# Models for irreducible representations of the symplectic algebra using
Dirac-type operators

Guner Muarem  Campus Middelheim
Middelheimlaan 1
M.G.221
2020 Antwerpen
België<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
(Date: 27 September 2023)

###### Abstract.

In this paper we will study both the finite and infinite-dimensional
representations of the symplectic Lie algebra $\mathfrak{sp}(2n)$ and develop
a polynomial model for these representations. This means that we will
associate a certain space of homogeneous polynomials in a matrix variable,
intersected with the kernel of $\mathfrak{sp}(2n)$-invariant differential
operators related to the symplectic Dirac operator with every irreducible
representation of $\mathfrak{sp}(2n)$. We will show that the systems of
symplectic Dirac operators can be seen as generators of parafermion algebras.
As an application of these new models, we construct a symplectic analogue of
the Rarita-Schwinger operator using the theory of transvector algebras.

###### Key words and phrases:

Clifford analysis, symplectic Dirac operators, representation theory, higher
spin, completely pointed modules, parastatistics

##  1\. Introduction

In the paper by Van Lancker, Sommen & Constales [32] the authors constructed
polynomial models for the orthogonal algebra $\mathfrak{so}(n)$. More
specifically, they associated with every $\mathfrak{so}(n)$-irreducible
representation $\mathbb{V}_{\lambda}$, a highest weight (vector) which was
realised in terms of homogeneous polynomials in a matrix variable. Of course,
this idea can be generalised for any Lie algebra $\mathfrak{g}$ and gives rise
to the following situation: Given an irreducible $\mathfrak{g}$-representation
of highest weight $(\lambda_{1},\dotsc,\lambda_{N})_{\mathfrak{g}}$ we want to
associate a polynomial model in terms of multi-homogeneous polynomials of
degree $(\lambda_{1},\dots,\lambda_{N})$ intersected with the kernel of some
$\mathfrak{g}$-invariant differential operators $G_{j}$ for $j\in J$, where
$J$ is a finite index set. Schematically, we are looking for the following
association:

$\displaystyle\begin{matrix}\text{Irreducible
$\mathfrak{g}$-representation}\\\
(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{g}}\end{matrix}\quad\longleftrightarrow\quad\begin{matrix}\text{Polynomial
model}\\\ \mathcal{P}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2n\times
N}\cap\ker(G_{j})\end{matrix}$

Note that the case of $\mathfrak{g}=\mathfrak{so}(n)$ was completely covered
by the aforementioned paper. The setting for these polynomial models is the
one of Clifford analysis: a hypercomplex function theory which revolves around
the Dirac operator on $\mathbb{R}^{n}$. This operator acts on functions with
values in the finite-dimensional spinor space $\mathbb{S}$ and is given by the
expression $\underline{\partial}_{x}=\sum_{j=1}^{n}e_{j}\partial_{x_{j}}$,
where $e_{j}$ are the so-called Clifford algebra generators satisfying
$\\{e_{j},e_{k}\\}=-2\delta_{jk}$. Note that this means that the elements of
the Clifford algebra are anti-commuting complex units. We refer the reader to
[12] for a thorough study of Clifford analysis. The key property of the Dirac
operator is that it factorises the Laplacian
$\Delta=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^{n}$ in the sense
that $\underline{\partial}_{x}^{2}=-\Delta$. This leads to the observation
that Clifford analysis is also a refinement of harmonic analysis. It is a
well-established fact in representation theory that the spaces
$\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})$ of harmonic polynomials, which
are homogeneous of degree $k$, form irreducible representations of the Lie
algebra $\mathfrak{so}(n)$ of highest weight
$(k,0,\dots,0)_{\mathfrak{so}(n)}$. The spinor representations (for $n$ even)
of highest weight
$(k+\frac{1}{2},\frac{1}{2},\dots,\pm\frac{1}{2})_{\mathfrak{so}(n)}$ can be
modelled by the spaces $\mathcal{M}_{k}(\mathbb{R}^{n},\mathbb{S})$ of
$k$-homogeneous solutions of the Dirac operator known as monogenics. When
working with (spinor-valued) functions depending on matrix variables one needs
to work with systems of Dirac and Laplace operators (and associated operators,
see later) which are related to the theory of Howe dualities [24]. In [32]
(and already started in [20]) this idea was used to associate with weights of
the form $(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{so}(n)}$ with dominant
weight condition $\lambda_{1}\geq\cdots\lambda_{N}$ certain spaces of
simplicial harmonics and monogenics (we come back to this in Section 2),
related to these aforementioned systems of differential operators (see also
[8]). The goal of this paper is to focus on the case where
$\mathfrak{g}=\mathfrak{sp}(2n)$ in the setting of symplectic Clifford
analysis, see [9]. The notion of the Dirac operator naturally generalises to
the symplectic framework: the symplectic Dirac operator $D_{s}$ is now
obtained as a contraction between the Weyl algebra generators
$\\{z_{1},\dots,z_{n},\partial_{z_{1}},\dots,\partial_{z_{n}}\\}$ and the
vector fields $\\{\partial_{x_{1}},\dots,\partial_{y_{n}}\\}$ using the
canonical symplectic form $\omega_{0}$ on $\mathbb{R}^{2n}$. In other words,
we can write
$D_{s}=\langle\underline{z},\underline{\partial}_{y}\rangle-\langle\underline{\partial}_{x},\underline{\partial}_{z}\rangle$,
where $\langle\cdot,\cdot\rangle$ is the usual Euclidean inner product on
$\mathbb{R}^{n}$. Note that this definition of the symplectic Dirac operator
is slightly different than the one of [9] but was already used in [14, 15]. We
first of all stress that obtaining polynomial models for
$\mathfrak{g}=\mathfrak{sp}(2n)$ will be slightly more intricate as the
symplectic spinor space $\mathbb{S}^{\infty}$ is infinite-dimensional.
Therefore, we will work in two steps:

  1. (i)

We will first consider the finite-dimensional representations. The
$\mathfrak{sp}(2n)$-invariant differential operators $G_{j}$ (from the scheme
above) then will lead to the notion of symplectic simplicial harmonics (see
Section 3). In some sense, these results are anticipated by the general theory
of Howe dualities (see [24]).

  2. (ii)

Afterwards, we will focus on an interesting class of infinite-dimensional
representations of highest weight
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty},$
where $\boxtimes$ denotes the Cartan product and

$\mathbb{S}^{\infty}=\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

These types of representations are connected with the theory of the symplectic
Dirac operator. As a consequence, this will lead to the novel notion of
symplectic simplicial monogenics (see Section 4) and a connection with
parastatistics.

As an application, we will define ‘higher spin’ Dirac operators (the Rarita-
Schwinger operator describing particles with spin-$\frac{3}{2}$ is an example
of such an operator) in the symplectic framework. We will call these higher
metaplectic Dirac operators. By means of example, we will introduce the
symplectic Rarita-Schwinger operator in Section 5.

##  2\. Simplicial harmonics and monogenics

###  2.1. The harmonic Fischer decomposition

Consider the regular action of the Lie group $\mathsf{SO}(n)$ on the space of
polynomials $\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$:

$H:\mathsf{SO}(n)\to\operatorname{\mathsf{Aut}}(\mathcal{P}(\mathbb{R}^{n},\mathbb{C})),\quad
g\mapsto\left(H(g)[P](\underline{x}):=P(g^{-1}\underline{x})\right).$

On the level of the Lie algebra $\mathfrak{so}(n)$, the so-called derived
action $dH$ on the polynomial space $\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$
is given by the angular momenta operators (known from quantum mechanics) which
are defined by means of $L_{ab}=x_{a}\partial_{x_{b}}-x_{b}\partial_{x_{a}}$
for $1\leq a<b\leq n$. In order to decompose the space of polynomials
$\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$ into irreducible representations for
the orthogonal Lie algebra, we first note that the space can be written as a
direct sum of $k$-homogeneous polynomials, i.e.
$\mathcal{P}(\mathbb{R}^{n},\mathbb{C})=\bigoplus_{k=0}^{\infty}\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C}).$
The space of $k$-homogeneous polynomials
$\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})$ further decomposes under the
action of $\mathsf{SO}(n)$ in terms of solutions of the Laplacian on
$\mathbb{R}^{n}$. The space of $k$-homogeneous harmonics is denoted by

$\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C}):=\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})\cap\ker(\Delta).$

Each of the spaces $\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})$ (for $n\geq
3)$ forms an irreducible $\mathfrak{so}(n)$-representation of highest weight
$(k)_{\mathfrak{so}(n)}$ (see [20]). Using these
$\mathfrak{so}(n)$-irreducible modules
$\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})$ we can further decompose the
space of $k$-homogeneous polynomials (see [18])

(1)
$\displaystyle\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})=\bigoplus_{j=0}^{\lfloor\frac{k}{2}\rfloor}|\underline{x}|^{2j}\mathcal{H}_{k-2j}(\mathbb{R}^{n},\mathbb{C}).$

The decomposition from above is not multiplicity-free: every irreducible
representation $\mathcal{H}_{j}(\mathbb{R}^{n},\mathbb{C})$ appears infinitely
many times on each row. In order to make this decomposition multiplicity-free,
one can use a hidden symmetry (apart from the obvious
$\mathsf{SO}(n)$-invariance). The multiplication operator
$|\underline{x}|^{2}$ allows us the raise the degree by two. On the other
hand, the Laplace operator $\Delta$, reduces the degree by two. Lastly, the
Euler operator $\mathbb{E}=\sum_{j=1}^{n}x_{j}\partial_{x_{j}}$ measures the
degree of homogeneity of the corresponding
$\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})$. This can be visualised as
follows:

${0}$${\mathcal{H}_{k}}$${\left|\underline{x}\right|^{2}\mathcal{H}_{k}}$${\left|\underline{x}\right|^{4}\mathcal{H}_{k}}$${\left|\underline{x}\right|^{6}\mathcal{H}_{k}\
\
\cdots}$$\scriptstyle{\Delta}$$\scriptstyle{\left|\underline{x}\right|^{2}}$$\scriptstyle{\Delta}$$\scriptstyle{\mathbb{E}}$$\scriptstyle{\left|\underline{x}\right|^{2}}$$\scriptstyle{\Delta}$$\scriptstyle{\mathbb{E}}$$\scriptstyle{\left|\underline{x}\right|^{2}}$$\scriptstyle{\mathbb{E}}$$\scriptstyle{\Delta}$

One can check that when these three operators are re-scaled as

$\displaystyle X:=-\frac{1}{2}\Delta,\qquad
Y:=\frac{1}{2}|\underline{x}|^{2},\qquad
H:=[X,Y]=-\left(\mathbb{E}+\frac{n}{2}\right),$

we have $\operatorname{Alg}(X,Y,H)\cong\mathfrak{sl}(2)$. This algebra is the
‘hidden symmetry’ which was mentioned above. As a result, the space
$\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$ has a multiplicity-free decomposition
under the joint action of the (harmonic) Howe dual pair
$\mathsf{SO}(n)\times\mathfrak{sl}(2)$.

###  2.2. The refined monogenic Fischer decomposition

Let $\\{e_{1},\dots,e_{n}\\}$ be an orthonormal basis for $\mathbb{R}^{n}$ and
let $\mathcal{F}(\mathbb{R}^{n},\mathbb{C})$ be a function space (e.g. the
space of complex-valued polynomials). Then the Dirac operator on
$\mathbb{R}^{n}$ is defined as
$\underline{\partial}_{x}=\sum_{j=1}^{n}e_{j}\partial_{x_{j}}$. The
multiplication operator is given by $\underline{x}=\sum_{j=1}^{n}e_{j}x_{j}.$
A solution $f$ of the Dirac operator $\underline{\partial}_{x}$ is called a
monogenic function. The space of all monogenic polynomials of degree
$k\in\mathbb{N}$ is denoted

$\displaystyle\mathcal{M}_{k}(\mathbb{R}^{n},\mathbb{S}):=\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{S})\cap\ker\underline{\partial}_{x}.$

We already mentioned that Clifford analysis is a refinement of harmonic
analysis. This refinement is also reflected in terms of the harmonic Howe
duality $\mathsf{SO}(n)\times\mathfrak{sl}(2)$ found in the previous section.
The Dirac operator is in a spin-invariant operator, so that the group part of
the refined Howe duality is $G=\operatorname{\mathsf{Spin}}(n)$.

The ‘hidden symmetry’ is then determined by the algebra generated by the Dirac
operator $\underline{\partial}_{x}$ and its adjoint $\underline{x}$. It is a
well-known fact that is no longer a Lie algebra, but a Lie superalgebra
$\mathfrak{osp}(1|2)$. This means that we obtain the refined Howe duality
$\operatorname{\mathsf{Spin}}(n)\times\mathfrak{osp}(1|2)$ which governs the
decomposition of the space
$\mathcal{P}(\mathbb{R}^{n},\mathbb{C})\otimes\mathbb{S}$ of spinor-valued
polynomials. However, we already know that the space of polynomials decomposes
into the space of harmonics. This then leads to the following decomposition of
the space of spinor-valued harmonics
$\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{S}):=\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})\otimes\mathbb{S}$:

######  Theorem 2.1 (Monogenic refinement [2]).

Let $H_{k}(\underline{x})\in\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{S})$ be a
$k$-homogeneous harmonic polynomial with spinor values. Then one has

$H_{k}(\underline{x})={M}_{k}(\underline{x})+\underline{x}{M}_{k-1}(\underline{x}),$

where $M_{j}(\underline{x})\in\mathcal{M}_{j}(\mathbb{R}^{n},\mathbb{S})$ (we
ignore the parity of the spinors here).

The definition of $k$-homogeneous monogenics can now be given an algebraic
interpretation. The refinement theorem above can be formulated, purely in
terms of $\mathfrak{so}(n)$ weights, as the tensor product (for $n$ even):

$\left(k\right)_{\mathfrak{so}(n)}\otimes\left(\frac{1}{2},\dots,\frac{1}{2},\pm\frac{1}{2}\right)_{\mathfrak{so}(n)}.$

The tensor product rules for $\mathfrak{so}(n)$ (see for example [19]) allow
us to decompose this tensor product (up to isomorphism) as follows:

$\displaystyle\left(k\right)_{\mathfrak{so}(n)}\otimes\left(\frac{1}{2},\dots,\pm\frac{1}{2}\right)_{\mathfrak{so}(n)}\cong\left(k+\frac{1}{2},\dots,\pm\frac{1}{2}\right)_{\mathfrak{so}(n)}\oplus\left(k-\frac{1}{2},\dots,\mp\frac{1}{2}\right)_{\mathfrak{so}(n)}.$

In order to make the isomorphims from above into an equality, we need the so-
called embedding factors. These are $\mathfrak{so}(n)$-invariant differential
operators. The first component in the tensor product decomposition is
trivially embedded using the identity operator, whereas the second component
is embedded by the operator $\underline{x}$. This now leads to the monogenic
refinement of the Fischer decomposition as described in Theorem 2.1. In other
words, $\mathcal{M}_{k}(\mathbb{R}^{n},\mathbb{S})$ is exactly the Cartan
product (for $n$ even)

$\left(k\right)_{\mathfrak{so}(n)}\boxtimes\left(\frac{1}{2},\dots,\frac{1}{2},\pm\frac{1}{2}\right)_{\mathfrak{so}(n)}=\left(k+\frac{1}{2},\dots,\frac{1}{2},\pm\frac{1}{2}\right)_{\mathfrak{so}(n)}=(k)^{\prime}_{\mathfrak{so}(n)}.$

We now obtain the full Fischer decomposition of the space of spinor-valued
polynomials:

######  Theorem 2.2 (Complete monogenic FD).

Under the joint action of $\mathsf{Spin}(n)\times\mathfrak{osp}(1|2)$, the
space of spinor valued polynomials decomposes as

$\mathcal{P}(\mathbb{R}^{n},\mathbb{S})=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{k}\underline{x}^{j}\mathcal{M}_{k-j}(\mathbb{R}^{n},\mathbb{S}).$

###  2.3. Simplicial harmonic and monogenics

In the previous section we considered polynomials in the vector variable
$\underline{x}=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ and considered the
multiplicity-free decomposition governed by the Howe duality
$\mathsf{SO}(n)\times\mathfrak{sl}(2)$. We will now work with $N$ vector
variables $(\underline{u}_{1},\dots,\underline{u}_{N})\in\mathbb{R}^{n\times
N}$. Note that this is also often called a matrix variable (see Chapter 3,
Section 4 in [20] for more details), we denote it by
$\underline{U}=(\underline{u}_{1},\dots,\underline{u}_{N})\in\mathbb{R}^{n\times
N}$. The Howe duality $\mathsf{SO}(n)\times\mathfrak{sl}(2)$ which governs the
multiplicity-free decomposition of the space of polynomials
$\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$ is only the first example of the
general Howe duality $\mathsf{SO}(n)\times\mathfrak{sp}(2N)$ (see e.g. [21])
associated with the regular action

$H[g][P](\underline{U}):=P(g^{-1}\underline{u}_{1},\dots,g^{-1}\underline{u}_{N})$

of $\mathsf{SO}(n)$ on the space of polynomials
$\mathcal{P}(\mathbb{R}^{n\times N},\mathbb{C})$ in a matrix variable.

Note that in the spinor-valued case, the action is naturally extended by the
multiplicative action of spinor space. The space
$\mathcal{P}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{n\times
N},\mathbb{C})$ of polynomials which are $\lambda_{j}$-homogeneous in each
variable $\underline{u}_{j}$ also has a decomposition in terms of
$\mathsf{SO}(n)$-irreducible representations under the regular action above.
The irreducible modules appearing here, are in fact a generalisation of the
spaces $\mathcal{H}_{k}(\mathbb{R}^{n},\mathbb{C})$ from the previous section
and are defined as follows (see [32]):

######  Definition 2.3.

Let $N$ be an integer such that $1\leq N\leq\lfloor\frac{n}{2}\rfloor$. A
function $f:\mathbb{R}^{n\times
N}\to\mathbb{C},(\underline{u}_{1},\dots,\underline{u}_{N})\mapsto
f(\underline{u}_{1},\dots,\underline{u}_{N})$ is called simplicial harmonic if
it satisfies the system of equations

$\displaystyle\langle\underline{\partial}_{u_{i}},\underline{\partial}_{u_{j}}\rangle
f=0$ $\displaystyle\quad\text{for all }1\leq i,j\leq N$
$\displaystyle\langle\underline{u}_{i},\underline{\partial}_{u_{j}}\rangle
f=0$ $\displaystyle\quad\text{for all }1\leq i<j\leq N.$

The vector space of simplicial harmonic polynomials which are homogeneous of
degree $\lambda_{i}$ in the vector variable $\underline{u}_{i}$ is denoted by
$\mathcal{H}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{n\times
N},\mathbb{C})$. Note that we must have $\lambda_{1}\geq\cdots\lambda_{N}\geq
0$ in order to satisfy the dominant weight condition.

######  Definition 2.4.

Let $N$ be an integer such that $1\leq N\leq\lfloor\frac{n}{2}\rfloor$. A
spinor-valued function $f:\mathbb{R}^{n\times
N}\to\mathbb{S},(\underline{u}_{1},\dots,\underline{u}_{N})\mapsto
f(\underline{u}_{1},\dots,\underline{u}_{N})$ is called simplicial monogenic
if it satisfies the system of equations

$\displaystyle\underline{\partial}_{u_{i}}f=0$ $\displaystyle\quad\text{for
all }1\leq i\leq N$
$\displaystyle\langle\underline{u}_{i},\underline{\partial}_{u_{j}}\rangle
f=0$ $\displaystyle\quad\text{for all }1\leq i<j\leq N.$

The vector space of simplicial harmonic polynomials which are homogeneous of
degree $\lambda_{i}$ in the vector variable $\underline{u}_{i}$ is denoted by
$\mathcal{S}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{n\times
N},\mathbb{S})$.

######  Theorem 2.5 (Van Lancker, Sommen & Constales [32]).

Let $N$ be an integer such that $1\leq N\leq\lfloor\frac{n}{2}\rfloor$ and
suppose that the dominant weight condition is satisfied (for the Lie algebra
$\mathfrak{so}(n))$. Then we have the following properties:

  1. (i)

The spaces
$\mathcal{H}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{n},\mathbb{C})$ are
irreducible of highest weight
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{so}(n)}$.

  2. (ii)

The spaces
$\mathcal{S}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{n},\mathbb{C})$ are
irreducible of highest weight
$(\lambda_{1},\dots,\lambda_{N})^{\prime}_{\mathfrak{so}(n)}$.

In [3, 11] these models were proven to be useful to construct so-called higher
spin Dirac operators. In the rest of this paper we will focus on the
symplectic Lie algebra $\mathfrak{sp}(2n)$ instead of $\mathfrak{so}(n)$. This
implies that the symmetric bilinear inner product on $\mathbb{R}^{n}$ will be
replaced by a skew-symmetric one on $\mathbb{R}^{2n}$. More specifically, it
has the following $(2n\times 2n)$-block matrix representation

$\Omega_{0}=\begin{pmatrix}0&I_{n}\\\ -I_{n}&0\end{pmatrix}.$

Later on, we will also use the notation

(2)
$\displaystyle\langle\underline{v},\underline{w}\rangle_{s}:=\underline{v}^{T}\Omega_{0}\underline{w}$

for all $\underline{v},\underline{w}\in\mathbb{R}^{2n}$. As a consequence, the
finite-dimensional Clifford algebra becomes an infinite-dimensional Weyl
algebra and the Dirac operator is replaced by the symplectic Dirac operator.
Due to the infinite-dimensionality of the symplectic spinors, this transition
is rather intricate.

######  Remark 2.6.

We further mention that there exist generalisations of the Fischer
decomposition from equation (1) and the refinement from Theorem 2.1 for matrix
variables, see [26, 31].

##  3\. The notion of symplectic simplicial harmonics

So far we discussed finite-dimensional irreducible representations of the Lie
algebra $\mathfrak{so}(n)$ and characterised these modules in terms of
simplicial harmonics and simplicial monogenics (see Theorem 2.5). We will now
do exactly the same thing for the symplectic Lie algebra $\mathfrak{sp}(2n)$.
We first focus on the finite-dimensional representations of
$\mathfrak{sp}(2n)$. Recall that the finite-dimensional irreducible
representations of the symplectic algebra are uniquely characterised by their
labels satisfying $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$.
In order to get some intuition with these finite-dimensional
$\mathfrak{sp}(2n)$-irreducible representations, we start by calculating their
dimension using the Weyl dimension formula (see [6] for more background).

######  Proposition 3.1.

Let $g_{i}:=n-i+1$ and $m_{i}:=\lambda_{i}+g_{i}$ for $1\leq i\leq n$ then

$\dim(\lambda_{1},\dots,\lambda_{n})_{\mathfrak{sp}(2n)}=\prod_{i}\left(\frac{m_{i}}{g_{i}}\right)\prod_{i<j}\left(\frac{m_{i}-m_{j}}{g_{i}-g_{j}}\right)\prod_{i<j}\left(\frac{m_{i}+m_{j}}{g_{i}+g_{j}}\right).$

Using this formula, one can check that the dimension of module
$(k)_{\mathfrak{sp}(2n)}$ is equal to

$\dim(k)_{\mathfrak{sp}(2n)}=\binom{k+2n-1}{2n-1}.$

One might recognize this number as the dimension of the space of
$k$-homgeneous polynomials $\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C})$. This
is no coincidence:

######  Proposition 3.2.

The space of $k$-homogeneous polynomials
$\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C})$ is an irreducible
$\mathfrak{sp}(2n)$-module of weight $(k)_{\mathfrak{sp}(2n)}$ and the
corresponding highest weight vector is $w_{k}(\underline{x})=x_{1}^{k}$.

###### Proof.

The generators of Lie algebra $\mathfrak{sp}(2n)$ can be realised on the space
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})$ as the following differential
operators:

(3)
$\displaystyle\begin{cases}Y_{jj}=x_{j}\partial_{y_{j}}\qquad&j=1,\dots,n\qquad
n\\\ Z_{jj}=y_{j}\partial_{x_{j}}\qquad&j=1,\dots,n\qquad n\\\
X_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}\qquad&j,k=1,\dots,n\qquad
n^{2}\\\
Y_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}\qquad&j<k=1,\dots,n\qquad\frac{n(n-1)}{2}\\\
Z_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}\qquad&j<k=1,\dots,n\qquad\frac{n(n-1)}{2}\end{cases}$

The positive roots are $\Phi^{+}=\\{X_{jk},Y_{jk},Y_{jj}\mid 1\leq j<k\leq
n\\}$ and the Cartan algebra is given by
$X_{jj}=x_{j}\partial_{x_{j}}-y_{j}\partial_{y_{j}}$ for $1\leq j\leq n$. One
easily checks that $w_{k}=x_{1}^{k}$ is a solution of all the positive roots.
Moreover, the action of the Cartan algebra $\mathfrak{h}$ on $w_{k}$ gives
$H_{1}x_{1}^{k}=kx_{1}^{k}$ and $H_{2}x_{1}^{k}=\cdots=H_{n}x_{1}^{k}=0$. ∎

######  Remark 3.3.

At this point it is worthwhile to compare the orthogonal and symplectic
framework. In the orthogonal setting, the space of $k$-homogeneous polynomials
is reducible as an $\mathfrak{so}(n)$-module and further decomposes into
harmonic polynomials (the latter are irreducible modules). In other words, we
have

$\displaystyle(k)_{\mathfrak{so}(n)}\longleftrightarrow\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})\cap\ker(\Delta).$

In the symplectic framework, we have

$\displaystyle(k)_{\mathfrak{sp}(2n)}\longleftrightarrow\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C}).$

In some sense, this is to be expected as the Laplace operator $\Delta$ can be
written as $\langle\underline{\partial}_{x},\underline{\partial}_{x}\rangle$
whereas the natural operator in the symplectic setting would be
$\langle\underline{\partial}_{x},\underline{\partial}_{x}\rangle_{s}$ which is
of course zero, so that no extra condition is to be imposed.

######  Remark 3.4.

Note that the remark above implies that there is no natural way to associate a
second-order differential operator (the Laplacian) to the symplectic Dirac
operator $D_{s}$ on $\mathbb{R}^{n}$. However, in [23] it was shown that if
one allows a compatible complex structure $\mathbb{J}$ on $\mathbb{R}^{2n}$
one can define a second symplectic Dirac operator $D_{t}$ (which is basically
a rotation of the operators $D_{s}$ using the complex structure) such that
$[D_{s},D_{t}]=-i\Delta$. Recently, this led to a hermitian refinement of
symplectic Clifford analysis in which the notion of complex harmonics was used
to describe the solution space of the symplectic Dirac operators $D_{s}$ and
$D_{t}$ using a Fischer decomposition, see [16].

###  3.1. The case of $N=2$

We now focus on the case where $N=2$ which allows us to take a look at the
more general module $(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}$ with
$\lambda_{1}\geq\lambda_{2}\geq 0$. We denote the two vector variables
$(\underline{x},\underline{u})\in\mathbb{R}^{2n\times 2}$ by

$\displaystyle\underline{x}$
$\displaystyle=(x_{1},\dots,x_{n},y_{1},\dots,y_{n})\in\mathbb{R}^{2n}$
$\displaystyle\underline{u}$
$\displaystyle=(u_{1},\dots,u_{n},v_{1},\dots,v_{n})\in\mathbb{R}^{2n}.$

At this point, one might expect that
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\leftrightarrow\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n},\mathbb{C})$.
This is however not the case as
$\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n},\mathbb{C})$ is not
irreducible as $\mathfrak{sp}(2n)$-module, as illustrated in the following
counterexample:

######  Example 3.5.

We start by calculating the dimension of the space
$\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n},\mathbb{C})$:

$\displaystyle\dim(\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n},\mathbb{C}))={\lambda_{1}+2n-1\choose
2n-1}{\lambda_{2}+2n-1\choose 2n-1}.$

Let us now take $n=4$ and consider by means of example the module
$(2,1)_{\operatorname{\mathfrak{sp}}(8)}$. The dimension of the polynomial
algebra is given by

$\dim(\mathcal{P}_{2,1}(\mathbb{R}^{4},\mathbb{C}))={9\choose 7}{8\choose
7}=288.$

However, one can check that $\dim(2,1)_{\operatorname{\mathfrak{sp}}(8)}=160$.
This means that the suggested model
$\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n},\mathbb{C})$ is too big
and some kind of reduction has to be done.

The tool for this dimensional reduction is the Howe duality
$\operatorname{\mathsf{Sp}}(2n)\times\mathfrak{so}(4)$.

######  Proposition 3.6.

Let $N=2$ and $(\underline{x},\underline{u})\in\mathbb{R}^{2n\times 2}$. Then
the irreducible representation given by the weight
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}$ can be realised in terms of
homogeneous polynomials as follows:

$\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n\times
2},\mathbb{C})\cap\ker(\langle\underline{x},\underline{\partial}_{u}\rangle,\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s})$

where the operators $\langle\underline{x},\underline{\partial}_{u}\rangle$ and
$\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s}$ are
$\operatorname{\mathfrak{sp}}(2n)$-invariant operators and
$\langle\cdot,\cdot\rangle_{s}$ is the ‘symplectic inner product’ (see
equation 2).

###### Proof.

We construct $\mathfrak{sp}(2n)$-invariant operators by contraction of the
coordinates and corresponding derivatives by using $\langle\cdot,\cdot\rangle$
or $\langle\cdot,\cdot\rangle_{s}$. By a paring argument, we expect ${4\choose
2}=6$ symplectic invariant operators. They are given by:
$\langle\underline{u},\underline{x}\rangle_{s}$,
$\langle\underline{u},\underline{\partial}_{x}\rangle$,
$\langle\underline{x},\underline{\partial}_{u}\rangle$,
$\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s}$,
$\langle\underline{u},\underline{\partial}_{u}\rangle+n$ and
$\langle\underline{x},\underline{\partial}_{x}\rangle+n$. Note that for the
mixed contractions we use the usual Euclidean inner product, because otherwise
the resulting operator is not symplectic invariant. These six
$\operatorname{\mathfrak{sp}}(2n)$-invariant operators span a copy of the
orthogonal algebra $\mathfrak{so}(4)$ (see for example Wallach [21], Theorem
5.6.14). Recall that Cartan algebra $\mathfrak{h}\subseteq\mathfrak{so}(4)$ is
given by

$\mathfrak{h}=\begin{cases}H_{1}=\langle\underline{u},\underline{\partial}_{u}\rangle+n\\\
H_{2}=\langle\underline{x},\underline{\partial}_{x}\rangle+n\end{cases}$

We now claim that the highest weight vector is given by

${w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})=x_{1}^{\lambda_{1}-\lambda_{2}}(x_{1}u_{2}-x_{2}u_{1})^{\lambda_{2}}}.$

Note that this highest weight vector is independent of
$(\underline{v},\underline{y})$. To confirm this, we check the following:

  1. (i)

The vector $w_{\lambda_{1},\lambda_{2}}$ is annihilated by the positive roots
of the $\mathfrak{so}(4)$. We have

$\displaystyle
x_{1}{\partial_{u_{1}}}{\left(x_{1}^{\lambda_{1}-\lambda_{2}}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}}\right)}$
$\displaystyle=-\lambda_{2}x_{2}x_{1}^{\lambda_{1}-\lambda_{2}+1}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}-1}$
$\displaystyle
x_{2}{\partial_{u_{2}}}{\left(x_{1}^{\lambda_{1}-\lambda_{2}}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}}\right)}$
$\displaystyle=\lambda_{2}x_{2}x_{1}^{\lambda_{1}-\lambda_{2}+1}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}-1}$

together with the fact that $w_{\lambda_{1},\lambda_{2}}$ is independent of
the variables $u_{3},\dots,u_{n}$ and $v_{3},\dots,v_{n}$ we can conclude that
$\langle\underline{x},\partial_{\underline{u}}\rangle
w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})=0$. One now similarly
checks that
$\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s}w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})=0$.

  2. (ii)

Moreover, $w_{\lambda_{1},\lambda_{2}}$ must be annihilated by the positive
roots vectors $\Phi^{+}_{\operatorname{\mathfrak{sp}}(2n)}$. A representation
of the Lie algebra $\mathfrak{sp}(2n)$ on the space
$\mathcal{P}(\mathbb{R}^{2n\times 2},\mathbb{C})$ is given by:

$\displaystyle\begin{cases}X_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}+u_{j}\partial_{u_{k}}-v_{k}\partial_{v_{j}}\qquad&j,k=1,\dots,n\\\
Y_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}+u_{j}\partial_{v_{k}}+u_{k}\partial_{v_{j}}\qquad&j<k=1,\dots,n\\\
Z_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}+u_{j}\partial_{v_{j}}\qquad&j<k=1,\dots,n\\\
Y_{jj}=x_{j}\partial_{y_{j}}+u_{j}\partial_{v_{j}}\qquad&j=1,\dots,n\\\
Z_{jj}=y_{j}\partial_{x_{j}}+v_{j}\partial_{x_{j}}\qquad&j=1,\dots,n\end{cases}$

Due to the fact that $w_{\lambda_{1},\lambda_{2}}$ only depends on the
variables $(x_{1},x_{2},u_{1},u_{2})$, we immediately see that
$w_{\lambda_{1},\lambda_{2}}$ is a solution of the differential operators
$Y_{jj}$ for all $1\leq j\leq n$. For the operators $X_{jk}$ with $1\leq
j<k\leq n$ we have

$\displaystyle X_{12}w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})$
$\displaystyle=(x_{1}\partial_{x_{2}}-y_{2}\partial_{y_{1}}+u_{1}\partial_{u_{2}}-v_{2}\partial_{v_{1}})w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})$
$\displaystyle=x_{1}\partial_{x_{2}}+u_{1}\partial_{u_{2}}w_{\lambda_{1,2}}(\underline{x},\underline{u})$
$\displaystyle=-u_{1}x_{1}x_{1}^{\lambda_{1}-\lambda_{2}}+u_{1}x_{1}x_{1}^{\lambda_{1}-\lambda_{2}}$
$\displaystyle=0$

and completely similar for the other operators $X_{jk}$. Repeating the
strategy from above, we also obtain $Y_{jk}w_{\lambda_{1},\lambda_{2}}=0$ for
all $1\leq j<k\leq n$.

  3. (iii)

The action of the Cartan algebra $\mathfrak{h}=\\{X_{jj}:j=1,\dots,n\\}$ of
$\operatorname{\mathfrak{sp}}(2n)$ must give the eigenvalues
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}$ (recall that the redundant
zeros are omitted here). For ${X}_{11}$ we obtain

$\displaystyle
x_{1}\left((\lambda_{1}-\lambda_{2})x_{1}^{\lambda_{1}-\lambda_{2}-1}(x_{1}u_{2}-x_{2}u_{1})^{\lambda_{2}}+x_{1}^{\lambda_{1}-\lambda_{2}}\lambda_{2}(x_{1}u_{2}-x_{2}u_{1})^{\lambda_{2}-1}u_{2}\right)$
$\displaystyle+u_{1}\left(x_{1}^{\lambda_{1}-\lambda_{2}}\lambda_{2}(x_{1}u_{2}-x_{2}u_{1})^{\lambda_{2}-1}(-x_{2})\right)$

This can be reduced to
$\lambda_{1}w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})$ as
wanted. By computing the action of $X_{22}$ on
$w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})$ we obtain

$\lambda_{2}u_{2}x_{1}^{\lambda_{1}-\lambda_{2}+1}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}-1}-\lambda_{2}u_{1}x_{2}x_{1}^{\lambda_{1}-\lambda_{2}}\left(u_{2}x_{1}-u_{1}x_{2}\right){}^{\lambda_{2}-1}=\lambda_{2}w_{\lambda_{1},\lambda_{2}}.$

Moreover, $X_{jj}w_{\lambda_{1},\lambda_{2}}=0$ for $j>2$.

∎

We now come to the notion of simplicial harmonics in the symplectic framework
for $N=2$ (compare with Definition 2.3).

######  Definition 3.7.

For $\lambda_{1}\geq\lambda_{2}\geq 0$, we define the space of symplectic
simplicial harmonics as

$\mathcal{H}^{s}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n\times
2},\mathbb{C}):=\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{2n\times
2},\mathbb{C})\cap\ker(\langle\underline{x},\underline{\partial}_{u}\rangle,\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s}).$

###  3.2. The general case of $N$ vector variables

In this subsection, we generalise Proposition 3.6 to the case of $N$ vector
variables. We start by determining the associated Howe dual partner:

######  Lemma 3.8.

Let
$\underline{U}=(\underline{u}_{1},\dots,\underline{u}_{N})\in\mathbb{R}^{2n\times
N}$. There are ${2N\choose 2}$ symplectic invariant operators which are
obtained by contraction between

  1. (i)

variables: $\langle\underline{u}_{i},\underline{u}_{j}\rangle_{s}$ with
$i,j\in\\{1,\dots,N\\}$ and $i\neq j$;

  2. (ii)

differential operators:
$\langle\underline{\partial}_{u_{i}},\underline{\partial}_{u_{j}}\rangle_{s}$
with $i,j\in\\{1,\dots,N\\}$ and $i\neq j$;

  3. (iii)

variables and differential operators:
$\langle\underline{u}_{i},\underline{\partial}_{u_{j}}\rangle$ with
$i,j\in\\{1,\dots,N\\}$.

These $\operatorname{\mathfrak{sp}}(2n)$-invariant operators give rise to a
copy of the Lie algebra $\mathfrak{so}(2N)$.

###### Proof.

This follows from straightforward computations of commutators. See for
instance [21]. ∎

We now have the following generalisation of Definition 3.7:

######  Definition 3.9.

We define the space of symplectic simplicial harmonics as

$\mathcal{H}^{s}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2n\times
N},\mathbb{C}):=\mathcal{P}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2n\times
N},\mathbb{C})\cap\ker(\langle\underline{u}_{r},\underline{\partial}_{u_{s}}\rangle,\langle\underline{\partial}_{u_{p}},\underline{\partial}_{u_{q}}\rangle_{s})$

for $1\leq r<s\leq N$ and $1\leq p\neq q\leq N$.

We now check that the space of symplectic simplicial harmonics provides a
polynomial model for the irreducible representations of highest weight
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}$.

######  Theorem 3.10 (Scalar polynomial model).

The spaces
$\mathcal{H}^{s}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2n\times
N},\mathbb{C})$ provide a model for the finite-dimensional representations of
$\mathfrak{sp}(2n)$ of highest weight
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}$.

###### Proof.

The proof is completely analogous to the proof of Proposition 3.6. However, we
first note that $\mathfrak{sp}(2n)$ has the following representation (under
the regular action) on the space of polynomials
$\mathcal{P}(\mathbb{R}^{2n\times N},\mathbb{C})$:

$\displaystyle\begin{cases}Y_{ii}=\sum_{N=1}^{n}x_{N,i}\partial_{y_{N,i}}\qquad&i=1,\dots,n\\\
Z_{ii}=\sum_{N=1}^{n}y_{N,i}\partial_{x_{N,i}}\qquad&i=1,\dots,n\\\
X_{ij}=\sum_{N=1}^{n}x_{N,i}\partial_{x_{N,j}}-y_{N,j}\partial_{y_{N,i}}\qquad&i,j=1,\dots,n\\\
Y_{ij}=\sum_{N=1}^{n}x_{N,i}\partial_{y_{N,N}}+x_{N,j}\partial_{y_{N,i}}\qquad&i<j=1,\dots,n\\\
Z_{ij}=\sum_{N=1}^{n}y_{N,i}\partial_{x_{N,j}}+y_{N,j}\partial_{x_{N,i}}\qquad&i<j=1,\dots,n\end{cases}$

We now define

$\Xi_{N}:=\begin{pmatrix}x_{11}&x_{12}&\cdots&x_{1N}\\\
x_{21}&x_{22}&\cdots&x_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{N1}&x_{N2}&\cdots&x_{NN}\\\ \end{pmatrix}\in\mathbb{R}^{N\times N}$

The highest weight vector for
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}$ is then given by

$\displaystyle w_{\lambda_{1},\dots,\lambda_{N}}:$
$\displaystyle=\Xi_{1}^{\lambda_{1}-\lambda_{2}}\det(\Xi_{2})^{\lambda_{2}-\lambda_{3}}\cdots\det(\Xi_{N})^{\lambda_{N}}=\prod_{j=1}^{N}\det(\Xi_{j})^{\lambda_{j}-\lambda_{j+1}},$

where $\lambda_{j}=0$ for $j>N$. Proceeding like in Proposition 3.6) now
concludes the proof. ∎

##  4\. Symplectic simplicial monogenics

Now that we know that the spaces of symplectic simplicial harmonics
$\mathcal{H}^{s}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2n\times
N},\mathbb{C})$ form a polynomial model for finite-dimensional irreducible
$\mathfrak{sp}(2n)$ representations of highest weight
$(\lambda_{1},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}$ we consider a class of
infinite-dimensional $\mathfrak{sp}(2n)$-representations. This special class
of representations is induced by the symplectic Dirac operator which we will
now define. We start with the symplectic version of the Clifford algebra:

######  Definition 4.1.

Let $(V,\omega)$ be a symplectic vector space. The symplectic Clifford algebra
$\mathsf{Cl}_{s}(V,\omega)$ is defined as the quotient algebra of the tensor
algebra $T(V)$ of $V$, by the two-sided ideal

$\mathcal{I}(V,{\omega}):=\\{v\otimes u-u\otimes v+\omega(v,u):u,v\in V\\}.$

In other words $\mathsf{Cl}_{s}(V,\omega):=T(V)/\mathcal{I}(V,{\omega})$ is
the algebra generated by $V$ in terms of the relation $[v,u]=-\omega(v,u)$,
where we have omitted the tensor product symbols.

######  Remark 4.2.

Note that the symplectic Clifford is often introduced as the Weyl algebra
generated by $2n$ commuting variables and their associated partial derivatives
where e.g. $[\partial_{z_{i}},z_{j}]=\delta_{ij}$.

###  4.1. The symplectic Dirac operator

Following Habermann [23] on the flat symplectic space
$(\mathbb{R}^{2n},\omega_{0})$, we define the symplectic Dirac operator by
means of

$\displaystyle
D_{s}:\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}\to\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty},\quad
f\mapsto(\langle\underline{z},\underline{\partial}_{y}\rangle-\langle\underline{\partial}_{x},\underline{\partial}_{z}\rangle)f,$

where $\langle\cdot,\cdot\rangle$ is the usual Euclidean inner product on
$\mathbb{R}^{n}$. Note that $\underline{z}\in\mathbb{R}^{n}$ plays a different
role than $(\underline{x},\underline{y})\in\mathbb{R}^{2n}$ here (it is used
to model the symplectic spinors as polynomials in $\underline{z}$). The
adjoint (with respect to the symplectic Fischer inner product, see [9]) is
given by
$X_{s}=\langle\underline{x},\underline{z}\rangle+\langle\underline{y},\underline{\partial}_{z}\rangle$.

######  Lemma 4.3 ([14]).

The symplectic Lie algebra $\mathfrak{sp}(2n)$ has the following realisation
on the space of symplectic spinor-valued polynomials
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathcal{P}(\mathbb{R}^{n},\mathbb{C})$:

(4)
$\displaystyle\begin{cases}X_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}-(z_{k}\partial_{z_{j}}+\frac{1}{2}\delta_{jk})&\quad
j,k=1,\dots,n\quad\qquad n^{2}\\\
Y_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}-\partial_{z_{j}}\partial_{z_{k}}&\quad
j<k=1,\dots,n\quad\frac{n(n-1)}{2}\\\
Z_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}+z_{j}z_{k}&\quad
j<k=1,\dots,n\quad\frac{n(n-1)}{2}\\\
Y_{jj}=x_{j}\partial_{y_{j}}-\frac{1}{2}\partial_{z_{j}}^{2}&\quad
j=1,\dots,n\qquad\qquad n\\\
Z_{jj}=y_{j}\partial_{x_{j}}+\frac{1}{2}z_{j}^{2}&\quad
j=1,\dots,n\qquad\qquad n\end{cases}$

The Cartan algebra $\mathfrak{h}$ is given by the operators $X_{jj}$ for
$1\leq j\leq n$ and the positive root vectors are $X_{jk},Y_{jk}$ and $Y_{jj}$
for $1\leq j<k\leq n$.

######  Corollary 4.4.

The symplectic Dirac operator $D_{s}$ is a $\mathfrak{sp}(2n)$-invariant
operator with respect to the realisation from Lemma 4.3.

We define the space of $k$-homogeneous symplectic monogenics as

$\displaystyle\mathcal{M}_{k}^{s}:=\ker_{k}(D_{s})=\ker(D_{s})\cap\left(\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}\right).$

We will see in the next subsection that these spaces corresponds to the
(direct sum) of the highest weights:

$\left(k-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(k-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

Our goal will be to associate with the infinite-dimensional representation

$\left(\lambda_{1}-\frac{1}{2},\dots,\lambda_{N}-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(\lambda_{1}-\frac{1}{2},\dots,\lambda_{N}-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}$

a certain space of $(\lambda_{1},\dots,\lambda_{N})$-homogeneous polynomials
intersected with some $\mathfrak{sp}(2n)$-invariant differential operators
(associated with the symplectic Dirac operator). In order to obtain a
characterisation of this space in terms of a Cartan product of two
$\mathfrak{sp}(2n)$-irreducible representations, we need some more background.
This is provided in the following subsection.

###  4.2. Completely pointed modules

We will now introduce the algebraic background for computing tensor products
of finite-dimensional $\mathfrak{sp}(2n)$-representations with the infinite-
dimensional $\mathfrak{sp}(2n)$-module $\mathbb{S}^{\infty}$. We first recall
some basic terminology considering the root system of $\mathfrak{sp}(2n)$. We
denote by $\epsilon_{i}=(0,\cdots,1,\cdots,0)$ the standard basis of
$\mathbb{R}^{n}$. The roots of the symplectic algebra $\mathfrak{sp}(2n)$ fall
into two categories: $2n$ so-called long roots $\pm 2\epsilon_{i}$ and
$2(n^{2}-n)$ short roots $\pm\epsilon_{i}\pm\epsilon_{j}$, where $1\leq i\leq
n$ and $i\neq j$. This means that the full set of roots $\Phi$ equals

$\Phi=\\{\pm(\epsilon_{i}\pm\epsilon_{j}):1\leq i<j\leq n\\}\cup\\{\pm
2\epsilon_{i}:1\leq i\leq n\\}.$

The simple roots are chosen to be

$\Delta=\\{\alpha_{i}:=\epsilon_{i}-\epsilon_{i+1}:1\leq i\leq
n-i\\}\cup\\{\alpha_{n}:=2\epsilon_{n}\\}.$

The fundamental weights are now given by
$\omega_{j}=\epsilon_{1}+\cdots+\epsilon_{j}.$ The two bases of the dual of
the Cartan algebra $\mathfrak{h}^{*}$ relate as follows

$\displaystyle\omega_{1}=\epsilon_{1},\quad\omega_{2}=\epsilon_{1}+\epsilon_{2},\quad\dots\quad\omega_{n}=\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{n}.$

We denote by $\mathbb{V}_{\lambda}$ to be the irreducible highest weight
module with highest weight $\lambda$. Note that we will sometimes denote this
by $\mathbb{V}(\lambda)$ for some specific value of $\lambda$. We will now
introduce a specific type of module of which the modules $\mathbb{S}^{\infty}$
will appear to be examples.

######  Definition 4.5.

Let $\mathfrak{g}$ be a semisimple Lie algebra with Cartan algebra
$\mathfrak{h}$ and let $\mathbb{V}$ be $\mathfrak{g}$-module.

  1. (i)

We say that $\mathbb{V}$ is a module with bounded multiplicities if there
exists a $k\in\mathbb{N}_{0}$ such that for every decomposition of the form
$\mathbb{V}=\bigoplus_{\lambda}\mathbb{V}_{\lambda},$ we have that
$\dim\mathbb{V}_{\lambda}\leq k$. The minimal $k$ for which this upper bound
holds is called the degree of $\mathbb{V}$ and it is denoted by
$\deg(\mathbb{V})=k$.

  2. (ii)

The module $\mathbb{V}$ is then called completely pointed if
$\deg(\mathbb{V})=1$.

The Lie algebras $\mathfrak{g}$ which have infinite-dimensional modules with
bounded multiplicities are rather limited as they only occur if and only if
$\mathfrak{g}=\mathfrak{sl}(n)$ or
$\mathfrak{g}=\operatorname{\mathfrak{sp}}(2n)$ (see for example [4]). The
following theorem restricts the possibilities for the completely pointed
modules in the case of symplectic algebra.

######  Theorem 4.6 (Britten, Hooper & Lemire [4]).

Let $\mathbb{V}_{\lambda}$ be a highest weight module of the symplectic Lie
algebra $\mathfrak{sp}(2n)$. Then it is completely pointed if and only if
$\lambda=0,\omega_{1},-\frac{1}{2}\omega_{n}$ or
$\omega_{n-1}-\frac{3}{2}\omega_{n}$ (or $\omega_{n}$ if $n=2$ or $3$).

  1. (i)

If $\lambda=0$ and $\lambda=\omega_{1}$ the module is finite-dimensional, more
specifically: $\mathbb{V}_{0}=\mathbb{C}v$ is the trivial representation and
$\mathbb{V}_{\omega_{1}}\cong\mathbb{C}^{2n}$ is the fundamental
representation. Moreover, in the special case $n=2,3$ the module
$\mathbb{V}_{\omega_{n}}$ is also finite-dimensional.

  2. (ii)

$\mathbb{V}_{\lambda}$ for $\lambda=-\frac{1}{2}\omega_{n}$ and
$\lambda=\omega_{n-1}-\frac{3}{2}\omega_{n}$ are infinite-dimensional
$\mathfrak{sp}(2n)$-modules. Their highest weight are given by

$\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\quad\text{
and }\quad\left(-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

These completely pointed modules have the following interpretation:

######  Theorem 4.7.

The infinite-dimensional symplectic spinor space $\mathbb{S}^{\infty}$ is a
reducible $\mathfrak{sp}(2n)$-module, its irreducible components exactly
coincide with the two completely pointed weight modules from above. The
corresponding highest weight vector is given by $1\oplus z_{n}$ (when
considering the realisation from Lemma 4.3).

###### Proof.

Recall from Lemma 4.3 that the $n$ Cartan elements (we only focus on the
spinor part here) of the Lie algebra $\mathfrak{sp}(2n)$ are given by

$H_{j}:=X_{jj}=-\left(z_{j}\partial_{z_{j}}+\frac{1}{2}\right)\qquad(1\leq
j\leq n).$

We now directly verify that the completely pointed module
$\mathbb{V}\left(-\frac{1}{2}\omega_{n}\right)$ has highest weight vector $1$
and $\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}\right)$ has highest
weight vector $z_{n}$. This implies that the highest weight vector of
$\mathbb{S}^{\infty}$ is $w=1\oplus z_{n}$. ∎

The following theorem will allow us to compute the tensor product of finite-
dimensional symplectic modules with the infinite-dimensional symplectic
modules $\mathbb{S}^{\infty}_{\pm}$.

######  Theorem 4.8 (Britten & Lemire [5]).

Let $\mathbb{V}(\lambda)$ be a finite-dimensional
$\mathfrak{sp}(2n)$-irreducible representation of highest weight
$(\lambda_{1},\dots,\lambda_{n})_{\mathfrak{sp}(2n)}$. Then, the tensor
products with the infinite-dimensional completely pointed modules decompose as
follows:

$\displaystyle\mathbb{V}\left(-\frac{1}{2}\omega_{n}\right)\otimes\mathbb{V}(\lambda)$
$\displaystyle\cong\bigoplus_{\kappa\in
T_{\lambda}}\mathbb{V}\left(-\frac{1}{2}\omega_{n}+\kappa\right)$
$\displaystyle\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}\right)\otimes\mathbb{V}(\lambda)$
$\displaystyle\cong\bigoplus_{\kappa\in
T_{\lambda}}\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}+\kappa\right)$

where $\kappa$ is of the following form

$\displaystyle
T_{\lambda}=\left\\{\kappa\mid\kappa=\lambda-\sum_{i=1}^{n}d_{i}\epsilon_{i}\right\\}$

subject to the properties:

  1. (i)

$d_{i}\in\mathbb{N}$,

  2. (ii)

$\sum_{i=1}^{n}d_{i}\in 2\mathbb{N}$,

  3. (iii)

$0\leq d_{i}\leq\nu_{i}$ for $i=1,\dots,n-1$,

  4. (iv)

$0\leq d_{n}\leq 2\nu_{n}+1$.

###  4.3. The symplectic monogenic Fischer decomposition

As we know from Proposition 3.2 that
$(k)_{\mathfrak{sp}(2n)}\leftrightarrow\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})$
this now gives rise to the following result, which is in fact the symplectic
counterpart of Theorem 2.2 proven in [10].

######  Theorem 4.9 (Symplectic monogenic FD, [10]).

Under the joint action of $\mathsf{Mp}(2n)\times\mathfrak{sl}(2)$ the space of
polynomials with values in the space of symplectic spinors decomposes as
follows:

(5)
$\displaystyle\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{\infty}X_{s}^{j}\mathcal{M}_{k}^{s},$

where
$\mathcal{M}_{k}^{s}=\ker(D_{s})\cap(\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathcal{P}(\mathbb{R}^{n},\mathbb{C}))$.

######  Remark 4.10.

The module $\mathcal{M}^{s}_{k}$ is $\mathfrak{sp}(2n)$-reducible and
decomposes into two irreducible parts (an even and odd part)
$\mathcal{M}_{\ell}^{s}=\left(\mathcal{M}_{\ell}^{s}\right)^{+}\oplus\left(\mathcal{M}_{\ell}^{s}\right)^{-}$,
which are irreducible $\mathfrak{sp}(2n)$-modules of highest weight

$\displaystyle\left(\mathcal{M}_{k}^{s}\right)^{+}$
$\displaystyle\leftrightarrow\left(k-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}$
$\displaystyle\left(\mathcal{M}_{k}^{s}\right)^{-}$
$\displaystyle\leftrightarrow\left(k-\frac{1}{2},-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

As in the orthogonal case, we now see that the space of $k$-homogeneous
symplectic monogenics can be defined in a purely algebraic way:

######  Definition 4.11.

The space of $k$-homogeneous polynomials solutions of the symplectic Dirac
operator $D_{s}$ are called symplectic monogenics and are denote by
$\mathcal{M}_{k}^{s}$. They are algebraically determined by

$\mathcal{M}_{k}^{s}:=(k)_{\mathfrak{sp}(2n)}\boxtimes\left(\left(-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(-\frac{1}{2},-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}\right)$

######  Corollary 4.12.

The $\mathfrak{sp}(2n)$-module $\mathcal{M}_{k}^{s}$ has highest weight vector
$x_{1}^{k}\otimes(1\oplus z_{n})$.

###### Proof.

This follows from Propositions 3.2 and 4.7. ∎

###  4.4. The case of $N$ vector variables

We will now be interested in systems of $N$ symplectic Dirac operators. We
will begin by studying the case of $N=2$. Recall from Section 3 that the
finite-dimensional representations of highest weight
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}$ are irreducible and can be
explicitly realised as the space of symplectic simplicial harmonics. We now
want to gain knowledge on the infinite-dimensional irreducible representations
of highest weight

$(\lambda_{1},\lambda_{2})^{\prime}_{\mathfrak{sp}(2n)}:=\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

As one might intuitively predict, the space
$(\lambda_{1},\lambda_{2})^{\prime}_{\mathfrak{sp}(2n)}$ will be defined by
means of the Cartan product
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty}$,
but the question obviously arises as how to define these spaces as a kernel
space for differential operators. We start by proving that the intuitive guess
is indeed correct.

######  Lemma 4.13.

The Cartan product of
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty}$ is
given by the direct sum of:

$\displaystyle\mathbb{V}\left(-\frac{1}{2}\omega_{n}+\lambda_{1}\epsilon_{1}+\lambda_{2}\epsilon_{2}\right)$
$\displaystyle=\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}$
$\displaystyle\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}+\lambda_{1}\epsilon_{1}+\lambda_{2}\epsilon_{2}\right)$
$\displaystyle=\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}$

###### Proof.

In order to check this, we need to calculate (a part of) the tensor product
decomposition of
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\otimes\mathbb{S}^{\infty}$.
This can be done using Theorem 4.8 where
$\mathbb{V}(\lambda)=(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}$. We use
the Young convention, so in other words we write
$\lambda=\lambda_{1}\epsilon_{1}+\lambda\epsilon_{2}$. Then $\kappa$ from
Theorem 4.8 is given by

$\kappa=(\lambda_{1}-d_{1})\epsilon_{1}+(\lambda_{2}-d_{2})\epsilon_{2}-d_{n}\epsilon_{n}$

Taking $d_{1}=d_{2}=d_{n}=0$ (as this will give the highest weight) leaves us
with

$\kappa=\lambda\epsilon_{1}+\lambda_{2}\epsilon_{2}.$

In order words $\mathbb{V}\left(-\frac{1}{2}\omega_{n}+\kappa\right)\oplus$
$\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}+\kappa\right)$ is the
Cartan product. ∎

We will now characterise the Cartan product
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty}$ as
the polynomial kernel space of certain differential operators. This will be
done by extending the Howe dual partner $\mathfrak{so}(2N)$ from Section 3.9.
We already know from the symplectic Fischer decomposition is that this dual
parter is $\mathfrak{sl}(2)$ for $N=1$. As we are working with two vector
variables here, we have two symplectic Dirac operators (one for each vector
variable) and two adjoint operators which we define as follows:

$\displaystyle
D_{s,x}=\langle\underline{z},\underline{\partial}_{y}\rangle-\langle\underline{\partial}_{z},\underline{\partial}_{x}\rangle\qquad\text{and}\qquad
X_{s,x}=\langle\underline{y},\underline{\partial}_{z}\rangle+\langle\underline{x},\underline{z}\rangle$
$\displaystyle D_{s,u}=\langle
z,\underline{\partial}_{v}\rangle-\langle\underline{\partial}_{z},\underline{\partial}_{u}\rangle\qquad\text{and}\qquad
X_{s,u}=\langle v,\underline{\partial}_{z}\rangle+\langle
u,\underline{z}\rangle$

Recall that $\langle\cdot,\cdot\rangle$ is the usual Euclidean inner product
here. We now turn to the question whether these operators close as a Lie
algebra.

######  Lemma 4.14.

The algebra generated by the Dirac operators $D_{s,x}$ and $D_{s,u}$ and their
adjoints $X_{s,x}$ and $X_{s,u}$ gives rise to a copy of the Lie algebra
$\mathfrak{so}(5)$.

###### Proof.

This follows from calculating the commutators between these operators, we have
for instance:

$\displaystyle[X_{s,u},D_{s,u}]=\mathbb{E}_{u}+\mathbb{E}_{v}+n$
$\displaystyle[X_{s,x},D_{s,x}]=\mathbb{E}_{x}+\mathbb{E}_{y}+n$
$\displaystyle[X_{s,u},D_{s,x}]=\langle\underline{u},\partial_{\underline{x}}\rangle$
$\displaystyle[X_{s,x},D_{s,u}]=\langle\underline{x},\partial_{\underline{u}}\rangle$
$\displaystyle[D_{s,u},D_{s,x}]=\langle\partial_{\underline{x}},\partial_{\underline{u}}\rangle_{s}$
$\displaystyle[X_{s,u},X_{s,x}]=\langle{\underline{x}},{\underline{u}}\rangle_{s}$

∎

This now leads to the following generalisation of symplectic monogenics:

######  Definition 4.15.

We define the space of symplectic simplicial monogenics for $N=2$ as follows:

$\displaystyle{\mathcal{S}_{\lambda_{1},\lambda_{2}}^{s}(\mathbb{\mathbb{R}}^{4n},\mathbb{S}^{\infty}):=\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{\mathbb{R}}^{4n},\mathbb{S}^{\infty})\cap\ker(D_{s,x},D_{s,u},\langle\underline{x},\underline{\partial}_{u}\rangle,\langle\underline{\partial}_{x},\underline{\partial}_{x}\rangle_{s}).}$

We obtain the following characterisation of the Cartan product
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty}$:

######  Theorem 4.16.

The space of symplectic simplicial monogenics
$\mathcal{S}_{\lambda_{1},\lambda_{2}}^{s}(\mathbb{\mathbb{R}}^{4n},\mathbb{S}^{\infty})$
is the direct sum of two irreducible $\mathfrak{sp}(2n)$-module of highest
weight

$\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}$

with highest weight vector
$x_{1}^{\lambda_{1}-\lambda_{2}}(x_{1}u_{2}-x_{2}u_{1})^{\lambda_{2}}\otimes(1\oplus
z_{n})$.

###### Proof.

The highest weight vector is obtained as the tensor product of the highest
weight vector from Theorem 3.10(iii) (scalar part) and
$w_{\lambda_{1},\lambda_{2}}(\underline{x},\underline{u})\otimes(1\oplus
z_{n})$. The rest is a straightforward computation. ∎

######  Remark 4.17.

In order to obtain the full decomposition of
$(\lambda_{1},\lambda_{2})_{\mathfrak{sp}(2n)}\otimes\mathbb{S}^{\infty}$
(this will lead to a new Fischer decomposition, which has to be interpreted as
the $N=2$ generalisation of 4.9) we need to take into account the other values
for the parameters $d_{1},d_{2},d_{n}\in\mathbb{N}$ (see Lemma 4.13). One must
pay some attention here: in the $N=1$ case, the space
$\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{C})$ is
$\mathfrak{sp}(2n)$-irreducible and the operator $X_{s}$ (and its powers)
served as the symplectic embedding factors making the isomorphism into an
equality. In the case of two vector variables, the space
$\mathcal{P}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{4n},\mathbb{C})$ is not
irreducible (see Example 3.5). By the general theory of Howe dualities, this
space further decomposes into the symplectic simplicial harmonics
$\mathcal{H}_{\lambda_{1},\lambda_{2}}^{s}(\mathbb{R}^{4n},\mathbb{C})$. Thus,
when $N>1$ the ‘analogy’ with the classical (orthogonal case) is restored in a
sense: the space
$\mathcal{P}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{R}^{2nN},\mathbb{C})$ has
a ‘symplectic harmonic’ Fischer decomposition with symplectic harmonic Howe
dual pair $\mathfrak{sp}(2n)\times\mathfrak{so}(2N)$. One could then look for
a ‘monogenic refinement’ which is then governed by the Howe dual pair
$\mathfrak{sp}(2n)\times\mathfrak{so}(2N+1)$. For example, in the case of
$N=2$ the latter would result in computing the tensor product
$\mathcal{H}^{s}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{4n},\mathbb{C})\otimes\mathbb{S}^{\infty}$
where we already know that
$\mathcal{H}^{s}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{4n},\mathbb{C})\boxtimes\mathbb{S}^{\infty}=\mathcal{S}^{s}_{\lambda_{1},\lambda_{2}}(\mathbb{R}^{4n},\mathbb{C})$.
This would yield the ‘symplectic translations’ of the results from [31, 26].
We plan to do this in a subsequent paper.

In the case of $N$ vector variables we obtain:

######  Lemma 4.18.

The Cartan product of
$(\lambda_{1},\lambda_{2},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}\boxtimes\mathbb{S}^{\infty}$
is given by the direct sum of:

$\displaystyle\mathbb{V}\left(-\frac{1}{2}\omega_{n}+\sum_{j=1}^{N}\lambda_{j}\epsilon_{j}\right)$
$\displaystyle=\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},\dots,\lambda_{N}-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}$
$\displaystyle\mathbb{V}\left(\omega_{n-1}-\frac{3}{2}\omega_{n}+\sum_{j=1}^{N}\lambda_{j}\right)$
$\displaystyle=\left(\lambda_{1}-\frac{1}{2},\lambda_{2}-\frac{1}{2},\dots,\lambda_{N}-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}$

###### Proof.

This is again a consequence of Theorem 4.8 where one now takes
$\mathbb{V}(\lambda)=(\lambda_{1},\lambda_{2},\dots,\lambda_{N})_{\mathfrak{sp}(2n)}$
and $\kappa=\sum_{j=1}^{N}\lambda_{j}\epsilon_{j}$. ∎

######  Theorem 4.19.

Considering a system of $N$ symplectic Dirac operators and their (Fischer)
adjoints. These operators give rise to a copy of the orthogonal Lie algebra
$\mathfrak{so}(2N+1)$ of dimension $N(1+2N)$.

###### Proof.

We use the short-hand notation $D_{a}=D_{s,u_{a}}$ and $X_{a}=X_{s,u_{a}}$ for
$1\leq a\leq N$. Then, the following commutation relations hold:

$\displaystyle[[D_{a},X_{b}],X_{c}]$ $\displaystyle=-2\delta_{ac}X_{b}$
$\displaystyle[[D_{a},X_{b}],D_{c}]$ $\displaystyle=2\delta_{bc}X_{a}$
$\displaystyle[[D_{a},D_{b}],X_{c}]$
$\displaystyle=2\delta_{bc}D_{a}-2\delta_{ac}D_{b}$
$\displaystyle[[X_{a},X_{b}],X_{c}]$
$\displaystyle=2\delta_{bc}X_{a}-2\delta_{ac}X_{b}$
$\displaystyle[[D_{a},D_{b}],D_{c}]$ $\displaystyle=0$

for all $a,b,c\in\\{1,\dots,N\\}$. ∎

######  Remark 4.20.

The theorem above allows us to link the system of $N$ symplectic Dirac
operators and their adjoints to the theory of so-called parastatistic algebras
from physics in the following way. The parafermion algebra was introduced by
Green [22] by a system of $N$ parafermion creation and annihilation operators
$f_{j}^{\pm}$ for $j=1,\dots,N$ satisfying

$\displaystyle[[f_{j}^{\xi},f_{j}^{\eta}],f_{\ell}^{\epsilon}]=|\epsilon-\eta|\delta_{k\ell}f_{j}^{\xi}-|\epsilon-\xi|\delta_{j\ell}f_{k}^{\eta},$

where $j,k,\ell\in\\{1,\dots,N\\}$ and $\xi,\epsilon,\eta\in\\{+,-\\}$. It is
well-established that this algebra is in fact isomorphic to the orthogonal Lie
algebra $\mathfrak{so}(2N+1)$ (see for instance [28]). In other words, the
operators $D_{a}$ and $X_{a}$ for $a\in\\{1,\dots,N\\}$ can be interpreted as
parafermion creation and annihilation operators.

The Cartan product from Lemma 4.18 now gets the following interpretation:

######  Definition 4.21.

We define the space of symplectic simplicial monogenics for $N$ vector
variables as follows:

$\displaystyle{\mathcal{S}_{\lambda_{1},\dots,\lambda_{N}}^{s}(\mathbb{\mathbb{R}}^{2n\times
N},\mathbb{S}^{\infty}):=\mathcal{P}_{\lambda_{1},\dots,\lambda_{N}}(\mathbb{\mathbb{R}}^{2n\times
N},\mathbb{S}^{\infty})\cap\ker(D_{s,u_{a}},\langle\underline{u}_{r},\underline{\partial}_{u_{s}}\rangle,\langle\underline{\partial}_{u_{p}},\underline{\partial}_{u_{q}}\rangle_{s}).}$

for $1\leq r<s\leq N$ and $1\leq p\neq q\leq N$.

##  5\. The symplectic Rarita-Schwinger operator

###  5.1. Transvector algebras

Let $\mathfrak{g}$ be a Lie algebra and let
$\mathfrak{s}\subseteq\mathfrak{g}$ be a reductive subalgebra. We then have
the decomposition $\mathfrak{g}=\mathfrak{s}\oplus\mathfrak{t}$, where
$\mathfrak{t}$ carries an $\mathfrak{s}$-action for the commutator (i.e.
$[\mathfrak{s},\mathfrak{t}]\subseteq\mathfrak{t}$). Fix a Cartan subalgebra
$\mathfrak{h}$ for $\mathfrak{s}$ and a triangular decomposition
$\mathfrak{s}=\mathfrak{s}^{-}\oplus\mathfrak{h}\oplus\mathfrak{s}^{+}$, where
$\mathfrak{s}^{\pm}$ consists of the positive (resp. negative) roots
$e_{\alpha}$ (resp. $e_{-\alpha}$) with $\alpha\in\Phi^{+}(\mathfrak{s})$.
Define a left ideal $J\subseteq\mathcal{U}(\mathfrak{g})$ in the universal
enveloping algebra $\mathcal{U}(\mathfrak{g})$ by means of
$\mathcal{U}(\mathfrak{g})\mathfrak{s}^{+}$ and further define the normaliser
as

$\operatorname{Norm}(J):=\\{u\in\mathcal{U}(\mathfrak{g})\mid Ju\subseteq
J\\}.$

We now have that $J$ is a two-sided ideal of $\operatorname{Norm}(J)$ so that
the quotient algebra
$\mathcal{S}(\mathfrak{g},\mathfrak{s})=\operatorname{Norm}(J)/J$ which is
known as the Mickelsson algebra.

Let us now consider an extension of $\mathcal{U}(\mathfrak{g})$ to a suitable
localisation $\mathcal{U}^{\prime}(\mathfrak{g})$ given by

$\mathcal{U}^{\prime}(\mathfrak{g})=\mathcal{U}(\mathfrak{g})\otimes_{\mathcal{U}(\mathfrak{h})}\operatorname{Frac}(\mathcal{U}(\mathfrak{h}))\
,$

where $\operatorname{Frac}(\mathcal{U}(\mathfrak{h}))$ is the field of
fractions in the (universal enveloping algebra of the) Cartan algebra. The
ideal $J^{\prime}$ can be introduced for this extension too (in a completely
similar way) and the quotient algebra
$\mathcal{Z}(\mathfrak{g},\mathfrak{s}):=\operatorname{Norm}(J^{\prime})/J^{\prime}$
is the Mickelsson-Zhelobenko algebra. These two algebras are naturally
identified, since one has

$\mathcal{Z}(\mathfrak{g},\mathfrak{s})=\mathcal{S}(\mathfrak{g},\mathfrak{s})\otimes_{\mathcal{U}(\mathfrak{h})}\operatorname{Frac}(\mathcal{U}(\mathfrak{h})).$

We refer to this algebra as a ‘transvector algebra’.

######  Definition 5.1.

The extremal projector for the Lie algebra
$\mathfrak{sl}(2)=\operatorname{Alg}\\{X,Y,H\\}$ is the idempotent operator
$\pi$ (meaning that $\pi^{2}=\pi$) given by the expression

(6)
$\displaystyle\pi:=1+\sum_{j=1}^{\infty}\frac{(-1)^{j}}{j!}\frac{\Gamma(H+2)}{\Gamma(H+2+j)}Y^{j}X^{j},$

where

$\frac{\Gamma(H+2)}{\Gamma(H+2+j)}=\frac{(H+1)!}{(H+j+1)!(H+j)!}$

is defined using the gamma function $\Gamma$. Note that we will sometimes
write $\pi_{\mathfrak{sl}(2)}$ in order to refer to the underlying Lie algebra
$\mathfrak{sl}(2)$.

######  Remark 5.2.

This operator satisfies $X\pi=\pi Y=0$ and can therefore be used a natural
object to project on the kernel $\ker(X)$. The operator is defined on an
extension $\mathcal{U}^{\prime}(\mathfrak{sl}(2))$ of the universal enveloping
algebra so that formal series containing the operator $H$ in the denominator
are well-defined.

###  5.2. Higher spin Dirac operators

The Rarita-Schwinger operator was first introduced in theoretical physics by
W. Rarita and J. Schwinger [27] in 1941 to describe fermionic particles of
spin-$\frac{3}{2}$, as a generalisation of the Dirac operator which describes
particles of spin-$\frac{1}{2}$. As a matter of fact, this operator is an
example of a so-called higher spin Dirac operator (HSD, [MR1708920]) given by

$\mathcal{Q}_{\lambda}:\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{V}_{\lambda})\to\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{V}_{\lambda}),$

where $\mathbb{V}_{\lambda}$ is an irreducible
$\operatorname{\mathsf{Spin}}(n)$-representation (we assume that $n$ is odd
here to avoid the parity of the spinors $\mathbb{S}$ and write
$m:=\lfloor\frac{n}{2}\rfloor$ for notational convenience) with highest weight

$\lambda=\left(\lambda_{1}+\frac{1}{2},\dots,\lambda_{m-1}+\frac{1}{2},\frac{1}{2}\right)_{\mathfrak{so}(n)}.$

These are often called higher spin representations. Recall from Theorem 2.5
that these higher spin representations can be explicitly realised as the
vector space of $\mathbb{S}$-valued polynomials in several vector variables
intersected with the kernel of some differential operators (the simplicial
monogenics).

  * •

We note that the Dirac operator

$\underline{\partial}_{x}:\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{S})\to\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{S})$

is the easiest example of the operator $\mathcal{Q}_{\lambda}$ where we take
$\lambda_{1}=\cdots=\lambda_{m-1}=0$, so that we have
$\mathbb{V}_{\lambda}=\mathbb{S}$.

  * •

The next type of operator is obtained when taking the highest weight
$\lambda=\left(k+\frac{1}{2},\dots,\frac{1}{2}\right)$, such that
$\mathbb{V}_{\lambda}=\mathcal{M}_{k}(\mathbb{R}^{n},\mathbb{S})$ (see Section
2.2). The corresponding higher spin Dirac operator is called the Rarita-
Schwinger operator

$\displaystyle\mathcal{R}_{k}:\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S}))\to\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S})).$

Note that the classical Rarita-Schwinger operator from theoretical physics
corresponds to the operator $\mathcal{R}_{1}$.

The monogenic refinement of the FD (see Theorem 2.1) in the vector variable
$\underline{u}\in\mathbb{R}^{n}$ can be used to obtain an explicit expression
for the first-order differential operator $\mathcal{R}_{k}$. The decomposition

$H_{k}(\underline{u})={M}_{k}(\underline{u})+\underline{u}{M}_{k-1}(\underline{u}).$

gives rise to the following diagram:

${\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S}))}$${\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S}))}$

$\cong$

$\cong$

${\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k})}$${\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k})}$${\oplus}$${\oplus}$${\mathcal{C}^{\infty}(\mathbb{R}^{n},\underline{u}\mathcal{M}_{k-1})}$${\mathcal{C}^{\infty}(\mathbb{R}^{n},\underline{u}\mathcal{M}_{k-1})}$$\mathcal{R}_{k}$$\underline{\partial}_{x}$
Figure 1. The Rarita-Schwinger operator arising from the Fischer
decomposition.

Note that there are other arrows that one could study in this scheme (giving
rise to e.g. the twistor operator) [7]. However, we will not consider these
here.

######  Remark 5.3.

Every function $f\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k})$ with
values in $\mathcal{M}_{k}$ can be thought of as a function depending on two
vector variables $f(\underline{x};\underline{u})$ such that fixed
$\underline{x}\in\mathbb{R}^{n}$ the function
$f(\underline{x};\underline{u})=f_{\underline{x}}(\underline{u})\in\mathcal{M}_{k}(\mathbb{R}^{n},\mathbb{S})$,
i.e. $f(\underline{x};\underline{u})\in\ker\underline{\partial}_{u}$. One
often calls the variable $\underline{u}$ a dummy variable. This illustrates
the importance of the polynomial models for $\operatorname{\mathsf{Spin}}(n)$
(see Theorem 2.5).

We summarise these results as follows:

######  Theorem 5.4 (Bureš et. al., [7]).

Let
$f(\underline{x};\underline{u})\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k})$.
The Rarita-Schwinger operator is the unique
$\operatorname{\mathsf{Spin}}(n)$-invariant first-order differential operator

$\displaystyle\mathcal{R}_{k}:\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S}))$
$\displaystyle\to\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathcal{M}_{k}(\mathbb{R}^{2n},\mathbb{S}))$
$\displaystyle f(\underline{x};\underline{u})$
$\displaystyle\mapsto\left(1+\frac{\underline{u}\
\underline{\partial}_{u}}{n+2k-2}\right)\underline{\partial}_{x}f.$

######  Remark 5.5.

Note that the more general higher spin Dirac operators can be defined using a
transvector algebra (see the work of Eelbode & Raeymaekers [17]). We will
follow a similar approach in the symplectic setting instead of the more ad hoc
approach in Figure 1.

###  5.3. Higher metaplectic Dirac operators

Inspired by the definition of higher spin Dirac operators
$\mathcal{Q}_{\lambda}$, we now investigate operators

$\mathcal{Q}_{k}^{s}:\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathbb{V}_{\lambda})\to\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathbb{V}_{\lambda}),$

with values in the (infinite-dimensional) $\mathsf{Mp}(2n)$-representations
$\mathbb{V}_{\lambda}$ where

$\lambda=\left(\lambda_{1}-\frac{1}{2},\dots,\lambda_{n}-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(\lambda_{1}-\frac{1}{2},\dots,\lambda_{n}-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

We refer to the operator $\mathcal{Q}_{k}^{s}$ as a higher metaplectic Dirac
operator (or HMD in short). To our knowledge, these operators are not yet
studied in the literature.

######  Example 5.6.

By taking $\lambda_{1}=0=\cdots=\lambda_{n}=0$ we have
$\mathbb{V}_{\lambda}=\mathbb{S}^{\infty}$ and we recover the symplectic Dirac
operator
$D_{s}:\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathbb{S}^{\infty})\to\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathbb{S}^{\infty}).$

We now take $\lambda_{1}=k$ and $\lambda_{2}=\cdots=\lambda_{n}=0$. Then the
corresponding highest weight is given by

$\lambda=\left(k-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\oplus\left(k-\frac{1}{2},\dots,-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}.$

In this case, we know that the space of $k$-homogeneous symplectic monogenics
$\mathcal{M}_{k}^{s}(\mathbb{R}^{2n},\mathbb{S})$ is a model for
$\mathbb{V}_{\lambda}$. We consider the elements of
$\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathcal{M}_{k}^{s})$ as functions in
two vector variables $f(\underline{x};\underline{u})$ which will allow us to
use the results obtained in Section 4.4, in which we defined the symplectic
Dirac operators $D_{s,x}$ and $D_{s,u}$. Recall from Lemma 4.14 that the Lie
algebra generated by the symplectic Dirac operators $D_{s,x}$ and $D_{s,u}$
and their adjoints $X_{s,x}$ and $X_{s,u}$ is isomorphic to
$\mathfrak{so}(5)$. Straightforward computation of commutators leads to:

######  Lemma 5.7.

The Lie algebra $\mathfrak{sl}(2)=\operatorname{Alg}(D_{s,u},X_{s,u})$ is
reductive in

$\mathfrak{so}(5)=\operatorname{Alg}(D_{s,x},D_{s,u},X_{s,x},X_{s,u}).$

This means that we can write
$\mathfrak{so}(5)=\mathfrak{sl}(2)\oplus\mathfrak{t}$ with $\mathfrak{t}$ a
subspace satisfying $[\mathfrak{sl}(2),\mathfrak{t}]\subseteq\mathfrak{t}$.

This means that we can construct the transvector algebra
$\mathcal{Z}(\mathfrak{so}(5),\mathfrak{sl}(2))$. Taking into account the
suitable rescaling the Lie algebra
$\mathfrak{sl}(2)\cong\operatorname{Alg}(X,Y,H)$ can be realised as follows
(note that this has no effect on the lemma above or the related commutators):

$X=\sqrt{2}D_{s,u},\quad Y=\sqrt{2}X_{s,u},\quad
H=-(\mathbb{E}_{u}+\mathbb{E}_{v}+n)$

Recall that the extremal projector $\pi_{\mathfrak{sl}(2)}$ of the Lie algebra
$\mathfrak{sl}(2)$ is formally given by the expression

$\pi_{\mathfrak{sl}(2)}=1+\sum_{j=1}^{+\infty}\frac{(-1)^{j}}{j!}\frac{\Gamma(H+2)}{\Gamma(H+2+j)}Y^{j}X^{j}.$

######  Lemma 5.8.

The transvector projection of the operator $D_{s,x}$ is given by

$\pi_{\mathfrak{sl}(2)}D_{s,x}=\left(1-\frac{YX}{(H+2)}\right)D_{s,x}.$

###### Proof.

We easily find that
$[D_{s,x},D_{s,u}]=-\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s}$
and
$[\langle\underline{\partial}_{x},\underline{\partial}_{u}\rangle_{s},D_{s,x}]=0$
so that infinite sum reduces to the finite sum from the lemma. ∎

We now come to the following definition of the symplectic Rarita-Schwinger
operator:

######  Definition 5.9.

We define the symplectic Rarita-Schwinger operator $\mathcal{R}_{k}^{s}$ as

$\displaystyle\mathcal{R}_{k}^{s}:\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathcal{M}_{k}^{s})$
$\displaystyle\to\mathcal{C}^{\infty}(\mathbb{R}^{2n},\mathcal{M}_{k}^{s})$
$\displaystyle f(\underline{x},\underline{u})$
$\displaystyle\mapsto\left(1+\frac{2X_{s,u}D_{s,u}}{k+n+2}\right)D_{s,x}f(\underline{x},\underline{u}).$

######  Remark 5.10.

Note that the name ‘symplectic Rarita-Schwinger operator’ already appears in
the literature; see for example the work of Krýsl [25] and the references
therein. However, their symplectic Rarita-Schwinger operator is defined on
symplectic spinor-valued forms whereas our operator acts on symplectic spinor-
valued polynomials.

##  6\. Conclusion and further research

In this paper we introduced the notion of symplectic simplicial harmonics and
monogenics and related them to the irreducible representations of the
symplectic Lie algebra. These results could also be useful for constructing
solutions for the symplectic Dirac operator. There are two useful mechanisms
in classical Clifford analysis to construct monogenic functions: first of all,
there is the Fueter theorem, and secondly the Cauchy-Kowalewskaya extension
(CK-extension in short). In the symplectic framework, the Fueter theorem was
proven in [13]. These new polynomial models might be useful to get a grasp on
the symplectic CK-extension (which has not been proven yet). Let us focus on
the orthogonal case to illustrate what we mean. The association between the
irreducible representation $\mathbb{V}_{\lambda}$ and the polynomial model as
a kernel space of certain differential operators corresponds with the first
dotted horizontal line in Figure 2.

polynomial
model$\mathbb{V}_{\lambda}$$\bigoplus_{\mu}\overline{\mathbb{V}}_{\mu}$(lower
dimension)polynomial model$n$$n-1$branchingCK-extension(inverse branching)
Figure 2. Branching, polynomials models and the CK-theorem.

The second step, visualised as the left downwards arrow, is to study the
behaviour of the $\mathfrak{so}(n)$-irreducible representation
$\mathbb{V}_{\lambda}$ as a representation for the subalgebra
$\mathfrak{so}(n-1)$. The representation $\mathbb{V}_{\lambda}$ will ofcourse
no longer be irreducible, but will decompose in lower dimensional irreducible
representations (which may occur multiple times) for the Lie algebra
$\mathfrak{so}(n-1)$. These lower dimensional summands of irreducible
representations will also admit polynomial models (this gives rise to the
horizontal dotted arrow at the bottom). This means that we are now left with
two polynomial models. The problem how these two models interact is then
solved by the CK-extension. In other words: the CK-extension can be viewed as
an inverse branching procedure.

Moreover, one could develop the function theory for the symplectic Rarita-
Schwinger operator (and for the more general HMD) like it was done in the
orthogonal case (see e.g. [29]). Last of all, we wish to mention that the
connection between Dirac-type operators and parastatistic algebras has
recently emerged in [1]. In this paper, the authors consider a system of $N$
Grassmann Dirac operators and their duals. Recall that the classical Dirac
operator $\underline{\partial}_{x}=\sum_{j=1}^{n}e_{j}\partial_{x_{j}}$ is a
contraction between Clifford algebra generators and partial derivatives. When
one replaces the derivatives $\partial_{x_{j}}$ by Grassmann derivatives
(partial derivatives with respect to an anti-commuting or Grassmann variable)
$\partial_{\theta_{j}}$ the resulting Dirac-type operator
$\underline{\partial}_{\theta}=\sum_{j=1}^{n}e_{j}\partial_{\theta_{j}}$ is
called the Grassmann Dirac operator and acts on spinor-valued forms. The
striking difference with the usual Dirac operator $\underline{\partial}_{x}$
is that $\underline{\partial}_{\theta}$ and its dual give rise to a copy of
the Lie algebra $\mathfrak{sl}(2)$ instead of the Lie super algebra
$\mathfrak{osp}(1|2)$, we refer to [30] for the details. When considering $N$
Grassmann Dirac operators and their duals, the authors of [1] show that the
algebra closes as $\mathfrak{so}(2N+1)$. Note that we found the same algebra
in Theorem 4.19 but in terms of symplectic Dirac operators acting on
symplectic spinor-valued polynomials. This yields a nice application of
supersymmetry, in which both the orthogonal and symplectic framework can be
combined. This is planned in future work.

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