A Fock Model and the Segal–Bargmann Transform for $\mathfrak{osp}(m,2|2n)$

A Fock Model and the Segal–Bargmann Transform for the Minimal Representation
of the Orthosymplectic Lie Superalgebra $\boldsymbol{\mathfrak{osp}(m,2|2n)}$

Sigiswald BARBIER, Sam CLAEREBOUT and Hendrik DE BIE S. Barbier, S. Claerebout
and H. De Bie Department of Electronics and Information Systems, Faculty of
Engineering and Architecture,
Ghent University, Krijgslaan 281, 9000 Gent, Belgium
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Received March 17, 2020, in final form August 12, 2020; Published online
August 26, 2020

The minimal representation of a semisimple Lie group is a ‘small’ infinite-
dimensional irreducible unitary representation. It is thought to correspond to
the minimal nilpotent coadjoint orbit in Kirillov’s orbit philosophy. The
Segal–Bargmann transform is an intertwining integral transformation between
two different models of the minimal representation for Hermitian Lie groups of
tube type. In this paper we construct a Fock model for the minimal
representation of the orthosymplectic Lie superalgebra
$\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which
intertwines the Schrödinger model for the minimal representation of the
orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock
model.

Segal–Bargmann transform; Fock model; Schrödinger model; minimal
representations; Lie superalgebras; spherical harmonics; Bessel–Fischer
product

17B10; 17B60; 22E46; 58C50

##  1 Introduction

The classical Segal–Bargmann transform

$\displaystyle\operatorname{SB}f(z):=\exp\left(-\tfrac{1}{2}z^{2}\right)\int_{{\mathbb{R}}^{m}}\exp(2x\cdot
z)\exp\big{(}{-}x^{2}\big{)}f(x)\mathrm{d}x$

is a unitary isomorphism from the space of square integrable functions on
$\mathbb{R}^{m}$ to the Fock space of entire functions on $\mathbb{C}^{m}$
which are square integrable which respect to the weight function
$\exp\big{(}{-}|z|^{2}\big{)}$. The Segal–Bargmann transform is defined in
such a way that it maps the creation (resp. anihilation) operators on the
Schrödinger space to coordinate multiplication (resp. differentiation) on the
Fock space. This implies in particular that the harmonic oscillator becomes
the much simpler Euler operator on the Fock space [7]. The Segal–Bargmann
transform can also be interpreted as an intertwining operator between two
models of the metaplectic representation (also known as the Segal–Shale–Weil
or oscillator representation) of the metaplectic group, a double cover of the
symplectic group. See [14] for more on the classical Segal–Bargmann transform
and the metaplectic representation.

The (even part of the) metaplectic representation is a prominent example of a
minimal representation. Another extensively studied example is the minimal
representation of ${\rm O}(p,q)$ [21, 22, 23, 20]. The minimal representation
of a semisimple Lie group is the irreducible unitary representation that
according to the orbit method should correspond to the minimal nilpotent
coadjoint orbit [15]. The Segal–Bargmann transform has been generalized to
this setting of minimal representations. Namely for a Hermitian Lie group of
tube type, there exists an explicit integral transform which intertwines the
Schrödinger and Fock model of the minimal representation [18].

Lie superalgebras and Lie supergroups are generalisations of Lie algebras and
Lie groups. They were introduced to mathematically describe supersymmetry.
Their representation theory is an active field of study with still a lot of
open questions, for instance a description of the unitary irreducible
representations. Since most ingredients of the orbit method still exist in the
super setting, it is believed that the orbit method should also in the super
setting be a good tool for the study of irreducible representations [19,
Chapter 6.3]. For example, the irreducible unitary representations of
nilpotent Lie supergroups have been classified that way [25, 26].

An ambitious aim in this light would be to construct minimal representations
and the intertwining Segal–Bargmann transform for Lie supergroups. Recently a
first step in that direction has already been taken. Namely the construction
of a minimal representation of the orthosymplectic Lie supergroup ${\rm
OSp}(p,q|2n)$ was accomplished in [6]. In the bosonic case (i.e., $n=0$) this
realisation corresponds to the Schrödinger model for ${\rm O}(p,q)$ of [20].

In this article we achieve two further goals. First we construct a Fock model
for the minimal representation of the Lie superalgebra
$\mathfrak{osp}(m,2|2n)$. We also define an integral transform which
intertwines the Schrödinger model of the minimal representation of
$\mathfrak{osp}(m,2|2n)$ with this Fock model. Note that only for $q=2$ the
Lie group ${\rm O}(p,q)$ is Hermitian of tube type and thus only in that case
do we have a Segal–Bargmann transform. For that reason we have only
constructed a Segal–Bargmann transform in the super setting for
$\mathfrak{osp}(m,2|2n)$. Our main results hold for $m-2n\geq 4$. This
restriction comes from [6], where key properties of the integral we use in the
definition of the Segal–Bargmann transform are only proven for the case
$m-2n\geq 4$.

We will work in this paper always on the algebraic level. So we will work with
representations of the Lie superalgebra $\mathfrak{osp}(m,2|2n)$ instead of
the Lie supergroup ${\rm OSp}(m,2|2n)$, and we will act on super-vector spaces
defined using superpolynomials instead of using global sections of a
supermanifold. This allows us to circumvent the delicate technicalities
associated with supergroups and supermanifolds. Note that in the bosonic case,
the spaces we work with are dense in certain Hilbert spaces. Using standard
techniques one can then integrate the representation to group level and extend
to the whole Hilbert space, see for example [17, Theorem 2.30] or [18, Theorem
2.17]. These techniques no longer work/exist in the super case. There does
exist an abstract way to integrate a so-called admissible
$(\mathfrak{g},K)$-module to group level [1] which was for example used in [6]
to integrate the minimal representation of $\mathfrak{osp}(p,q|2n)$ to ${\rm
OSp}(p,q|2n)$. However, this approach is not very concrete.

Explicit examples such as the one constructed in this paper could help to
develop such integration tools and to find the correct definitions in the
super setting. For example our representations ought to be ‘unitary’ in some
sense. A definition of unitary representations does exists in the supersetting
[8, Definition 2]. However, a large class of Lie superalgebras, including
$\mathfrak{osp}(p,q|2n)$, do not allow for any super unitary representation in
this sense [25, Theorem 6.2.1]. This highly unsatisfactory situation has
inspired the search for a new or extended definition of a unitary
representation [13, 27]. At the moment it is still unclear what the right
definition should be, but we believe that the construction of explicit
examples which ought to be ‘unitary’ could be useful for this endeavour.

###  1.1 Structure of the paper

We structure the paper as follows. In Section 2 we fix notations and introduce
the spaces and algebras we will use throughout the paper. In Section 3 we
recall the Schrödinger model of the minimal representation of
$\mathfrak{osp}(m,2|2n)$ defined in [6]. We also introduce an integral which
leads to an $\mathfrak{osp}(m,2|2n)$-invariant, non-degenerate, superhermitian
form.

The next three sections contain the main body of this paper. In Section 4 we
construct the Fock space as a quotient of the space of superpolynomials. We
then define the Bessel–Fischer product, which gives us a non-degenerate,
superhermitian form on our Fock space (Propositions 4.7 and 4.12). In the
bosonic case $(n=0)$, this Bessel–Fischer product is equivalent to the inner
product coming from an integral on the Fock space [18, Proposition 2.6]. Since
we do no longer have this integral in the super setting, we construct a direct
proof for the superhermitian property. This seems new even in the bosonic
case. We also show that our Fock space has a reproducing kernel (Theorem
4.11).

In Section 5 we endow this Fock space with an $\mathfrak{osp}(m,2|2n)$-module
structure leading to a Fock model of the minimal representation of
$\mathfrak{osp}(m,2|2n)$. We prove that this is an irreducible representation
and obtain a very explicit description (Theorem 5.3). In particular, we have
complete branching rules for the subalgebras $\mathfrak{osp}(m|2n)$ and
$\mathfrak{osp}(m-1|2n)$.

In Section 6, we define an integral transform which maps the space of
functions used in the Schrödinger realisation to the space of functions of the
Fock realisation (Definition 6.1). We show that this integral is an
intertwining isomorphism which preserves the superhermitian form (Theorems 6.3
and 6.6). We also give an explicit inverse (Definition 6.7). As an application
we use the Segal–Bargmann transform to define generalised Hermite functions.

In Appendix A we gather some definitions and results on special functions
which are used throughout the paper. We have also put the technical and
lengthy proof of Theorem 6.3 in Appendix B.

##  2 Preliminaries and notations

In this paper Jordan and Lie algebras will be defined over complex numbers
$\mathbb{C}$ if they have a $\mathbb{C}$ in subscript, otherwise they are
defined over the field of real numbers $\mathbb{R}$. Function spaces will
always be defined over the complex field $\mathbb{C}$. We use the convention
$\mathbb{N}=\\{0,1,2,\ldots\\}$.

A super-vector space is defined as a $\mathbb{Z}_{2}$-graded vector space,
i.e., $V=V_{0}\oplus V_{1}$, with $V_{0}$ and $V_{1}$ vector spaces. An
element $v$ of a super-vector space $V$ is called homogeneous if it belongs to
$V_{i}$, $i\in\mathbb{Z}_{2}$. We call $i$ the parity of $v$ and denote it by
$|v|$. An homogeneous element $v$ is even if $|v|=0$ and odd if $|v|=1$. When
we use $|v|$ in a formula, we are considering homogeneous elements, with the
implicit convention that the formula has to be extended linearly for arbitrary
elements. We denote the super-vector space $V$ with $V_{0}=\mathbb{R}^{m}$ and
$V_{1}=\mathbb{R}^{n}$ as $\mathbb{R}^{m|n}$. We will always assume $m\geq 2$.

###  2.1 Superpolynomials

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$.

######  Definition 2.1.

The space of superpolynomials over $\mathbb{K}$ is defined as

$\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}:=\mathcal{P}\big{(}\mathbb{K}^{m}\big{)}\otimes_{\mathbb{C}}\Lambda\big{(}\mathbb{K}^{2n}\big{)},$

where $\mathcal{P}\big{(}\mathbb{K}^{m}\big{)}$ denotes the space of complex-
valued polynomials over the field $\mathbb{K}$ in $m$ variables and
$\Lambda\big{(}\mathbb{K}^{2n}\big{)}$ denotes the Grassmann algebra in $2n$
variables. The variables of $\mathcal{P}(\mathbb{K}^{m})$ and
$\Lambda\big{(}\mathbb{K}^{2n}\big{)}$ are called even and odd variables,
respectively. They satisfy the commutation relations

$\displaystyle x_{i}x_{j}=(-1)^{|i||j|}x_{j}x_{i},$

where $|i|:=|x_{i}|$, $i\in\\{0,1,\ldots,m+2n-1\\}$.

Let $\langle{\cdot\,,\cdot}\rangle_{\beta}$ be a supersymmetric, non-
degenerate, even bilinear form on $\mathbb{K}^{m|2n}$ with basis
$\\{x_{i}\\}_{i=0}^{m+2n-1}$. We denote the matrix components by
$\beta_{ij}:=\langle{x_{i},x_{j}}\rangle_{\beta}$ and denote the components of
the inverse matrix by $\beta^{ij}$, i.e., $\beta^{ij}$ is defined such that
$\sum_{j}\beta_{ij}\beta^{jk}=\delta_{ik}$. Set
$x^{j}=\sum_{i}x_{i}\beta^{ij}$. The differential operator $\partial^{i}$ is
defined as the unique derivation in
$\operatorname{End}\big{(}\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\big{)}$
such that $\partial^{i}(x_{j})=\delta_{ij}$, with $\delta_{ij}$ the Kronecker
delta. We also define $\partial_{j}=\sum_{i}\partial^{i}\beta_{ji}$. Then it
holds that $\partial_{i}(x^{j})=\delta_{ij}$.

When we are working with both real and complex polynomials at the same time,
we will denote $\partial_{i}$ and $\partial^{i}$ for the real variable $x_{i}$
as $\partial_{x^{i}}$ and $\partial_{x_{i}}$, respectively. Similarly, we will
denote $\partial_{i}$ and $\partial^{i}$ for the complex variable $z_{i}$ as
$\partial_{z^{i}}$ and $\partial_{z_{i}}$, respectively.

We will make frequent use of the following operators:

$\displaystyle
R^{2}:=\sum_{i,j}\beta^{ij}x_{i}x_{j},\qquad\mathbb{E}:=\sum_{i,j}\beta^{ij}x_{i}\partial_{j}\qquad\mbox{and}\qquad\Delta:=\sum_{ij}\beta^{ij}\partial_{i}\partial_{j}.$
(2.1)

Here, the operator $R^{2}$ is called the square of the radial coordinate and
acts through multiplication. The operators $\mathbb{E}$ and $\Delta$ are
called the Euler operator and the Laplacian, respectively. We have the
following lemma.

######  Lemma 2.2.

The operators $R^{2}$, $\mathbb{E}$ and $\Delta$ satisfy

$\displaystyle[\Delta,R^{2}]=4\mathbb{E}+2M,\qquad[\Delta,\mathbb{E}]=2\Delta,\qquad[R^{2},\mathbb{E}]=-2R^{2},$

where $M=m-2n$ is the superdimension. In particular,
$\big{(}R^{2},\mathbb{E}+\frac{M}{2},-\frac{\Delta}{2}\big{)}$ forms an
$\mathfrak{sl}_{\mathbb{K}}(2)$-triple. Furthermore, they commute in
$\operatorname{End}\big{(}\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\big{)}$
with the operators

$L_{ij}:=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i}.$

###### Proof.

A straightforward calculation or see, for example, [12]. ∎

If we are working with two sets of variables we will add a variable indicator
to avoid confusion. For example, we denote

$\displaystyle
R^{2}_{x}=\sum_{i,j}\beta^{ij}x_{i}x_{j},\qquad\mathbb{E}_{x}=\sum_{i,j}\beta^{ij}x_{i}\partial_{x^{j}},\qquad\Delta_{x}=\sum_{ij}\beta^{ij}\partial_{x^{i}}\partial_{x^{j}}$

and $L_{ij}^{x}=x_{i}\partial_{x^{j}}-(-1)^{|i||j|}x_{j}\partial_{x^{i}}$ for
the real variables and

$\displaystyle
R^{2}_{z}=\sum_{i,j}\beta^{ij}z_{i}z_{j},\qquad\mathbb{E}_{z}=\sum_{i,j}\beta^{ij}z_{i}\partial_{z^{j}},\qquad\Delta_{z}=\sum_{ij}\beta^{ij}\partial_{z^{i}}\partial_{z^{j}}$

and $L^{z}_{ij}=z_{i}\partial_{z^{j}}-(-1)^{|i||j|}z_{j}\partial_{z^{i}}$ for
the complex variables.

###  2.2 The orthosymplectic Lie superalgebra

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$.

######  Definition 2.3.

The orthosymplectic Lie superalgebra $\mathfrak{osp}_{\mathbb{K}}(m|2n,\beta)$
is defined as the subalgebra of $\mathfrak{gl}_{\mathbb{K}}(m|2n)$ preserving
a supersymmetric non-degenerate even bilinear form $\beta$. Thus
$\mathfrak{osp}_{\mathbb{K}}(m|2n,\beta)$ is spanned by
$X\in\mathfrak{gl}_{\mathbb{K}}(m|2n)$ such that

$\displaystyle\langle{X(u),v}\rangle_{\beta}+(-1)^{|u||X|}\langle{u,X(v)}\rangle_{\beta}=0,$

for all $u,v\in\mathbb{K}^{m|2n}$.

The orthosymplectic Lie superalgebra has a differential operator realisation
on $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$. A basis in this realisation
is given by

$\displaystyle
L_{ij}:=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i},\qquad\mbox{for
}i<j,$ $\displaystyle L_{ii}:=2x_{i}\partial_{i},\qquad\mbox{for }|i|=1.$

###  2.3 Spherical harmonics

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. The space of
homogeneous superpolynomials of degree $k$ is denoted by

$\displaystyle\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}:=\big{\\{}p\in\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\mathbb{E}p=kp\big{\\}}.$

The space of spherical harmonics of degree $k$ is defined by

$\displaystyle\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}:=\big{\\{}f\in\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\Delta
f=0\big{\\}},$

i.e., it is the space of homogeneous polynomials of degree $k$ which are in
the kernel of the Laplacian.

The Fischer decomposition gives a decomposition of the space of
superpolynomials using these spherical harmonics [12, Theorem 3].

######  Proposition 2.4.

If $m-2n\neq-2\mathbb{N}$, then $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$
decomposes as

$\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{\infty}R^{2j}\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}.$

In [24] the following generalisation of the Fischer decomposition was obtained
that still holds for the exceptional case $M\in-2\mathbb{N}$.

######  Proposition 2.5 (generalised Fischer decomposition).

The superspace $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$ decomposes as

$\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{\infty}\big{(}R^{2}\Delta
R^{2}\big{)}^{j}\operatorname{\widetilde{\mathcal{H}}}_{k}\big{(}\mathbb{K}^{m|2n}\big{)},$

where

$\displaystyle\operatorname{\widetilde{\mathcal{H}}}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\big{\\{}f\in\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\Delta
R^{2}\Delta f=0\big{\\}}$

is the space of generalised spherical harmonics of degree $k$.

From [9, Theorem 5.2] we obtain the following.

######  Proposition 2.6.

If $M\not\in-2\mathbb{N}$, then
$\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$ is an irreducible
$\mathfrak{osp}_{\mathbb{K}}(m|2n)$-module.

The dimension of the spherical harmonics of degree $k$ is given in [12,
Corollary 1].

######  Proposition 2.7.

The dimension of $\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$, for $m\neq
0$ is given by

$\displaystyle\dim\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k-2}\big{(}\mathbb{K}^{m|2n}\big{)},$

with

$\displaystyle\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\sum_{i=0}^{\min(k,2n)}\binom{2n}{i}\binom{k-i+m-1}{m-1}.$

We will also use the following formula for the dimension of the spherical
harmonics of degree $k$.

######  Proposition 2.8.

The dimension of $\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$, for $m>1$
is given by

$\displaystyle\dim\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m-1|2n}\big{)}+\dim\mathcal{P}_{k-1}\big{(}\mathbb{K}^{m-1|2n}\big{)}.$

###### Proof.

If we prove

$\displaystyle\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k-2}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m-1|2n}\big{)}-\dim\mathcal{P}_{k-1}\big{(}\mathbb{K}^{m-1|2n}\big{)}=0,$

then the proposition follows from Proposition 2.7. First suppose $2n\leq k-2$,
then the above equation becomes

$\displaystyle\sum_{i=0}^{2n}\binom{2n}{i}\left(\binom{k-i+m-1}{m-1}-\binom{k-i+m-3}{m-1}\right.$
$\displaystyle\left.\qquad{}-\binom{k-i+m-2}{m-2}-\binom{k-i+m-3}{m-2}\right)=0,$

which is true since the recursive formula of binomial coefficients gives us
that

$\displaystyle\binom{k-i+m-1}{m-1}-\binom{k-i+m-3}{m-1}-\binom{k-i+m-2}{m-2}-\binom{k-i+m-3}{m-2}=0,$

for all $i\in\\{0,\ldots,2n\\}$. For $2n>k-2$ we have the following extra
terms

$\displaystyle\binom{2n}{k-1}\left(\binom{m}{m-1}-\binom{m-1}{m-2}-\binom{m-2}{m-2}\right)+\binom{2n}{k}\left(\binom{m-1}{m-1}-\binom{m-2}{m-2}\right),$

where we ignore the last term if $2n=k-1$. Using basic binomial properties,
these terms are clearly equal to zero. ∎

For $n=0$ a more insightful reasoning as to why this formula holds is given by
Proposition 5 in [11].

###  2.4 The spin factor Jordan superalgebra $\boldsymbol{J}$

To each Hermitian Lie group of tube type corresponds an Euclidean Jordan
algebra. These Jordan algebras were a crucial ingredient in the unified
approach to construct minimal representations and the Segal–Bargmann transform
[17, 18]. More concretely, one can associate with each Jordan algebra certain
Lie algebras (the structure algebra and the TKK-algebra), and the Jordan
algebra is also used in the construction of spaces on which these Lie algebras
act. In this paper we will not use anything directly from Jordan theory, but
introducing $\mathfrak{osp}(m,2|2n)$ via the spin factor Jordan superalgebra
leads to a natural decomposition of $\mathfrak{osp}(m,2|2n)$ as well as to
some interesting subalgebras that we will use.

######  Definition 2.9.

A Jordan superalgebra is a supercommutative superalgebra $J$ satisfying the
Jordan identity

$\displaystyle(-1)^{|x||z|}[L_{x},L_{yz}]+(-1)^{|y||x|}[L_{y},L_{zx}]+(-1)^{|z||y|}[L_{z},L_{xy}]=0\qquad\text{for
all }x,y,z\in J.$

Here the operator $L_{x}$ is (left) multiplication with $x$ and
$[\cdot\,,\cdot]$ is the supercommutator, i.e.,
$[L_{x},L_{y}]:=L_{x}L_{y}-(-1)^{|x||y|}L_{y}L_{x}$.

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. We will define the
spin factor Jordan superalgebra associated with a supersymmetric, non-
degenerate, even, bilinear form. Let $V_{\mathbb{K}}$ be a super-vector space
over $\mathbb{K}$ with $\dim(V_{\mathbb{K}})=(m-1|2n)$ and a supersymmetric,
non-degenerate, even, bilinear form
$\langle\cdot\;,\cdot\rangle_{\tilde{\beta}}$ where, for
$\mathbb{K}=\mathbb{R}$, the even part has signature $(m-1,0)$. Recall that we
always assume $m\geq 2$. We choose a homogeneous basis
$(e_{i})_{i=1}^{m+2n-1}$ of $V_{\mathbb{K}}$. For $u=\sum_{i}u^{i}e_{i}$ and
$v=\sum_{i}v^{i}e_{i}$ we then have

$\displaystyle\langle
u,v\rangle_{\tilde{\beta}}=\sum_{i,j}u^{i}{\tilde{\beta}}_{ij}v^{j}\qquad\mbox{with}\quad{\tilde{\beta}}_{ij}:=\langle
e_{i},e_{j}\rangle_{\tilde{\beta}}.$

######  Definition 2.10.

The spin factor Jordan superalgebra is defined as
$J_{\mathbb{K}}:=\mathbb{K}e_{0}\oplus V_{\mathbb{K}}$ with $|e_{0}|=0$. The
Jordan product is given by

$\displaystyle(\lambda e_{0}+u)(\mu e_{0}+v)=(\lambda\mu+\langle
u,v\rangle_{\tilde{\beta}})e_{0}+\lambda v+\mu u,$

for $u,v\in V_{\mathbb{K}}$ and $\lambda,\mu\in\mathbb{K}$.

Thus $e_{0}$ is the unit of $J_{\mathbb{K}}$. We extend the homogeneous basis
$(e_{i})_{i=1}^{m+2n-1}$ of $V_{\mathbb{K}}$ to a homogeneous basis
$(e_{i})_{i=0}^{m+2n-1}$ of $J_{\mathbb{K}}$ and extend the bilinear form
$\langle\cdot\;,\cdot\rangle_{\tilde{\beta}}$ as follows. Set $\beta_{00}=-1$,
$\beta_{i0}=0=\beta_{0i}$ and $\beta_{ij}={\tilde{\beta}}_{ij}$ for
$i,j\in\\{1,\ldots,m+2n-1\\}$. Then the corresponding form
$\langle\cdot\;,\cdot\rangle_{\beta}$ is a supersymmetric, non-degenerate,
even bilinear form on the super-vector space $J_{\mathbb{K}}$ where, for
$\mathbb{K}=\mathbb{R}$, the even part has signature $(m-1,1)$.

Define $(\beta^{ij})_{ij}$ as the inverse of $(\beta_{ij})_{ij}$. Let
$\big{(}e^{i}\big{)}_{i}$ be the right dual basis of $(e_{i})_{i}$ with
respect to the form $\langle\cdot\;,\cdot\rangle_{\beta}$, i.e.,

$\displaystyle\langle e_{i}\;,e^{j}\rangle_{\beta}=\delta_{ij}.$

Then

$\displaystyle e^{j}=\sum_{i}e_{i}\beta^{ij}.$

In this paper we will assume that the orthosymplectic metric is standardized
such that

$\displaystyle\beta=(\beta_{ij})_{i,j=0}^{m+2n-1}=\left(\begin{array}[]{cc|cc}-1&&&\\\
&I_{m-1}&&\\\ \hline\cr&&&-I_{n}\\\ &&I_{n}&\end{array}\right),$

with $I_{d}$ the $d$-dimensional identity matrix.

From now on the real spin factor Jordan superalgebra will always be denoted by
$J$ or $J_{\mathbb{R}}$ and its complexified version by $J_{\mathbb{C}}$.

###  2.5 The TKK algebra

With each Jordan (super)algebra one can associate a $3$-graded Lie
(super)algebra via the TKK-construction. There exist different TKK-
constructions in the literature, see [5] for an overview, but for the spin
factor Jordan superalgebra $J_{\mathbb{K}}$ all constructions lead to the
orthosymplectic Lie superalgebras $\mathfrak{osp}_{\mathbb{K}}(m,2|2n)$. We
will quickly review the Koecher construction. First consider
$\operatorname{Inn}(J_{\mathbb{K}})$, the subalgebra of
$\mathfrak{gl}(J_{\mathbb{K}})$ of inner derivations. It is generated by the
operators $[L_{u},L_{v}]$, $u,v\in J_{\mathbb{K}}$. If we add the left
multiplication operators $L_{u}$ to the inner derivations we obtain the inner
structure algebra:

$\displaystyle\mathfrak{istr}(J_{\mathbb{K}}):=\\{L_{u}|u\in
J_{\mathbb{K}}\\}\oplus\operatorname{Inn}(J_{\mathbb{K}})=\operatorname{span}_{\mathbb{K}}\\{L_{u},[L_{u},L_{v}]\colon
u,v\in J_{\mathbb{K}}\\}.$

Let $J_{\mathbb{K}}^{+}$ and $J_{\mathbb{K}}^{-}$ be two copies of
$J_{\mathbb{K}}$. As a vector space we define the TKK-algebra of
$J_{\mathbb{K}}$ as

$\displaystyle\operatorname{TKK}(J_{\mathbb{K}}):=J_{\mathbb{K}}^{-}\oplus\mathfrak{istr}(J_{\mathbb{K}})\oplus
J_{\mathbb{K}}^{+}.$

The Lie bracket on $\operatorname{TKK}(J_{\mathbb{K}})$ is defined as follows.
We interpret $\mathfrak{istr}(J_{\mathbb{K}})$ as a subalgebra of
$\mathfrak{gl}(J_{\mathbb{K}})$ and for homogeneous $x,y\in
J_{\mathbb{K}}^{+}$, $u,v\in J_{\mathbb{K}}^{-}$, $a,b\in J_{\mathbb{K}}$ we
set

$\displaystyle[x,u]=2L_{xu}+2[L_{x},L_{u}],\qquad$
$\displaystyle[x,y]=[u,v]=0,$ $\displaystyle[L_{a},x]=ax,\qquad$
$\displaystyle[L_{a},u]=-au,$
$\displaystyle[[L_{a},L_{b}],x]=[L_{a},L_{b}]x,\qquad$
$\displaystyle[[L_{a},L_{b}],u]=[L_{a},L_{b}]u.$

From [6, Proposition 3.1] we obtain the following

######  Theorem 2.11.

We have

$\displaystyle\mathfrak{istr}(J_{\mathbb{K}})=\mathfrak{osp}_{\mathbb{K}}(J_{\mathbb{K}})\oplus\mathbb{K}L_{e_{0}}\cong\mathfrak{osp}_{\mathbb{K}}(m-1,1)\oplus\mathbb{K}L_{e_{0}},\qquad\operatorname{TKK}(J_{\mathbb{K}})\cong\mathfrak{osp}_{\mathbb{K}}(m,2|2n),$

where the direct sum decomposition is as algebras.

Using the bilinear form

$\displaystyle\overline{\beta}=\left(\begin{array}[]{cccc|c}1&&&&\\\
&(\beta_{ij})_{i,j=1}^{m-1}&&&\\\ &&\beta_{00}&&\\\ &&&-1&\\\
\hline\cr&&&&(\beta_{ij})_{i,j=m}^{m+2n-1}\end{array}\right),$

we have the following explicit isomorphism of
$\operatorname{TKK}(J_{\mathbb{K}})$ with the differential operator
realisation of $\mathfrak{osp}_{\mathbb{K}}(m,2|2n)$:

$\displaystyle e_{i}^{-}\mapsto L_{\tilde{i},(m+1)}+L_{\tilde{i},0},\qquad$
$\displaystyle e_{0}^{-}\mapsto L_{m,(m+1)}+L_{m,0},$ $\displaystyle
L_{e_{i}}\mapsto L_{\tilde{i},m},\qquad$ $\displaystyle L_{e_{0}}\mapsto
L_{0,(m+1)},$ $\displaystyle[L_{e_{i}},L_{e_{j}}]\mapsto
L_{\tilde{i},\tilde{j}},$ $\displaystyle e_{i}^{+}\mapsto
L_{\tilde{i},(m+1)}-L_{\tilde{i},0},\qquad$ $\displaystyle e_{0}^{+}\mapsto-
L_{m,(m+1)}+L_{m,0}.$

Here $\tilde{i}=i$ if $|i|=0$ and $\tilde{i}=i+2$ if $|i|=1$.

##  3 The Schrödinger model of $\boldsymbol{\mathfrak{osp}(m,2|2n)}$

In the bosonic case (i.e., $n=0$), the Schrödinger model of the minimal
representation of a Hermitian Lie group $G$ of tube type can be seen as a
representation realized on the Hilbert space $L^{2}(C)$ where $C$ is a minimal
orbit for the structure group of $G$, see [18]. In the super setting the
classical definition of minimal orbits no longer works. Indeed,
supermanifolds, in contrast to ordinary manifolds, are not completely
determined by their points. In [6] an orbit was instead defined as the
quotient supermanifold of the structure group by a stabilizer subgroup
together with an embedding. Using this definition, a minimal orbit $C$ was
constructed which can be characterized by $R^{2}=0$, with $R^{2}$ as
introduced in (2.1). We refer to Section 4 of [6] for a detailed description
of this minimal orbit. We will now recall the Schrödinger model constructed in
[6]. A critical role in this construction is played by the Bessel operators.

###  3.1 The Bessel operator

The Bessel operator $\operatorname{\mathcal{B}}_{\lambda}(x_{k})$ is a linear
operator acting on $\mathcal{P}\big{(}\mathbb{R}^{m|2n}\big{)}$. It depends on
a complex parameter $\lambda$ and an explicit expression is given by

$\displaystyle\operatorname{\mathcal{B}}_{\lambda}(x_{k}):=(-\lambda+2\mathbb{E})\partial_{k}-x_{k}\Delta,$

where $\mathbb{E}$ and $\Delta$ are the Euler operator and Laplacian
introduced in (2.1).

From Proposition 4.9 of [6] we obtain the following.

######  Proposition 3.1.

The Bessel operators map $\langle R^{2}\rangle$ into
$\big{\langle}R^{2}\big{\rangle}$ if and only if $\lambda=2-M$, where $M=m-2n$
is the superdimension of $\mathbb{R}^{m|2n}$ and
$\big{\langle}R^{2}\big{\rangle}$ is the ideal in
$\mathcal{P}\big{(}\mathbb{R}^{m|2n}\big{)}$ generated by $R^{2}$.

Therefore we will only use the Bessel operator with the parameter $2-M$ in
this paper and we set
$\operatorname{\mathcal{B}}(x_{i}):=\operatorname{\mathcal{B}}_{2-M}(x_{i})$.
We obtain the following two properties of the Bessel operator from Proposition
4.2 in [4].

######  Proposition 3.2 (supercommutativity).

For all $i\in\\{0,\ldots,m+2n-1\\}$ we have

$\displaystyle\operatorname{\mathcal{B}}(x_{i})\operatorname{\mathcal{B}}(x_{j})=(-1)^{|i||j|}\operatorname{\mathcal{B}}(x_{j})\operatorname{\mathcal{B}}(x_{i}).$

######  Proposition 3.3 (product rule).

For all $i\in\\{0,\ldots,m+2n-1\\}$ we have

$\displaystyle\operatorname{\mathcal{B}}(x_{i})(\phi\psi)=\operatorname{\mathcal{B}}(x_{i})(\phi)\psi+(-1)^{|i||\phi|}\phi\operatorname{\mathcal{B}}(x_{i})(\psi)$
$\displaystyle\hphantom{\operatorname{\mathcal{B}}(x_{i})(\phi\psi)=}{}+2(-1)^{|i||\phi|}\mathbb{E}(\phi)\partial_{i}(\psi)+2\partial_{i}(\phi)\mathbb{E}(\psi)-2x_{i}\sum_{r,s}(-1)^{|\phi||r|}\beta^{rs}\partial_{r}(\phi)\partial_{s}(\psi).$

As a direct result from the product rule we have

$\displaystyle[\operatorname{\mathcal{B}}(x_{i}),x_{j}]=\beta_{ij}(M-2+2\mathbb{E})-2L_{ij},$
(3.1)

for all $i,j\in\\{0,\ldots,m+2n-1\\}$.

In what follows we will mostly use the following slightly modified version of
the Bessel operator.

######  Definition 3.4.

The modified Bessel operator
$\operatorname{\widetilde{\mathcal{B}}}_{\lambda}(x_{k})$ is given by

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(x_{0}):=-\operatorname{\mathcal{B}}(x_{0}),\qquad\operatorname{\widetilde{\mathcal{B}}}(x_{k}):=\operatorname{\mathcal{B}}(x_{k}),$

for $k\in\\{1,\ldots,m+2n-1\\}$.

###  3.2 Radial superfunctions

Let us denote the supervariables $x_{m},\ldots,x_{m+2n-1}$ by
$\theta_{1},\ldots,\theta_{2n}$. We keep the notations for
$x_{0},\ldots,x_{m-1}$. Define $\theta^{I}$ as
$\theta_{1}^{i_{1}}\theta_{2}^{i_{2}}\cdots\theta_{n}^{i_{2n}}$ for
$I=(i_{1},i_{2},\dots,i_{2n})\in\mathbb{Z}_{2}^{2n}$. Consider a function
$h\colon\mathbb{R}\rightarrow\mathbb{R}$,
$h\in\mathcal{C}^{2n}(\mathbb{R}\setminus\\{0\\})$ and a superfunction
$f=f_{0}+\sum\limits_{I\neq 0}f_{I}\theta^{I}$, with
$f_{0},f_{I}\in\mathcal{C}^{\infty}\big{(}\mathbb{R}^{m}\big{)}$ and where
$f_{0}$ has non-zero values. Then a new superfunction

$\displaystyle h(f):=\sum_{j=0}^{2n}\dfrac{1}{j!}\left(\sum_{I\neq
0}f_{I}\theta^{I}\right)^{j}h^{(j)}(f_{0})$ (3.2)

is defined in [10, Definition 3]. Here $h^{(j)}$ denotes the $j$th derivative
of $h$.

Set the supervariable

$\displaystyle r^{2}:=\sum_{i=1}^{m+2n-1}x^{i}x_{i},$

so $R^{2}=-x_{0}^{2}+r^{2}$. We can use equation (3.2) with
$f=\big{(}x_{0}^{2}+r^{2}\big{)}/2$ and $h$ the square root to define the
superfunction $|X|$ as

$\displaystyle|X|:=\sqrt{\dfrac{x_{0}^{2}+r^{2}}{2}}.$

Using equation (3.2) again, but now with $f=|X|$ we define $\exp(|X|)$ and
$\Lambda_{2,j}^{\mu,\nu}(|X|)$ as radial superfunctions. Here
$\Lambda_{2,j}^{\mu,\nu}$ is the generalised Laguerre function introduced in
Appendix A.2.

###  3.3 The $\boldsymbol{(\mathfrak{g},\mathfrak{k})}$-module
$\boldsymbol{W}$

We introduce the notations

$\displaystyle\mathfrak{g}:=\operatorname{TKK}(J)\cong\mathfrak{osp}(m,2|2n),$
$\displaystyle\mathfrak{k}:=\\{(x,I,-x)\colon I\in\operatorname{Inn}(J),\,x\in
J\\},$
$\displaystyle\mathfrak{k}_{0}:=\mathfrak{k}\cap\mathfrak{istr}(J)=\operatorname{Inn}(J).$

######  Theorem 3.5.

The following isomorphisms

$\displaystyle\mathfrak{k}_{0}\cong\mathfrak{osp}(m-1,0|2n),\qquad\mathfrak{k}\cong\mathfrak{osp}(m,0|2n)\oplus\mathbb{R},$

hold as algebras.

###### Proof.

This follows from a straightforward verification by, for example, looking at
the matrix realisation of $\mathfrak{g}$. ∎

In Section 5.2 of [6] the so-called Schrödinger model of $\mathfrak{g}$ was
constructed for $M\geq 4$. Since this model will play an important role in
this paper, we will give a short recall of its construction. Firstly, we
consider a representation $\pi$ of $\mathfrak{g}$ acting on smooth
superfunctions on $J^{-}$. Explicitly, $\pi$ of
$\mathfrak{g}=\operatorname{TKK}(J)=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$
is given as follows:

  * •

$\pi(e_{l},0,0)=-2\imath x_{l}$,

  * •

$\pi(0,L_{e_{k}},0)=-x_{0}\partial_{k}+x_{k}\partial_{0}$,

  * •

$\pi(0,[L_{e_{i}},L_{e_{j}}],0)=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i}$,

  * •

$\pi(0,L_{e_{0}},0)=\frac{2-M}{2}-\mathbb{E}$,

  * •

$\pi(0,0,e_{l})=-\frac{1}{2}\imath\operatorname{\widetilde{\mathcal{B}}}(x_{l})$,

with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$, $l\in\\{0,1,\ldots,m+2n-1\\}$ and
where $\imath$ is the imaginary unit. For $n=0$, our convention corresponds to
the Schrödinger realisation given in [18]. It only differs from the
Schrödinger realisation given in [6] by a change of basis. It can be shown
that all the operators occuring in $\pi$ are tangential to $R^{2}$, with
$R^{2}$ as in equation (2.1). So we can consider the representation obtained
by quotienting out $R^{2}$. This quotient representation has an interesting
subrepresentation consisting of $\mathfrak{k}$-finite vectors. It is generated
by $\exp(-2|X|)$, with $\exp(-2|X|)$ the smooth radial superfunction on
$J^{-}$ as defined in Section 3.2. This subrepresentation wil be our
Schrödinger model. So the Schrödinger model is given as follows:

$\displaystyle
W:=U(\mathfrak{g})\exp(-2|X|)\mod\big{\langle}R^{2}\big{\rangle},$

where the $\mathfrak{g}$-module structure is given by the Schrödinger
representation $\pi$.

Theorem 5.3 of [6] gives the following decomposition of $W$.

######  Theorem 3.6 (decomposition of $W$).

Assume $M\geq 4$.

  * $(1)$

The decomposition of $W$ as a $\mathfrak{k}$-module is given by

$\displaystyle W=\bigoplus_{j=0}^{\infty}W_{j},\qquad\mbox{with}\quad
W_{j}=U(\mathfrak{k})\Lambda_{2,j}^{M-3,-1}(2|X|),$

where $W_{j}$ and thus also $W$ are $\mathfrak{k}$-finite.

  * $(2)$

$W$ is a simple $\mathfrak{g}$-module.

  * $(3)$

An explicit decomposition of $W_{j}$ into irreducible
$\mathfrak{k}_{0}$-modules is given by

$\displaystyle
W_{j}=\bigoplus_{k=0}^{j}\bigoplus_{l=0}^{1}\Lambda_{2,j-k}^{M-3+2k,2l-1}(2|X|)\big{(}\mathcal{H}_{k}\big{(}\mathbb{R}^{m-1|2n}\big{)}\otimes\mathcal{H}_{l}(\mathbb{R})\big{)}.$

Furthermore, if $m$ is even we also have the following
$\mathfrak{k}$-isomorphism

$\displaystyle
W_{j}\cong\mathcal{H}_{j}\big{(}\mathbb{R}^{m|2n}\big{)}\otimes\mathcal{H}_{\frac{M-2}{2}+j}\big{(}\mathbb{R}^{2}\big{)}.$

###  3.4 The integral and sesquilinear form

In Section 8 of [6] an integral which restricts to $W$ was constructed. We
will use a renormalized version of that integral, restricted to $x_{0}>0$. To
give the integral explicitly we consider spherical coordinates in
$\mathbb{R}^{m-1}$ by setting $x_{i}=\omega_{i}s$ with
$\omega_{i}\in\mathbb{S}^{m-2}$, $i\in\\{1,\ldots,m-1\\}$ and

$\displaystyle s^{2}:=\sum_{i=1}^{m-1}x_{i}^{2}.$

We also introduce the following notations

$\displaystyle\theta^{2}:=\sum_{i,j=m}^{m+2n-1}\beta^{ij}x_{i}x_{j},\qquad
1+\eta:=\sqrt{1-\dfrac{\theta^{2}}{2s^{2}}},\qquad
1+\xi:=\sqrt{1+\dfrac{\theta^{2}}{2x_{0}^{2}}}$

and the morphism

$\displaystyle\phi^{\sharp}(f):=\sum_{j=0}^{n}\dfrac{\theta^{2j}}{j!}\left(\dfrac{1}{4x_{0}}\partial_{x_{0}}-\dfrac{1}{4s}\partial_{s}\right)^{j}(f),$

for
$f\in\mathcal{C}^{\infty}\big{(}\mathbb{R}^{m}\setminus\\{0\\}\big{)}\otimes\Lambda\big{(}\mathbb{R}^{2n}\big{)}$.
We remark that in [6, Lemma 8.2] it is shown that $\phi^{\sharp}$ is actually
an algebra isomorphism. The Berezin integral on
$\Lambda\big{(}\mathbb{R}^{2n}\big{)}$ is defined as

$\displaystyle\int_{B}:=\partial^{m+2n-1}\partial^{m+2n-2}\cdots\partial^{m}.$

######  Definition 3.7.

Suppose $M\geq 4$. For $f\in W$ we define the integral $\int_{W}$ by

$\displaystyle\int_{W}f:=\dfrac{1}{\gamma}\int_{0}^{\infty}\int_{\mathbb{S}^{m-2}}\int_{B}\rho^{m-3}(1+\eta)^{m-3}(1+\xi)^{-1}\phi^{\sharp}(f)_{|s=x_{0}=\rho}\mathrm{d}\rho\mathrm{d}\omega,$
(3.3)

where $\gamma\in\mathbb{C}$ is the renormalisation factor such that
$\int_{W}\exp(-4|X|)=1$.

We can show that the integral is well defined modulo
$\big{\langle}R^{2}\big{\rangle}$. This follows from Section 8 of [6] together
with the fact that $\gamma$ is non-zero.

######  Proposition 3.8.

For $M=m-2n\geq 4$ we have

$\displaystyle\gamma=\dfrac{2^{5-2M}}{n!}\left(\dfrac{3-m}{2}\right)_{n}\dfrac{\pi^{\frac{m-1}{2}}}{\Gamma{\big{(}\frac{m-1}{2}\big{)}}}\Gamma(M-2),$

where we used the Pochhammer symbol $(a)_{k}=a(a+1)(a+2)\cdots(a+k-1)$. Note
that $\big{(}\frac{3-m}{2}\big{)}_{n}=0$ implies $M=m-2n\leq 1$. Therefore
$\gamma$ is non-zero.

###### Proof.

Let us denote the integral $\int_{W}$ not normalized by $\gamma$ as
$\int_{W^{\prime}}$, i.e., $\int_{W^{\prime}}=\gamma\int_{W}$. We wish to
calculate

$\displaystyle\gamma=\int_{W^{\prime}}\exp(-4|X|).$

In [3, Lemma 6.3.5] a similar integral is calculated if one observes that
$\widetilde{K}_{-\frac{1}{2}}(t)=\frac{\sqrt{\pi}}{2}\exp(-t)$, (equation
(A.1) in Appendix A.1) where $\widetilde{K}_{\alpha}$ is the K-Bessel function
introduced in Appendix A.1. Using the same calculations as in the proof of [3,
Lemma 6.3.5], we then obtain

$\displaystyle\gamma=\dfrac{1}{n!}\left(\dfrac{3-m}{2}\right)_{n}\dfrac{\pi^{\frac{m-3}{2}}}{\Gamma{\big{(}\frac{m-1}{2}\big{)}}}\dfrac{\Gamma\big{(}\frac{M}{2}\big{)}\Gamma\big{(}\frac{M}{2}-1\big{)}\Gamma\big{(}\frac{M-1}{2}\big{)}^{2}}{\Gamma(M-1)},$

provided we take into account that we need extra factors $2$ in certain places
and that we restrict ourselves to $x_{0}>0$. Note that in [3, Lemma 6.3.5] the
tilde in $\widetilde{K}_{-\frac{1}{2}}$ sometimes mistakenly disappears. If we
use Legendre’s duplication formula:

$\displaystyle\Gamma\left(\dfrac{z+1}{2}\right)\Gamma\left(\dfrac{z}{2}\right)=2^{1-z}\sqrt{\pi}\Gamma(z),$

on $\Gamma\big{(}\frac{M}{2}\big{)}\Gamma\big{(}\frac{M-1}{2}\big{)}$ and
$\Gamma\big{(}\frac{M-1}{2}\big{)}\Gamma\big{(}\frac{M}{2}-1\big{)}$ the
result follows. ∎

######  Definition 3.9.

For $f,g\in W$ we define the sesquilinear form
$\langle{\cdot\,,\cdot}\rangle_{W}$ as

$\displaystyle\langle{f,g}\rangle_{W}:=\int_{W}f\overline{g}.$

Theorem 8.13 and Lemma 8.14 in [6] give us the following two properties.

######  Proposition 3.10.

Suppose $M=m-2n\geq 4$. The Schrödinger representation $\pi$ on $W$ is skew-
supersymmetric with respect to $\langle{\cdot\,,\cdot}\rangle_{W}$, i.e.,

$\displaystyle\langle{\pi(X)f,g}\rangle_{W}=-(-1)^{|X||f|}\langle{f,\pi(X)g}\rangle_{W},$

for all $X\in\mathfrak{g}$ and $f,g\in W$.

######  Proposition 3.11.

Suppose $M=m-2n\geq 4$. The form $\langle{\cdot\,,\cdot}\rangle_{W}$ defines a
sesquilinear, non-degenerate form on $W$, which is superhermitian, i.e.,

$\displaystyle\langle{f,g}\rangle_{W}=(-1)^{|f||g|}\overline{\langle{g,f}\rangle}_{W},$

for all $f,g\in W$.

Note that for both Theorem 8.13 and Lemma 8.14 in [6] there is an extra
condition saying that $M$ must be even. However, since we are working in the
exceptional case that corresponds with $q=2$ in [6], the proofs still hold
without this extra condition.

##  4 The Fock space

In [18, Section 2.3] an inner product on the polynomial space
$\mathcal{P}(\mathbb{C}^{m})$ was introduced, namely the Bessel–Fischer inner
product

$\displaystyle\langle{p,q}\rangle_{\mathcal{B}}:=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}\bar{q}(z)\big{|}_{z=0},$

where $\bar{q}(z)=\overline{q(\bar{z})}$ is obtained by conjugating the
coefficients of the polynomial $q$. In [18, Proposition 2.6] it is proven
that, for polynomials, the Bessel–Fischer inner product is equal to the
$L^{2}$-inner product of the Fock space. Since there is no immediate extension
of this $L^{2}$-inner product to the super setting, we will use the
Bessel–Fischer inner product on polynomials as the starting point to
generalize the Fock space to superspace.

###  4.1 Definition and properties

######  Definition 4.1.

We define the polynomial Fock space as the superspace

$\displaystyle\operatorname{\mathcal{F}}:=\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}/\big{\langle}R^{2}\big{\rangle}.$

From the generalised Fischer decomposition of Proposition 2.5 it follows that

$\displaystyle\operatorname{\mathcal{F}}\cong\bigoplus_{l=0}^{\infty}\operatorname{\widetilde{\mathcal{H}}}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}.$

In particular, if the superdimension $M=m-2n$ is such that
$M\not\in-2\mathbb{N}$, then the Fischer decomposition from Proposition 2.4
gives us

$\displaystyle\operatorname{\mathcal{F}}\cong\bigoplus_{l=0}^{\infty}\mathcal{H}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}.$

######  Definition 4.2.

For $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ we define the
Bessel–Fischer product of $p$ and $q$ as

$\displaystyle\langle{p,q}\rangle_{\mathcal{B}}:=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}\bar{q}(z)\big{|}_{z=0},$

where $\bar{q}(z)=\overline{q(\bar{z})}$ is obtained by conjugating the
coefficients of the polynomial $q$ and
$\operatorname{\widetilde{\mathcal{B}}}$ is the complex version of the
modified Bessel operators introduced in Definition 3.4.

Explicitly for $p=\sum_{\alpha}a_{\alpha}z^{\alpha}$ and
$q=\sum_{\beta}b_{\beta}z^{\beta}$ we have

$\displaystyle\langle{p,q}\rangle_{\mathcal{B}}=\sum_{\alpha,\beta}a_{\alpha}\bar{b}_{\beta}(-1)^{\alpha_{0}}\operatorname{\mathcal{B}}(z_{0})^{\alpha_{0}}\cdots\operatorname{\mathcal{B}}(z_{m+2n-1})^{\alpha_{m+2n-1}}z_{0}^{\beta_{0}}\cdots
z_{m+2n-1}^{\beta_{m+2n-1}}\big{|}_{z=0}.$

Note that it is only an inner product in the bosonic case. However, in [13] a
new definition of Hilbert superspaces was introduced where the preserved form
is no longer an inner product, but rather a non-degenerate, sesquilinear,
superhermitian form. We will prove that the Bessel–Fischer product is such a
form when restricted to $\operatorname{\mathcal{F}}$ with
$M-2\not\in-2\mathbb{N}$.

######  Proposition 4.3 (sesquilinearity).

For $p,q,r,s\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ and
$\alpha,\beta,\gamma,\delta\in\mathbb{C}$ we have

$\displaystyle\langle{\alpha p+\gamma r,\beta q+\delta
s}\rangle_{\mathcal{B}}=\alpha\bar{\beta}\langle{p,q}\rangle_{\mathcal{B}}+\alpha\bar{\delta}\langle{p,s}\rangle_{\mathcal{B}}+\gamma\bar{\beta}\langle{r,q}\rangle_{\mathcal{B}}+\gamma\bar{\delta}\langle{r,s}\rangle_{\mathcal{B}}.$

###### Proof.

This follows from the linearity of the Bessel operators. ∎

######  Proposition 4.4 (orthogonality).

For $p_{k}\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ and
$p_{l}\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$ with $l\neq k$ we
have $\langle{p_{k},p_{l}}\rangle_{\mathcal{B}}=0$.

###### Proof.

This follows from the fact that Bessel operators lower the degree of
polynomials by one. ∎

######  Proposition 4.5.

For $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ and
$i\in\\{0,1,\ldots,m+2n-1\\}$ we have

$\displaystyle\langle{z_{i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\big{\langle}p,\operatorname{\widetilde{\mathcal{B}}}(z_{i})q\big{\rangle}_{\mathcal{B}}.$

###### Proof.

This follows immediately from the definition of the Bessel–Fischer product. ∎

To prove that the Bessel–Fischer product is superhermitian we will use
induction on the degree of the polynomials. We will need the following lemma
in the induction step of the proof.

######  Lemma 4.6.

Suppose we have proven that
$\langle{p,q}\rangle_{\mathcal{B}}=(-1)^{|p||q|}\overline{\langle{q,p}\rangle}_{\mathcal{B}}$
for all $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree lower
than or equal to $k\in\mathbb{N}$. Then for
$p,q\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$, $l\leq k$ we have

$\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}\qquad\text{and}\qquad\langle{L_{0i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,L_{0i}q}\rangle_{\mathcal{B}},$

for all $i,j\in\\{1,\ldots,m+2n-1\\}$.

###### Proof.

We will prove the $L_{ij}$ case explicitly. The $L_{0i}$ case is entirely
analogous. First note that if we combine the given superhermitian property
with Proposition 4.5, we get

$\displaystyle\big{\langle}\operatorname{\widetilde{\mathcal{B}}}(z_{i})p,q\big{\rangle}_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,z_{i}q}\rangle_{\mathcal{B}}.$

for all $p\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree $k$ or
lower and all $q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree $k-1$
or lower. Assume $p,q\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$,
$l\leq k$. We obtain

$\displaystyle\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{\partial_{j}p,\operatorname{\widetilde{\mathcal{B}}}(z_{i})q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{|i|(|p|+|j|)}\big{(}(M-2+2(l-1))\langle{\partial_{j}p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{|i|(|p|+|j|)}\big{(}\langle{\operatorname{\widetilde{\mathcal{B}}}(z_{j})p,\partial_{i}q}\rangle_{\mathcal{B}}+\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{p,z_{j}\partial_{i}q}\rangle_{\mathcal{B}}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}\big{)}.$

This gives

$\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}-(-1)^{|i||j|}\langle{z_{j}\partial_{i}p,q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{p,z_{j}\partial_{i}q}\rangle_{\mathcal{B}}-(-1)^{(|i|+|j|)|p|}\langle{p,z_{i}\partial_{j}q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}-(-1)^{|j||p|}\big{(}\langle{z_{i}\Delta
p,\partial_{j}q}\rangle_{\mathcal{B}}-\langle{\partial_{i}p,z_{j}\Delta
q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}-(-1)^{|j||p|}\big{(}\langle{z_{i}\Delta
p,\partial_{j}q}\rangle_{\mathcal{B}}-\langle{\partial_{i}p,z_{j}\Delta
q}\rangle_{\mathcal{B}}\big{)},$

from which the desired result follows if we prove

$\displaystyle 0=(-1)^{|i|(|p|+|j|)}\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}-(-1)^{|i|(|p|+|j|)}\langle{\partial_{j}p,z_{i}\Delta
q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{0=}{}-(-1)^{|j||p|}\langle{z_{i}\Delta
p,\partial_{j}q}\rangle_{\mathcal{B}}+(-1)^{|j||p|}\langle{\partial_{i}p,z_{j}\Delta
q}\rangle_{\mathcal{B}}.$ (4.1)

For the first term in right hand side of this equation we have

$\displaystyle\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}=(-1)^{|p||j|}\big{\langle}\Delta
p,\operatorname{\widetilde{\mathcal{B}}}(z_{j})\partial_{i}q\big{\rangle}_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{j}\Delta
p,\partial_{i}q}\rangle_{\mathcal{B}}}{}=(-1)^{|p||j|}\big{(}(M-2+2(l-1))\langle{\Delta
p,\partial_{j}\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\Delta
p,z_{j}\partial_{i}\Delta q}\rangle_{\mathcal{B}}\big{)},$

such that, using similar calculations for the other three terms, equation
(4.1) can be rewritten as

$\displaystyle\langle{L_{ij}\Delta p,\Delta
q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{\Delta p,L_{ij}\Delta
q}\rangle_{\mathcal{B}}.$

Since $\Delta p$ and $\Delta q$ are polynomials of degree lower than $l$ the
lemma follows from a straightforward induction argument on $l$. ∎

######  Proposition 4.7 (superhermitianity).

The Bessel–Fischer product is superhermitian, i.e.,

$\displaystyle\langle{p,q}\rangle_{\mathcal{B}}=(-1)^{|p||q|}\overline{\langle{q,p}\rangle}_{\mathcal{B}},$

for $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$.

###### Proof.

Because of the orthogonality we only need to prove the property for $p$, $q$
homogeneous polynomials of the same degree. Because of the sesquilinearity we
may assume that $p$ and $q$ are monomials. We will use induction on the degree
$k$ of the polynomials, the case $k=0$ being trivial. Suppose we have proven
the theorem for all $p,q\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$. We
now look at $\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}$ for arbitrary
$i,j\in\\{0,1,\ldots,m+2n-1\\}$. To simplify the following calculations we
will restrict ourselves to $i,j\neq 0$, the cases $i=0$ or $j=0$ being
similar. Denote $c=(M-2+2k)$. Using the commutation of equation (3.1) and
Proposition 4.5 we find

$\displaystyle\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}$
$\displaystyle=(-1)^{|i||p|}\langle{p,\operatorname{\mathcal{B}}(z_{i})z_{j}q}\rangle_{\mathcal{B}}$
$\displaystyle=(-1)^{|i||p|+|i||j|}\langle{p,z_{j}\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}\langle{p,[\operatorname{\mathcal{B}}(z_{i}),z_{j}]q}\rangle_{\mathcal{B}}$
$\displaystyle=(-1)^{|i||p|+|i||j|}\langle{p,z_{j}\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}.$

Using the induction hypothesis together with Proposition 4.5 this becomes

$\displaystyle\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}=}{}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}.$

Switching the roles of $z_{i}p$ and $z_{j}q$ we also obtain

$\displaystyle\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=(-1)^{(|i|+|j|)|q|+|i||j|}\langle{\operatorname{\mathcal{B}}(z_{i})q,\operatorname{\mathcal{B}}(z_{j})p}\rangle_{\mathcal{B}}+(-1)^{|j||q|}c\beta_{ji}\langle{q,p}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}-2(-1)^{|j||q|}\langle{q,L_{ji}p}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}}{}=(-1)^{|i||q|+|i||p|+|q||p|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|j||q|+|i||j|+|p||q|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}+2(-1)^{|i||q|+|q||p|+|i||j|}\langle{L_{ij}p,q}\rangle_{\mathcal{B}},$

where we used the induction hypothesis on all three terms of the right hand
side. If we use Lemma 4.6 on the last term and multiply both sides of this
equation with $(-1)^{(|p|+|i|)(|q|+|j|)}$ we get

$\displaystyle(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=(-1)^{|i||j|+|i||p|+|j||p|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}+(-1)^{|j||q|+|i||q|+|p||j|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}}{}=\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}},$

where we made use of $\beta_{ij}=0$ for $|i|\neq|j|$ in the last step. ∎

Lemma 4.6 can now be extended to the following proposition.

######  Corollary 4.8.

We have

$\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}\qquad\text{and}\qquad\langle{L_{0i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,L_{0i}q}\rangle_{\mathcal{B}},$

for all $i,j\in\\{1,\ldots,m+2n-1\\}$ and
$p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$.

We have the following proposition.

######  Proposition 4.9.

For all $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ it holds that

$\displaystyle\big{\langle}R^{2}p,q\big{\rangle}_{\mathcal{B}}=0=\big{\langle}p,R^{2}q\big{\rangle}_{\mathcal{B}}.$

Thus the Bessel–Fischer product is well-defined on
$\operatorname{\mathcal{F}}$.

###### Proof.

Because of the superhermitian property we only need to prove
$\big{\langle}p,R^{2}q\big{\rangle}=0$ holds for all
$p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. The complex version of
Proposition 3.1 implies that for arbitrary
$p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ there exists a
$q^{\prime}\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ such that

$\displaystyle\big{\langle}p,R^{2}q\big{\rangle}_{\mathcal{B}}=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}R^{2}q(z)|_{z=0}=R^{2}q^{\prime}(z)|_{z=0}.$

Since the constant term of $R^{2}q^{\prime}(z)$ is always zero this proves the
theorem. ∎

Note that the above proposition also shows that the Bessel–Fischer product is
degenerate on the space $\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. In the
following subsection we look at the non-degeneracy of the Bessel–Fischer
product when we restrict it to $\operatorname{\mathcal{F}}$.

###  4.2 The reproducing kernel and non-degeneracy

In the bosonic case a reproducing kernel for the Fock space was constructed in
Section 2.4 of [18]. We will prove the non-degeneracy of the Bessel–Fischer
product on $\operatorname{\mathcal{F}}$ by first constructing a generalisation
of this reproducing kernel in superspace.

######  Lemma 4.10.

Suppose $M-2\not\in-2\mathbb{N}$ and $k\in\mathbb{N}$. Define the
superfunction $\mathbb{K}^{k}(z,w)$ by

$\displaystyle\mathbb{K}^{k}(z,w):=\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$

where we used the Pochhammer symbol $(a)_{k}=a(a+1)(a+2)\cdots(a+k-1)$ and
$z|w$ is defined as

$\displaystyle z|w:=2z_{0}w_{0}+2\sum_{i,j=1}^{m+2n-1}z_{i}\beta^{ij}w_{j}.$

For all $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ we then have

$\displaystyle\big{\langle}p,\mathbb{K}^{k}(z,w)\big{\rangle}_{\mathcal{B}}=p(w)\mod\big{\langle}R^{2}_{w}\big{\rangle}.$

###### Proof.

We have

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{l})(z|w)^{k}$
$\displaystyle=(-1)^{\delta_{0l}}\left((M-2+2\mathbb{E})\partial_{l}-z_{l}\Delta\right)(z|w)^{k}$
$\displaystyle=2w_{l}k(M+2k-4)(z|w)^{k-1}-4z_{l}k(k-1)R^{2}_{w}(z|w)^{k-2}$
$\displaystyle=4w_{l}k\left(k+\dfrac{M}{2}-2\right)(z|w)^{k-1}-R^{2}_{w}4z_{l}k(k-1)(z|w)^{k-2}.$

Now suppose $p$ is a monomial, i.e.,
$p(z)=a\prod\limits_{l=0}^{m+2n-1}z_{l}^{\alpha_{l}}$, with $|\alpha|=k$ and
$a\in\mathbb{C}$. Iterating the previous calculation and working modulo
$R^{2}_{w}$ we obtain

$\displaystyle\langle{p,(z|\overline{w})^{k}}\rangle_{\mathcal{B}}$
$\displaystyle=\left.p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}(z|w)^{k}\right|_{z=0}=a\prod_{l=0}^{m+2n-1}\left.\operatorname{\widetilde{\mathcal{B}}}(z_{l})^{\alpha_{l}}(z|w)^{k}\right|_{z=0}$
$\displaystyle=4^{k}(k(k-1)\cdots
1)\left(k+\dfrac{M}{2}-2\right)\cdots\left(\dfrac{M}{2}-1\right)a\prod_{l=0}^{m+2n-1}w_{l}^{\alpha_{l}}$
$\displaystyle=4^{k}k!\left(\dfrac{M}{2}-1\right)_{k}p(w),$

which gives us the desired result. ∎

######  Theorem 4.11 (reproducing kernel of $\operatorname{\mathcal{F}}$).

Suppose $M-2\not\in-2\mathbb{N}$ and define the superfunction
$\mathbb{K}(z,w)$ by

$\displaystyle\mathbb{K}(z,w):=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2}\big{(}\sqrt{(z|\overline{w})}\big{)}=\sum_{k=0}^{\infty}\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$

where $\tilde{I}_{\alpha}$ is the I-Bessel introduced in Appendix A.1. For all
$p\in\operatorname{\mathcal{F}}$ we have

$\displaystyle\langle{p,\mathbb{K}(z,w)}\rangle_{\mathcal{B}}=p(w).$

###### Proof.

By Lemma 4.10, the orthogonality property and the Fischer decomposition we
find that

$\displaystyle\sum_{k=0}^{\infty}\mathbb{K}^{k}(z,w)=\sum_{k=0}^{\infty}\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$

has the desired property. ∎

The non-degeneracy of the Bessel–Fischer product on
$\operatorname{\mathcal{F}}$ is now an almost immediate result.

######  Proposition 4.12 (non-degenerate case).

For $M-2\not\in-2\mathbb{N}$ the Bessel–Fischer product is non-degenerate on
the polynomial Fock space $\operatorname{\mathcal{F}}$, i.e., if
$\langle{p,q}\rangle_{\mathcal{B}}=0,$ for all
$q\in\operatorname{\mathcal{F}}$, then $p=0$.

###### Proof.

Suppose $p\in\operatorname{\mathcal{F}}$ is such that
$\langle{p,q}\rangle_{\mathcal{B}}=0$ for all
$q\in\operatorname{\mathcal{F}}$. Using the reproducing kernel we obtain
$p(w)=\langle{p,\mathbb{K}(z,w)}\rangle_{\mathcal{B}}=0$. Hence $p=0$. ∎

Note that the previous proposition only works when $M-2\not\in-2\mathbb{N}$.
For the $M-2\in{-}2\mathbb{N}$ case the Bessel–Fischer product will always be
degenerate.

######  Proposition 4.13 (degenerate case).

For $M-2\in-2\mathbb{N}$ the Bessel–Fischer product is degenerate on the
polynomial Fock space $\operatorname{\mathcal{F}}$, i.e., there exists a
$q\in\operatorname{\mathcal{F}}$, $q\neq 0$ such that
$\langle{p,q}\rangle_{\mathcal{B}}=0,$ for all
$p\in\operatorname{\mathcal{F}}$.

###### Proof.

Suppose $M-2\in-2\mathbb{N}$ and
$q\in\mathcal{H}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\subset\operatorname{\widetilde{\mathcal{H}}}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\subset\operatorname{\mathcal{F}}$.
Then

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{i})q=\pm((M-2+2\mathbb{E})\partial_{i}q-z_{i}\Delta
q)=0,$

for all $i\in\\{0,1,\ldots,m+2n-1\\}$. For all
$p\in\operatorname{\mathcal{F}}$ it follows then immediately from the
definition that $\langle{p,q}\rangle_{\mathcal{B}}=0$. Since

$\displaystyle\dim\mathcal{H}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\geq\dim\mathcal{P}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m-1|2n}\big{)}\geq
1,$

we conclude that such a $q\neq 0$ exists. ∎

##  5 The Fock model of $\boldsymbol{\mathfrak{osp}(m,2|2n)}$

In [18] the Fock representation $\rho:=\pi_{\mathbb{C}}\circ c$ is obtained by
twisting the complexification of the Schrödinger representation
$\pi_{\mathbb{C}}$ with the Cayley transform $c$. This Cayley transform is an
isomorphism of $\mathfrak{g}_{\mathbb{C}}$ which induces a Lie algebra
isomorphism between $\mathfrak{k}_{\mathbb{C}}$ and
$\mathfrak{istr}(J_{\mathbb{C}})$. We will use a similar approach in our
construction of the Fock model. We start by defining a Cayley transform $c$ in
our setting.

###  5.1 Definition and properties

Let $\mathfrak{g}_{\mathbb{C}}$ and $\mathfrak{k}_{\mathbb{C}}$ be the
complexified versions of $\mathfrak{g}$ and $\mathfrak{k}$ introduced in
Section 3.3, i.e.,

$\displaystyle\mathfrak{g}_{\mathbb{C}}=\operatorname{TKK}(J_{\mathbb{C}})\cong\mathfrak{osp}_{\mathbb{C}}(m+2|2n),$
$\displaystyle\mathfrak{k}_{\mathbb{C}}=\\{(z,I,-z)\colon
I\in\operatorname{Inn}(J_{\mathbb{C}}),z\in
J_{\mathbb{C}}\\}\cong\mathfrak{osp}_{\mathbb{C}}(m|2n)\oplus\mathbb{C},$

Let $\imath$ denote the complex unit. Define the Cayley transform
$c\in\operatorname{End}(\mathfrak{g}_{\mathbb{C}})$ as

$\displaystyle
c:=\exp\left(\frac{\imath}{2}\operatorname{ad}(e_{0}^{-})\right)\exp\big{(}\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}\big{)},$

with $e_{0}^{-}$ and $e_{0}^{+}$ the copies of $e_{0}$ in $J^{-}$ and $J^{+}$,
respectively.

######  Proposition 5.1.

Using the decomposition
$\mathfrak{g}_{\mathbb{C}}=J^{-}_{\mathbb{C}}\oplus\mathfrak{istr}(J_{\mathbb{C}})\oplus
J^{+}_{\mathbb{C}}$ we obtain the following explicit expression for the Cayley
transform

  * •

$c(a,0,0)=\left(\dfrac{a}{4},\imath L_{a},a\right)$,

  * •

$c(0,L_{a}+I,0)=\left(\imath\dfrac{a}{4},I,-\imath a\right)$,

  * •

$c(0,0,a)=\left(\dfrac{a}{4},-\imath L_{a},a\right)$,

with $a\in J$ and $I\in\operatorname{Inn}(J_{\mathbb{C}})$. It induces a Lie
superalgebra isomorphism:

$\displaystyle c\colon\
\mathfrak{k}_{\mathbb{C}}\rightarrow\mathfrak{istr}(J_{\mathbb{C}}),\qquad(a,I,-a)\mapsto
I+2\imath L_{a}.$

###### Proof.

Expanding
$\exp\big{(}\frac{\imath}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\big{)}\exp\big{(}\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}\big{)}$
we obtain

$\displaystyle
c=1+\dfrac{\imath}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}-\dfrac{1}{8}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}+\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{1}{2}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{1}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}$
$\displaystyle\hphantom{c=}{}-\dfrac{\imath}{4}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{\imath}{8}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}$
$\displaystyle\hphantom{c=}{}+\dfrac{1}{16}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}.$

We have the following straightforward calculations

$\displaystyle c(a,0,0)=a^{-}+0+0+2\imath L_{a}+a^{+}-a^{-}-\imath
L_{a}+0+\dfrac{a^{-}}{4}=\left(\dfrac{a}{4},\imath L_{a},a\right),$
$\displaystyle c(0,L_{a}+I,0)=(L_{a}+I)+\imath\dfrac{a^{-}}{2}+0-\imath
a^{+}+0-L_{a}+0-\imath\dfrac{a^{-}}{4}+0=\left(\imath\dfrac{a}{4},I,-\imath
a\right),$ $\displaystyle c(0,0,a)=a^{+}-\imath
L_{a}+\dfrac{a^{-}}{4}+0+0+0+0+0+0=\left(\dfrac{a}{4},-\imath L_{a},a\right),$

proving the theorem. ∎

We complexify the Schrödinger representation $\pi$ given in Section 3.3 to
obtain a representation $\pi_{\mathbb{C}}$ of
$\mathfrak{g}_{\mathbb{C}}=J^{-}_{\mathbb{C}}\oplus\mathfrak{istr}(J_{\mathbb{C}})\oplus
J^{+}_{\mathbb{C}}$ acting on $\operatorname{\mathcal{F}}$. Explicitly,
$\pi_{\mathbb{C}}$ is given by

  * •

$\pi_{\mathbb{C}}(e_{l},0,0)=-2\imath z_{l}$,

  * •

$\pi_{\mathbb{C}}(0,L_{e_{k}},0)=-z_{0}\partial_{k}+z_{k}\partial_{0}$,

  * •

$\pi_{\mathbb{C}}(0,[L_{e_{i}},L_{e_{j}}],0)=z_{i}\partial_{j}-(-1)^{|i||j|}z_{j}\partial_{i}$,

  * •

$\pi_{\mathbb{C}}(0,L_{e_{0}},0)=\frac{2-M}{2}-\mathbb{E}$,

  * •

$\pi_{\mathbb{C}}(0,0,e_{l})=-\frac{1}{2}\imath\operatorname{\widetilde{\mathcal{B}}}(z_{l})$,

with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$ and $l\in\\{0,1,2,\ldots,m+2n-1\\}$.
Since $L_{ij}$, $\mathbb{E}$ and
$\operatorname{\widetilde{\mathcal{B}}}(z_{l})$ map
$\big{\langle}R^{2}\big{\rangle}$ into $\big{\langle}R^{2}\big{\rangle}$ this
representation is well defined on $\operatorname{\mathcal{F}}$. As in the
bosonic case, we will define the Fock representation $\rho$ as the composition
of $\pi_{\mathbb{C}}$ with the Cayley transform $c$,

$\displaystyle\rho:=\pi_{\mathbb{C}}\circ c.$

So $\rho$ of $\mathfrak{g}=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$ acting
on $\operatorname{\mathcal{F}}$ is given as follows

  * •

$\rho(e_{0},0,0)=-\dfrac{\imath}{2}\big{(}z_{0}+\operatorname{\widetilde{\mathcal{B}}}(z_{0})+M-2+2\mathbb{E}\big{)}$,

  * •

$\rho(e_{k},0,0)=-\dfrac{\imath}{2}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})+2(z_{0}\partial_{k}-z_{k}\partial_{0})\big{)}$,

  * •

$\rho(0,L_{e_{0}},0)=\dfrac{1}{2}\big{(}z_{0}-\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$,

  * •

$\rho(0,L_{e_{k}},0)=\dfrac{1}{2}\big{(}z_{k}-\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}$,

  * •

$\rho(0,[L_{e_{i}},L_{e_{j}}],0)=z_{i}\partial_{j}-(-1)^{|i||j|}z_{j}\partial_{i}$,

  * •

$\rho(0,0,e_{k})=-\dfrac{\imath}{2}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})-2(z_{0}\partial_{k}-z_{k}\partial_{0})\big{)}$,

  * •

$\rho(0,0,e_{0})=-\dfrac{\imath}{2}\big{(}z_{0}+\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2-M-2\mathbb{E}\big{)}$,

with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$.

######  Proposition 5.2 ($\mathfrak{osp}(m,2|2n)$-invariance).

The Fock representation $\rho$ on $\operatorname{\mathcal{F}}$ is skew-
supersymmetric with respect to the Bessel–Fischer product, i.e.,

$\displaystyle\langle{\rho(X)p,q}\rangle_{\mathcal{B}}=-(-1)^{|X||p|}\langle{p,\rho(X)q}\rangle_{\mathcal{B}},$

for all $X\in\mathfrak{g}$ and $p,q\in\operatorname{\mathcal{F}}$.

###### Proof.

Suppose $p,q\in\operatorname{\mathcal{F}}$. From Corollary 4.8 we obtain

$\displaystyle\langle{\rho(e_{i},0,-e_{i})p,q}\rangle_{\mathcal{B}}=\langle{-2\imath
L_{0i}p,q}\rangle_{\mathcal{B}}=-(-1)^{|i||p|}\langle{p,2\overline{\imath}L_{0i}q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{\rho(e_{i},0,-e_{i})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|i||p|}\langle{p,\rho(e_{i},0,-e_{i})q}\rangle_{\mathcal{B}}$

and

$\displaystyle\langle{\rho(0,[L_{e_{i}},L_{e_{j}}],0)p,q}\rangle_{\mathcal{B}}=\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{\rho(0,[L_{e_{i}},L_{e_{j}}],0)p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{(|i|+|j|)|p|}\langle{p,\rho(0,[L_{e_{i}},L_{e_{j}}],0)q}\rangle_{\mathcal{B}},$

for $i,j\in\\{1,\ldots,m+2n-1\\}$. Furthermore, we have

$\displaystyle\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}=\big{\langle}{-}\imath\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}p,q\big{\rangle}_{\mathcal{B}}=-\imath\big{(}\langle{z_{k}p,q}\rangle_{\mathcal{B}}+\big{\langle}\operatorname{\widetilde{\mathcal{B}}}(z_{k})p,q\big{\rangle}_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-\imath(-1)^{|k||p|}\big{(}\big{\langle}p,\operatorname{\widetilde{\mathcal{B}}}(z_{k})q\big{\rangle}_{\mathcal{B}}+\langle{p,z_{k}q}\rangle_{\mathcal{B}}\big{)}$
$\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|k||p|}\big{\langle}p,\overline{\imath}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}q\big{\rangle}_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|k||p|}\langle{p,\rho(e_{k},0,e_{k})q}\rangle_{\mathcal{B}}$

and similarly

$\displaystyle\langle{\rho(0,L_{e_{k}},0)p,q}\rangle_{\mathcal{B}}=\left\langle\dfrac{1}{2}\big{(}z_{k}-\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}p,q\right\rangle_{\mathcal{B}}=-(-1)^{|k||p|}\langle{p,\rho(0,L_{e_{k}},0)q}\rangle_{\mathcal{B}},$

for $k\in\\{0,\ldots,m+2n-1\\}$. Because of Proposition 4.4 we may assume $p$
and $q$ are homogeneous polynomials the same degree. We now have

$\displaystyle\langle{\rho(e_{0},0,-e_{0})p,q}\rangle_{\mathcal{B}}$
$\displaystyle=\langle{-2\imath(M-2+2\mathbb{E})p,q}\rangle_{\mathcal{B}}=-\langle{p,-2\imath(M-2+2\mathbb{E})q}\rangle_{\mathcal{B}}$
$\displaystyle=-\langle{p,\rho(e_{0},0,-e_{0})q}\rangle_{\mathcal{B}},$

which proves the theorem. ∎

###  5.2 The $\boldsymbol{(\mathfrak{g},\mathfrak{k})}$-module
$\boldsymbol{F}$

We define

$\displaystyle F:=U(\mathfrak{g})1\mod\big{\langle}R^{2}\big{\rangle},$

where the $\mathfrak{g}$ module structure is given by the Fock representation
$\rho$. We also introduce the notation

$\displaystyle
F_{k}:=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\mod\big{\langle}R^{2}\big{\rangle}$

and reintroduce $r^{2}$ from Section 3.2 as its complexified version:

$\displaystyle r^{2}:=\sum_{i=1}^{m+2n-1}z^{i}z_{i}.$

We have $R^{2}=-z_{0}^{2}+r^{2}$ and since we are working modulo
$\big{\langle}R^{2}\big{\rangle}$ this implies $z_{0}^{2}=r^{2}$. In the
following, we will work modulo $\big{\langle}R^{2}\big{\rangle}$ but omit
$\big{\langle}R^{2}\big{\rangle}$ from our notation. For
$M-1\not\in-2\mathbb{N}$ we now find

$\displaystyle F_{k}$
$\displaystyle=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}=\bigoplus_{l=0}^{\big{\lfloor}\frac{k}{2}\big{\rfloor}}z_{0}^{2l}\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\oplus\bigoplus_{l=0}^{\big{\lfloor}\frac{k+1}{2}\big{\rfloor}-1}z_{0}^{2l+1}\mathcal{H}_{k-2l-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}$
$\displaystyle=\bigoplus_{l=0}^{\big{\lfloor}\frac{k}{2}\big{\rfloor}}r^{2l}\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\oplus\bigoplus_{l=0}^{\big{\lfloor}\frac{k+1}{2}\big{\rfloor}-1}z_{0}r^{2l}\mathcal{H}_{k-2l-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}$
$\displaystyle\cong\mathcal{P}_{k}(\mathbb{C}^{m-1|2n})\oplus
z_{0}\mathcal{P}_{k-1}\big{(}\mathbb{C}^{m-1|2n}\big{)},$

where we made use of the Fischer decomposition (Theorem 2.4) in the last step.
In particular, by Proposition 2.8,

$\displaystyle\dim
F_{k}=\dim\mathcal{P}_{k}\big{(}\mathbb{C}^{m-1|2n}\big{)}+\dim\mathcal{P}_{k-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}=\dim\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}.$

If $M\not\in-2\mathbb{N}$, then for $p\in F_{k}$ the Fischer decomposition on
$\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ also gives

$\displaystyle
p=\sum_{l=0}^{\left\lfloor\frac{k}{2}\right\rfloor}R^{2l}h_{k-2l}=h_{k}\mod\big{\langle}R^{2}\big{\rangle},$

with $h_{k-2l}\in\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m|2n}\big{)}$. This
implies $F_{k}\cong\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ for $M\geq
2$.

######  Theorem 5.3 (decomposition of $F$).

For $M\geq 3$, we have the following:

  * $(1)$

$F_{k}$ is an irreducible $\mathfrak{k}$-module.

  * $(2)$

$F$ is an irreducible $\mathfrak{g}$-module and its $\mathfrak{k}$-type
decomposition is given by

$\displaystyle F=\bigoplus_{k=0}^{\infty}F_{k}.$

  * $(3)$

An explicit decomposition of $F_{k}$ into irreducible
$\mathfrak{k}_{0}$-modules is given by

$\displaystyle
F_{k}=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\mod\big{\langle}R^{2}\big{\rangle}.$

###### Proof.

We have the following elements of the action of $\mathfrak{g}$:

$\displaystyle\rho^{+}_{0}:=\rho\left(c^{-1}\left(\frac{-e_{0}}{2},0,0\right)\right)=\imath
z_{0},$
$\displaystyle\rho^{-}_{0}:=\rho\big{(}c^{-1}(0,0,-2e_{0})\big{)}=\imath\operatorname{\widetilde{\mathcal{B}}}(z_{0}),$
$\displaystyle\rho_{0i}:=\rho\left(-\frac{e_{i}}{2},0,\frac{e_{i}}{2}\right)=\imath
L_{0i},$ $\displaystyle\rho_{ij}:=\rho(0,[L_{e_{i}},L_{e_{j}}],0)=L_{ij},$

with $i,j\in\\{1,\ldots,m+2n-1\\}$. The elements $\rho_{ij}$ for
$i,j\in\\{1,\ldots,m+2n-1\\}$, $i\leq j$ give rise to an irreducible
representation of $\mathfrak{k}_{0}$ on
$\mathcal{H}_{k}\big{(}\mathbb{C}^{m-1|2n}\big{)}$ as a result of Proposition
2.6. These elements leave $r^{2}=z_{0}^{2}$ invariant and therefore also leave
powers of $z_{0}$ invariant, which proves $(3)$.

Again by Proposition 2.6 the elements $\rho_{ij}$ and $\rho_{0i}$ give rise to
an irreducible representation of $\mathfrak{k}$ on
$\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}\cong F_{k}$, which proves
$(1)$. For the first two elements we have

$\displaystyle\rho^{+}_{0}\big{(}z_{0}^{k}\big{)}=\imath z_{0}^{k+1},$
$\displaystyle\rho^{-}_{0}\big{(}z_{0}^{k}\big{)}=\imath\operatorname{\widetilde{\mathcal{B}}}(z_{0})z_{0}^{k}=\imath
k(M+2k-4)z_{0}^{k-1}-\imath k(k-1)z_{0}^{k-1}=\imath k(M+k-3)z_{0}^{k-1},$

which shows that $\rho^{+}_{0}$ allows us to go to polynomials of higher
degrees while $\rho^{-}_{0}$ allows us to go the other direction for $M\geq
3$. Therefore we obtain $(2)$. ∎

The following isomorphism is a direct result of this theorem.

######  Corollary 5.4.

Suppose $M\geq 3$ and let

$\displaystyle\operatorname{\mathcal{F}}=\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}/\big{\langle}R^{2}\big{\rangle}$

be the polynomial Fock space defined in Definition 4.1. We have
$\operatorname{\mathcal{F}}\cong F$.

##  6 The Segal–Bargmann transform

In this section we construct the Segal–Bargmann transform and show that it is
an isomorphism from $W$ as defined in Section 3.3 to $F$ as defined in Section
5.2. It will make use of the integral $\int_{W}$ we defined in Definition 3.7.
This integral is only defined for $M\geq 4$. Therefore we will always assume
$M\geq 4$ throughout this section.

###  6.1 Definition and properties

Let $\widetilde{I}_{\alpha}(t)$ be the I-Bessel function as introduced in
Appendix A.1. We define an entire function
$\operatorname{\mathbb{I}}_{\alpha}$ on $\mathbb{C}$ by

$\displaystyle\operatorname{\mathbb{I}}_{\alpha}(t):=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2+\alpha}\big{(}2\sqrt{t}\big{)}=\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\Gamma\big{(}l+\frac{M}{2}-1+\alpha\big{)}}t^{l}.$

Clearly we have $\operatorname{\mathbb{I}}_{0}(0)=1$ and

$\displaystyle\partial_{t}^{j}\operatorname{\mathbb{I}}_{0}(t)=\Gamma\left(\dfrac{M}{2}-1\right)\partial_{t}^{j}\left(\widetilde{I}_{\frac{M}{2}-2}\big{(}2\sqrt{t}\big{)}\right)=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2+j}\big{(}2\sqrt{t}\big{)}=\operatorname{\mathbb{I}}_{j}(t).$

We are now able to state the Segal–Bargmann transform that extends the one
from the bosonic case obtained in [18].

######  Definition 6.1.

For $f\in W$ the Segal–Bargmann transform is defined as

$\displaystyle\operatorname{SB}f(z):=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x),$

where $x|z$ is defined as

$\displaystyle x|z:=2x_{0}z_{0}+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}z_{j}$

and we view $\operatorname{\mathbb{I}}_{\alpha}(x|z)$ as a radial
superfunction in the sense of equation (3.2).

Note that $\operatorname{\mathbb{I}}_{0}(4(x|z))$ is the reproducing kernel
$\mathbb{K}(x,z)$ of the Fock space we found in Theorem 4.11.

######  Proposition 6.2.

For $M\geq 4$ the Segal–Bargmann transform $\operatorname{SB}$ is well
defined.

###### Proof.

We wish to prove that the integral

$\displaystyle\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x)$

is convergent for all $f\in W$ and that $\operatorname{SB}(R^{2})=0$. As shown
in the proof of Theorem 8.13 in [6], the elements of $W$ can be decomposed
into elements of the form

$\displaystyle P_{k}\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(2|X|),$

with $P_{k}$ a homogeneous polynomial of degree $k$. Here
$\widetilde{K}_{\alpha}$ is the K-Bessel function introduced in Appendix A.1
interpreted as a radial superfunction as in Section 3.2. Furthermore
$\alpha_{1},\alpha_{2}\in\mathbb{N}$ are subject to the relations
$k\geq\alpha_{1}+2\alpha_{2}$ and $M\geq 2\alpha_{1}+2$. Also, observe that

$\displaystyle|X|=\sqrt{\dfrac{x_{0}^{2}+r^{2}}{2}}=\sqrt{x_{0}^{2}+\dfrac{R^{2}}{2}}=\sqrt{x_{0}^{2}}\quad\mod\big{\langle}R^{2}\big{\rangle},$
(6.1)

which is equal to $x_{0}$ within the domain of integration of $\int_{W}$.
Because of all this and equation (A.1) it suffices to prove

$\displaystyle\int_{W}P_{k}\operatorname{\mathbb{I}}_{0}(x|z)\widetilde{K}_{-\frac{1}{2}}(2|X|)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(2|X|)$

is convergent for all $k\in\mathbb{N}$ and
$\alpha_{1},\alpha_{2}\in\mathbb{N}$ subject to the above mentioned relations.
We will use the explicit description of $\int_{W}$ given in equation (3.3).
The morphism $\phi^{\sharp}$ leaves the degree of a polynomial unchanged.
Hence, we can expand

$\displaystyle\big{(}\phi^{\sharp}(P_{k})\big{)}_{|s=x_{0}=\rho}=\sum_{j=0}^{k}\rho^{k-j}a_{j}(\theta)b_{j}(\omega),$

where $a_{j}(\theta)$ is a polynomial in
$\mathcal{P}\big{(}\mathbb{R}^{0|2n}\big{)}$ of degree $j$ and $b_{j}(\omega)$
is a function depending on the spherical coordinates $\omega$. For
$c\in\mathbb{Z}$ we obtain

$\displaystyle(1+\eta)^{c}=\sum_{j=0}^{n}\dfrac{1}{j!}\left(\dfrac{-c}{2}\right)_{j}\dfrac{\theta^{2j}}{2^{j}s^{2j}},\qquad(1+\xi)^{c}=\sum_{j=0}^{n}\dfrac{(-1)^{j}}{j!}\left(\dfrac{-c}{2}\right)_{j}\dfrac{\theta^{2j}}{2^{j}x_{0}^{2j}}$

and

$\displaystyle\phi^{\sharp}\big{(}\widetilde{K}_{\alpha}(c|X|)\big{)}=\widetilde{K}_{\alpha}(c|X|)=\sum_{j=0}^{n}\dfrac{(-1)^{j}c^{2j}\theta^{2j}}{j!8^{j}}\widetilde{K}_{\alpha+j}(c\rho),$

from the proof of Lemma 8.6 in [6]. Here $\eta$, $\xi$ and $\theta^{2}$ were
defined in Section 3.4. We introduce the notations

$\displaystyle\omega|z:=2\sum_{i=1}^{m-1}\omega_{i}z_{i}\qquad\mbox{and}\qquad\theta|z:=2\sum_{i,j=m}^{m+2n-1}x_{i}\beta^{ij}z_{j}.$

We can use equation (3.2) with $h=\operatorname{\mathbb{I}}_{0}$ and $f=x|z$
as a function in the $x$ variables to obtain

$\displaystyle\operatorname{\mathbb{I}}_{0}(x|z)=\operatorname{\mathbb{I}}_{0}(2x_{0}z_{0}+s\omega|z+\theta|z)=\sum_{j=0}^{2n}\dfrac{1}{j!}(\theta|z)^{j}\operatorname{\mathbb{I}}_{j}(2x_{0}z_{0}+s\omega|z).$

Using the properties of $\phi^{\sharp}$ described in Lemma 8.3 of [6] and the
expansion of $(1+\xi)$ and $(1+\eta)$, we now find

$\displaystyle\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=\sum_{l_{1}=0}^{2n}\dfrac{1}{l_{1}!}(\theta|z)^{l_{1}}\operatorname{\mathbb{I}}_{l_{1}}(2\rho(1+\xi)z_{0}+\rho(1+\eta)\omega|z)_{|s=x_{0}=\rho}$
$\displaystyle\hphantom{\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}}{}=\sum_{l_{1}=0}^{2n}\dfrac{1}{l_{1}!}(\theta|z)^{l_{1}}\operatorname{\mathbb{I}}_{l_{1}}\left(\rho\sum_{l_{2}=0}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}(2(-1)^{l_{2}}z_{0}+\omega|z)\right).$

We use equation (3.2) again, this time with
$h=\operatorname{\mathbb{I}}_{l_{1}}$ and $f$ equal to the sum over $l_{2}$.
Note that the $l_{2}=0$ term corresponds with $f_{0}$. We obtain

$\displaystyle\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=\sum_{l_{1},l_{3}=0}^{2n}\dfrac{1}{l_{1}!l_{3}!}(\theta|z)^{l_{1}}\left(\rho\sum_{l_{2}=1}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}(2(-1)^{l_{2}}z_{0}+\omega|z)\right)^{l_{3}}$
$\displaystyle\hphantom{\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=}{}\times\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(\rho(2z_{0}+\omega|z)).$

Combining all this, we see that

$\displaystyle\dfrac{1}{\gamma}\int_{0}^{\infty}\int_{\mathbb{S}^{m-2}}\int_{B}\rho^{m-3}(1+\eta)^{m-3}(1+\xi)^{-1}$
$\displaystyle\qquad{}\times\phi^{\sharp}(P_{k}\operatorname{\mathbb{I}}_{0}(x|z)\widetilde{K}_{-\frac{1}{2}}(2|X|)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(|X|))_{|s=x_{0}=\rho}\mathrm{d}\rho\mathrm{d}\omega,$

converges if

$\displaystyle\int_{0}^{\infty}\int_{B}\rho^{m-3}\sum_{j_{1}=0}^{k}\sum_{j_{2},j_{3},j_{4},j_{5}=0}^{n}\sum_{l_{1},l_{3}=0}^{2n}\dfrac{1}{j_{2}!}\left(\dfrac{3-m}{2}\right)_{j_{2}}\dfrac{\theta^{2j_{2}}}{2^{j_{2}}\rho^{2{j_{2}}}}\dfrac{(-1)^{j_{3}}}{j_{3}!}\left(\dfrac{1}{2}\right)_{j_{3}}\dfrac{\theta^{2{j_{3}}}}{2^{j_{3}}\rho^{2{j_{3}}}}$
$\displaystyle\qquad{}\times\rho^{k-j_{1}}a_{j_{1}}(\theta)\dfrac{1}{l_{1}!l_{3}!}(\theta|z)^{l_{1}}\left(\rho\sum_{l_{2}=1}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}c_{1}\right)^{l_{3}}\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(c_{2}\rho)$
$\displaystyle\qquad{}\times\dfrac{(-1)^{j_{4}}\theta^{2{j_{4}}}}{{j_{4}}!2^{j_{4}}}\widetilde{K}_{-\frac{1}{2}+{j_{4}}}(2\rho)\dfrac{(-1)^{j_{5}}\theta^{2{j_{5}}}}{{j_{5}}!2^{j_{5}}}\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5}}(2\rho)\mathrm{d}\rho$

converges for all $c_{1},c_{2}\in\mathbb{C}$. This in turn converges if

$\displaystyle\int_{0}^{\infty}\int_{B}\rho^{m-3+k-j_{1}+l_{3}-2l_{3}l_{2}-2j_{2}-2j_{3}}a_{j_{1}}(\theta)(\theta|z)^{l_{1}}\theta^{2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}}\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(c\rho)$
$\displaystyle\qquad\times\widetilde{K}_{-\frac{1}{2}+{j_{4}}}(2\rho)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5}}(2\rho)\mathrm{d}\rho$

converges for all $0\leq j_{1}\leq k$, $0\leq j_{2},j_{3},j_{4},j_{5}\leq n$,
$0\leq l_{1},l_{3}\leq 2n$, $1\leq l_{2}\leq n$ and all $c\in\mathbb{C}$. The
Berezin integral is zero unless
$j_{1}+l_{1}+2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}=2n$. The integral

$\displaystyle\int_{0}^{\infty}\rho^{\sigma-1}\widetilde{I}_{\beta_{1}}\big{(}\sqrt{a\rho}\big{)}\widetilde{K}_{\beta_{2}}(2\rho)\widetilde{K}_{\beta_{3}}(2\rho)\mathrm{d}\rho,$

with $\beta_{1}\geq 0$ converges if
$\sigma>2\max\\{\beta_{2},0\\}+2\max\\{\beta_{3},0\\}$. This follows from the
asymptotic behaviour of the Bessel functions, see Appendix A.1. Therefore we
get the following condition

$\displaystyle m-2+k-j_{1}+l_{3}-2l_{3}l_{2}-2j_{2}-2j_{3}$
$\displaystyle\qquad{}>2\max\left\\{-\frac{1}{2}+{j_{4}},0\right\\}+2\max\left\\{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5},0\right\\},$

with $j_{1}+l_{1}+2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}=2n$. Taking into
account $k\geq\alpha_{1}+2\alpha_{2}$ and $M\geq 2\alpha_{1}+2$ the condition
reduces to $M>2$. We still need that $\operatorname{SB}\big{(}R^{2}\big{)}=0$,
but this follows easily from
$\big{(}\phi^{\sharp}\big{(}R^{2}\big{)}\big{)}_{|s=x_{0}=\rho}=\big{(}{-}x_{0}^{2}+s^{2}\big{)}_{|s=x_{0}=\rho}=0$.
∎

We can now show that $\operatorname{SB}$ intertwines the Schrödinger model
with the Fock model.

######  Theorem 6.3 (intertwining property).

For $M\geq 4$ the Segal–Bargmann transform intertwines the action $\pi$ on $W$
with the action $\rho$ on $F$, i.e.,

$\displaystyle\operatorname{SB}\circ\pi(X)=\rho(X)\circ\operatorname{SB},$

for all $X\in\mathfrak{g}$.

###### Proof.

The proof of this theorem is a technical and long but rather straightforward
calculation. We refer to Appendix B for more details. ∎

######  Proposition 6.4.

The Segal–Bargmann transform $\operatorname{SB}$ induces a
$\mathfrak{g}$-module isomorphism between $W$ and $F$.

###### Proof.

From the way we normalized the integral $\int_{W}$ and equation (6.1) it is
clear that

$\displaystyle\operatorname{SB}(\exp(-2|X|))(0)=\int_{W}\exp(-4|X|)=1.$

Therefore the Segal–Bargmann transform maps a non-zero element of $W$ to a
non-zero element of $F$. It also intertwines the actions $\pi$ and $\rho$.
Since Theorems 3.6 and 5.3 give us that $W$ and $F$ are irreducible
$\mathfrak{g}$-modules, we conclude that $\operatorname{SB}$ is an isomorphism
of $\mathfrak{g}$-modules. ∎

######  Lemma 6.5.

We have $\operatorname{SB}(\exp(-2|X|)(z)=1$.

###### Proof.

Since $\exp(-2|X|)$ is in $W_{0}$ Proposition 6.4 implies that
$\operatorname{SB}(\exp(-2|X|))(z)$ is in $F_{0}=\mathbb{C}$. Hence
$\operatorname{SB}(\exp(-2|X|))(z)$ is a constant. From the way we normalized
the integral $\int_{W}$ we have $\operatorname{SB}(\exp(-2|X|))(0)=1$ and
therefore $\operatorname{SB}(\exp(-2|X|))(z)=1$. ∎

######  Theorem 6.6 (unitary property).

For $M\geq 4$ the Segal–Bargmann transform preserves the sesquilinear forms,
i.e.,

$\displaystyle\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{f,g}\rangle_{W},$

for all $f,g\in W$.

###### Proof.

We first look at the case $f=\exp(-2|X|)$. Because of Lemma 6.5 and the
superhermitian property of the Bessel–Fischer product we have

$\displaystyle\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{1,\operatorname{SB}g}\rangle_{\mathcal{B}}=\overline{\langle{\operatorname{SB}g,1}\rangle}_{\mathcal{B}}=\operatorname{SB}(\overline{g(x)})\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}1\big{|}_{z=0}$
$\displaystyle\hphantom{\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=\int_{W}\exp(-2x_{0})\big{(}\operatorname{\mathbb{I}}_{0}(x|\operatorname{\widetilde{\mathcal{B}}}(z))\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}1\big{)}\overline{g(x)}\big{|}_{z=0},$

for all $g\in W$. Here
$\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}$
and $\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$
should be considered as infinite power sums of the Bessel operator with

$\displaystyle
x|\operatorname{\widetilde{\mathcal{B}}}(z):=2x_{0}\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\widetilde{\mathcal{B}}}(z_{j})=2\sum_{i,j=0}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\mathcal{B}}(z_{j}).$

Since they act on a constant with respect to the variable $z$ we get

$\displaystyle\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}1=\operatorname{\mathbb{I}}_{0}(0)\exp(0)1=1,$

which gives

$\displaystyle\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\int_{W}\exp(-2x_{0})\overline{g(x)}=\langle{\exp(-2|X|),g}\rangle_{W},$

if we use equation (6.1). Now suppose $f,g\in W$. Since $W$ is an irreducible
$\mathfrak{g}$-module (Theorem 3.6), there exists a $Y\in U(\mathfrak{g})$
such that $f=\pi(Y)\exp(-2|X|)$. Therefore we can reduce the general case to
the previous case using the intertwining property (Theorem 6.3) and the fact
that the sesquilinear forms are skew symmetric for $\pi$ and $\rho$
(Propositions 3.10 and 5.2):

$\displaystyle\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{\operatorname{SB}(\pi(Y)\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{\rho(Y)\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=-\langle{\operatorname{SB}(\exp(-2|X|)),\rho(Y)\operatorname{SB}g}\rangle_{\mathcal{B}}=-\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}(\pi(Y)g)}\rangle_{\mathcal{B}}$
$\displaystyle\hphantom{\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=-\langle{\exp(-2|X|),\pi(Y)g}\rangle_{W}=\langle{\pi(Y)\exp(-2|X|),g}\rangle_{W}=\langle{f,g}\rangle_{W},$

which proves the theorem. ∎

###  6.2 The inverse Segal–Bargmann transform

######  Definition 6.7.

For $p\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ the inverse
Segal–Bargmann transform is defined as

$\displaystyle\operatorname{SB}^{-1}p(x):=\exp(-2|X|)\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}p(z)\big{|}_{z=0},$

with

$\displaystyle
x|\operatorname{\widetilde{\mathcal{B}}}(z):=2x_{0}\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\widetilde{\mathcal{B}}}(z_{j})=2\sum_{i,j=0}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\mathcal{B}}(z_{j}).$

Note that both
$\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}$
and $\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$ are
infinite power sums of the Bessel operator. However, they are well defined
operators on polynomials in $z$ since the Bessel operator lowers the degree of
the polynomials and therefore the power operators become zero after a finite
number of terms. Thus $\operatorname{SB}^{-1}$ is a well-defined operator on
$\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. Because of Proposition 3.1 it
maps $\big{\langle}R^{2}\big{\rangle}$ to zero and thus
$\operatorname{SB}^{-1}$ can be restricted to $F$. Moreover, it is also well
defined as the inverse of the Segal–Bargmann transform. This follows from the
following proposition.

######  Proposition 6.8.

The inverse Segal–Bargmann transform is well defined as the inverse of the
Segal–Bargmann transform defined in Definition 6.1.

###### Proof.

From Proposition 6.4, we know that $\operatorname{SB}$ has an inverse. Suppose
the operator $A$ is this inverse. Using Theorem 6.6 we then have the following
calculation:

$\displaystyle\langle{Ap,\psi}\rangle_{W}$
$\displaystyle=\langle{\operatorname{SB}(Ap),\operatorname{SB}\psi}\rangle_{\mathcal{B}}=(-1)^{|p||\psi|}\overline{\langle{\operatorname{SB}\psi,p}\rangle}_{\mathcal{B}}=(-1)^{|p||\psi|}\overline{\operatorname{SB}\psi}\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}p(z)\big{|}_{z=0}$
$\displaystyle=(-1)^{|p||\psi|}\left.\left(\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\int_{W}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2x_{0})\overline{\psi(x)}p(z)\right)\right|_{z=0}$
$\displaystyle=\int_{W}\big{(}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2x_{0})p(z)\big{)}\big{|}_{z=0}\overline{\psi(x)}$
$\displaystyle=\big{\langle}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2|X|)p(z)\big{|}_{z=0},\psi\big{\rangle}_{W},$

where we used equation (6.1) in the last step. Since the sesquilinear form
$\langle{\cdot\,,\cdot}\rangle_{W}$ is non-degenerate we obtain
$A=\operatorname{SB}^{-1}$. ∎

We can make the inverse Segal–Bargmann transform more explicit on the space of
homogeneous polynomials. To the best of our knowledge, this explicit
expression is also new for the bosonic case.

######  Proposition 6.9.

For $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$, $k\in\mathbb{N}$ the
inverse Segal–Bargmann transform $\operatorname{SB}^{-1}$ is given by

$\displaystyle\operatorname{SB}^{-1}p(x)=\exp(-2|X|)\sum_{j=0}^{k}\dfrac{(-1)^{j}}{j!(k-j)!}\dfrac{\Gamma\big{(}\frac{M}{2}-1\big{)}}{\Gamma\big{(}k-j+\frac{M}{2}-1\big{)}}\big{\langle}z_{0}^{j}(x|z)^{k-j},\overline{p}\big{\rangle}.$

###### Proof.

For $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ we have

$\displaystyle\operatorname{SB}^{-1}p(x)$
$\displaystyle=\exp(-2|X|)\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}p(z)\big{|}_{z=0}$
$\displaystyle=\exp(-2|X|)\langle{\operatorname{\mathbb{I}}_{0}(x|z)\exp(-z_{0}),\overline{p}}\rangle_{\mathcal{B}}.$

Because of the orthogonality of the Bessel–Fischer product we only need to
look at the homogeneous polynomial term of degree $k$ in the expansion of
$\operatorname{\mathbb{I}}_{0}(x|z)\exp(-z_{0})$. Explicitly, we need the
homogeneous term of degree $k$ in

$\displaystyle\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\Gamma\big{(}l+\frac{M}{2}-1\big{)}}(x|z)^{l}\sum_{j=0}^{\infty}\dfrac{(-1)^{j}}{j!}z_{0}^{j},$

which is

$\displaystyle\Gamma\left(\frac{M}{2}-1\right)\sum_{j=0}^{k}\dfrac{1}{(k-j)!\Gamma\big{(}k-j+\frac{M}{2}-1\big{)}}(x|z)^{k-j}\dfrac{(-1)^{j}}{j!}z_{0}^{j}$

as desired. ∎

###  6.3 The generalized Hermite functions

As a standard application of the Segal–Bargmann transform, we can construct
generalized Hermite functions which extend the ones of the bosonic case given
in [18].

######  Definition 6.10.

The generalized Hermite functions on $W$ are defined by

$\displaystyle
h_{\alpha}(x):=\exp(2|X|)\left(\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}\right)^{\alpha}\exp(-4|X|),\qquad\text{with}\quad\left(\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}\right)^{\alpha}:=\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(e_{i})^{\alpha_{i}},$

for $\alpha\in\mathbb{N}^{m|2n}$. The generalized Hermite polynomials
$H_{\alpha}$ are defined by the equation

$\displaystyle h_{\alpha}(x)=H_{\alpha}(x)\exp(-2|X|).$

######  Proposition 6.11 (Hermite to monomial property).

We have

$\displaystyle\operatorname{SB}h_{\alpha}=(2z)^{\alpha}.$

###### Proof.

We will use the fact that $\int_{W}$ is supersymmetric with respect to the
Bessel operators [6, Proposition 8.9]:

$\displaystyle\int_{W}(\operatorname{\mathcal{B}}(x_{k})f)g=(-1)^{|f||k|}\int_{W}f(\operatorname{\mathcal{B}}(x_{k})g).$

Combining this with equation (B.3) of Lemma B.1 we find

$\displaystyle\operatorname{SB}h_{\alpha}(z)$
$\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})h_{\alpha}(x)$
$\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(x_{i})^{\alpha_{i}}\exp(-4|X|)$
$\displaystyle=\exp(-z_{0})\int_{W}\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(x_{i})^{\alpha_{i}}\left(\operatorname{\mathbb{I}}_{0}(x|z)\right)\exp(-4|X|)$
$\displaystyle=\exp(-z_{0})\int_{W}\prod_{i}\dfrac{1}{2^{\alpha_{i}}}(4z_{i})^{\alpha_{i}}\left(\operatorname{\mathbb{I}}_{0}(x|z)\right)\exp(-4|X|)$
$\displaystyle=\exp(-z_{0})\int_{W}(2z)^{\alpha}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-4|X|)$
$\displaystyle=(2z)^{\alpha}\operatorname{SB}(\exp(-2|X|))(z).$

Because of Lemma 6.5 the theorem follows. ∎

##  Appendix A Special functions

###  A.1 Bessel functions

The I-Bessel function $I_{\alpha}(t)$ (or modified Bessel function of the
first kind) is defined by

$\displaystyle
I_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{\alpha}\sum_{k=0}^{\infty}\dfrac{1}{k!\Gamma(k+\alpha+1)}\left(\dfrac{t}{2}\right)^{2k}$

and the K-Bessel function $K_{\alpha}$ (or modified Bessel function of the
third kind) by

$\displaystyle
K_{\alpha}(t):=\dfrac{\pi}{2\sin(\pi\alpha)}(I_{-\alpha}(t)-I_{\alpha}(t)),$

for $\alpha,t\in\mathbb{C}$, see [2, Section 4.12]. In this paper we will need
the following renormalisations

$\displaystyle\widetilde{I}_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{-\alpha}I_{\alpha}(t),\qquad\widetilde{K}_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{-\alpha}K_{\alpha}(t).$

Remark that we have the following special case [2, equation (4.12.4)]

$\displaystyle\widetilde{K}_{-\frac{1}{2}}(t)=\frac{\sqrt{\pi}}{2}\exp(-t).$
(A.1)

The asymptotic behaviour of the I-Bessel function can be deducted from
equations (4.12.7) and (4.12.8) of [2],

$\displaystyle\widetilde{I}_{\alpha}(t)=\dfrac{1}{\sqrt{2\pi}}\big{(}{-}t^{2}\big{)}^{-\frac{2\alpha+1}{4}}\left(\exp\left(-i\left(\dfrac{(2\alpha+1)\pi}{4}-\sqrt{-t^{2}}\right)\right)\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right)\right.$
$\displaystyle\left.\hphantom{\widetilde{I}_{\alpha}(t)=}{}+\exp\left(i\left(\dfrac{(2\alpha+1)\pi}{4}-\sqrt{-t^{2}}\right)\right)\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right)\right),$

for $|t|\rightarrow+\infty$. The asymptotic behaviour of the K-Bessel function
for $t\in\mathbb{R}$ is given in Appendix B.2 of [6],

$\displaystyle\mbox{for }t\rightarrow
0\colon\quad\widetilde{K}_{\alpha}(t)=\begin{cases}\displaystyle\frac{\Gamma(\alpha)}{2}\left(\frac{t}{2}\right)^{-2\alpha}+o\big{(}t^{-2\alpha}\big{)}&\mbox{if
}\alpha>0,\vspace{1mm}\\\
\displaystyle-\log\left(\frac{t}{2}\right)+o\left(\log\left(\frac{t}{2}\right)\right)&\mbox{if
}\alpha=0,\vspace{1mm}\\\ \displaystyle\frac{\Gamma(-\alpha)}{2}+o(1)&\mbox{if
}\alpha<0,\end{cases}$ $\displaystyle\mbox{for
}t\rightarrow+\infty\colon\quad\widetilde{K}_{\alpha}(t)=\dfrac{\sqrt{\pi}}{2}\left(\dfrac{t}{2}\right)^{-\alpha-\frac{1}{2}}e^{-t}\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right).$

###  A.2 Generalised Laguerre functions

Consider the generating function

$\displaystyle
G_{2}^{\mu,\nu}(t,x):=\dfrac{1}{(1-t)^{\frac{\mu+\nu+2}{2}}}\widetilde{I}_{\frac{\mu}{2}}\left(\dfrac{tx}{1-t}\right)\widetilde{K}_{\frac{\nu}{2}}\left(\dfrac{x}{1-t}\right),$

for complex parameters $\mu$ and $\nu$. The generalised Laguerre functions
$\Lambda_{2,j}^{\mu,\nu}(x)$ are defined in [16] as the coefficients in the
expansion

$\displaystyle
G_{2}^{\mu,\nu}(t,x)=\sum_{j=0}^{\infty}\Lambda_{2,j}^{\mu,\nu}(x)t^{j}.$

##  Appendix B Proof of Theorem 6.3

To give the proof of Theorem 6.3 we first need a few technical lemmas.

######  Lemma B.1 (properties of $\operatorname{\mathbb{I}}_{0}$).

For $k\in\\{1,\ldots,m+2n-1\\}$ we have

$\displaystyle\partial_{z^{k}}\operatorname{\mathbb{I}}_{0}(x|z)=2x_{k}\operatorname{\mathbb{I}}_{1}(x|z),\qquad$
$\displaystyle\partial_{x^{k}}\operatorname{\mathbb{I}}_{0}(x|z)=2z_{k}\operatorname{\mathbb{I}}_{1}(x|z),$
(B.1)
$\displaystyle\partial_{z^{0}}\operatorname{\mathbb{I}}_{0}(x|z)=-2x_{0}\operatorname{\mathbb{I}}_{1}(x|z),\qquad$
$\displaystyle\partial_{x^{0}}\operatorname{\mathbb{I}}_{0}(x|z)=-2z_{0}\operatorname{\mathbb{I}}_{1}(x|z),$
(B.2)
$\displaystyle\mathbb{E}_{z}\operatorname{\mathbb{I}}_{0}(x|z)=(x|z)\operatorname{\mathbb{I}}_{1}(x|z),\qquad$
$\displaystyle\mathbb{E}_{x}\operatorname{\mathbb{I}}_{0}(x|z)=(x|z)\operatorname{\mathbb{I}}_{1}(x|z),$

and for $i\in\\{0,\ldots,m+2n-1\\}$ we have

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{i})\operatorname{\mathbb{I}}_{0}(x|z)=4x_{i}\operatorname{\mathbb{I}}_{0}(x|z),\qquad\operatorname{\widetilde{\mathcal{B}}}(x_{i})\operatorname{\mathbb{I}}_{0}(x|z)=4z_{i}\operatorname{\mathbb{I}}_{0}(x|z).$
(B.3)

The last equation expresses that $\operatorname{\mathbb{I}}_{0}(x|z)$ is an
eigenfunction of $\operatorname{\widetilde{\mathcal{B}}}$.

###### Proof.

Equation (B.1) follows from the chain rule. Equation (B.2) follows immediately
from equation (B.1) and the definition of the Euler operator. Using the same
calculation as in the proof of Lemma 4.10, we obtain

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})(x|z)^{l}=4x_{k}l\left(l+\frac{M}{2}-2\right)(x|z)^{l-1}.$

We then find

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{\mathbb{I}}_{0}(x|z)$
$\displaystyle=\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\,\Gamma\left(l+\frac{M}{2}-1\right)}\operatorname{\widetilde{\mathcal{B}}}(z_{k})(x|z)^{l},$
$\displaystyle=4x_{k}\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=1}^{\infty}\dfrac{l\left(l+\frac{M}{2}-2\right)}{l!\,\Gamma\left(l+\frac{M}{2}-1\right)}(x|z)^{l-1}$
$\displaystyle=4x_{k}\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l-1=0}^{\infty}\dfrac{1}{(l-1)!\,\Gamma\left((l-1)+\frac{M}{2}-1\right)}(x|z)^{l-1}$
$\displaystyle=4x_{k}\operatorname{\mathbb{I}}_{0}(x|z).$

The calculations for the case
$\operatorname{\widetilde{\mathcal{B}}}(x_{k})\operatorname{\mathbb{I}}_{0}(x|z)$
is analogous. ∎

######  Lemma B.2 (properties of $\exp$).

For $k\in\\{1,\ldots,m+2n-1\\}$ we have

$\displaystyle\mathbb{E}_{z}\exp(-z_{0})=-z_{0}\exp(-z_{0}),\qquad$
$\displaystyle\mathbb{E}_{x}\exp(-2x_{0})=-2x_{0}\exp(-2x_{0}),$ (B.4)
$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\exp(-z_{0})=z_{k}\exp(-z_{0}),\qquad$
$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(x_{k})\exp(-2x_{0})=4x_{k}\exp(-2x_{0}),$
(B.5)
$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{0})\exp(-z_{0})=(2-M+z_{0})\exp(-z_{0}).\quad$
(B.6)

###### Proof.

Equation (B.4) is immediate. For equation (B.6) we have

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{0})\exp(-z_{0})$
$\displaystyle=(2-M-2\mathbb{E})\partial_{0}\exp(-z_{0})+z_{0}\Delta\exp(-z_{0})$
$\displaystyle=(2-M-2\mathbb{E})\exp(-z_{0})-z_{0}\partial_{0}\partial_{0}\exp(-z_{0})$
$\displaystyle=(2-M+2z_{0})\exp(-z_{0})-z_{0}\exp(-z_{0})$
$\displaystyle=(2-M+z_{0})\exp(-z_{0}),$

while for equation (B.5) we compute

$\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\exp(-z_{0})=0-z_{k}\Delta\exp(-z_{0})=z_{k}\partial_{0}\partial_{0}\exp(-z_{0})=z_{k}\exp(-z_{0}).$

A similar calculation shows
$\operatorname{\widetilde{\mathcal{B}}}(x_{k})\exp(-2x_{0})=4x_{k}\exp(-2x_{0})$.
∎

######  Lemma B.3 (properties of $\operatorname{SB}$).

We have

$\displaystyle\mathbb{E}_{z}\operatorname{SB}f(z)=-z_{0}\operatorname{SB}f(z)+\int_{W}\left(x|z\right)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$
(B.7)
$\displaystyle\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)=\dfrac{1}{2}(2-M+z_{0})\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)=}{}-\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$
(B.8)
$\displaystyle\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)=}{}-2\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$
(B.9) $\displaystyle
L_{0k}^{z}\operatorname{SB}f(z)=-z_{k}\operatorname{SB}f(z)+2\int_{W}\left(z_{0}x_{k}+z_{k}x_{0}\right)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$
(B.10)

###### Proof.

Equation (B.7) follows from (B.2) and (B.4). To prove equation (B.8) we apply
the product rule (Proposition 3.3) to
$-\mathcal{B}(z_{0})(\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z))$ and
substituting equations (B.3), (B.6), and the following easily verified
identities

$\displaystyle\mathbb{E}_{z}(\exp(-z_{0}))\partial_{z^{0}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{0}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$
$\displaystyle\partial_{z^{0}}(\exp(-z_{0}))\mathbb{E}_{z}(\operatorname{\mathbb{I}}_{0}(x|z))=\left(x|z\right)\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$
$\displaystyle
z_{0}\sum_{r,s}\beta^{rs}\partial_{z^{r}}(\exp(-z_{0}))\partial_{z^{s}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{0}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z).$

Equation (B.9) follows in the same way using equations (B.3), (B.5), and

$\displaystyle\mathbb{E}_{z}(\exp(-z_{0}))\partial_{z^{k}}(\operatorname{\mathbb{I}}_{0}(x|z))=-2z_{0}x_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$
$\displaystyle\partial_{z^{k}}(\exp(-z_{0}))\mathbb{E}_{z}(\operatorname{\mathbb{I}}_{0}(x|z))=0,$
$\displaystyle
z_{k}\sum_{r,s}\beta^{rs}\partial_{z^{r}}(\exp(-z_{0}))\partial_{z^{s}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{k}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$

while equation (B.10) follows immediately from

$\displaystyle
L_{0k}^{z}(\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z))=L_{0k}^{z}(\exp(-z_{0}))\operatorname{\mathbb{I}}_{0}(x|z)+\exp(-z_{0})L_{0k}^{z}(\operatorname{\mathbb{I}}_{0}(x|z))$
$\displaystyle\qquad{}=-z_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z)+\exp(-z_{0})(z_{0}(2x_{k}\operatorname{\mathbb{I}}_{1}(x|z))-z_{k}(-2x_{0}\operatorname{\mathbb{I}}_{1}(x|z)))$
$\displaystyle\qquad{}=-z_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z)+2\exp(-z_{0})(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z).$

This proves the lemma. ∎

We can now prove Theorem 6.3. For convenience we restate it here.

######  Theorem B.4 (intertwining property).

For $M\geq 4$ the Segal–Bargmann transform intertwines the action $\pi$ on $W$
with the action $\rho$ on $F$, i.e.,

$\displaystyle\operatorname{SB}\circ\pi(X)=\rho(X)\circ\operatorname{SB},$
(B.11)

for all $X\in\mathfrak{g}$.

###### Proof.

We will use the decomposition
$\mathfrak{g}=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$ to prove equation
(B.11) case by case.

Case 1: $X=(e_{0},0,0)$. We wish to prove

$\displaystyle\operatorname{SB}(2x_{0}f)(z)=\dfrac{1}{2}z_{0}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)+\left(\dfrac{M}{2}-1\right)\operatorname{SB}f(z)+\mathbb{E}_{z}\operatorname{SB}f(z).$

If we substitute (B.7) and (B.8) into the equation the result follows.

Case 2: $X=(e_{k},0,0)$, $k\neq 0$. Substituting (B.9) and (B.10) in

$\displaystyle\operatorname{SB}(2x_{k}f)(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)+L_{0k}^{z}\operatorname{SB}f(z)$

proves equation (B.11) for $X=(e_{k},0,0)$.

Case 3: $X=(0,L_{e_{0}},0)$. We wish to prove

$\displaystyle\dfrac{1}{2}\operatorname{SB}((2-M)f)-\operatorname{SB}(\mathbb{E}_{x}f)(z)=\dfrac{1}{2}z_{0}\operatorname{SB}f(z)-\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z).$

Using (B.8) this becomes

$\displaystyle\operatorname{SB}((2-M)f)=\operatorname{SB}(\mathbb{E}_{x}f)(z)-2\operatorname{SB}(x_{0}f)(z)+\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$

From the proof of [6, Proposition 8.9] it follows that
$\int_{W}(\mathbb{E}_{x}+M-2)f=0$ when $\int_{W}f$ is well defined. This gives

$\displaystyle\operatorname{SB}((2-M)f)$
$\displaystyle=\exp(-z_{0})\int_{W}(2-M)\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x)$
$\displaystyle=\exp(-z_{0})\int_{W}\mathbb{E}_{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x))$
$\displaystyle=\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x)-2\operatorname{SB}(x_{0}f)(z)+\operatorname{SB}(\mathbb{E}_{x}f)(z),$

where we used (B.2) and (B.4) to obtain the last equality.

Case 4: $X=(0,L_{e_{k}},0)$, $k\neq 0$. In a similar fashion to (B.10), we
find

$\displaystyle-\operatorname{SB}\big{(}L_{0k}^{x}f\big{)}(z)$
$\displaystyle=-\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})L_{0k}^{x}f(x)$
$\displaystyle=\exp(-z_{0})\int_{W}L_{0k}^{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$
$\displaystyle=2\int_{W}(x_{0}z_{k}+z_{0}x_{k})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x)-2\operatorname{SB}(x_{k}f)(z).$

Because of (B.9), this is equivalent to

$\displaystyle-\operatorname{SB}\big{(}L_{0k}^{x}f\big{)}(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)-\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z),$

proving equation (B.11) for $X=(0,L_{e_{k}},0)$.

Case 5: $X=(0,[L_{e_{i}},L_{e_{j}}],0)$, $i,j\neq 0$. Using Proposition 3.10,
we obtain

$\displaystyle\operatorname{SB}\big{(}L_{ij}^{x}f\big{)}(z)$
$\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})L_{ij}^{x}f(x)$
$\displaystyle=-\exp(-z_{0})\int_{W}L_{ij}^{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$
$\displaystyle=-2\exp(-z_{0})\int_{W}\big{(}x_{i}z_{j}-(-1)^{|i||j|}x_{j}z_{i}\big{)}\operatorname{\mathbb{I}}_{1}(x|z)\exp(-2x_{0})f(x)$
$\displaystyle=\exp(-z_{0})\int_{W}L_{ij}^{z}(\operatorname{\mathbb{I}}_{0}(x|z))\exp(-2x_{0})f(x)$
$\displaystyle=L_{ij}^{z}\operatorname{SB}f(z),$

thus equation (B.11) holds.

Case 6: $X=(0,0,e_{0})$. Using (B.7) and (B.8), we can rewrite

$\displaystyle\dfrac{1}{2}z_{0}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)+\left(1-\dfrac{M}{2}\right)\operatorname{SB}f(z)-\mathbb{E}_{z}\operatorname{SB}f(z)$

as

$\displaystyle
2z_{0}\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)+(2-M)\operatorname{SB}f(z)-2\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$

In a similar fashion to (B.8), we find

$\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\operatorname{\widetilde{\mathcal{B}}}(x_{0})f(x)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)}{}=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\widetilde{\mathcal{B}}}(x_{0})\left(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\right)f(x)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)}{}=2z_{0}\operatorname{SB}f(z)+(2-M)\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)=}{}-2\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$

So we conclude that equation (B.11) holds for $X=(0,0,e_{0})$.

Case 7: $X=(0,0,e_{k})$, $k\neq 0$. To show

$\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)-L_{0k}^{z}\operatorname{SB}f(z),$

we substitute (B.9) and (B.10) into the equation. So we obtain

$\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=2z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=}{}-4\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$

In a similar fashion to (B.9), we find

$\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\operatorname{\widetilde{\mathcal{B}}}(x_{k})f(x)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)}{}=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\widetilde{\mathcal{B}}}(x_{k})(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)}{}=2z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$
$\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=}{}-4\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$

This proves the theorem. ∎

### Acknowledgements

SB is supported by a BOF Postdoctoral Fellowship from Ghent University. HDB is
supported by the Research Foundation Flanders (FWO) under Grant EOS 30889451.
The authors would like to thank the anonymous referees for carefully reading
the paper and for their useful comments.

## References

  * [1] Alldridge A., Fréchet globalisations of Harish-Chandra supermodules, Int. Math. Res. Not. 2017 (2017), 5182–5232, arXiv:1403.4055.
  * [2] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  * [3] Barbier S., A minimal representation of the orthosymplectic Lie superalgebra, Ph.D. Thesis, Ghent University, 2018, available at http://hdl.handle.net/1854/LU-8560881.
  * [4] Barbier S., Coulembier K., Polynomial realisations of Lie (super)algebras and Bessel operators, Int. Math. Res. Not. 2017 (2017), 3148–3179, arXiv:1512.01387.
  * [5] Barbier S., Coulembier K., On structure and TKK algebras for Jordan superalgebras, Comm. Algebra 46 (2018), 684–704, arXiv:1609.00271.
  * [6] Barbier S., Frahm J., A minimal representation of the orthosymplectic Lie supergroup, Int. Math. Res. Not., to appear, arXiv:1710.07271.
  * [7] Bargmann V., On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214.
  * [8] Carmeli C., Cassinelli G., Toigo A., Varadarajan V.S., Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 263 (2006), 217–258, arXiv:hep-th/0501061.
  * [9] Coulembier K., The orthosymplectic superalgebra in harmonic analysis, J. Lie Theory 23 (2013), 55–83, arXiv:1208.3827.
  * [10] Coulembier K., De Bie H., Sommen F., Orthosymplectically invariant functions in superspace, J. Math. Phys. 51 (2010), 083504, 23 pages, arXiv:1006.4744.
  * [11] De Bie H., Genest V.X., van de Vijver W., Vinet L., A higher rank Racah algebra and the ${\mathbb{Z}}^{n}_{2}$ Laplace–Dunkl operator, J. Phys. A: Math. Theor. 51 (2018), 025203, 20 pages, arXiv:1610.02638.
  * [12] De Bie H., Sommen F., Spherical harmonics and integration in superspace, J. Phys. A: Math. Theor. 40 (2007), 7193–7212, arXiv:0705.3148.
  * [13] de Goursac A., Michel J.-P., Superunitary representations of Heisenberg supergroups, Int. Math. Res. Not., to appear, arXiv:1601.07387.
  * [14] Folland G.B., Harmonic analysis in phase space, Annals of Mathematics Studies, Vol. 122, Princeton University Press, Princeton, NJ, 1989.
  * [15] Gan W.T., Savin G., On minimal representations definitions and properties, Represent. Theory 9 (2005), 46–93.
  * [16] Hilgert J., Kobayashi T., Mano G., Möllers J., Special functions associated with a certain fourth-order differential equation, Ramanujan J. 26 (2011), 1–34, arXiv:0907.2608.
  * [17] Hilgert J., Kobayashi T., Möllers J., Minimal representations via Bessel operators, J. Math. Soc. Japan 66 (2014), 349–414, arXiv:1106.3621.
  * [18] Hilgert J., Kobayashi T., Möllers J., Ørsted B., Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups, J. Funct. Anal. 263 (2012), 3492–3563, arXiv:1203.5462.
  * [19] Kirillov A.A., Lectures on the orbit method, Graduate Studies in Mathematics, Vol. 64, Amer. Math. Soc., Providence, RI, 2004.
  * [20] Kobayashi T., Mano G., The Schrödinger model for the minimal representation of the indefinite orthogonal group ${\rm O}(p,q)$, Mem. Amer. Math. Soc. 213 (2011), vi+132 pages, arXiv:0712.1769.
  * [21] Kobayashi T., Ørsted B., Analysis on the minimal representation of ${\rm O}(p,q)$. I. Realization via conformal geometry, Adv. Math. 180 (2003), 486–512, arXiv:math.RT/0111083.
  * [22] Kobayashi T., Ørsted B., Analysis on the minimal representation of ${\rm O}(p,q)$. II. Branching laws, Adv. Math. 180 (2003), 513–550, arXiv:math.RT/0111085.
  * [23] Kobayashi T., Ørsted B., Analysis on the minimal representation of ${\rm O}(p,q)$. III. Ultrahyperbolic equations on ${\mathbb{R}}^{p-1,q-1}$, Adv. Math. 180 (2003), 551–595, arXiv:math.RT/0111086.
  * [24] Lávička R., Šmíd D., Fischer decomposition for polynomials on superspace, J. Math. Phys. 56 (2015), 111704, 9 pages, arXiv:1508.03426.
  * [25] Neeb K.-H., Salmasian H., Lie supergroups, unitary representations, and invariant cones, in Supersymmetry in Mathematics and Physics, Lecture Notes in Math., Vol. 2027, Springer, Heidelberg, 2011, 195–239, arXiv:1012.2809.
  * [26] Salmasian H., Unitary representations of nilpotent super Lie groups, Comm. Math. Phys. 297 (2010), 189–227, arXiv:0906.2515.
  * [27] Tuynman G.M., The left-regular representation of a super Lie group, J. Lie Theory 29 (2019), 1–78.