreover, we deﬁne the µ-degree dµ(λ) := (rn, s1) and the total µ-degree |dµ(λ)| = rn + s1. We denote νλ = λ − rn$n − s1$1 ∈ B(µ).

+ Pn−1 Theorem 3.3. Let i ∈ {1, . . . , n − 1} and let µ ∈ PH be of the form µ = j=1 µjωj with µi = 0. Then + + P(G,H)(µ) = B(G,H)(µ) + P(G,H)(0). 9 Proof. Immediate from the preceding discussion by writing λ = λ−rn$n −s1$1 +(rn$n + s1$1).

Let H∗ denote the complexiﬁcation of L∗. Let TH∗ ⊂ H∗ denote the maximal torus of H∗ consisting of diagonal elements. Then ACTH∗ ⊂ G is a maximal torus of G, where AC is the complexiﬁcation of A. Fix a Weyl chamber for this torus that projects to the positive

Weyl chamber of H∗ determined by the upper triangular matrices. Let λe be a dominant integral weight and suppose that the corresponding irreducible representation πG contains λe H πµ upon restriction to H. The restriction λe∗ of λe to TH∗ is also dominant and integral. H∗ The corresponding irreducible representation π of H∗ occurs with multiplicity one in the λe∗ H restriction πµ |H∗ . This sets up a bijection

B(µ) → {σ ∈ P + | dim Hom (V H∗ ,V H ) ≥ 1}, H∗ H∗ σ µ

see [15, Thm.3.1]. The torus ACTH∗ is conjugated to TG and projection along AC corresponds to projecting along $1 and $n.

Deﬁnition 3.4. The elements of B(µ) can be regarded as the highest weights of the irre- H ducible constituents of πµ |H∗ . As such, the element ν ∈ B(µ) is denoted by ν∗.

+ Pn−1 Proposition 3.5. Let i ∈ {1, . . . , n − 1} and let µ ∈ PH be of the form µ = j=1 µjωj with 0 + G G G µ = 0. Let λ, λ ∈ P (µ). If dim Hom (V 0 ,V ⊗ V ) ≥ 1 for i = 1 or i = n, then i (G,H) G λ λ $i 0 |dµ(λ )| ≤ |dµ(λ)|.

Proof. It is enough to show that |dµ(λ − αj)| ≤ |dµ(λ)| for all j = 1, . . . , n, whenever + + λ − αj ∈ P(G,H)(µ). Note that λ − αi 6∈ P(G,H)(µ). In general we have

• (α1, 0) = ($1, $1) + ($1, 0) − ($2, $1),

• (αj, 0) = −($j−1, $j−1) + ($j, $j) + ($j, $j−1) − ($j+1, $j), for j = 2, . . . , n − 1,

• (αn, 0) = −($n−1, $n−1) + ($n, $n−1) + ($n, 0), which shows that either the µ-degree does not change or the total µ-degree drops by one after subtracting a simple root. + Pn−1 Corollary 3.6. Let i ∈ {1, . . . , n − 1} and let µ ∈ PH be of the form µ = j=1 µjωj with + n n µi = 0 and let λ ∈ P(G,H)(µ) be associated to (r, s) ∈ N0 × N0 . Then there exist polynomials µ qλ,ν ∈ C[φ$1 , φ$n ], ν ∈ B(µ), such that

µ X µ µ (3.4) Φλ(g) = qλ,ν(φ$1 (g), φ$n (g))Φν (g). ν∈B(µ) The polynomials are uniquely determined by Φµ. Moreover, the total degree of qµ is r +s . λ λ,νλ n 1 The total degrees of all other polynomial coefficients in (3.4) are of total degree ≤ rn + s1. 0 0 µ 0 0 If the degree (rn, s1) of a polynomial coefficient qλ,ν in (3.4) satisfies rn + s1 = rn + s1 then ν + rn$n + s1$1 ≤ λ in the usual ordering. 10 Proof. The proof is with induction on rn + s1, similar to the proof of Proposition 2.3.

In view of Theorem 3.3 and Proposition 3.5 we only have to show that the coeﬃcient ci in φ Φµ = c Φµ + l.o.t. is non-zero for i = 1, n. This follows from surjectivity of the Cartan $i λ i λ+$i projection V U ⊗ V U → V U . λ $i λ+$i

4. Vector-valued orthogonal polynomials + Pn−1 Let i ∈ {1, . . . , n−1} and let µ ∈ PH be of the form µ = j=1 µjωj with µi = 0. Fix a total + ordering on B(µ) = {ν1, . . . , νN } that is compatible with the usual partial ordering on PG . L The space of L∗-equivariant endomorphisms EndL∗ (Vµ ) consists of block matrices, the blocks being multiples of the identity that correspond to the L∗-intertwiner of the L∗-isotypical type N L N νi,∗. These multiples are collected in a vector in C . The linear map i : EndL∗ (Vµ ) → C that we obtain in this way is unitary for the Hermitian inner products (x, y) 7→ tr(x∗y) and (z, ζ) 7→ z∗T µζ, where T µ is the diagonal matrix whose entries are the dimensions of V L∗ . νi,∗ We identify the space of invariant functions (2.6) restricted to A with CN -valued functions on A. In particular we denote µ µ Ψλ(a) = i(Φλ(a)). µ Definition 4.1. (1) Let Ψ0 : A → Mat(C,N ×N) be the matrix-valued function on A whose i-th column in a is given by Ψµ (a). νi µ (2) Let Wfpol : A → Mat(C,N × N) be the matrix-valued function on A defined by µ µ ∗ µ µ Wfpol(a) := Ψ0 (a) T Ψ0 (a). µ Lemma 4.2. The function Wfpol is positive definite almost everywhere on A.

µ µ µ Proof. Since T has real entries, Wfpol is certainly self-adjoint. By [15, L.6.1], Ψ0 is invertible on a dense open subset of A, and the result follows. µ N µ µ µ µ We deﬁne Qλ ∈ C [x, y] by Ψ0 Qλ(φ$1 , φ$n ) = Ψλ, which makes sense because Ψ0 is invertible almost everywhere.

µ Lemma 4.3. The entries of the function Wfpol are polynomials in φ$1 , φ$n . The (i, j)-entry of W µ (a) is given by tr(Φµ (a)∗Φµ (a)), which is the restriction Proof. fpol νi νj of an L × L invariant sum of matrix coeﬃcients on U to A, hence it can be written as a

polynomial in φ$1 , φ$n . µ µ Deﬁnition 4.4. Let Wpol ∈ Mat(C,N × N)[z, z] be deﬁned by Wpol(φ$1 (a), φ$n (a)) = µ 2 2 n−1 Wfpol(a) and wn ∈ C(C ) by wn(z, z) = (1 − |z| ) . N Let Q1,Q2 ∈ C [z, z] and consider the pairing Z µ ∗ µ µ (Q1,Q2) 7→ hQ1,Q2iµ = Q1 (z, z) Wpol(z, z)Q2 (z, z)wn(z, z)dλ(z). D µ This provides CN [z, z] with a Hermitian inner product. Let CN [z, z] denote its completion. 11 µ µ Theorem 4.5. The map Φλ 7→ Qλ extends to an isometry µ 2 L L×L N (L (U, du) ⊗ End(Vµ )) → C [x, y] .

Proof. This is clear from the preceding discussion. It follows that the space spanned by the spherical functions of type µ can be identified with a space of vector-valued orthogonal polynomials in two variables as long as µ satisfies the multiplicity freeness assumption. The spherical functions are also determined as simultaneous eigenfunctions of a commutative subquotient of U(g), see [11, L.2.2]. This implies that the vector-valued orthogonal polynomials are also determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In the case µ = 0 we have control over this algebra by means of the Harish-Chandra homomorphism for spherical varieties, see [9]. In the general case this algebra is not yet fully understood.

5. Examples

µ In this section we are concerned with explicit expressions of the functions Ψ0 : A → Mat(C,N × N), as these functions determine the polynomial part of the matrix weight. The µ entries of Ψ0 are matrix coeﬃcients of U-representations that are restricted to A. More precisely, the (i, j)-th entry is the matrix coeﬃcient

a 7→ mνj (a), vνi,∗ ,vνi,∗ where v ∈ V L ⊂ V U is a highest weight vector of L of weight ν . These calculations νi,∗ µ νj ∗ i,∗ can be done systematically by computer algebra. Indeed, in [18, §4.1] such an algorithm is provided for a diﬀerent pair of groups. We have adapted this algorithm to our situation. First we show that we only have to calculate the restriction (Ψµ )| because of the transformation νi A1 behavior of Ψµ under A . νi 2

Lemma 5.1. There exist integers mi,j, 1 ≤ i, j ≤ N such that

Ψµ (a (s)a) = diag(eimi,1s, . . . , eimi,N s)Ψµ (a) νi 2 νi

for a2(s) ∈ A2 and a ∈ A.

+ n n Proof. Let λ ∈ P(U,L)(µ) be associated to the pair (r, s) ∈ N0 × N0 . The pair (r, s)

corresponds to an element (λ, µe) ∈ Λ(b G, BGL(C,n)). The GL(n, C)-module of highest weight L µe restricted to L ⊂ GL(n, C) is just Vµ . Let M = L∗ · A2. The MC-subrepresentations GL(n,C) of π restricted to L∗ are clearly A2-stable. Hence A2 acts with a character on the µe L L∗-subrepresentations of Vµ . For the actual computations we have restricted ourselves to a couple of examples in a

small class of representations of L, namely those of highest weight kωn−1. The reason is that

this greatly simpliﬁes the determination of the highest weight vectors of the irreducible L∗- L representations of Vµ , which is step 7 of the algorithm in [19, §4.1]. Let Ei,j ∈ g denote the 12 element with zeros everywhere except for a one in the entry (i, j). We denote the fundamental weights of H∗ by η1, . . . , ηn−2, where ηi = ωi+1|TH∗ for i = 1, . . . , n − 2.

L Lemma 5.2. Let µ = kωn−1. Let vµ ∈ Vµ be a highest weight vector. Then

k X πL| = πL∗ µ L∗ jηn−2 j=0

j and the highest weight vectors are given by En,1vµ. The dimension of the representation of n+j−2 L∗ of highest weight jηn−2 is given by j .

Proof. Since En,1 commutes with the root vectors of the positive roots of H∗ the ele- j ment En,1vµ is a highest weight vector of H∗ of weight (k − j)ωn−2 unless it is zero. The representation space of πH∗ is isomorphic to Sj( n−1) which is of dimension n−2+j. jηn−2 C j n n The set B(kωn−1) is given by the tuples (r, s) ∈ N0 × N0 such that rj + sj+1 = 0 for j = 1, . . . , n − 1, rn−1 + sn = k and rn + s1 = 0. We obtain

B(kωn−1) = {νj = (k − j)$n−1 + j$n|j = 0, . . . , k}.

kωn−1 To ﬁnd Ψ0 (a) with a ∈ A we invoke Lemma 5.1 and [17, Prop.2.3]. Let (νj, kωn−1) be n j determined by (r, s) ∈ N0 × N0. This pair determines an element (νj, µe ) ∈ Λ(b G, BGL(C,n)), j j with µe = k$n−1 − sn$n. The bottom of the µe -well for the pair (SL(n + 1, C), GL(n, C)) is given by

j B(µe ) = {−p$1 + (k − j − p)$n−1 + (j + p)$n|p = −j, . . . , −1}∪

{(k − j − p)$n−1 + (j + 2p)$n|p = 0, . . . , p},

see e.g. [17, Prop.2.3]. This implies that the MC-types (see Remark 2.5) are given by j+2p 2 ($1 − $n) + (k − j − p)ηn−2. This observation allows us to reﬁne Lemma 5.2.

Lemma 5.3. Let µ = kωn−1. Then there exists m1, . . . , mk+1 ∈ Z such that

kωn−1 Ψ0 (a1(t)a2(s)) =

im1s imk+1s kωn−1 i(n−1)s i(n−1)ks diag(e , . . . , e )Ψ0 (a1(t))diag(1, e , . . . , e ).

Proof. The considerations preceding the lemma indicates what the MC-types are and hence L with which characters the torus A2 acts on the L∗-isotypical components of Vµ . This allows us to calculate the integers of Lemma 5.2 and the result follows. Our algorithm [16] µ calculates Ψ0 |A1 for µ = kωn−1. For the values (n, k) = (3, 2), (4, 2) and (5, 2) we ﬁnd   cos2(t) cos(t) 1 (5.1) Ψ2ωn−1 (a (t)) =  cos(t) 1 (1 + cos2(t)) cos(t)  . 0 1  2  1 cos(t) cos2(t) 13 The diagonal matrix T 2ωn−1 that is needed to calculate the weight matrix is given by 1 diag( 2 n(n − 1), n − 1, 1), the dimensions of the irreducible L∗-repesentations that occur in V L . For n = 3, 4, 5 we obtain kωn−1

2ωn−1 Wpol (z, z) =  1 2 4 1 2 1 + 2 (n − 1)(2|z| + n|z| ) 2 z(1 + (n − 1)|z| )  1 z(1 + (n − 1)|z|2) 1 (n − 1 + 2(1 + n2)|z|2 + (n − 1)|z|4  2 4 1 2 1 2 2 n(n + 1)z 2 z(n − 1 + |z| ) 1 2  2 n(n + 1)z 1 z(n − 1 + |z|2)  2  1 2 4 2 n(n − 1) + (n − 1)|z| + |z|

2 n−1 and wn(z, z) = (1 − |z| ) . Using computer algebra we calculate

 3n 0 0  Z n+2 (5.2) W 2ωn−1 (z, z)w (z, z)dλ(z, z) =  0 3n(n+1) 0  pol n  4(n+2)  D n2(n+1) 0 0 2(n+2)

L 2 for n = 2,..., 23. The diagonal entries are equal to (dim V2ω1 ) times the reciprocals of the U U U dimensions of Vν0 ,Vν1 and Vν2 , in agreement with (2.7) (use Weyl’s dimension formula to obtain dim V L = n+1, dim V U = n(n + 1)2(n + 2)/12, dim V U = n(n + 1)(n + 2)/3 and 2ωn−1 2 ν0 ν1 dim V U = n+2). This supports the conjecture that Ψ2ωn−1 is given by (5.1) for all n. ν2 2 0 Remark 5.4. (1) Note that we can actually vary n ∈ (1, ∞) in W because for all such n the matrix W remains positive definite almost everywhere. At this point it would be interesting to bring the algebra of differential operators in the game that have the spherical functions as simultaneous eigenfunctions, see e.g. [11, §3]. Indeed, one could calculate the expressions of some of these differential operators to investigate the n-dependency. This example would then be a candidate to produce examples of multi-variable matrix valued orthogonal polynomials that allow for a shift, see [19]. (2) To obtain more examples of matrix weights the algorithm needs to be extended so that more general L-representations can be dealt with. However, the representation spaces in which we have to perform calculations soon become too large to deal with for the computer. A possible way to circumvent this issue is to use the approximated spherical functions from [17].

Acknowledgments. The author thanks Stanislav Smirnov for his contribution to the GAP- code [16]. His work was done in the project WHK/SHK-project: Representation theory, Computer Algebra and Special Functions, which had been granted to the author by the University of Paderborn. 14 References

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15 Institut fur¨ Mathematik, Universitat¨ Paderborn, Warburger Str. 100, D-33098 Pader- born, Germany E-mail address: [email protected]