# A Hermitian refinement of symplectic Clifford analysis

David Eelbode111UAnwerpen, Middelheimlaan 1, 2020 Antwerp Belgium
<EMAIL_ADDRESS>Guner Muarem222UAnwerpen, Middelheimlaan 1, 2020
Antwerp Belgium<EMAIL_ADDRESS>
###### Abstract

In this paper we develop the Hermitian refinement of symplectic Clifford
analysis, by introducing a complex structure $\mathbb{J}$ on the canonical
symplectic manifold $(\mathbb{R}^{2n},\omega_{0})$. This gives rise to two
symplectic Dirac operators $D_{s}$ and $D_{t}$ (in the sense of Habermann
[8]), leading to a $\mathfrak{u}(n)$-invariant system of equations on
$\mathbb{R}^{2n}$. We discuss the solution space for this system, culminating
in a Fischer decomposition for the space of polynomials on $\mathbb{R}^{2n}$
with values in the symplectic spinors. To make this decomposition explicit, we
will construct the associated embedding factors using a transvector algebra.

##  1 Introduction

This paper is to be situated in the framework of so-called symplectic
(Clifford) analysis, a function theory which focuses on solutions for the
symplectic Dirac operator $D_{s}$ (see [8] for basic information). This is a
first-order differential operator which generalises the classical Dirac
operator333In itself already a generalisation of the operator introduced by
P.A.M. Dirac to describe the behaviour of electrons. on $\mathbb{R}^{n}$ in
the following sense: instead of the typical spin group invariance
characterising the standard Dirac operator, which is orthogonal in nature
because $\mathsf{Spin}(n)$ is a double cover of the orthogonal group
$\mathsf{SO}(n)$, one will get symplectic invariance. It is still a Dirac
operator though, because the symplectic Dirac operator $D_{s}$ acts on
functions taking values in the symplectic generalisation of the spinor space
$\mathbb{S}$ from the orthogonal setting, and because its solutions can be
related to solutions for the Laplace operator (this point will also be treated
in this paper, although we immediately note that it is more subtle than the
orthogonal case, where one simply has that the square of the classical Dirac
operator gives the Laplace operator). In this paper, we will consider a
symmetry reduction from the symplectic Lie group $\mathsf{Sp}(2n)$ to its
subgroup $\mathsf{U}(n)$. To do this, we will introduce a complex structure
$\mathbb{J}$ on the symplectic manifold $(\mathbb{R}^{2n},\omega_{0})$. Just
like in the orthogonal case, introducing a complex structure $\mathbb{J}$ on
$\mathbb{R}^{2n}$ will then lead to a refinement of symplectic Clifford
analysis.

Before introducing this symplectic analogue, we will shortly review the basics
of the Hermitian refinement of (classical) Clifford analysis, breaking down
the symmetry of the orthogonal group $\mathsf{SO}(2n)$ to the unitary group
$\mathsf{U}(n)$. This leads to a system of two Hermitian Dirac operators which
are invariant under the unitary group $\mathsf{U}(n)$ (more precisely: its
double cover, realised as a subgroup of the spin group). It is well-known that
the algebraic structure generated by these Hermitian Dirac operators and their
(Fischer) dual operators gives rise to the Lie superalgebra
$\mathfrak{osp}(2|2)$, which then leads to a Howe dual pair
$\mathsf{U}(n)\times\mathfrak{osp}(2|2)$ with an associated Fischer
decomposition for the space of spinor-valued complex harmonics (see [2], or
Theorem 2.8).

The symplectic version of this last theorem will be the main result of this
paper. To do so we will define the operator $D_{t}$, related to the symplectic
Dirac operator $D_{s}$ under a suitable complex structure. Next we describe
the unitary invariance of this pair of operators $(D_{s},D_{t})$ and determine
the algebra generated by them, leading to the dual pair
$\mathsf{U}(n)\times\mathfrak{su}(1,2)$. Finally, we will use this dual pair
to obtain a Fischer decomposition for the space of spinor-valued bi-harmonics
on the space $\mathbb{R}^{2n}\cong\mathbb{C}^{n}$.

##  2 Hermitian Clifford analysis

The key ingredient in the transition from the classical case to Hermitean
refinement is a complex structure $\mathbb{J}$, given by

$\mathbb{J}=\begin{pmatrix}0&I_{n}\\\ -I_{n}&0\end{pmatrix}\in\mathsf{SO}(2n)\
.$

On the standard orthonormal basis $\\{e_{1},\ldots,e_{2n}\\}$ for
$\mathbb{R}^{2n}$ this complex structure $\mathbb{J}$ thus acts as follows:
$\mathbb{J}(e_{k})=-e_{k+n}$ and $\mathbb{J}(e_{k+n})=e_{k}$, for all $1\leq
k\leq n$. Note that the coordinates will be referred to as
$(x_{1},\ldots,x_{n},y_{1}\ldots,y_{n})\in\mathbb{R}^{2n}$ in this paper. The
(classical) Dirac operator $\underline{\partial}_{x}$ and its (Fischer) dual
$\underline{x}$ on $\mathbb{R}^{2n}$ are given by

$\displaystyle\underline{x}=\sum_{k=1}^{n}\big{(}e_{k}x_{k}+e_{n+k}y_{k}\big{)}\quad\mbox{and}\quad\underline{\partial}_{x}=\sum_{k=1}^{n}\big{(}e_{k}\partial_{x_{k}}+e_{n+k}\partial_{y_{k}}\big{)}\
,$

and using the complex structure $\mathbb{J}$ one can define a twisted (or
rotated) version:

$\displaystyle\underline{x}^{\mathbb{J}}:$
$\displaystyle=\mathbb{J}[\underline{x}]=\sum_{k=1}^{n}(y_{k}e_{2k-1}-x_{k}e_{2k})$
$\displaystyle{\partial^{\mathbb{J}}_{\underline{x}}}:$
$\displaystyle=\mathbb{J}[\underline{\partial}_{x}]=\sum_{k=1}^{n}(e_{2k-1}\partial_{y_{k}}-e_{2k}\partial_{x_{k}})\
.$

Note that $\mathbb{J}$ acts on the basis vectors $e_{k}$ here. In the
definition below, $\mathbb{S}$ stands for the space of Dirac spinors on
$\mathbb{R}^{2n}$ (i.e. the sum of positive and negative Weyl spinors),
typically realised using a minimal left ideal in the complex Clifford algebra
$\mathbb{C}_{2n}$.

######  Definition 2.1.

An $\mathbb{S}$-valued function $f$ on $\mathbb{R}^{2n}$ satisfying the
equations
$\underline{\partial}_{x}f={\partial^{\mathbb{J}}_{\underline{x}}}f=0$ is
called Hermitian monogenic (or $h$-monogenic). One further defines the space

$\displaystyle\mathcal{M}_{k}^{h}(\mathbb{R}^{2n},\mathbb{S}):=\mathcal{P}_{k}(\mathbb{R}^{2n},\mathbb{S})\cap\big{(}\ker(\underline{\partial}_{x})\cap\ker(\partial^{\mathbb{J}}_{\underline{x}})\big{)}\
,$

containing $k$-homogeneous polynomials which are $h$-monogenic.

There is another way to introduce a pair of Hermitian Dirac operators, using
the Witt basis for $\mathbb{R}^{2n}\cong\mathbb{C}^{n}$:

######  Definition 2.2.

The Witt basis vectors for $\mathbb{C}^{n}$ are defined as

$\displaystyle\mathfrak{f}_{k}=\frac{1}{2}\left(e_{k}-ie_{n+k}\right)\quad\mbox{and}\quad\mathfrak{f}_{k}^{\dagger}=-\frac{1}{2}\left(e_{k}+ie_{n+k}\right)\
.$

The Hermitian Dirac operators and their adjoints are then introduced as
follows:

$\displaystyle(\underline{\partial}_{z},\underline{\partial}_{z}^{\dagger})=\left(\sum_{k=1}^{n}\mathfrak{f}_{k}^{\dagger}\partial_{z_{k}},\sum_{k=1}^{n}\mathfrak{f}_{k}\partial_{\overline{z}_{k}}\right)\quad\mbox{and}\quad(\underline{z},\underline{z}^{\dagger})=\left(\sum_{k=1}^{n}\mathfrak{f}_{k}z_{k},\sum_{k=1}^{n}\mathfrak{f}_{k}^{\dagger}\overline{z}_{k}\right)\
.$ (2.1)

The Dirac operator on $\mathbb{R}^{2n}$ can then be written as
$\underline{\partial}_{x}=2(\underline{\partial}_{z}^{\dagger}-\underline{\partial}_{z})$,
and similarly for its adjoint
$\underline{x}=\underline{z}-\underline{z}^{\dagger}$. This shows why
Hermitian Clifford analysis is a refinement of classical Clifford analysis:
$h$-monogenics always belong to the kernel of the Dirac operator
$\underline{\partial}_{x}$, but not necessarily the other way around.

######  Remark 2.3.

The operators $\underline{\partial}_{z}$ and
$\underline{\partial}_{z}^{\dagger}$ can be used to factorise the Laplacian
$\Delta$ on $\mathbb{R}^{2n}$, since
$\Delta=4(\underline{\partial}_{z}^{\dagger}\underline{\partial}_{z}+\underline{\partial}_{z}\underline{\partial}_{z}^{\dagger})=4\\{\underline{\partial}_{z}^{\dagger},\underline{\partial}_{z}\\}$.
This implies that every $h$-monogenic function is harmonic.

The Witt basis can be used to define a simple model for the spinor space
$\mathbb{S}$ in the orthogonal setting. Indeed, one has that
$\mathbb{S}=\mathbb{C}_{2n}I$ where $I$ is the primitive idempotent
$I=I_{1}\cdots I_{n}$ with $I_{k}=\mathfrak{f}_{k}\mathfrak{f}_{k}^{\dagger}$
for all $1\leq k\leq n$. Due to the fact that $\mathfrak{f}_{k}I=0$, the
spinor space $\mathbb{S}$ reduces to the action of the (complex) Grassmann
algebra $\Lambda^{\dagger}_{n}$ spanned by the elements
$\\{\mathfrak{f}^{\dagger}_{k}\\}_{k=1}^{n}$, which means that we can write
$\mathbb{S}=\Lambda^{\dagger}_{n}I$. This implies that the spinor space
decomposes into ‘homogeneous parts’ as

$\mathbb{S}=\bigoplus_{r=0}^{n}\mathbb{S}_{(r)}=\bigoplus_{r=0}^{n}(\Lambda^{\dagger}_{n})^{r}I$

where $(\Lambda^{\dagger}_{n})^{r}$ consists of all $r$-vectors in
$\Lambda^{\dagger}_{n}$. This decomposition is certainly not spin-invariant,
but it does carry a (left) multiplicative $\mathfrak{u}(n)$-action, see e.g.
[2]. The invariant subspaces $\mathbb{S}_{(r)}$ can be characterised as
eigenspaces of the so-called spin-Euler operator $\beta=n-\sum_{k}I_{k}$,
measuring the ‘degree of homogeneity’ $r$.

Now that we have introduced the Hermitian Dirac operators
$\underline{\partial}_{z}$ and $\underline{\partial}_{z}^{\dagger}$ and
discussed the decomposition of the spinor space $\mathbb{S}$ into
$\mathfrak{u}(n)$-irreducible representations, we focus on the algebra
generated by these Dirac operators and their (Fischer) duals. Recall that in
the orthogonal case the operators $\underline{\partial}_{x}$ and
$\underline{x}$ give rise to the Lie superalgebra $\mathfrak{osp}(1|2)$,
whereas in the Hermitian setting one obtains the algebra
$\mathfrak{osp}(2|2)$. For further details, such as the definitions for the
generators and their (anti-)commutation relations, we refer to [2]. There, one
can also find the following theorem:

######  Theorem 2.4 (Hermitian Howe duality).

The space $\mathcal{P}(\mathbb{R}^{2n},\mathbb{S})$ of spinor-valued
polynomials on $\mathbb{R}^{2n}$ can be decomposed in a multiplicity-free way,
using the Howe dual pair
$\operatorname{\mathsf{U}}(n)\times\mathfrak{osp}(2|2)$.

Using the identification
$(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\leftrightarrow(x_{1},\dots,x_{n},y_{1},\dots,y_{n})\in\mathbb{R}^{2n}$,
where we have put $z_{k}=x_{k}+iy_{k}$, we get a canonical identification

$\displaystyle
P(\underline{x},\underline{y})\in\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\
\leftrightarrow\
P(\underline{z},\overline{\underline{z}})\in\mathcal{P}(\mathbb{C}^{n},\mathbb{C})\
.$

Further structure can then be imposed on this space of polynomials:

######  Definition 2.5.

Let $a,b$ be two non-negative integers. A polynomial
$P(\underline{z},\overline{\underline{z}})$ in
$\mathcal{P}(\mathbb{C}^{n},\mathbb{C})$ is said to be bi-homogeneous of
degree $(a,b)$ if

$\displaystyle
P(\lambda\underline{z},\overline{\lambda}\overline{\underline{z}})=\lambda^{a}\overline{\lambda}^{b}P(\underline{z},\overline{\underline{z}})$

for all $\underline{z}\in\mathbb{C}^{n}$ and $\lambda\in\mathbb{C}$. We denote
the space of the bi-homogeneous polynomials of degree $(a,b)$ by
$\mathcal{P}_{a,b}(\mathbb{C}^{n},\mathbb{C})$ or
$\mathcal{P}_{a,b}(\mathbb{R}^{2n},\mathbb{C})$.

######  Remark 2.6.

Recall that in the classical case, the degree of homogeneity of a polynomial
$P_{k}\in\mathcal{P}_{k}(\mathbb{R}^{n},\mathbb{C})$ was measured by the Euler
operator $\mathbb{E}=\sum_{j=1}^{n}x_{j}\partial_{x_{j}}$. In the Hermitian
setting, one defines the so-called $h$-Euler operators as

$\displaystyle\mathbb{E}_{z}=\sum_{j=1}^{n}z_{j}\partial_{z_{j}}\quad\text{
and
}\quad\mathbb{E}_{z}^{\dagger}=\sum_{j=1}^{n}\overline{z}_{j}\partial_{\overline{z}_{j}}\
,$

the eigenvalues of which are precisely the bi-degree parameters $a$ and $b$.

######  Definition 2.7.

We define $\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})$ as the space of
harmonic polynomials on $\mathbb{R}^{2n}$ of bi-degree $(a,b)$. This means
that
$\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C}):=\mathcal{P}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\cap\ker(\Delta)$.

Denoting the space of $k$-harmonic polynomials on $\mathbb{R}^{2n}$ by means
of $\mathcal{H}_{k}(\mathbb{R}^{2n},\mathbb{C})$, we then have that

$\displaystyle\mathcal{H}_{k}(\mathbb{R}^{2n},\mathbb{C})=\bigoplus_{k=a+b}\mathcal{H}_{a,b}(\mathbb{C}^{n})\
,$

which expresses a irreducible representation for $\mathsf{SO}(2n)$ at the
left-hand side (harmonic polynomials of degree $k$) as a direct sum of
irreducible representations for the subgroup $\mathsf{U}(n)$ at the right-hand
side. A key result in understanding the Hermitean refinement of the Fischer
decomposition is the following theorem:

######  Theorem 2.8.

Suppose
$H_{a,b}^{(r)}(\underline{z},\overline{\underline{z}})\in\mathcal{H}_{a,b}(\mathbb{C}^{n},\mathbb{C})\otimes\mathbb{S}_{(r)}$
with $0<r<n$ and $a,b>0$, then
$H_{a,b}^{(r)}(\underline{z},\overline{\underline{z}})$ can be decomposed as

$\displaystyle H_{a,b}^{(r)}(\underline{z},\overline{\underline{z}})$
$\displaystyle=M_{a,b}^{(r)}(\underline{z},\overline{\underline{z}})+\underline{z}M_{a-1,b}^{(r+1)}(\underline{z},\overline{\underline{z}})+\underline{z}^{\dagger}M_{a,b-1}^{(r-1)}(\underline{z},\overline{\underline{z}})$
$\displaystyle+\left((a+r-1)\underline{z}\underline{z}^{\dagger}-(b+m-r-1)\underline{z}^{\dagger}\underline{z}\right)M_{a-1,b-1}^{(r)}(\underline{z},\overline{\underline{z}}),$

where all
$M_{a,b}^{(r)}(\underline{z},\overline{\underline{z}})\in\ker(\underline{\partial}_{z})\cap\ker(\underline{\partial}^{\dagger}_{z})$,
i.e. are $h$-monogenic.

######  Remark 2.9.

There are a few trivial cases for the decomposition above, which explains the
restrictions on the parameters $a,b$ and $r$ (see [2] for more details).

##  3 Hermitian symplectic Dirac operators

Just like in the previous section, we start by introducing the rotated (or
‘twisted’) symplectic Dirac operator and its dual, using the complex structure
$\mathbb{J}$ from above. For that purpose, we recall their definitions (see
[8]):

$\displaystyle
D_{s}=\sum_{j=1}^{n}\big{(}iq_{j}\partial_{y_{j}}-\partial_{q_{j}}\partial_{x_{j}}\big{)}\quad\mbox{and}\quad
X_{s}=\sum_{j=1}^{n}\big{(}y_{j}\partial_{q_{j}}+iq_{j}x_{j}\big{)}\ .$

These operators act on symplectic spinor-valued functions
$f(x_{1},\ldots,y_{n})$, whereby the symplectic spinor space can be realised
as the Schwartz space $\mathcal{S}(\mathbb{R}^{n},\mathbb{C})$ of rapidly
decreasing functions on $\mathbb{R}^{n}$. These then correspond to the smooth
vectors of the oscillator representation for the metaplectic group
$\mathsf{Mp}(2n)$, see for instance [8, 11]. We will use the variable
$\underline{q}=(q_{1},\ldots,q_{n})\in\mathbb{R}^{n}$ as the argument for
these functions, and use the symbol $\mathbb{S}^{\infty}$ to refer to this
spinor space (where the infinity symbol stresses the fact that the underlying
metaplectic representation is infinite-dimensional).

######  Remark 3.1.

We denote the even and odd part of $\mathbb{S}^{\infty}$, containing even and
odd functions respectively, by means of $\mathbb{S}^{\infty}_{+}$ and
$\mathbb{S}^{\infty}_{-}$. These spaces then define the (positive and
negative) spinor representations for the symplectic group $\mathsf{Sp}(2n)$,
or its associated Lie algebra. The highest weights of these representations
are given by

$\displaystyle\mathbb{S}^{\infty}_{+}\leftrightarrow\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{sp}(2n)}\quad\text{
and
}\quad\mathbb{S}^{\infty}_{-}\leftrightarrow\left(-\frac{1}{2},\dots,-\frac{1}{2},-\frac{3}{2}\right)_{\mathfrak{sp}(2n)}\
,$

and they provide examples of so-called completely pointed modules, see [3, 4].

Now that we have defined the values of the functions that will be considered
in this paper, we can use the complex structure $\mathbb{J}$, acting on the
column vector $(q_{j},-i\partial_{q_{j}})^{T}$, to arrive at the following
operators:

######  Definition 3.2.

The twisted symplectic Dirac operator $D_{t}$ and its (Fischer) dual $X_{t}$
are given by:

$\displaystyle
D_{t}:=\mathbb{J}[D_{s}]=\sum_{j=1}^{n}\big{(}iq_{j}\partial_{x_{j}}+\partial_{y_{j}}\partial_{q_{j}}\big{)}\quad\mbox{and}\quad
X_{t}:=\mathbb{J}[X_{s}]$
$\displaystyle=\sum_{j=1}^{n}\big{(}x_{j}\partial_{q_{j}}-iy_{j}q_{j}\big{)}\
.$

######  Remark 3.3.

The operator $D_{t}$ was often ignored up to this point in the context of
symplectic Clifford analysis for $D_{s}$, mostly because the former operator
does not have the $\mathfrak{sp}(2n)$-invariance properties of the latter. The
operator $D_{t}$ is indeed not invariant under the realisation for
$\mathfrak{sp}(2n)$ which appears in the setting of $D_{s}$ (see the first set
of operators in the lemma below). However, when switching to another
realisation for this symplectic Lie algebra (the second set of operators in
the lemma below), $D_{t}$ will be invariant. As is to be expected, $D_{s}$
will in turn not be invariant under this new realisation. We would thus argue
that both operators are symplectically invariant, just like Habermann mentions
in [8], under the caveat that one has to specify for which realisation. Recall
that this invariance amounts to saying that the generators below commute with
the symplectic Dirac operators, so e.g.
$[D_{s},X_{jk}]=[D_{t},\tilde{Y}_{jk}]=0$.

######  Lemma 3.4.

We have the following two realisations of the Lie algebra $\mathfrak{sp}(2n)$
on the space
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}$ of
symplectic spinor-valued polynomials

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

and

$\displaystyle\begin{cases}\tilde{X}_{jk}=x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}}+q_{k}\partial_{q_{j}}+\frac{1}{2}\delta_{jk}&1\leq
j\leq k\leq n\\\
\tilde{Y}_{jk}=x_{j}\partial_{y_{k}}+x_{k}\partial_{y_{j}}-iq_{j}q_{k}&j<k=1,\dots,n\\\
\tilde{Z}_{jk}=y_{j}\partial_{x_{k}}+y_{k}\partial_{x_{j}}-i\partial_{q_{j}}\partial_{q_{k}}&j<k=1,\dots,n\\\
\tilde{Y}_{jj}=x_{j}\partial_{y_{j}}-\frac{i}{2}q_{j}^{2}&j=1,\dots,n\\\
\tilde{Z}_{jj}=y_{j}\partial_{x_{j}}-\frac{i}{2}\partial_{q_{j}}^{2}&j=1,\dots,n\end{cases}$

The symplectic Dirac operator $D_{s}$ is invariant under the first
realisation, the operator $D_{t}$ under the second.

###### Proof.

This follows from straightforward commutation relations. ∎

######  Remark 3.5.

The capital letters used to denote these operators are referring to the
letters used in a typical matrix realisation (see e.g. [7]), so that the Lie
algebra homomorphism becomes immediately clear.

We will now perform a symmetry reduction, such that the operators $D_{s}$ and
$D_{t}$ both become invariant under one and the same group (or associated Lie
algebra). To see that this must be the unitary group, we will first switch to
matrices and characterise all symplectic matrices which commute with the
complex structure $\mathbb{J}$.

######  Lemma 3.6.

We have that $\mathsf{Sp}_{\mathbb{J}}(2n):=\\{M\in\mathsf{Sp}(2n)\mid
M\mathbb{J}=\mathbb{J}M\\}$ defines a realisation for the unitary Lie group
$\mathsf{U}(n)$.

###### Proof.

In order to see this, assume that $M$ is of the block-form
$\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)$, where $A,B,C$
and $D$ are $(n\times n)-$matrices. There are two conditions to be met here:
$M^{T}\mathbb{J}M=\mathbb{J}$ (being symplectic) and
$M\mathbb{J}=\mathbb{J}M$. The latter tells us that $A=D$ and $B=-C$, and
plugging the second condition in the first immediately yields that
$M^{T}M=\textup{Id}$ gives the identity matrix. This is precisely the
condition required to be a unitary matrix, with the well-known group
isomorphism
$\Phi:\mathsf{Sp}_{\mathbb{J}}(2n,\mathbb{R})\to\mathsf{U}(n):M\mapsto A+iB$.
∎

On the level of operators acting on functions, we get the following:

######  Lemma 3.7.

The Lie algebra $\mathfrak{u}(n)$ can be realised in terms of the differential
operators acting on the space
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}$:

$\displaystyle\begin{cases}A_{jk}=y_{j}\partial_{x_{k}}-x_{k}\partial_{y_{j}}+y_{k}\partial_{x_{j}}-x_{j}\partial_{y_{k}}+i(q_{j}q_{k}-\partial_{q_{j}}\partial_{q_{k}})&\quad
1\leq j<k\leq n\\\
B_{jj}=y_{j}\partial_{x_{j}}-x_{j}\partial_{y_{j}}+\frac{i}{2}\left(q_{j}^{2}-\partial_{q_{j}}^{2}\right)&\quad
1\leq j\leq n\\\
C_{jk}=x_{j}\partial_{x_{k}}-x_{k}\partial_{x_{j}}+y_{j}\partial_{y_{k}}-y_{k}\partial_{y_{j}}+q_{j}\partial_{q_{k}}-q_{k}\partial_{q_{j}}&\quad
1\leq j<k\leq n\end{cases}$

###### Proof.

One can use the previous lemma to obtain the matrix realisation, and use lemma
3.4 to obtain the corresponding differential operators. Alternatively, one can
also employ the fact that the operator $\sum_{j}B_{jj}$ corresponds to the
symplectic structure $\mathbb{J}$ from the previous lemma, and use this to
obtain the required linear combinations which commute with $\sum_{j}B_{jj}$. ∎

######  Corollary 3.8.

The operators $D_{s}$ and $D_{t}$ are invariant differential operators with
respect to the realisation for $\mathfrak{u}(n)$ from above.

###### Proof.

It is straightforward to check that $D_{s}$ and $D_{t}$ commute with all the
operators from Lemma 3.7. ∎

This means that we can now refine the $\mathfrak{sp}(2n)$-invariant equation
$D_{s}f=0$, by introducing a system of two $\mathfrak{u}(n)$-invariant
equations: $D_{s}f=$ and $D_{t}f=0$, for an $\mathbb{S}^{\infty}$-valued
function $f$ (note that we will mostly stick to polynomials in this paper). On
the analogy with the classical case, we have the following:

######  Definition 3.9.

A function which is a solution for both $D_{s}$ and $D_{t}$ is called a
Hermitian symplectic monogenic (or $h$-symplectic monogenic for short).

These two symplectic Dirac operators $(D_{s},D_{t})$ will serve as the
analogue of the operators
$(\underline{\partial}_{x},\underline{\partial}_{x}^{\mathbb{J}})$. One can
develop the analogy with the classical situation even further by introducing
the symplectic analogues of the operators $\underline{\partial}_{z}$ and
$\underline{\partial}_{z}^{\dagger}$, see (2). Recall that these Hermitian
Dirac operators $\underline{\partial}_{z}$ and
$\underline{\partial}_{z}^{\dagger}$ were introduced using the Witt basis
vectors. In the symplectic setting, we will do something similar by
introducing the operators $D_{z}$ and $D_{z}^{\dagger}$, known in the
literature as the symplectic Dolbeault operators [10]. These are defined using
the lowering operator $L_{j}=q_{j}+\partial_{q_{j}}$ and raising operator
$R_{j}=q_{j}-\partial_{q_{j}}$ $(1\leq j\leq n)$ acting on symplectic spinors.

######  Definition 3.10.

The symplectic Dolbeault operators are defined as follows:

$\displaystyle D_{z}:=\frac{1}{2}(D_{s}+iD_{t})$
$\displaystyle=-\sum_{j=1}^{n}L_{j}\partial_{z_{j}}=-\langle\underline{L},\underline{\partial}_{z}\rangle$
$\displaystyle D_{z}^{\dagger}:=\frac{1}{2}(D_{s}-iD_{t})$
$\displaystyle=\sum_{j=1}^{n}R_{j}\partial_{{\overline{z}}_{j}}=\langle\underline{R},\underline{\partial}_{\overline{z}}\rangle$
$\displaystyle X_{z}:=\frac{1}{2}(X_{s}+iX_{t})$
$\displaystyle=\frac{i}{2}\sum_{j=1}^{n}L_{j}{\overline{z}_{j}}=\frac{i}{2}\langle\underline{L},\underline{\overline{z}}\rangle$
$\displaystyle X_{z}^{\dagger}:=\frac{1}{2}(X_{s}-iX_{t})$
$\displaystyle=\frac{i}{2}\sum_{j=1}^{n}R_{j}{{z}_{j}}=\frac{i}{2}\langle\underline{R},\underline{z}\rangle\
,$

where we used the short-hand notation notation $\langle\cdot,\cdot\rangle$ for
the inner product.

The general philosophy in passing from the orthogonal setting to the
symplectic one often amounts to replacing an anti-commutator
$\\{\cdot,\cdot\\}$ by a commutator $[\cdot,\cdot]$. As a simple example we
mention the definition for the Clifford algebra versus the Weyl algebra. A
similar thing happens for the factorisation of the Laplace operator. Whereas
the formula
$\Delta=4\\{\underline{\partial}_{z},\underline{\partial}_{z}^{\dagger}\\}$
expresses $\Delta$ as an anti-commutator, a simple computation leads to the
commutator relation $[D_{s},D_{t}]=-i\Delta$ in the symplectic setting. An
immediate consequence is the following:

######  Lemma 3.11.

We have the inclusion $\ker(D_{s})\cap\ker(D_{t})\subseteq\ker(\Delta)$.

This shows that also Hermitian symplectic Clifford analysis is a refinement of
harmonic analysis for the Laplace operator $\Delta$ on $\mathbb{R}^{2n}$. The
dual operators $X_{s}$ and $X_{t}$ obviously satisfy a similar relation
$[X_{s},X_{t}]=-ir^{2}$, where $r^{2}$ is the multiplication operator
$r^{2}=\sum_{j}(x_{j}^{2}+y_{j}^{2})=\sum_{j}|z_{j}|^{2}$. Putting everything
together, this leads to a third realisation for the Lie algebra
$\mathfrak{sl}(2)$ in our framework, next to the algebras
$\operatorname{Alg}(D_{s},X_{s})$ and $\operatorname{Alg}(D_{t},X_{t})$. The
former was proved in [6] and used to obtain the symplectic Fischer
decomposition, the latter is based on similar calculations. Indeed, it is
well-known that $\operatorname{Alg}(\Delta,r^{2})\cong\mathfrak{sl}(2)$, after
a suitable normalisation of these operators. To summarise, we then have the
following:

Complex data | Orthogonal setting | Symplectic setting
---|---|---
none | $-\Delta=\underline{\partial}_{x}^{2}$ | $\Delta\neq D_{s}^{2}$
complex structure $\mathbb{J}$ |  Real viewpoint $\Delta=\underline{\partial}_{x}^{2}=\mathbb{J}[\underline{\partial}_{x}]^{2}$ Complex viewpoint $\Delta=4\\{\underline{\partial}_{z},\underline{\partial}_{z}^{\dagger}\\}$ |  Real viewpoint $\Delta=[D_{s},D_{t}]$ Complex viewpoint $\Delta=[D_{z},D_{z}^{\dagger}]$

We will now describe a wide class of examples of $h$-symplectic monogenics, by
making the link with holomorphic functions in several variables. Let
$f(\underline{z}):\mathbb{C}^{n}\to\mathbb{C}$ be a complex-valued function in
several complex variables which is of the class $\mathcal{C}^{1}(\Omega)$
(i.e. continuously differentiable). We say that $f$ is holomorphic in several
variables if $\partial_{\overline{z}_{j}}f(\underline{z})=0$ for all $1\leq
j\leq n$, where
$\partial_{\overline{z}_{j}}:=\frac{1}{2}(\partial_{x_{j}}+i\partial_{y_{j}})$
is the Cauchy-Riemann operator in the relevant variable. Note that we will
often write functions $f(\underline{z})$ in terms of real variables, i.e. as
$f(\underline{x},\underline{y})$. We can then prove the following:

######  Theorem 3.12.

Let $f(\underline{x},\underline{y})$ be a holomorphic function in several
complex variables. Then the functions of the form

$\displaystyle
F(\underline{x},\underline{y};\underline{q}):=f(\underline{x},\underline{y})e^{-\frac{1}{2}|\underline{q}|}$

are solutions of $D_{s}$ and $D_{t}$, i.e. symplectic $h$-monogenics.

###### Proof.

Letting the symplectic Dirac operator $D_{s}$ act on
$F(\underline{x},\underline{y};\underline{q})$ gives:

$\displaystyle
D_{s}\left(e^{-\frac{1}{2}|\underline{q}|^{2}}f(\underline{x},\underline{y})\right)$
$\displaystyle=e^{-\frac{1}{2}|\underline{q}|^{2}}\sum_{k}q_{k}(\partial_{x_{k}}+i\partial_{y_{k}})f(\underline{x},\underline{y})$

This is indeed equal to zero if
$f(\underline{z})=f(\underline{x},\underline{y})$ is holomorphic in several
variables. A similar calculation for $D_{t}$ completes the proof. ∎

######  Remark 3.13.

We note that Habermann [8] studied the following system for the symplectic
Dirac operators $D_{s}$ and $D_{t}$ on $\mathbb{R}^{2}$:

$\displaystyle\begin{cases}(iq\partial_{y}-\partial_{q}\partial_{x})f(x,y;q)&=0\\\
(iq\partial_{x}+\partial_{q}\partial_{y})f(x,y;q)&=0\end{cases}$

and concluded that the solution are of the form $f=e^{-\frac{1}{2}q^{2}}H$
where $H$ is a holomorphic function. Proposition 3.12 thus generalises these
solutions to $\mathbb{R}^{2n}$. However, it turns out that there are much more
solutions when the dimension increases from 2 to $2n$ (with $n>1$). If $n=3$
for instance, it is easily seen that

$\displaystyle
F(\underline{x},\underline{y};\underline{q})=e^{-\frac{1}{2}\left(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}\right)}\left(2\left(2q_{3}^{2}-1\right)\left(x_{2}-iy_{2}\right)-4q_{2}q_{3}\left(x_{3}-iy_{3}\right)\right)$

is indeed symplectic $h$-monogenic, although not of the holomorphic type. In
what follows we will therefore describe the elements in
$\ker(D_{s})\cap\ker(D_{t})$ more systematically.

##  4 The symplectic Hermitian Howe duality

To determine the Howe dual pair $\mathsf{U}(n)\times\mathfrak{g}$ which can be
seen as the symplectic version of the dual pair
$\mathsf{U}(n)\times\mathfrak{osp}(2|2)$ from Theorem 2.4, we start by
describing the Lie algebra generated by the operators $D_{s}$ and $D_{t}$ and
their (Fischer) duals $X_{s}$ and $X_{t}$. Note that we have already obtained
the following three realisations for the Lie algebra $\mathfrak{sl}(2)$:

$\displaystyle\operatorname{Alg}\\{D_{s},X_{s},\mathbb{E}+n\\}\cong\operatorname{Alg}\\{D_{t},X_{t},\mathbb{E}+n\\}\cong\operatorname{Alg}(\Delta,r^{2},\mathbb{E}+n)\cong\mathfrak{sl}(2)\
.$

The three copies of $\mathfrak{sl}(2)$ do not commute mutually however,
because for instance

$\displaystyle[D_{t},X_{s}]$
$\displaystyle=\sum_{j=1}^{n}\big{(}i(x_{j}\partial_{y_{j}}-y_{j}\partial_{x_{j}})+\partial_{q_{j}}^{2}-q_{j}^{2}\big{)}\neq
0\ .$

The operator above will thus be required to close the Lie algebra, so let us
give it a name:

######  Definition 4.1.

The operator $\mathcal{O}$, acting on $\mathbb{S}^{\infty}$-valued functions
$f(\underline{x},\underline{y};\underline{q})$ on $\mathbb{R}^{2n}$, is
defined by means of

$\displaystyle\mathcal{O}:=\sum_{j=1}^{n}\big{(}i(x_{j}\partial_{y_{j}}-y_{j}\partial_{x_{j}})+\partial_{q_{j}}^{2}-q_{j}^{2}\big{)}\
.$

It turns out that by adding the operator $\mathcal{O}$ to the operators from
above in our $\mathfrak{sl}(2)$-triplets, we will indeed get an algebra of
dimension 8 which closes under the commutator bracket $[\cdot,\cdot]$. Let us
first introduce this Lie algebra in terms of matrices:

######  Definition 4.2.

The Lie algebra $\mathfrak{su}(1,2)$ is a (quasi-split) real form of the
complex Lie algebra $\mathfrak{sl}(3)$, and is defined in terms of matrices as
follows

$\displaystyle\mathfrak{su}(1,2)=\left\\{\begin{pmatrix}\alpha&\beta&ic\\\
\gamma&\overline{\alpha}-\alpha&-\overline{\beta}\\\
id&-\overline{\gamma}&-\overline{\alpha}.\end{pmatrix}\in\mathbb{C}^{3\times
3}:c,d\in\mathbb{R}\text{ and }\alpha,\beta,\gamma\in\mathbb{C}\right\\}$

A basis for this Lie algebra is given by the the following set of matrices:

$\displaystyle\begin{matrix}H_{1}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}&X_{1}=\begin{pmatrix}0&1&0\\\ 0&0&-1\\\
0&0&0\end{pmatrix}&X_{2}=\begin{pmatrix}0&i&0\\\ 0&0&i\\\
0&0&0\end{pmatrix}&X_{3}=\begin{pmatrix}0&0&i\\\ 0&0&0\\\
0&0&0\end{pmatrix}\\\ H_{2}=\begin{pmatrix}i&0&0\\\ 0&-2i&0\\\
0&0&i\end{pmatrix}&Y_{1}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\
0&-1&0\end{pmatrix}&Y_{2}=\begin{pmatrix}0&0&0\\\ i&0&0\\\
0&i&0\end{pmatrix}&Y_{3}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
i&0&0\end{pmatrix}\end{matrix}$

The commutators $[M_{1},M_{2}]$ between two matrices $M_{1}$ and $M_{2}$ in
$\mathfrak{su}(1,2)$ give rise to the table below (with $M_{1}$ in a row and
$M_{2}$ in a column).

$[\cdot,\cdot]$ | $H_{1}$ | $H_{2}$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $Y_{1}$ | $Y_{2}$ | $Y_{3}$
---|---|---|---|---|---|---|---|---
$H_{1}$ | $0$ | $0$ | $X_{1}$ | $X_{2}$ | $2X_{3}$ | $-Y_{1}$ | $-Y_{2}$ | $-2Y_{3}$
$H_{2}$ | 0 | 0 | $3X_{2}$ | $-3X_{1}$ | $0$ | $-3Y_{2}$ | $3Y_{1}$ | 0
$X_{1}$ | $-X_{1}$ | $-3X_{2}$ | 0 | $2X_{3}$ | $0$ | $H_{1}$ | $H_{2}$ | $-Y_{2}$
$X_{2}$ | $-X_{2}$ | $3X_{1}$ | $-2X_{3}$ | $0$ | $0$ | $H_{2}$ | $-H_{1}$ | $-Y_{1}$
$X_{3}$ | $-2X_{3}$ | $0$ | $0$ | $0$ | $0$ | $-X_{2}$ | $-X_{1}$ | $-H_{1}$
$Y_{1}$ | $Y_{1}$ | $3Y_{2}$ | $-H_{1}$ | $-H_{2}$ | $X_{2}$ | $0$ | $-2Y_{3}$ | $0$
$Y_{2}$ | $Y_{2}$ | $-3Y_{1}$ | $-H_{2}$ | $H_{1}$ | $X_{1}$ | $2Y_{3}$ | $0$ | $0$
$Y_{3}$ | $2Y_{3}$ | $0$ | $Y_{2}$ | $Y_{1}$ | $H_{1}$ | $0$ | $0$ | $0$

######  Theorem 4.3.

The algebra generated by the symplectic Dirac operators $D_{s},D_{t}$ and
their adjoints $X_{s},X_{t}$ gives rise to a copy of the Lie algebra
$\mathfrak{su}(1,2)$.

###### Proof.

It follows from straightforward calculations that the eight operators can be
identified with the matrices from above as follows:

$\displaystyle(H_{1},H_{2},X_{1},X_{2},X_{3},Y_{1},Y_{2},Y_{3})\leftrightarrow(\mathbb{E}+n,i\mathcal{O},iX_{t},iX_{s},-\frac{i}{2}r^{2},-D_{t},D_{s},\frac{i}{2}\Delta)\
.$

Together with the table above, this proves the theorem. ∎

##  5 Decomposition of the symplectic spinor space

In Section 2 we studied the decomposition of the (standard) spinor space
$\mathbb{S}$ in terms of the spin-Euler operator $\beta$. In this section, we
will do the same thing for the infinite-dimensional symplectic spinor space
$\mathbb{S}^{\infty}$. For this we will need the powers $\rho^{k}$ of the
determinant representation (where $k\in\mathbb{R}$), given by

$\displaystyle\rho^{k}:\mathfrak{u}(n)\to\mathbb{C}:M\mapsto\det(M)^{k}\ ,$

with highest weight $(k,k,\dots,k)_{\mathfrak{u}(n)}$. Note that these weights
correspond to the action of a Cartan algebra for the complexification of
$\mathfrak{u}(n)$, where the elements $H_{j}:=iB_{jj}$ are chosen (see Lemma
3.7). As for $\mathbb{C}$-valued polynomials, we first mention the following
well-known result:

######  Theorem 5.1.

The space $\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})$ of bi-homogeneous
harmonics of degree $(a,b)$ is an irreducible $\mathfrak{u}(n)$-modules of
highest weight $(a,0,\dots,0,-b)_{\mathfrak{u}(n)}$ and has highest weight
vector
$w_{a,b}(\underline{z},\overline{\underline{z}})=z_{1}^{a}\overline{z}_{n}^{b}$.
Moreover, the dimension of the module is given by

$\dim(\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C}))=\frac{n+a+b-1}{n-1}{a+n-2\choose
n-2}{b+n-2\choose n-2}.$

Next, we consider the branching of the infinite-dimensional symplectic
spinors, when considered as a $\mathfrak{u}(n)$-module.

######  Theorem 5.2.

The symplectic spinor space $\mathbb{S}^{\infty}$ decomposes as follows:

$\displaystyle\mathbb{S}^{\infty}\bigg{\downarrow}^{\mathfrak{sp}(2n)}_{\mathfrak{u}(n)}=\bigoplus_{k=0}^{\infty}\mathbb{S}^{\infty}_{(r)}=\bigoplus_{r=0}^{\infty}\left(-\frac{1}{2},\dots,-\frac{1}{2}-r\right)_{\mathfrak{u}(n)}\
.$

The space $\mathbb{S}^{\infty}_{(r)}$ hereby has dimension ${n+r-1\choose r}$.

###### Proof.

The branching of the metaplectic representation
$\mathbb{S}^{\infty}=\mathbb{S}^{\infty}_{+}\oplus\mathbb{S}^{\infty}_{-}$ to
the maximal compact subalgebra $\mathfrak{u}(n)\subseteq\mathfrak{sp}(2n)$ was
treated in [9], and is given by

$\displaystyle\mathbb{S}_{+}^{\infty}$
$\displaystyle\to\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{u}(n)}\otimes\bigoplus_{r\in
2\mathbb{N}}(0,0,\dots,-r)_{\mathfrak{u}(n)}$
$\displaystyle\mathbb{S}_{-}^{\infty}$
$\displaystyle\to\left(-\frac{1}{2},\dots,-\frac{1}{2}\right)_{\mathfrak{u}(n)}\otimes\bigoplus_{r\in
2\mathbb{N}+1}(0,0,\dots,-r)_{\mathfrak{u}(n)}$

Working out these tensor products leads to the desired result. The dimension
of these spaces can be calculated using the Weyl dimension formula:

$\displaystyle\dim\mathbb{S}^{\infty}_{(r)}=\prod_{j=2}^{n}\frac{j+k-1}{j-1}={n+r-1\choose
r}\ ,$

which concludes the proof. ∎

######  Remark 5.3.

Note that this dimension coincides with the dimension of the space
$\mathcal{P}_{r}(\mathbb{R}^{2n},\mathbb{C})$ of $r$-homogeneous polynomials.
This is not a coincidence, since the spaces $\mathbb{S}^{\infty}_{(r)}$ are
known as the eigenspaces for the bosonic quantum oscillator.

In the following scheme we summarise the comparison between classical spinors
in $\mathbb{S}$ and symplectic spinors in $\mathbb{S}^{\infty}$:

Orthogonal | Symplectic
---|---
$\mathbb{S}=\bigoplus_{r=0}^{n}\mathbb{S}_{(r)}$ | $\mathbb{S}^{\infty}=\bigoplus_{r=0}^{\infty}\mathbb{S}_{(r)}^{\infty}$
$\dim\mathbb{S}_{(r)}=\dim\bigwedge^{r}V={\binom{n}{r}}$ | $\dim\mathbb{S}^{\infty}_{(r)}=\dim\operatorname{Sym}^{r}(V)={n+r-1\choose r}$
spin-Euler $\beta$ | Hermite operator $\Delta_{q}-|\underline{q}|^{2}$

######  Lemma 5.4.

The unitary representation $\mathbb{S}_{(r)}^{\infty}$ is generated by the
weight vector

$\displaystyle
w(\underline{q})=e^{-\frac{1}{2}|\underline{q}|^{2}}\mathsf{H}_{r}(q_{n})\ ,$

where $\mathsf{H}_{r}(q_{n})$ is a Hermite polynomial of degree $r$ in the
variable $q_{n}$.

###### Proof.

For all indices $1\leq j<n$ we directly obtain that

$\displaystyle
H_{j}w(\underline{q})=\frac{1}{2}\left(\partial_{q_{j}}^{2}-q_{j}^{2}\right)w(\underline{q})=-\frac{1}{2}w(\underline{q})\
.$

The action of the last Cartan element $H_{n}$ is slightly more complicated:

$\displaystyle\frac{1}{2}\left(\partial_{q_{n}}^{2}-q_{n}^{2}\right)w(\underline{q})=\frac{1}{2}e^{-\frac{1}{2}|\underline{q}|^{2}}\left(4(r-1)r\mathsf{H}_{r-2}\left(q_{n}\right)-\mathsf{H}_{r}\left(q_{n}\right)-4q_{n}r\mathsf{H}_{r-1}\left(q_{n}\right)\right)\
.$

Using the recursion relation for Hermite polynomials, which says that

$\displaystyle\mathsf{H}_{r}(q_{n})=2q_{n}\mathsf{H}_{r-1}(q_{n})-2(r-1)\mathsf{H}_{r-2}(q_{n})\
,$

we can rewrite this action as

$\frac{1}{2}\left(\partial_{q_{n}}^{2}-q_{n}^{2}\right)w(\underline{q})=-\frac{1}{2}e^{-\frac{1}{2}|\underline{q}|^{2}}(2r+1)\mathsf{H}_{r}(q_{n})=-\frac{1}{2}\left(r+\frac{1}{2}\right)w(\underline{q})\
.$

One further verifies that $w(\underline{q})$ is a solution for the positive
root vectors, which finishes the proof. ∎

##  6 The symplectic Hermitian Fischer decomposition

Now that we have obtained the reduction of the symplectic spinors
$\mathbb{S}^{\infty}$ in terms of the spaces $\mathbb{S}^{\infty}_{(r)}$ and
have identified our candidate for the Howe dual pair as
$\mathsf{U}(n)\times\mathfrak{su}(1,2)$, we are ready to formulate the Fischer
decomposition for the space
$\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}_{(r)}^{\infty}$.
In terms of $\mathfrak{u}(n)$-irreducible representations, we need to compute
the tensor product

$\displaystyle(a,0,\dots,0,-b)_{\mathfrak{u}(n)}\otimes\left(-\frac{1}{2},\dots,-\frac{2r+1}{2}\right)_{\mathfrak{u}(n)}\
.$ (6.1)

Our strategy is the following:

  1. (i)

First we will compute
$(a,0,\dots,0,-b)_{\mathfrak{u}(n)}\otimes(0,\cdots,0,-r)_{\mathfrak{u}(n)}$
using a suitable ‘dualisation argument’ and then determine the tensor product
of the resulting summands with the determinant representation
$\rho^{-\frac{1}{2}}$.

  2. (ii)

Of course, the procedure in step (i) yield a lot of information, but it only
gives the decomposition of the tensor product up to an isomorphism. In order
to make the isomorphism into an equality (as was in Theorem 2.8) we need the
so-called embedding factors which will turn out to be unitary invariant
differential operators.

For two irreducible representations for $\mathfrak{u}(n)$ with highest weights
$\alpha$ and $\beta$, the tensor product decomposes as
$\alpha\otimes\beta\cong\bigoplus_{\gamma}\mu_{\gamma}\gamma$, where
$\mu_{\gamma}$ is the multiplicity. As a particular case, we mention the
following result (from [13]):

$\displaystyle(m,0,\dots,0)_{\mathfrak{u}(n)}\otimes(\beta_{1},\dots,\beta_{n})_{\mathfrak{u}(n)}\cong\bigoplus\>(\beta_{1}+\nu_{1},\dots,\beta_{n}+\nu_{n})_{\mathfrak{u}(n)}$

where the summation is such that $\nu_{1}+\cdots+\nu_{n}=m$. Moreover, we have
the conditions $0\leq\nu_{1}\leq m$ and
$0\leq\nu_{k}\leq\beta_{k-1}-\beta_{k}$ for $2\leq k\leq n$. Invoking the fact
that $(V\otimes W)^{\ast}\cong V^{\ast}\otimes W^{\ast}$ (the dualisation
mentioned above), it then suffices to put

$\displaystyle
m=r\quad\mbox{and}\quad(\beta_{1},\ldots,\beta_{n})=(b,0,\ldots,0,-a)$

in the formula above, after which one can take the dual (or contragradient)
and a tensor product with the (one-dimensional) determinant representation
$\rho^{-\frac{1}{2}}$ to arrive at the theorem below. Note that from now on,
we restrict ourselves to the case $n\geq 3$.

######  Theorem 6.1.

We have the following tensor product decomposition:

$\mathcal{H}_{a,b}\otimes\mathbb{S}_{(r)}^{\infty}\cong\bigoplus_{j=0}^{a}\bigoplus_{i=0}^{b}\left(a-j-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\cdots,-i-\frac{1}{2},i+j-b-r-\frac{1}{2}\right)_{\mathfrak{u}(n)}\
,$

whereby the indices $i$ and $j$ satisfy the additional constraint $i+j\leq r$.

######  Remark 6.2.

If $a,b\geq r$, the direct sum at the right-hand side has a maximal number of
summands, equal to the triangular number $t_{r+1}=\frac{1}{2}(r+1)(r+2)$. This
should be contrasted with the classical case (the case
$\mathcal{H}_{a,b}\otimes\mathbb{S}_{(r)}$ in Hermitean Clifford analysis),
where the maximal number of summands is always bounded by $4$.

As usual in the framework of Fischer decompositions, we will now try to make
the information in the theorem above more explicit.

######  Remark 6.3.

Let us for instance calculate
$\mathcal{H}_{1,2}(\mathbb{C}^{3},\mathbb{C})\otimes\mathbb{S}_{(3)}^{\infty}$.
Using the formula from above we have:

$\displaystyle(1,0,-2)_{\mathfrak{u}(3)}\otimes(0,0,-3)_{\mathfrak{u}(3)}^{\prime}$
$\displaystyle\cong(1,0,-5)_{\mathfrak{u}(3)}^{\prime}\oplus(1,-1,-4)_{\mathfrak{u}(3)}^{\prime}\oplus(1,-2,-3)_{\mathfrak{u}(3)}^{\prime}$
$\displaystyle\oplus(0,0,-4)_{\mathfrak{u}(3)}^{\prime}\oplus(0,-1,-3)_{\mathfrak{u}(3)}^{\prime}\oplus(0,-2,-2)_{\mathfrak{u}(3)}^{\prime},$

where we have introduced the shorthand notation
$(a,b,c)^{\prime}:=\left(a-\frac{1}{2},b-\frac{1}{2},c-\frac{1}{2}\right)$.

Before we can express the summands in the decomposition for
$\mathbb{S}^{(r)}$-valued harmonics in terms of $h$-monogenics, we introduce
the following:

######  Definition 6.4.

For positive integers $a,b$ and $r$ with $b\leq r$ we use the notation
$\mathcal{M}_{a,b}^{(r)}$ for the irreducible representation highest weight

$\displaystyle\left(a-\frac{1}{2},-\frac{1}{2},\ldots,-b-\frac{1}{2},-r-\frac{1}{2}\right)\
.$

We restrict ourselves to the case $n\geq 3$ here.

The dimension of these spaces follows from the Weyl dimension formula, for
which we refer to a standard textbook:

######  Lemma 6.5.

Let $a,b,r\geq 0$ be arbitrary integers with $b\leq r$, then the dimension of
the $\mathfrak{u}(n)$-irreducible representation $\mathcal{M}_{a,b}^{(r)}$ is
given by

$\displaystyle\dim\mathcal{M}_{a,b}^{(r)}=\frac{\Gamma(a+n-2)\Gamma(b+n-2)\Gamma(r+n-1)}{\Gamma(a+1)\Gamma(b+1)\Gamma(r+2)}\frac{(a+b+n-2)(a+r+n-1)(r-b+1)}{(n-1)!(n-2)!(n-3)!}\
.$

Although this is not absolutely necessary, one can now use the formula above
to verify Theorem 6.1 by a dimensional argument. Indeed, using symbolic
computation software (the authors used Mathematica), one can verify that the
following numerical equality holds:

######  Lemma 6.6.

For arbitrary positive integers $a,b,r\geq 0$ for which $b\leq r$, we have
that

$\displaystyle\sum_{i=0}^{b}\sum_{j=0}^{a}\dim\mathcal{M}_{a,b}^{(r)}(\mathbb{R}^{2n})$
$\displaystyle={r+n-1\choose
n-1}\frac{(a+n+n-1)\Gamma(a+n-1)\Gamma(b+n-1)}{(n-1)!(n-2)!\Gamma(a+1)\Gamma(b+1)}\
,$

whereby the indices $i$ and $j$ satisfy the additional constraint $i+j\leq r$.

Our goal for the rest of this section is to show that these spaces are
precisely the spaces of (symplectic) $h$-monogenics of a degree $(a,b;r)$, so
that we can also define these spaces in an equivalent way as follows:

$\displaystyle\mathcal{M}_{a,b}^{(r)}(\mathbb{R}^{2n})=\mathcal{P}_{a,b}(\mathbb{R}^{2n},\mathbb{S}^{\infty}_{(r)})\cap\big{(}\ker(D_{s})\cap\ker(D_{t})\big{)}\
.$ (6.2)

######  Remark 6.7.

Note that the condition $b\leq r$ on the parameters may seem a bit strange,
since this implies that spaces of $h$-monogenics will not always exist.
However, a similar phenomenon also occurs in the classical case. One can for
instance see that there are no $h$-monogenic polynomials of degree $(a,0)$
taking values in the homogeneous spinor space $\mathbb{S}_{(0)}$. So even in
Hermitean Clifford analysis, the existence of the spaces
$\mathcal{M}_{a,b}^{(j)}$ is governed by restrictions on the parameters
$(a,b;j)$.

The inclusion from left to right in formula (6.2) is rather immediate, the
opposite direction requires more work. To see why

$\displaystyle\mathcal{M}_{a,b}^{(r)}\subseteq\mathcal{P}_{a,b}(\mathbb{R}^{2n},\mathbb{S}_{(r)}^{\infty})\cap\big{(}\ker(D_{s})\cap\ker(D_{t})\big{)}\
,$ (6.3)

it suffices to note that we have an explicit model for the highest weight
vector which generates the representation under the action of the negative
root vectors in $\mathfrak{u}(n)$. Indeed, recalling the definition for the
(raising) operators $R_{j}=q_{j}-\partial_{q_{j}}$ and the Hermite polynomials
$\mathsf{H}_{k}(t)$, it is not hard to see that the function

$\displaystyle w_{a,b;r}(\underline{x},\underline{y};\underline{q})$
$\displaystyle:=z_{1}^{a}\big{(}\overline{z}_{n-1}R_{n}-\overline{z}_{n}R_{n-1}\big{)}^{b}\mathsf{H}_{r-b}(q_{n})e^{-\frac{1}{2}|\underline{q}|^{2}}\
,$

satisfies the necessary requirements. It suffices to consider the action of
the positive root operators

$\displaystyle
C_{jk}+iA_{jk}=2\big{(}z_{j}\partial_{z_{k}}-\overline{z}_{k}\partial_{\overline{z}_{j}})+(q_{j}+\partial_{q_{j}})(\partial_{q_{k}}-q_{k})\
,$

where $1\leq j<k\leq n$, and the Cartan elements

$\displaystyle
H_{j}=H_{j}=-i(z_{j}\partial_{z_{j}}-\overline{z}_{j}\partial_{{\overline{z}}_{j}})+\frac{i}{2}\left(q_{j}^{2}-\partial_{q_{j}}^{2}\right)$

for $1\leq j\leq n$. The former all act trivially on
$w_{a,b;r}(\underline{x},\underline{y};\underline{q})$, whereas the latter
indeed give the desired eigenvalues. Next, we prove that this function is also
a solution the symplectic Dirac operators. One has that

$\displaystyle D_{z}w_{a,b;r}(\underline{x},\underline{y};\underline{q})$
$\displaystyle=az_{1}^{a-1}L_{1}(\overline{z}_{n-1}R_{n}-\overline{z}_{n}R_{n-1})^{b}\mathsf{H}_{r-b}(q_{n})e^{-\frac{1}{2}|\underline{q}|}=0\
,$

since the lowering operator $L_{1}$ acts on the Gaussian only (here we have
used the fact that $n\geq 3$, so that we can commute $L_{1}$ with the
operators $R_{n}$ and $R_{n-1}$). A similar calculation for the operator
$D_{z}^{\dagger}$ then proves that
$w_{a,b;r}(\underline{z},\overline{\underline{z}})$ is indeed $h$-monogenic.
To prove the opposite inclusion in equation (6.2), we will argue as follows:
given the fact that symplectic $h$-monogenics are always harmonic, we
certainly know that

$\displaystyle\mathcal{P}_{a,b}(\mathbb{R}^{2n},\mathbb{S}^{\infty}_{(r)})\cap\big{(}\ker(D_{s})\cap\ker(D_{t})\big{)}\subseteq\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}_{(r)}\
,$

whereby we assume that $b\leq r$. Now, we know how the right-hand side
decomposes (see Theorem 6.1), so any polynomial
$P(\underline{x},\underline{y})$ of degree $(a,b)$ on $\mathbb{R}^{2n}$ taking
values in $\mathbb{S}^{(r)}$ which satisfies $D_{s}P=D_{t}P=0$ can be written
as a linear combination of elements belonging to the spaces predicted by this
theorem. However, in what follows we will prove that there is a unique summand
in the decomposition containing $h$-monogenics, and that all the other
summands are orthogonal (in a suitable sense) to the space of $h$-monogenics.
This will thus force $P(\underline{x},\underline{y})$ to be an element of that
unique component, which will turn out to be the one generated by
$w_{a,b;r}(\underline{x},\underline{y};\underline{q})$.

The main idea behind the proof announced in the previous paragraph involves
the construction of the so-called ‘embedding factors’. These are (in full
generality) the $\mathfrak{u}(n)$-invariant linear maps injecting $\gamma$
into $\alpha\otimes\beta\cong\bigoplus\mu_{\gamma}\gamma$. For example, the
vector variable $\underline{z}$ appearing in Theorem 2.8 can be seen as an
example of such an embedding map, sending $h$-monogenics of certain degree
into the tensor product $\mathcal{H}_{a,b}\otimes\mathbb{S}_{(r)}$. In the
setting of Howe dual pairs, these embedding factors are typically given by the
operators dual (in the sense of the Fischer inner product) to the differential
operators defining the (monogenic) solution spaces. This is also the case in
the present situation, since one can show the following:

######  Lemma 6.8.

Let $0\leq\ell\leq b$ and $0\leq k\leq b$, then the following properties hold:

$\displaystyle
X_{z}^{\ell}\mathcal{M}_{a,b-\ell}^{(r+\ell)}\in\mathcal{H}_{a,b}\otimes\mathbb{S}^{\infty}_{(r)}\quad\mbox{and}\quad(X_{z}^{\dagger})^{k}\mathcal{M}_{a-k,b}^{(r-k)}\in\mathcal{H}_{a,b}\otimes\mathbb{S}^{\infty}_{(r)}\
.$

Note that these statements only hold if the parameters characterising the
$\mathcal{M}$-spaces make sense, so $b-\ell\leq r+\ell$ and $b\leq r-k$
respectively.

###### Proof.

It is clear that the operator $X_{z}^{\ell}$ raises the bi-degree from
$(a,b-\ell)$ to $(a,b)$, and that the spinor-degree is lowered from $r+\ell$
to $r$. It thus suffices to show that the result is indeed harmonic. To that
end, we note that $[\Delta_{z},X_{z}]=-D_{z}$ (where we have put
$\Delta_{z}=\sum\partial_{z_{j}}\bar{\partial}_{z_{j}}$) and
$[D_{z},X_{z}]=0$. A similar calculation for the operator $X_{z}^{\dagger}$
then proves the lemma. ∎

It is now tempting to think that one has a handle on all embedding factors,
through combinations of powers of $X_{z}$ and $X_{z}^{\dagger}$.
Unfortunately, the situation is slightly more complicated, because a simple
calculation shows that for instance the product operator
$X_{z}X_{z}^{\dagger}$ cannot be used to embed $\mathcal{M}_{a,b}^{(r)}$ into
the space of $\mathbb{S}^{\infty}_{(r)}$-valued harmonics. It turns out that
one needs a slightly more complicated algebraic structure to characterise the
embedding factors, expressed in terms of a so-called transvector algebra. Note
that this (quadratic) algebra already appeared in the setting of Howe
dualities in harmonic analysis for the Laplace operator, see for instance [5].
The construction of this type of algebra hinges upon the existence of a
reductive subalgebra inside a Lie algebra, hence the following result:

######  Lemma 6.9.

The Lie subalgebra $\mathfrak{sl}(2)$ is reductive in $\mathfrak{su}(1,2)$.

###### Proof.

We need to prove that $\mathfrak{su}(1,2)$ decomposes as
$\mathfrak{sl}(2)\oplus\mathfrak{t}$, such that the subspace $\mathfrak{t}$
caries the adjoint action of $\mathfrak{sl}(2)$. If we now define
$\mathfrak{t}:=\textup{span}_{\mathbb{R}}\big{(}\mathcal{O},X_{s},X_{t},D_{s},D_{t}\big{)}$
and $\mathfrak{sl}(2)$ as the subalgebra generated by the remaining operators
$\Delta,r^{2}$ and $\mathbb{E}+n$, the result follows from instance from the
isomorphism constructed in Theorem 4.3. ∎

For the most general definition of a transvector algebra, or the (intimitely
related) Mickelsson algebra, we refer to the literature: see for instance [12,
14]. In the present paper we will proceed by constructing this algebra through
its generators, which is typically done in terms of the extremal projection
operator. Denoting our transvector algebra by means of
$\mathcal{Z}(\mathfrak{su}(1,2),\mathfrak{sl}(2))$, we note that its
generators are constructed in terms of the extremal projector

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

whereby the (suitably normalised) operators $X,Y$ and $H$ are respectively
given by $X=\frac{1}{2}\Delta$, $Y=-\frac{1}{2}r^{2}$ and $H=-(\mathbb{E}+n)$.
Note that there is no need to care about convergence issues for the formal
series above, since we will never need more than a finite number of terms.
Also the factors in the denominator are harmless, because the previous
observation will always allow us to get rid of the denominators after
multiplication with a suitable factor (technically speaking, this amounts to
going from a transvector to a Mickelsson algebra). The upshot is that although
the operators in the subalgebra $\mathfrak{t}\subseteq\mathfrak{su}(1,2)$ do
not map harmonics to harmonics, one does have that
$\pi_{\mathfrak{sl}(2)}(\mathfrak{t})\subseteq\mbox{End}(\ker(\Delta))$. So
the operators in $\pi_{\mathfrak{sl}(2)}(\mathfrak{t})$ and products thereof
will be our embedding factors. Let us then prove the following technical
result:

######  Lemma 6.10.

  1. (i)

The transvector projections of the operators $X_{s}$ and $X_{t}$, acting on
harmonic functions, can be rescaled using the factor $H+2$, such that

$\displaystyle-(H+2)\pi_{\mathfrak{sl}(2)}X_{s}$
$\displaystyle=\left(YX-(H+2)\right)X_{s}=(\mathbb{E}+n-2)X_{s}-\frac{1}{2}r^{2}D_{t}$
$\displaystyle-(H+2)\pi_{\mathfrak{sl}(2)}X_{t}$
$\displaystyle=\left(YX-(H+2)\right)X_{t}=(\mathbb{E}+n-2)X_{t}+\frac{1}{2}r^{2}D_{s}\
.$

In other words, only the first two terms of the extremal projection operator
$\pi_{\mathfrak{sl}(2)}$ are required to do the projection.

  2. (ii)

These rescaled operators commute in $\operatorname{End}(\ker(\Delta))$.

###### Proof.

The first statement (i) follows from the fact that $[\Delta,[\Delta,X_{s}]]=0$
(and similarly for $X_{t}$). This thus means that the infinite sum in the
extremal projector reduces to a finite sum of two terms only. The explicit
expression then follows from straightforward calculations, hereby taking into
account that everything is to be seen as an operator acting on $\ker(\Delta)$.
For the second statement (ii) we will make use of the commutator equality
$[AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B$ which allows us to say that

$\displaystyle[\pi_{\mathfrak{sl}(2)}X_{s},\pi_{\mathfrak{sl}(2)}X_{t}]$
$\displaystyle=\pi_{\mathfrak{sl}(2)}[X_{s},\pi_{\mathfrak{sl}(2)}]X_{t}+\pi_{\mathfrak{sl}(2)}^{2}[X_{s},X_{t}]+\pi_{\mathfrak{sl}(2)}[\pi_{\mathfrak{sl}(2)},X_{t}]X_{s}\
.$

Note that we can commute any factor involving the Euler operator $\mathbb{E}$
through the product $r^{2}\Delta$, because it is homogeneous of degree zero,
and that $[r^{2},X_{s}]=0$ such that

$\displaystyle[\pi_{\mathfrak{sl}(2)},X_{s}]$
$\displaystyle=-\frac{1}{4}\left[(\mathbb{E}+n-2)^{-1}r^{2}\Delta,X_{s}\right]=-\frac{1}{4}r^{2}\left[\Delta(\mathbb{E}+n-2)^{-1}r^{2},X_{s}\right]\
.$

This is enough to see why
$\pi_{\mathfrak{sl}(2)}[X_{s},\pi_{\mathfrak{sl}(2)}]=0$ (and similarly for
the last term in the righ-hand side above). Indeed, we can hereby make use of
the fact $\pi_{\mathfrak{sl}(2)}r^{2}=0$, which is one of the defining
properties for the extremal projection operator. This also gets rid of the
middle term, as $[X_{s},X_{t}]\sim r^{2}$. Finally, it suffices to multiply
the (trivial) commutator
$[\pi_{\mathfrak{sl}(2)}X_{s},\pi_{\mathfrak{sl}(2)}X_{t}]$ from the left with
$(H+2)(H+1)$ to see that also the rescaled operators will commute. ∎

One can obviously also consider the projections of $X_{z}$ and
$X_{z}^{\dagger}$. Briefly denoting the rescaled projection operator by means
of $\tilde{\pi}_{\mathfrak{sl}(2)}=-(H+2)\pi_{\mathfrak{sl}(2)}$, we get the
following:

######  Definition 6.11.

The operators $\mathbb{X}_{z}$ and $\mathbb{X}_{z}^{\dagger}$ are defined as
follows:

$\displaystyle\mathbb{X}_{z}$
$\displaystyle:=\tilde{\pi}_{\mathfrak{sl}(2)}X_{z}=(\mathbb{E}+n-2)X_{z}+\frac{i}{2}|\underline{z}|^{2}D_{z}$
$\displaystyle\mathbb{X}_{z}^{\dagger}$
$\displaystyle:=\tilde{\pi}_{\mathfrak{sl}(2)}X_{z}^{\dagger}=(\mathbb{E}+n-2)X_{z}^{\dagger}-\frac{i}{2}|\underline{z}|^{2}D_{z}^{\dagger}\
.$

In view of Lemma 6.10 it is clear that also
$[\mathbb{X}_{z},\mathbb{X}_{z}^{\dagger}]=0$. In particular, one can thus
consider the polynomial algebra
$\mathbb{C}[\mathbb{X}_{z},\mathbb{X}_{z}^{\dagger}]$ in these operators,
defining a subalgebra of the endomorphism algebra End$(\ker(\Delta))$. Note
that in switching from $(X_{s},X_{t})$ to $(X_{z},X_{z}^{\dagger})$ we
introduced a complex unit, which means that
$\mathbb{C}[\mathbb{X}_{z},\mathbb{X}_{z}^{\dagger}]$ should be seen as a
subalgebra of the complexified transvector algebra
$Z(\mathfrak{su}(1,2),\mathfrak{sl}(2))\otimes\mathbb{C}$ here. Monomials in
the algebra $\mathbb{C}[\mathbb{X}_{z},\mathbb{X}_{z}^{\dagger}]$ are
precisely the embedding operators we are after, hence the following
definition:

######  Definition 6.12.

For all positive integers $k,\ell\in\mathbb{N}$ we define the maps

$\displaystyle\mathcal{I}_{k,\ell}:=\mathbb{X}^{k}(\mathbb{X}^{\dagger})^{\ell}\in\mbox{End}\big{(}\ker(\Delta)\big{)}\
,$

whereby $\mathcal{I}_{0,0}=\textup{Id}$ is the identity map. Each of these
maps is invariant with respect to $\mathfrak{u}(n)$, in the sense that
$\mathcal{I}_{k,\ell}$ commutes with all the generators.

It is clear that the map $\mathcal{I}_{k,\ell}$ raises the bi-degree of
homogeneity in $(\underline{z},\underline{z}^{\dagger})$ from $(a,b)$ to
$(a+\ell,b+k)$, and it maps spinors in $\mathbb{S}^{\infty}_{(r)}$ to elements
in $\mathbb{S}^{\infty}_{(r-k+\ell)}$. As such, these maps can indeed be used
as embedding maps:

$\displaystyle\mathcal{I}_{k,\ell}:\mathcal{M}_{a-\ell,b-k}^{(r+k-\ell)}(\mathbb{R}^{2n})\longrightarrow\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}_{(r)}\
,$

where we momentarily ignore the conditions on the parameters (depending on the
values of the parameters, the space at the left-hand side of the arrow may not
exist). To finish the goal we have set right before Remark 6.7, the following
will come in handy:

######  Lemma 6.13.

For all $(k,\ell)\neq(0,0)$, we have the following:

$\displaystyle\mathcal{I}_{k,\ell}:\ker(D_{s})\cap\ker(D_{t})\longrightarrow\big{(}\ker(D_{s})\cap\ker(D_{t})\big{)}^{\perp}\
,$

whereby the orthogonality is expressed with respect to the Fischer inner
product, defined on
$\mathcal{P}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}$ by means
of

$\displaystyle\langle
P(\underline{x},\underline{y})\otimes\psi(\underline{q}),Q(\underline{x},\underline{y})\otimes\varphi(\underline{q})\rangle:=\big{(}P(\underline{\partial}_{y},-\underline{\partial}_{x})Q(\underline{x},\underline{y})\big{)}\bigg{|}_{x=y=0}\int_{\mathbb{R}^{n}}\overline{\psi}(\underline{q})\varphi(\underline{q})d\underline{q}\
.$

###### Proof.

Because the operator $\mathcal{I}_{k,\ell}$ is a monomial in the operators
$\mathbb{X}_{z}$ and $\mathbb{X}_{z}^{\dagger}$ (note that we exclude the
identity operator $\mathcal{I}_{0,0}$ here), it is sufficient to investigate
$\mathcal{I}_{1,0}$ and $\mathcal{I}_{0,1}$ separately. These operators are
projections (up to a multiplicative constant, coming from the Euler factor
$H+n$) of suitable linear combinations of $X_{s}$ and $X_{t}$, so we can even
restrict our attention to these operators instead. The result then follows
from the fact that $X_{s}$ (resp. $X_{t}$) is the Fischer adjoint of $D_{s}$
(resp. $D_{t}$). ∎

This allows us to prove the following:

######  Lemma 6.14.

For all positive integers $b\leq r$, the space
$\mathcal{M}_{a,b}^{(r)}(\mathbb{R}^{2n})$ can be defined as the space of
$\mathbb{S}^{\infty}_{(r)}$-valued $h$-monogenic polynomials on
$\mathbb{R}^{2n}$ of bi-degree $(a,b)$.

###### Proof.

Since any $\mathbb{S}^{\infty}_{(r)}$-valued polynomial
$P_{a,b}(\underline{x},\underline{y};\underline{q})$ of bi-degree $(a,b)$ in
the kernel of the operators $D_{s}$ and $D_{t}$ is also harmonic, we know that
it must belong to one of the spaces appearing at the right-hand side of
Theorem 6.1, embedded using one of the mappings $\mathcal{I}_{k,\ell}$
constructed above. However, we have just shown that the only map that
preserves the property of being symplectic $h$-monogenic is
$\mathcal{I}_{0,0}$, which means that
$P_{a,b}(\underline{x},\underline{y};\underline{q})\in\mathcal{M}_{a,b}^{(r)}(\mathbb{R}^{2n})$.
∎

Putting everything together then leads to the following refinement of Theorem
6.1:

######  Theorem 6.15 (Symplectic Hermitian Fischer decomposition).

We have the following explicit decomposition for the space of spinor-valued
harmonics:

$\displaystyle\mathcal{H}_{a,b}(\mathbb{C}^{n},\mathbb{C})\otimes\mathbb{S}_{(r)}^{\infty}$
$\displaystyle=\bigoplus_{j=0}^{a}\bigoplus_{i=0}^{b}\mathcal{I}_{b-i,j}\mathcal{M}_{a-j,i}^{(b+r-(i+j))}(\mathbb{R}^{2n})\
,$

where the indices are constrained by the condition $i+j\leq r$. This is a
decomposition of $\mathbb{S}^{\infty}_{(r)}$-valued harmonics into symplectic
$h$-monogenics.

###### Proof.

The decomposition itself follows from 6.1, together with the observation that
the maps $\mathcal{I}_{k,\ell}$ belong to End$(\ker(\Delta))$ and therefore
have the desired mapping properties. The last statement (i.e. the connection
with symplectic $h$-monogenics) follows from the previous lemma. ∎

Despite the obvious similarities between the theorem above and the classical
Fischer decompositions (for the harmonic and monogenic case), there are also a
few crucial differences. First of all, note that when the space
$\mathcal{M}_{a,b}^{(r)}$ is included in the tensor product
$\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}_{(r)}$,
then this space is not given by the Cartan product

$\displaystyle(a,0,\ldots,0,-b)\boxtimes\mathbb{S}^{\infty}_{(r)}\cong\left(a-\frac{1}{2},-\frac{1}{2},\ldots,-\frac{1}{2},-b-r-\frac{1}{2}\right)\
.$

This is clearly different from the aforementioned Fischer decompositions, in
which the monogenics were always embedded as a Cartan product. Moreover, the
vector space of ($r$-homogeneous) spinor-valued harmonic polynomials does not
always contain symplectic $h$-monogenics. Put differently, for certain choices
of indices $(a,b;r)$ one has that

$\displaystyle\mathcal{H}_{a,b}(\mathbb{R}^{2n},\mathbb{C})\otimes\mathbb{S}^{\infty}_{(r)}\subseteq\big{(}\ker(D_{s})\cap\ker(D_{t})\big{)}^{\perp}\
.$

As a simple example, consider
$\mathcal{H}_{0,1}(\mathbb{R}^{2n})\otimes\mathbb{S}^{\infty}_{(0)}=\mathbb{X}_{z}\mathcal{M}_{0,0}^{(1)}$.
This is reminiscent of the situation in (standard) Hermitean Clifford
analysis, where one for instance has that
$\mathcal{H}_{1,0}(\mathbb{R}^{2n})\otimes\mathbb{S}_{(0)}=\underline{z}\mathcal{M}_{0,0}^{(1)}$.

##  7 Conclusion and further research

In this paper we introduced a complex structure $\mathbb{J}$ on
$\mathbb{R}^{2n}$ which gave rise to a symplectic version of Hermitian
Clifford analysis, centered around the system of equations $D_{s}f=D_{t}f=0$.
In this new setting, the Lie algebra $\mathfrak{su}(1,2)$ then appeared
naturally as a dual partner for the unitary group $\mathsf{U}(n)$. This leads
to the first two rows in the overview below. In the classical case (middle
column), the Hermitean system was further refined, leading to a system of 4
‘quaternionic’ invariant equations (governed by the group
$\mathsf{Sp}(n)=\mathsf{Sp}(2n,\mathbb{C})\cap\mathsf{U}(2n)$, see [1]). The
symplectic analogue of this function theory, the empty slot in the overview,
will be the topic of further investigation.

Complex structures | Orthogonal case | Symplectic case
---|---|---
none | $\mathsf{SO}(n)\times\mathfrak{osp}(1|2)$ |  $\mathsf{Sp}(2n)\times\mathfrak{sl}(2)$
Kähler $\mathbb{J}$ | $\mathsf{U}(n)\times\mathfrak{osp}(2|2)$ | $\mathsf{U}(n)\times\mathfrak{su}(1,2)$
Hyperkähler $\mathbb{I,J,K}$ | $\mathsf{Sp}(n)\times\mathfrak{osp}(4|2)$ | ?

### Acknowledgments

Guner Muarem was supported by the FWO-EoS project G0H4518N.

## References

  * [1] F. Brackx, H. De Schepper, D. Eelbode, R. Lávička, and V. Souček.  Fischer decomposition for $\mathfrak{osp}(4|2)$-monogenics in quaternionic Clifford analysis.  Math. Methods Appl. Sci. 39(16):4874–4891, 2016.
  * [2] F. Brackx, H. De Schepper, D. Eelbode, and V. Souček.  The Howe dual pair in Hermitean Clifford analysis.  Rev. Mat. Iberoam., 26(2):449–479, 2010.
  * [3] D. J. Britten, J. Hooper, and F. W. Lemire.  Simple $C_{n}$ modules with multiplicities $1$ and applications.  Canad. J. Phys., 72(7-8):326–335, 1994.
  * [4] D. J. Britten and F. W. Lemire.  On modules of bounded multiplicities for the symplectic algebras.  Trans. Amer. Math. Soc., 351(8):3413–3431, 1999.
  * [5] H. De Bie, D. Eelbode and M. Roels.  The harmonic transvector algebra in two vector variables.  J. of Algebra, 473(1): 247–282, 2017.
  * [6] H. De Bie, M. Holikova and P. Somberg.  Basic Aspects of Symplectic Clifford Analysis for the Symplectic Dirac Operator.  Adv. Appl. Cliff. Alg., 27: 1103–1132, 2017.
  * [7] W. Fulton and J. Harris.  Representation theory: a first course, volume 129.  Springer Science & Business Media, 2013.
  * [8] K. Habermann and L. Habermann.  Introduction to symplectic Dirac operators, volume 1887 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2006.
  * [9] R. C. King and B. G. Wybourne.  Analogies between finite-dimensional irreducible representations of $\mathsf{SO}(2n)$ and infinite-dimensional irreducible representations of $\mathfrak{sp}(2n)$.  J. Math. Phys., 41(7):5002–5019, 2000.
  * [10] E. O. Korman.  Symplectic Dolbeault operators on Kähler manifolds.  Ann. Global Anal. Geom., 44(3): 339–358, 2013.
  * [11] B. Kostant.  Symplectic spinors.  In Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica & Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Rome, 1973), 139–152. Academic Press, London, 1974.
  * [12] J. Mickelsson.  Step algebras for semi-simple subalgebras of Lie algebras.  Rep. Math. Phys. 4: 307–318, 1973.
  * [13] D. Zhelobenko.  Compact Lie groups and their representations. Vol. 40. American Mathematical Soc., 1973.
  * [14] D.P. Zhelobenko.  Transvector algebras in representation theory and dynamic symmetry Group-Theoretical Methods in Physics, vol. II, 71–93, Utrecht, Yurmala VNU Sci. Press, 1985.


# Representation Learning in Sequence to Sequence Tasks: Multi-filter Gaussian
Mixture Autoencoder

Yunhao Yang, Zhaokun Xue
Department of Computer Science, University of Texas at Austin
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Heterogeneity of sentences exists in sequence to sequence tasks such as
machine translation. Sentences with largely varied meanings or grammatical
structures may increase the difficulty of convergence while training the
network. In this paper, we introduce a model to resolve the heterogeneity in
the sequence to sequence task. The Multi-filter Gaussian Mixture Autoencoder
(MGMAE)111Code: https://github.com/yunhaoyang234/MGMAE utilizes an autoencoder
to learn the representations of the inputs. The representations are the
outputs from the encoder, lying in the latent space whose dimension is the
hidden dimension of the encoder. The representations of training data in the
latent space are used to train Gaussian mixtures. The latent space
representations are divided into several mixtures of Gaussian distributions. A
filter (decoder) is tuned to fit the data in one of the Gaussian distributions
specifically. Each Gaussian is corresponding to one filter so that the filter
is responsible for the heterogeneity within this Gaussian. Thus the
heterogeneity of the training data can be resolved. Comparative experiments
are conducted on the Geo-query dataset and English-French translation. Our
experiments show that compares to the traditional encoder-decoder model, this
network achieves better performance on sequence to sequence tasks such as
machine translation and question answering.

##  1 Introduction

Semantic parsing comprises translating sentences into various kinds of formal
representations, with the goal of executing these to compute a result in the
context of an environment such as a knowledge base. It generally converts
natural languages into machine-understandable representations. The
representations should preserve the meanings of the words/sentences. Semantic
parsing can be applied to many aspects such as machine translation, question
answering, automated reasoning, and code generation. Generally, an encoder-
decoder model is used for semantic parsing.

The encoder-decoder model is one of the techniques for semantic parsing. The
encoder-decoder model is a technique of applying recurrent neural networks to
sequence-to-sequence prediction problems. It was originally developed for
machine translation problems. Then, the model achieves good performances at
other related sequence-to-sequence prediction problems.

However, the encoder-decoder model lacks the potential on generating
representations. Then, an autoencoder is introduced for representation
learning. An autoencoder is a type of deep neural network that is trained in a
supervised manner. It is used to reconstruct the input in a pre-defined
wayKramer (1991). Internally, it has a hidden layer that used to represent the
inputGoodfellow et al. (2016). In between the encoder and decoder, there is a
latent space. In the latent space, words or sentences are typically
represented by lower-dimension vectors, where the words or sentences with
similar meanings are close to each other. Thus, better results may be obtained
by analyzing the data on the latent space, such as the distance between
vectors (sentence similarities).

Because of these properties, autoencoder is commonly used in representation
learning. Representation learning, also known as feature learning, is a set of
techniques that allows a system to automatically determine the representations
needed for feature detection or classification from raw dataBengio et al.
(2014).

In sequence to sequence tasks, an autoencoder is a model where its inputs and
outputs are the same. This technique is known as self-supervised learning,
where the input data is the target data. One goal for self-supervised learning
is to learn the representations of the input data.

The representations are the outputs of the encoder, lying in the latent space.
Latent space refers to an abstract multi-dimensional space containing feature
values that cannot be interpreted directly. Instead, the values in the latent
space are meaningful internal representations that are encoded from some
externally observed data. The latent space intends to provide data that is
machine recognizable/understandable to a computer through a quantitative
spatial representation or modeling.

Latent space can be considered as a set of hidden features or internal
representations over the input data, thus a small variation in the latent
space may be caused by a large difference in the input data. The process of
learning a latent space is to tune the parameters within the encoder and
decoder to obtain a better representation of the data. A well-learned latent
space would help the machine learning model better understanding the features
and easily identifying the principal components of the data.

##  2 Related Work

Encoder-decoder network is one of the classical model for solving sequence to
sequence tasks such as machine translation. In machine translation, an encoder
on the language can create inner representations and a decoder then unveils
these coding into desired outputCvetko (2020). The purpose of machine
translation relevant tasks is to obtain a semantic isomorphism (eg. similar
syntax structure). If the semantic qualities of the input can be condensed to
be an intermediate representation that is also good for the syntactic
unfolding in the next stage, then this intermediate representation is the
modular link between the input and the target outputOshri and Khandwala
(2015).

Encoder-decoder network in sequence to sequence problems generally utilizes
deep recurrent neural networks and train the networks in a supervised
mannerSherstinsky (2020). This approach is a general end-to-end approach to
sequence learning that makes minimal assumptions on the sequence structure.
The method applies a multilayered Long Short-Term Memory (LSTM)Hochreiter and
Schmidhuber (1997) encoder to map the input sequence to a vector of fixed
dimensionality (hidden layer dimension), and then an LSTM decoder is used to
construct the target sequence from the vectorSutskever et al. (2014).

Another approach is Convolutional AutoencoderKalchbrenner et al. (2014). The
network uses Dynamic k-Max Pooling, a global pooling operation over linear
sequences. The encoder consists of a set of one-dimensional convolution layers
that reduce the dimension of the input data. The decoder consists of a set of
transposed one-dimensional convolution layers that are symmetric to the layers
in the encoder. The decoder is designed for reconstructing the input data
based on the lower-dimensional vectors extracted from the latent space.

In addition to the ordinary supervised learning approach, some research
develops an autoencoder structure to exploit sequence representations in an
unsupervised fashion. Research has explored this approach for machine
translation, under the utilization of pre-trained cross-lingual word
embedding, they seek to exploit a language-independent multilingual sentence
representation to easily generalize to a new languageMullov et al. (2021).

Autoencoder is often used to learn the representations of the raw input, which
the representations preserve or exploit useful features. Several researches
have utilized autoencoder for representation learningLi et al. (2021),
specifically for series dataKortmann et al. (2021). Research has shown that
latent space representations generated from autoencoder can better indicate
the key attributes of the input dataYang and Whinston (2021). Thus the learned
representations can be used as a starting stage for the sequence to sequence
tasks.

Since the latent space representations can better demonstrate the features of
the raw input, the representations can also be divided into different groups
where each group consists of homogeneous data. There are some researches
exploring hierarchical structures on the latent space to obtain richer
representationsGoyal et al. (2017); Bouchacourt et al. (2018). There are
several other researches performing clustering or learning a mixture model
within the latent space encoding, including Hard K-means Clustering Yang et
al. (2017), Soft K-means Clustering Jabi et al. (2019), Gaussian Mixture Model
VAEDilokthanakul et al. (2016). These researches have shown that dividing
latent space can improve the quality of the representations in some aspects.
Another research utilizes this conclusion to develop a multi-filter Gaussian-
mixture variational autoencoder. Where the latent space is trained as a
mixture of Gaussians, then each Gaussian is corresponding to a filter that is
used for decoding. The experiments have shown that the multi-filter structure
does enhance the performance in Camera ISPYang et al. (2021).

##  3 Methodology

The architecture is inspired by the Patch Subspace Variational Autoencoder
(PS-VAE) for Camera ISPYang et al. (2021). There is an autoencoder to learn
the representations of the inputs, and several filters to decode the
representations in a defined way. A Gaussian mixture model is used to
determine which filter the input goes. This model is denoted by Multi-filter
Gaussian Mixture Autoencoder (MGMAE). A visualization of MGMAE is in Figure 1.

Figure 1: Network Architecture: the top is the autoencoder whose inputs and
outputs are the same, where the representations of the inputs are generated.
The representations are divided into mixtures of Gaussians and determined
which filter to go. Each Gaussian corresponds to a filter (decoder). The
filter is used to decode the representations in a defined way and it is
trained using only the data that belongs to the corresponded Gaussian.

###  3.1 Representation Learning via Autoencoder

One direction of generating representations that preserve the semantic meaning
is being able to reconstruct the generated vector back into its original text.
This can be done by utilizing an autoencoder.

The autoencoder consists of a similar structure with the encoder-decoder
model. The encoder takes in a sequence of inputs and returns a sequence of
outputs and a final hidden state. It consists of an embedding layer that
transforms the input text into an embedded vector, a bidirectional long short-
term memory classifier that generates the outputs and hidden states. The
decoder executes one step of computation at a time until reaching the ”End of
Sentence” token or hitting the maximum sentence length. The input for each
step is the output from the previous decoding step. At the first step, the
start-of-sentence token is fed into the decoder as the beginning of the
decoding sequence. The first hidden state of the decoder is the final hidden
state from the encoder. The decoder consists of a long short-term memory
network, a linear layer that converts the output from hidden dimension to
output dimension, and a LogSoftmax layer. A dot product attention
mechanismLuong et al. (2015) is appended to the decoder, after the LSTM
network. The attention weights are computed using the hidden state
$\overline{h}$ generated by the LSTM network and the sequence of the encoder’s
outputs $h$.

To train the autoencoder, a Negative Likelihood Loss is utilized to update
gradients. The outputs of the decoder are gone through a LogSoftmax layer and
then used to compute the Negative Likelihood Loss:

$\mathcal{L}(x,y)=\\{l_{1},...,l_{N}\\}^{T},\quad l_{n}=-w_{y_{n}}x_{n,y_{n}}$
(1)

where $x$ is the output tensor and $y$ is the label tensor.

After the autoencoder is fine tuned, the encoder is utilized to generate
representations. The final hidden state outputted from the encoder is
considered the representation of the input sequence. The space where the
representations laying in is denoted as the latent space.

###  3.2 Gaussian Mixture Model on Latent Space

A Gaussian Mixture Model (GMM) is a parametric probability density function
represented as a weighted sum of Gaussian component densitiesReynolds (2015).
GMM parameters are estimated from the training data using the iterative
expectation-maximization (EM) algorithm. The Gaussian mixture model is a
weighted sum of $M$ component Gaussian densities:

$p(x|\lambda)=\sum_{i=1}^{M}w_{i}g(x|\mu_{i},\Sigma_{i})$ (2)

in equation 2, $x$ is the vector from the latent space, which is the output
from the encoder. The latent space dimension is denoted $L$. $w_{i}$ for
$i=1,...,M$ are the mixture weights, $M$ is the number of Gaussian mixtures,
as well as the number of filters in the multi-filter model. The component
Gaussian densities are computed by:

$g(x|\mu_{i},\Sigma_{i})=\frac{1}{(2\pi)^{L/2}\sqrt{|\Sigma_{i}|}}\exp(\frac{d^{\prime}\Sigma^{-1}d}{2})$
(3)

where $d=x-\mu_{i}$, $\mu_{i}$ and $\Sigma_{i}$ are the mean vector and
covariance matrix respectively. The mean and covariance are represented by the
notation $\lambda=\\{\mu_{i},\Sigma_{i},w_{i}\\}$.

###  3.3 Multi-filter Decoding

MGMAE has a certain number of filters for decoding purposes. Each filter
shares an identical structure with the decoder in the autoencoder. The filters
are designed for constructing the target outputs from the representations in
the latent space. For instance, the filters generate answers in the question
answering system or construct target language in machine translation.

The Gaussian mixture model has divided the data into $k$ clusters, where each
cluster is a Gaussian distribution. Based on the clustering result, we set up
$k$ filters that each cluster is associated with one filter. All the filters
are identical in network structure but their parameters are independent. Each
filter is trained only using the data assigned to its corresponding cluster.
The loss used to train the filters is the Negative Likelihood Loss. The losses
for each filter are computed separately and their gradients are updated
separately as well. While during the testing stage, only the samples
classified into the corresponding cluster will go through this filter.

In this way, we can form Gaussian mixture groups and construct a set of
feature-specific filters that resolves the heterogeneity in each group.

##  4 Experiments and Results

In order to demonstrate the effectiveness of the multi-filter Gaussian Mixture
architecture, some comparative experiments are conducted. The experiments are
run on two types of sequence to sequence tasks: question answering and machine
translation. The goal of the experiments is to show that the multi-filter
architecture does achieve better performance compare to the ordinary encoder-
decoder model. Thus we can prove the effectiveness of our multi-filter
Gaussian mixture model.

###  4.1 Question Answering in Geo-query

First, the model is tested on solving question answering problems. This set of
experiments uses the Geo-query datasetZelle and Mooney (1996) that includes
geographical questions and answers.

There are two metrics for evaluation: the token level accuracy, and the
denotation accuracy. The token level accuracy is based on simple token-level
comparison against the reference logical form. And the denotation accuracy is
the percentage of denotation match, as in Jia and Liang (2016) and Liang et
al. (2011).

In the comparative experiment, the models are trained and tested under the
same environment settings. The training set consists of 480 question-answer
pairs and the development set consists of 120 pairs for testing purpose. All
the models are tuned in 10 epochs, with word-embedding dimension 150, hidden
dimension (which is the latent space dimension) 200, learning rate 0.001,
dropout rate 0.2, and the LSTM network in the encoder is set to bidirectional.
Under this experimental setting, several MGMAEs are evaluated, from single-
filter MGMAE to four-filter MGMAE. Thus we can explore how the number of
filters affects the performance and what should be the optimal number of
filters. In addition, The MGMAEs are compared with the ordinary encoder-
decoder model to show whether there is any enhancement. Experiment for each
model is conducted five times and the averaged results are shown in Table 1.

Table 1: Performance comparison between the original encoder-decoder model and MGMAEs in different number of filters. Filter Number | Token | Denotation
---|---|---
1 | 74.2 | 43.6
2 | 77.9 | 48.6
3 | 76.5 | 45.1
4 | 73.6 | 42.5
Enc-Dec | 77.4 | 43.8

The results have shown how the number of filters affects performance. On the
Geo-query dataset, two-filter MGMAE achieves the best performance, then the
accuracies decrease when the number of filters increases. The single-filter
MGMAE does not work as well as the ordinary encoder-decoder model, since the
encoder in MGMAE is only trained by the input data, while the encoder in the
encoder-decoder model is tuned by the input-output sequences. However, the
two-filter MGMAE significantly improves the performance and gains better
results compared to the encoder-decoder model. Which indicates the
effectiveness of the multi-filter architecture. Figure 2 shows the latent
space representations for the training samples. The left figure clearly
indicates that two Gaussians superbly separate the samples, thus we get the
best result for two-filter MGMAE. By contrast, three Gaussians do not well
divide the samples, which may result in cluster misassignment in the test data
and thus deprecate the performance.

Figure 2: latent space clustering in Geo-query training data. The left figure
shows the clustering result for two Gaussian mixtures, the right figures shows
the result for three mixtures.

###  4.2 Machine Translation

To justify the performance of our MGMAE in machine translation tasks, we use
the English-French sentence pairs from Tab-delimited Bilingual Sentence Pairs
as the training and testing dataset. In this set of experiments, models in
both control group and experimental group are trained and tested using this
dataset. Since this is a set of comparative experiments, we concentrates on
comparing the results from different models under the same experimental
settings, rather than tuning hyper-parameters to achieve the best performance.

To evaluate and compare the results, BLEU (bilingual evaluation understudy)
score is utilized. BLEU score is a quick and inexpensive metric for evaluating
the quality of text which has been machine-translated from one natural
language to anotherPapineni et al. (2002). To compute BLEU score, first we use
the test corpus’ reference length $r$ and candidate translation corpus’s
length $c$ to calculate the brevity penalty:

$BP=\exp(1-r/c)$ (4)

then, the BLEU score is computed by:

$BLEU=BP\cdot\exp(\sum_{n=1}^{N}w_{n}\log p_{n})$ (5)

where $p_{n}$ is the N-gram precision. In the experiments, we use $N=4$ and
uniform weights $w_{n}=1/4$.

Table 2: Performance comparison between the original encoder-decoder model and MGMAEs in machine translation. The table shows the average BLEU score ($\pm$ standard deviation) across five identical experiments. Filter Number | BLEU
---|---
1 | 41.7 ($\pm 2.4$)
2 | 44.1 ($\pm 5.7$)
3 | 42.9 ($\pm 4.4$)
4 | 39.7 ($\pm 6.3$)
Enc-Dec | 42.4 ($\pm 1.9$)

In this set of experiments, we use a training set consists of 10,000 bilingual
sentence pairs and the development set consists of 2,000 pairs for evaluation.
All the models are tuned under the same experimental settings: a bidirectional
LSTM in the encoder, LSTMs in the decoder and filters, with word-embedding
dimension 150, hidden dimension 200, and dropout rate 0.2. The models are
optimized in 20 epochs with a learning rate of 0.001.

Table 2 indicates that our MGMAE can perform better than the encoder-decoder
model as long as we choose the correct number of filters. Two-filter and
three-filter MGMAE attain some good accuracies, this is revealed in Figure 3
as well. Two-filter MGMAE works the best and its latent space is well divided.
The filters are trained by a group of clear-defined features and tested by the
samples consists of highly similar features. Clusters in the four-filter model
are mixed up, which increases the possibility of misclassification, thus the
samples may be assigned to the wrong filter.

Figure 3: latent space clustering in Bilingual Pairs training data. The
figures from top left to bottom right present the latent space clustering
results for one, two, three, and four Gaussian mixtures respectively.

##  5 Discussion and Future Direction

Table 3: Silhouette Coefficient vs. model’s performance in Geo-query question-answering and Bilingual Sentence Pairs (English to French) datasets. Filter Number | Geo Silhouette | Geo Token | Geo Denotation | En-Fr Silhouette | En-Fr BLEU
---|---|---|---|---|---
2 | 43.7 | 78.1 | 49.4 | 26.4 | 44.2
3 | 31.5 | 75.8 | 44.7 | 21.6 | 41.0
4 | 22.3 | 73.2 | 42.2 | 17.3 | 38.5
5 | 19.7 | 71.6 | 39.4 | 15.7 | 34.6
6 | 18.9 | 68.4 | 32.7 | 15.4 | 34.7

The experiments demonstrate that our MGMAE is able to enhance the performance
on sequence to sequence tasks. One of the key component that significantly
affect the performance is the number of filters in the MGMAE. To determine the
optimal number of mixtures for the GMM parameter, we use Silhouette
Coefficient to evaluate the clustering performance given a specific number of
mixtures.

Table 3 presents how the Silhouette coefficient related to the model’s
performance. We can observe a positive relationship between the Silhouette
coefficient and the Token-level accuracy, Denotation accuracy, and BLEU score.
This indicates that the quality of the Gaussian mixture is positively related
to the prediction accuracies of the model. A better clustering result means
the data is clearly separable under a certain number of clusters. Then, each
filter can be trained using a fixed set of unique features with less
overlapping to the features in other clusters. Thus we can obtain better
outcomes.

An advantage of our model is that the MGMAE is trained to generate
representations only by using the input data, which means the representations
are independent of the target output data. Therefore, this model may be
applied to tasks such as multi-lingual translation which translates from one
language to multiple languages. In such tasks, the autoencoder and
representations are only trained once, then we only need to optimize the
parameters in the filters for multiple tasks.

###  5.1 Future Research

A future direction is extending this approach to probability distributions
other than Gaussian distribution, such as Exponential mixture modelMosler and
Seidel (2001) or Poisson mixture modelBurda et al. (2012). Instead of assuming
all the samples are normally distributed, different mixture models are
selected based on the distribution of latent space representations of the
training data.

Another potential improvement is applying soft-clustering to the
representations, replacing hard-clustering (what we are doing in this paper).
During predictions, instead of output a discrete result, the mixture model can
return posterior probabilities of each component given the data. The posterior
probabilities are considered the weights to assign filters. The
representations $R$ are decoded by all the filters and the outputs $P$ are
integrated based on the posterior probability $Pr(i|G)$, rather than using one
filter with highest probability to decode.

$P=\sum_{i}Pr(i|G)\cdot f_{i}(R)$ (6)

where $G$ is the input to the Gaussian mixture model and $P(i|G)$ is the
posterior probability that the representation is in cluster $i$, and $f_{i}$
is the decoding function in the filter that corresponding to cluster $i$. This
may largely reduce the consequence of misclassifying clusters.

##  6 Conclusion

We introduce a Multi-filter Gaussian Mixture Autoencoder for solving sequence
to sequence problems. Our MGMAE resolves the heterogeneity of the data by
divide samples into relatively homogeneous groups and utilizing multiple
filters so that each filter is corresponding to one group. In this approach,
each filter concentrates on a smaller set of features. Based on the results
from the two sets of comparative experiments, we concluded that the multi-
filter architecture is able to enhance the performance in question answering
and machine translation.

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