Appendix A: Representation Theory Point of View

In this appendix, we present an alternative construction of \(Y_D (a,\textrm)b\). Fix an integer n and two elements \(a\in V_\) and \(b\in V_\). As in Sect. 4.2, due to the integrability assumption from Definition 4.1(a), without loss of generality we can assume that all eigenspaces \(V_\Delta \) of H are finite dimensional; since we can choose finite-dimensional \(\mathfrak (D,\mathbb )\)–representations \(W'\subseteq V_\) and \(W''\subseteq V_\) such that \(a\in W'\) and \(b\in W''\).

Then the map \(\mu _n^1\), defined by Eq. (4.3), restricts to an element \(\mu _n^1\) \(\in \) \(\,}}(V_\) \(\otimes \) \(V_,\) \(V_)\) as in Eq. (4.5). Recall from the proof of Lemma 4.3 that \(\mu ^1_n\) is \(\mathfrak (D-1,\mathbb )\)–equivariant, where the \(\mathfrak (D-1,\mathbb )\)–subalgebra of \(\mathfrak (D,\mathbb )\) is spanned by \(\Omega _\) for \(2 \leqslant \alpha , \beta \leqslant D\). Thus, \(\mu _n^1\) is an invariant element of \(\,}}(V_ \otimes V_, V_)\) under the subalgebra \(\mathfrak (D-1,\mathbb )\), i.e., the action of \(\mathfrak (D-1,\mathbb )\) on it is trivial. Hence, we can apply the following lemma.

Lemma A.1

Let F be a representation of \(\mathfrak (D,\mathbb )\), which is decomposable into a direct sum of finite-dimensional irreducible \(\mathfrak (D,\mathbb )\)–representations. Assume that \(v \in F\) is invariant under the action of the subalgebra \(\mathfrak (D-1,\mathbb )\). Then v is contained in a subrepresentation of the type \(\mathop \nolimits _^ Q_m \otimes \mathbb _m^} [\textrm]\), where \(\mathbb _m^} [\textrm]\) are the \(\mathfrak (D,\mathbb )\)–representations of degree m homogeneous harmonic polynomials, and \(Q_m\) are multiplicity spaces with only a finite number of them nonzero.

Proof

Recall that the Lie algebra \(\mathfrak (D,\mathbb )\) is of type \(D_l\) for \(D=2\,l\) and of type \(B_l\) for \(D=2\,l+1\). In either case it has rank l. Denote by \(R(\Lambda )\) the irreducible highest weight \(\mathfrak (D,\mathbb )\)–module with highest weight \(\Lambda \), and denote by \(\pi _1,\dots ,\pi _l\) the fundamental weights. Then \(R(\pi _1) \cong \mathbb ^D\) is the vector representation. Its symmetric powers are (see, e.g., [15, 30]):

$$\begin S^m \mathbb ^D \cong \bigoplus _} R((m-2k)\pi _1) . \end$$

(A.1)

Therefore, an irreducible \(\mathfrak (D,\mathbb )\)–module \(R(\Lambda )\) is contained in \(S^* \mathbb ^D \cong \mathbb [\textrm]\) if and only if \(\Lambda =m\pi _1\) for some \(m\in \mathbb _\). It is well known that the space \(\mathbb _m^}[\textrm]\) is an irreducible \(\mathfrak (D,\mathbb )\)–representation with highest weight \(m\pi _1\), i.e.,

$$\begin \mathbb _m^}[\textrm] \cong R(m\pi _1) . \end$$

(A.2)

In fact, the decomposition (A.1) corresponds to the expression (2.10) of a degree m homogeneous polynomial in terms of \(\textrm^2\) and harmonic polynomials. The proof now follows immediately from the next lemma. \(\square \)

Lemma A.2

Let R be an irreducible \(\mathfrak (D,\mathbb )\)–module. Assume that there exists a nonzero vector \(v\in R\) annihilated by the subalgebra \(\mathfrak (D-1,\mathbb )\). Then \(R\cong \mathbb _m^}[\textrm]\) for some \(m\in \mathbb _\). Moreover, the vector v is unique up to a scalar multiple.

Proof

For \(D=2l\), in accordance with the branching rule (see, e.g., [15, Theorem 8.1.4]), the restriction of \(R(\lambda )\) to the subalgebra \(\mathfrak (2l-1,\mathbb ) \) is given by

$$\begin R(\lambda )\big |^_(2l-1,\mathbb ) } \cong \bigoplus _\nu R'(\nu ) . \end$$

(A.3)

Here \(R'(\nu )\) is the irreducible finite-dimensional representation of \(\mathfrak (2l-1,\mathbb ) \) with highest weight \(\nu \), and the sum is taken over all weights \(\nu \) satisfying the inequalities

$$\begin \lambda _1\geqslant \nu _1\geqslant \lambda _2\geqslant \nu _2\geqslant \cdots \geqslant \lambda _ \geqslant \nu _\geqslant |\lambda _| , \end$$

(A.4)

with all the \(\nu _i\) being simultaneously integers or half-integers together with the \(\lambda _i\).

For completeness, let us also consider the case \(D=2l+1\). Then, in accordance with the branching rule (see, e.g., [15, Theorem 8.1.3]), the restriction of \(R(\lambda )\) to the subalgebra \(\mathfrak (2l,\mathbb ) \) is given by

$$\begin R(\lambda )\big |^_(2l,\mathbb ) } \cong \bigoplus _ R'(\nu ) , \end$$

(A.5)

where \(R'(\nu )\) is the irreducible finite-dimensional representation of \(\mathfrak (2l,\mathbb ) \) with highest weight \(\nu \), and the sum is taken over all weights \(\nu \) satisfying the inequalities

$$\begin \lambda _1\geqslant \nu _1\geqslant \lambda _2\geqslant \nu _2\geqslant \cdots \geqslant \lambda _\geqslant \nu _\geqslant \lambda _\geqslant |\nu _l| , \end$$

(A.6)

with all the \(\nu _i\) being simultaneously integers or half-integers together with the \(\lambda _i\).

Now let us assume that the restriction \(R(\lambda )\big |^_(D-1,\mathbb )}\) contains the trivial representation \(R'(0)\). Then inequalities (A.4) and (A.6) imply that for \(\nu =(0,\dots ,0)\), one has \(\lambda =(\lambda _1,0,\dots ,0)\), where \(\lambda _1\) is a nonnegative integer. Therefore \(\lambda =m\pi _1\) for some \(m \in \mathbb _\). This means that \(R\cong R(m\pi _1) \cong \mathbb _m^}[\textrm]\).

Finally, the uniqueness of v follows from the fact that the decompositions (A.3) and (A.5) are multiplicity free; in particular, the trivial \(\mathfrak (D-1,\mathbb )\)–module \(R'(0)\) appears only once in them. \(\square \)

We remark that the unique (up to a scalar multiple) \(\mathfrak (D-1,\mathbb )\)–invariant element \(h_m(\textrm)\in \mathbb _m^}[\textrm]\) can be obtained as the image of the projection of \((z^1)^m \in S^m\mathbb ^D\) onto the summand \(R(m\pi _1)\cong \mathbb _m^}[\textrm]\) in the decomposition (A.1). This implies that \(h_m(\textrm)\) can be normalized so that \( h_m(x\textrm_1) = x^m \).

Remark A.1

A more explicit expression for \(h_m(\textrm)\) can be derived from [2, Eqs. (3.25), (3.29)]. In more details, a generating series for \(h_m(\textrm)\) is the expansion

$$\begin \iota __1,\textrm} \bigl ((\textrm_1+\textrm)^2\bigr )^&:= e^\cdot \partial _}} (\textrm^2)^ \Bigr |_=\textrm_1} \\&= \mathop \limits _^ \left( -D+2\\ m\end}\right) h_m(\textrm) \,. \end$$

We can further write

$$\begin h_m(\textrm) = \frac^2)^}-D+2\\ m\end}\right) } \, C^_m \bigl ( -z^1 (\textrm^2)^ \bigr ) \end$$

in terms of the Gegenbauer polynomials \(C^_m(x)\), which are defined by the expansion

$$\begin (1-2xt+t^2)^ = \mathop \limits _^ C^_m(x) \, t^m , \qquad 0\leqslant |x| <1 , \; |t| \leqslant 1 , \; \alpha >0 . \end$$

Let us fix a basis \(\(\textrm)\}_^_m^D}\) for \(\mathbb _m^}[\textrm]\) such that \(h_(\textrm)=h_m(\textrm)\) and

$$\begin h_(x\textrm_1) = \delta _ x^m ; \end$$

(A.7)

the last property can be achieved by subtracting from each \(h_\) a suitable multiple of \(h_\). As a consequence of Lemmas A.1 and A.2, we obtain that in the space \(\,}}(V_ \otimes V_,V_)\) there is a system of linearly independent elements \(\\}_\) such that:

(a):

For every fixed n, k and m, the subsystem \(\\}_^_m^D}\) is a basis of an irreducible \(\mathfrak (D,\mathbb )\)–subrepresentation isomorphic to \(\mathbb _m^}[\textrm]\), corresponding to the fixed basis \(\(\textrm)\}_^_m^D}\).

(b):

There is a decomposition \( \mu _n^1\big |_ \otimes V_} = \displaystyle \sum \limits _ \gamma ^_ f^n_ \) where the \(\gamma \)’s are complex numbers.

Then we can define

$$\begin Y_D(a,\textrm)b \,:= \, \mathop \limits _ \gamma ^_ \, f^n_ (a \otimes b) \, \bigl (\textrm^2\bigr )^} \, h_ (\textrm) . \end$$

(A.8)

By construction, this expression is \(\textrm(D,\mathbb )\)–equivariant; hence, it does not depend on the choice of basis \(\(\textrm)\}_^_m^D}\). Moreover, \(Y_D(a,x\textrm_1)=Y_1(a,x)\) by the assumption (A.7). Thus, (A.8) agrees with our previous definition, due to the uniqueness from Lemma 4.8.