# Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules
associated to the ideals of minimal nilpotent orbits for simple Lie groups of
type $A$

Hiroyoshi TAMORI

###### Abstract.

We classify completely prime primitive ideals whose associated varieties are
the closure of the minimal nilpotent orbit of
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible
$(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators.
Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.
Mathematics Subject Classification $2010$: 22E46.
Key Words and Phrases: Minimal representation; Completely prime primitive
ideal; Simple Lie group; Type A.

##  1\. Introduction

Let $G$ be a simple real Lie group and
$\mathfrak{g}_{0}:=\mathop{\mathrm{Lie}}\nolimits(G)$. Fix a maximally compact
subalgebra $\mathfrak{k}_{0}$ of $\mathfrak{g}_{0}$. Assume that the
complexification $\mathfrak{g}$ of $\mathfrak{g}_{0}$ is simple. Joseph proved
that if $\mathfrak{g}$ is not of type A, then the universal enveloping algebra
$U(\mathfrak{g})$ has a unique completely prime ideal $J_{0}$ whose associated
variety, which is a subset of
$\mathfrak{g}^{\vee}:=\mathop{\mathrm{Hom}}\nolimits_{\mathbb{C}}(\mathfrak{g},\mathbb{C})$,
is the closure of the minimal nilpotent coadjoint orbit $\mathcal{O}^{\min}$
[10], [4, Theorem 3.1]. Minimal representations of $G$ are defined as
admissible irreducible representations where the annihilators of the
associated $(\mathfrak{g},\mathfrak{k})$-modules are equal to the Joseph ideal
$J_{0}$.

Minimal representations are classified and are known to be infinitesimally
equivalent to unitary representations [20, Corollary 5.1]. In the Kirillov-
Kostant orbit philosophy, they are considered to be attached to $G$-orbits in
$\mathcal{O}^{\min}\cap\\{X\in\mathfrak{g}^{\vee}\mid
X(\mathfrak{g}_{0})\subset\mathbb{R}\\}$, and are considered to be a part of
building blocks of the unitary dual of $G$.

Moreover, minimal representations are of interest in physics. For example, the
oscillator representation of the metaplectic group (whose irreducible
components are minimal) has a realization as the bound states of the quantum
harmonic oscillator, and the minimal representation of the indefinite
orthogonal group $\mathrm{O}(p,2)(p\geq 6)$ has a realization as some solution
space of the wave equation in the Minkowski space (see [13, Theorem 1.4] for
example).

On the other hand, there exist representations called “minimal” even when
$\mathfrak{g}$ is simple of type $A$, that is, when the definition of the
Joseph ideal (hence the one of minimality) is not given. Here the vague term
“minimal” means that there is an interest in physics as in the previous
examples, the $\mathfrak{k}$-types are as simple as possible (called pencil,
see Corollary 3.5 (2) for the definition), or there is a relation with the
minimal nilpotent orbit. Such representations include the ladder
representation of $\mathrm{O}(2,4)$ (which is locally isomorphic to
$\mathrm{SU}(2,2)$) expressing the bound states of the Hydrogen atom [12,
Remark 3.6.2 (3)] and the irreducible unitary representations of the double
cover $\widetilde{\mathrm{SL}}(3,\mathbb{R})$ of $\mathrm{SL}(3,\mathbb{R})$
given by Torasso (see [21, Theoreme VII.1], [19]).

The aim of this paper is to define minimality in terms of annihilators of
$(\mathfrak{g},\mathfrak{k})$-modules so that the above representations are
minimal, and to classify minimal representations for connected simple real Lie
groups of type $A_{n-1}$ $(n\geq 2)$. Hence this study can be regarded as a
small step of classifying irreducible $(\mathfrak{g},\mathfrak{k})$-modules
associated with a fixed completely prime primitive ideal.

Our first claim is that the two-sided ideals in $U(\mathfrak{g})$ whose
associated graded ideals coincide with the ideal defined by
$\mathcal{O}^{\min}$ are parametrized by the complex numbers $\mathbb{C}$ (see
Theorem 2.8). If we consider a simple Lie algebra not of type $A$, that
condition characterizes the Joseph ideal [4, Theorem 3.1].

Let us write $J_{a}$ for the ideal corresponding to $a\in\mathbb{C}$ (see
Definition 2.7 for the precise definition). We say that an irreducible
$(\mathfrak{g},\mathfrak{k})$-module is _$a$ -minimal_ if its annihilator
equals $J_{a}$, and is _minimal_ if it is $a$-minimal for some
$a\in\mathbb{C}$ (see Definition 3.1).

The associated variety gives us a necessary condition for the existence of
minimal $(\mathfrak{g},\mathfrak{k})$-modules. If a minimal
$(\mathfrak{g},\mathfrak{k})$-module exists, then
$\mathcal{O}^{\min}\cap\\{X\in\mathfrak{g}^{\vee}\mid
X(\mathfrak{g}_{0})\subset\mathbb{R}\\}$ is not empty. Hence it suffices to
consider the real forms $\mathfrak{su}(p,q)$ $(p,q>0)$ and
$\mathfrak{sl}(n,\mathbb{R})$ by [18, Proposition 4.1].

The main theorems in this paper are Theorems 5.1 and 5.3, which classify
$a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules for any $a\in\mathbb{C}$ and
$\mathfrak{g}_{0}=\mathfrak{su}(p,q)$ $(p,q>0)$,
$\mathfrak{sl}(n,\mathbb{R})$. The number of the isomorphism classes of
$a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules is given in Table 1.1.

Table 1.1. The number of the isomorphism classes of $a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules $\mathfrak{g}_{0}$ | $a$ | number
---|---|---
$\mathfrak{su}(p,1)(p\geq 2)$ | $\mathbb{C}$ | $2$
$\mathfrak{su}(1,q)(q\geq 2)$ | $\mathbb{C}$ | $2$
$\mathfrak{su}(p,q)(p,q\geq 2)$ | $(p+q)/2+\mathbb{Z}$ | $2$
| $\mathbb{C}\setminus((p+q)/2+\mathbb{Z})$ | $0$
$\mathfrak{sl}(n,\mathbb{R})(n\geq 4)$ | $\mathbb{C}$ | $2$
$\mathfrak{sl}(3,\mathbb{R})$ | $\mathbb{Z}$ | $3$
| $\mathbb{C}\setminus\mathbb{Z}$ | $2$

We remark that many of minimal $(\mathfrak{g},\mathfrak{k})$-modules do not
admit nondegenerate invariant Hermitian forms and are not unitarizable, which
does not occur when $\mathfrak{g}_{0}$ is not of type $A$. For the
classification for unitarizable minimal $(\mathfrak{g},\mathfrak{k})$-modules,
see Remarks 5.2 (3) and 5.4 (2).

For $\mathfrak{su}(1,q)$ $(q>0)$ and $\mathfrak{sl}(n,\mathbb{R})$, there
exist $a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules for any
$a\in\mathbb{C}$. They are related with the induction from maximal
($\theta$-stable or real) parabolic subalgebras of $\mathfrak{g}$
corresponding to the partition $(1,n-1)$. From the viewpoint of the orbit
method, the situation matches to the fact that the nonzero coadjoint orbits of
minimal dimension consist of $\mathcal{O}^{\min}$ and semisimple orbits
through nonzero characters of the parabolic subalgebras.

Our construction of a minimal $(\mathfrak{g},\mathfrak{k})$-module for
$\mathfrak{g}_{0}=\mathfrak{su}(p,q)$ $(p,q>0)$ and the one for
$\mathfrak{sl}(n,\mathbb{R})$ are different. We realize it as the irreducible
quotient of a Verma module for $\mathfrak{su}(p,q)$, while we construct it as
the infinite-dimensional irreducible subquotient of the kernel of an
intertwining differential operator between parabolically induced modules for
$\mathfrak{g}_{0}=\mathfrak{sl}(n,\mathbb{R})$. The latter construction for
Torasso’s representation ($n=3$) was given in [15, Theorem 1.6], and the one
for genuine $a$-minimal representations of the universal cover of
$\mathrm{SL}(3,\mathbb{R})$ was recently given in [16, Theorem 1.7 (5)]. We
give a general framework for the construction in Section 4.

The proof of exhaustion uses the facts that highest weights of
$\mathfrak{k}$-types of a minimal $(\mathfrak{g},\mathfrak{k})$-module lie in
some specified lattice, and that $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-modules are isomorphic to each other if they
have a common $\mathfrak{k}$-type (see Proposition 3.3). The argument is the
same as the classification when $\mathfrak{g}_{0}$ is not of type $A$ [20]. We
also need to show the injectivity of some intertwining differential operator
for the nonexistence of minimal
$(\mathfrak{sl}(3,\mathbb{C}),\mathfrak{so}(3,\mathbb{R}))$-modules with some
$\mathfrak{k}$-types (see Proposition 6.1).

Notation: We set $\mathbb{N}:=\\{0,1,2,\ldots\\}$. Lie groups will be denoted
by Latin capital letters, their Lie algebras by corresponding lower case
German letters with subscript zero. We omit the subscript zero to denote their
complexifications. Given a simple Lie algebra $\mathfrak{g}$, a Cartan
subalgebra $\mathfrak{h}$, and a finite-dimensional $\mathfrak{g}$-module $V$,
we write $V_{\mu}$ for the weight space of $V$ of weight $\mu$ with respect to
$\mathfrak{h}$.

##  2\. Primitive ideals and irreducible highest weight modules

Let $n\geq 2$ and $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$. In this section,
we classify two-sided ideals $J$ in the enveloping algebra $U(\mathfrak{g})$
whose associated graded ideals $\mathop{\mathrm{gr}}\nolimits J$ are the ideal
$\mathcal{I}(\mathcal{O}^{\min})$ defined by the minimal nilpotent coadjoint
orbit $\mathcal{O}^{\min}$ of $\mathfrak{g}^{\vee}$. Moreover, we obtain the
classification of irreducible highest weight modules whose annihilators are
equal to one of these ideals.

For $A\in\mathfrak{g}$ and $1\leq i,j\leq n$, we write $A_{i,j}$ for the
$(i,j)$-th component of the matrix $A$. We write $\mathfrak{h}$ for the Cartan
subalgebra of $\mathfrak{g}$ consisting of diagonal matrices. Let $e_{i}$
$(1\leq i\leq n)$ be an element of the dual $\mathfrak{h}^{\vee}$ defined by
$e_{i}(H)=H_{i,i}$ for $H\in\mathfrak{h}$. Then the set of roots for
$(\mathfrak{g},\mathfrak{h})$ is written as $\Sigma=\Set{e_{i}-e_{j}}{1\leq
i,j\leq n}$. We fix a set of positive roots
$\Sigma^{+}=\Set{e_{i}-e_{j}}{i<j}$. For an element $\lambda$ in
$\mathfrak{h}^{\vee}$, we define $\lambda_{i}\in\mathbb{C}$ $(1\leq i\leq n)$
by $\lambda=\sum_{i}\lambda_{i}e_{i}$ and $\sum_{i}\lambda_{i}=0$. Let us
write $(\lambda_{1},\ldots,\lambda_{n})$ for $\lambda$.

We first describe the irreducible decomposition of the second symmetric power
$S^{2}(\mathfrak{g})$. Let $I_{n}$ be the unit $n$-by-$n$ matrix, $E_{i,j}$
the matrix whose $(i,j)$-th component is one and the others are zero. Set

$T_{i,j}:=E_{i,j}-\frac{\delta_{i,j}}{n}I_{n}.$

Here $\delta_{i,j}$ denotes the Kronecker delta. We remark that $A=\sum_{1\leq
i,j\leq n}A_{i,j}T_{i,j}$ holds for every $A\in\mathfrak{g}$. We define a
nondegenerate invariant bilinear form $B$ on $\mathfrak{g}$ by

(2.1) $\displaystyle
B(X,Y):=\mathop{\mathrm{Tr}}\nolimits(XY)\quad(X,Y\in\mathfrak{g}).$

Put

(2.2) $\displaystyle\Omega:=\sum_{1\leq i,j\leq n}T_{i,j}T_{j,i}\in
S^{2}(\mathfrak{g}).$

######  Proposition 2.1.

  1. (1)

As a $\mathfrak{g}$-module, the second symmetric power $S^{2}(\mathfrak{g})$
irreducibly decomposes to

(2.3) $\displaystyle\begin{cases}F(2e_{1}-2e_{n})\oplus
F(e_{1}+e_{2}-e_{n-1}-e_{n})\oplus F(e_{1}-e_{n})\oplus F(0)&\text{ if $n\geq
4$,}\\\ F(2e_{1}-2e_{n})\oplus F(e_{1}-e_{n})\oplus F(0)&\text{ if $n=3$,}\\\
F(2e_{1}-2e_{n})\oplus F(0)&\text{ if $n=2$.}\end{cases}$

Here $F(\lambda)$ denotes the irreducible $\mathfrak{g}$-submodule of
$S^{2}(\mathfrak{g})$ with highest weight $\lambda$ and we have
$F(0)=\mathbb{C}\Omega$.

  2. (2)

For $n\geq 4$, the $\mathfrak{g}$-submodule
$F(e_{1}+e_{2}-e_{n-1}-e_{n})\oplus F(e_{1}-e_{n})\oplus F(0)$ is equal to

(2.4)
$\displaystyle\mathop{\mathrm{span}}\nolimits\Set{T_{i,j}T_{k,l}-T_{i,l}T_{k,j}}{1\leq
i,j,k,l\leq n}.$

  3. (3)

For $n\geq 3$, the linear map

$\mathfrak{g}\to F(e_{1}-e_{n});A\mapsto\sum_{i,j,k}A_{i,j}T_{i,k}T_{k,j}$

is a $\mathfrak{g}$-isomorphism.

###### Proof.

By $[T_{r,s},T_{i,j}]=\delta_{i,s}T_{r,j}-\delta_{r,j}T_{i,s}$, we have

(2.5) $\displaystyle[T_{r,s},T_{i,j}T_{k,l}-T_{i,l}T_{k,j}]$
$\displaystyle=(\delta_{i,s}T_{r,j}-\delta_{r,j}T_{i,s})T_{k,l}+T_{i,j}(\delta_{k,s}T_{r,l}-\delta_{r,l}T_{k,s})$
(2.6)
$\displaystyle\quad-(\delta_{i,s}T_{r,l}-\delta_{r,l}T_{i,s})T_{k,j}-T_{i,l}(\delta_{k,s}T_{r,j}-\delta_{r,j}T_{k,s})$
(2.7)
$\displaystyle=\delta_{i,s}(T_{r,j}T_{k,l}-T_{r,l}T_{k,j})-\delta_{r,j}(T_{i,s}T_{k,l}-T_{i,l}T_{k,s})$
(2.8)
$\displaystyle\quad+\delta_{k,s}(T_{i,j}T_{r,l}-T_{i,l}T_{r,j})-\delta_{r,l}(T_{i,j}T_{k,s}-T_{i,s}T_{k,j}).$

Therefore the subspace (2.4) is invariant under the action of $\mathfrak{g}$
and has a highest weight vector $T_{1,n-1}T_{2,n}-T_{1,n}T_{2,n-1}$ of weight
$e_{1}+e_{2}-e_{n-1}-e_{n}$ for $n\geq 2$.

Let us consider a vector
$\sum_{k}T_{i,k}T_{k,j}=\sum_{k}(T_{i,k}T_{k,j}-T_{i,j}T_{k,k})$, which
belongs to (2.4). We see

(2.9) $\displaystyle[T_{r,s},\sum_{k}T_{i,k}T_{k,j}]$
$\displaystyle=\sum_{k}\\{(\delta_{i,s}T_{r,k}-\delta_{r,k}T_{i,s})T_{k,j}+T_{i,k}(\delta_{k,s}T_{r,j}-\delta_{r,j}T_{k,s})\\}$
(2.10)
$\displaystyle=\delta_{i,s}\sum_{k}T_{r,k}T_{k,j}-\delta_{r,j}\sum_{k}T_{i,k}T_{k,s}.$

Therefore $[\mathfrak{g},\Omega]=0$ and the map in (3) is a
$\mathfrak{g}$-homomorphism into $S^{2}(\mathfrak{g})$. For $n\geq 3$, the
submodule (2.4) has highest weight $e_{1}-e_{n}$ since
$\sum_{k}T_{1,k}T_{k,n}$ is nonzero.

Furthermore, (2.4) has highest weight $0$ as it contains $\Omega$, and it does
not contain a nonzero multiple of $T_{1,n}T_{1,n}$, which is a highest weight
vector of weight $2e_{1}-2e_{n}$ in $S^{2}(\mathfrak{g})$.

By the Weyl dimension formula, we see the dimension of (2.3) is equal to the
square of $n(n-1)/2$, which is the dimension of $S^{2}(\mathfrak{g})$. Hence
(2.3) equals the direct sum of $F(2e_{1}-2e_{n})$ and (2.4), and our claim
follows. ∎

For $1\leq i<j<k<l\leq n$, we put

$\mathcal{M}_{i,j,k,l}:=\Set{a_{1}(T_{i,j}+T_{k,l})+a_{2}(T_{i,k}+T_{j,l})+a_{3}(T_{i,l}+T_{j,k})\in\mathfrak{g}}{\sum_{r=1}^{3}a_{r}=0}.$

Define a injective linear map

$T\colon\Set{A\in\mathfrak{gl}(n,\mathbb{C})}{A_{i,j}=0\text{ for $i\geq
j$}}\to
S^{2}(\mathfrak{g})_{0};A\mapsto\sum_{i,j}A_{i,j}(T_{i,i}T_{j,j}-T_{i,j}T_{j,i}).$

######  Proposition 2.2.

Assume $n\geq 4$. Then the zero weight space
$F(e_{1}+e_{2}-e_{n-1}-e_{n})_{0}$ is equal to

(2.11)
$\displaystyle\mathop{\mathrm{span}}\nolimits\Set{T(\mathcal{M}_{i,j,k,l})}{1\leq
i<j<k<l\leq n}.$

###### Proof.

Define an $\mathfrak{su}(n)$-invariant Hermitian inner product $(\cdot,\cdot)$
on $\mathfrak{g}$ by
$(X,Y):=B(X,{}^{t}\overline{Y})=\mathop{\mathrm{Tr}}\nolimits(X{}^{t}\overline{Y})$
($X,Y\in\mathfrak{g}$), where $\overline{Y}$ denotes the complex conjugate of
$Y$. It induces an $\mathfrak{su}(n)$-invariant Hermitian inner product on
$\mathfrak{g}\otimes\mathfrak{g}$ and one on $S^{2}(\mathfrak{g})$ via the
symmetrization, which we write $(\cdot,\cdot)$ by abuse of notation. By the
assumption $n\geq 4$ and Proposition 2.1 and its proof, the zero weight space
$F(e_{1}+e_{2}-e_{n-1}-e_{n})_{0}$ is the orthogonal complement of
$F(e_{1}-e_{n})_{0}+F(0)=\mathop{\mathrm{span}}\nolimits\Set{\sum_{k}T_{i,k}T_{k,i}}{1\leq
i\leq n}$ in the zero weight space
$\mathop{\mathrm{span}}\nolimits\Set{T_{i,i}T_{j,j}-T_{i,j}T_{j,i}}{i<j}$ of
(2.4). In particular, $\dim
F(e_{1}+e_{2}-e_{n-1}-e_{n})_{0}=n(n-1)/2-n=n(n-3)/2$.

Let us prove that $T(\mathcal{M}_{i,j,k,l})$ is contained in
$F(e_{1}+e_{2}-e_{n-1}-e_{n})_{0}$ for any $i<j<k<l$. By

$(T_{i,j},T_{r,s})=\mathop{\mathrm{Tr}}\nolimits\left(\left(E_{i,j}-\frac{\delta_{i,j}}{n}I_{n}\right)\left(E_{s,r}-\frac{\delta_{r,s}}{n}I_{n}\right)\right)=\delta_{i,r}\delta_{j,s}-\frac{\delta_{i,j}\delta_{r,s}}{n},$

we have

(2.12) $\displaystyle
2\Bigl{(}T_{i,i}T_{j,j}-T_{i,j}T_{j,i},\sum_{s}T_{r,s}T_{s,r}\Bigr{)}$ (2.13)
$\displaystyle=\sum_{s}\\{(T_{i,i},T_{r,s})(T_{j,j},T_{s,r})+(T_{j,j},T_{r,s})(T_{i,i},T_{s,r})$
(2.14)
$\displaystyle\qquad\quad-(T_{i,j},T_{r,s})(T_{j,i},T_{s,r})-(T_{j,i},T_{r,s})(T_{i,j},T_{s,r})\\}$
(2.15)
$\displaystyle=\sum_{s}\left\\{2\left(\delta_{i,r}\delta_{i,s}-\frac{\delta_{r,s}}{n}\right)\left(\delta_{j,r}\delta_{j,s}-\frac{\delta_{r,s}}{n}\right)-\delta_{i,r}\delta_{j,s}-\delta_{j,r}\delta_{i,s}\right\\}$
(2.16)
$\displaystyle=2\left(\delta_{i,r}-\frac{1}{n}\right)\left(\delta_{j,r}-\frac{1}{n}\right)-\delta_{i,r}-\delta_{j,r}$
(2.17)
$\displaystyle=\frac{2}{n^{2}}-\left(\frac{2}{n}+1\right)(\delta_{i,r}+\delta_{j,r})$

for any $i<j$ and $1\leq r\leq n$. Since $\sum_{r,s}A_{r,s}=0$ and
$\sum_{r}({A}_{r,r_{0}}+A_{r_{0},r})=0$ for any $A\in\mathcal{M}_{i,j,k,l}$
and $1\leq r_{0}\leq n$, it follows that $T(\mathcal{M}_{i,j,k,l})$ is
orthogonal to $F(e_{1}-e_{n})_{0}+F(0)$, and is included in
$F(e_{1}+e_{2}-e_{3}-e_{4})_{0}$.

Let us prove that the linear map $\Pi$ from
$\sum_{i<j<k<l}\mathcal{M}_{i,j,k,l}$ to $\bigoplus_{1\leq i<j\leq
n-2}\mathbb{C}\oplus\bigoplus_{1\leq i\leq n-3}\mathbb{C}$ defined by
$\Pi(A):=((A_{i,j})_{(i,j)},(A_{i,n-1})_{i})$ is bijective. If it holds, we
obtain $\dim\sum_{i<j<k<l}\mathcal{M}_{i,j,k,l}=(n-2)(n-3)/2+n-3=n(n-3)/2=\dim
F(e_{1}+e_{2}-e_{3}-e_{4})_{0}$, and the proof is complete.

For any $(a_{i,j})\in\bigoplus_{1\leq i<j\leq n-2}\mathbb{C}$ and
$(a_{i,n-1})\in\bigoplus_{1\leq i\leq n-3}\mathbb{C}$, the image of

$\sum_{1\leq i<j\leq
n-2}a_{i,j}(T_{i,j}+T_{n-1,n}-T_{i,n}-T_{j,n-1})+\sum_{1\leq i\leq
n-3}a_{i,n-1}(T_{i,n-1}+T_{n-2,n}-T_{i,n}-T_{n-2,n-1})$

is equal to $((a_{i,j}),(a_{i,n-1}))$. Hence $\Pi$ is surjective.

What is left is to prove the injectivity of $\Pi$. Assume
$A\in\sum_{i<j<k<l}\mathcal{M}_{i,j,k,l}$ and $\Pi(A)=0$. We see
$\sum_{j=1}^{i-1}A_{j,i}+\sum_{j=i+1}^{n}A_{i,j}=0$ for any $1\leq i\leq n$.
Hence for any $1\leq i\leq n-3$, we have
$A_{i,n}=-\sum_{j=i+1}^{n-1}A_{i,j}-\sum_{j=1}^{i-1}A_{j,i}=0$. Moreover, the
sum of any two of $A_{n-2,n-1},A_{n-2,n},A_{n-1,n}$ is zero. Therefore $A=0$,
and $\Pi$ is injective. ∎

We next classify irreducible highest weight modules annihilated by some
$\mathfrak{g}$-submodules of $\bigoplus_{0\leq i\leq 2}S^{i}(\mathfrak{g})$.
For $\lambda\in\mathfrak{h}^{\vee}$, we write $L(\lambda)$ for the irreducible
highest weight $\mathfrak{g}$-module with highest weight $\lambda$. Let
$l_{\lambda}$ be a nonzero highest weight vector in $L(\lambda)$. We remark
that the infinitesimal character of $L(\lambda)$ is the orbit of the Weyl
group through $\lambda+\rho$, where $\rho=\sum_{1\leq i\leq
n}((n+1-2i)/2)e_{i}$ is half the sum of positive roots for $\Sigma^{+}$.

For $a\in\mathbb{C}$, let $F^{a}$ be the $\mathfrak{g}$-submodule of
$\mathfrak{g}+S^{2}(\mathfrak{g})$ defined by

(2.18) $\displaystyle
F^{a}:=\begin{cases}F(e_{1}+e_{2}-e_{n-1}-e_{n})+F(e_{1}-e_{n})^{a}&\text{ if
$n\geq 4$,}\\\ F(e_{1}-e_{n})^{a}&\text{ if $n=2,3$,}\end{cases}$

where

$F(e_{1}-e_{n})^{a}:=\Set{\sum_{i,j,k}A_{i,j}T_{i,k}T_{k,j}-\frac{a(n-2)}{n}A}{A\in\mathfrak{g}}\subset\mathfrak{g}+S^{2}(\mathfrak{g}).$

The normalizing factor $(n-2)/n$ is put in order to simplify our assertions.
We remark that $F^{a}=0$ when $n=2$.

For a positive integer $m$, put $\mathbf{1}_{m}$ to be the $m$-tuple

$\mathbf{1}_{m}:=(\underbrace{1,\ldots,1}_{m}).$

We regard $\mathbf{1}_{0}$ as the empty tuple. For $1\leq i\leq n$ and
$a\in\mathbb{C}$, let us define an element of $\mathfrak{h}^{\vee}$ by

(2.19) $\displaystyle\lambda(i,a)$
$\displaystyle:=\frac{1}{n}\left(\left(-a-\frac{n}{2}\right)\mathbf{1}_{i-1},(n-1)a-\frac{n(n+1-2i)}{2},\left(-a+\frac{n}{2}\right)\mathbf{1}_{n-i}\right).$

Let $\mathop{\mathrm{sym}}\nolimits$ be the symmetrization map, that is, the
$\mathfrak{g}$-isomorphism defined by

$\mathop{\mathrm{sym}}\nolimits\colon S(\mathfrak{g})\to U(\mathfrak{g});\
A_{1}A_{2}\cdots
A_{m}\mapsto\frac{1}{m!}\sum_{\eta\in\mathfrak{S}_{m}}A_{\eta(1)}A_{\eta(2)}\cdots
A_{\eta(m)},$

where $m\in\mathbb{N}$ and $\mathfrak{S}_{m}$ denotes the group of
permutations of $\\{1,2,\ldots,m\\}$.

We use the following lemma to compute annihilators of highest weight modules:

######  Lemma 2.3.

Let $\lambda\in\mathfrak{h}^{\vee}$, and $F$ be a $\mathfrak{g}$-submodule of
$U(\mathfrak{g})$ with respect to the adjoint action. Then
$\mathop{\mathrm{sym}}\nolimits(F)$ is contained in the annihilator
$\mathop{\mathrm{Ann}}\nolimits L(\lambda)$ if and only if the zero weight
space $\mathop{\mathrm{sym}}\nolimits(F_{0})$ annihilates $l_{\lambda}$.

###### Proof.

It suffices to show the “if” part. Suppose
$\mathop{\mathrm{sym}}\nolimits(F_{0})l_{\lambda}=0$. We write
$\mathbb{C}_{\lambda}$ for the character of the Borel subalgebra
$\mathfrak{b}$ where $\mathfrak{h}$ acts by $\lambda$ and the nilradical acts
trivially. Then $L(\lambda)$ is isomorphic to the quotient of the Verma module
$M(\lambda):=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_{\lambda}$ by
the maximal proper submodule. Let $m_{\lambda}\in M(\lambda)$ be a nonzero
highest weight vector of weight $\lambda$. Let
$\mathop{\mathrm{proj}}\nolimits_{2}$ be the projection to the second
component of the decomposition
$U(\mathfrak{g})=U(\mathfrak{g})\mathfrak{n}\oplus U(\mathfrak{b}^{-})$. Since
$F$ is a $\mathfrak{g}$-module, $\mathop{\mathrm{sym}}\nolimits(F)M(\lambda)$
is a $\mathfrak{g}$-submodule of $M(\lambda)$. By the Poincaré-Birkoff-Witt
theorem, we see
$\mathop{\mathrm{sym}}\nolimits(F)M(\lambda)=\mathop{\mathrm{sym}}\nolimits(F)U(\mathfrak{n}^{-})m_{\lambda}=U(\mathfrak{n}^{-})\mathop{\mathrm{sym}}\nolimits(F)m_{\lambda}=U(\mathfrak{n}^{-})\mathop{\mathrm{proj}}\nolimits_{2}(\mathop{\mathrm{sym}}\nolimits(F))m_{\lambda}$.
Hence the $\lambda$ weight space of
$\mathop{\mathrm{sym}}\nolimits(F)M_{\lambda}$ is
$\mathop{\mathrm{sym}}\nolimits(F_{0})m_{\lambda}$, which is zero by our
assumption. Therefore the submodule
$\mathop{\mathrm{sym}}\nolimits(F)M(\lambda)$ is proper, and
$\mathop{\mathrm{sym}}\nolimits(F)L(\lambda)$ is zero. ∎

######  Proposition 2.4.

Let $a\in\mathbb{C},\lambda\in\mathfrak{h}^{\vee}$. Then
$\mathop{\mathrm{sym}}\nolimits(F^{a})$ annihilates the highest weight module
$L(\lambda)$ if and only if $\lambda=\lambda(i,a)$ for some $1\leq i\leq n$.

For the proof of Proposition 2.4, we use the following

######  Lemma 2.5.

Assume $n\geq 4$. The symmetrization
$\mathop{\mathrm{sym}}\nolimits(F(e_{1}+e_{2}-e_{n-1}-e_{n}))$ annihilates
$L(\lambda)$ if and only if $\lambda=\lambda(i,a)$ for some $1\leq i\leq n$
and $a\in\mathbb{C}$.

###### Proof.

By Lemma 2.3, Proposition 2.2 and the definition of $\mathcal{M}_{i<j<k<l}$,
we see that
$\mathop{\mathrm{sym}}\nolimits(F(e_{1}+e_{2}-e_{n-1}-e_{n}))\subset\mathop{\mathrm{Ann}}\nolimits
L(\lambda)$ is equivalent to
$\mathop{\mathrm{sym}}\nolimits(T(\mathcal{M}_{i,j,k,l}))l_{\lambda}=0$ for
$i<j<k<l$, and to

(2.20)
$\displaystyle\mathop{\mathrm{sym}}\nolimits(T_{i,i}T_{j,j}-T_{i,j}T_{j,i}+T_{k,k}T_{l,l}-T_{k,l}T_{l,k})l_{\lambda}$
(2.21)
$\displaystyle=\mathop{\mathrm{sym}}\nolimits(T_{i,i}T_{k,k}-T_{i,k}T_{k,i}+T_{j,j}T_{l,l}-T_{j,l}T_{l,j})l_{\lambda}$
(2.22)
$\displaystyle=\mathop{\mathrm{sym}}\nolimits(T_{i,i}T_{l,l}-T_{i,l}T_{l,i}+T_{j,j}T_{k,k}-T_{j,k}T_{k,j})l_{\lambda}$

for $i<j<k<l$. Since we have

$\mathop{\mathrm{sym}}\nolimits(T_{r,r}T_{s,s}-T_{r,s}T_{s,r})l_{\lambda}=\\{\lambda_{r}\lambda_{s}-(\lambda_{r}-\lambda_{s})/2\\}l_{\lambda}=\\{(\lambda_{r}+1/2)(\lambda_{s}-1/2)+1/4\\}l_{\lambda}$

for $1\leq r<s\leq n$, the equations (2.21) are equivalent to

(2.23)
$\displaystyle(\lambda_{i}+1/2)(\lambda_{j}-1/2)+(\lambda_{k}+1/2)(\lambda_{l}-1/2)$
(2.24)
$\displaystyle=(\lambda_{i}+1/2)(\lambda_{k}-1/2)+(\lambda_{j}+1/2)(\lambda_{l}-1/2)$
(2.25)
$\displaystyle=(\lambda_{i}+1/2)(\lambda_{l}-1/2)+(\lambda_{j}+1/2)(\lambda_{k}-1/2)$

for $i<j<k<l$. The equations $\eqref{eq:lambdaijkl}$ hold if and only if

$(\lambda_{i}-\lambda_{l}+1)(\lambda_{j}-\lambda_{k})=(\lambda_{i}-\lambda_{j})(\lambda_{k}-\lambda_{l})=0\quad\text{
for $i<j<k<l$,}$

which implies the assertion by the condition $\sum_{i}\lambda_{i}=0$. ∎

###### Proof of Proposition 2.4.

By Lemma 2.3,
$\mathop{\mathrm{sym}}\nolimits(F^{a})\subset\mathop{\mathrm{Ann}}\nolimits
L(\lambda)$ is equivalent to
$\mathop{\mathrm{sym}}\nolimits(F^{a}_{0})l_{\lambda}=0$.

We see
$F(e_{1}-e_{n})^{a}_{0}=\\{\sum_{i,k}A_{i}T_{i,k}T_{k,i}-a(n-2)/n\sum_{i}A_{i}T_{i,i}\mid\sum_{i}A_{i}=0\\}$
and

(2.26)
$\displaystyle\mathop{\mathrm{sym}}\nolimits\Bigl{(}\sum_{i,k}A_{i}T_{i,k}T_{k,i}-\frac{a(n-2)}{n}\sum_{i}A_{i}T_{i,i}\Bigr{)}l_{\lambda}$
(2.27)
$\displaystyle=\Bigl{(}\sum_{i}A_{i}\lambda_{i}^{2}+\frac{1}{2}\sum_{i<k}A_{i}[T_{i,k},T_{k,i}]+\frac{1}{2}\sum_{i>k}A_{i}[T_{k,i},T_{i,k}]-\frac{a(n-2)}{n}\sum_{i}A_{i}\lambda_{i}\Bigr{)}l_{\lambda}$
(2.28) $\displaystyle=\sum_{i}A_{i}C_{i}l_{\lambda},$

where we put

$C_{i}:=\lambda_{i}^{2}+\left(\frac{n+1}{2}-i-\frac{a(n-2)}{n}\right)\lambda_{i}+\frac{1}{2}\sum_{k=1}^{i-1}\lambda_{k}-\frac{1}{2}\sum_{k=i+1}^{n}\lambda_{k}.$

Therefore
$\mathop{\mathrm{sym}}\nolimits(F(e_{1}-e_{n})^{a}_{0})l_{\lambda}=0$ if and
only if the scalar $C_{i}$ is independent of $1\leq i\leq n$, that is,

(2.29)
$\displaystyle(\lambda_{i+1}-\lambda_{i})\left(\lambda_{i+1}+\lambda_{i}+\frac{n}{2}-i-\frac{a(n-2)}{n}\right)=0\quad\text{
for $1\leq i\leq n-1$.}$

When $n=2$, (2.29) holds and any element in $\mathfrak{h}^{\vee}$ can be
written as $\lambda(1,a)$ for some $a\in\mathbb{C}$, and the assertion
follows. When $n=3$, the condition $\sum\lambda_{i}=0$ implies that (2.29) is
equivalent to
$\lambda=(-1+2a/3,(1/2-a/3)\mathbf{1}_{2}),(-1/2-a/3,2a/3,1/2-a/3),((-1/2-a/3)\mathbf{1}_{2},1+2a/3)$,
which is our claim by $F^{a}_{0}=F(e_{1}-e_{3})^{a}_{0}$ and (2.19).

When $n\geq 4$, the definition of $F^{a}$, Lemma 2.5 and the above argument
imply that $\mathop{\mathrm{sym}}\nolimits(F^{a}_{0})l_{\lambda}=0$ if and
only if $\lambda$ is equal to $\lambda(r,c)$ satisfying (2.29) for some $1\leq
r\leq n$ and $c\in\mathbb{C}$. Since $\lambda(r,c)$ satisfies (2.29) exactly
when $c=a$, we obtain the desired conclusion. ∎

######  Remark 2.6.

The infinitesimal character of $L(\lambda(i,a))$ is the orbit through
$\lambda(i,a)+\rho=\sum_{k=1}^{i-1}(n/2-k-a/n)e_{k}+a(n-1)/ne_{i}+\sum_{k=i+1}^{n}(n/2-k+1-a/n)e_{k}$
and does not depend on $1\leq i\leq n$. In particular, the Casimir element
$\mathop{\mathrm{sym}}\nolimits(\Omega)$ acts by

(2.30) $\displaystyle\|\lambda(n,a)+\rho\|^{2}-\|\rho\|^{2}$ (2.31)
$\displaystyle=\sum_{i=1}^{n-1}\left\\{\left(\frac{n+1}{2}-i-\frac{2a+n}{2n}\right)^{2}-\left(\frac{n+1}{2}-i\right)^{2}\right\\}+\frac{a^{2}(n-1)^{2}}{n^{2}}-\frac{(n-1)^{2}}{4}$
(2.32)
$\displaystyle=\frac{(n-1)(2a+n)}{2n}\left(\frac{2a+n}{2n}-2\left(\frac{n+1}{2}-\frac{n}{2}\right)\right)+\frac{(n-1)^{2}(2a+n)(2a-n)}{4n^{2}}$
(2.33) $\displaystyle=\frac{(n-1)(2a+n)(2a-n)}{4n},$

where the norm $\|\cdot\|$ on $\mathfrak{h}^{\vee}$ is induced from the
invariant bilinear form $B$ defined in (2.1).

######  Definition 2.7.

For $a\in\mathbb{C}$, we define $J_{a}$ to be the two-sided ideal in
$U(\mathfrak{g})$ generated by the subspace
$\mathop{\mathrm{sym}}\nolimits(F^{a})$ and the element
$\mathop{\mathrm{sym}}\nolimits(\Omega)-(n-1)(a+n/2)(a-n/2)/n$:

$J_{a}:=\left\langle\mathop{\mathrm{sym}}\nolimits(X),\mathop{\mathrm{sym}}\nolimits(\Omega)-\frac{(n-1)(2a+n)(2a-n)}{4n}\mid
X\in F^{a}\right\rangle.$

######  Theorem 2.8.

  1. (1)

Let $J$ be a two-sided ideal of $U(\mathfrak{g})$. The following are
equivalent:

    1. (i)

$J=J_{a}$ for some $a\in\mathbb{C}$.

    2. (ii)

the associated graded ideal $\mathop{\mathrm{gr}}\nolimits J$ is equal to the
ideal $\mathcal{I}(\overline{\mathcal{O}^{\min}})$ defined by the closure
$\overline{\mathcal{O}^{\min}}$ of the minimal nilpotent coadjoint orbit in
$\mathfrak{g}^{\vee}$.

    3. (iii)

$J$ is completely prime, primitive and has $\overline{\mathcal{O}^{\min}}$ as
its associated variety.

Moreover, we have $J_{a}=J_{a^{\prime}}$ if and only if $a=a^{\prime}$ when
$n\geq 3$, and $a=\pm a^{\prime}$ when $n=2$.

  2. (2)

Let $\lambda\in\mathfrak{h}^{\vee}$ and $a\in\mathbb{C}$. Then the annihilator
of $L(\lambda)$ is equal to $J_{a}$ if and only if $\lambda=\lambda(i,a)$ for
some $1\leq i\leq n$ satisfying

(2.34) $a\not\in\frac{n}{2}+\mathbb{N}$ if $i=1$ and
$a\not\in-\frac{n}{2}-\mathbb{N}$ if $i=n$.

######  Remark 2.9.

  1. (1)

When $\mathfrak{g}$ is simple Lie algebra not of type $A$, there uniquely
exists the two-sided ideal $J$ satisfying (ii) (or equivalently, (iii)) [4,
Theorem 3.1]. The ideal is called the Joseph ideal.

  2. (2)

The equivalence of (i) and (ii), and (2) for the case $a=n/2-i,n/2-i+1$ or
$i=1,n$ (that is, the case where $\lambda$ extends to a character of a
standard maximal parabolic subalgebra) can be deduced form the argument in [2,
Section 7.8] (there is some typo: $c^{\prime}$ should be $c^{\prime}(n+2)/n$
except for “$\phi=\phi^{a}+c^{\prime}\phi^{s}$”).

###### Proof of Theorem 2.8.

Let $a\in\mathbb{C}$. We first claim $\mathop{\mathrm{gr}}\nolimits
J_{a}=\mathcal{I}(\overline{\mathcal{O}^{\min}})$ and (2). By Proposition 2.4
and Remark 2.6, we have $J_{a}\subset\mathop{\mathrm{Ann}}\nolimits
L(\lambda)$ if and only if $\lambda=\lambda(i,a)$ for $1\leq i\leq n$. Hence
the proof of (2) is reduced to show that $J_{a}=\mathop{\mathrm{Ann}}\nolimits
L(\lambda(i,a))$ exactly when the condition (2.34) holds.

By [6, Theorem III.2.1] and Proposition 2.1, the ideal
$\mathcal{I}(\overline{\mathcal{O}^{\min}})$ is generated by
$F(e_{1}+e_{2}-e_{n-1}-e_{n})$ (if $n\geq 4$), $F(e_{1}-e_{n})$ (if $n\geq 3$)
and $F(0)$. Therefore the definition of $J_{a}$ and
$J_{a}\subset\mathop{\mathrm{Ann}}\nolimits L(\lambda(i,a))$ shows
$\mathcal{I}(\overline{\mathcal{O}^{\min}})\subset\mathop{\mathrm{gr}}\nolimits
J_{a}\subset\mathop{\mathrm{gr}}\nolimits\mathop{\mathrm{Ann}}\nolimits
L(\lambda(i,a))$.

Any ideal strictly containing $\mathcal{I}(\overline{\mathcal{O}^{\min}})$ has
finite-codimension in $S(\mathfrak{g})$ since $S^{k}(\mathfrak{g})$ decomposes
to the direct sum of the $k$-th Cartan power of $\mathfrak{g}$ and
$\mathcal{I}(\overline{\mathcal{O}^{\min}})\cap S^{k}(\mathfrak{g})$ as a
$\mathfrak{g}$-module for $k\in\mathbb{N}$ [6, Proposition III.1.1]. Moreover,
we see that $\mathop{\mathrm{gr}}\nolimits\mathop{\mathrm{Ann}}\nolimits
L(\lambda(i,a))$ has infinite-codimension in $S(\mathfrak{g})$ if and only if
$L(\lambda(i,a))$ is infinite-dimensional, or the condition (2.34) holds. In
particular, the ideal $J_{a}$ has infinite-codimension by
$J_{a}\subset\mathop{\mathrm{Ann}}\nolimits
L(1,a)\cap\mathop{\mathrm{Ann}}\nolimits L(n,a)$ and
$(n/2+\mathbb{N})\cap(-n/2+\mathbb{N})=\emptyset$. Therefore we obtain
$\mathcal{I}(\overline{\mathcal{O}^{\min}})=\mathop{\mathrm{gr}}\nolimits
J_{a}\subset\mathop{\mathrm{gr}}\nolimits\mathop{\mathrm{Ann}}\nolimits
L(\lambda(i,a))$ and the last inclusion becomes an equality exactly in the
case (2.34), which proves our claims.

We next prove (1). By
$\mathcal{I}(\overline{\mathcal{O}^{\min}})=\mathop{\mathrm{gr}}\nolimits
J_{a}$, the ideal $J_{a}$ is completely prime since
$\mathcal{I}(\overline{\mathcal{O}^{\min}})$ is prime in $S(\mathfrak{g})$,
and the associated variety $\mathop{\mathrm{Ass}}\nolimits J_{a}$ is
$\overline{\mathcal{O}^{\min}}$. Hence (i) implies (ii) and (iii).

Let us prove (ii)$\Rightarrow$(i). Assume (ii). By the above result by
Garfinkle, there exist scalars $a,c\in\mathbb{C}$ such that $F^{a}$ and
$\mathop{\mathrm{sym}}\nolimits(\Omega)-c$ is included in $J$. Take a maximal
(hence primitive) ideal $J^{\max}$ containing $J$. By the work of Duflo [9,
Satz 7.3], $J^{\max}$ is the annihilator of some irreducible highest weight
module. It follows from $F^{a}\subset J^{\max}$ and Proposition 2.4 that
$J^{\max}=\mathop{\mathrm{Ann}}\nolimits L(\lambda(i,a))$ for some $1\leq
i\leq n$. Since $J^{\max}$ contains
$\mathop{\mathrm{sym}}\nolimits(\Omega)-(n-1)(a-n/2)(a+n/2)/n$ and does not
contain scalars, we see $c=(n-1)(a-n/2)(a+n/2)/n$ and $J_{a}\subset J$. Then
$\mathop{\mathrm{gr}}\nolimits
J_{a}=\mathcal{I}(\overline{\mathcal{O}^{\min}})=\mathop{\mathrm{gr}}\nolimits
J$ implies $J=J_{a}$, and (i) follows.

Let us prove (iii)$\Rightarrow$(i). Assume (iii). By [17, Théorème IV.1],
there exist a standard parabolic subalgebra $\mathfrak{q}$ and a character
$\chi$ of $\mathfrak{q}$ such that $J=\mathop{\mathrm{Ann}}\nolimits
U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\chi$. By [9, Satz 10.9], we have

$\dim\mathfrak{g}/\mathfrak{q}=\mathop{\mathrm{Dim}}\nolimits(U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\chi)=\mathop{\mathrm{Dim}}\nolimits(U(\mathfrak{g})/J)/2=\dim\overline{\mathcal{O}^{\min}}/2=n-1,$

where $\mathop{\mathrm{Dim}}\nolimits$ denotes the Gelfand-Kirillov dimension.
Therefore $\mathfrak{q}$ is the maximal parabolic subalgebra corresponding to
the partition $(1,n-1)$ or the one corresponding to $(n-1,1)$, and $\chi$ is
equal to $\lambda(1,a)$ or $\lambda(n,a)$ on $\mathfrak{h}$ for some
$a\in\mathbb{C}$, respectively. By Lemma 2.10, we see $J_{a}\subset J$, and
(i) follows from $\mathop{\mathrm{gr}}\nolimits
J_{a}=\mathcal{I}(\mathcal{O}^{\min})\supset\mathop{\mathrm{gr}}\nolimits J$.

Finally, we prove the last assertion of (1). Suppose $J_{a}=J_{a^{\prime}}$.
It suffices to show $a=a^{\prime}$ when $n\geq 3$, and $a=\pm a^{\prime}$ when
$n=2$. By $\mathop{\mathrm{gr}}\nolimits
J_{a}=\mathcal{I}(\mathcal{O}^{\min})$, we see that $J_{a}$ does not contain a
nonzero element in $\mathfrak{g}$ nor a nonzero scalar. Therefore the
nonzeroness of $F^{a}$ and $F^{a^{\prime}}$ shows $a=a^{\prime}$ when $n\geq
3$. When $n=2$, the ideal $J_{a}$ is generated by
$\mathop{\mathrm{sym}}\nolimits(\Omega)-(a^{2}-1)/2$ and we see $a=\pm
a^{\prime}$. ∎

The next lemma is used not only in the proof of Theorem 2.8 but also in the
construction of minimal $(\mathfrak{g},\mathfrak{k})$-modules in Theorem 5.3.

######  Lemma 2.10.

Let $a\in\mathbb{C}$ and $\mathfrak{q}$ be the standard parabolic subalgebra
$\mathfrak{q}_{(1,n-1)}$ (resp. $\mathfrak{q}_{(n-1,1)}$) corresponding to the
partition $(1,n-1)$ (resp. $(n-1,1)$). Then the ideal $J_{a}$ annihilates the
generalized Verma module
$U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\mathbb{C}_{\lambda(1,a)}$ (resp.
$U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\mathbb{C}_{\lambda(n,a)}$).

###### Proof.

Let $J$ be the anti-diagonal $n$-by-$n$ matrix whose anti-diagonal entries are
one. The involution on $\mathfrak{g}$ defined by $X\mapsto-J{}^{t}XJ$ for
$X\in\mathfrak{g}$ preserves $\mathfrak{h}$. It maps $\mathfrak{q}_{(1,n-1)}$
onto $\mathfrak{q}_{(n-1,1)}$, and induces an isomorphism on
$\mathfrak{h}^{\vee}$ sending $\lambda(1,a)$ to $\lambda(n,-a)$ for
$a\in\mathbb{C}$. Hence it suffices to show our assertion for
$\mathfrak{q}=\mathfrak{q}_{(1,n-1)}$.

Since $J_{a}$ and
$\mathop{\mathrm{Ann}}\nolimits(U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\mathbb{C}_{\lambda(1,a)})$
have the same infinitesimal character, it suffices to prove
$\mathop{\mathrm{sym}}\nolimits(F^{a})\subset\mathop{\mathrm{Ann}}\nolimits(U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\mathbb{C}_{\lambda(1,a)})$.
Let $m_{\lambda(1,a)}$ be a nonzero highest weight vector of weight
$\lambda(1,a)$ in
$U(\mathfrak{g})\otimes_{U(\mathfrak{q})}\mathbb{C}_{\lambda(1,a)}$, and write
$(F^{a})^{-}$ for the space of lowest weight vectors in $F^{a}$. We have

$\mathop{\mathrm{sym}}\nolimits(F^{a})U(\mathfrak{g})m_{\lambda(1,a)}=U(\mathfrak{g})\mathop{\mathrm{sym}}\nolimits(F^{a})m_{\lambda(1,a)}=U(\mathfrak{g})\mathop{\mathrm{sym}}\nolimits((F^{a})^{-})m_{\lambda(1,a)}.$

In the last equality, we used
$\mathop{\mathrm{sym}}\nolimits([\mathfrak{n},(F^{a})^{-}])m_{\lambda(1,a)}=[\mathfrak{n},\mathop{\mathrm{sym}}\nolimits((F^{a})^{-})]m_{\lambda(1,a)}=\mathfrak{n}\mathop{\mathrm{sym}}\nolimits((F^{a})^{-})m_{\lambda(1,a)}$,
where $\mathfrak{n}$ denotes the subalgebra of $\mathfrak{g}$ consisting of
all positive root vectors. Therefore we are reduced to see
$\mathop{\mathrm{sym}}\nolimits((F^{a})^{-})m_{\lambda(1,a)}=0$. From
Proposition 2.1, the subspace $(F^{a})^{-}$ is spanned by
$T_{n,1}T_{n-1,2}-T_{n,2}T_{n-1,1}$ (if $n\geq 4$) and
$\sum_{k}T_{n,k}T_{k,1}-a(n-2)/nT_{n,1}$. By $T_{n-1,2}$ (if $n\geq 4$),
$T_{n,k}\in[\mathfrak{q},\mathfrak{q}]$ for $2\leq k\leq n-1$, we obtain

(2.35)
$\displaystyle\mathop{\mathrm{sym}}\nolimits(T_{n,1}T_{n-1,2}-T_{n,2}T_{n-1,1})m_{\lambda(1,a)}=0\quad\text{
if $n\geq 4$},$ (2.36)
$\displaystyle\mathop{\mathrm{sym}}\nolimits\Bigl{(}\sum_{k}T_{n,k}T_{k,1}-\frac{a(n-2)}{n}T_{n,1}\Bigr{)}m_{\lambda(1,a)}$
(2.37)
$\displaystyle=\left(\lambda(1,a)_{1}-\frac{1}{2}+\lambda(1,a)_{n}+\frac{n-1}{2}-\frac{a(n-2)}{n}\right)T_{n,1}m_{\lambda(1,a)}=0,$

and the proof is complete. ∎

##  3\. Minimal representations

In this section, we extend the definition of minimal
$(\mathfrak{g},\mathfrak{k})$-modules to real simple Lie algebras including
type $A$, and describe some properties on their $\mathfrak{k}$-types and a
criterion for isomorphism. In this section, we do not restrict ourselves to
simple Lie algebras of type $A$.

We will write $\mathfrak{g}_{0}$ for a real simple Lie algebra whose
complexification $\mathfrak{g}$ is simple. Fix a Cartan involution $\theta$ on
$\mathfrak{g}_{0}$, and write
$\mathfrak{g}_{0}=\mathfrak{k}_{0}+\mathfrak{p}_{0}$ for the Cartan
decomposition with respect to $\theta$.

######  Definition 3.1.

  1. (1)

An irreducible $(\mathfrak{g},\mathfrak{k})$-module is called _minimal_ if the
associated graded ideal of the annihilator is the ideal defined by the closure
of the minimal nilpotent coadjoint orbit in $\mathfrak{g}^{\vee}$.

  2. (2)

Let $a\in\mathbb{C}$ and assume $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$
($n\geq 2$). An irreducible $(\mathfrak{g},\mathfrak{k})$-module is called
_$a$ -minimal_ if the annihilator equals the ideal $J_{a}$ (see Definition
2.7).

  3. (3)

An irreducible admissible representation of a simple Lie group $G$ is called
minimal (resp. $a$-minimal) if the underlying
$(\mathfrak{g},\mathfrak{k})$-module is minimal (resp. $a$-minimal).

######  Remark 3.2.

Let $V$ be an irreducible $(\mathfrak{g},\mathfrak{k})$-module.

  1. (1)

When $\mathfrak{g}$ is not of type $A$, the
$(\mathfrak{g},\mathfrak{k})$-module $V$ is minimal if and only if the
annihilator of $V$ is the Joseph ideal (see Remark 2.9 (1)). When
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ ($n\geq 2$), $V$ is minimal if and
only if $V$ is $a$-minimal for some $a\in\mathbb{C}$, which holds if and only
if $\mathop{\mathrm{Ann}}\nolimits V$ is a completely prime primitive ideal
whose associated variety is the closure of the minimal nilpotent orbit by
Theorem 2.8 (1).

  2. (2)

Assume $V$ is minimal. Write
$\mathfrak{k}^{\perp}:=\\{X\in\mathfrak{g}^{\vee}\mid X(\mathfrak{k})=0\\}$.
Since $V$ is infinite-dimensional, we have
$0\subsetneq\mathop{\mathrm{Ass}}\nolimits
V\subset(\mathop{\mathrm{Ass}}\nolimits\mathop{\mathrm{Ann}}\nolimits
V)\cap\mathfrak{k}^{\perp}=\overline{\mathcal{O}^{\min}}\cap\mathfrak{k}^{\perp}=(\mathcal{O}^{\min}\cap\mathfrak{k}^{\perp})\cup\\{0\\}$.
Hence, for the existence of minimal $(\mathfrak{g},\mathfrak{k})$-modules, we
have a necessary condition

(3.1) $\displaystyle\mathcal{O}^{\min}\cap\mathfrak{k}^{\perp}\neq\emptyset.$

In particular, there exist no minimal $(\mathfrak{g},\mathfrak{k})$-modules
for $\mathfrak{g}_{0}=\mathfrak{su}(n),\mathfrak{sl}(n,\mathbb{H})$ $(n\geq
2)$ by [18, Proposition 4.1].

  3. (3)

Assume $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ ($n\geq 2$) and $V$ is
$a$-minimal. Then the contragredient $(\mathfrak{g},\mathfrak{k})$-module of
$V$ is $(-a)$-minimal. When $\mathfrak{g}_{0}$ is $\mathfrak{su}(p,q)$ with
$p+q=n$ (resp. $\mathfrak{sl}(n,\mathbb{R})$), the complex conjugate of $V$ is
$(-\overline{a})$-minimal (resp. $\overline{a}$-minimal). In particular, if
$n\geq 3$ and $V$ admits a nondegenerate invariant Hermitian form, then
$a\in\mathbb{R}$ (resp. $a\in\sqrt{-1}\mathbb{R}$) by Theorem 2.8 (1).

For a reductive Lie algebra $\mathfrak{l}$, we write
$\mathfrak{l}_{\mathop{\mathrm{ss}}\nolimits}$ and
$\mathfrak{z}(\mathfrak{l})$ for the derived subalgebra
$[\mathfrak{l},\mathfrak{l}]$ and the center, respectively.

From now, we also assume (3.1). Fix a Cartan subalgebra $\mathfrak{t}_{0}$ of
$\mathfrak{k}_{0}$. Define $\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}_{0}$
to be the centralizer $\mathfrak{z}_{\mathfrak{g}_{0}}(\mathfrak{t}_{0})$ of
$\mathfrak{t}_{0}$ in $\mathfrak{g}_{0}$. Then
$\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}_{0}$ is a maximally compact
Cartan subalgebra of $\mathfrak{g}_{0}$ [11, Proposition 6.60]. By the
assumption (3.1), we can take a positive system
$\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{+}$ of the
root system $\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})$
such that the highest root $\psi$ is noncompact imaginary and
$\theta\phi\in\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{+}$
for any
$\phi\in\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{+}$.
Then $\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{+}$
defines a positive system for
$(\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t})$
by

$\Sigma(\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t})^{+}:=\Set{\mu}{\phi\in\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{+},\phi=0\text{
on
}\mathfrak{z}(\mathfrak{k}),\mu=\phi|_{\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t}}}.$

Let $(\cdot,\cdot)$ be the inner product on the real vector space spanned the
set of roots
$\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})$ induced by
a nondegenerate invariant bilinear form $B$ on $\mathfrak{g}$. We normalize
the invariant form so that $(\psi,\psi)=2$. Let
$\mathfrak{k}^{1},\mathfrak{p}^{1}$ be the subspaces of
$\mathfrak{k},\mathfrak{p}$ spanned by weight spaces of $\mathfrak{t}$-weights
whose inner product with $\psi$ are $(\psi,\psi)/2=1$, respectively. We will
write $\rho(\mathfrak{k}^{1},\mathfrak{t})$ for half the sum of such roots in
$\Sigma(\mathfrak{k},\mathfrak{t})$. Set
$(\mathbb{C}\psi)^{\perp}:=\mathbb{C}\Set{H_{\phi}\in\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}}{\phi\in\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}),(\phi,\psi)=0}$,
where $H_{\phi}$ denotes the element in
$\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}$ corresponding to
$\phi\in(\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})^{\vee}$ under $B$. Let
$\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}$ be the set of
elements in $\mathfrak{t}^{\vee}$ annihilating
$(\mathbb{C}\psi)^{\perp}\cap\mathfrak{t}$. Here we regard
$\mathfrak{t}^{\vee}$ as a subspace of
$\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits}$ via the invariant form.

Let us check

(3.2)
$\displaystyle\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}=\begin{cases}\mathbb{C}\psi&\text{
if $\mathfrak{g}_{0}\not\cong\mathfrak{su}(p,q)$,}\\\
\mathbb{C}(e_{1}-e_{n})+\mathbb{C}(e_{1}+e_{n}-2/n\sum_{i}e_{i})&\text{ if
$\mathfrak{g}_{0}=\mathfrak{su}(p,q)$.}\end{cases}$

Here we put $n=p+q$ and $\\{e_{i}-e_{i+1}\\}_{1\leq i<n}$ denotes the simple
roots of $\Sigma(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})$ in
the latter case. If $\mathfrak{g}$ is not of type $A$, the extended Dynkin
diagram of $\mathfrak{g}$ shows that the $\theta$-stable subspace
$(\mathbb{C}\psi)^{\perp}$ contains vectors corresponding to
$(\mathop{\mathrm{rank}}\nolimits\mathfrak{g}-1)$ simple roots. Hence
$\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}=\mathbb{C}\psi$. By
the assumption (3.1), the remaining cases are
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{R}),\mathfrak{su}(p,q)$. For these
cases, the extended Dynkin diagram of $\mathfrak{g}$ shows
$\mathbb{C}(e_{1}-e_{n})\subset\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}\subset\mathbb{C}(e_{1}-e_{n})+\mathbb{C}(e_{1}+e_{n}-2/n\sum_{i}e_{i})$.
Calculating $\dim((\mathbb{C}\psi)^{\perp}\cap\mathfrak{t})$ for these cases,
we obtain (3.2).

Let us see some basic properties of minimal
$(\mathfrak{g},\mathfrak{k})$-modules:

######  Proposition 3.3.

Let $V$ be a minimal $(\mathfrak{g},\mathfrak{k})$-module. Then the following
hold.

  1. (1)

$V$ is $\mathfrak{k}$-multiplicity free.

  2. (2)

Assume that a nonzero root vector of root $\psi$ annihilates no nonzero
element in $V$. Then the highest weights of $\mathfrak{k}$-types in $V$ belong
to the subspace

$-\rho(\mathfrak{k}^{1},\mathfrak{t})+\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}.$

  3. (3)

Let $V^{\prime}$ be a $(\mathfrak{g},\mathfrak{k})$-module. Assume
$V^{\prime}$ has a common $\mathfrak{k}$-type as $V$ and
$\mathop{\mathrm{Ann}}\nolimits V\subset\mathop{\mathrm{Ann}}\nolimits
V^{\prime}$. Then $V^{\prime}$ is isomorphic to $V$.

###### Proof.

(1) and (3) follows from the same argument as [20, Proposition 3.1].

We next prove (2). Let $v$ be a highest weight vector in the
$\mathfrak{k}$-module $V$ of weight $\mu\in\mathfrak{t}^{\vee}$. Put
$d:=\dim\mathfrak{k}^{1}$ and write
$\\{\phi\in\Sigma(\mathfrak{k},\mathfrak{t})\mid(\phi,\psi)=1\\}=\\{\phi_{i}\\}_{i=1}^{d}$.
Since the Lie subalgebra
$\mathfrak{k}^{1}+\mathfrak{p}^{1}+\mathfrak{g}_{\psi}$ is a Heisenberg
subalgebra, we can take $\mathfrak{t}$-weight vectors
$z\in\mathfrak{p}_{\psi}$, $x_{i}\in\mathfrak{k}^{1}_{\phi_{i}}$ and
$y_{i}\in\mathfrak{p}^{1}_{-\psi-\phi_{i}}$ satisfying
$[x_{i},y_{j}]=\delta_{i,j}z$ for $1\leq i,j\leq d$.

Set
$\mathfrak{r}:=\mathfrak{z}_{\mathfrak{g}}(\mathfrak{g}_{\psi}+\mathfrak{g}_{-\psi}+\mathbb{C}H_{\psi})$
and define $\Xi\colon\mathfrak{r}\to S^{2}(\mathfrak{g})$ by

$\Xi(X):=Xz+\frac{1}{2}\sum_{1\leq i\leq d}([X,x_{i}]y_{i}-[X,y_{i}]x_{i}).$

By [5, Proposition 4.3], the linear map $\Xi$ is an injective
$\mathfrak{r}$-homomorphism and the image of $\Xi$ is contained in the sum $E$
of irreducible $\mathfrak{g}$-submodules of $S^{2}(\mathfrak{g})$ which are
not isomorphic to the trivial $\mathfrak{g}$-module and $\mathfrak{g}$. Since
the Joseph ideal and the ideals $J_{a}$ $(a\in\mathbb{C})$ contain the
symmetrization of $E$ by [6] and Definition 2.7, any element in the image of
$\mathop{\mathrm{sym}}\nolimits\circ\Xi$ annihilates the minimal
$(\mathfrak{g},K)$-module $V$. Hence, for
$H\in(\mathbb{C}\psi)^{\perp}\cap\mathfrak{t}\subset\mathfrak{r}$, we have

(3.3) $\displaystyle 0=\mathop{\mathrm{sym}}\nolimits\circ\Xi(H)v$
$\displaystyle=\mathop{\mathrm{sym}}\nolimits\left(zH+\frac{1}{2}\sum_{i=1}^{d}([H,x_{i}]y_{i}-[H,y_{i}]x_{i})\right)v$
(3.4)
$\displaystyle=zHv+\frac{1}{4}\sum_{i=1}^{d}(2\phi_{i}-\psi)(H)(x_{i}y_{i}+y_{i}x_{i})v=(\mu+\rho(\mathfrak{k}^{1}))(H)zv.$

Here we used $\psi(H)=0,[x_{i},y_{i}]=z$ and $x_{i}v=0$ at the last equality.
Therefore
$\mu+\rho(\mathfrak{k}^{1})\in\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}$
by the assumption. ∎

######  Remark 3.4.

We write $V^{\mathfrak{g}_{\psi}}$ for the set of elements in $V$ annihilated
by $\mathfrak{g}_{\psi}$. Since a minimal $(\mathfrak{g},\mathfrak{k})$-module
$V$ is irreducible and infinite-dimensional, the subspace
$V^{\mathfrak{g}_{\psi}}$ or $V^{\mathfrak{g}_{-\psi}}$ is zero from [22,
Lemma 3.2]. Hence by reversing the positivity and making $-\psi$ the highest
weight of $\mathfrak{g}$ if necessary, we obtain the assumption
$V^{\mathfrak{g}_{\psi}}=0$ of Proposition 3.3 (2). We remark that if the
center of $\mathfrak{k}$ is trivial, then
$V^{\mathfrak{g}_{\psi}}=V^{\mathfrak{g}_{-\psi}}=0$.

The following properties are used to classify minimal
$(\mathfrak{g},\mathfrak{k})$-modules for
$\mathfrak{g}\cong\mathfrak{sl}(n,\mathbb{C})$ in Section 5:

######  Corollary 3.5.

Let $V$ be a minimal $(\mathfrak{g},\mathfrak{k})$-module. Then the following
hold.

  1. (1)

Assume $\mathfrak{g}_{0}$ is Hermitian and
$\mathfrak{g}_{0}\not\cong\mathfrak{sl}(2,\mathbb{R})$. Then $V$ is an
irreducible highest or lowest weight module.

  2. (2)

Assume $V^{\mathfrak{g}_{\psi}}=0$. If $\mathfrak{g}_{0}$ is not isomorphic to
$\mathfrak{sl}(n,\mathbb{R})(n\geq 3\text{ odd or }n=2)$, then $V$ has pencil
$\mathfrak{k}$-types: there exist a dominant integral weight
$\mu_{0}\in\mathfrak{t}^{\vee}$ with respect to
$\Sigma(\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t})^{+}$
such that the $\mathfrak{k}$-type decomposition of $V$ is multiplicity free
and is given by the direct sum of the irreducible $\mathfrak{k}$-modules with
highest weight $\mu_{0}+k\psi$ for $k\in\mathbb{N}$.

###### Proof.

We may assume (3.1). Moreover, let us first assume
$\mathfrak{g}\not\cong\mathfrak{su}(1,2)$. For (1), it suffices to show that
$V^{\mathfrak{g}_{\psi}}$ or $V^{\mathfrak{g}_{-\psi}}$ is nonzero. Suppose,
contrary to our claim, $V^{\mathfrak{g}_{\psi}}=V^{\mathfrak{g}_{-\psi}}=0$.
By applying Proposition 3.3 (2) to the fixed positive system and the opposite
positive system where $-\psi$ is the highest noncompact imaginary root,
highest weights of $\mathfrak{k}$-types in $V$ belong to

$\\{-\rho(\mathfrak{k}^{1},\mathfrak{t})+\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}\\}\cap
w_{l}\\{\rho(\mathfrak{k}^{1},\mathfrak{t})+\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}\\},$

where $w_{l}$ denotes the longest element in the Weyl group of
$(\mathfrak{k},\mathfrak{t})$. Therefore $w_{l}\psi$ belongs to
$\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}$.

When $\mathfrak{g}_{0}\not\cong\mathfrak{su}(p,q)$, the equality (3.2) shows
$w_{l}\psi\in\mathbb{C}\psi$ and irreducible components of
$\mathfrak{k}$-module $\mathfrak{p}$ must be one-dimensional. Then
$\mathfrak{g}$ is three-dimensional, and is isomorphic to
$\mathfrak{sl}(2,\mathbb{C})$, a contradiction.

When $\mathfrak{g}_{0}=\mathfrak{su}(p,q)$, the equality (3.2) shows that
$e_{p}-e_{p+1}$ belongs to
$\mathbb{C}(e_{1}-e_{n})+\mathbb{C}(e_{1}+e_{n}-2/n\sum_{i}e_{i})$, which
shows $p+q=2,3$. It contradicts to our assumptions
$\mathfrak{g}_{0}\not\cong\mathfrak{su}(1,2)$ and (3.1).

We next prove (2) for $\mathfrak{g}_{0}\not\cong\mathfrak{su}(1,2)$. By (3.2),
the set of weights of $\mathfrak{p}$ in
$\mathfrak{t}_{\mathop{\mathrm{Heis}}\nolimits}^{\perp}$ are

(3.5) $\displaystyle\begin{cases}\\{\pm\psi\\}&\text{ if
$\mathfrak{g}\not\cong\mathfrak{sl}(n,\mathbb{R})(n\geq 3,\text{
odd}),\mathfrak{su}(1,2)$, }\\\ \\{\pm\psi/2,\pm\psi\\}&\text{ if
$\mathfrak{g}\cong\mathfrak{sl}(n,\mathbb{R})$ $(n\geq 3,\text{ odd})$.}\\\
\end{cases}$

Let $V(\mu)$ be the $\mathfrak{k}$-isotypic component in $V$ of the
irreducible $\mathfrak{k}$-module with highest weight $\mu$. By the assumption
and Proposition 3.3 (2) and (3.5), the subspace $\mathfrak{g}V(\mu)$ is
included in $V(\mu-\psi)+V(\mu)+V(\mu+\psi)$. Then it follows from the
infinite-dimensionality of $V$ and Proposition 3.3 (1) that the minimal
$(\mathfrak{g},\mathfrak{k})$-module $V$ has pencil $\mathfrak{k}$-types.

It remains to prove assertions for $\mathfrak{g}=\mathfrak{su}(1,2)$. By
taking the contragredient if necessary, we may assume that the action of
$\mathfrak{g}_{e_{1}-e_{3}}$ on $V$ is faithful (see Remark 3.4). From Remark
3.2 (1), there exists some $a\in\mathbb{C}$ such that $V$ is $a$-minimal. Let
$v$ be a nonzero highest weight vector of a $\mathfrak{k}$-type in $V$ with
highest weight $\mu$. By
$0=\mathop{\mathrm{sym}}\nolimits(\sum_{i}T_{1,i}T_{i,3}-a/2T_{1,3})v=(\mu_{1}+\mu_{3}-1/2-a/2)T_{1,3}v$
and $V^{\mathfrak{g}_{e_{1}-e_{3}}}=0$, we obtain $\mu_{2}=-(a+1)/2$. Hence
the difference of two highest weights of $\mathfrak{k}$-types in $V$ belongs
to the line $\mathbb{R}(e_{1}-e_{3})$. Since the set of dominant integral
weights in $\mathbb{R}(e_{1}-e_{3})$ with respect to
$\Sigma(\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t})^{+}$
is $\mathbb{N}(e_{1}-e_{3})$, the minimal $(\mathfrak{g},\mathfrak{k})$-module
$V$ has pencil $\mathfrak{k}$-types and $V$ is a lowest weight module. ∎

######  Remark 3.6.

  1. (1)

There is another proof of Corollary 3.5 (1) given by Vogan. Since
$\mathfrak{g}$ is not of type $A_{1}$, the dimension of every nonzero
$\mathop{\mathrm{Int}}\nolimits(\mathfrak{k})$-orbit in $\mathfrak{p}$ is
greater than one. Then by [23, Theorem 4.6], the associated variety of a
minimal $(\mathfrak{g},\mathfrak{k})$-module $V$ is irreducible, and included
in an irreducible component of $\mathfrak{p}$. Therefore if $\mathfrak{g}_{0}$
is Hermitian, then $V$ is a highest (or lowest) weight module.

  2. (2)

Even when $\mathfrak{g}_{0}\cong\mathfrak{sl}(n,\mathbb{R})(n\geq 3\text{
odd})$, minimal $(\mathfrak{g},\mathfrak{k})$-modules has pencil
$\mathfrak{k}$-types, as we can see from the classification of minimal
$(\mathfrak{g},\mathfrak{k})$-modules given in Theorem 5.3.

##  4\. Covariant differentials

In this section, we construct an intertwining differential operator between
parabolically induced representations where the infinite-dimensional
composition factors of the kernel are all minimal. When
$\mathfrak{g}_{0}=\mathfrak{sl}(n,\mathbb{R})$, the $\mathfrak{k}$-finite
vectors in the kernel will be determined in Section 5 for some specific cases.
Since the method will be applied to a new construction of many minimal
representations in a subsequent paper, we do not restrict ourselves to simple
Lie algebras of type $A$.

Let $G$ be a connected simple Lie group with finite center, and fix an Iwasawa
decomposition $G=KA_{\min}N_{\min}$. Assume that the complexification
$\mathfrak{g}$ is simple and
$\mathfrak{g}\not\cong\mathfrak{sl}(2,\mathbb{C})$. Write
$\Sigma(\mathfrak{g},\mathfrak{a}_{\min})$ for the restricted root system for
$(\mathfrak{g}_{0},\mathfrak{a}_{\min,0})$, and
$\Sigma(\mathfrak{g},\mathfrak{a}_{\min})^{+}$ for the positive system defined
as the set of $\mathfrak{a}_{\min}$-weights of $\mathfrak{n}_{\min}$. Let
$Q=MAN$ be a standard parabolic subgroup of $G$ and its Langlands
decomposition. Take a maximally split Cartan subalgebra
$\mathfrak{h}^{\mathop{\mathrm{s}}\nolimits}_{0}$ of $\mathfrak{g}_{0}$
containing $\mathfrak{a}_{\min,0}$. Then the center
$\mathfrak{z}(\mathfrak{m})$ of $\mathfrak{m}$ is contained in $\mathfrak{k}$,
and we have
$\mathfrak{h}^{\mathop{\mathrm{s}}\nolimits}=(\mathfrak{m}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{h}^{\mathop{\mathrm{s}}\nolimits})\oplus\mathfrak{z}(\mathfrak{m})\oplus\mathfrak{a}$.
Via this direct sum decomposition, we will regard
$\mathfrak{z}(\mathfrak{m})^{\vee}$ as a subspace of
$(\mathfrak{h}^{\mathop{\mathrm{s}}\nolimits})^{\vee}$. Moreover, any element
$\mu\in\mathfrak{z}(\mathfrak{m})^{\vee}$ extends to a character of
$\mathfrak{m}$ via
$\mathfrak{m}=\mathfrak{m}_{\mathop{\mathrm{ss}}\nolimits}\oplus\mathfrak{z}(\mathfrak{m})$.
We use the same symbol $\mu$ for the character.

Let $(\sigma,V)$ be a finite-dimensional irreducible representation of $M$ and
$\nu$ an element in the dual $\mathfrak{a}^{\vee}$. We further assume that the
$\mathfrak{m}$-module $V$ irreducibly decomposes to a multiple of a character.
By abuse of notation, we write $d\sigma\in\mathfrak{z}(\mathfrak{m})^{\vee}$
for the character. Set

$\lambda:=-d\sigma-\nu\in\mathfrak{z}(\mathfrak{m})^{\vee}+\mathfrak{a}^{\vee}\subset(\mathfrak{h}^{\mathop{\mathrm{s}}\nolimits})^{\vee}.$

We use the same symbol for the character of $\mathfrak{q}$ via
$\mathfrak{q}/(\mathfrak{m}_{\mathop{\mathrm{ss}}\nolimits}+\mathfrak{n})\cong\mathfrak{z}(\mathfrak{m})\oplus\mathfrak{a}$.
By letting $M$ act by $\sigma$, $A$ act by $\exp(\nu)$ and $N$ act trivially,
we obtain an irreducible representation $(\sigma_{\nu},V)$ of $Q$. We write
$C^{\infty}(G,\sigma_{\nu})$ to be the space consisting of all smooth
functions from $G$ to $V$. The representation of $G$ induced by $\sigma_{\nu}$
is defined as

$C^{\infty}(G,\sigma_{\nu})^{Q}:=\\{f\in C^{\infty}(G,\sigma_{\nu})\mid
f(gq)=\sigma_{\nu}(q^{-1})f(g)\text{ for $g\in G,q\in Q$}\\},$

where the action of $G$ is induced by the left transition $L$ on
$C^{\infty}(G,\sigma_{\nu})$:

$(L(h)f)(g):=f(h^{-1}g)$ for $h,g\in G,f\in C^{\infty}(G,\sigma_{\nu})$.

We define a bilinear map $\Psi\colon U(\mathfrak{g})\times
C^{\infty}(G,\sigma_{\nu})\to C^{\infty}(G,\sigma_{\nu})$ by the right
differentiation:

(4.1) $\displaystyle\Psi(X_{1}X_{2}\cdots
X_{n},f)(g):=\left.\frac{d^{n}}{dt_{1}\cdots
dt_{n}}\right|_{t_{1}=\cdots=t_{n}=0}f(ge^{t_{1}X_{1}}\cdots e^{t_{n}X_{n}})$

for $n\in\mathbb{N},X_{1},\ldots,X_{n}\in\mathfrak{g}_{0},f\in
C^{\infty}(G,\sigma_{\nu}),g\in G$.

We write $(\sigma_{\nu})^{\vee}$ for the contragredient representation of $Q$.
Then the left ideal generated by
$\mathop{\mathrm{Ann}}\nolimits_{U(\mathfrak{q})}(\sigma_{\nu})^{\vee}$ is
given by

(4.2) $\displaystyle
I(\mathfrak{q},\lambda):=U(\mathfrak{g})\mathop{\mathrm{Ann}}\nolimits_{U(\mathfrak{q})}(\lambda).$

We remark that $I(\mathfrak{q},\lambda)$ is stable under the adjoint action of
$Q$. Let $\mathop{\mathrm{proj}}\nolimits$ be the canonical projection

(4.3) $\displaystyle\mathop{\mathrm{proj}}\nolimits\colon U(\mathfrak{g})\to
U(\mathfrak{g})/I(\mathfrak{q},\lambda).$

Then $\Psi$ factors through $\mathop{\mathrm{proj}}\nolimits$ to induce a
$(G\times Q)$-intertwining operator

(4.4) $\displaystyle U(\mathfrak{g})/I(\mathfrak{q},\lambda)\otimes
C^{\infty}(G,\sigma_{\nu})^{Q}\to C^{\infty}(G,\sigma_{\nu})$

by [14, Lemma 2.14]. Here the $(G\times Q)$-actions on
$U(\mathfrak{g})/I(\mathfrak{q},\lambda)$ and $C^{\infty}(G,\sigma_{\nu})$ are
given by

(4.5)
$\displaystyle(h,q)\mathop{\mathrm{proj}}\nolimits(X)=\mathop{\mathrm{proj}}\nolimits(\mathop{\mathrm{Ad}}\nolimits(q)(X)),\quad(h,q)f(g)=\sigma_{\nu}(q)f(h^{-1}gq)$

for $g,h\in G,q\in Q,X\in U(\mathfrak{g})$ and $f\in
C^{\infty}(G,\sigma_{\nu})$.

Define $\iota$ to be the anti-involution of $U(\mathfrak{g})$ by

$\iota(X):=-X\text{ for $X\in\mathfrak{g}$.}$

######  Definition 4.1.

We define a $\mathfrak{g}$-submodule $F$ of $\bigoplus_{0\leq i\leq
2}S^{i}(\mathfrak{g})$ as follows.

  * •

When $\mathfrak{g}$ is not of type $A$, put $F$ to be the
$\mathfrak{g}$-module complement of the trivial $\mathfrak{g}$-submodule in
$S^{2}(\mathfrak{g})$.

  * •

When $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, fix $a\in\mathbb{C}$ and put
$F$ to be $F^{a}$ (see (2.18)).

Define $W$ to be the $Q$-submodule of
$U(\mathfrak{g})/I(\mathfrak{q},\lambda)$ defined by

$W:=\mathop{\mathrm{proj}}\nolimits\circ\iota\circ\mathop{\mathrm{sym}}\nolimits(F)\subset
U(\mathfrak{g})/I(\mathfrak{q},\lambda).$

From (4.4), the tensor-hom adjunction and the triviality of the action of $Q$
on $C^{\infty}(G,\sigma_{\nu})^{Q}$, we obtain a $G$-intertwining second-order
differential operator

(4.6) $\displaystyle D=D(Q,\sigma_{\nu})\colon
C^{\infty}(G,\sigma_{\nu})^{Q}\to
C^{\infty}(G,W^{\vee}\otimes\sigma_{\nu})^{Q}$

defined by $\langle w,Df(g)\rangle:=\Psi(w,f)(g)$ for $w\in W,f\in
C^{\infty}(G,\sigma_{\nu})^{Q},g\in G$. Here $\langle\cdot,\cdot\rangle$
denotes the pairing of $W$ and its dual $W^{\vee}$.

In the rest of this section, we study properties of the kernel of $D$. The
kernel of $D$ picks out functions annihilated by
$\mathop{\mathrm{sym}}\nolimits(F)$:

######  Lemma 4.2.

In the setting of Definition 4.1, we have $\mathop{\mathrm{Ker}}\nolimits
D=\\{f\in C^{\infty}(G,\sigma_{\nu})^{Q}\mid
dL(\mathop{\mathrm{sym}}\nolimits(F))f=0\\}$. Here $dL$ denotes the
differential of the left transition $L$.

###### Proof.

Let $f\in C^{\infty}(G,\sigma_{\nu})^{Q}$. Then $Df=0$ is equivalent to
$\Psi(\iota(X),f)(g)=0$ for $X\in\mathop{\mathrm{sym}}\nolimits(F),g\in G$.
For $n\in\mathbb{N},X_{1},\ldots,X_{n}\in\mathfrak{g}_{0},f\in
C^{\infty}(G,\sigma_{\nu})$ and $g\in G$, we have

(4.7) $\displaystyle\Psi(\iota(X_{1}X_{2}\cdots
X_{n}),f)(g)=\left.\frac{d^{n}}{dt_{l}\cdots
dt_{n}}\right|_{t_{1}=\cdots=t_{n}=0}f(ge^{-t_{n}X_{n}}\cdots
e^{-t_{1}X_{1}})$ (4.8) $\displaystyle=\left.\frac{d^{n}}{dt_{1}\cdots
dt_{n}}\right|_{t_{1}=\cdots=t_{n}=0}f(e^{-t_{n}\mathop{\mathrm{Ad}}\nolimits(g)(X_{n})}\cdots
e^{-t_{1}\mathop{\mathrm{Ad}}\nolimits(g)(X_{1})}g)$ (4.9)
$\displaystyle=dL(\mathop{\mathrm{Ad}}\nolimits(g)(X_{1}X_{2}\cdots
X_{n}))f(g).$

Therefore $\Psi(\iota(X),f)(g)=dL(\mathop{\mathrm{Ad}}\nolimits(g)(X))f(g)$
for $X\in U(\mathfrak{g}),f\in C^{\infty}(G,\sigma_{\nu})$, $g\in G$. Hence
$Df=0$ is equivalent to $dL(Y)f(g)=0$ for
$Y\in\mathop{\mathrm{sym}}\nolimits(F),g\in G$, and the proof is complete. ∎

In the following proposition, the term “$(a)$-minimal” means “minimal” if
$\mathfrak{g}$ is not of type $A$, and “$a$-minimal” if
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ $(n\geq 3)$.

######  Proposition 4.3.

In the setting of Definition 4.1, the following hold.

  1. (1)

Any $(a)$-minimal subrepresentation of $C^{\infty}(G,\sigma_{\nu})^{Q}$ is
contained in $\mathop{\mathrm{Ker}}\nolimits D$.

  2. (2)

Any infinite-dimensional irreducible subquotient of
$\mathop{\mathrm{Ker}}\nolimits D$ is an $(a)$-minimal representation.

###### Proof.

Since $\mathop{\mathrm{sym}}\nolimits(F)$ is contained in the Joseph ideal by
[6] (when $\mathfrak{g}$ is not of type $A$) and in $J_{a}$ by Definition 2.7
(when $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$), the assertion (1) follows
from Lemma 4.2.

Let us prove (2). Let $\tau$ be an infinite-dimensional irreducible
subquotient of $\mathop{\mathrm{Ker}}\nolimits D$. Write $F^{\prime}$ for the
image of $F$ under the projection from $\bigoplus_{0\leq i\leq
2}S^{i}(\mathfrak{g})$ to $S^{2}(\mathfrak{g})$. Then $F^{\prime}$ and the
trivial $\mathfrak{g}$-submodule of $S^{2}(\mathfrak{g})$ generates
$\mathcal{I}(\overline{\mathcal{O}^{\min}})$ by [6].

By Lemma 4.2, the annihilator $\mathop{\mathrm{Ann}}\nolimits\tau$ contains
$\mathop{\mathrm{sym}}\nolimits(F)$. Hence
$\mathop{\mathrm{gr}}\nolimits(\mathop{\mathrm{Ann}}\nolimits\tau)$ contains
$F^{\prime}$. Since $\mathop{\mathrm{Ann}}\nolimits\tau$ has an infinitesimal
character, the associated graded ideal
$\mathop{\mathrm{gr}}\nolimits(\mathop{\mathrm{Ann}}\nolimits\tau)$ contains
the trivial $\mathfrak{g}$-submodule of $S^{2}(\mathfrak{g})$. Therefore
$\mathop{\mathrm{gr}}\nolimits(\mathop{\mathrm{Ann}}\nolimits\tau)$ contains
$\mathcal{I}(\overline{\mathcal{O}^{\min}})$. Since the ideal
$\mathop{\mathrm{Ann}}\nolimits\tau$ has infinite codimension in
$U(\mathfrak{g})$, we have
$\mathop{\mathrm{gr}}\nolimits(\mathop{\mathrm{Ann}}\nolimits\tau)=\mathcal{I}(\overline{\mathcal{O}^{\min}})$.
Therefore $\tau$ is $(a)$-minimal by Theorem 2.8 (1) and Remark 2.9 (1). Here
we used the fact that $\mathop{\mathrm{sym}}\nolimits(F^{a})\subset J_{b}$
$(b\in\mathbb{C})$ implies $b=a$ for
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$ $(n\geq 3)$. ∎

The following two lemmas give necessary conditions for
$\mathop{\mathrm{Ker}}\nolimits D\neq 0$.

######  Lemma 4.4.

When $\mathop{\mathrm{Ker}}\nolimits D\neq 0$, the annihilator of the
irreducible highest weight module $L(\lambda)$ equals the Joseph ideal or the
augmentation ideal $U(\mathfrak{g})\mathfrak{g}$ if $\mathfrak{g}$ is not of
type $A$, and contains $J_{-a}$ if $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$.

###### Proof.

Assume $\mathop{\mathrm{Ker}}\nolimits D\neq 0$. Then there exist
$f\in\mathop{\mathrm{Ker}}\nolimits D$ and $g_{0}\in G$ with $f(g_{0})\neq 0$.
Let us show that $\iota\circ\mathop{\mathrm{sym}}\nolimits(F)$ annihilates
$L(\lambda)$. As Lemma 2.3, it suffices to show $X\in I(\mathfrak{q},\lambda)$
for any weight vector $X\in\iota\circ\mathop{\mathrm{sym}}\nolimits(F)$ of
weight zero. We write $\overline{\mathfrak{n}}$ for the nilradical of the
opposite parabolic of $\mathfrak{q}$. By the Poincaré-Birkhoff-Witt theorem
and the decomposition $\mathfrak{g}=\overline{\mathfrak{n}}+\mathfrak{q}$, we
see $X\in c+I(\mathfrak{q},\lambda)$ for some $c\in\mathbb{C}$. By
$f\in\mathop{\mathrm{Ker}}\nolimits D$ and the proof of Lemma 4.2, we have
$0=dL(\mathop{\mathrm{Ad}}\nolimits(g_{0})\circ\iota(X))f(g_{0})=\Psi(X,f)(g_{0})=cf(g_{0})$.
Therefore $X\in I(\mathfrak{q},\lambda)$, and
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F)$ annihilates $L(\lambda)$.

When $\mathfrak{g}$ is not of type $A$, it follows from [2, Proposition 5.3]
that a proper primitive ideal containing
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F)=\mathop{\mathrm{sym}}\nolimits(F)$
is the Joseph ideal or $U(\mathfrak{g})\mathfrak{g}$, and the assertion
follows. When $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, it follows from the
above argument,
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F)=\mathop{\mathrm{sym}}\nolimits(F^{-a})$
and Proposition 2.4 that $\lambda$ equals $\lambda(i,-a)$ for some $i$. By
Remark 2.6, the ideal $J_{-a}$ annihilates $L(\lambda)$. ∎

Let $\Delta(\mathfrak{g},\mathfrak{a}_{\min})$ be the set of simple restricted
roots, $\Delta_{Q}$ the subset of $\Delta(\mathfrak{g},\mathfrak{a}_{\min})$
defining the standard parabolic subgroup $Q$. When $Q$ is minimal, we have
$\Delta_{Q}=\emptyset$. Define

$\Delta_{Q}^{\nu}:=\\{\alpha\in\Delta(\mathfrak{g},\mathfrak{a}^{\mathop{\mathrm{s}}\nolimits})\mid\mathop{\mathrm{proj}}\nolimits(\mathfrak{g}_{-\alpha})=\mathop{\mathrm{proj}}\nolimits(F_{-\alpha})\\},$

where $\mathop{\mathrm{proj}}\nolimits$ is defined in (4.3) and the subscript
$-\alpha$ means the $\mathfrak{a}_{\min}$-weight space of weight $-\alpha$. We
write $Q^{\nu}=M^{\nu}A^{\nu}N^{\nu}$ for the standard parabolic subgroup
corresponding to $\Delta_{Q}^{\nu}$ and its Langlands decomposition. By
$\Delta_{Q}\subset\Delta_{Q}^{\nu}$, we see $Q\subset Q^{\nu}$. Therefore we
have the decomposition
$\mathfrak{a}=\mathfrak{a}^{\nu}\oplus(\mathfrak{a}\cap\mathfrak{m}^{\nu})$
and
$\mathfrak{z}(\mathfrak{m})=(\mathfrak{z}(\mathfrak{m})\cap\mathfrak{m}^{\nu}_{\mathop{\mathrm{ss}}\nolimits})\oplus\mathfrak{z}(\mathfrak{m}^{\nu})$.

######  Lemma 4.5.

Assume that $\mathop{\mathrm{Ker}}\nolimits D$ is nonzero. Then the character
$\nu$ (resp. $d\sigma$) is zero on $\mathfrak{a}\cap\mathfrak{m}^{\nu}$ (resp.
$\mathfrak{z}(\mathfrak{m})\cap\mathfrak{m}^{\nu}_{\mathop{\mathrm{ss}}\nolimits}$)
to define a character of $\mathfrak{a}^{\nu}$ (resp.
$\mathfrak{z}(\mathfrak{m}^{\nu})$), and there exists an irreducible
representation $\sigma^{\prime}$ of $M^{\nu}$ satisfying the following
conditions.

  1. (1)

The representation space of $\sigma^{\prime}$ agrees with that of $\sigma$.

  2. (2)

The differentiated action of
$(\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits}$ is trivial.

  3. (3)

$\sigma^{\prime}(m)=\sigma(m)$ for $m\in M$.

Moreover, we have $\mathop{\mathrm{Ker}}\nolimits D(Q,\sigma_{\nu})\subset
C^{\infty}(G,\sigma^{\prime}_{\nu})^{Q^{\nu}}\subset
C^{\infty}(G,\sigma_{\nu})^{Q}$.

###### Proof.

By the assumption, we can take $f\in\mathop{\mathrm{Ker}}\nolimits D$,
$g_{0}\in G$ with $f(g_{0})\neq 0$. Then

(4.10) $\displaystyle
f(gm)=\sigma(m^{-1})f(g),\quad\left.\frac{d}{dt}\right|_{t=0}f(ge^{tX})=0$

for $g\in G,m\in
M,X\in(\mathfrak{m}_{0})_{\mathop{\mathrm{ss}}\nolimits}\cup\mathfrak{n}_{0}\cup(\mathfrak{g}_{0})_{-\alpha}$
$(\alpha\in\Delta_{Q}^{\nu})$. Moreover, the Lie subalgebra of
$\mathfrak{g}_{0}$ generated by
$(\mathfrak{m}_{0})_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{n}_{0}$ and
$(\mathfrak{g}_{0})_{-\alpha}$ $(\alpha\in\Delta_{Q}^{\nu})$ contains
$(\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits}$:

(4.11)
$\displaystyle(\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits}\subset\langle(\mathfrak{m}_{0})_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{n}_{0},(\mathfrak{g}_{0})_{-\alpha}\mid\alpha\in\Delta_{Q}^{\nu}\rangle.$

By (4.10) and (4.11), we obtain
$(d\sigma+\nu)(H)f(g_{0})=\left.\frac{d}{dt}\right|_{t=0}f(g_{0}\exp(-tH))=0$
for
$H\in(\mathfrak{a}_{0}\cap\mathfrak{m}^{\nu}_{0})+(\mathfrak{z}(\mathfrak{m}_{0})\cap(\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits})\subset(\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits}$.
Therefore $\nu$ (resp. $d\sigma$) can be seen as an element in
$(\mathfrak{a}^{\nu})^{\vee}$ (resp.
$\mathfrak{z}(\mathfrak{m}^{\nu})^{\vee}$).

We next define a representation $\sigma^{\prime}$ of $M^{\nu}$. Let $g\in
M^{\nu},v\in V$. Since $f\neq 0$ and $\sigma$ is finite-dimensional and
irreducible, the representation space $V$ is spanned by $f(G)$. Hence we can
write $v=\sum_{j=1}^{m}a_{j}f_{j}(h_{j})$ for some
$m\in\mathbb{N},a_{j}\in\mathbb{C},f_{j}\in\mathop{\mathrm{Ker}}\nolimits D$
and $h_{j}\in G$ $(1\leq j\leq m)$. Then we define $\sigma^{\prime}(g)v$ to be
$\sum_{j=1}^{m}a_{j}f_{j}(h_{j}g^{-1})$.

Let us check that $\sigma^{\prime}(g)v$ is well-defined. It suffices to show
$\sum_{j=1}^{m}a_{j}f_{j}(h_{j}g^{-1})=0$ assuming
$\sum_{j=1}^{m}a_{j}f_{j}(h_{j})=0$. Since $\mathfrak{m}^{\nu}_{0}$ is
generated by $\mathfrak{m}_{0}$ and $(\mathfrak{g}_{0})_{\pm\alpha}$
$(\alpha\in\Delta_{Q}^{\nu})$, we can write $g=g_{1}\cdots g_{n}$ for some
$n\in\mathbb{N}$ and

$g_{i}\in
M\cup\exp(\mathfrak{n}_{0})\cup\bigcup_{\alpha\in\Delta^{\nu}_{Q}}\exp((\mathfrak{g}_{0})_{-\alpha})\quad(1\leq
i\leq n).$

By (4.10), we see $\sum_{j=1}^{m}a_{j}f_{j}(h_{j}g^{-1})=\sigma(\prod_{1\leq
i\leq n,g_{i}\in M}g_{i})\sum_{j=1}^{m}a_{j}f_{j}(h_{j})=0$. Hence
$\sigma^{\prime}(g)v$ does not depend on $f$ and the expression of $v$.

We easily see that $\sigma^{\prime}(g)\in\operatorname{GL}(V)$ and
$\sigma^{\prime}$ is a group homomorphism from $M^{\nu}$ to
$\operatorname{GL}(V)$. In a small neighborhood of the neutral element in
$M^{\nu}$, any element $g$ can be smoothly decomposed into $g_{1}g_{2}$ for
some
$g_{1}\in\exp(\mathfrak{z}(\mathfrak{m}^{\nu}_{0})),g_{2}\in\exp((\mathfrak{m}^{\nu}_{0})_{\mathop{\mathrm{ss}}\nolimits})$,
and $\sigma^{\prime}(g)=\sigma(g_{1})$ by (4.10) and (4.11). Therefore
$\sigma^{\prime}$ is a smooth representation of $M^{\nu}$. The conditions (1),
(2) and (3) and $f\in C^{\infty}(G,\sigma^{\prime}_{\nu})^{Q^{\nu}}$ follow
directly from the definition of $\sigma^{\prime}$. ∎

##  5\. Classification of minimal representations

Let $\mathfrak{g}_{0}$ be a real form of $\mathfrak{sl}(n,\mathbb{C})$ $(n\geq
2)$. By Remark 3.2 (2), if there is a minimal
$(\mathfrak{g},\mathfrak{k})$-module, then $\mathfrak{g}_{0}$ is isomorphic to
$\mathfrak{su}(p,q)$ $(p,q>0,p+q=n)$ or $\mathfrak{sl}(n,\mathbb{R})$. In this
section, we classify $a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules for
$\mathfrak{g}_{0}=\mathfrak{su}(p,q)$ $(p,q>0,p+q=n)$,
$\mathfrak{sl}(n,\mathbb{R})$ $(n\geq 3)$ for any $a\in\mathbb{C}$.

We remark that when
$\mathfrak{g}_{0}=\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{su}(1,1)$, any
primitive ideal with infinite-codimension can be written as $J_{a}$ for some
$a\in\mathbb{C}$. Therefore an irreducible
$(\mathfrak{g},\mathfrak{k})$-module is minimal if and only if it is infinite-
dimensional, and the classification of minimal
$(\mathfrak{g},\mathfrak{k})$-module is well-known (see [7, Sections
II.1.2–3], for example). As for the classification of unitarizable ones, see
[7, Theorems II.1.1.3 and 1.1.5].

Let us first consider the case $\mathfrak{g}_{0}=\mathfrak{su}(p,q)$
$(p,q>0,n=p+q\geq 3)$ and
$\mathfrak{k}_{0}=\mathfrak{su}(p)\oplus\mathfrak{su}(q)$. We take a compact
Cartan subalgebra $\mathfrak{t}_{0}$ of $\mathfrak{g}_{0}$ as diagonal
matrices. We use the notation in Section 2.

######  Theorem 5.1.

Let $\mathfrak{g}_{0}=\mathfrak{su}(p,q)$ $(p,q>0,n=p+q\geq 3)$,
$a\in\mathbb{C}$. The following $(\mathfrak{g},\mathfrak{k})$-modules are
$a$-minimal and any $a$-minimal $(\mathfrak{g},\mathfrak{k})$-module is
isomorphic to some of them.

  1. (1)

In the case $p=1$,

(5.1) $\displaystyle L(\lambda(1,a))\quad$ $\displaystyle a\not\in
n/2+\mathbb{N},\quad$ $\displaystyle L(\lambda(1,-a))^{\vee}\quad$
$\displaystyle a\not\in-n/2-\mathbb{N},$ (5.2) $\displaystyle
L(\lambda(2,a))\quad$ $\displaystyle a\in(n-2)/2+\mathbb{N},\quad$
$\displaystyle L(\lambda(2,-a))^{\vee}\quad$ $\displaystyle
a\in-(n-2)/2-\mathbb{N}.$

  2. (2)

In the case $q=1$,

(5.3) $\displaystyle L(\lambda(n,a))\quad$ $\displaystyle
a\not\in-n/2-\mathbb{N},\quad$ $\displaystyle L(\lambda(n,-a))^{\vee}\quad$
$\displaystyle a\not\in n/2+\mathbb{N},$ (5.4) $\displaystyle
L(\lambda(n-1,a))\quad$ $\displaystyle a\in-(n-2)/2-\mathbb{N},\quad$
$\displaystyle L(\lambda(n-1,-a))^{\vee}\quad$ $\displaystyle
a\in(n-2)/2+\mathbb{N}.$

  3. (3)

In the case $p,q>1$,

(5.5) $\displaystyle L(\lambda(p,a))\quad$ $\displaystyle
a\in-(p-q)/2-\mathbb{N},\quad$ $\displaystyle L(\lambda(p,-a))^{\vee}\quad$
$\displaystyle a\in(p-q)/2+\mathbb{N},$ (5.6) $\displaystyle
L(\lambda(p+1,a))\quad$ $\displaystyle a\in-(p-q)/2+\mathbb{N},\quad$
$\displaystyle L(\lambda(p+1,-a))^{\vee}\quad$ $\displaystyle
a\in(p-q)/2-\mathbb{N}.$

Furthermore, these minimal $a$-modules are not isomorphic to each others
except for the ones caused by

(5.7) $\displaystyle\lambda(p,-(p-q)/2)$
$\displaystyle=\lambda(p+1,-(p-q)/2).$

The $\mathfrak{k}$-types of an $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module $L(\lambda(i,a))$ (resp.
$L(\lambda(i,-a))^{\vee}$) are multiplicity free and the set of highest
weights is given by $\lambda(i,a)+\mathbb{N}(-e_{p}+e_{p+1})$ (resp.
$-w_{l}\lambda(i,-a)+\mathbb{N}(e_{1}-e_{n})$). Here $w_{l}$ denotes the
longest element of the Weyl group for $(\mathfrak{k},\mathfrak{t})$.

######  Remark 5.2.

  1. (1)

Since $-\lambda(i,a)_{j}=-\overline{\lambda(i,\overline{a})_{j}}$ for any
$1\leq j\leq n$, we have
$L(\lambda(i,a))^{\vee}\cong\overline{L(\lambda(i,\overline{a}))}$.

  2. (2)

As in the proof of Lemma 2.10, there exists an isomorphism
$\mathfrak{su}(p,q)\cong\mathfrak{su}(q,p)$ where the
$\mathfrak{su}(p,q)$-module $L(\lambda(i,a))$ maps to the
$\mathfrak{su}(q,p)$-module $L(\lambda(n+1-i,-a))$.

  3. (3)

By [3, Theorem 7.4] or [8], when $L(\lambda(i,a))$ is $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module, it is unitarizable if and only if

    1. (i)

$p=1,i=1$ and $a\in(-\infty,2-n/2]\cup\Set{n/2-i}{1\leq i\leq n-3}$,

    2. (ii)

$q=1,i=n$ and $a\in\Set{-n/2+i}{1\leq i\leq n-3}\cup(-2+n/2,\infty]$ or

    3. (iii)

$i\neq 1,n$.

  4. (4)

Recall that $\mathrm{O}(2,4)$ is locally isomorphic to $\mathrm{SU}(2,2)$. The
$(\mathfrak{g},\mathfrak{k})$-module of the irreducible unitary representation
of $\mathrm{O}(2,4)$ constructed in [1, 12] is isomorphic to
$L(\lambda(2,0))\oplus L(\lambda(2,0))^{\vee}$, and the two irreducible
components are $0$-minimal.

###### Proof of Theorem 5.1.

By Corollary 3.5 (1), any $a$-minimal $(\mathfrak{g},\mathfrak{k})$-module is
a highest or lowest weight module. From Remark 3.2 (3), it suffices to
classify $a$-minimal highest weight $(\mathfrak{g},\mathfrak{k})$-modules.
Theorem 2.8 (2) shows that the irreducible highest weight
$\mathfrak{g}$-module $L(\lambda)$ is an $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module if and only if the highest weight
$\lambda$ is written as $\lambda(i,a)$ for some $1\leq i\leq n$, satisfies
(2.34), and is dominant integral with respect to
$\Sigma(\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits},\mathfrak{k}_{\mathop{\mathrm{ss}}\nolimits}\cap\mathfrak{t})^{+}$.
From the expression (2.19) of $\lambda(i,a)$ and the fact that
$\lambda(i,a)=\lambda(i^{\prime},a^{\prime})$ if and only if $a=a^{\prime}$
and $(i^{\prime},a^{\prime})=(i,a),(i-1,n/2-i+1),(i+1,n/2-i)$ hold, we obtain
the classification of $a$-minimal $(\mathfrak{g},\mathfrak{k})$-modules.

The assertion on $\mathfrak{k}$-types follows from Corollary 3.5 (2). ∎

The rest of this section is devoted to the case
$\mathfrak{g}_{0}=\mathfrak{sl}(n,\mathbb{R})(n\geq 3)$. We will construct an
$a$-minimal $(\mathfrak{g},\mathfrak{k})$-module as the $\mathfrak{k}$-finite
vectors of the kernel of an intertwining differential operator between
parabolically induced representations. Furthermore, the construction will be
applied to show the nonexistence of some $a$-minimal
$(\mathfrak{sl}(3,\mathbb{C}),\mathfrak{so}(3,\mathbb{C}))$-modules (see
Proposition 6.1).

Let $\mathfrak{h}_{0}$ be the split Cartan subalgebra consisting of diagonal
matrices in $\mathfrak{g}_{0}$. Take a positive system
$\Sigma(\mathfrak{g},\mathfrak{h})^{+}$ of restricted roots as in Section 2.
Fix a standard parabolic subalgebra $\mathfrak{q}_{0}$ of $\mathfrak{g}_{0}$.
Let $G$ be the universal cover $\widetilde{\mathrm{SL}}(n,\mathbb{R})$ of
$\mathrm{SL}(n,\mathbb{R})$, $K$ the inverse image of $\mathrm{SO}(n)$,
$Q=MAN$ the standard parabolic subgroup with Lie algebra $\mathfrak{q}_{0}$
and its Langlands decomposition, $\sigma$ an irreducible representation of the
component group of $M$, and $\nu$ a character of the Lie algebra
$\mathfrak{a}$. In Section 4, we defined an intertwining differential operator

$\displaystyle D=D(Q,\sigma_{\nu})\colon C^{\infty}(G,\sigma_{\nu})^{Q}\to
C^{\infty}(G,(\mathop{\mathrm{proj}}\nolimits\circ\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a}))^{\vee}\otimes\sigma_{\nu})^{Q}.$

We write $(\mathop{\mathrm{Ker}}\nolimits
D)_{K},C^{\infty}(G,\sigma_{\nu})^{Q}_{K}$ for the space of $K$-finite vectors
of $\mathop{\mathrm{Ker}}\nolimits D,C^{\infty}(G,\sigma_{\nu})^{Q}$,
respectively.

Let us state the classification result for
$\mathfrak{g}_{0}=\mathfrak{sl}(n,\mathbb{R})$. Put a subset $Z$ of
$\mathbb{N}$ to be

$Z:=\begin{cases}\mathbb{N}\cap(|a|-n/2-2\mathbb{N})&\text{ if
$a\in\mathbb{R}$,}\\\ \emptyset&\text{ otherwise.}\end{cases}$

######  Theorem 5.3.

  1. (1)

Let $Q_{1}=M_{1}A_{1}N_{1}$ be the standard parabolic subgroup of $G$ whose
Lie algebra is a real form of $\mathfrak{q}_{(1,n-1)}$. Write
$\mathop{\mathrm{triv}}\nolimits$ for the trivial character of $M_{1}$, and
$\mathop{\mathrm{sgn}}\nolimits$ for the nontrivial character of $M_{1}$. Then
the infinite-dimensional subquotients of
$C^{\infty}(G,\mathop{\mathrm{triv}}\nolimits_{-\lambda(1,-a)})^{Q_{1}}_{K}$,
$C^{\infty}(G,\mathop{\mathrm{sgn}}\nolimits_{-\lambda(1,-a)})^{Q_{1}}_{K}$,
whose $\mathfrak{k}$-type decompositions are

(5.8) $\displaystyle\bigoplus_{k\in 2\mathbb{N}\setminus
Z}\mathcal{H}^{k}(\mathbb{R}^{n}),\bigoplus_{k\in(2\mathbb{N}+1)\setminus
Z}\mathcal{H}^{k}(\mathbb{R}^{n}),$

are $a$-minimal, respectively. Here $\mathcal{H}^{k}(\mathbb{R}^{n})$ denotes
the space of harmonic polynomials on $\mathbb{R}^{n}$ of homogeneous degree
$k$, which is irreducible as a $\mathfrak{k}$-module.

  2. (2)

Assume $n=3$ and $a\in\mathbb{Z}$. Let $B=MAN$ be the standard Borel subgroup
of $G$. Write $\sigma^{0}$ for the irreducible two-dimensional representation
of $M\cong\\{\pm 1,\pm\mathbf{i},\pm\mathbf{j},\pm\mathbf{k}\\}$. Then
$\mathop{\mathrm{Ker}}\nolimits D(B,\sigma^{0}_{-\lambda(2,-a)})_{K}$ is
$a$-minimal and isomorphic to

$\bigoplus_{k\geq 0}S^{2|a|+1+4k}\mathbb{C}^{2}$

as $\mathfrak{k}$-modules, where $S^{k}\mathbb{C}^{2}$ denotes the $k$-th
symmetric power of the two-dimensional irreducible $\mathfrak{k}$-module
$\mathbb{C}^{2}$.

Moreover, any $a$-minimal $(\mathfrak{g},\mathfrak{k})$-module is isomorphic
to one of the above.

######  Remark 5.4.

  1. (1)

When $n=3$ and $a=0$, the representation $\mathop{\mathrm{Ker}}\nolimits
D(B,\sigma^{0}_{-\lambda(2,0)})$ is infinitesimally equivalent to the one
given by Torasso [21], and the realization as the kernel of an intertwining
differential operator was given by Kubo and Ørsted [15]. The $K$-type formula
in Theorem 5.3 (2) (more precisely, Proposition 6.1 in a different
parametrization) was obtained by their recent work [16, Theorem 1.7 (5)].

For $a=\pm 1$, the $a$-minimal representation of
$\widetilde{\mathrm{SL}}(3,\mathbb{R})$ which does not descend to the
representation of $\mathrm{SL}(3,\mathbb{R})$ appeared in [23, Example 12.4].

  2. (2)

If an $a$-minimal $(\mathfrak{g},\mathfrak{k})$-module is unitarizable, then
$a\in\sqrt{-1}\mathbb{R}$ by Remark 3.2 (3). Conversely, assume
$a\in\sqrt{-1}\mathbb{R}$. Theorem 5.3 says that an $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module is associated to a composition factor of
unitary principal series or Torasso’s representation, which is unitary.
Therefore it is unitarizable.

###### Proof of Theorem 5.3 (1).

By Theorem 2.8 (2), the finite-dimensional $\mathfrak{g}$-modules whose
annihilators include $J_{a}$ are the space of homogeneous polynomials
$\mathcal{P}^{-a-n/2}(\mathbb{R}^{n})$ of degree $-a-n/2$ if
$-a-n/2\in\mathbb{N}$ and the contragredient of
$\mathcal{P}^{a-n/2}(\mathbb{R}^{n})$ if $a-n/2\in\mathbb{N}$, and do not
exist otherwise. Here the $\mathfrak{g}$-action on
$\mathcal{P}^{|a|-n/2}(\mathbb{R}^{n})$ is induced by the symmetric power of
the contragredient of the $n$-dimensional defining representation of
$\mathfrak{g}$ when $|a|-n/2\in\mathbb{N}$. As $\mathfrak{k}$-modules, they
are isomorphic to

(5.9) $\displaystyle\bigoplus_{k\in Z}\mathcal{H}^{k}(\mathbb{R}^{n})$

by the theory of harmonic polynomials.

By Lemma 2.10,
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})=\mathop{\mathrm{sym}}\nolimits(F^{-a})$
annihilates the generalized Verma module
$U(\mathfrak{g})\otimes_{U(\mathfrak{q}_{1})}\mathbb{C}_{\lambda(1,-a)}$.
Hence we have $\mathop{\mathrm{Ker}}\nolimits
D(Q_{1},\sigma_{-\lambda(1,-a)})=C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}$
for any
$\sigma=\mathop{\mathrm{triv}}\nolimits,\mathop{\mathrm{sgn}}\nolimits$ and
Proposition 4.3 (2) shows that any infinite-dimensional composition factor is
$a$-minimal. By the Frobenius reciprocity and the branching law for the pair
$(\mathfrak{so}(n),\mathfrak{so}(n-1))$ (see [11, Proposition 9.16] for
example), the $\mathfrak{k}$-type decomposition of
$C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ is given by

(5.10) $\displaystyle\bigoplus_{k\in
2\mathbb{N}+\epsilon}\mathcal{H}^{k}(\mathbb{R}^{n})\text{,\quad where
}\epsilon=\begin{cases}0&\text{ if
$\sigma=\mathop{\mathrm{triv}}\nolimits$,}\\\ 1&\text{ if
$\sigma=\mathop{\mathrm{sgn}}\nolimits$.}\end{cases}$

There exists an irreducible infinite-dimensional subquotient $V$ of
$C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ as $Z$ does not contain
$2\mathbb{N}+\epsilon$. Let $\psi$ be the highest weight of
$\mathfrak{k}$-module $\mathfrak{p}$ as in Section 3. Since the highest weight
$\mathcal{H}^{k}(\mathbb{R}^{n})$ is $k\psi/2$, we see
$\mathfrak{g}\cdot\mathcal{H}^{k}(\mathbb{R}^{n})\subset\mathcal{H}^{k-2}(\mathbb{R}^{n})+\mathcal{H}^{k}(\mathbb{R}^{n})+\mathcal{H}^{k+2}(\mathbb{R}^{n})$
as subspaces of $C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$. Therefore
the $\mathfrak{k}$-type decomposition of $V$ is written as $\bigoplus_{k\in
2\mathbb{N}+k_{0}}\mathcal{H}^{k}(\mathbb{R}^{n})$ for some
$k_{0}\in\mathbb{N}$, and the multiplicity of $V$ is one by (5.10).

Assume that $C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ has a
composition factor $V^{\prime}$ other than $V$. The $\mathfrak{k}$-type
decomposition of $V$ shows that $V^{\prime}$ is finite-dimensional. By the
above argument, $\|a\|-n/2\in\mathbb{N}$ and $V^{\prime}$ is isomorphic to
$\mathcal{P}^{-a-n/2}(\mathbb{R}^{n})$ if $-a-n/2\in\mathbb{N}$ and the
contragredient of $\mathcal{P}^{a-n/2}(\mathbb{R}^{n})$ if
$a-n/2\in\mathbb{N}$. By (5.9) and (5.10), the multiplicity of $V^{\prime}$ is
one and there are no composition factors other than $V,V^{\prime}$.

Conversely, if $\|a\|-n/2\in\mathbb{N}$ and
$C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ contains a
$\mathfrak{k}$-type $\mathcal{H}^{k}(\mathbb{R}^{n})$ for some $k\in Z$, then
there exists a finite-dimensional composition factor by Proposition 3.3 (3)
and (5.9).

By the above argument, if $(2\mathbb{N}+\epsilon)\cap Z=\emptyset$,
$C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ is irreducible (hence
$a$-minimal). If $(2\mathbb{N}+\epsilon)\cap Z\neq\emptyset$, the composition
factors of $C^{\infty}(G,\sigma_{-\lambda(1,-a)})^{Q_{1}}_{K}$ consist of an
$a$-minimal $(\mathfrak{g},\mathfrak{k})$-module whose $\mathfrak{k}$-type
decomposition is as in (5.8) and one of the above finite-dimensional
$\mathfrak{g}$-modules, and their multiplicities are one, which proves (1). ∎

##  6\. Genuine minimal representations of
$\widetilde{\mathrm{SL}}(3,\mathbb{R})$

Let $\mathfrak{g}_{0}=\mathfrak{sl}(3,\mathbb{R})$. We use the notation in
Sections 4 and 5. In this section, we explicitly describe the space of
$\mathfrak{k}$-finite vectors of the kernel of
$D(B,\sigma^{0}_{-\lambda(2,-a)})$ and prove the rest of Theorem 5.3.

We first fix notations for a covering map from $\mathrm{SU}(2)$ onto
$\mathrm{SO}(3)$. Let
$\mathbb{H}=\mathbb{R}1+\mathbb{R}\mathbf{i}+\mathbb{R}\mathbf{j}+\mathbb{R}\mathbf{k}$
be the quaternion algebra. We regard $\mathbb{H}$ as a vector space over
$\mathbb{C}=\mathbb{R}1+\mathbb{R}\mathbf{i}$ by the right multiplication,
take $\Set{1,\mathbf{j}}$ as a basis, and identify $\mathbb{H}$ with
$\mathbb{C}^{2}$. The standard Hermitian inner product on $\mathbb{C}^{2}$
induces an inner product on $\mathbb{H}$ given by
$(\mathbf{x},\mathbf{y})=(\mathbf{x}\overline{\mathbf{y}}+\mathbf{y}\overline{\mathbf{x}})/2$
for $\mathbf{x},\mathbf{y}\in\mathbb{H}$, where $\overline{\mathbf{y}}$
denotes the conjugate of $\mathbf{y}$. Since the left multiplication by a unit
quaternion is a $\mathbb{C}$-vector space homomorphism and preserves the norm
on $\mathbb{H}$, the group of unit quaternions are identified with
$\mathrm{SU}(2)$. We also identify the imaginary part of $\mathbb{H}$ with
$\mathbb{R}^{3}$ by
$b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\leftrightarrow{}^{t}\\!(b,c,d)$ for
$b,c,d\in\mathbb{R}$. Since the imaginary part of $\mathbb{H}$ is stable under
the conjugation by a unit quaternion, we obtain a two-to-one homomorphism

(6.1) $\displaystyle K=\mathrm{SU}(2)\to\mathrm{SO}(3);\
\mathbf{y}\mapsto(\mathbf{x}\mapsto\mathbf{y}\mathbf{x}\overline{\mathbf{y}}).$

The differentiation and the complexification give the isomorphism of Lie
algebras

(6.2)
$\displaystyle\mathfrak{sl}(2,\mathbb{C})\cong\mathfrak{so}(3,\mathbb{C});\frac{1}{2}\begin{pmatrix}a\sqrt{-1}&-b+c\sqrt{-1}\\\
b+c\sqrt{-1}&-a\sqrt{-1}\end{pmatrix}\leftrightarrow\begin{pmatrix}0&c&b\\\
-c&0&-a\\\ -b&a&0\end{pmatrix}\ \ (a,b,c\in\mathbb{C}).$

We next fix notation for the irreducible representations of $K$. Let
$m\in\mathbb{N}$ and $\mathcal{P}_{m}[t]$ the space of polynomials of degree
up to $m$ with variable $t$. The linear action of $K$ on $\mathcal{P}_{m}[t]$
defined by

(6.3) $\displaystyle g^{-1}q(t):=(\beta t+\overline{\alpha})^{m}q((\alpha
t-\overline{\beta})/(\beta t+\overline{\alpha}))$

for $g=\begin{pmatrix}\alpha&-\overline{\beta}\\\
\beta&\overline{\alpha}\end{pmatrix}\in K,q\in\mathcal{P}_{m}[t]$ gives us an
irreducible $(m+1)$-dimensional representation. Then the differentiated action
$\pi_{m}$ of $\mathfrak{sl}(2,\mathbb{C})$ on $\mathcal{P}_{m}[t]$ is given by

(6.4) $\displaystyle\pi_{m}\left(\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right)$ $\displaystyle=m-2t\frac{d}{dt},\
\pi_{m}\left(\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\right)$
$\displaystyle=-\frac{d}{dt},\ \pi_{m}\left(\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\right)$ $\displaystyle=-mt+t^{2}\frac{d}{dt}.$

For $m\in\mathbb{N}$, we define a nondegenerate $K$-invariant pairing
$\langle,\rangle_{m}$ of the $m$-th symmetric power $S^{m}\mathbb{C}^{2}$ and
$\mathcal{P}_{m}[t]$ by the linear extension of

(6.5) $\displaystyle\left\langle\begin{pmatrix}1\\\
0\end{pmatrix}^{i}\begin{pmatrix}0\\\
1\end{pmatrix}^{m-i},t^{j}\right\rangle_{m}:=\begin{cases}i!(m-i)!/m!&\text{
if $i=j$,}\\\ 0&\text{ otherwise}\end{cases}$

for $0\leq i,j\leq m$.

We regard $\sigma^{0}$ as the restriction of the defining representation
$\mathbb{C}^{2}$ of $K$ to $M$. By the Frobenius reciprocity, the restriction
map induces an isomorphism from
$C^{\infty}(G,\sigma^{0}_{-\lambda(2,-a)})^{B}$ to
$C^{\infty}(K,\sigma^{0})^{M}$ as $K$-modules. Since $\mathcal{P}_{m}[t]$ is
isomorphic to the dual of $S^{m}\mathbb{C}^{2}$ as $K$-modules, we have a
linear injection

(6.6) $\displaystyle\Phi_{m}\colon
S^{m}\mathbb{C}^{2}\otimes(\mathcal{P}_{m}[t]\otimes\sigma^{0})^{M}\to
C^{\infty}(K,\sigma^{0})^{M};\
\Phi_{m}\left(v\otimes\begin{pmatrix}q_{1}(t)\\\
q_{2}(t)\end{pmatrix}\right)(k):=\begin{pmatrix}\langle
v,kq_{1}(t)\rangle_{m}\\\ \langle v,kq_{2}(t)\rangle_{m}\end{pmatrix}$

for
${}^{t}\\!(q_{1}(t),q_{2}(t))\in(\mathcal{P}_{m}[t]\otimes\sigma^{0})^{M},v\in
S^{m}\mathbb{C}^{2},k\in K$. Then the direct sum of $\Phi_{m}$ for $m\geq 0$
gives an isomorphism

(6.7) $\displaystyle
C^{\infty}(G,\sigma^{0}_{-\lambda(2,-a)})^{Q}_{K}\cong\bigoplus_{m\geq
0}S^{m}\mathbb{C}^{2}\otimes(\mathcal{P}_{m}[t]\otimes\sigma^{0})^{M}.$

For $a\in\mathbb{Z}$ and $k\in\mathbb{N}$, we put

(6.8) $\displaystyle m(a,k):=2|a|+1+4k$

and define a polynomial $q(a,k:t^{2})\in\mathcal{P}_{m(a,k)}[t]$ by

(6.9) $\displaystyle q(a,k;t^{2})$
$\displaystyle:={}_{2}F_{1}(-|a|-1/2-2k,-a/2-|a|/2-k:a/2-|a|/2+1/2-k;t^{2})$
(6.10)
$\displaystyle=\sum_{l=0}^{a/2+|a|/2+k}\frac{(-|a|-1/2-2k)_{l}(-a/2-|a|/2-k)_{l}}{l!(a/2-|a|/2+1/2-k)_{l}}t^{2l}.$

Here ${}_{2}F_{1}$ denotes the Gauss hypergeometric function and we use the
Pochhammer symbol $(x)_{l}=x(x+1)\cdots(x+l-1)$.

The subspace $\mathop{\mathrm{Ker}}\nolimits
D(B,\sigma^{0}_{-\lambda(2,-a)})_{K}$ of
$C^{\infty}(G,\sigma^{0}_{-\lambda(2,-a)})^{B}$ is described explicitly in the
following

######  Proposition 6.1.

When $a\not\in\mathbb{Z}$, $\mathop{\mathrm{Ker}}\nolimits
D(B,\sigma^{0}_{-\lambda(2,-a)})_{K}=0$. When $a\in\mathbb{Z}$,
$\mathop{\mathrm{Ker}}\nolimits D(B,\sigma^{0}_{-\lambda(2,-a)})_{K}$ equals

(6.11) $\displaystyle\begin{cases}\bigoplus_{k\geq
0}\Phi_{m(a,k)}\left(S^{m(a,k)}\mathbb{C}^{2}\otimes\begin{pmatrix}-t^{m(a,k)}q(a,k;t^{-2})\\\
q(a,k;t^{2})\end{pmatrix}\right)&\text{ if $a$ is even,}\\\ \bigoplus_{k\geq
0}\Phi_{m(a,k)}\left(S^{m(a,k)}\mathbb{C}^{2}\otimes\begin{pmatrix}q(a,k;t^{2})\\\
-t^{m(a,k)}q(a,k;t^{-2})\end{pmatrix}\right)&\text{ if $a$ is
odd,}\end{cases}$

and is isomorphic to $\bigoplus_{k\geq 0}S^{m(a,k)}\mathbb{C}^{2}$ as
$\mathfrak{k}$-modules.

We use the following lemma in the proof of Proposition 6.1. Let $\mathfrak{n}$
be the nilradical of $\mathfrak{b}$.

######  Lemma 6.2.

Let $a\in\mathbb{C},\lambda\in\mathfrak{h}^{\vee}$. Then
$[\mathfrak{n},\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})]$ is included
in $I(\mathfrak{b},\lambda)$ (see (4.2) for the definition) if and only if
$\lambda=\lambda(2,-a)$.

###### Proof.

The lowest weight vector
$\mathop{\mathrm{sym}}\nolimits(\sum_{i=1}^{3}T_{3,i}T_{i,1}+a/3T_{3,1})$ in
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})=\mathop{\mathrm{sym}}\nolimits(F^{-a})$
is $T_{3,2}T_{2,1}+(-\lambda_{2}+a/3-1/2)T_{3,1}$ modulo
$I(\mathfrak{b},\lambda)$. Since we have

(6.12) $\displaystyle[T_{1,2},T_{3,2}T_{2,1}+(-\lambda_{2}+a/3-1/2)T_{3,1}]$
$\displaystyle\equiv T_{3,2}(T_{1,1}-T_{2,2})-(-\lambda_{2}+a/3-1/2)T_{3,2}$
(6.13) $\displaystyle\equiv(\lambda_{1}-a/3+1/2)T_{3,2}\bmod
I(\mathfrak{b},\lambda),$ (6.14)
$\displaystyle[T_{2,3},T_{3,2}T_{2,1}+(-\lambda_{2}+a/3-1/2)T_{3,1}]$
$\displaystyle\equiv(T_{2,2}-T_{3,3})T_{2,1}+(-\lambda_{2}+a/3-1/2)T_{2,1}$
(6.15) $\displaystyle\equiv(-\lambda_{3}+a/3+1/2)T_{2,1}\bmod
I(\mathfrak{b},\lambda),$

the condition
$[\mathfrak{n},\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})]\subset
I(\mathfrak{b},\lambda)$ is equivalent to $\lambda_{1}=a/3-1/2$ and
$\lambda_{3}=a/3+1/2$, which says $\lambda=\lambda(2,-a)$. ∎

###### Proof of Proposition 6.1.

Since $I(\mathfrak{b},\lambda(2,a))$ is closed under the adjoint action of
$\mathfrak{n}$, Lemma 6.2 and its proof show that
$\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})+I(\mathfrak{b},\lambda(2,-a))$
is spanned by $T_{3,2}T_{2,1}+(-\lambda(2,-a)_{2}+a/3-1/2)T_{3,1}$, which
equals

$X:=(T_{2,1}-T_{1,2})(T_{3,2}-T_{2,3})+(a+1/2)(T_{3,1}-T_{1,3})\in
U(\mathfrak{k})$

modulo $I(\mathfrak{b},\lambda(2,a))$.

Let ${}^{t}\\!(q_{1}(t),q_{2}(t))$ be a nonzero element in
$(\mathcal{P}_{m}[t]\otimes\sigma^{0})^{M}$. By (6.3), the condition
${}^{t}\\!(q_{1}(t),q_{2}(t))\in(\mathcal{P}_{m}[t]\otimes\sigma^{0})^{M}$ is
equivalent to

(6.16) $\displaystyle\begin{pmatrix}i&\\\
&-i\end{pmatrix}\begin{pmatrix}q_{1}(t)\\\ q_{2}(t)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\mathbf{i}^{-1}q_{1}(t)\\\
\mathbf{i}^{-1}q_{2}(t)\end{pmatrix}=\begin{pmatrix}(-i)^{m}q_{1}(-t)\\\
(-i)^{m}q_{2}(-t)\end{pmatrix},$ (6.17) $\displaystyle\begin{pmatrix}&-1\\\
1&\end{pmatrix}\begin{pmatrix}q_{1}(t)\\\ q_{2}(t)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\mathbf{j}^{-1}q_{1}(t)\\\
\mathbf{j}^{-1}q_{2}(t)\end{pmatrix}=\begin{pmatrix}t^{m}q_{1}(-t^{-1})\\\
t^{m}q_{2}(-t^{-1})\end{pmatrix}.$

Hence our assumption on $q_{1}(t),q_{2}(t)$ is written as

  1. (1)

$m$ is odd,

  2. (2)

$q_{1}(t)$ is odd when $m\equiv 1\bmod 4$, $q_{1}(t)$ is even when $m\equiv
3\bmod 4$, and

  3. (3)

$q_{2}(t)=-t^{m}q_{1}(-t^{-1})\neq 0$.

Let $F\in C^{\infty}(G,\sigma^{0}_{-\lambda(2,-a)})^{B}$. By the decomposition
$G=KB$, we see that $F\in\mathop{\mathrm{Ker}}\nolimits D$ if and only if
$DF=0$ on $K$. By the definitions of $D$ (see Definition 4.1) and $\Phi_{m}$
(6.6), the irreducible $K$-submodule
$\Phi_{m}(S^{k}(\mathbb{C}^{2})\otimes{}^{t}\\!(q_{1}(t),q_{2}(t)))$ is
included in $\mathop{\mathrm{Ker}}\nolimits D(B,\sigma^{0}_{-\lambda(2,-a)})$
if and only if $\pi_{m}(X)q_{1}(t)=0$ and $\pi_{m}(X)q_{2}(t)=0$.

Under the isomorphism (6.2), the vector $4X$ corresponds to

$\begin{pmatrix}0&-\sqrt{-1}\\\
-\sqrt{-1}&0\end{pmatrix}\begin{pmatrix}\sqrt{-1}&0\\\
0&-\sqrt{-1}\end{pmatrix}+(1+2a)\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

By (6.4), the action $\pi_{m}(4X)$ on $\mathcal{P}_{m}[t]$ is given by the
differential operator

$(-mt+t^{2}\frac{d}{dt}-\frac{d}{dt})(m-2t\frac{d}{dt})+(1+2a)(mt-t^{2}\frac{d}{dt}-\frac{d}{dt}).$

Let $q(t)=\sum_{l=0}^{k}a_{l}t^{l}\in\mathcal{P}_{m}[t]$. Since

(6.18) $\displaystyle\pi_{m}(4X)q(t)$
$\displaystyle=\sum_{l}a_{l}(\\{(m-2l)(-m+l)+(1+2a)(m-l)\\}t^{l+1}$ (6.19)
$\displaystyle\qquad\qquad+\\{(m-2l)(-l)+(1+2a)(-l)\\}t^{l-1})$ (6.20)
$\displaystyle=\sum_{l}a_{l}\\{(m-l)(-m+2l+1+2a)t^{l+1}-l(m-2l+1+2a)t^{l-1}\\},$

the condition $\pi_{m}(X)q(t)=0$ is equivalent to

$a_{l}\frac{l}{2}\left(\frac{l}{2}+k_{0}-\frac{m}{2}\right)=a_{l-2}\left(\frac{l}{2}-\frac{m+2}{2}\right)\left(\frac{l}{2}-k_{0}-1\right)$

for any $l$. Here we put $a_{l}=0$ for any $l\neq 0,\ldots,k$ and

$k_{0}:=(m-1-2a)/4.$

By the assumption that $m$ is odd, there is a nonzero polynomial solution to
$\pi_{m}(X)q(t)=0$ if and only if $2k_{0}\in\\{0,2,\ldots,m-1\\}$ if and only
if $m\in 2|a|+1+4\mathbb{N}$. In this case, the solution space is spanned by

(6.21) $\displaystyle{}_{2}\\!F_{1}(-m/2,-k_{0};k_{0}+1-m/2;t^{2})$
$\displaystyle=\sum_{k=0}^{k_{0}}\frac{(-m/2)_{k}(-k_{0})_{k}}{(1)_{k}(k_{0}+1-m/2)_{k}}t^{2k},$
(6.22) $\displaystyle t^{m}{}_{2}\\!F_{1}(-m/2,-k_{0};k_{0}+1-m/2;t^{-2})$
$\displaystyle=\sum_{k=0}^{k_{0}}\frac{(-m/2)_{k}(-k_{0})_{k}}{(1)_{k}(k_{0}+1-m/2)_{k}}t^{m-2k}.$

By the above arguments, $q_{1},q_{2}\in\mathcal{P}_{m}[t]$ satisfies the
assumptions (1), (2), (3) and $\pi_{m}(X)q_{1}(t)=\pi_{m}(X)q_{2}(X)=0$ if and
only if $a\in\mathbb{Z}$, $m=m(a,k)$ for some $k\in\mathbb{N}$ and

(6.23)
$\displaystyle(q_{1}(t),q_{2}(t))\in\begin{cases}\mathbb{C}(-t^{m(a,k)}q(a,k:t^{-2}),q(a,k:t^{2}))&\text{
when $a$ is even,}\\\ \mathbb{C}(q(a,k:t^{2}),-t^{m(a,k)}q(a,k:t^{-2}))&\text{
when $a$ is odd,}\end{cases}$

which is the desired conclusion. ∎

Let us finish the proof of Theorem 5.3. Let
$\mathfrak{g}_{0}=\mathfrak{sl}(n,\mathbb{R})$.

###### Proof of the rest of Theorem 5.3.

The assertion (2) follows from Proposition 6.1.

Let us prove the exhaustion of $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-modules. Let us take a positive system on
$\Phi(\mathfrak{g},\mathfrak{h}^{\mathop{\mathrm{c}}\nolimits})$ such that the
highest root $\psi$ is noncompact imaginary as in Section 3. By
$\mathfrak{p}\cong S^{2}(\mathbb{C}^{n})$ as $\mathfrak{k}$-modules, we see
$\rho(\mathfrak{k}^{1},\mathfrak{t})=n\psi/2$. Therefore the description (3.2)
and Proposition 3.3 (2) show that the highest weight $\lambda$ of a
$\mathfrak{k}$-type of an $a$-minimal $(\mathfrak{g},\mathfrak{k})$-module
belongs to $\mathbb{N}\psi/2$ if $n\geq 4$ and to $\mathbb{N}\psi/4$ if $n=3$.
When $\lambda$ belongs to $\mathbb{N}\psi/2$, the $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module has a common $\mathfrak{k}$-type with the
one appearing in Theorem 5.3 (1) by (5.10). Therefore they are isomorphic by
Proposition 3.3 (3).

Hence what is left is to prove that when
$\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$, any $a$-minimal
$(\mathfrak{g},\mathfrak{k})$-module $V$ which does not exponentiate to a
representation of $\mathrm{SO}(3)$ (called genuine) is isomorphic to the one
in (2). By Casselman’s subrepresentation theorem, we can embed $V$ into
$C^{\infty}(G,\sigma_{-\lambda})^{B}_{K}$ for some
$\sigma\in\widehat{M},\lambda\in\mathfrak{a}^{\vee}$. Proposition 4.3 (1)
shows that $V$ is contained in $\mathop{\mathrm{Ker}}\nolimits
D(B,\sigma_{-\lambda})$. Since $V$ is genuine, $\sigma$ is the two-dimensional
irreducible representation $\sigma^{0}$ of $M$. Moreover, since the identity
component of any standard parabolic subgroup strictly larger than $B$ contains
the center of $G$, Lemma 4.5 shows that
$[\mathfrak{n},\iota\circ\mathop{\mathrm{sym}}\nolimits(F^{a})]$ is included
in $I(\mathfrak{b},\lambda)$ so that $\mathop{\mathrm{Ker}}\nolimits
D(B,\sigma^{0}_{-\lambda})$ does not descend to the representation of
$\mathrm{SL}(3,\mathbb{R})$. By Lemma 6.2, we must have
$\lambda=\lambda(2,-a)$. Therefore $V$ is isomorphic to the minimal $a$-module
in Proposition 6.1, and the proof of Theorem 5.3 is complete. ∎

## Acknowledgements

The author is deeply grateful to Prof. Toshiyuki Kobayashi for helpful
comments and warm encouragement. The author expresses his sincere thanks to
Prof. David Vogan for insightful comments and letting him know the work by
Mœglin [17] and the alternative proof of Corollary 3.5. This work was
supported by JSPS KAKENHI Grant Number 17J01075, 20J00024 and the Program for
Leading Graduate Schools, MEXT, Japan.

Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo,
Hokkaido, 060-0810, Japan

E-mail<EMAIL_ADDRESS>
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