# Restriction theorems and root systems for symmetric superspaces

Shifra Reif, Siddhartha Sahi and Vera Serganova

###### Abstract.

In this paper we consider those involutions $\theta$ of a finite-dimensional
Kac-Moody Lie superalgebra ${\mathfrak{g}}$, with associated decomposition
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$, for which a Cartan
subspace ${\mathfrak{a}}$ in ${\mathfrak{p}}_{\bar{0}}$ is self-centralizing
in ${\mathfrak{p}}$. For such $\theta$ the restriction map $C_{\theta}$ from
${\mathfrak{p}}$ to ${\mathfrak{a}}$ is injective on the algebra
$P({\mathfrak{p}})^{\mathfrak{k}}$ of ${\mathfrak{k}}$-invariant polynomials
on ${\mathfrak{p}}$. There are five infinite families and five exceptional
cases of such involutions, and for each case we explicitly determine the
structure of $P({\mathfrak{p}})^{\mathfrak{k}}$ by giving a complete set of
generators for the image of $C_{\theta}$. We also determine precisely when the
restriction map $R_{\theta}$ from $P({\mathfrak{g}})^{\mathfrak{g}}$ to
$P({\mathfrak{p}})^{\mathfrak{k}}$ is surjective. Finally we introduce the
notion of a generalized restricted root system, and show that in the present
setting the ${\mathfrak{a}}$-roots $\Delta({\mathfrak{a}},{\mathfrak{g}})$
always form such a system.

##  1\. Introduction

In this paper we consider the analog of the Chevalley restriction theorem for
an involution $\theta$ of a finite dimensional Kac-Moody Lie superalgebra
${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}+{\mathfrak{g}}_{\bar{1}}$. That is to
say we study the structure of the algebra $P({\mathfrak{p}})^{\mathfrak{k}}$
of ${\mathfrak{k}}$-invariant polynomials on ${\mathfrak{p}}$, where
${\mathfrak{k}}$ is the fixed subalgebra of $\theta$ and ${\mathfrak{p}}$ is
the $(-1)$-eigenspace, by analyzing its restriction to a Cartan subspace
${\mathfrak{a}}$ in
${\mathfrak{p}}_{\bar{0}}={\mathfrak{p}}\cap{\mathfrak{g}}_{\bar{0}}$. This
defines a map $C_{\theta}$ from $P({\mathfrak{p}})^{\mathfrak{k}}$ to the
polynomial algebra $P({\mathfrak{a}})$, which is injective precisely when the
centralizer of ${\mathfrak{a}}$ in
${\mathfrak{p}}_{1}={\mathfrak{p}}\cap{\mathfrak{g}}_{\bar{1}}$ is $0$. This
is equivalent to ${\mathfrak{g}}$ admitting an Iwasawa decomposition of the
form ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{a}}\oplus{\mathfrak{n}}$,
and so we will say such a $\theta$ is an Iwasawa involution. In this case
$P({\mathfrak{p}})^{\mathfrak{k}}$ is isomorphic to the image of $C_{\theta}$.
In the rest of the paper we assume that $\theta$ is an Iwasawa involution.

In Theorem A we determine the image of $C_{\theta}$ for an Iwasawa involution
$\theta$. More precisely we show that the image is equal to a certain
subalgebra $I_{\theta}({\mathfrak{a}})$ of $P({\mathfrak{a}})$ defined below.
Our result is completely explicit: there are five infinite families of Iwasawa
involutions and five exceptional cases, and in each case we provide a complete
set of generators for $I_{\theta}({\mathfrak{a}})$ except for one subfamily
(see Section 6.4).

We consider also the algebra $P({\mathfrak{g}})^{\mathfrak{g}}$ of
${\mathfrak{g}}$-invariant polynomials on ${\mathfrak{g}}$ and its restriction
to ${\mathfrak{p}}$. This defines a map $R_{\theta}$ from
$P({\mathfrak{g}})^{\mathfrak{g}}$ to $P({\mathfrak{p}})^{\mathfrak{k}}$ which
is in general neither injective nor surjective. In Theorem B we show that
$R_{\theta}$ is in fact surjective _except_ for a subfamily of one infinite
family and two of the exceptional cases. Again the results are completely
explicit: we determine the image of the composite restriction map
$C_{\theta}\circ R_{\theta}$ and show that it coincides with
$I_{\theta}({\mathfrak{a}})$ except in these cases.

We also determine the structure of the set
$\Delta({\mathfrak{a}},{\mathfrak{g}})$ of roots of ${\mathfrak{a}}$ in
${\mathfrak{g}}$. Once more we proceed in an explicit manner: we choose a
Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}$ which contains
${\mathfrak{a}}$, and we study the restriction of the ${\mathfrak{h}}$-roots
to ${\mathfrak{a}}$. The set $\Delta({\mathfrak{h}},{\mathfrak{g}})$ of
${\mathfrak{h}}$-roots is a generalized root system (GR system) in the sense
of Serganova [Serga2], however this is not always the case for
$\Delta({\mathfrak{a}},{\mathfrak{g}})$. Therefore we introduce the notion of
a generalized restricted root system (GRR system) by relaxing one of the
requirements of a GR system. In Theorem C we show that
$\Delta({\mathfrak{a}},{\mathfrak{g}})$ is always a GRR system.

We now describe our results more precisely, starting with the definition of
$I_{\theta}({\mathfrak{a}})$. Let $W$ be the Weyl group of
$\Delta_{0}=\Delta({\mathfrak{a}},{\mathfrak{g}}_{\bar{0}})$ and let
$P({\mathfrak{a}})^{W}$ be the algebra of $W$-invariant polynomials on
${\mathfrak{a}}$. We say that a root
$\alpha\in\Delta({\mathfrak{a}},{\mathfrak{g}})$ is _singular_ if no multiple
of $\alpha$ is a root in $\Delta_{0}$. The root space
${\mathfrak{g}}_{\alpha}$ of a singular root is purely odd, and in fact
${\mathfrak{g}}_{\alpha}$ has dimension $\left(0|2k_{\alpha}\right)$ for some
integer $k_{\alpha}$. We define $I_{\theta}({\mathfrak{a}})$ to be the
subalgebra consisting of $W$-invariant polynomials $f\in
P({\mathfrak{a}})^{W}$ which satisfy the following derivative condition for
every singular $\alpha$

$\left(D_{h_{\alpha}}\right)^{k}\left(f\right)\in\langle\alpha\rangle,\quad
k=1,3,\ldots,2k_{\alpha}-1.$

Here $h_{\alpha}$ is the image of $\alpha$ under the map
${\mathfrak{a}}^{*}\to{\mathfrak{a}}$ induced by an invariant bilinear form,
$D_{h_{\alpha}}$ is the partial derivative along $h_{\alpha}$, and
$\langle\alpha\rangle$ is the principal ideal of $P({\mathfrak{a}})$ generated
by $\alpha$.

The three main results of this paper, which hold in the setting of an Iwasawa
involution $\theta$ of a Kac-Moody Lie superalgera ${\mathfrak{g}}$, are as
follows.

######  Theorem A.

The restriction map $C_{\theta}:P({\mathfrak{p}})^{\mathfrak{k}}\rightarrow
P({\mathfrak{a}})$ is injective and its image is $I_{\theta}({\mathfrak{a}})$.

######  Theorem B.

The restriction map $R_{\theta}:P({\mathfrak{g}})^{\mathfrak{g}}\rightarrow
P({\mathfrak{p}})^{\mathfrak{k}}$ is surjective unless
$({\mathfrak{g}},{\mathfrak{k}})$ is one of the pairs
$\left(\mathfrak{osp}(2m|2n_{1}+2n_{2}),\mathfrak{osp}(m|2n_{1})\oplus\mathfrak{osp}(m|2n_{2})\right)$,
$(F_{4},\mathfrak{gosp}(2|4))$ or
$(D(2,1,a),\mathfrak{osp}(2|2)\oplus\mathfrak{so}(2))$.

######  Theorem C.

The restricted roots $\Delta({\mathfrak{a}},{\mathfrak{g}})$ form a GRR
system.

We refer the reader to Section 2.2 for the precise definition of a GRR system.

Theorem A was previously obtained in [AHZ] in a slightly different setup, with
a completely different proof. Our argument is very explicit: we directly
verify that Theorem A holds for every Iwasawa involution. This approach allows
us to prove Theorems B and reveals a great deal of information about Iwasawa
involutions, generalized restricted root systems, and root multiplicities. We
hope that this information will be useful for researchers in the general
theory of Lie superalgebras, as well as those interested in the applications
of Lie superalgebras. We briefly describe one specific application that we
have in mind.

In [SV] Sergeev and Veselov have developed the theory of a deformed GR system,
which is a GR system together with a bilinear form and a $W$-invariant
multiplicity function on the roots satisfying certain “admissibility”
constraints. They have used this theory to construct families of completely
integrable systems, which are supersymmetric analogs of the classical
Calogero-Moser-Sutherland models of mathematical physics. It is natural to ask
whether there exists a similar deformation theory for a GRR system, which
might lead to new integrable systems. In a sequel to this paper we intend to
study this question in conjunction with the classification problem for GRR
systems, and we expect that the results of the present paper will play a key
role in these investigations.

The paper is organized as follows. In Section 2 we recall some generalities on
roots and restricted roots for a Lie superalgebra, give the definition of a
GRR system and prove Theorem C. In Sections 3 and 4, we show that $C_{\theta}$
is injective and that its image is contained in $I_{\theta}({\mathfrak{a}})$.
Finally in Section 5, we show by explicit computation that the image of
$C_{\theta}$ is precisely $I_{\theta}({\mathfrak{a}})$. The same computations
also give proofs of Theorems B and C. Our analysis reveals a fair amount of
detailed information about the sets of restricted roots, and we organize this
in the appendix for future reference.

### Acknowledgements

The project was made possible by a SQuaRE at the American Institute for
Mathematics. The authors thank AIM for providing a supportive and
mathematically rich environment.” This project is partially supported by
Israel Science Foundations Grant 1957/21. S. Reif gratefully acknowledges
support from the Institute for Advanced Study while working on this project.
The research of S. Sahi was partially supported by NSF grants DMS-1939600 and
2001537, and Simons Foundation grant 509766. Vera Serganova was supported in
part by NSF grant 2001191 and by Tromso research foundation.

##  2\. Supersymmetric spaces

###  2.1. Restricted Root Systems

Let ${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$ be
a Kac-Moody Lie superalgebra with a non-degenerate even bilinear form
$(\cdot,\cdot)$, namely ${\mathfrak{g}}$ is one of $\mathfrak{gl}(m|n)$,
$\mathfrak{osp}(m|2n)$, $D(2,1,a)$, $F_{4}$ and $G_{3}$. Let $\theta$ be an
involution on ${\mathfrak{g}}$ which fixes $(\cdot,\cdot)$. Then
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$ where ${\mathfrak{k}}$ is
the subalgebra of ${\mathfrak{g}}$ of $\theta$-fixed points and
${\mathfrak{p}}:=\\{x\in{\mathfrak{g}}:\theta(x)=-x\\}$. Let
${\mathfrak{a}}\subseteq{\mathfrak{p}}_{\bar{0}}$ be the Cartan subspace of
the symmetric pair $({\mathfrak{g}}_{0},{\mathfrak{k}}_{0})$. We assume that
there exists an Iwasawa decomposition
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{a}}\oplus{\mathfrak{n}}$ where
${\mathfrak{n}}:=\operatorname{span}_{\mathbb{C}}\\{x\in{\mathfrak{g}}:[a,x]=c_{a}x,c_{a}\in\mathbb{R}_{>0}\\}$
for some regular element $a\in{\mathfrak{a}}$. This is equivalent to
${\mathfrak{p}}^{\mathfrak{a}}={\mathfrak{a}}$. Let ${\mathfrak{h}}$ be a
Cartan subalgebra of ${\mathfrak{g}}$ containing ${\mathfrak{a}}$ and let
${\mathfrak{t}}:={\mathfrak{h}}\cap{\mathfrak{k}}$. We denote by
$p_{\mathfrak{a}}:{\mathfrak{h}}\rightarrow{\mathfrak{a}}$ the projection with
kernel ${\mathfrak{t}}$. Since $\theta$ preserves $(\cdot,\cdot)$, it
restricts to a nondegenerate invariant form on ${\mathfrak{a}}$ and
${\mathfrak{t}}$.

We denote by $\Delta({\mathfrak{g}},{\mathfrak{h}})$ the set of roots of
${\mathfrak{g}}$ and define the set of restricted roots to be
$\Delta({\mathfrak{g}},{\mathfrak{a}}):=\\{\bar{\alpha}\mid_{\mathfrak{a}}\
:\bar{\alpha}\in\Delta({\mathfrak{g}},{\mathfrak{h}})\\}\subset{\mathfrak{a}}^{*}$.
For a restricted root $\alpha$, the root space is defined by
${\mathfrak{g}}_{\alpha}:=\\{x\in{\mathfrak{g}}:[h,x]=\alpha(h)x,\ \forall
h\in{\mathfrak{a}}\\}$ and $\dim{\mathfrak{g}}_{\alpha}$ is called the
_multiplicity_ of $\alpha$. The root space ${\mathfrak{g}}_{\alpha}$ need not
be purely even or odd space. A root $\alpha$ is called _singular_ if
${\mathfrak{g}}_{c\alpha}\cap{\mathfrak{g}}_{\bar{0}}=\\{0\\}$ is for any
$c\in\mathbb{C}$. Note that in $\Delta({\mathfrak{g}},{\mathfrak{h}})$ a root
is not multipliable if and only if it is isotropic. Hence all singular roots
are restriction of isotropic roots.

Since the restriction of $(\cdot,\cdot)$ to ${\mathfrak{a}}$ is
nondengenerate, we have a form on ${\mathfrak{a}}^{*}$ and we define the baby
Weyl group to be $W=\langle
s_{\alpha}\mid\alpha\in\Delta({\mathfrak{g}},{\mathfrak{a}})_{\bar{0}}\rangle$
where $\Delta({\mathfrak{g}},{\mathfrak{a}})_{\bar{0}}$ is the set of roots
which are restrictions of even roots, and $s_{\alpha}$ is the reflection
corresponding to $\alpha$. Choose a nonzero vector $h_{\alpha}$ in $({\rm
Ker}\alpha)^{\perp}\subset{\mathfrak{a}}$. Note that $({\rm
Ker}\alpha)^{\perp}$ is one dimensional so $h_{\alpha}$ is unique up to a
scalar.

######  Remark 2.1.

The projection
$p_{\mathfrak{a}}([{\mathfrak{g}}_{\alpha},{\mathfrak{g}}_{-\alpha}])$ is
orthogonal to ${\rm Ker}\alpha$. Indeed, letting $x\in{\mathfrak{g}}_{\alpha}$
and $y\in{\mathfrak{g}}_{-\alpha}$, since $({\mathfrak{a}},{\mathfrak{t}})=0$,
we have

$(p_{\mathfrak{a}}([x,y]),a)=([x,y],a)=(x,[y,a])=\alpha(a)(x,y)=0$

for all $a\in{\rm Ker}\alpha$.

Let $B_{\theta}(x,y)=(x,\theta y)$ be a twisted form defined on
${\mathfrak{g}}$. Then $B_{\theta}$ is a supersymmetric even non-degenerate
bilinear form, see [AHZ, Prop. 2.10]. Note that for every root space
$\theta{\mathfrak{g}}_{\alpha}={\mathfrak{g}}_{-\alpha}$ because $[h,\theta
x]=-\theta([h,x])=-\alpha(h)\theta x$ for any
$h\in{\mathfrak{a}},x\in{\mathfrak{g}}_{\alpha}$. Hence $B_{\theta}$ restricts
to a non-degenerate supersymmetric form on ${\mathfrak{g}}_{\alpha}$.

######  Remark 2.2.

If $\alpha$ is a singular root, ${\mathfrak{g}}_{\alpha}$ is a purely odd
space with a skew-symmetric bilinear form $B_{\theta}$. In particular
$\dim{\mathfrak{g}}_{\alpha}$ is even. Moreover, if $B_{\theta}(x,y)\neq 0$
then $[x,\theta y]\neq 0$.

###  2.2. The Symmetric Algebra and the Algebra of Invariants

For a super-vector space $V=V_{\bar{0}}\oplus V_{\bar{1}}$, we denote by
$S(V)$ the symmteric algebra on $V$, that is
$S=S(V_{\bar{0}})\otimes\bigwedge(V_{\bar{1}})$ as vector spaces. Note that
$S(V)=\bigoplus_{d=0}^{\infty}S(V)_{d}$ where $S(V)_{d}$ is the component of
degree $d$.

Let $m=\dim V_{\bar{0}}$ and $n=\dim V_{\bar{1}}$. Denote by $P_{d}$ be the
set of partitions $\lambda$ of $d$ that fit in the $(m|n)$-hook, namely
$\lambda_{m+1}\leq n$. Recall that $S^{d}(V)=\bigoplus_{\lambda\in
P_{d}}V_{\lambda}$ as a $\mathfrak{gl}(V)$-module where $V_{\lambda}$ is the
irreducible module with highest weight

$\lambda_{1}\varepsilon_{1}+\ldots+\lambda_{m}\varepsilon_{m}+\max(\lambda^{\prime}_{1}-m,0)\delta_{1}+\ldots+\max(\lambda^{\prime}_{n}-m,0)\delta_{n}.$

Note that for any superspace $V$, the polynomial algebra $P(V)$ is naturally
isomorphic to the symmetric algebra $S(V^{*})$. Given a subspace $W\subset V$,
the restriction map from $S(V^{*})$ to $S(W^{*})$ is well defined.

Given an Iwasawa involution $\theta$, the algebra of invariants is the
following subalgebra of $S({\mathfrak{a}}^{*})$:

$I_{\theta}({\mathfrak{a}})=\left\\{f\in
S({\mathfrak{a}}^{*})^{W}\mid\left(D_{h_{\alpha}}\right)^{k}\left(f\right)\in\left\langle\alpha\right\rangle\text{
for }\alpha\text{ singular
},k=1,3,\ldots,\dim{\mathfrak{g}}_{\alpha}-1\right\\}.$

We shall abbreviate $I({\mathfrak{a}}^{*})=I_{\theta}({\mathfrak{a}})$ in the
rest of the paper.

######  Example 2.3.

Let $\tilde{\mathfrak{g}}:={\mathfrak{g}}\times{\mathfrak{g}}$ and let
$\tilde{\theta}:\tilde{\mathfrak{g}}\rightarrow\tilde{\mathfrak{g}}$ be such
that $\tilde{\theta}(x,y)=(y,x)$. Then we get a symmetric pair
$(\tilde{\mathfrak{g}},\tilde{\mathfrak{k}})$ where
$\tilde{\mathfrak{k}}=\\{(x,x):x\in{\mathfrak{g}}\\}\cong{\mathfrak{g}}$. Here
$\tilde{\mathfrak{p}}=\\{(x,-x):x\in{\mathfrak{g}}\\}$ and the Cartan subspace
$\tilde{\mathfrak{a}}=\\{(h,-h):h\in{\mathfrak{h}}\\}$ is identified with the
Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}$. In this case the
restricted roots are the roots of ${\mathfrak{g}}$, the dimension of each root
space is $2$ and each root space is either purely even or purely odd. A root
of ${\mathfrak{g}}$ with an odd root space is inmultipliable if and only if it
is isotropic. Hence the algebra of invariants is

$I(\tilde{\mathfrak{a}}^{*})\cong I({\mathfrak{h}}^{*}):=\left\\{f\in
S({\mathfrak{a}}^{*})^{W}:D_{h_{\alpha}}(f)\in\langle\alpha\rangle\text{ for
$\alpha$ isotropic root}\right\\}.$

By [Serge2], this is precisely the image of the restriction map
$S({\mathfrak{g}}^{*})^{\mathfrak{g}}$ to $S({\mathfrak{h}}^{*})$.

######  Remark 2.4.

The algebra $S(\mathfrak{g})$ is in fact $\operatorname{gr}U({\mathfrak{g}})$,
namely the associated graded of the universal enveloping algebra of
${\mathfrak{g}}$. Since the adjoint action of ${\mathfrak{g}}$ gives an
isomorphic ${\mathfrak{g}}$-module structures on $U({\mathfrak{g}})$ and
$S({\mathfrak{g}})$, we also get that
$\operatorname{gr}Z({\mathfrak{g}})=S({\mathfrak{g}})^{\mathfrak{g}}$. Let
$D({\mathfrak{g}},{\mathfrak{k}}):=U({\mathfrak{g}})^{\mathfrak{k}}/({\mathfrak{k}}(U({\mathfrak{g}})\cap
U({\mathfrak{g}})^{\mathfrak{k}})$. Since
${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$ as a
${\mathfrak{k}}$-module, we also get that
$\operatorname{gr}D({\mathfrak{g}},{\mathfrak{k}})=S({\mathfrak{p}})^{\mathfrak{k}}\cong
S({\mathfrak{g}})^{\mathfrak{k}}/({\mathfrak{k}}S({\mathfrak{g}})\cap
S({\mathfrak{g}})^{\mathfrak{k}})$. The functor $\operatorname{gr}$ sends the
projection map $Z({\mathfrak{g}})\rightarrow D({\mathfrak{g}},{\mathfrak{k}})$
to the projection $S({\mathfrak{g}})^{\mathfrak{g}}\rightarrow
S({\mathfrak{p}})^{\mathfrak{k}}$ and reflects surjectivity, that is the
former map is surjective if the latter map is surjective. Using the invariant
bilinear form, we note that the latter map is equivalent to the restriction
map $S({\mathfrak{g}}^{*})^{\mathfrak{g}}\rightarrow
S({\mathfrak{p}}^{*})^{\mathfrak{k}}$.

###  2.3. Generalized Restricted Root Systems

We show that a restricted root system is a generalized root system in the
following sense.

######  Definition 2.5.

Let $V$ be a finite-dimensional vector space with a non-degenerate bilinear
form $\langle\cdot,\cdot\rangle$. A finite set $R=R_{sing}\cup R_{reg}\subset
V\backslash\\{0\\}$ is called a GRR system if

  1. (1)

$R$ spans $V$ and $R_{reg}=-R_{reg}$, $R_{sing}=-R_{sing}$.

  2. (2)

If $\alpha\in R_{reg}$ then $\langle\alpha,\alpha\rangle\neq 0$. For every
$\beta\in R$,
$\frac{2\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\in\mathbb{Z}$
and
$s_{\alpha}(\beta)=\beta-\frac{2\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\alpha\in
R$. Moreover, $s_{\alpha}(R_{reg})=R_{reg}$ and
$s_{\alpha}(R_{sing})=R_{sing}$.

  3. (3)

If $\alpha\in R_{sing}$ then for any $\beta\in R$ such that
$\langle\alpha,\beta\rangle\neq 0$, at least one of the vectors
$\beta+\alpha,\beta-\alpha$ belongs to $R$. Moreover, if $\alpha,\beta\in
R_{sing}$ and $\alpha+\beta\in R$ then $\alpha+\beta\in R_{reg}$.

This generalizes the definition of [Serga2], where in the third condition it
is required that exactly one of $\alpha+\beta,\alpha-\beta$ belongs to $R$.
The notion of GRR system also includes the root system $BC(m,n)$ given in [SV,
Sec. 2]. Note that $R_{reg}$ is a nonreduced root system.

We now prove that $R=\Delta({\mathfrak{g}},{\mathfrak{a}})$ is a GRR system
for every Iwasawa decomposition. Here $R_{sing}$ are the singular roots and
the inner product is given by the inner product of ${\mathfrak{a}}$.

###### Proof of Theorem C.

The property (1) follows from the same property of
$\Delta({\mathfrak{g}},{\mathfrak{h}})$. Let $\alpha\in R_{re}$. Assume that
$\alpha$ has an even root vector. This in particular implies that
$(\alpha,\alpha)\neq 0$. Moreover, there exists
$x\in{\mathfrak{g}}_{\alpha}\cap{\mathfrak{g}}_{\bar{0}}$ such that $(x,\theta
x)\neq 0$. Indeed, otherwise for every
$y,z\in{\mathfrak{g}}_{\alpha}\cap{\mathfrak{g}}_{\bar{0}}$, one has
$(y,\theta z)=\frac{1}{4}\left((y+z,\theta y+\theta z)-(y-z,\theta y-\theta
z)\right)=0$, which contradicts the fact that $(\cdot,\cdot)$ is non
degenerate. Now, since $(x,\theta x)\neq 0$, the subalgebra
$\mathfrak{s}_{\alpha}=\operatorname{span}_{\mathbb{R}}\\{x,\theta x,[x,\theta
x]\\}$ is isomorphic to $\mathfrak{sl}_{2}(\mathbb{R})$ so (2) follows for
$\alpha$ with an even root vector. We are left to prove (2) for $k\alpha$
where $\alpha$ has an even root vector. Since
$\bigoplus_{r\in\mathbb{C}}{\mathfrak{g}}_{r\alpha}$ is a finite-dimensional
$s_{\alpha}$-module, of $k$ must be in $\frac{1}{2}\mathbb{Z}$ and $(2)$
holds.

For (3), we first show that $(\alpha,\beta)$ is a nonzero multiple of
$\beta(h_{\alpha})$ for any $\alpha,\beta\in R$. Let
$x,y\in{\mathfrak{g}}_{\alpha}$ be such that $(x,\theta y)\neq 0$. Then
$H_{\alpha}=p_{\mathfrak{a}}([x,\theta y])$ is a nonzero scalar multiple of
$h_{\alpha}$. Indeed, for any $h\in{\mathfrak{a}}$,

$(H_{\alpha},h)=(p_{\mathfrak{a}}([x,\theta y]),h)=([x,\theta y],h)=(x,[\theta
y,h])=\alpha(h)(x,\theta y).$

Now, $(x,\theta y)\neq 0$ and $\alpha(h)$ should be nonzero for some $h$.
Hence $H_{\alpha}\neq 0$. Moreover, the above equality shows that
$H_{\alpha}\in({\rm Ker}\alpha)^{\perp}$ and so proportional to $h_{\alpha}$.
Let $t_{\alpha}\in{\mathfrak{a}}$ be the dual element to $\alpha$, that is
$(t_{\alpha},h)=\alpha(h)$ for all $h\in{\mathfrak{a}}$. Then $t_{\alpha}$ is
also proportional to $h_{\alpha}$ and we get that

$(\alpha,\beta)=(t_{\alpha},t_{\beta})\neq 0$

if and only if $(h_{\alpha},h_{\beta})\neq 0$. This is equivalent to if and
only if $(H_{\alpha},H_{\beta})\neq 0$. These inequalities are equivalent to
$\alpha(H_{\beta})\neq 0$ because

$(H_{\alpha},H_{\beta})=([x,\theta y],H_{\beta})=(x,[\theta
y,H_{\beta}])=\alpha(H_{\beta})(x,\theta y)$

and $(x,\theta y)\neq 0$. By symmetry this is equivalent to
$\beta(H_{\alpha})\neq 0$. Thus, $(\alpha,\beta)$ is a nonzero multiple of
$\beta(h_{\alpha})$.

Suppose that $a\in R_{im}$, $\beta\in R$ but $\beta\pm\alpha\notin R$. This
means that for any $z\in{\mathfrak{g}}_{\beta}$,
$[{\mathfrak{g}}_{\alpha},z]=[{\mathfrak{g}}_{-\alpha},z]=0$. In particular
$[x,z]=[y,z]=[\theta x,z]=[\theta y,z]=0$. This implies that

$0=[[x,\theta y]-\theta([x,\theta y]),z]=2[p_{\mathfrak{a}}([x,\theta
y]),z]=2[H_{\alpha},z]=2\beta(H_{\alpha})z.$

Thus $\beta(H_{\alpha})=0$ as required.

Suppose $\alpha,\beta\in R_{sing}$. Then
${\mathfrak{g}}_{\alpha},{\mathfrak{g}}_{\beta}$ are purely odd. Since
${\mathfrak{g}}$ is a Kac-Moody, if $\alpha+\beta\in R$, then
${\mathfrak{g}}_{\alpha+\beta}=[{\mathfrak{g}}_{\alpha},{\mathfrak{g}}_{\beta}]$.
Hence ${\mathfrak{g}}_{\alpha+\beta}$ must contain a nonzero even vector and
so $\alpha+\beta\in R_{reg}$. ∎

##  3\. The Image of the Chavalley Restriction Map

We begin proving Theorem A. We start by showing that the image of the
Chevalley restriction map lies in the space $I({\mathfrak{a}}^{*})$. This
means that the image satisfies two types of invariance conditions:
$W$-invariance and a condition related to roots whose root spaces are purely
odd.

To show that the image is $W$-invariant, we note that
${\mathfrak{a}}\subset{\mathfrak{p}}_{\bar{0}}$. This means we can first
restrict functions from ${\mathfrak{p}}$ to ${\mathfrak{p}}_{\bar{0}}$ and
then to ${\mathfrak{a}}$. The restriction from ${\mathfrak{p}}$ to
${\mathfrak{p}}_{\bar{0}}$ gives a ${\mathfrak{k}}_{\bar{0}}$-invariant
function. The invariance under a reflection $s_{\alpha}$, where $\alpha$ is a
root with an even root vector, can be shown using the representation theory of
the corresponding $\mathfrak{sl}(2)$-triple.

We now prove that the second type of invariance condition holds. Suppose that
the root space $\alpha$ is purely odd but one of its multiples has an even
root vector. Then the derivative condition follows from the following lemma.

Let
$\mathfrak{p}_{\alpha}:=\left(\left(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{-\alpha}\right)\cap\mathfrak{p}\right)\oplus\mathfrak{a}$
and
${\mathfrak{k}}_{\alpha}=\left({\mathfrak{g}}_{\alpha}\oplus{\mathfrak{g}}_{-\alpha}\right)\cap{\mathfrak{k}}$.
We prove the derivative condition by first restricting to
$S({\mathfrak{p}}_{\alpha})^{{\mathfrak{k}}_{\alpha}}$.

######  Proposition 3.1.

For a singular root $\alpha$ of multiplicity $2n$, the projection $f_{0}$ of
$f\in S(\mathfrak{p}^{*})^{\mathfrak{k}}$ to $S(\mathfrak{a}^{*})$ satisfies

$\left(D_{h_{\alpha}}\right)^{k}\left(f_{0}\right)\in\left\langle\alpha\right\rangle$

for $k=1,3,\ldots,2n-1$, that is
$\left(D_{h_{\alpha}}\right)^{k}\left(f_{0}\right)$ is zero on ${\rm
Ker}\alpha$.

Let us examine the action of ${\mathfrak{k}}_{\alpha}$ on
${\mathfrak{p}}_{\alpha}$. By Remark 2.2, we can take $e_{1},\ldots
e_{n},e_{1}^{\prime},\ldots,e_{n}^{\prime}$ to be a basis of
${\mathfrak{g}}_{\alpha}$ such that
$B_{\theta}(e_{i},e_{j}^{\prime})\in\mathbb{C}^{*}\delta_{ij}$,
$B_{\theta}\left(e_{i},e_{j}\right)=B_{\theta}\left(e_{i}^{\prime},e_{j}^{\prime}\right)=0$
and they are normalized such that

$\left[e_{i}+\theta e_{i},e_{j}^{\prime}-\theta
e_{j}^{\prime}\right]=\delta_{ij}h_{\alpha}.$

We use the fact that for any $e,e^{\prime}\in{\mathfrak{g}}_{\alpha}$,
$[e,e^{\prime}]=0$, for otherwise, $[e,e^{\prime}]$ is an even root vector of
$2\alpha$. Note that for any $h\in\mathfrak{a}$,

$\left[e_{i}+\theta e_{i},h\right]=-\alpha(h)\left(e_{i}-\theta e_{i}\right).$

Let
$\xi_{1},\ldots,\xi_{n},\eta_{1},\ldots,\eta_{n}\in{\mathfrak{p}}^{*}_{\alpha}$
be such that

$\xi_{i}\left(e_{j}-\theta e_{j}\right)=\eta_{i}\left(e_{j}^{\prime}-\theta
e_{j}^{{}^{\prime}}\right)=\delta_{ij},$ $\xi_{i}\left(e_{j}^{\prime}-\theta
e_{j}^{\prime}\right)=\eta_{i}\left(e_{j}-\theta e_{j}\right)=0,$

and

$\xi_{i}\left(\mathfrak{a}\right)=\eta_{i}\left(\mathfrak{a}\right)=0.$

Let $f\in S(\mathfrak{p}_{\alpha}^{*})$. It follows from the above that

(1) $\left(e_{i}+\theta e_{i}\right).f=\alpha\frac{\partial
f}{\partial\xi_{i}}-\eta_{i}D_{h_{\alpha}}\left(f\right).$

###### Proof of Proposition 3.1.

Let $f\in S(\mathfrak{p}^{*})^{\mathfrak{k}}$. Then

$f|_{{\mathfrak{p}}_{\alpha}}=f_{0}+\sum
f_{i_{1},\ldots,i_{m}}^{j_{1},\ldots,j_{m^{\prime}}}\xi_{i_{1}}\cdots\xi_{i_{m}}\eta_{j_{1}}\cdots\eta_{j_{m^{\prime}}}$

where $f_{0},f_{i_{1},\ldots,i_{m}}^{j_{1},\ldots,j_{m^{\prime}}}\in
S(\mathfrak{a}^{*})$ and the sum runs on all subsets
$\left\\{i_{1},\ldots,i_{m}\right\\},\left\\{j_{1},\ldots,j_{m^{\prime}}\right\\}\subseteq\left\\{1,\ldots,n\right\\}.$
By $\mathfrak{k}$-invariance, $\left(e_{k}+\theta
e_{k}\right).\left(f\right)=0$. Denote $f_{k}:=f_{1,\ldots,k}^{1,\ldots,k}$,
$k\geq 1$ and $f_{0}^{0}:=f_{0}$. By (1), the
$S(\mathfrak{a}^{*})$-coefficient of
$\xi_{1}\cdots\xi_{k-1}\eta_{1}\ldots\eta_{k}$ in $L_{e_{k}+\theta
e_{k}}\left(f\right)$ is

(2) $\alpha f_{k}-D_{\text{$h_{\alpha}$}}f_{k-1}=0.$

for $k\leq n$.

We claim that for $0\leq j\leq n-1$ , one has

$\left(D_{h_{\alpha}}\right)^{2j+1}f_{0}\in\left\langle\alpha\right\rangle.$

In fact, we claim that $\left(D_{h_{\alpha}}\right)^{2j+1}f_{0}$ is a linear
combination of terms of the form $\alpha_{1}^{i}f_{1,\ldots,k}^{1,\ldots,k}$
for $i$ odd and $k\leq 2j+1$. We prove this claim by induction on $j$. For
$j=0$, it follows from (2). Suppose it holds for $j-1$ . Let us show that it
holds for $j$. One has

$\displaystyle\left(D_{h_{\alpha}}\right)^{2j+1}f_{0}$
$\displaystyle=\left(D_{h_{\alpha}}\right)^{2}\left(\left(D_{h_{\alpha}}\right)^{2j-1}f_{0}\right)$
$\displaystyle\stackrel{{\scriptstyle\text{induction}}}{{=}}\left(D_{h_{\alpha}}\right)^{2}\left(\sum_{i\text{
odd, }k\leq 2j-1}a_{i,k}\alpha^{i}f_{k}\right)$
$\displaystyle=\left(D_{h_{\alpha}}\right)\left(\sum_{i\text{ odd, }k\leq
2j-1}a_{i,k}\left(\alpha(h_{\alpha})i\alpha^{i-1}f_{k}+\alpha^{i+1}f_{k+1}\right)\right)$
$\displaystyle=\sum_{i\text{ odd, }k\leq
2j-1}a_{i,k}\left(\alpha(h_{\alpha})^{2}i(i-1)\alpha^{i-2}f_{k}+\alpha(h_{\alpha})\left(2i+1\right)\alpha^{i}f_{k+1}+\alpha^{i+2}f_{k+2}\right).$

Thus
$\left(D_{h_{\alpha}}\right)^{2j+1}f_{0}\in\left\langle\alpha\right\rangle$
and the assertion follows. ∎

######  Remark 3.2.

Note that one can not continue the argument for $j\geq n$ since (2) does not
give information about $D_{h_{\alpha}}f_{n}$.

######  Example 3.3.

Suppose that $n=3$, that is, the multiplicity of $\mathfrak{g}_{\alpha}$ is 6.
Denote $D:=D_{h_{\alpha}}$ and $c:=\alpha(h_{\alpha})$. Then

$Df_{0}=\alpha f_{1}$ $D^{2}f_{0}=c^{2}f_{1}+\alpha^{2}f_{2}$
$D^{3}f_{0}=\left(c^{2}+2c\right)\alpha f_{2}+\alpha^{3}f_{3}$
$D^{4}f_{0}=\left(c^{3}+2c^{2}\right)f_{2}+\left(c^{2}+2c+3\right)\alpha^{2}f_{3}+\alpha^{4}f_{4}$
$D^{5}f_{0}=\left(3c^{3}+4c_{4}^{2}+6c\right)\alpha
f_{3}+\left(c^{2}+2c+3\right)\alpha^{3}f_{4}+\alpha^{5}f_{5}.$

##  4\. Injectivity of the Chevalley restriction map

The injectivity of the Chevalley restriction map follows from the following
general lemma.

######  Lemma 4.1.

Let ${\mathfrak{g}}$ be a finite-dimensional Lie superalgebra and $V$ be a
finite dimensional ${\mathfrak{g}}$-module and assume that
${\mathfrak{g}}_{\bar{1}}V_{\bar{0}}=V_{\bar{1}}$. Then the restriction map
$\mathbb{C}[V]^{\mathfrak{g}}\to\mathbb{C}[V_{\bar{0}}]^{{\mathfrak{g}}_{\bar{0}}}$
is injective.

###### Proof.

Choose a basis $\xi_{1},\dots,\xi_{n}$ for $V_{\bar{1}}^{*}$. We use the
identity

$\mathbb{C}[V]=\mathbb{C}[V_{\bar{0}}]\otimes\Lambda(\xi_{1},\dots,\xi_{n})$

and introduce a $\mathbb{Z}$-grading on $\mathbb{C}[V]$ by setting
$\mathbb{C}[V_{0}]$ to have degree zero and all to have $\xi_{i}$ degree $1$.
The condition ${\mathfrak{g}}_{1}V_{\bar{0}}=V_{\bar{1}}$ implies that in the
dual representation $V^{*}$, for any nonzero $\xi\in V^{*}_{\bar{1}}$, we have
${\mathfrak{g}}_{\bar{1}}\xi\neq 0$. Indeed,

$\langle{\mathfrak{g}}_{\bar{1}}\xi,V_{\bar{0}}\rangle=\langle\xi,{\mathfrak{g}}_{\bar{1}}V_{\bar{0}}\rangle=\langle\xi,V_{\bar{1}}\rangle.$

Assume that $f\in\mathbb{C}[V]^{\mathfrak{g}}$ and $f(V_{\bar{0}})=0$. Suppose
by contradiction that $f\neq 0$. Write $f=\sum_{i>0}f_{i}$ (the sum of graded
terms) and let $p$ be the smallest such that $f_{p}\neq 0$. If
$X\in{\mathfrak{g}}_{\bar{1}}$, then $X=X^{\prime}+X^{\prime\prime}$ with
$X^{\prime}\mathbb{C}[V]_{i}\subset\mathbb{C}[V]_{i-1}$ and
$X^{\prime\prime}\mathbb{C}[V]_{i}\subset\mathbb{C}[V]_{i+1}$. Note that for
any $\xi\in V^{*}_{\bar{1}}$ and $X\in{\mathfrak{g}}_{\bar{1}}$, one has
$X\xi=X^{\prime}\xi$. Since $f$ is ${\mathfrak{g}}_{\bar{1}}$-invariant we get
$X^{\prime}f_{p}=0$ for all $X\in{\mathfrak{g}}_{\bar{1}}$. Our goal is to
show that $f_{p}=0$, leading to a contradiction. We consider $\mathbb{C}[V]$
as a free $\mathbb{C}[V_{\bar{0}}]$-module and note that $X^{\prime}$ is an
endomorphism of $\mathbb{C}[V_{\bar{0}}]$-modules.

We can localize and work over the field of fractions
$\mathbb{C}(V_{\bar{0}})$. Since ${\mathfrak{g}}_{\bar{1}}\xi\neq 0$ for any
nonzero $\xi\in V^{*}_{\bar{1}}$, we can choose a
$\mathbb{C}(V_{\bar{0}})$-basis $\eta_{1},\dots\eta_{n}$ in
$\mathbb{C}(V_{\bar{0}})\otimes V^{*}_{1}$ such that
$\frac{\partial}{\partial\eta_{i}}=X_{i}^{\prime}$ for some
$X_{i}\in{\mathfrak{g}}_{\bar{1}}$. Indeed, suppose $X_{1}\in\mathfrak{g}_{1}$
is such that $X^{\prime}_{1}\xi_{1}\neq 0$ and set $\hat{\xi}_{1}=\xi_{1}$.
Choose $\hat{\xi}_{2}\in{\rm Ker}X^{\prime}_{1}$ and let
$X^{\prime}_{2}\in\mathfrak{g}_{1}$ be such that
$X^{\prime}_{2}\hat{\xi}_{2}\neq 0$. We continue choosing
$\hat{\xi}_{i}\in{\rm Ker}X_{i-1}^{\prime}$ and taking $X_{i}$ such that
$X_{i}\hat{\xi}_{i}\neq 0$. We get that $X_{i}^{\prime}\hat{\xi}_{i}$ is a
nonzero element of $\mathbb{C}(V_{\bar{0}})$ and
$X_{i}^{\prime}\hat{\xi}_{j}=0$ for $j\geq i$. We get that
$X_{i}^{\prime}\hat{\xi}_{j}=a_{ij}\in\mathbb{C}(V_{\bar{0}})$ where
$a_{ii}\neq 0$ and $a_{ij}=0$ for $i<j$. Hence, we can pick the basis
$\eta_{1},\ldots,\eta_{n}$ such that $X_{i}^{\prime}\eta_{j}=\delta_{ij}$ and
so $X_{i}^{\prime}=\frac{\partial}{\partial\eta_{i}}$. However any $f_{p}$
annihilated by all $\frac{\partial}{\partial\eta_{i}}$ must be zero if $p>0$.
This leads to $f_{p}=0$. ∎

######  Corollary 4.2.

Let ${\mathfrak{g}}$ be a Lie superalgebra of some algebraic group $G$ and $V$
be a finite-dimensional $G$-module. Assume that $W\subset V_{0}$ is a subspace
such that $W+{\mathfrak{g}}W=V$. Then the restriction map
$\mathbb{C}[V]^{\mathfrak{g}}\to\mathbb{C}[W]$ is injective.

###### Proof.

We note that $G_{0}W$ is Zarisky dense in $V_{0}$. Hence the restriction map
$\mathbb{C}[V_{0}]^{{\mathfrak{g}}_{0}}\to\mathbb{C}[W]$ is injective. The
composition with the injective map
$\mathbb{C}[V]^{\mathfrak{g}}\to\mathbb{C}[V_{\bar{0}}]^{{\mathfrak{g}}_{\bar{0}}}$
is also injective. ∎

We conclude the following proposition:

######  Proposition 4.3.

Let ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$ have Iwasawa
decomposition. Then the restriction map
$\mathbb{C}[{\mathfrak{p}}]^{\mathfrak{k}}\to\mathbb{C}[\mathfrak{a}]$ is
injective.

###### Proof.

We imply the previous corollary with $V={\mathfrak{p}}$,
${\mathfrak{g}}={\mathfrak{k}}$ and $W=\mathfrak{a}$. The condition
$\mathfrak{a}+{\mathfrak{k}}\mathfrak{a}={\mathfrak{p}}$ follows from
$[{\mathfrak{g}}_{\alpha},\mathfrak{a}]={\mathfrak{g}}_{\alpha}$ for any
restricted root $\alpha$. ∎

##  5\. Surjectivity of the Chevalley restriction map.

We describe $I({\mathfrak{a}}^{*})$ in all cases and show that the Chevalley
restriction map $C_{\theta}$ surjects onto $I({\mathfrak{a}}^{*})$. This
completes the proof of Theorem A. The computation of the set of restricted
roots in each case, as well as of $h_{\alpha}$ for every singular root is
given in the Appendix. We also give the image of the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ and prove Theorem B.

For most cases, we use the following argument to show that $C_{\theta}$ is
surjective. Take the following commutative diagram of restriction maps,

${S({\mathfrak{g}}^{*})^{\mathfrak{g}}}$${I({\mathfrak{h}}^{*})}$${S({\mathfrak{p}}^{*})^{\mathfrak{k}}}$${I({\mathfrak{a}}^{*})}$$\scriptstyle{C}$$\scriptstyle{R_{\theta}}$$\scriptstyle{R}$$\scriptstyle{C_{\theta}}$

Note that $C$ is an isomorphism by [Serge2] and $C_{\theta}$ is an embedding.
Thus, $C_{\theta}$ is surjective if $R$ is surjective. Moreover, $R_{\theta}$
is surjective if and only if $R$ is surjective.

###  5.1. Case
${\mathfrak{g}}=\mathfrak{gl}(m|2n),\mathfrak{k}=\mathfrak{osp}(m|2n)$.

Here

$I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon_{1},\ldots,\varepsilon_{m},\delta_{1},\ldots,\delta_{n}]^{S_{m}\times
S_{n}}\mid\left(\frac{2\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\delta_{1}}\right)f\in\left<\varepsilon_{1}-\delta_{1}\right>\right\\},$

(see Case 6.2 in the Appendix).

By [SV, Thm. 2], $I({\mathfrak{a}}^{*})$ is generated by
$\phi_{k}=\varepsilon_{1}^{k}+\ldots+\varepsilon_{m}^{k}-2\delta_{1}^{k}-\dots-2\delta_{n}^{k}$,
$k\in\mathbb{Z}_{\geq 1}$, which are the projections of
$\bar{\phi}_{k}=\bar{\varepsilon}_{1}^{k}+\ldots+\bar{\varepsilon}_{m}^{k}-\bar{\delta}_{1}^{k}-\dots-\bar{\delta}_{2n}^{k}$
fromß $I({\mathfrak{h}}^{*})$. Hence the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ is surjective and so is
$R_{\theta}$.

In is natural to consider the same involution on the subalgebras of
${\mathfrak{g}}$. Note that the center of $\mathfrak{gl}(m|n)$ belongs to
${\mathfrak{a}}$. If $m\neq 2n$, we can also consider the pair
${\mathfrak{g}}=\mathfrak{sl}(m|2n)$, ${\mathfrak{k}}=\mathfrak{osp}(m|2n)$.
Let $I_{\mathfrak{gl}}({\mathfrak{a}}^{*})$ be the ring
$I({\mathfrak{a}}^{*})$ above and $I_{\mathfrak{sl}}({\mathfrak{a}}^{*})$ be
the ring corresponding to the pair ${\mathfrak{g}}=\mathfrak{sl}(m|2n)$,
${\mathfrak{k}}=\mathfrak{osp}(m|2n)$. Then

$I_{\mathfrak{sl}}({\mathfrak{a}}^{*})\cong
I_{\mathfrak{gl}}({\mathfrak{a}}^{*})/\langle\varepsilon_{1}+\ldots+\varepsilon_{m}-\delta_{1}-\ldots-\delta_{n}\rangle.$

The algebra $\mathfrak{sl}(n|n)$ does not have a bilinear form so we consider
instead the pair ${\mathfrak{g}}=\mathfrak{psl}(2n|2n)$,
${\mathfrak{k}}=\mathfrak{osp}(2n|2n)$. Denote the corresponding ring of
invariants by $I_{\mathfrak{psl}}({\mathfrak{a}}^{*})({\mathfrak{a}}^{*})$.
For $n\geq 2$, we have that

$I_{\mathfrak{psl}}({\mathfrak{a}}^{*})\cong
I_{\mathfrak{sl}}({\mathfrak{a}}^{*})\cap\mathbb{C}\left[\varepsilon_{i}-\varepsilon_{j},\varepsilon_{i}-\delta_{k},\delta_{l}-\delta_{k}\mid
i,j=1,\ldots,2n;k,l=1,\ldots,n\right].$

For $n=1$, we first note that the sets of even and odd roots of
$\mathfrak{psl}(2|2)$ are $\bar{\Delta}_{\bar{0}}=\\{\pm
2\bar{\varepsilon},\pm 2\bar{\delta}\\}$ and
$\bar{\Delta}_{\bar{1}}=\\{\pm\bar{\varepsilon}\pm\bar{\delta}\\}$,
respectively. Hence the restricted roots of the pair
${\mathfrak{g}}=\mathfrak{psl}(2|2)$, ${\mathfrak{k}}=\mathfrak{osp}(2|2)$ are
$\pm 2\varepsilon$ with multiplicity $(1|0)$ and $\pm\varepsilon$ with
multiplicity $(0|2)$. Hence, in this case
$I({\mathfrak{a}}^{*})=\mathbb{C}[\varepsilon^{2}]$.

For these cases, the restriction map from $I(\mathfrak{h}^{*})$ to
$I({\mathfrak{a}}^{*})$ is surjective by [Serge2, 0.6.2] by a similar argument
as the one for ${\mathfrak{g}}=\mathfrak{gl}(m|2n)$.

###  5.2. Case ${\mathfrak{g}}=\mathfrak{gl}(2m+a|2n+b)$,
${\mathfrak{k}}=\mathfrak{gl}(m|n)\oplus\mathfrak{gl}(m+a|n+b)$, $a,b\geq 0$.

Here

$\displaystyle
I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon_{1}^{2},\ldots,\varepsilon_{m}^{2},\delta_{1}^{2},\ldots,\delta_{n}^{2}]^{S_{m}\times
S_{n}}\mid\left(\frac{\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\delta_{1}}\right)f\in\left<\varepsilon_{1}-\delta_{1}\right>\right\\}$

(see Case 6.2 in the Appendix).

By [SV, Thm. 2], $I({\mathfrak{a}}^{*})$ is generated by
$\phi_{2k}=\varepsilon_{1}^{2k}+\ldots+\varepsilon_{m}^{2k}-\delta_{1}^{2k}-\dots-\delta_{n}^{2k}$,
$k\in\mathbb{Z}_{\geq 1}$, which are the projections of
$\bar{\phi}_{2k}=\frac{1}{2}(\bar{\varepsilon}_{1}^{2k}+\ldots+\bar{\varepsilon}_{2m+a}^{2k}-\bar{\delta}_{1}^{2k}-\dots-\bar{\delta}_{2n+b}^{2k})$
from $I({\mathfrak{h}}^{*})$. Hence the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ is surjective and so is
$R_{\theta}$.

###  5.3. Case ${\mathfrak{g}}=\mathfrak{osp}(2m+a|4n+2b)$,
$\mathfrak{k}=\mathfrak{osp}(m|2n)\oplus\mathfrak{osp}(m+a|2n+2b)$, $a\geq 1$,
$b\geq 0$.

Here

$I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon_{1}^{2},\ldots,\varepsilon_{m}^{2},\delta_{1}^{2}\ldots\delta_{n}^{2}]^{S_{m}\times
S_{n}}\mid\left(\frac{2\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\delta_{1}}\right)f\in\left<\varepsilon_{1}-\delta_{1}\right>\right\\}$

(see Case 6.2 in the Appendix). Similarly to the case in Section 5.1,
$I({\mathfrak{a}}^{*})$ is generated by
$\phi_{2k}=\varepsilon_{1}^{2k}+\ldots+\varepsilon_{m}^{2k}-2\delta_{1}^{2k}-\dots-2\delta_{n}^{2k}$,
$k\in\mathbb{Z}_{\geq 1}$, which are the projections of
$\bar{\phi}_{2k}=\bar{\varepsilon}_{1}^{2k}+\ldots+\bar{\varepsilon}_{2m+a}^{2k}-\bar{\delta}_{1}^{2k}-\dots-\bar{\delta}_{2n+b}^{2k}$
in $I({\mathfrak{h}}^{*})$. Hence the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ is surjective and so is
$R_{\theta}$.

###  5.4. Case
${\mathfrak{g}}=\mathfrak{osp}(2m|4n+2b),\mathfrak{k}=\mathfrak{osp}(m|2n)\oplus\mathfrak{osp}(m|2n+2b)$,
$b\geq 0$.

This case is also part of Case 6.2, described in the Appendix. However, here
the root $\varepsilon_{i}$ is a singular root. The Weyl group $W$ acts on
$\varepsilon_{1},\ldots,\varepsilon_{m}$ as a group of type $D$ and we have
the additional condition that
$\left(\frac{\partial}{\partial\varepsilon_{1}}\right)^{k}f\in\left<\varepsilon_{1}\right>$
for $k=1,3\ldots,2b-1$. Thus
$I({\mathfrak{a}}^{*})=I_{1}\left(\mathfrak{a}^{*}\right)\oplus
I_{2}\left(\mathfrak{a}^{*}\right)$ where

$\displaystyle I_{1}\left(\mathfrak{a}^{*}\right)=$
$\displaystyle\left\\{f\in\mathbb{C}\left[\varepsilon_{1}^{2},\ldots,\varepsilon_{m}^{2},\delta_{1}^{2},\ldots,\delta_{n}^{2}\right]^{S_{m}\times
S_{n}}\mid\left(\frac{2\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\delta_{1}}\right)f\in\left\langle\varepsilon_{1}-\delta_{1}\right\rangle\right\\},$
$\displaystyle I_{2}\left(\mathfrak{a}^{*}\right)=$
$\displaystyle\left\\{f\in\left(\varepsilon_{1}\cdots\varepsilon_{m}\right)^{2b+1}\mathbb{C}\left[\varepsilon_{1}^{2},\ldots,\varepsilon_{m}^{2},\delta_{1}^{2},\ldots,\delta_{n}^{2}\right]^{S_{m}\times
S_{n}}\mid\left(\frac{2\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\delta_{1}}\right)f\in\left\langle\varepsilon_{1}-\delta_{1}\right\rangle\right\\}.$

We prove surjectivity by showing that the dimensions of
$S(\mathfrak{p}^{*})_{d}^{\mathfrak{k}}$ and $I(\mathfrak{a}^{*})_{d}$ are
equal for any degree $d$. We express these dimensions in term of certain
partitions: a partitions $\lambda$ of $d$ is called _regular_ if it is
contained in the fat $\left(m|2n\right)$-hook and either

  * •

All the parts $\lambda_{i}$ are even; or

  * •

$\lambda_{1},\ldots,\lambda_{m}$ are odd, $\lambda_{m}\geq 2n+2b+1$ and
$\lambda_{i}$ is even for $i\geq m+1$.

In the first case, we call $\lambda$ an _even regular_ partition, and in the
second case, an _odd regular_ partition. Note that the diagram of an odd
regular partition must contain the $m\times(2n+2b+1)$ rectangle.

######  Example 5.1.

Let $m=2$, $n_{1}=2$ and $b=1$. Then the following paritions are regular:

The following partition is not regular (but is regular for $b=0$):

######  Lemma 5.2.

The dimension of $S(\mathfrak{p}^{*})_{d}^{\mathfrak{k}}$ is at least the
numbers of regular partitions whose parts sum to $d$.

###### Proof.

In this case $\mathfrak{p^{*}}$ is isomorphic to
$\mathbb{C}^{m|2n}\otimes\mathbb{C}^{m|2n+2b}$ as $\mathfrak{k}$-modules. We
view $\mathbb{C}^{m|2n}\otimes\mathbb{C}^{m|2n+2b}$ as a
$\mathfrak{g}^{\prime}:=\mathfrak{gl}(m|2n)\oplus\mathfrak{gl}(m|2n+2b)$-module
which was restricted to $\mathfrak{k}$. By [CW, Thm. 3.2], we have the
folowing decomposition of $\mathfrak{g}^{\prime}$-modules

$S\left(\mathbb{C}^{m|2n}\otimes\mathbb{C}^{m|2n+2b}\right)_{d}\cong\bigoplus_{\lambda}V_{m|2n}^{\lambda}\otimes
V_{m|2n+2b}^{\lambda},$

where the sum is over all paritions $\lambda$ of $d$ satisfying
$\lambda_{m+1}\leq 2n\leq 2n+2b$. Here the highest weight of the simple
modules $V_{m|2n}^{\lambda}$ and $V_{m|2n+2b}^{\lambda}$ is

$\lambda=\lambda_{1}\varepsilon_{1}+\ldots\lambda_{m}\varepsilon_{m}+\left\langle\lambda^{\prime}_{1}-m\right\rangle\delta_{1}+\ldots+\left\langle\lambda^{\prime}_{2n}-m\right\rangle\delta_{2n},$

where $\left\langle r\right\rangle:=\max\left\\{0,r\right\\}$ (for
$V_{m|2n+2b}^{\lambda}$ there is zero multiple of
$\delta_{2n+1}+\ldots+\delta_{2n+2b}$).

Suppose that $\lambda$ is an even regular partition.

By [SS, Prop. 4.6]111In the notation of [SS], the partitions do not have to be
even but the correspondence between partions to weights is “doubled”, see [SS,
(32)], the dimension of the invariants under $\mathfrak{osp}\left(m|2n\right)$
(resp. $\mathfrak{osp}\left(m|2n+b\right)$) in $V_{m|2n}^{\lambda}$ (resp.
$V_{m|2n+2b}^{\lambda}$) is one. Hence $V_{m|2n}^{\lambda}\otimes
V_{m|2n+b}^{\lambda}$ contains a nonzero $\mathfrak{k}$-fixed vector.

We are left to construct a nonzero $\mathfrak{k}$-fixed vector for every odd
regular partition $\mu$. Let $\mathfrak{str}_{m|2n}$ and
$\mathfrak{str}_{m|2n+2b}$ be the super-trace module over
$\mathfrak{gl}\left(m|2n\right)$ and $\mathfrak{gl}\left(m|2n+2b\right)$,
respectively. Then

$\displaystyle V_{m|2n}^{\lambda}\otimes\mathfrak{str}_{m|2n}\cong
V_{m|2n}^{\mu},\quad V_{m|2n+2b}^{\lambda}\otimes\mathfrak{str}_{m|2n+2b}\cong
V_{m|2n+2b}^{\mu}.$

where $\lambda$ is the partition obtained from $\mu$ by removing a box from
the first $n$-rows and adding a box to the first $2n$ (resp. $2n+2b$) columns.
Note that this can be done since $\lambda_{m}\geq 2n+2b+1\geq 2n+1$).
Moreover, tensoring with the super-trace module yields an isomorphic module
over $\mathfrak{osp}\left(m|2n\right)$ (resp.
$\mathfrak{osp}\left(m|2n+2b\right)$) and $\lambda$ is an even regular
partition. Hence

$\dim\left(V_{m|2n}^{\mu}\right)^{\mathfrak{osp}\left(m|2n\right)}=\dim\left(V_{m|2n+2b}^{\mu}\right)^{\mathfrak{osp}\left(m|2n+2b\right)}=1,$

and $V_{m|2n}^{\mu}\otimes V_{m|2n+2b}^{\mu}$ contains a
$\mathfrak{k}$-invariant vector as desired. ∎

######  Lemma 5.3.

The dimension of $I({\mathfrak{a}}^{*})_{d}$ is at most the numbers of regular
partitions whose parts sum to $d$.

To prove the lemma, we show that the leading term of an $f\in
I({\mathfrak{a}}^{*})_{d}$ corresponds to a regular partition. However, the
order on monomials and the correspondence to partitions is not the standard
one. We demonstrate the idea of the proof in the following example.

######  Example 5.4.

Let $m=3$, $n=3$ and $b=0$. We take the order on monomials corresponding to
the lexical order on $x_{1}>x_{2}>y_{1}>y_{2}>y_{3}>x_{2}$. The monomial
$x_{1}^{11}x_{2}^{11}y_{1}^{10}y_{2}^{8}y_{3}^{4}x_{3}^{3}$ corresponds to the
following regular partition

$(11,11|10,8,4|3)$$11$$11$$3$$10$$8$$4$

The monomial $x_{1}^{11}x_{2}^{7}y_{1}^{10}y_{2}^{8}y_{3}^{0}x_{3}^{3}$
supposedly corresponds to the following shape but we show that it can not be a
leading monomial of an element in $I(\mathfrak{a}^{*})$.

$11$$7$$3$$10$$8$$(11,7|10,8,0|3)$

Our proof shows that in this case, the powers of $y_{1},y_{2},y_{3}$ are
greater or equal to $2$ and that the power of $x_{2}$ is at least the power of
$x_{3}$ plus 6.

###### Proof of Lemma 5.3.

By [SV, Prop. 2], the number of even regular partitions of total size $d$ is
equal to the dimension of $I_{1}\left({\mathfrak{a}}^{*}\right)_{d}$. We are
left to show that the dimension of $I_{2}({\mathfrak{a}}^{*})_{d}$ is equal to
the number of odd regular partitions of total size $d$.

Let $f\in I_{2}(\mathfrak{a}^{*})_{d}$ and denote by

$\left(\lambda_{1},\ldots,\lambda_{m-1}\ |\ \mu_{1},\ldots,\mu_{n}\ |\
\lambda_{m}\right):=x_{1}^{\lambda_{1}}\cdots
x_{m-1}^{\lambda_{m-1}}y_{1}^{\mu_{1}}\cdots
y_{n}^{\mu_{n}}x_{m}^{\lambda_{m}}$

the leading monomial of $f$ with respect to the order that corresponds to the
lexical order

$x_{1}>\ldots>x_{m-1}>y_{1}>\ldots>y_{n}>x_{m}.$

First we claim that $\mu_{i}\geq 2$ for all $i$. Suppose that $\mu_{j}=0$ for
some $j$. Since $\left(\frac{2\partial}{\partial
x_{m}}+\frac{\partial}{\partial y_{j}}\right)f\mid_{x_{m}=y_{j}}=0$, there
should be another monomial in $f$ of the form

$\left(\lambda_{1},\ldots,\lambda_{m-1}\ |\ \mu_{1},\ldots,r,\ldots,\mu_{n}\
|\ \lambda_{m}-r\right),$

where $r\in 2\mathbb{Z}_{\geq 2}$ is the power of $x_{j}$. This contradicts
the maximality of the first monomial.

Second, we claim that $\lambda_{m-1}\geq\lambda_{m}+2n$. Indeed, since
$\left(\frac{2\partial}{\partial x_{m}}+\frac{\partial}{\partial
y_{n}}\right)f\mid_{x_{m}=y_{n}}=0$ and by maximality of
$\left(\lambda_{1},\ldots,\lambda_{m-1}|\mu_{1},\ldots,\mu_{n}|\lambda_{m}\right)$,
we get that

$\left(\lambda_{1},\ldots,\lambda_{m-1}\ |\ \mu_{1},\ldots,\mu_{n}-2r_{0}\ |\
\lambda_{m}+2r_{0}\right)$

is also a monomial in $f$ for some $r_{0}\geq 1$. Next note that since
$\left(\frac{2\partial}{\partial x_{m}}+\frac{\partial}{\partial
y_{n-1}}\right)f\mid_{x_{m}=y_{n-1}}=0$ and by maximality of
$\left(\lambda_{1},\ldots,\lambda_{m-1}|\mu_{1},\ldots,\mu_{n}|\lambda_{m}\right),$we
get that

$\left(\lambda_{1},\ldots,\lambda_{m-1}\ |\
\mu_{1},\ldots,\mu_{n-1}-2r_{1},\mu_{n}-2r_{0}\ |\
\lambda_{m}+2r_{0}+2r_{1}\right)$

is also a monomial of $f$ for some $r_{1}\geq 1$. Repeating this argument with
$\left(\frac{2\partial}{\partial x_{m}}+\frac{\partial}{\partial
y_{i}}\right)$ for $i=n_{1}-3,\ldots,1$ we get that

$\left(\lambda_{1},\ldots,\lambda_{m-1}\ |\
\mu_{1}-2r_{n-1},\ldots,\mu_{n}-2r_{0}\ |\
\lambda_{m}+2\sum_{i=0}^{n-1}r_{i}\right)$

is a monomial of $f$ for $r_{0},\ldots,r_{n-1}\geq 1$. Suppose by
contradiction that $\lambda_{m-1}\leq 2n$. By the $S_{m}$-symmetry,

$\left(\lambda_{1},\ldots,\lambda_{m}+2\sum_{i=0}^{n-1}r_{i}\ |\
\mu_{1}-2r_{n-1},\ldots,\mu_{n}-2r_{0}\ |\ \lambda_{m-1}\right)$

is also a monomial of $f$ which is bigger than the leading monomial. Thus,
$\lambda_{m-1}\geq\lambda_{m}+2\sum_{i=0}^{n-1}r_{i}\geq\lambda_{m}+2n$.

Now, given that
$\left(\lambda_{1},\ldots,\lambda_{m-1}|\mu_{1},\ldots,\mu_{n}|\lambda_{m}\right)$
is such that $\mu_{1},\ldots,\mu_{n}\geq 2$ and
$\lambda_{m-1}\geq\lambda_{m}+2n$, we can associate the following partition to
$f$: the first $m$ rows are given by
$\left(\lambda_{1},\ldots,\lambda_{m-1},\lambda_{m}+2n\right)$ and the
following rows are given by the transpose of the partition
$\left(\frac{\mu_{1}}{2}-1,\frac{\mu_{1}}{2}-1,\ldots,\frac{\mu_{n}}{2}-1,\frac{\mu_{n}}{2}-1\right)$
(see Example 5.4). ∎

By [Serge2, 0.6.5], the elements of $I({\mathfrak{h}}^{*})$ are either such
that all variables have an even degree or divisible by $\prod_{1\leq i\leq
2m,1\leq j\leq 4m+2b}(\bar{\varepsilon}_{i}-\bar{\delta}_{j})$. Hence the
image of the restriction from $I({\mathfrak{h}}^{*})$ to
$I({\mathfrak{a}}^{*})$ is generated be the elements.

$\displaystyle\varepsilon_{1}^{2k}+\ldots+\varepsilon_{2m}^{2k}-\delta_{1}^{2k}-\ldots-\delta_{n}^{2k},$
$\displaystyle(\varepsilon_{1}\cdots\varepsilon_{m})^{2b+1}\prod(\varepsilon_{i}^{2}-\delta_{j}^{2})^{2}(\varepsilon_{1}^{2k}+\ldots+\varepsilon_{2m}^{2k})(\delta_{1}^{2l}+\ldots+\delta_{n}^{2l}),\quad
k,l\in\mathbb{Z}_{\geq 0}.$

We note that the minimal-degree element in $I({\mathfrak{a}}^{*})$ for which
the degree $\varepsilon_{1}$ is odd has degree $m(2n+2b+1)$. However the
minimal-degree element in the image of $I({\mathfrak{h}}^{*})$ for which the
degree $\varepsilon_{1}$ is odd has degree $m(4n+2b+1)$. Hence the restriction
map from $I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ is not surjective
and $R_{\theta}$ is not surjective either.

###  5.5. Case
${\mathfrak{g}}=\mathfrak{osp}(2m|2n),\mathfrak{k}=\mathfrak{gl}(m|n)$.

Let $m_{1}:=\left\lfloor\frac{m}{2}\right\rfloor$, then

$I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon_{1}^{2},\ldots,\varepsilon_{m_{1}}^{2},\delta_{1}^{2}\ldots\delta_{n}^{2}]^{S_{m_{1}}\times
S_{n}}\mid\left(\frac{\partial}{\partial\varepsilon_{1}}+\frac{2\partial}{\partial\delta_{1}}\right)f\in\left<\varepsilon_{1}-\delta_{1}\right>\right\\}$

(see Case 6.2 and Case 6.2 in the Appendix).

Similarly to the case in Section 5.1, $I({\mathfrak{a}}^{*})$ is generated by
$\phi_{2k}=2\varepsilon_{1}^{2k}+\ldots+2\varepsilon_{m_{1}}^{2k}-\delta_{1}^{2k}-\dots-\delta_{n}^{2k}$
which are the projections of
$\bar{\phi}_{2k}=\bar{\varepsilon}_{1}^{2k}+\ldots+\bar{\varepsilon}_{m}^{2k}-\bar{\delta}_{1}^{2k}-\dots-\bar{\delta}_{n}^{2k}$
in $I({\mathfrak{h}}^{*})$. Hence the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ is surjective and so is
$R_{\theta}$.

###  5.6. Case ${\mathfrak{g}}=F_{4},\mathfrak{k}=\mathfrak{gosp}(2|4)$.

Here

$I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon_{1}^{2},\varepsilon_{2}^{2},\delta^{2}]^{S_{2}}\mid\left(\frac{\partial}{\partial\varepsilon_{1}}+\frac{3}{2}\frac{\partial}{\partial\delta}\right)f\in\left<\varepsilon_{1}-\delta\right>\right\\},$

(see Case 6.2 in the Appendix). By [SV, Prop. 2], $\dim
I({\mathfrak{a}}^{*})_{2d}$ is at most the number of partitions of $d$ which
fit into the $(1|2)$-fat hook.

Let us show that $\dim S({\mathfrak{p}}^{*})^{\mathfrak{k}}_{2d}$ is at least
the number of such partitions. The ${\mathfrak{k}}$-module
${\mathfrak{p}}^{*}$ is isomorphic to $V\oplus V^{*}$ where $V$ is the module
described in [SSS, Sec. 3.2]. Hence $S({\mathfrak{p}}^{*})\cong S(V)\otimes
S(V)^{*}$. By [SSS, Prop. 3.6], $S(V)_{d}=\oplus_{\lambda\in E_{d}}L(\lambda)$
where $E_{d}$ is a set of weights in bijection with $\\{(q,2r,s)\mid
q+2r+s=d;\ q,r\geq 0,s\geq 2\\}\cup\\{d\\}$. This set is in bijection with the
set of partitions of $d$ in the $(1|2)$-fat hook by

$s$$2r$$q$$d$

Applying Schur’s Lemma to $S(V)_{d}\otimes S(V)_{d}^{*}\subseteq
S({\mathfrak{p}}^{*})_{2d}$, we obtain that the number of linearly independent
invariant vectors is at least the number irreducible summands of $S(V)_{d}$,
that is, the number of partitions of $d$ which fit into the $(1|2)$-fat hook.

By [SV, Thm. 2], $I({\mathfrak{a}}^{*})$ is generated by
$\phi_{2k}=\varepsilon_{1}^{2k}+\varepsilon_{2}^{2k}-\frac{2}{3}\delta^{2k}$,
$k\in\mathbb{Z}_{\geq 0}$. We note that $\dim I({\mathfrak{a}}^{*})_{4}=2$
whereas it follows from [Serge2, 0.6.8] that $\dim
I({\mathfrak{h}}^{*})_{4}=1$ and so the restriction map from
$I({\mathfrak{h}}^{*})$ to $I({\mathfrak{a}}^{*})$ can not be surjective. Thus
$R_{\theta}$ is not surjective in this case.

The image of the restriction from $I({\mathfrak{h}}^{*})$ to
$I({\mathfrak{a}}^{*})$ is generated by the images of the generators of
$I({\mathfrak{h}}^{*})$ (which are also listed in the next section), namely

$\phi_{2},\quad 4\phi_{6}-15\phi_{4}\phi_{2}\ \text{ and }\
(\delta^{2}-\varepsilon_{1}^{2})^{2}(\delta^{2}-\varepsilon_{2}^{2})^{2}(\varepsilon_{1}^{2i}+\varepsilon_{2}^{2i})\delta^{2j},i,j\in\mathbb{Z}_{\geq
0}.$

###  5.7. Case ${\mathfrak{g}}=F_{4},\mathfrak{k}=\mathfrak{sl}(1|4)$.

Here

$I(\mathfrak{a}^{*})=\left\\{f\in\mathbb{C}\left[\varepsilon^{2},\delta^{2}\right]\
|\
\left(\frac{\partial}{\partial\varepsilon}+\frac{3\partial}{\partial\delta}\right)^{i}f\in\left\langle\varepsilon-\delta\right\rangle,i=1,3\right\\}$

(see Case 6.2 in the Appendix).

Let us show that $I({\mathfrak{h}}^{*})$ surjects onto $I({\mathfrak{a}}^{*})$
and thus $R_{\theta}$ is also surjective. Similarly to the case in Section
5.8, $\dim\mathbb{C}\left[\varepsilon^{2},\delta^{2}\right]_{2d}=d+1$. Then
the condition $D_{h_{\alpha}}f\in\langle\alpha\rangle$ imposes one linear
relation. The condition $D_{h_{\alpha}}^{3}f\in\langle\alpha\rangle$ imposes
another linear relation for $d\geq 4$. We get that $\dim
I({\mathfrak{a}}^{*})_{2}=1$ and $\dim I({\mathfrak{a}}^{*})_{2d}=d-1$ for
$d\geq 2$. We claim that $\dim
I({\mathfrak{h}}^{*})_{2}\mid_{\mathfrak{a}}\geq 1$ and $\dim
I({\mathfrak{h}}^{*})_{2d}\mid_{\mathfrak{a}}\geq d-1$ for $d\geq 2$.

By [Serge2, 0.6.8], every element in $I({\mathfrak{h}}^{*})$ is of the form
$f=f_{0}+\prod(\bar{\delta}\pm\bar{\varepsilon}_{1}\pm\bar{\varepsilon}_{2}\pm\bar{\varepsilon}_{3})\cdot
f_{1}$ where $f_{0}\in\mathbb{C}[L_{2},L_{6}]$ and
$f_{1}\in\mathbb{C}[{\mathfrak{h}}^{*}]^{W}$, and

$\displaystyle L_{2}$
$\displaystyle:=3(\bar{\varepsilon}_{1}^{2}+\bar{\varepsilon}_{2}^{2}+\bar{\varepsilon}_{3}^{2})-\bar{\delta}^{2},$
$\displaystyle L_{6}$
$\displaystyle:=\bar{\delta}^{6}+\bar{\varepsilon}_{1}^{6}+\bar{\varepsilon}_{2}^{6}+\bar{\varepsilon}_{3}^{6}$
$\displaystyle+(\bar{\varepsilon}_{1}-\bar{\varepsilon}_{2})^{6}+(\bar{\varepsilon}_{2}-\bar{\varepsilon}_{3})^{6}+(\bar{\varepsilon}_{1}-\bar{\varepsilon}_{3})^{6}+(\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2})^{6}+(\bar{\varepsilon}_{2}+\bar{\varepsilon}_{3})^{6}+(\bar{\varepsilon}_{1}+\bar{\varepsilon}_{3})^{6}$
$\displaystyle-\frac{1}{64}\sum(\bar{\delta}\pm\bar{\varepsilon}_{1}\pm\bar{\varepsilon}_{2}\pm\bar{\varepsilon}_{3})^{6}.$

The restriction from ${\mathfrak{h}}^{*}$ to ${\mathfrak{a}}^{*}$ is given by
$\varepsilon=\bar{\varepsilon}_{1}\mid_{\mathfrak{a}}$,
$\delta=\bar{\delta}\mid_{\mathfrak{a}}$ and
$\bar{\varepsilon}_{2}\mid_{\mathfrak{a}}=\bar{\varepsilon}_{3}\mid_{\mathfrak{a}}=0$.
For $d\leq 3$, we see that the restrictions of $L_{2}\in
I({\mathfrak{h}}^{*})_{2}$, $L_{2}^{2}\in I({\mathfrak{h}}^{*})_{4}$ and
$L_{2}^{3},L_{6}\in I({\mathfrak{h}}^{*})_{6}$ to ${\mathfrak{a}}$ are
linearly independent and give the correct dimensions. We continue by induction
on $d$, that is, assume that $\dim
I({\mathfrak{h}}^{*})_{2(d-1)}\mid_{\mathfrak{a}}\geq d-2$. Then
$\dim\left(\left(L_{2}\cdot
I({\mathfrak{h}}^{*})_{2(d-1)}\right)\mid_{\mathfrak{a}}\right)\geq d-2$. Take
$f_{1}\in\mathbb{C}[{\mathfrak{h}}^{*}]_{2d-8}^{W}$ such that
$f_{1}\mid_{\mathfrak{a}}$ is not divisible by $L_{2}\mid_{\mathfrak{a}}$.
Then
$f=\prod(\bar{\delta}\pm\bar{\varepsilon}_{1}\pm\bar{\varepsilon}_{2}\pm\bar{\varepsilon}_{3})\cdot
f_{1}$ is in $I({\mathfrak{h}})^{*}_{2d}\mid_{\mathfrak{a}}$ but not in
$\left(L_{2}\cdot I({\mathfrak{h}}^{*})_{2(d-1)}\right)\mid_{\mathfrak{a}}$.
Thus, $\dim I({\mathfrak{h}}^{*})_{2d}\mid_{\mathfrak{a}}\geq d-1$.

The generators of $I({\mathfrak{a}}^{*})$ are the restrictions of the
generators of $I({\mathfrak{h}}^{*})$, namely of

$3\varepsilon^{2}-\delta^{2},\quad
3\varepsilon^{6}-\delta^{6}-16(\delta^{2}-\varepsilon^{2})^{3},\quad(\delta^{2}-\varepsilon^{2})^{4}\varepsilon^{2i}\delta^{2j},i,j\in\mathbb{Z}_{\geq
0}.$

###  5.8. Case
${\mathfrak{g}}=D(2,1,a),\mathfrak{k}=\mathfrak{osp}(2|2)\oplus\mathfrak{so}(2)$.

Here

$I({\mathfrak{a}}^{*})=\left\\{f\in\mathbb{C}[\varepsilon^{2},\delta^{2}]\mid\left((a+1)\frac{\partial}{\partial\varepsilon}+\frac{\partial}{\partial\delta}\right)f\in\left<\varepsilon-\delta\right>\right\\},$

(see Case 6.2 in the Appendix).

To show surjectivity in this case, we first note that $\dim
I({\mathfrak{a}}^{*})_{2d}=d$. Indeed,
$\dim\mathbb{C}[\varepsilon^{2},\delta^{2}]_{2d}=d+1$ with basis
$\varepsilon^{2d},\varepsilon^{2d-2}\delta^{2},\ldots,\delta^{2d}$. The
derivative condition is equivalent to

$\left.\left((a+1)\frac{\partial}{\partial\varepsilon}+\frac{\partial}{\partial\delta}\right)f\right|_{\varepsilon=\delta}=0.$

This gives one linear condition between the basis elements.

Let us show that $\dim S({\mathfrak{p}})^{\mathfrak{k}}_{2d}\geq d$. The
${\mathfrak{k}}$-module ${\mathfrak{p}}^{*}$ is isomorphic to $V\oplus V^{*}$
where $S(V)_{d}$ is the described in [SSS, Prop. 3.6]: for $a\neq 0,-1$ such
that $-\frac{1}{a}\notin\mathbb{Q}_{\leq}0$, $S(V)_{d}$ is a direct sum of $d$
irreducible modules. Hence $S({\mathfrak{p}}^{*})\cong S(V)\otimes S(V)^{*}$.
By Schur’s lemma, the number of linearly independent invariant vectors is the
number of irreducible summands of $S(V)_{d}$ which is $d$, as required. Since
$\dim S({\mathfrak{p}})^{\mathfrak{k}}_{d}=d$ for dense set of
$a\in\mathbb{C}$, it follows that for an arbitrary $a$, $\dim
S({\mathfrak{p}})^{\mathfrak{k}}_{d}\geq d$.

######  Proposition 5.5.

The ring $I({\mathfrak{a}}^{*})$ is generated by the deformed Newton sums
$\phi_{2k}=\varepsilon^{2k}-(a+1)\delta^{2k}$, $k\geq 1$.

###### Proof.

For $a\notin\mathbb{Q}_{\leq-1}$, this follows from [SV, Thm. 2]. We prove the
proposition for general $a\neq-1$. Let $W_{d}$ be the subspace of
$I({\mathfrak{a}}^{*})_{2d}$ generated by the deformed Newton sums. We show
that $\dim W_{d}\geq\dim I({\mathfrak{a}}^{*})_{2d}$ by induction on $d$. For
$d=1$, the claim holds since
$I({\mathfrak{a}}^{*})_{2}=\operatorname{span}\\{\varepsilon^{2}-(a+1)\delta^{2}\\}$.
Suppose that $\dim I({\mathfrak{a}}^{*})_{2d-2}=d-1=\dim
I({\mathfrak{a}}^{*})_{2d-2}$. Let
$V_{d}:=(\varepsilon^{2}-(a+1)\delta^{2})I({\mathfrak{a}}^{*})_{2d-2}\subseteq
W_{d}$. Then $\dim V_{d}=d-1$. We show that $\dim W_{d}\geq d$ by showing that
$\varepsilon^{2d}-(a+1)\delta^{2d}$ is in $W_{d}$ but not in $V_{d}$. Suppose
otherwise, then

$\varepsilon^{2d}-(a+1)\delta^{2d}=(\varepsilon^{2}-\delta^{2})(a_{0}\varepsilon^{2d-2}+a_{1}\varepsilon^{2d-4}\delta^{2}+\ldots+a_{d}\delta^{2d-2})$

for some $a_{0},\ldots,a_{d}\in\mathbb{C}$ such that
$a_{0}\varepsilon^{2d-2}+a_{1}\varepsilon^{2d-4}\delta^{2}+\ldots+a_{d}\delta^{2d-2}\in
I({\mathfrak{a}}^{*})_{2d-2}$. This equality is impossible unless
$1=a_{0}=a_{1}=\ldots=a_{d}=-a-1$. Hence, for $a\neq 2$, the subspace of
$I({\mathfrak{a}}^{*})_{2d}$ generated by the deformed Newton sums is at least
$d$ and so the subspace is equal to $d$. For $a=2$, the proposition is known
by [SV, Thm. 2].

∎

Note that by [Serge2, 0.6.6], $\dim I({\mathfrak{h}}^{*})_{6}=2$. Since $\dim
I({\mathfrak{a}}^{*})_{6}=3$, $I({\mathfrak{h}}^{*})$ does not surject onto
$I({\mathfrak{a}}^{*})$ in this case. Hence $R_{\theta}$ is not surjective in
this case. The image of the restriction from $\dim I({\mathfrak{h}}^{*})$ to
$I({\mathfrak{a}}^{*})$ in this case is generated by $\phi_{2}$ and elements
of the form $(\varepsilon^{2}-\delta^{2})^{2}\varepsilon^{2i}\delta^{2j}$,
$i,j\in\mathbb{Z}$.

###  5.9. Case ${\mathfrak{g}}=G_{3},\mathfrak{k}=D(2,1,3)$.

Here

$I({\mathfrak{a}}^{*})=\mathbb{C}[\varepsilon^{2}_{1},\varepsilon^{2}_{2},\varepsilon^{2}_{3}]^{S_{3}}/\left<\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}=1\right>$

(see Case 6.2 in the Appendix).

We claim that in this case $I({\mathfrak{h}}^{*})$ surjects onto
$I({\mathfrak{a}}^{*})$. Indeed, $I({\mathfrak{a}}^{*})$ is equal to the set
of invariant polynomials under the Weyl group of the Lie algebra $G_{2}$. This
set is generated by two algebraically independent elements of degree $2$ and
$6$, (see for example [Hu, 3.7]). Since
$\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2},\
\varepsilon_{1}^{2}\varepsilon_{2}^{2}\varepsilon_{3}^{2}$, are algebraically
independent and have the suitable degrees, they generate
$I({\mathfrak{a}}^{*})$. These elements are the images of
$3\bar{\delta}^{2}-2(\bar{\varepsilon}_{1}^{2}+\bar{\varepsilon}_{2}^{2}+\bar{\varepsilon}_{3}^{2})$
and
$-(\bar{\delta}^{2}-\bar{\varepsilon}_{1}^{2})(\bar{\delta}^{2}-\bar{\varepsilon}_{2}^{2})(\bar{\delta}^{2}-\bar{\varepsilon}_{3}^{2})$
which are in $I({\mathfrak{h}}^{*})$ as $\bar{\delta}\mid_{{\mathfrak{a}}}=0$.
Since the generators of $I({\mathfrak{a}}^{*})$ are restrictions of elements
in $I({\mathfrak{h}}^{*})$, the restriction map from $I({\mathfrak{h}}^{*})$
to $I({\mathfrak{a}}^{*})$ is surjective.

###  5.10. Case ${\mathfrak{g}}=F_{4},\mathfrak{k}=\mathfrak{sl}_{2}\oplus
D(2,1;2)$.

Here

$I(\mathfrak{a}^{*})=\left\\{f\in\mathbb{C}\left[\varepsilon_{1}^{2},\varepsilon_{2}^{2},\varepsilon_{3}^{2}\right]^{S_{3}}\
|\
\left(\frac{\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\varepsilon_{2}}+\frac{\partial}{\partial\varepsilon_{3}}\right)f\in\left\langle\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\right\rangle\right\\},$

(see Case 6.2 in the Appendix).

Let us show that $I({\mathfrak{h}}^{*})$ surjects onto
$I({\mathfrak{a}}^{*})$. This will in particular imply that $R_{\theta}$ is
surjective. Let
$\hat{D}=\frac{\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\varepsilon_{2}}+\frac{\partial}{\partial\varepsilon_{3}}-3\frac{\partial}{\partial\delta}$
and recall that $I({\mathfrak{h}}^{*})$ consists of polynomials $\hat{f}$ in
$\mathbb{C}\left[\bar{\varepsilon}_{1}^{2},\bar{\varepsilon}_{2}^{2},\bar{\varepsilon}_{3}^{2},\bar{\delta}^{2}\right]^{S_{3}}$
for which
$\hat{D}\hat{f}\in\left\langle\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}+\bar{\varepsilon}_{3}+\bar{\delta}\right\rangle$
(here $\bar{\delta}\mid_{{\mathfrak{a}}}=0$). Let $f\in
I(\mathfrak{a}^{*})_{2d}$. Then $\hat{D}f=p_{1}\cdot F$ for some
$F\in\mathbb{C}\left[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\right]_{2d-2}^{S_{3}}$.

Let $p_{n}=\varepsilon_{1}^{n}+\varepsilon_{2}^{n}+\varepsilon_{3}^{n}$.
Recall that $p_{3},p_{2},p_{1}$ freely generate
$\mathbb{C}\left[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\right]^{S_{3}}$
and that $p_{6},p_{4},p_{2}$ freely generate
$\mathbb{C}\left[\varepsilon_{1}^{2},\varepsilon_{2}^{2},\varepsilon_{3}^{2}\right]^{S_{3}}$.
Note that $\hat{D}p_{n}=np_{n-1}$. Write

$\hat{f}:=f+\sum_{r_{0}+3r_{6}+2r_{4}+r_{2}=d}a_{\vec{r}}\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}$

where $r_{0}\geq 1$, $a_{\vec{r}}\in\mathbb{C}$. Then $\hat{f}\in
I(\mathfrak{h}^{*})$ if and only if $\hat{D}\hat{f}\mid_{\delta=-p_{1}}=0$.
This means that

$\hat{D}\hat{f}\mid_{\delta=-p_{1}}=p_{1}F+\sum_{r_{0}+3r_{6}+2r_{4}+r_{2}=d}a_{\vec{r}}\hat{D}\left(\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}\right)\mid_{\delta=-p_{1}}=0.$

By the following technical lemma, one can find $a_{\vec{r}}$’s that will make
the above expression zero for every
$F\in\mathbb{C}\left[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\right]_{2d-2}^{S_{3}}$
which proves the existence of a preimage in $I({\mathfrak{h}}^{*})$.

######  Lemma 5.6.

The elements

$q_{\vec{r}}:=p_{1}^{-1}\cdot\hat{D}\left(\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}\right)\mid_{\delta=-p_{1}},\text{
where }r_{0}\geq 1,r_{0}+3r_{6}+2r_{4}+r_{2}=d,$

span
$\mathbb{C}\left[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\right]_{2d-2}^{S_{3}}$.

###### Proof.

We show that every basis element $p_{3}^{n_{3}}p_{2}^{n_{2}}p_{1}^{n_{1}}$ in
$\mathbb{C}\left[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\right]_{2d-2}^{S_{3}}$
appears as a leading monomial of some $q_{\vec{r}}$. Indeed,

$\displaystyle\hat{D}\left(\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}\right)$
$\displaystyle=-6r_{0}\delta^{2r_{0}-1}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}+6r_{6}\delta^{2r_{0}}p_{6}^{r_{6}-1}p_{5}p_{4}^{r_{4}}p_{2}^{r_{2}}$
$\displaystyle+4r_{4}\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}-1}p_{3}p_{2}^{r_{2}}+2r_{2}\delta^{2r_{0}}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}-1}p_{1}.$

So

$\displaystyle q_{\vec{r}}$
$\displaystyle=6r_{0}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}}p_{1}^{2r_{0}-2}+6r_{6}p_{6}^{r_{6}-1}p_{5}p_{4}^{r_{4}}p_{2}^{r_{2}}p_{1}^{2r_{0}-1}$
$\displaystyle+4r_{4}p_{6}^{r_{6}}p_{4}^{r_{4}-1}p_{3}p_{2}^{r_{2}}p_{1}^{2r_{0}-1}+2r_{2}p_{6}^{r_{6}}p_{4}^{r_{4}}p_{2}^{r_{2}-1}p_{1}^{2r_{0}}.$

The following relations imply that the leading monomial of $q_{\vec{r}}$ is
$p_{3}^{2r_{6}+r_{4}}p_{2}^{r_{2}}p_{1}^{r_{4}+2r_{0}-2}$.

$\displaystyle p_{6}$
$\displaystyle=\frac{1}{3}p_{3}^{2}+p_{3}p_{2}p_{1}+\frac{1}{3}p_{3}p_{1}^{3}+\frac{1}{4}p_{2}^{3}-\frac{3}{4}p_{2}^{2}p_{1}^{2}-\frac{1}{4}p_{2}p_{1}^{4}+\frac{1}{12}p_{1}^{6}$
$\displaystyle p_{5}$
$\displaystyle=\frac{5}{6}p_{3}p_{2}+\frac{5}{6}p_{3}p_{1}^{2}-\frac{5}{6}p_{2}p_{1}^{3}+\frac{1}{6}p_{1}^{5}$
$\displaystyle p_{4}$
$\displaystyle=\frac{4}{3}p_{3}p_{1}+\frac{1}{2}p_{2}^{2}-p_{1}^{2}p_{2}+\frac{1}{6}p_{1}^{4}.$

Thus, given $p_{3}^{n_{3}}p_{2}^{n_{2}}p_{1}^{n_{1}}$ with
$3n_{3}+2n_{2}+n_{1}=2d-2$, it appears as a leading monomial for $q_{\vec{r}}$
for
$\vec{r}=(r_{0},r_{6},r_{4},r_{2})=(\left\lfloor\frac{n_{1}+2}{2}\right\rfloor,\left\lfloor\frac{n_{3}}{2}\right\rfloor,p(n_{3}),n_{2})$,
where $p(n_{3})\in\\{0,1\\}$ is the parity of $n_{3}$. ∎

The ring $I({\mathfrak{a}}^{*})$ is generated by the images of the generators
of $I({\mathfrak{h}}^{*})$, namely

$\displaystyle
3(\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2}),$
$\displaystyle\varepsilon_{1}^{6}+\varepsilon_{2}^{6}+\varepsilon_{3}^{6}+\sum_{i<j}(\varepsilon_{i}-\varepsilon_{j})^{6}-\frac{1}{32}\sum(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3})^{6},$
$\displaystyle\prod(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3})^{2}(\varepsilon_{1}^{2k}+\varepsilon_{2}^{2k}+\varepsilon_{3}^{2k}),\
k\in\mathbb{Z}_{\geq 1}.$

##  6\. Apendix: restricted roots systems for even symmetric pairs.

In this appendix we list properties of restricted root systems for all cases
that correspond to Iwasawa involutions.

###  6.1. Cartan Subspace

We use the standard basis
$\bar{\varepsilon}_{1},\ldots,\bar{\varepsilon}_{\bar{m}},\bar{\delta}_{1},\ldots,\bar{\delta}_{\bar{n}}$
for denoting the roots of ${\mathfrak{g}}$. For ${\mathfrak{g}}\neq
D(2,1,a),F_{4}$,

$(\bar{\varepsilon}_{i},\bar{\varepsilon}_{j})=\delta_{ij}=-(\bar{\delta}_{i},\bar{\delta}_{j}),\quad(\bar{\varepsilon}_{i},\bar{\delta}_{j})=0.$

For ${\mathfrak{g}}=D(2,1,a)$, we have
$(\bar{\varepsilon}_{1},\bar{\varepsilon}_{1})=-a-1$,
$(\bar{\varepsilon}_{2},\bar{\varepsilon}_{2})=1$ and
$(\bar{\varepsilon}_{3},\bar{\varepsilon}_{3})=a.$ For ${\mathfrak{g}}=F_{4}$,
we have $(\bar{\varepsilon}_{i},\bar{\varepsilon}_{j})=2\delta_{ij}$,
$(\bar{\delta},\bar{\delta})=-6$ and $(\bar{\varepsilon}_{i},\bar{\delta})=0$.
Note that for ${\mathfrak{g}}=G_{3}$, we use the basis
$\bar{\varepsilon}_{1},\bar{\varepsilon}_{2},\bar{\varepsilon}_{3},\bar{\delta}$
such that
$\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}+\bar{\varepsilon}_{3}=0$,
$(\bar{\varepsilon}_{i},\bar{\varepsilon}_{j})=-1$ for $i\neq j$ and
$(\bar{\varepsilon}_{i},\bar{\varepsilon}_{i})=2$ (see for example [GK,
10.9]).

We use the notation
$\varepsilon_{1},\ldots,\varepsilon_{m},\delta_{1},\ldots,\delta_{n}$ to
denote a basis to ${\mathfrak{a}}^{*}$. The restriction function from
${\mathfrak{h}}^{*}$ to ${\mathfrak{a}}^{*}$ is described in each case. We
denote by
$\bar{e}_{1},\ldots,\bar{e}_{\bar{m}},\bar{d}_{1},\ldots,\bar{d}_{\bar{n}}$
the dual basis to
$\bar{\varepsilon}_{1},\ldots,\bar{\varepsilon}_{\bar{m}},\bar{\delta}_{1},\ldots,\bar{\delta}_{\bar{n}}$,
and by $e_{1},\ldots,e_{m},d_{1},\ldots,d_{n}$ the dual basis to
$\varepsilon_{1},\ldots,\varepsilon_{m},\delta_{1},\dots,\delta_{n}$.

###  6.2. Restricted Root Systems.

Below is the list of type of the restricted root system following [SV] and the
restriction function from ${\mathfrak{h}}^{*}$ to ${\mathfrak{a}}^{*}$.  Case
${\mathfrak{g}}$ ${\mathfrak{k}}$ Type 1 $\mathfrak{gl}(m|2n)$
$\mathfrak{osp}(m|2n)$ $A(m-1|n-1)$ 2 $\mathfrak{gl}(2m+a|2n+b)$
$\mathfrak{gl}(m|n)\oplus\mathfrak{gl}(m+a|n+b)$ $BC(m|n)$ $a,b\geq 0$ 3
$\mathfrak{osp}(2m+a|4n+2b)$
$\mathfrak{osp}(m|2n)\oplus\mathfrak{osp}(m+a|2n+2b)$ $BC(m|n)$ $a,b\geq 0$ 4
$\mathfrak{osp}(4m+2|2n)$ $\mathfrak{gl}(2m+1|n)$ $BC(m|n)$ 5
$\mathfrak{osp}(4m|2n)$ $\mathfrak{gl}(2m|n)$ $BC(m|n)$ 6 $F_{4}$
$\mathfrak{gosp}(2|4)$ $BC(2|1)$ 7 $F_{4}$ $\mathfrak{sl}(1|4)$ $BC(1|1)$ 8
$D(2,1,a)$ $\mathfrak{osp}(2|2)\oplus\mathfrak{so}(2)$ $BC(1|1)$ 9 $G_{3}$
$D(2,1,3)$ $G_{2}$ 10 $F_{4}$ $\mathfrak{sl}_{2}\oplus D(2,1;2)$ exotic

We describe in each case the restriction function from ${\mathfrak{h}}^{*}$ to
${\mathfrak{a}}^{*}$. The standard basis elements which are not written are
being restricted to zero. In the following table $1\leq i\leq m$ and $1\leq
j\leq n$.  Case Restriction from ${\mathfrak{h}}^{*}$ to ${\mathfrak{a}}^{*}$
6.2 $\varepsilon_{i}=\bar{\varepsilon}_{i}\mid_{\mathfrak{a}}$
$\delta_{j}=\bar{\delta}_{j}\mid_{\mathfrak{a}}=\bar{\delta}_{n+j}\mid_{\mathfrak{a}}$
6.2
$\varepsilon_{i}=\bar{\varepsilon}_{i}\mid_{\mathfrak{a}}=-\bar{\varepsilon}_{m+i}\mid_{\mathfrak{a}}$
$\delta_{j}=\bar{\delta}_{j}\mid_{\mathfrak{a}}=-\bar{\delta}_{n_{1}+j}\mid_{\mathfrak{a}}$
6.2 $\varepsilon_{i}=\bar{\varepsilon}_{i}\mid_{\mathfrak{a}}$
$\delta_{j}=\bar{\delta}_{j}\mid_{\mathfrak{a}}=-\bar{\delta}_{n+j}\mid_{\mathfrak{a}}$
6.2
$\varepsilon_{i}=\bar{\varepsilon}_{i}\mid_{\mathfrak{a}}=-\bar{\varepsilon}_{2m-i+2}\mid_{\mathfrak{a}}$
$\delta_{j}=\bar{\delta}_{j}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon_{i}=\bar{\varepsilon}_{i}\mid_{\mathfrak{a}}=-\bar{\varepsilon}_{2m-i+1}\mid_{\mathfrak{a}}$
$\delta_{j}=\bar{\delta}_{j}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon_{1}=\frac{\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}}{2}\mid_{\mathfrak{a}}$,
$\varepsilon_{2}=\frac{\bar{\varepsilon}_{1}-\bar{\varepsilon}_{2}}{2}\mid_{\mathfrak{a}}$
$\delta=-\frac{\bar{\delta}}{2}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon=\frac{\bar{\varepsilon}_{1}}{2}\mid_{\mathfrak{a}}$
$\delta=\frac{\bar{\delta}}{2}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon=\bar{\varepsilon}_{1}\mid_{\mathfrak{a}}$,
$\delta=-\bar{\varepsilon}_{2}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon_{1}=\bar{\varepsilon}_{1}\mid_{\mathfrak{a}}$,
$\varepsilon_{2}=\bar{\varepsilon}_{2}\mid_{\mathfrak{a}}$,
$\varepsilon_{3}=\bar{\varepsilon}_{3}\mid_{\mathfrak{a}}$ 6.2
$\varepsilon_{1}=\bar{\varepsilon}_{1}\mid_{\mathfrak{a}}$,
$\varepsilon_{2}=\bar{\varepsilon}_{2}\mid_{\mathfrak{a}}$,
$\varepsilon_{3}=\bar{\varepsilon}_{3}\mid_{\mathfrak{a}}$

We write below the multiplicity of each restricted root. When the root space
is pure, we write a positive number for the dimension of an even space and a
negative number for the dimension of an odd space. We let
$k:=-\frac{(\delta_{j},\delta_{j})}{(\varepsilon_{i},\varepsilon_{i})}$ be the
deformation parameter used in Section 6.3 to compute $D_{\alpha}$.

####  6.2.1. Multiplicities in type $A(m-1|n-1)$

Case | $\pm(\varepsilon_{i}-\varepsilon_{j})$ | $\pm(\delta_{i}-\delta_{j})$ | $\pm(\varepsilon_{i}-\delta_{j})$ | $k$
---|---|---|---|---
6.2 | $1$ | $4$ | $-2$ | $\frac{1}{2}$

To compute
$k=-\frac{(\delta_{j},\delta_{j})}{(\varepsilon_{i},\varepsilon_{i})}$, we
note that the dual basis element to $\varepsilon_{i}$ in ${\mathfrak{a}}$ is
$\bar{e}_{i}$. The dual basis element to $\delta_{j}$ in ${\mathfrak{a}}$ is
$\frac{\bar{d}_{j}+\bar{d}_{n+j}}{2}$. Hence
$(\varepsilon_{i},\varepsilon_{i})=(\bar{e}_{i},\bar{e}_{i})=1$ and
$(\delta_{j},\delta_{j})=\left(\frac{\bar{d}_{j}+\bar{d}_{n+j}}{2},\frac{\bar{d}_{j}+\bar{d}_{n+j}}{2}\right)=-\frac{1}{2}$.

####  6.2.2. Multiplicities in type $BC(m|n)$

Case | $\pm\varepsilon_{i}\pm\varepsilon_{j}$ | $\pm\varepsilon_{i}$ | $\pm 2\varepsilon_{i}$ | $\pm\delta_{i}\pm\delta_{j}$ | $\pm\delta_{i}$ | $\pm 2\delta_{i}$ | $\pm\varepsilon_{i}\pm\delta_{j}$ | $k$
---|---|---|---|---|---|---|---|---
6.2 | $2$ | $(2a|2b)$ | $1$ | $2$ | $(2b|2a)$ | $1$ | $-2$ | $1$
6.2 | $1$ | $(a|2b)$ |  | $4$ | $(4b|2a)$ | $3$ | $-2$ | $\frac{1}{2}$
6.2 | $4$ | $2$ | $1$ | $1$ | $-2$ | $1$ | $-2$ | $2$
6.2 | $4$ |  | $1$ | $1$ |  | $1$ | $-2$ | $2$
6.2 | $3$ |  | $1$ |  |  | $1$ | $-2$ | $\frac{3}{2}$
6.2 |  |  | $5$ |  |  | $1$ | $-4$ | $3$
6.2 |  |  | $1$ |  |  | $1$ | $-2$ | $\frac{1}{a+1}$

Note that in Case 6.2, $m=2,n=1$ and in Cases 6.2 and 6.2, $m=n=1$. The
computation of $k$ form is done in a similar fashion as Case 6.2.

####  6.2.3. Multiplicities in type $G_{2}$

Case | long root | short root
---|---|---
6.2 | $1$ | $(1|2)$

####  6.2.4. An exotic case

Case | $\pm\varepsilon_{i}\pm\varepsilon_{j}$ | $\pm\varepsilon_{i}$ | $\frac{1}{2}\left(\pm\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\right)$
---|---|---|---
6.2 | $1$ | $1$ | $-2$

In this case $(\varepsilon_{i},\varepsilon_{i})=1$ and
$(\varepsilon_{i},\varepsilon_{j})=0$ for $i\neq j$.

###  6.3. Computation of $D_{h_{\alpha}}$

For Cases 6.2-6.2, we compute the derivative condition in the following
manner. Suppose that $\varepsilon_{1},\delta_{1}\in{\mathfrak{a}}^{*}$ are
such that $(\varepsilon_{1},\varepsilon_{1})=1$, $(\delta_{1},\delta_{1})=-k$,
$(\varepsilon_{1},\delta_{1})=0$ and $\alpha=\varepsilon_{1}-\delta_{1}$ is a
former isotropic root. Let $e_{1},d_{1}$ be the dual elements in
${\mathfrak{a}}$ to $\varepsilon_{1},\delta_{1}$. Then $e_{1}+d_{1}\in{\rm
Ker}\alpha$ and $h_{\alpha}=e_{1}+\frac{1}{k}d_{1}\in({\rm
Ker}\alpha)^{\perp}$. Hence the condition $D_{h_{\alpha}}f\in{\rm Ker}\alpha$
becomes
$\left(\frac{\partial}{\partial\varepsilon_{1}}+\frac{\partial}{k\partial\delta_{1}}\right)f\in\langle\varepsilon_{1}-\delta_{1}\rangle$.

For Case 6.2, take $\alpha=\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}$.
Then ${\rm Ker}\alpha=\text{span}\\{e_{1}-e_{2},e_{2}-e_{3}\\}$ and
$h_{\alpha}=e_{1}+e_{2}+e_{3}\in({\rm Ker}\alpha)^{\perp}$. Hence the
derivative condition becomes
$(\frac{\partial}{\partial\varepsilon_{1}}+\frac{\partial}{\partial\varepsilon_{2}}+\frac{\partial}{\partial\varepsilon_{3}})f\in\langle\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\rangle$.

###  6.4. Generators for $I({\mathfrak{a}}^{*})$

We list the generators in each case.  Case Generators 6.2
$\varepsilon_{1}^{k}+\ldots+\varepsilon_{m}^{k}-2\delta_{1}^{k}-\dots-2\delta_{n}^{k}$
6.2
$\varepsilon_{1}^{2k}+\ldots+\varepsilon_{m}^{2k}-\delta_{1}^{2k}-\dots-\delta_{n}^{2k}$
6.2, $a\neq 0$
$\varepsilon_{1}^{2k}+\ldots+\varepsilon_{m}^{2k}-2\delta_{1}^{2k}-\dots-2\delta_{n}^{2k}$
6.2, 6.2
$2\varepsilon_{1}^{2k}+\ldots+2\varepsilon_{m_{1}}^{2k}-\delta_{1}^{2k}-\dots-\delta_{n}^{2k}$
6.2 $\varepsilon_{1}^{2k}+\varepsilon_{2}^{2k}-\frac{2}{3}\delta^{2k}$ 6.2
$3\varepsilon^{2}-\delta^{2},\
3\varepsilon^{6}-\delta^{6}-16(\delta^{2}-\varepsilon^{2})^{3},\
(\delta^{2}-\varepsilon^{2})^{4}\varepsilon^{2i}\delta^{2j}$ 6.2
$\varepsilon^{2k}-(a+1)\delta^{2k}$ 6.2
$\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2},\
\varepsilon_{1}^{2}\varepsilon_{2}^{2}\varepsilon_{3}^{2}$ 6.2
$3(\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\varepsilon_{3}^{2})$,
$\varepsilon_{1}^{6}+\varepsilon_{2}^{6}+\varepsilon_{3}^{6}+\sum_{i<j}(\varepsilon_{i}-\varepsilon_{j})^{6}-\frac{1}{32}\sum(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3})^{6}$,
$\prod(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3})^{2}(\varepsilon_{1}^{2k}+\varepsilon_{2}^{2k}+\varepsilon_{3}^{2k})$
Here $k\in\mathbb{Z}_{\geq 1}$ and $i,j\in\mathbb{Z}_{\geq 0}$ in Case 6.2.

We remark that for the Case 6.2, $a=0$, we have
$I({\mathfrak{a}}^{*})=I_{1}({\mathfrak{a}}^{*})\oplus
I_{2}({\mathfrak{a}}^{*})$ as shown in Section 5.4. The subring
$I_{1}({\mathfrak{a}}^{*})$ is generated by
$\varepsilon_{1}^{2k}+\ldots+\varepsilon_{m}^{2k}-\delta_{1}^{2k}-\dots-\delta_{n}^{2k}$,
$k\in\mathbb{Z}_{\geq 1}$ and $I_{2}({\mathfrak{a}}^{*})$ is a module over
$I_{1}({\mathfrak{a}}^{*})$. However, we do not know the generators of
$I_{2}({\mathfrak{a}}^{*})$ as an $I_{1}({\mathfrak{a}}^{*})$-module.

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