Journal of Lie Theory Volume 11 (2001) 441–458 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Invariant Theory for the Orthogonal Group via Star Products

Didier Arnal, Oumarou Boukary Baoua, Chal Benson, and Gail Ratcliff^{∗}

Communicated by J. Faraut

Abstract. In this paper we apply star products to the invariant theory for multiplicity free actions. The space of invariants for a compact linear multiplicity free action has two canonical bases which are orthogonal with respect to two different inner products. One of these arises in connection with the star product. We use this fact to determine the elements in the canonical bases for the invariants under the action of SO(n, ℝ) × 𝕋 on ℂ^{n}. The formulae obtained improve prior results due to the last two authors and Jenkins.

1. Introduction

Let K be a compact Lie group acting unitarily on a finite dimensional Hermitian vector space V . One says that the action of K on V is multiplicity free when the space of polynomials ℂ[V] decomposes into pair-wise inequivalent irreducible K -modules. Although this condition is a very restrictive, there is a rich family of examples which have, moreover, been completely classified [13, 5, 14]. The concept of multiplicity free action plays an organizational role in Classical Invariant Theory. We refer the reader to [11] for motivation and further references.

Suppose now that the action of K on V is multiplicity free and let

ℂ[V] = ∑_{λ∈Λ} P_{λ}

denote the decomposition of ℂ[V] into K -irreducible subspaces. Here Λ is a countably infinite set of indices. Since the trivial representation can occur only once in ℂ[V], there are no non-constant K -invariant polynomials in ℂ[V]. It is, however, of interest to consider K -invariants in ℂ[V_{ℝ}], the ring of polynomial functions on the underlying real space. In this context, there are two canonical vector space bases for the algebra ℂ[V_{ℝ}]^{K} of K -invariants, each indexed by Λ:

• a basis {p_{λ} : λ ∈ Λ} consisting of homogeneous polynomials p_{λ}, and

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag • a basis {q_{λ} : λ ∈ Λ} consisting of polynomials orthogonal with respect to

the Fock inner product ⟨·, ·⟩_{F} on ℂ[V_{ℝ}].

The polynomials {q_{λ}} are obtained from the p_{λ}’s via Gram-Schmidt orthogonalization using the Fock inner product. Precise definitions will be given below in Section 3.

We will show that the p_{λ}’s are themselves orthogonal with respect to the “star” inner product on ℂ[V_{ℝ}], denoted ⟨·, ·⟩_{∗}, which is obtained from the Fock inner product on ℂ[V] using ℂ[V_{ℝ}] ≃ ℂ[V] ⊗ ℂ[\overline{V}]. This is a classical process which “doubles the variables”. On the other hand, there is a more abstract point of view on the star inner product, which we adopt in Section 2 as the definition. In fact, ⟨f, g⟩_{∗} = tr(AB^{∗}) where A, B are the operators on Fock space with Berezin (or Wick) symbols Â(z) = f(z)e^{−|z|^{2}/2} and B̂(z) = g(z)e^{−|z|^{2}/2}. This shows that the star inner product arises naturally in the context of the (Berezin) star product.

The current work extends results that are contained in the second author’s Ph.D. dissertation [2]. Our goal is to show how the theory of star products can be used as a tool to study invariant theory for multiplicity free actions. The orthogonality of the polynomials {p_{λ}} with respect to the star inner product follows from the observation that the projection operator onto P_{λ} has symbol dim(P_{λ})p_{λ}(z)e^{−|z|^{2}/2}. As a consequence, we obtain an orthogonalization procedure, Proposition 3.5, that determines the p_{λ}’s from the fundamental invariants, a finite subset of {p_{λ} : λ ∈ Λ} which generates ℂ[V_{ℝ}]^{K} as an algebra.

The explicit determination of the p_{λ}’s and q_{λ}’s presents significant combinatorial difficulties. In Section 4 we consider a specific multiplicity free action, that of K = SO(n, ℝ) × 𝕋 on V = ℂ^{n} (n ≥ 3). This example is the subject of [4], which contains recurrence relations for the p_{λ}’s. In Proposition 4.1, we derive these relations using star product techniques. Using this result and orthogonality of the p_{λ}’s with respect to the star inner product, we are able to obtain explicit formulae for the p_{λ}’s in this example. (See Theorem 4.5.) The generalized binomial coefficients, which express the polynomials {q_{λ}} in term of the p_{λ}’s, are also given in [6] via recurrence relations. Here, in Theorem 4.8, we give an explicit formula for these numbers. Our proof again uses properties of the star inner product. The same techniques are used to prove Proposition 4.9, which gives product formulas for the p_{λ}’s.

We see that the use of star products leads to new and more explicit formulae for the canonical invariants for the action of SO(n, ℝ)×𝕋. We are optimistic that these methods can be put to good use in connection with further examples of multiplicity free actions.

We conclude this section by outlining a connection between the problems addressed here and analysis on the Heisenberg group. This provides motivation for the current work and was our original source of interest. If one extends V to form the Heisenberg group H_{V} = V × ℝ with suitable product, then K acts via automorphisms on H_{V} which fix the center ℝ. One says that the action of K on H_{V} yields a Gelfand pair when the K -invariant integrable functions L_{K}^{1}(H_{V}) form a commutative algebra under convolution. This is the case if and only if the action of K on V is multiplicity free. There is, moreover, a well developed theory of spherical functions associated to such Gelfand pairs. The polynomials {q_{λ} : λ ∈ Λ} determine a dense set of full measure in the space of bounded spherical functions. We refer the reader to [3] for details on this connection.

2. Preliminaries on the star product

This section contains background material concerning Berezin symbols and star products. Our viewpoint here is concrete and combinatorial. We will discuss only the most classical version of these ideas, involving Fock space and its reproducing kernel. (See Section 2.7 in [10] and the original source [8] for further details). For generalizations to the setting of a Hilbert space with reproducing kernel, we refer the reader to [9] and [15]. For connections to the orbit method in representation theory, see [1], [2] and [17].

Throughout this paper, V denotes a complex vector space with finite dimension n, (positive definite) Hermitian inner product ⟨·, ·⟩, and norm | · |. Often it will be convenient to identify (V, ⟨·, ·⟩) with ℂ^{n} equipped with the standard Hermitian structure ⟨z, w⟩ = z · \overline{w}. This can always be done by choosing any orthonormal basis for V .

Fock space: The space ℂ[V] of holomorphic polynomials on V can be equipped with the Fock (or Fischer) inner product

⟨f, g⟩_{F} = (\frac{1}{2π} )^{n} ∫_{V} f(z)\overline{g(z)}e^{−|z|^{2}/2}dz.  (2.1)

Here “dz” denotes Lebesgue measure on the underlying real vector space V_{ℝ} ≃ ℝ^{2n}, normalized by the inner product on V . Fock space F = F_{n} is the Hilbert space completion of ℂ[V] with respect to ⟨·, ·⟩_{F}. This is the set of holomorphic functions on V which are square integrable with respect to the Gaussian measure e^{−|z|^{2}/2}dz. Equation (2.1) extends from ℂ[V] to F. See [10] for the completeness property of this space. We remark that the Bargmann transform yields an isometry from L^{2}(ℝ^{n}) to F. This fact, however, will play no role in the current work.

Identifying V with ℂ^{n} yields an isomorphism ℂ[V] ≃ ℂ[z_{1}, …, z_{n}]. The monomials z^{α} = z_{1}^{α_{1}} ⋯ z_{n}^{α_{n}} form a complete system in F with

⟨z^{α}, z^{β}⟩_{F} = \{ \begin{matrix}2^{|α|}α! & for β = α \\ 0 & for β ≠ α \end{matrix} .  (2.2)

Here we have used the standard multi-index notation: |α| = α_{1} + ⋯ + α_{n} = deg(z^{α}), α! = α_{1}! ⋯ α_{n}!.

The function

E_{z}(u) = e^{\frac{1}{2} ⟨u,z⟩}  (2.3)

is the reproducing kernel for Fock space. That is,

⟨f, E_{z}⟩_{F} = f(z)  (2.4)

for all f ∈ F. To establish this, one can identify V with ℂ^{n} and form the Taylor series

E_{z}(u) = exp (\frac{u · \overline{z}}{2} ) = ∑_{j=1}^{∞} \frac{1}{j!2^{j}} (u · \overline{z})^{j} = ∑_{j=1}^{∞} \frac{1}{j!2^{j}} ∑_{|β|=j} \frac{j!}{β!}u^{β}\overline{z}^{β}  (2.5)

= ∑_{β} \frac{1}{2^{|β|}β!}u^{β}\overline{z}^{β}.

Using Equation (2.2) we now see that ⟨z^{α}, E_{z}⟩_{F} = z^{α}. This proves (2.4), as the monomials form a complete orthogonal system in F.

Formula (2.1) also yields a Hermitian inner product on ℂ[V_{ℝ}], the complex valued polynomial functions on V_{ℝ}. To avoid confusion between the spaces F and ℂ[V_{ℝ}], we will refer to this as the Fock inner product on ℂ[V_{ℝ}]. For f, g ∈ ℂ[V_{ℝ}], observe that (2π)^{n}⟨f, g⟩_{F} coincides with the L^{2}-inner product of f(z)e^{−|z|^{2}/4} with g(z)e^{−|z|^{2}/4}. Such functions arise naturally as spherical functions on the Heisenberg group modulo its center. This fact motivates our interest in the Fock inner product on ℂ[V_{ℝ}].

Berezin symbols and the star product: The Berezin symbol for an operator A on F is the real analytic function Â on V_{ℝ} defined by

Â(z) = \frac{⟨AE_{z}, E_{z}⟩_{F}}{⟨E_{z}, E_{z}⟩_{F}} = \frac{(AE_{z})(z)}{E_{z}(z)} = (AE_{z})(z)e^{− \frac{1}{2} |z|^{2}}.

Here A may be unbounded on F but we require that the “coherent states” {E_{z} : z ∈ V } belong to the domains of both A and A^{∗}. When each of the operators A, B and AB satisfy these conditions we define the star product Â ⋆ B̂ of the two symbols Â and B̂ via

Â ⋆ B̂ = (AB) .

The operators that play a role in this paper are polynomial coefficient differential operators and finite rank projections. We will need to use the following facts concerning Berezin symbols and star products:

(2.6) For f ∈ ℂ[V], define the multiplication operator m_{f} on F by m_{f}(g) = fg. This operator has symbol m̂_{f} = f.

(2.7) Given f ∈ ℂ[V] ≃ ℂ[z_{1}, …, z_{n}], we let f(∂_{z}) be the operator obtained by replacing each variable z_{j} with ∂/∂z_{j}. This is an unbounded operator on F with

f(∂_{z})^{∗} = m_{\overline{f}(z/2)}, m_{f}^{∗} = \overline{f}(2∂_{z}).

Here \overline{f} ∈ ℂ[V] is the holomorphic polynomial obtained by conjugating the coefficients of f. We have (f(∂_{z})) = f(\overline{z}/2).

(2.8) Combining (2.6) and (2.7) we see that f ∈ ℂ[V_{ℝ}] ≃ ℂ[z_{1}, …, z_{n}, \overline{z_{1}}, …, \overline{z_{n}}] is the symbol of the Wick ordered operator f(z, 2∂_{z}).

(2.9) The symbol π̂ for the orthogonal projection operator π onto a finite dimensional subspace of ℂ[V] with orthonormal basis {v_{j}}_{j=1}^{m} is

π̂(z) = (∑_{j=1}^{m} v_{j}(z)\overline{v_{j}(z)}) e^{−|z|^{2}/2}.

(2.10) If A is a trace class operator, then ∫_{V} Â(z)dz = (2π)^{n}tr(A).

(2.11) If Â is holomorphic or B̂ is antiholomorphic, then (AB) = Â ⋆ B̂ = ÂB̂.

Note that, if we extend {v_{j}}_{j=1}^{m} in (2.9) to an orthonormal basis {v_{j}}_{j=1}^{∞} in ℂ[V], then we can write the decomposition (2.5) of E_{z} in the basis {v_{j}} as E_{z}(u) = ∑_{j=1}^{∞} \overline{v_{j}(z)}v_{j}(u). Following [10], this series converges uniformly on any compact subset of V . Thus we obtain π(E_{z}) = ∑_{j=1}^{m} \overline{v_{j}(z)}v_{j}, which immediately yields (2.9).

For completeness, we recall one further result which will not be used in the sequel. One has

(Af)(z) = (\frac{1}{2π} )^{n} ∫_{V} f(w)Ã(w, z)e^{⟨z−w,w⟩/2}dw  (2.12)

where Ã(w, z) = ⟨AE_{w}, E_{z}⟩_{F}/⟨E_{w}, E_{z}⟩_{F}. The multiplier Ã(w, z) is the unique function on V × V which is holomorphic in w and anti-holomorphic in z, with Ã(z, z) = Â(z). Thus (2.12) shows that an operator A is completely determined by its symbol Â.

The integral formula for the star product is then given by

(Â ⋆ B̂)(z) = (\frac{1}{2π} )^{n} ∫_{V} Ã(w, z)B̃(z, w)e^{−|z−w|^{2}/2}dw.

The star inner product: For f, g ∈ ℂ[V_{ℝ}] we define

⟨f, g⟩_{∗} = (\frac{1}{2π} )^{n} ∫_{V} (f(z)e^{−|z|^{2}/2}) ⋆ (\overline{g(z)}e^{−|z|^{2}/2}) dz.  (2.13)

This gives another Hermitian inner product on ℂ[V_{ℝ}], although it is not a priori clear that it is positive definite or even well-defined. To show this, we first make the identifications V ≃ ℂ^{n}, ℂ[V_{ℝ}] ≃ ℂ[z_{1}, …, z_{n}, \overline{z}_{1}, …, \overline{z}_{n}] and apply formula (2.13) to a pair of monomials z^{α}\overline{z}^{β}, z^{α′} \overline{z}^{β′}. We will write m_{j} for the operator m_{z_{j}} and m_{α} for m_{z^{α}}.

The projection operator π_{0}(f) = f(0) from F onto the constant polynomials in ℂ[V] has symbol π̂_{0} = E_{z}(0)e^{−|z|^{2}/2} = e^{−|z|^{2}/2}. Hence also (m_{α}π_{0}) = z^{α}e^{−|z|^{2}/2} by (2.6) and (2.11). Now (2.7) gives (2^{|β|}∂_{z}^{β}) = \overline{z}^{β}, and by (2.11) we obtain

z^{α}\overline{z}^{β}e^{−|z|^{2}/2} = 2^{|β|}(m_{α}π_{0}) (∂_{z}^{β}) = 2^{|β|} (m_{α}π_{0}∂_{z}^{β}) .

Thus we can write

⟨z^{α}\overline{z}^{β}, z^{α′} \overline{z}^{β′}⟩_{∗} = (\frac{1}{2π} )^{n} ∫_{V} 2^{|β|}2^{|α′|} (m_{α}π_{0}∂_{z}^{β}m_{β′}π_{0}∂_{z}^{α′}) dz
= 2^{|β|}2^{|α′|}tr (m_{α}π_{0}∂_{z}^{β}m_{β′}π_{0}∂_{z}^{α′})

using (2.10). The operator m_{α}π_{0}∂_{z}^{β}m_{β′}π_{0}∂_{z}^{α′} is zero unless β = β′, in which case it sends z^{α′} to β!α′!z^{α} and annihilates all other monomials in z. This gives, finally,

⟨z^{α}\overline{z}^{β}, z^{α′} \overline{z}^{β′}⟩_{∗} = δ_{α,α′}δ_{β,β′} (2^{|α|}α!) (2^{|β|}β!) .  (2.14)

One can use (2.14) to compute the star inner product ⟨f, g⟩_{∗} for any pair of polynomials f, g ∈ ℂ[V_{ℝ}]. It is now clear that the star inner product is a well defined Hermitian inner product on ℂ[V_{ℝ}]. Moreover, (2.14) provides a concrete combinatorial viewpoint on the star inner product. In view of Equation (2.2), we can recast this as follows. The star inner product on ℂ[V_{ℝ}] arises from ⟨·, ·⟩_{F} on ℂ[V] by doubling the variables. More precisely, we have:

Lemma 2.1. The star inner product ⟨·, ·⟩_{∗} on ℂ[V_{ℝ}] coincides with ⟨·, ·⟩_{F} ⊗ ⟨·, ·⟩_{F^{∗}} under the identification ℂ[V_{ℝ}] = ℂ[V ⊕ \overline{V}] ≃ ℂ[V] ⊗ ℂ[\overline{V}] ⊂ F ⊗ F^{∗}.

One may choose to regard F ⊗F^{∗} as the space of Hilbert-Schmidt operators on F. The subspace ℂ[V_{ℝ}] consists of finite rank operators with image contained in ℂ[V]. Lemma 2.1 shows that the star norm on ℂ[V_{ℝ}] agrees with the restriction of the Hilbert-Schmidt norm |||T |||^{2} = tr(T^{∗}T) on F ⊗F^{∗}. The star inner product on ℂ[V_{ℝ}] is thus a classical object of study.

3. Polynomial invariants of a multiplicity free action

Now let K be a closed Lie subgroup of the unitary group U(V) and consider the natural representations of K on ℂ[V] and ℂ[V_{ℝ}] defined by the rule

(k · f)(z) = f(k^{−1}z).

We remark that the Bargmann transform L^{2}(ℝ^{n}) → F intertwines the representation of K on ℂ[V] with the restriction of the metaplectic representation to K ⊂ U(n) ⊂ Sp(2n, ℝ). The space ℂ[V] decomposes as a countably infinite direct sum of finite dimensional K -irreducible subspaces:

ℂ[V] = ∑_{λ∈Λ} P_{λ}.  (3.1)

One says that the action of K on V is multiplicity free when the irreducible K - modules {P_{λ} : λ ∈ Λ} are pair-wise inequivalent. In this case the decomposition given in (3.1) is canonical.

Proposition 3.1 below relates multiplicity free actions to the star product. First note that for k ∈ U(V) one has E_{k^{−1}z}(u) = e^{⟨u,k^{−1}z⟩/2} = e^{⟨ku,z⟩/2} = E_{z}(ku). So for operators A on F we have

(k · Â)(z) = Â(k^{−1}z) = (AE_{k^{−1}z}) (k^{−1}z)e^{−|k^{−1}z|^{2}/2}

= (AU_{k}^{−1}E_{z}) (k^{−1}z)e^{−|z|^{2}/2} = (U_{k}AU_{k}^{−1}E_{z}) (z)e^{−|z|^{2}/2} = (U_{k}AU_{k}^{−1}) (z),

where U_{k} denotes the unitary operator on F given by (U_{k}f)(z) = f(k^{−1}z).

An operator A on F is said to be K -invariant when U_{k}A = AU_{k} for all k ∈ K. The above calculation shows that A is K -invariant if and only if k · Â = Â for all k ∈ K. Since composites of K -invariant operators are K - invariant, it follows that the star product Â ⋆ B̂ of two K -invariant symbols is K -invariant. As noted in (2.8), f ∈ ℂ[V_{ℝ}] is the symbol for the Wick ordered operator f(z, 2∂_{z}). So the symbol map gives an isomorphism between the vector space PD(V)^{K} of K -invariant polynomial coefficient differential operators on V and the the space ℂ[V_{ℝ}]^{K} of K -invariant polynomials on V_{ℝ}. Moreover, this correspondence becomes an algebra isomorphism if we equip ℂ[V_{ℝ}] with the star product. It is shown in [12] that the algebra PD(V)^{K} is commutative if and only if the action of K on V is multiplicity free. These remarks prove the following:

Proposition 3.1. The space ℂ[V_{ℝ}]^{K} of K -invariant polynomials on V_{ℝ} forms an algebra under the star product. This algebra is abelian if and only if the action of K on V is multiplicity free.

We now suppose that the action of K on V is multiplicity free. Since the spaces P_{m}(V) of homogeneous polynomials of degree m are U(V)-invariant, each P_{λ} is a subspace of some P_{m}(V). We let |λ| denote the degree of homogeneity of the polynomials in P_{λ}, so that P_{λ} ⊂ P_{|λ|}(V), and write

d_{λ} = dim(P_{λ}).

Homogeneous invariants: For each λ ∈ Λ, let p_{λ} ∈ ℂ[V_{ℝ}] be defined by

p_{λ}(z) = \frac{1}{d_{λ}} ∑_{j=1}^{d_{λ}} v_{j}(z)\overline{v_{j}(z)},  (3.2)

where {v_{j}}_{j=1}^{d_{λ}} is any orthonormal basis for P_{λ}. In [3] it is shown that this definition does not depend on the choice of orthonormal basis for P_{λ} and that {p_{λ} | λ ∈ Λ} is a vector space basis for the space ℂ[V_{ℝ}]^{K}. Note that p_{λ} is homogeneous of degree 2|λ|. In view of (2.9), we obtain:

Lemma 3.2. The symbol for the projection operator π_{λ} onto the subspace P_{λ} is given by

π̂_{λ}(z) = d_{λ}p_{λ}(z)e^{−|z|^{2}/2}.

Lemma 3.3. The polynomials {p_{λ} | λ ∈ Λ} are pair-wise orthogonal with respect to the star inner product, and ⟨p_{λ}, p_{λ}⟩_{∗} = 1/d_{λ}.

Proof. Using Lemma 3.2, we compute:

⟨p_{λ}, p_{μ}⟩_{∗} = (\frac{1}{2π} )^{n} \frac{1}{d_{λ}d_{μ}} ∫_{V} π̂_{λ} ⋆ π̂_{μ}(z)dz = (\frac{1}{2π} )^{n} \frac{1}{d_{λ}d_{μ}} ∫_{V} (π_{λ}π_{μ}) (z)dz
= tr(π_{λ}π_{μ})/d_{λ}d_{μ} = δ_{λ,μ}d_{λ}/d_{λ}d_{μ} = δ_{λ,μ}/d_{λ}.

Here we have also used (2.10) and the fact that \overline{p_{μ}(z)} = p_{μ}(z).

Fundamental invariants: The index set Λ in (3.1) can be concretely realized as follows: Choose a maximal torus in K and a system of positive roots. These choices produce a simple ordering ≺ on the weights of the representations of K. We let Λ be the set of highest weights for the irreducible representations of K which occur in ℂ[V], so that the irreducible component P_{λ} has highest weight λ ∈ Λ.

Following [12] we call the primitive elements of Λ fundamental highest weights. These are finite in number and freely generate Λ. Letting λ_{1}, λ_{2}, …, λ_{r} denote the fundamental highest weights listed in increasing order using ≺, we have

Λ = {a_{1}λ_{1} + ⋯ + a_{r}λ_{r} | a_{1}, …, a_{r} ∈ ℤ_{+}},

where ℤ_{+} denotes the non-negative integers. The fundamental invariants γ_{1}, …, γ_{r} are defined as

γ_{j} = p_{λ_{j}}.

One has ℂ[V_{ℝ}]^{K} = ℂ[γ_{1}, …, γ_{r}]. For λ = a_{1}λ_{1} + ⋯ + a_{r}λ_{r} ∈ Λ we define

γ^{λ} = γ_{1}^{a_{1}}γ_{2}^{a_{2}} ⋯ γ_{r}^{a_{r}}.

In [7] it is shown that p_{λ} = c_{λ}γ^{λ} + s_{λ} where c_{λ} is a positive constant and s_{λ} ∈ Span{p_{μ} : μ ∈ Λ, |μ| = |λ|, μ ≺ λ}. (See Equation (2.5) in [7].) Thus we have:

Lemma 3.4. For λ ∈ Λ,

Span{γ^{μ} : μ ∈ Λ, |μ| = |λ|, μ ⪯ λ} = Span{p_{μ} : μ ∈ Λ, |μ| = |λ|, μ ⪯ λ}.

Lemmas 3.3 and 3.4 yield an orthogonalization procedure which determines the p_{λ}’s.

Proposition 3.5. For each m ≥ 0 the polynomials {p_{λ} : |λ| = m} are obtained from {γ^{λ} : |λ| = m} via Gram-Schmidt orthogonalization using the star inner product, where

• the indices Λ_{m} = {λ : |λ| = m} are ordered using the weight ordering ≺, and

• the p_{λ}’s are normalized so that ⟨p_{λ}, p_{λ}⟩_{∗} = 1/d_{λ}.

One can also replace the ordinary products γ^{λ} in Proposition 3.5 by star products. Specifically, for λ = a_{1}λ_{1} + ⋯ + a_{r}λ_{r} we let

γ^{⋆λ} = γ_{1}^{⋆a_{1}} ⋆ γ_{2}^{⋆a_{2}} ⋆ ⋯ ⋆ γ_{r}^{⋆a_{r}}

where γ_{j}^{⋆a_{j}} denotes the a_{j} -fold star product γ_{j} ⋆ ⋯ ⋆ γ_{j}. Since ℂ[V_{ℝ}]^{K} is commutative under the star product, 3.1) the definition of γ^{⋆λ} is independent of the ordering used for the factors γ_{j}. By (2.8) we see that γ^{⋆λ} is the symbol of γ_{1}(z, 2∂_{z})^{a_{1}} ⋯ γ_{r}(z, 2∂_{z})^{a_{r}}, whereas γ^{λ} is the symbol of (γ_{1}^{a_{1}} ⋯ γ_{r}^{a_{r}})(z, 2∂_{z}). In particular,

γ^{⋆λ} = γ^{λ} + r_{λ}

where r_{λ} ∈ {γ^{μ} : |μ| < |λ| or (|μ| = |λ| and μ ≺ λ)}. The following result is now a corollary of Proposition 3.5.

Proposition 3.6. The polynomials {p_{λ} : λ ∈ Λ} are obtained from {γ^{⋆λ} : λ ∈ Λ} via Gram-Schmidt orthogonalization using the star inner product, where

• Λ is ordered according to the rule

λ < λ′ ⟺ (|λ| < |λ′|) or (|λ| = |λ′| and λ ≺ λ′).

• the p_{λ}’s are normalized so that ⟨p_{λ}, p_{λ}⟩_{∗} = 1/d_{λ}.

Inhomogeneous invariants: The space ℂ[V_{ℝ}]^{K} has a second canonical basis, denoted {q_{λ} | λ ∈ Λ}. These are the polynomials obtained from {p_{λ} | λ ∈ Λ} via Gram-Schmidt orthogonalization using the Fock inner product on ℂ[V_{ℝ}], where • the index set Λ is ordered so that λ precedes λ′ whenever |λ| < |λ′|, and

• the q_{λ}’s are normalized so that q_{λ}(0) = 1.

It is shown in [3] that the polynomials q_{λ} are well defined, and independent of the ordering used within each Λ_{m} = {λ ∈ Λ : |λ| = m}. If we give Λ the ordering from Proposition 3.6 then Lemma 3.4 implies that we can use the polynomials {γ^{λ}} in place of the p_{λ}’s in the orthogonalization procedure. This is the content of Theorem 2.1 in [7].

The polynomial q_{λ} is inhomogeneous of total degree 2|λ|. The homogeneous component of degree 2|λ| in q_{λ} is (−1)^{|λ|}p_{λ}. We express q_{λ} in the basis {p_{μ} | μ ∈ Λ} by

q_{λ} = ∑_{μ∈Λ}(−1)^{|μ|}\left[{λ \atop μ}\right]p_{μ}.

The values \left[{λ \atop μ}\right] are called generalized binomial coefficients. We have \left[{λ \atop λ}\right] = 1, and \left[{λ \atop μ}\right] = 0 for |μ| > |λ| or |μ| = |λ| with μ ≠ λ. The generalized binomial coefficients are non-negative rational numbers. (See [6, 7].)

There is a remarkable “Pieri formula,” due to Yan, which provides another link between the bases {p_{λ} : λ ∈ Λ} and {q_{λ} : λ ∈ Λ}:

\frac{1}{m!} (\frac{|z|}{2}^{2})^{m} d_{μ}p_{μ} = ∑_{|λ|=|μ|+m} \left[{λ \atop μ}\right]d_{λ}p_{λ}.  (3.3)

We refer the reader to [16] and [6] for proofs of this result. Using Lemma 3.3, we can rephrase this Pieri formula as

\left[{λ \atop μ}\right] = \frac{d_{μ}}{m!} ⟨(\frac{|z|}{2}^{2})^{m} p_{μ}, p_{λ}⟩_{⋆}  (3.4)

where m = |λ| − |μ|.

4. The action of K = SO(n, ℝ) × 𝕋

The rest of this paper concerns the standard action of K = SO(n, ℝ) × 𝕋 on V = ℂ^{n} for n ≥ 3, namely

(A, c) · z = cAz

for A ∈ SO(n, ℝ), c ∈ 𝕋, z ∈ V .

The decomposition of ℂ[V] under the action of K is determined by the classical theory of spherical harmonics. The polynomial

ε(z) = z_{1}^{2} + ⋯ + z_{n}^{2}

is invariant under the action of SO(n, ℝ) on ℂ[V]. We let

Δ = ε(∂_{z}) = (\frac{∂}{∂z_{1}} )^{2} + ⋯ + (\frac{∂}{∂z_{n}} )^{2}

and define the space of harmonic polynomials H = ker(Δ). Thus

H_{m} = P_{m}(V) ∩ H

is the space of harmonic polynomial which are homogeneous of degree m. For non-negative integers k, ℓ,

P_{k,ℓ} = m_{ε}^{ℓ}H_{k} = ε^{ℓ}H_{k}

is an irreducible K -invariant subspace of P_{k+2ℓ}(V). The space P_{k,ℓ} has dimension d_{k,ℓ} = h_{k} = dim(H_{k}), independent of ℓ. Explicitly, h_{0} = 1, h_{1} = n and

h_{k} = dim(P_{k}(V)) − dim(P_{k−2}(V)) = \binom{k + n − 1}{k} − \binom{k + n − 3}{k − 2}

for k ≥ 2.

The action of K is multiplicity free because the P_{k,ℓ}’s are pair-wise inequivalent K -modules. Indeed, P_{k,ℓ} and P_{k′,ℓ′} have the same dimension only when k = k′, and the circle 𝕋 acts in these spaces by the characters c ↦ c^{−(2k+ℓ)} and c ↦ c^{−(2k′+ℓ′)} respectively. We write the decomposition (3.1) for this example as

ℂ[V] = ∑_{(k,ℓ)∈Λ} P_{k,ℓ}

where Λ denotes the set of all pairs of non-negative integers.

Homogeneous invariants: Here we will derive formulae for the polynomials {p_{k,ℓ} : (k, ℓ) ∈ Λ}. Recall that these form a canonical homogeneous basis for ℂ[V_{ℝ}]^{K}. Let γ_{1}, γ_{2} ∈ ℂ[V_{ℝ}]^{K} be the polynomials defined as

γ_{1}(z) = np_{1,0}(z) = |z|^{2}/2, γ_{2}(z) = np_{0,1}(z) = |ε(z)|^{2}/4.

These are (up to multiples) the fundamental invariants in this example. We will express each p_{k,ℓ} as a polynomial in γ_{1}, γ_{2}. In fact, Theorem 5.12 in [4] solves this problem by providing recurrence relations for the p_{k,ℓ}’s. We repeat this result below.

Proposition 4.1. Writing p_{k} for p_{k,0} one has

(a) p_{k,ℓ} = p_{k}γ_{2}^{ℓ}/c_{k,ℓ} where

c_{k,ℓ} = 4^{ℓ}ℓ! ∏_{j=1}^{ℓ}(k + n/2 + ℓ − j) = 4^{ℓ}(ℓ!)^{2}\binom{k + n/2 + ℓ − 1}{ℓ}.

(b) p_{k} is determined by the recurrence relation

h_{k}p_{k} = \frac{γ_{1}^{k}}{k!} − ∑_{j=1}^{⌊k/2⌋} h_{k−2j}p_{k−2j,j} = \frac{γ_{1}^{k}}{k!} − ∑_{j=1}^{⌊k/2⌋} \frac{h_{k−2j}p_{k−2j}γ_{2}^{j}}{c_{k−2j,j}} for k ≥ 2,

with initial conditions p_{0} = 1, p_{1} = γ_{1}/n.

We remark that the polynomials “p_{k ℓ}” in [4] are not normalized by dividing by h_{k} = dim(P_{k,ℓ}). This accounts for the dimension factors in Proposition 4.1 that are absent in [4]. Here we will show how Proposition 4.1 can be derived using the machinery of star products. First we establish the following lemma.

Lemma 4.2. The orthogonal projection operator π_{k,ℓ} of F onto P_{k,ℓ} satisfies

π_{k,ℓ} = \frac{1}{c_{k,ℓ}} m_{ε}^{ℓ}π_{k,0}Δ^{ℓ}.

Proof. Let T_{kℓ} be the operator defined on F by

T_{kℓ} = \frac{1}{c_{k,ℓ}} m_{ε}^{ℓ}π_{k,0}Δ^{ℓ}.

First note that an easy calculation yields

Δ(gε^{ℓ}) = 4ℓ(k + n/2 + ℓ − 1)gε^{ℓ−1}

for g ∈ H_{k}. By induction we obtain Δ^{ℓ}(gε^{ℓ}) = c_{k,ℓ}g. (This is Equation (5.14) in [4].) Thus we have T_{k,ℓ}(gε^{ℓ}) = gε^{ℓ} for g ∈ H_{k}, and hence Im(T_{k,ℓ}) = P_{k,ℓ}. In addition, Δ^{ℓ}m_{ε}^{ℓ}π_{k,0} = c_{k,ℓ}π_{k,0} shows us that T_{k,ℓ}^{2} = T_{k,ℓ}. Next we note that T_{k,ℓ} is formally self-adjoint, since (2.7) shows Δ^{∗} = m_{ε}/4 and m_{ε}^{∗} = 4Δ.

Proof. We turn now to the Proof of Proposition 4.1. Lemma 4.2 shows that

π̂_{k,ℓ} = \frac{1}{c_{k,ℓ}} m̂_{ε}^{⋆ℓ} ⋆ π̂_{k,0} ⋆ Δ̂^{⋆ℓ}

where m̂_{ε}^{⋆ℓ} and Δ̂^{⋆ℓ} denote ℓ-fold star products. From (2.6) and (2.7) we see that

m̂_{ε}(z) = ε(z), Δ̂(z) = ε(∂_{z}) (z) = ε(\overline{z}/2) = \frac{\overline{ε(z)}}{4}.

As m̂_{ε} is holomorphic and Δ̂ is antiholomorphic, (2.11) shows that the star product above is an ordinary product:

π̂_{k,ℓ} = \frac{1}{c_{k,ℓ}} m̂_{ε}^{ℓ}π̂_{k,0}Δ̂^{ℓ} = \frac{1}{c_{k,ℓ}} π̂_{k,0}ε^{ℓ} (\frac{\overline{ε}}{4})^{ℓ} = \frac{1}{c_{k,ℓ}} π̂_{k,0}γ_{2}^{ℓ}.  (4.1)

From Lemma 3.2 we have π̂_{k,ℓ} = h_{k}p_{k,ℓ}e^{−|z|^{2}/2}, and hence p_{k,ℓ} = p_{k}γ_{2}^{ℓ}/c_{k,ℓ}. To establish the second assertion in Proposition 4.1 we use the fact that

∑_{ℓ+2j=k} h_{ℓ}p_{ℓ,j} = \frac{γ_{1}^{k}}{k!} .  (4.2)

This follows from the decomposition π_{k} = ∑_{ℓ+2j=k} π_{ℓ,j} of the projection operator π_{k} onto P_{k}(V) by passing to symbols. We use the orthonormal basis {z^{α}/(2^{k}α!)^{1/2} : |α| = k} for P_{k}(V)to obtain

π̂_{k}(z)e^{|z|^{2}/2} = \frac{1}{2^{k}} ∑_{|α|=k} \frac{z^{α}\overline{z}^{α}}{α!} = \frac{1}{2^{k}k!} ∑_{|α|=k} \frac{k!}{α!}z^{α}\overline{z}^{α} = \frac{1}{k!} (\frac{z · \overline{z}}{2} )^{k} = \frac{γ_{1}(z)}{k!}^{k}.

Corollary 4.3. For all (k, ℓ) we have:

∫_{V} γ_{1}^{k}γ_{2}^{ℓ}e^{−|z|^{2}/2}dz = k!(2π)^{n} ∑_{j≤k/2} \frac{c_{k−2j,ℓ+j}}{c_{k−2j,j}} [\binom{k − 2j + n − 1}{k − 2j} − \binom{k − 2j + n − 3}{k − 2j − 2}] .

Proof. By (4.1), (4.2), we see that

e^{−|z|^{2}/2}γ_{1}^{k}/k! = ∑_{j≤k/2} π̂_{k−2j,j} = ∑_{j≤k/2} \frac{1}{c_{k−2j,j}} π̂_{k−2j,0}γ_{2}^{j},

and thus

e^{−|z|^{2}/2}γ_{1}^{k}γ_{2}^{ℓ} = k! ∑_{j≤k/2} \frac{1}{c_{k−2j,j}} π̂_{k−2j,0}γ_{2}^{j+ℓ} = k! ∑_{j≤k/2} \frac{c_{k−2j,j+ℓ}}{c_{k−2j,j}} π̂_{k−2j,j+ℓ}.

Integrating over V , we then obtain

∫_{V} γ_{1}^{k}γ_{2}^{ℓ}e^{−|z|^{2}/2}dz = k!(2π)^{n} ∑_{j≤k/2} \frac{c_{k−2j,ℓ+j}}{c_{k−2j,j}} tr(π_{k−2j,j+ℓ}),

and the result follows.

We remark that, in [4], this formula is given by

∫_{V} γ_{1}^{k}γ_{2}^{ℓ}e^{−|z|^{2}/2} = k!(2π)^{n}4^{ℓ}(ℓ!)^{2}\binom{k + 2l + n − 1}{k}\binom{ℓ + n/2 − 1}{ℓ}.

We can find no independent proof of the combinatorial identity given by the right hand sides of these equations.

Lemma 3.4 shows that for fixed k, ℓ one has

Span{γ_{1}^{k−2j}γ_{2}^{ℓ+j} : j = 0, …, ⌊k/2⌋} = Span{p_{k−2j,ℓ+j} : j = 0, …, ⌊k/2⌋}. (4.3)

This can also be derived using Proposition 4.1 and induction on m = k + 2ℓ. (See Lemma 5.17 in [4]].) Propositions 3.5 and 3.6 specialize to the current example as:

Proposition 4.4. For each m ≥ 0 the polynomials {p_{k,ℓ} : k + 2ℓ = m} are obtained from {γ_{1}^{k}γ_{2}^{ℓ} : k + 2ℓ = m} via Gram-Schmidt orthogonalization using the star inner product, where

• the indices Λ_{m} = {(k, ℓ) : k + 2ℓ = m} are ordered so that (k, ℓ) precedes (k′, ℓ′) when k < k′, and

• the p_{k,ℓ}’s are normalized so that ⟨p_{k,ℓ}, p_{k,ℓ}⟩_{∗} = 1/h_{k}.

Alternatively, one can obtain the polynomials {p_{k,ℓ} : (k, ℓ) ∈ Λ} from {γ_{1}^{⋆k} ⋆ γ_{2}^{⋆ℓ} : (k, ℓ) ∈ Λ} via Gram-Schmidt orthogonalization using the star inner product, where • Λ is given the ordering

(k, ℓ) < (k′, ℓ′) ⟺ (k + 2ℓ < k′ + 2ℓ′) or (k + 2ℓ = k′ + 2ℓ′ and k < k′), and

• the p_{k,ℓ}’s are normalized so that ⟨p_{k,ℓ}, p_{k,ℓ}⟩_{∗} = 1/h_{k}.

The polynomials γ_{1}^{⋆k} ⋆ γ_{2}^{⋆ℓ} in Proposition 4.4 can be computed by repeated application of the following identities:

γ_{1} ⋆ (γ_{1}^{k}γ_{2}^{ℓ}) = γ_{1}^{k+1}γ_{2}^{ℓ} + (k + 2ℓ)γ_{1}^{k}γ_{2}^{ℓ},  (4.4)
γ_{2} ⋆ (γ_{1}^{k}γ_{2}^{ℓ}) = γ_{1}^{k}γ_{2}^{ℓ+1} + 4ℓγ_{1}^{k+1}γ_{2}^{ℓ} + 2kγ_{1}^{k−1}γ_{2}^{ℓ+1} +  (4.5)

k(k − 1)γ_{1}^{k−2}γ_{2}^{ℓ+1} + 4ℓ(k + ℓ + n/2 − 1)γ_{1}^{k}γ_{2}^{ℓ}.

To establish these formulae, first note that by (2.8)

γ_{1} = ∑_{j} z_{j} \frac{\overline{z_{j}}}{2} = [∑_{j} m_{j}∂_{j}] , γ_{2} = ε(z)ε (\frac{\overline{z}}{2}) = [m_{ε}Δ] , and
γ_{1}^{k}γ_{2}^{ℓ} = (∑_{j} z_{j} \frac{\overline{z_{j}}}{2} )^{k} ε(z)^{ℓ}ε (\frac{\overline{z}}{2})^{ℓ} = ∑_{|α|=k} \frac{k!}{α!}ε(z)^{ℓ}z^{α} (\frac{\overline{z}}{2})^{α} ε (\frac{\overline{z}}{2})^{ℓ}

= [ ∑_{|α|=k} \frac{k!}{α!}m_{ε}^{ℓ}m_{α}∂_{z}^{α}Δ^{ℓ}] .

Thus we have:

γ_{1} ⋆ (γ_{1}^{k}γ_{2}^{ℓ}) = [ ∑_{j,|α|=k} \frac{k!}{α!}m_{j}∂_{j}m_{ε}^{ℓ}m_{α}∂_{z}^{α}Δ^{ℓ}] .

Since the commutator of ∂_{j} and m_{ε}^{ℓ}m_{α} is multiplication by

∂_{j}(ε^{ℓ}z^{α}) = 2ℓz_{j}ε^{ℓ−1}z^{α} + α_{j}ε^{ℓ}z^{α−e_{j}},

we obtain

γ_{1} ⋆ (γ_{1}^{k}γ_{2}^{ℓ}) = [ ∑_{j,|α|=k} \frac{k!}{α!}(2ℓm_{j}^{2}m_{ε}^{ℓ−1}m_{α}∂_{z}^{α}Δ^{ℓ} + α_{j}m_{ε}^{ℓ}m_{α}∂_{z}^{α}Δ^{ℓ} + m_{ε}^{ℓ}m_{α+e_{j}}∂_{z}^{α+e_{j}}Δ^{ℓ})]

= 2ℓγ_{1}^{k}γ_{2}^{ℓ} + kγ_{1}^{k}γ_{2}^{ℓ} + [ ∑_{|β|=k+1} ∑_{j:β_{j}≥1} \frac{k!}{(β − e_{j})!}m_{ε}^{ℓ}m_{β}∂_{z}^{β}Δ^{ℓ}]

= (k + 2ℓ)γ_{1}^{k}γ_{2}^{ℓ} + [ ∑_{|β|=k+1} \frac{k!}{β!} (∑_{j} β_{j}) m_{ε}^{ℓ}m_{β}∂_{z}^{β}Δ^{ℓ}]

= (k + 2ℓ)γ_{1}^{k}γ_{2}^{ℓ} + [ ∑_{|β|=k+1} \frac{(k + 1)!}{β!}m_{ε}^{ℓ}m_{β}∂_{z}^{β}Δ^{ℓ}]

= (k + 2ℓ)γ_{1}^{k}γ_{2}^{ℓ} + γ_{1}^{k+1}γ_{2}^{ℓ}.

This proves (4.4). The argument for (4.5) is more complicated but uses the same techniques.

Next we will show that the orthogonalization procedure in Proposition 4.4 can be carried out to obtain explicit formulae for the p_{k,ℓ}’s in terms of γ_{1} and γ_{2}. We will prove:

Theorem 4.5.

h_{k}p_{k,ℓ} = \frac{1}{c_{k,ℓ}} ∑_{j=0}^{⌊k/2⌋} \frac{(−1)^{j}}{c_{k−j−1,j}} \frac{γ_{1}^{k−2j}γ_{2}^{ℓ+j}}{(k − 2j)!} .

The values c_{k,ℓ} are defined for k, ℓ ≥ 0 as in Proposition 4.1. In particular, we have c_{k,0} = 1 for all k ≥ 0. In addition, we define c_{−1,0} = 1, so that the formula reduces to p_{0,ℓ} = γ_{2}^{ℓ}/c_{0,ℓ} when k = 0. This is consistent with Proposition 4.1(a). Our proof of Theorem 4.5 uses two further lemmas.

Lemma 4.6. For non-negative integers k, ℓ, k′, ℓ′ one has

(a) ⟨γ_{1}^{k}γ_{2}^{ℓ}, p_{k′,ℓ′}⟩_{∗} = 0 if k′ + 2ℓ′ ≠ k + 2ℓ,

(b) ⟨γ_{1}^{k}γ_{2}^{ℓ}, p_{k′,ℓ′}⟩_{∗} = 0 if k′ + 2ℓ′ = k + 2ℓ and ℓ > ℓ′

(c) ⟨γ_{1}^{k}γ_{2}^{ℓ}, p_{k′,ℓ′}⟩_{∗} = k!c_{k′,ℓ′}/c_{k′,ℓ′−ℓ} i f k′ + 2ℓ′ = k + 2ℓ and ℓ ≤ ℓ′

Proof. Since γ_{1}^{k}γ_{2}^{ℓ} ∈ P_{k+2ℓ}(V)⊗\overline{P_{k+2ℓ}(V)} and p_{k′,ℓ′} ∈ P_{k′+2ℓ′}(V)⊗\overline{P_{k′+2ℓ′}(V)} ⊂ ℂ[V_{ℝ}], (a) follows immediately from Equation (2.14).

Equation (4.3) shows that γ_{1}^{k}γ_{2}^{ℓ} can be written as a linear combination of {p_{k−2j,ℓ+j} : j = 0, …, ⌊k/2⌋}. When ℓ > ℓ′ we have ⟨p_{k−2j,ℓ+j}, p_{k′,ℓ′}⟩_{∗} = 0 for all j = 0, …, ⌊k/2⌋ by Lemma 3.3. This proves (b).

To show (c), we first use Proposition 4.1 to write

\frac{γ_{1}^{k}γ_{2}^{ℓ}}{k!} = ∑_{j=0}^{⌊k/2⌋} \frac{h_{k−2j}p_{k−2j}γ_{2}^{j+ℓ}}{c_{k−2j,j}} = ∑_{j=0}^{⌊k/2⌋} h_{k−2j} \frac{c_{k−2j,j+ℓ}}{c_{k−2j,j}} p_{k−2j,j+ℓ}.

By Lemma 3.3, ⟨p_{k−2j,j+ℓ}, p_{k′,ℓ′}⟩_{∗} = δ_{k−2j,k′}δ_{j+ℓ,ℓ′}/h_{k−2j}, and thus

⟨\frac{γ_{1}^{k}γ_{2}^{ℓ}}{k!} , p_{k′,ℓ′}⟩_{∗} = ∑_{j=0}^{⌊k/2⌋} h_{k−2j} \frac{c_{k−2j,j+ℓ}}{c_{k−2j,j}} ⟨p_{k−2j,j+ℓ}, p_{k′,ℓ′}⟩_{∗} = \frac{c_{k′,ℓ′}}{c_{k′,ℓ′−ℓ}} .

Lemma 4.7. The polynomial Q_{m}(x) defined for m ≥ 2 as

Q_{m}(x) = ∑_{j=0}^{m}(−1)^{j}\binom{m}{j}(x − j − 2) ⋯ (x − j − m)

is identically zero.

Proof. Write (x − j − 2) ⋯ (x − j − m) = ∑_{k=0}^{m−1} A_{k}(x)j^{k} where A_{k}(x) is a polynomial in x independent of j. We then have

Q_{m}(x) = ∑_{k=0}^{m−1} (∑_{j=0}^{m}(−1)^{j}\binom{m}{j}j^{k}) A_{k}(x) = 0

since ∑_{j=0}^{m}(−1)^{j}\binom{m}{j}j^{k} = 0 for k = 0, …, m − 1. The latter fact is a standard combinatorial identity which can be proved by noting that

∑_{j=0}^{m}(−1)^{j}\binom{m}{j}j^{k}x^{j} = (x\frac{d}{dx})^{k} [(1 − x)^{m}]

vanishes at x = 1 for k = 0, …, m − 1.

Proof (of Theorem 4.5). The discussion following the statement of Theorem 4.5 shows that we can assume k > 0. Let p̃_{k,ℓ} = \frac{1}{c_{k,ℓ}} ∑_{j=0}^{⌊k/2⌋} \frac{(−1)^{j}}{c_{k−j−1,j}} \frac{γ_{1}^{k−2j}γ_{2}^{ℓ+j}}{(k−2j)!} . In view of Lemma 3.3, it suffices to show that

⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = δ_{k,k′}δ_{ℓ,ℓ′}

for all k′, ℓ′.

Parts (a) and (b) in Lemma 4.6 show that ⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = 0 when k′ + 2ℓ′ ≠ k + 2ℓ and when k′ + 2ℓ′ = k + 2ℓ but ℓ > ℓ′. Indeed, ⟨γ_{1}^{k−2j}γ_{2}^{ℓ+j}, p_{k′,ℓ′}⟩_{∗} = 0 in all such cases for j = 0, …, ⌊k/2⌋.

Now suppose that k′ + 2ℓ′ = k + 2ℓ and ℓ ≤ ℓ′. Let m = ℓ′ − ℓ so that k′ = k − 2m and ℓ′ = ℓ + m. Parts (b) and (c) in Lemma 4.6 yield

⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = \frac{1}{c_{k,ℓ}} ∑_{j=0}^{⌊k/2⌋} \frac{(−1)^{j}}{c_{k−j−1,j}(k − 2j)!}⟨γ_{1}^{k−2j}γ_{2}^{ℓ+j}, p_{k′,ℓ′}⟩_{∗}  (4.6)

= \frac{1}{c_{k,ℓ}} ∑_{j=0}^{m} \frac{(−1)^{j}}{c_{k−j−1,j}(k − 2j)!} \frac{(k − 2j)!c_{k′,ℓ′}}{c_{k′,ℓ′−(ℓ+j)}}

= \frac{c_{k−2m,ℓ+m}}{c_{k,ℓ}} ∑_{j=0}^{m} \frac{(−1)^{j}}{c_{k−j−1,j}c_{k−2m,m−j}}

When m ≥ 2, we substitute c_{k−j−1,j} = 4^{j}j!(k + n/2 − 2)!/(k + n/2 − 2 − j)! and c_{k−2m,m−j} = 4^{m−j}(m − j)!(k + n/2 − m − j − 1)!/(k + n/2 − 2m − 1)! in (4.6) and simplify to obtain

⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = \frac{c_{k−2m,ℓ+m}}{c_{k,ℓ}} \frac{(k + n/2 − 2m − 1)!}{4^{m}m!(k + n/2 − 2)!}Q_{m}(k + n/2) = 0

by Lemma 4.7.

When m = 1 Equation (4.6) reads

⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = \frac{c_{k−2,ℓ+1}}{c_{k,ℓ}} [\frac{1}{c_{k−1,0}c_{k−2,1}} − \frac{1}{c_{k−2,1}c_{k−2,0}} ] = 0.

Finally, when m = 0 Equation (4.6) reduces to

⟨p̃_{k,ℓ}, p_{k′,ℓ′}⟩_{∗} = ⟨p̃_{k,ℓ}, p_{k,ℓ}⟩_{∗} = \frac{c_{k,ℓ}}{c_{k,ℓ}} \frac{1}{c_{k−1,0}c_{k,0}} = 1

as desired.

Inhomogeneous invariants: Recall that the space of SO(n, ℝ) × 𝕋-invariant polynomials on ℂ^{n} has a canonical basis {q_{k,ℓ} : (k, ℓ) ∈ Λ} that is orthogonal with respect to the Fock inner product on ℂ[V_{ℝ}]. The polynomials q_{k,ℓ} are obtained via Gram-Schmidt orthogonalization from the p_{k,ℓ}’s using a suitable ordering and normalization.

The polynomials q_{k,ℓ} are expressed in terms of the p_{k,ℓ}’s in the form

q_{K,L} = ∑_{k+2ℓ≤K+2L}(−1)^{k+2ℓ}\left[{K, L \atop k, ℓ}\right]p_{k,ℓ}.

Our goal here is to obtain explicit formulae for the q_{k,ℓ}’s. Since Theorem 4.5 gives the p_{k,ℓ}’s explicitly in terms of γ_{1} and γ_{2}, we need only provide a formula for the generalized binomial coefficient \left[{K,L \atop k,ℓ}\right]. Recurrence relations for these coefficients were given in [6]. These are quite complicated and will not be repeated here, as we will prove:

Theorem 4.8. For (k, ℓ), (K, L) ∈ Λ one has

\left[{K, L \atop k, ℓ}\right] = \frac{c_{K,L}}{c_{k,ℓ}} ∑_{j=0}^{min(⌊k/2⌋,L−ℓ)} \binom{K + 2L − 2(ℓ + j)}{k − 2j}\frac{(−1)^{j}}{c_{k−j−1,j}c_{K,L−ℓ−j}}

when k + 2ℓ ≤ K + 2L and ℓ ≤ L. Otherwise \left[{K,L \atop k,ℓ}\right] = 0.

Proof. For (k, ℓ), (K, L) ∈ Λ with |(k, ℓ)| = k + 2ℓ ≤ K + 2L = |(K, L)|, Equation (3.4) yields

\left[{K, L \atop k, ℓ}\right] = ⟨\frac{γ_{1}^{m}}{m!} h_{k}p_{k,ℓ}, p_{K,L}⟩_{⋆}

where m = (K + 2L) − (k + 2ℓ). Using Theorem 4.5, this becomes

\left[{K, L \atop k, ℓ}\right] = \frac{1}{c_{k,ℓ}} ∑_{j=0}^{⌊k/2⌋} \frac{(−1)^{j}}{m!(k − 2j)!c_{k−j−1,j}} ⟨γ_{1}^{m+k−2j}γ_{2}^{ℓ+j}, p_{K,L}⟩_{∗}.

Parts (b) and (c) in Lemma 4.6 show that

⟨γ_{1}^{m+k−2j}γ_{2}^{ℓ+j}, p_{K,L}⟩_{∗} = \{ \begin{matrix}0 & if j > L − ℓ \\ \frac{(m+k−2j)!c_{K,L}}{c_{K,L−ℓ−j}} & if j ≤ L − ℓ \end{matrix} .

Thus we have \left[{K,L \atop k,ℓ}\right] = 0 when ℓ > L and

\left[{K, L \atop k, ℓ}\right] = \frac{1}{c_{k,ℓ}} ∑_{j=0}^{min(⌊k/2⌋,L−ℓ)}(−1)^{j} \frac{(m + k − 2j)!}{m!(k − 2j)!} \frac{c_{K,L}}{c_{k−j−1,j}c_{K,L−ℓ−j}}

= \frac{c_{K,L}}{c_{k,ℓ}} ∑_{j=0}^{min(⌊k/2⌋,L−ℓ)} \binom{K + 2L − 2(ℓ + j)}{k − 2j}\frac{(−1)^{j}}{c_{k−j−1,j}c_{K,L−ℓ−j}}

when ℓ ≤ L.

Product formulae: The method used to prove Theorem 4.8 can also be applied to products of the form p_{k,ℓ}p_{k′,ℓ′}. We first consider a product p_{k}p_{k′} = p_{k,0}p_{k′,0}. Since p_{k}p_{k′} is SO(n, ℝ) × 𝕋-invariant and homogeneous of degree 2(k + k′), we have

p_{k}p_{k′} = ∑_{K+2L=k+k′} A_{k,k′}^{K,L}p_{K,L}

for some numbers A_{k,k′}^{K,L}. In fact

A_{k,k′}^{K,L} = \frac{c_{K,L}h_{K}}{h_{k}h_{k′}} ∑_{\begin{matrix}0≤j≤⌊k/2⌋ \\ 0≤j′≤⌊k′/2⌋ \\ j+j′≤L \end{matrix}} \binom{k − 2j + k′ − 2j′}{k − 2j}\frac{(−1)^{j+j′}}{c_{k−j−1,j}c_{k′−j′−1,j′}c_{K,L−j−j′}} . (4.7)

Indeed, using Lemma 3.3, Theorem 4.5 and Lemma 4.6 we compute

A_{k,k′}^{K,L} = h_{K}⟨p_{k}p_{k′}, p_{K,L}⟩_{∗}

= \frac{h_{K}}{h_{k}h_{k′}} ∑_{j=0}^{⌊k/2⌋} ∑_{j′=0}^{⌊k′/2⌋} \frac{(−1)^{j+j′}⟨γ_{1}^{k−2j+k′−2j′}γ_{2}^{j+j′}, p_{K,L}⟩_{∗}}{(k − 2j)!(k′ − 2j′)!c_{k−j−1,j}c_{k′−j′−1,j′}}

= \frac{h_{K}}{h_{k}h_{k′}} ∑_{\begin{matrix}0≤j≤⌊k/2⌋ \\ 0≤j′≤⌊k′/2⌋ \\ j+j′≤L \end{matrix}} \frac{(−1)^{j+j′}(k − 2j + k′ − 2j′)!c_{K,L}}{(k − 2j)!(k′ − 2j′)!c_{k−j−1,j}c_{k′−j′−1,j′}c_{K,L−j−j′}} .

The formula for p_{k,ℓ}p_{k′,ℓ′} follows from the above by using Proposition 4.1(a). One obtains:

Proposition 4.9.

p_{k,ℓ}p_{k′,ℓ′} = \frac{1}{c_{k,ℓ}c_{k′ℓ′}} ∑_{K+2L=k+k′} \frac{c_{K,ℓ+ℓ′+L}}{c_{K,L}} A_{k,k′}^{K,L}p_{K,ℓ+ℓ′+L}

where A_{k,k′}^{K,L} is given by (4.7).

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D. Arnal and O. B. Baoua Département de Mathématiques Université de Metz Ile du Saulcy 57045 Metz cedex 01 France arnal@poncelet.univ-metz.fr boukary@poncelet.univ-metz.fr C. Benson and G. Ratcliff Dept. of Math. and Comp. Science Univ. of Missouri-St. Louis St. Louis, MO 63121 U.S.A. benson@arch.umsl.edu ratcliff@arch.umsl.edu

Received Mai 24, 2000 and in final form January 23, 2001