particular the group $\mathop{\rm U{}}\nolimits_{2}({\sf H})$. So we obtain a
large family of projective representations of $\mathop{\rm
U{}}\nolimits_{2}({\sf H})$ that define projective semibounded representations
of the Lie group $\mathop{\rm U{}}\nolimits_{2}({\sf
H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}$. In Theorem 517 we show that the
restricted groups are maximal “covariance groups” of these representations of
$\mathop{\rm U{}}\nolimits_{1}({\sf H})$. These results provide a new
perspective and relevant information on significant constructions which arise
naturally in Quantum Field Theory (see [PS86, Ne10b, KR87]).

###  5.1 Smooth and semibounded representations

######  Definition 51.

([Ne10a]) A unitary representation $\pi\colon G\to\mathop{\rm
U{}}\nolimits(\mathcal{H})$ is said to be smooth if the subspace
$\mathcal{H}^{\infty}\subseteq\mathcal{H}$ of smooth vectors is dense. This is
automatic for continuous representations of finite-dimensional groups, but not
for Banach–Lie groups. For any smooth unitary representation, the derived
representation

${\tt d}\pi\colon{\mathfrak{g}}=\mathop{\bf L{}}\nolimits(G)\to\mathop{{\rm
End}}\nolimits(\mathcal{H}^{\infty}),\quad{\tt
d}\pi(x)v:=\frac{d}{dt}\Big{|}_{t=0}\pi(\exp tx)v$

carries significant information in the sense that the closure $\partial\pi(x)$
of the operator ${\tt d}\pi(x)$ coincides with the infinitesimal generator of
the unitary one-parameter group $\pi(\exp tx)$. We call $(\pi,\mathcal{H})$
semibounded if the function

$s_{\pi}\colon{\mathfrak{g}}\to{\mathbb{R}}\cup\\{\infty\\},\quad
s_{\pi}(x):=\sup\big{(}{\rm Spec}(i\partial\pi(x))\big{)}$ (37)

is bounded on the neighborhood of some point in ${\mathfrak{g}}$. Then the set
$W_{\pi}$ of all such points is an open $\mathop{{\rm
Ad}}\nolimits(G)$-invariant convex cone in the Lie algebra ${\mathfrak{g}}$.
We call $\pi$ bounded if $s_{\pi}$ is bounded on some $0$-neighborhood, i.e.,
$W_{\pi}={\mathfrak{g}}$.

We shall also use the momentum set $I_{\pi}$. This is the weak-$*$-closed
convex hull of the image of the momentum map on the projective space of
$\mathcal{H}^{\infty}$:

$\Phi_{\pi}\colon{\mathbb{P}}({\cal
H}^{\infty})\to{\mathfrak{g}}^{\prime}\quad\hbox{ with
}\quad\Phi_{\pi}([v])(x)=\frac{1}{i}\frac{\langle{\tt
d}\pi(x)v,v\rangle}{\langle v,v\rangle}\quad\mbox{ for }[v]={\mathbb{C}}v.$

As a weak-$*$-closed convex subset, $I_{\pi}$ is completely determined by its
support functional

$s_{\pi}\colon{\mathfrak{g}}\to{\mathbb{R}}\cup\\{\infty\\},\quad
s_{\pi}(x)=-\inf\langle I_{\pi},x\rangle=\sup({\rm Spec}(i\partial\pi(x))).$

We now show that semibounded unitary representations decompose discretely into
bounded ones, and these have been studied in Section 3.

######  Theorem 52.

Let $G$ be a connected Hilbert–Lie group. Then every semibounded unitary
representation $(\pi,\mathcal{H})$ of $G$ is a direct sum of bounded ones.

###### Proof.

Proposition A2 implies that the open invariant cone
$W_{\pi}\subseteq{\mathfrak{g}}$ intersects the center
${\mathfrak{z}}({\mathfrak{g}})$, so that the restriction $\pi|_{Z(G)_{e}}$ is
also semibounded. We claim that it is a direct sum of bounded representations.
In fact, by [Ne08, Thm. 5.1], the momentum set

$C:=I_{\pi|_{Z(G)}}\subseteq{\mathfrak{z}}({\mathfrak{g}})^{\prime}$

is locally compact with respect to the weak-$*$ topology and we obtain a
representation of the $C^{*}$-algebra $C_{0}(C,{\mathbb{C}})$ on $\mathcal{H}$
that commutes with $\pi(G)$. Using the corresponding spectral measure on $C$,
we may thus decompose into representations $(\pi_{j})_{j\in J}$, for which $C$
is compact. Then $\pi_{j}|_{Z(G)}$ is bounded, so that
$W_{\pi_{j}}+{\mathfrak{z}}({\mathfrak{g}})=W_{\pi_{j}}$. As $W_{\pi_{j}}$
intersects the center, it follows that $0\in W_{\pi_{j}}$, i.e., that
$\pi_{j}$ is bounded. ∎

If ${\mathfrak{g}}$ is semisimple, then Proposition A2 shows as in the proof
above shows that every semibounded representation of ${\mathfrak{g}}$ is
bounded. Then $W_{\pi}$ intersects ${\mathfrak{z}}({\mathfrak{g}})=\\{0\\}$,
which is equivalent to boundedness of the representation.

######  Definition 53.

([JN19]) We call a smooth projective representation
$(\overline{\pi},\mathcal{H})$ of a Lie group $G$ bounded (resp. semibounded),
if the corresponding smooth representation $(\pi,\mathcal{H})$ of the central
Lie group extension

$G^{\sharp}=\\{(U,g)\in\mathop{\rm U{}}\nolimits(\mathcal{H})\times
G\colon\overline{U}=\overline{\pi}(g)\\},$

given by $\pi(U,g)=U$, is bounded (resp. semibounded).

The following theorem shows that we cannot expect to find more semibounded
representations of Hilbert–Lie groups by passing to central extensions. It
builds on quite subtle techniques involving in particular Lie group cohomology
and the Bruhat–Tits Fixed Point Theorem. For more details, we refer to the
proof of Theorem B6 in the appendix.

######  Theorem 54.

Let ${\mathfrak{g}}$ be a Hilbert–Lie algebra. Then every projective
semibounded unitary representation $(\overline{\pi},\mathcal{H})$ of a
corresponding simply connected Lie group $G$ lifts to a unitary representation
of $G$.

###### Proof.

We consider the corresponding semibounded representation $(\pi,\mathcal{H})$
of the central extension $G^{\sharp}$ of $G$. Then
$\partial\pi(1,0)=i\mathbf{1}$ implies that the momentum set
$I_{\pi}\subseteq({\mathfrak{g}}^{\sharp})^{\prime}$ does not annihilate
$(1,0)$. Hence it contains an element of the form $\lambda=(z,\alpha)$,
$z\not=0$. Then $\mathcal{O}_{\lambda}$ is semi-equicontinuous (see (B5) in
Appendix B), so that the orbit of $\alpha\in{\mathfrak{g}}^{\prime}$ under the
affine action is semi-equicontinuous (cf. Proposition B2 and Remark B7). Now
Theorem B6 implies that the central Lie algebra extension is trivial. As $G$
is simply connected, it now follows from[Ne02] that the central Lie group
extension $G^{\sharp}\to G$ splits. ∎

######  Remark 55.

Let $\widehat{G}=G\rtimes_{\alpha}{\mathbb{R}}$ for a continuous
${\mathbb{R}}$-action defined by $\alpha\colon{\mathbb{R}}\to\mathop{{\rm
Aut}}\nolimits(G)$. Then $\alpha$ preserves ${\mathfrak{z}}({\mathfrak{g}})$
and all simple ideals $({\mathfrak{g}}_{j})_{j\in J}$ (Lemma 210).

Suppose that $(\pi,\mathcal{H})$ is a positive energy representation of
$\widehat{G}$, i.e., $U_{t}:=\pi(e,t)=e^{itH}$ with $H=H^{*}$ bounded from
below. Then $U_{t}\in\pi(G)^{\prime\prime}$ for $t\in{\mathbb{R}}$ by the
Borchers–Arveson Theorem ([BGN20, Thm. 4.14]). If $\pi$ is semibounded, then
we may w.l.o.g. assume that $\mathbf{d}=(0,1)\in W_{\pi}$.

  * •

As $\pi(Z(G))$ is central in $\pi(G)^{\prime\prime}$, we must have
$\alpha_{t}(z)=z$ for $t\in{\mathbb{R}},z\in Z(G)_{e}$, provided $\ker\pi$ is
discrete.

  * •

Let $G^{0}\subseteq G$ be the closed subgroup of $\alpha$-fixed points, which
is a Hilbert–Lie group (cf. Subsection 4.3). If $\pi$ is semibounded with
$\mathbf{d}=(0,1)\in W_{\pi}$, then the restriction of $\pi$ to
$G^{0}\rtimes_{\alpha}{\mathbb{R}}=G^{0}\times{\mathbb{R}}$ is also
semibounded. The same holds for the restriction to the abelian subgroup
$Z\times{\mathbb{R}}$ for $Z:=Z(G^{0})_{e}$, so that [Ne08, Thm. 4.1] implies
the existence of a corresponding spectral measure. It follows in particular,
that the restriction to $Z\times{\mathbb{R}}$ is a direct sum of bounded
representations on $G^{0}$-invariant subspaces. We conclude with Proposition
A2, applied to the open cone $W_{\pi}$, that $(\pi,\mathcal{H})$ is a direct
sum of bounded representations of $G^{0}$.

If, in addition, the dimensions of the finite-dimensional
$\mathbf{d}$-eigenspaces are bounded from above, then the discussion above
shows that $\widehat{Z}$ is countable, so that Theorem 37 implies that
$\pi|_{G^{0}_{e}}$ is a direct sum of irreducible bounded representations.

###  5.2 Semibounded representations require semibounded $i\mathbf{d}$

Let $\mathbf{d}=-\mathbf{d}^{*}=iH$ be a, not necessarily bounded, operator on
the complex Hilbert space ${\sf H}$ and
${\mathfrak{g}}^{\infty}={\mathfrak{u}}_{2}({\sf H})^{\infty}$ be the
Fréchet–Lie algebra of smooth vectors for the ${\mathbb{R}}$-action defined by
$\alpha_{t}(x)=e^{t\mathbf{d}}xe^{-t\mathbf{d}}$. Then
${\mathfrak{g}}^{\infty}$ is dense in ${\mathfrak{u}}_{2}({\sf H})$ and we
have a Fréchet–Lie group
$\widehat{G}^{\infty}:=G^{\infty}\rtimes_{\alpha}{\mathbb{R}}$. If
$\mathbf{d}$ is diagonal, then

${\mathfrak{u}}({\sf H})_{\rm alg}={\mathfrak{u}}_{2}({\sf H})_{\rm
alg}\subseteq{\mathfrak{u}}_{2}({\sf H})^{\infty}.$

In this subsection we show that, if $\widehat{\mathfrak{g}}^{\infty}$ contains
open invariant cones contained in the half spaces

${\mathfrak{g}}^{\infty}\oplus(0,\infty)\mathbf{d},$

then the selfadjoint operator $i\mathbf{d}$ is semibounded. Intersecting with
${\mathfrak{g}}^{\infty}+\mathbf{d}\cong{\mathfrak{g}}^{\infty}\times\\{1\\}$,
we see that our requirement is equivalent to the existence of a proper open
convex subset of ${\mathfrak{g}}^{\infty}$, invariant under the affine action

$\beta_{g}x:=\mathop{{\rm Ad}}\nolimits_{g}x+g\mathbf{d}g^{-1}-\mathbf{d}$

on ${\mathfrak{g}}^{\infty}$.

######  Remark 56.

Consider the half Lie group $G:=\mathop{\rm U{}}\nolimits_{2}({\sf
H})\rtimes_{\alpha}{\mathbb{R}}$, which is a topological group and a smooth
manifold, such that all left multiplications are smooth (see [MN18] for a
detailed discussion of half Lie groups). Right multiplications with
$g\in\mathop{\rm U{}}\nolimits_{2}({\sf H})$ are of the form
$(h,t)\mapsto(h\alpha_{t}(g),t)$, hence are smooth if and only if
$g\in\mathop{\rm U{}}\nolimits_{2}({\sf H})^{\infty}$, the subgroup of
elements in $\mathop{\rm U{}}\nolimits_{2}({\sf H})$ with smooth
$\alpha$-orbits. Hence the Fréchet–Lie group $G^{\infty}=\mathop{\rm
U{}}\nolimits_{2}({\sf H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}$ consists of
all elements for which the right multiplication is smooth (see [BHM23] for a
systematic discussion of the Lie group structure on such subgroups).

######  Lemma 57.

Let $A$ and $B$ be selfadjoint operators on the complex Hilbert space ${\sf
H}$ and assume that $B$ is bounded. Then $A$ is semibounded if and only if
$A+B$ is.

###### Proof.

For unit vectors $v\in\mathcal{D}(A)=\mathcal{D}(A+B)$, we have

$\langle v,(A+B)v\rangle=\langle v,Av\rangle+\langle v,Bv\rangle\in\langle
v,Av\rangle+[-\|B\|,\|B\|].$

Thus $A$ is semibounded if and only if $A+B$ is. ∎

######  Theorem 58.

If ${\mathfrak{g}}^{\infty}={\mathfrak{u}}_{2}({\sf H})^{\infty}$ contains
proper open $\beta$-invariant convex subsets, then $i\mathbf{d}$ is
semibounded.

###### Proof.

Let $\Omega\subseteq{\mathfrak{g}}^{\infty}$ be a proper $\beta$-invariant
open convex subset. Replacing $\mathbf{d}$ by $\mathbf{d}+x$ for some
$x\in{\mathfrak{g}}^{\infty}$, we may w.l.o.g. assume that
$0\in\partial\Omega$. Note that $i\mathbf{d}$ is semibounded if and only if
$i(\mathbf{d}+x)$ has this property (Lemma 57).

The Hahn–Banach separation theorem implies the existence of a non-zero
continuous linear functional $\lambda\in{\mathfrak{g}}^{\infty,^{\prime}}$
with $\inf\lambda(\Omega)=\lambda(0)=0$. This implies that

$\lambda(g\mathbf{d}g^{-1}-\mathbf{d})\geq 0\quad\mbox{ for }\quad g\in
G^{\infty}.$

Derivatives along one-parameter subgroups of $G^{\infty}$ lead to the
conditions

$\lambda([x,\mathbf{d}])=0\quad\mbox{ and
}\quad\lambda([x,[x,\mathbf{d}]])\geq 0\quad\mbox{ for }\quad
x\in{\mathfrak{g}}^{\infty}.$ (38)

The first condition implies that
$\lambda\in({\mathfrak{g}}^{\infty})^{\prime}$ is invariant under the
${\mathbb{R}}$-action defined by conjugation with $e^{t\mathbf{d}}$. For the
Hilbert space ${\mathfrak{g}}$, we consider the decomposition

${\mathfrak{g}}^{\infty}={\mathfrak{g}}^{0}\oplus({\mathfrak{g}}^{\infty}\cap({\mathfrak{g}}^{0})^{\bot}).$

Then the definition of ${\mathfrak{g}}^{\infty}$ implies that
${\mathfrak{g}}^{\infty,0}={\mathfrak{g}}^{0}$, so that
${\mathfrak{g}}^{\infty}={\mathfrak{g}}^{0}\oplus\overline{[\mathbf{d},{\mathfrak{g}}^{\infty}]}$.
Now $\lambda$ vanishes on $[\mathbf{d},{\mathfrak{g}}^{\infty}]$, so that
$\lambda\in({\mathfrak{g}}^{\infty,0})^{\prime}$. Therefore
$\lambda(x)=\mathop{{\rm tr}}\nolimits(\mathbf{b}_{\lambda}x)$ for some
Hilbert–Schmidt operator $\mathbf{b}_{\lambda}\in{\mathfrak{g}}^{\infty,0}$.

Next we evaluate the second condition. First we observe that $\mathbf{d}$
commutes with $\mathbf{b}_{\lambda}$, hence preserves all eigenspaces of
$\mathbf{b}_{\lambda}$. For the spectral decomposition of
$i\mathbf{b}_{\lambda}$,

${\sf H}=\widehat{\bigoplus}_{\mu\in{\mathbb{R}}}{\sf H}^{\mu},\quad\mbox{ we
put }\quad\mathbf{d}_{\mu}:=\mathbf{d}|_{{\sf H}^{\mu}}.$

Then ${\sf H}^{\mu}$ is finite-dimensional for $\mu\not=0$. We claim that

$h(x,y):=\lambda([x^{*},[\mathbf{d},y]])=-\lambda([x^{*},[y,\mathbf{d}]])$

defines a positive semidefinite hermitian form on
${\mathfrak{g}}^{\infty}_{\mathbb{C}}$. In fact, the form is clearly
sesquilinear, and from $\overline{\lambda(x)}=\lambda(\overline{x})$ and
$\lambda([\mathbf{d},{\mathfrak{g}}^{\infty}])=\\{0\\}$ we derive

$\overline{h(y,x)}=\lambda(\overline{[y^{*},[\mathbf{d},x]]})=\lambda([-y,[x^{*},\mathbf{d}]])=\lambda([[x^{*},\mathbf{d}],y])=\lambda([x^{*},[\mathbf{d},y]])=h(x,y),$

so that $h$ is hermitian. We thus have

$h(x,x)=\lambda([x^{*},[\mathbf{d},x]])=\lambda([[-x^{*},[x,\mathbf{d}]])\geq
0\quad\mbox{ for all }\quad x\in{\mathfrak{g}}^{\infty}_{\mathbb{C}}$

because this holds for $x\in{\mathfrak{g}}^{\infty}$ by (38). Let
$\nu\not=\mu$ and $x\in B({\sf H}^{\mu},{\sf H}^{\nu})=B_{2}({\sf
H}^{\mu},{\sf H}^{\nu})\subseteq{\mathfrak{g}}^{\infty}_{\mathbb{C}}$ (recall
that at least one of these spectral subspaces is finite-dimensional). Then

$[i\mathbf{b}_{\lambda},x]=(\nu-\mu)x\quad\mbox{ and
}\quad[i\mathbf{b}_{\lambda},x^{*}]=(\mu-\nu)x^{*}$

lead to

$\displaystyle 0$
$\displaystyle\leq\lambda([x^{*},[\mathbf{d},x]])=\mathop{{\rm
tr}}\nolimits(\mathbf{b}_{\lambda}[x^{*},[\mathbf{d},x]])$
$\displaystyle=\mathop{{\rm
tr}}\nolimits(i\mathbf{b}_{\lambda}[x^{*},[-i\mathbf{d},x]])=-\mathop{{\rm
tr}}\nolimits([i\mathbf{b}_{\lambda},x^{*}][i\mathbf{d},x])$
$\displaystyle=-(\mu-\nu)\mathop{{\rm
tr}}\nolimits(x^{*}[i\mathbf{d},x])=(\nu-\mu)\mathop{{\rm
tr}}\nolimits(i\mathbf{d}xx^{*}-x^{*}xi\mathbf{d})$
$\displaystyle=(\nu-\mu)\mathop{{\rm
tr}}\nolimits(i\mathbf{d}_{\nu}xx^{*}-x^{*}xi\mathbf{d}_{\mu})$
$\displaystyle=(\nu-\mu)\big{(}\mathop{{\rm
tr}}\nolimits(i\mathbf{d}_{\nu}xx^{*})-\mathop{{\rm
tr}}\nolimits(i\mathbf{d}_{\mu}x^{*}x)\big{)}.$

We assume w.l.o.g. that $\nu>\mu$. Consider an operators $x=P_{v,w}$ of the
form $P_{v,w}(u):=\langle w,u\rangle v,$ where $w\in{\sf H}^{\mu}$ and
$v\in{\sf H}^{\nu}$ are unit eigenvectors for $\mathbf{d}$. Then

$\mathop{{\rm tr}}\nolimits(i\mathbf{d}_{\mu}x^{*}x)=\mathop{{\rm
tr}}\nolimits(i\mathbf{d}_{\mu}P_{w,w})=\langle
w,i\mathbf{d}_{\mu}w\rangle\quad\mbox{ and }\quad\mathop{{\rm
tr}}\nolimits(i\mathbf{d}_{\nu}xx^{*})=\langle v,i\mathbf{d}_{\nu}v\rangle.$

We therefore obtain

$\sup{\rm Spec}(i\mathbf{d}_{\mu})\leq\inf{\rm
Spec}(i\mathbf{d}_{\nu})\quad\mbox{ for }\quad\mu<\nu.$

Let us write this as $i\mathbf{d}_{\mu}\leq i\mathbf{d}_{\nu}$.

If the Hilbert–Schmidt operator $i\mathbf{b}_{\lambda}$ has a positive
eigenvalue, it has a maximal eigenvalue $\mu_{\rm max}>0$. Then
$i\mathbf{d}_{\mu_{\rm max}}\geq i\mathbf{d}_{\mu}$ for all other spectral
values $\mu$ of $i\mathbf{b}_{\lambda}$ and the fact that ${\sf H}^{\mu_{\rm
max}}$ is finite-dimensional implies that $i\mathbf{d}$ is bounded from above.
Likewise, the existence of a negative eigenvalue of $i\mathbf{b}_{\lambda}$
implies that $i\mathbf{d}$ is bounded from below. As $\lambda$ is non-zero,
one of these two cases occurs, so that $i\mathbf{d}$ is semibounded. ∎

######  Corollary 59.

A semibounded smooth representation of the Fréchet–Lie group $\mathop{\rm
U{}}\nolimits_{2}({\sf H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}$ exists if and
only if the hermitian operator $i\mathbf{d}$ on ${\sf H}$ is semibounded.

###### Proof.

Theorem 58 shows that the semiboundedness if $i\mathbf{d}$ is necessary for
the existence of a a semibounded smooth representation of $\mathop{\rm
U{}}\nolimits_{2}({\sf H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}.$

To see that it is also sufficient, we observe that, if $i\mathbf{d}$ is
semibounded, then the identical representation of $\mathop{\rm
U{}}\nolimits_{2}({\sf H})$ on ${\sf H}$ extends to a continuous
representation $\widehat{\pi}$ of $\mathop{\rm U{}}\nolimits_{2}({\sf
H})\rtimes_{\alpha}{\mathbb{R}}$, which defines a smooth representation of the
Lie group $\mathop{\rm U{}}\nolimits_{2}({\sf
H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}$. The corresponding space of smooth
vectors is the space ${\sf H}^{\infty}={\sf H}^{\infty}(\mathbf{d})$ of smooth
vectors of the selfadjoint operator $i\mathbf{d}$ (cf. [Ne10a, Thm. 7.2]). As
the restriction of this representation to $\mathop{\rm U{}}\nolimits_{2}({\sf
H})$ is bounded, the cone $W_{\pi}$ contains an open half-space
${\mathfrak{u}}_{2}({\sf H})\pm{\mathbb{R}}_{+}\mathbf{d}$, so that
$\widehat{\pi}$ defines a semibounded smooth representation of $\mathop{\rm
U{}}\nolimits_{2}({\sf H})^{\infty}\rtimes_{\alpha}{\mathbb{R}}$. ∎

###  5.3 Representations of the trace class group

From here on, in this section ${\mathfrak{g}}={\mathfrak{u}}_{2}({\sf
H}),{\mathfrak{g}}_{1}={\mathfrak{u}}_{1}({\sf H}),G=\mathop{\rm
U{}}\nolimits_{2}({\sf H})$ and $G_{1}=\mathop{\rm U{}}\nolimits_{1}({\sf
H})$. We write ${\mathfrak{t}}_{1}={\mathfrak{t}}\cap{\mathfrak{g}}_{1}$ for
the subalgebra of diagonal operators in ${\mathfrak{u}}_{1}({\sf H})$, where
${\mathfrak{t}}\subseteq{\mathfrak{g}}$ also is the subalgebra of diagonal
operator with respect to some orthonormal basis. From [Ne98, Thms. I.11,
III.8] and [Ne14] we have:

######  Theorem 510.

For each $\lambda\in\mathcal{P}_{b}$, there exists a unique bounded unitary
representation
$\rho_{\lambda}\colon{\mathfrak{g}}_{1}\to{\mathfrak{u}}(\mathcal{H}_{\lambda})$
whose weight set with respect to ${\mathfrak{t}}_{1}$ is given by

$\mathcal{P}_{\lambda}=\mathop{{\rm
conv}}\nolimits(\mathcal{W}\lambda)\cap(\lambda+{\mathbb{Z}}[\Delta]).$

Then $\mathcal{W}\lambda=\mathop{{\rm Ext}}\nolimits(\mathop{{\rm
conv}}\nolimits(\mathcal{P}_{\lambda}))$ and
$\rho_{\lambda}\sim\rho_{\mu}\Longleftrightarrow\mu\in\mathcal{W}\lambda.$

######  Definition 511.

We call $(\rho_{\lambda},\mathcal{H}_{\lambda})$ the representation with
extremal weight $\lambda\in\mathcal{P}_{b}$.

For more on convex sets and cones, we refer to Appendix B.

######  Proposition 512.

For $\lambda\in\mathcal{P}_{b}$, let $(\pi_{\lambda},\mathcal{H}_{\lambda})$
be the corresponding norm continuous unitary representation of
$G_{1}=\mathop{\rm U{}}\nolimits_{1}(\mathcal{H})$ and
$I_{\lambda}\subseteq{\mathfrak{g}}_{1}^{\prime}$ be the corresponding
momentum set. Then

$I_{\lambda}=I_{\mu}\qquad\Longleftrightarrow\qquad\overline{\mathop{{\rm
conv}}\nolimits\mathcal{W}\lambda}=\overline{\mathop{{\rm
conv}}\nolimits\mathcal{W}\mu},$

where closures refer to the weak-$*$-topology on
$i{\mathfrak{t}}_{1}^{\prime}\cong\ell^{\infty}(J,{\mathbb{R}})$.

###### Proof.

By Fenchel duality, the weak-$*$-closed convex subsets $I_{\lambda}$ and
$I_{\mu}$ are equal if and only if their support functionals $s_{\lambda}$ and
$s_{\mu}$ coincide ([Ne20, §2.1]). As the support functionals $s_{\lambda}$
and $s_{\mu}$ are continuous and $\mathop{{\rm
Ad}}\nolimits(G_{1})$-invariant, and $\mathop{{\rm
Ad}}\nolimits(G_{1}){\mathfrak{t}}_{1}$ is dense in ${\mathfrak{g}}_{1}$, this
is equivalent to
$s_{\lambda}|_{{\mathfrak{t}}_{1}}=s_{\mu}|_{{\mathfrak{t}}_{1}}$. These are
the support functionals of the momentum sets $I_{\pi_{\lambda}|_{T_{1}}}$ and
$I_{\pi_{\mu}|_{T_{1}}}$ of the restrictions to $T_{1}$. By Theorem 510 these
are the weak-$*$-closed convex hulls $-i\overline{\mathop{{\rm
conv}}\nolimits(\mathcal{W}\lambda)}$ and $-i\overline{\mathop{{\rm
conv}}\nolimits(\mathcal{W}\mu)}$ in ${\mathfrak{t}}_{1}^{\prime}$,
respectively. ∎

As $\rho_{\lambda}\cong\rho_{\mu}$ if and only if $\mu\in\mathcal{W}\lambda$
([Ne04, Thm. 5.12]), the momentum set $I_{\lambda}$ is a rather coarse
invariant. It is easy to see examples of non-equivalent representations with
the same momentum set. A simple example is obtained for

$J={\mathbb{Z}},\qquad\lambda=\chi_{{\mathbb{N}}}\quad\mbox{ and
}\quad\mu=\chi_{{\mathbb{N}}_{0}}$

(see also Remark 515).

We recall the following result from [Ne04, Thm. VII.18]. For further details
we refer to [Ne98, NeSt01].

######  Theorem 513.

(Extension Theorem) Let ${\sf H}=\ell^{2}(J,{\mathbb{C}})$, let
$\lambda:J\to{\mathbb{Z}}$ be a bounded weight and
$(\rho_{\lambda},\mathcal{H}_{\lambda})$ be the corresponding bounded unitary
representation of $G_{1}=\mathop{\rm U{}}\nolimits_{1}({\sf H})$. For the
derivation of ${\mathfrak{u}}_{1}({\sf H})$, defined by

$\mathbf{d}=\mathop{{\rm diag}}\nolimits(i\lambda_{j}),\quad\mbox{ and
}\quad\alpha_{t}(g)=e^{t\mathbf{d}}ge^{-t\mathbf{d}}$

we consider the restricted group

$\mathop{\rm U{}}\nolimits_{\rm res}({\sf H},\mathbf{d}):=\\{g\in\mathop{\rm
U{}}\nolimits({\sf H})\colon
g\mathbf{d}g^{-1}-\mathbf{d}\in{\mathfrak{u}}_{2}({\sf H})\\},$

endowed with its natural Banach–Lie group structure. Then $\rho_{\lambda}$
extends to a projective representation of the group $\mathop{\rm
U{}}\nolimits_{\rm res}({\sf H},\mathbf{d})$ on $\mathcal{H}_{\lambda}$. In
particular, it restricts to a projective representation of $\mathop{\rm
U{}}\nolimits_{2}({\sf H})$.

Here the representations $\rho_{\lambda}$ come from Theorem 510 and the
construction of the Lie group structure on $\mathop{\rm U{}}\nolimits_{\rm
res}({\sf H},d)$ can be found in [Ne04]. Similar to the discussion of central
extensions in Subsection 4.2, it is based on the observation that we have a
surjective homomorphism

$\mathop{\rm U{}}\nolimits_{1,2}({\sf H},\mathbf{d})\rtimes\mathop{\rm
U{}}\nolimits({\sf H})^{0}\to\mathop{\rm U{}}\nolimits_{\rm res}({\sf
H},\mathbf{d}),\quad(g_{1},g_{2})\mapsto g_{1}g_{2}.$

Here

$\mathop{\rm U{}}\nolimits_{1,2}({\sf H},\mathbf{d})=\\{g\in\mathop{\rm
U{}}\nolimits_{2}({\sf H})\colon(\forall\ell=1,\ldots,k)\
\|g_{\ell\ell}-\mathbf{1}\|_{1}<\infty\\},$

where $g=(g_{\ell,m})_{1\leq\ell,m\leq N}$ is the matrix description of $g$
with respect to the different eigenspaces ${\sf
H}_{m}:=\ker(\mathbf{d}-id_{m}\mathbf{1})$, $m=1,\ldots,N$, of $i\mathbf{d}$.
Accordingly

$\mathop{\rm U{}}\nolimits({\sf H})^{0}\cong\prod_{m=1}^{N}\mathop{\rm
U{}}\nolimits({\sf H}_{m}).$

The construction of the extension of the representation $\rho_{\lambda}$ of
$\mathop{\rm U{}}\nolimits_{1}({\sf H})$ to $\mathop{\rm U{}}\nolimits_{\rm
res}({\sf H},\mathbf{d})$ in [Ne04] shows that, on the subgroup $\mathop{\rm
U{}}\nolimits_{1,2}({\sf H},\mathbf{d})$, in which $\mathop{\rm
U{}}\nolimits_{1}(\mathcal{H})$ is dense, it extends by continuity to a
unitary representation. Moreover, the $\lambda$-weight space is one-
dimensional, of the form ${\mathbb{C}}\Omega_{\lambda}$, and
$\Omega_{\lambda}$ is fixed by the subgroup $\mathop{\rm U{}}\nolimits({\sf
H})^{0}$.

Theorem 517 below shows the extension of $\rho_{\lambda}$ to a projective
representation of $\mathop{\rm U{}}\nolimits_{\rm res}({\sf H},\mathbf{d})$ is
maximal.

We also note that a Lie algebra cocycle corresponding to the central extension
${\mathfrak{u}}_{2}(\mathcal{H})^{\sharp}$, defined by this projective
representation, can be evaluated with the methods explained in Example 415.
This leads to

$\omega(x,y)=i\lambda([x,y])=\omega_{\mathbf{d}}(x,y),$ (39)

if we consider $\lambda\in\ell^{\infty}(J,{\mathbb{R}})$ as a continuous
linear functional on ${\mathfrak{u}}_{1}({\sf H})$.

###  5.4 Covariance of bounded representations of the trace class ideal

Let $\lambda\colon J\to{\mathbb{R}}$ be bounded and integral, i.e.,
$\lambda_{j}-\lambda_{k}\in{\mathbb{Z}}$ for $j,k\in J$, and
$(\rho_{\lambda},\mathcal{H}_{\lambda})$ be the corresponding bounded unitary
representation of the simply connected covering group $\widetilde{\mathop{\rm
U{}}}\nolimits_{1}({\sf H})$ (Theorem 510 and [Ne98]). In this subsection we
determine the covariance group

$\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}:=\\{g\in\mathop{\rm
U{}}\nolimits({\sf
H})\colon\rho_{\lambda}\circ\alpha_{g}\sim\rho_{\lambda}\\},$

where $\mathop{\rm U{}}\nolimits({\sf H})$ acts by the automorphisms
$\alpha_{g}(h):=ghg^{-1}$ on $\mathop{\rm U{}}\nolimits_{1}({\sf H})$, and
also by canonical lifts to $\widetilde{\mathop{\rm U{}}}\nolimits_{1}({\sf
H})$.

######  Example 514.

(a) If $\lambda$ is constant $c$, then $\rho_{\lambda}(g)=\det^{\lambda}(g)$,
as a representation of $\widetilde{\mathop{\rm U{}}}\nolimits_{1}({\sf H})$ if
$c\not\in{\mathbb{Z}}$. This one-dimensional representation is $\mathop{\rm
U{}}\nolimits({\sf H})$-invariant, so that $\mathop{\rm U{}}\nolimits({\sf
H})_{\lambda}=\mathop{\rm U{}}\nolimits({\sf H})$ in this case.

(b) If $\lambda$ is finitely supported, then $\rho_{\lambda}$ extends to a
bounded unitary representation of $\mathop{\rm U{}}\nolimits({\sf H})$, so
that we also obtain in this case $\mathop{\rm U{}}\nolimits({\sf
H})_{\lambda}=\mathop{\rm U{}}\nolimits({\sf H})$ ([Ne98, Ne14b]).

(c) If there exists a constant $c$ such that $\lambda-c$ is finitely
supported, then we combine (a) and (b) to see that $\mathop{\rm
U{}}\nolimits({\sf H})_{\lambda}=\mathop{\rm U{}}\nolimits({\sf H})$ also
holds in this case.

######  Remark 515.

Let $(e_{j})_{j\in J}$ be an orthonormal basis of ${\sf H}$ and
$g\in\mathop{\rm U{}}\nolimits({\sf H})$ be a permutation of the basis
elements. We write $S_{J}\subseteq\mathop{\rm U{}}\nolimits({\sf H})$ for the
subgroup of all unitary operators mapping the orthonormal basis onto itself
and identify it with the group of all bijections of the set $J$; we write
$S_{(J)}\trianglelefteq S_{J}$ for the normal subgroup of those permutations
moving only finitely many elements.

Since $\rho_{\lambda}$ is uniquely determined by the weight set
$\mathcal{P}_{\lambda}\subseteq\widehat{T}_{1}$ for the diagonal torus
$T_{1}\subseteq\mathop{\rm U{}}\nolimits_{1}({\sf H})$ ([Ne98]), we have
$g\in\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}$ if and only if
$g\lambda\in\mathcal{W}\lambda=\mathop{{\rm Ext}}\nolimits(\mathop{{\rm
conv}}\nolimits(\mathcal{P}_{\lambda}))$ (note that $\mathcal{W}\cong
S_{(J)}$). This is always the case if $\mathop{{\rm supp}}\nolimits(\lambda)$
is finite or if there exists a constant $c\in{\mathbb{R}}$, for which
$\lambda-c$ has finite support. If this is not the case, then $\lambda$ takes
two different values infinitely often and there exists a $g\in S_{J}$ with
$g.\lambda\not\in\mathcal{W}\lambda$; see also Remark 516 below.

######  Remark 516.

A necessary condition for $g\in\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}$
is that the momentum set $I_{\rho_{\lambda}}$ satisfies
$\alpha_{g}^{*}I_{\rho_{\lambda}}=I_{\rho_{\lambda}}$ (cf. Proposition 512),
resp., that

$s_{\lambda}\circ\alpha_{g}=s_{\lambda}$ (40)

holds for the support functional
$s_{\lambda}:=s_{\rho_{\lambda}}\colon{\mathfrak{u}}_{1}({\sf
H})\to{\mathbb{R}}$. The following example shows that (40) does not imply
$g\in\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}$.

If $\lambda=\chi_{M}$ is the characteristic function of a subset $M\subseteq
J$ which is infinite and co-infinite, then any $g\in S_{J}$ satisfies
$g\lambda=\chi_{g.M}$ and

$\mathcal{W}\chi_{M}=\\{\chi_{E}\colon|E\setminus M|=|M\setminus E|<\infty\\}$

imply that $g\in\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}$ is equivalent to

$|gM\setminus M|=|M\setminus gM|<\infty.$

We now calculate the support functional. For $x\in
i{\mathfrak{t}}_{1}\cong\ell^{1}(J,{\mathbb{R}})$, we have

$s_{\lambda}(-ix)=\sup\langle\mathcal{W}\chi_{M},x\rangle=\sup\Big{\\{}\sum_{j\in
E}x_{j}\colon|E\setminus M|=|M\setminus E|<\infty\Big{\\}}.$

If $\mathop{{\rm supp}}\nolimits(x)$ is finite and $x=x^{+}-x^{-}$ with
$x^{\pm}:=(\pm x)\vee 0$ is the canonical decomposition, then

$s_{\lambda}(-ix)=\sum_{j\in J}x^{+}_{j}\quad\mbox{ for }\quad x\in
i{\mathfrak{t}}_{1}\cong\ell^{1}(J,{\mathbb{R}}).$

By continuity of the map $x\mapsto x^{+}$ and of $s_{\lambda}$, it follows
that $s_{\lambda}(-ix)=\mathop{{\rm tr}}\nolimits(x^{+})$ for $x\in
i{\mathfrak{t}},$ and by conjugation invariance of $s_{\lambda}$, we further
get

$s_{\lambda}(-ix)=\mathop{{\rm tr}}\nolimits(x^{+})\quad\mbox{ for }\quad
x\in\mathop{{\rm Herm}}\nolimits_{1}({\sf H}).$ (41)

This is true for every infinite $M$ which is also co-infinite. Since there are
many such sets which are not conjugate under the Weyl group $\mathcal{W}\cong
S_{(J)}$, it follows that the momentum set $I_{\rho_{\lambda}}$ does not
determine the representation up to equivalence. Concretely, the
weak-$*$-closure of $\mathop{{\rm conv}}\nolimits(\mathcal{W}\chi_{M})$ does
not distinguish subsets $M_{1}$ and $M_{2}$ of the same cardinality if the
complements $M_{1}^{c}$ and $M_{2}^{c}$ also have the same cardinality, i.e.,
if $\chi_{M_{2}}\in S_{J}.\chi_{M_{1}}$. Typical examples of such sets are
$M_{1}={\mathbb{N}}\subseteq J={\mathbb{Z}}$ and $M_{2}={\mathbb{N}}_{0}$.

######  Theorem 517.

Suppose that $\lambda$ is bounded and define a diagonal operator
$\mathbf{d}\in{\mathfrak{u}}({\sf H})$ by $\mathbf{d}e_{j}=i\lambda_{j}e_{j}$
for $j\in J$. Then

$\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}=\mathop{\rm U{}}\nolimits_{\rm
res}({\sf H},\mathbf{d}):=\\{g\in\mathop{\rm U{}}\nolimits({\sf
H})\colon\|g\mathbf{d}g^{-1}-\mathbf{d}\|_{2}<\infty\\}.$

###### Proof.

We know already from Theorem 513 that “$\supseteq$” holds. So it remains to
verify “$\subseteq$”. Recall from Theorem 513 that $\rho_{\lambda}$ extends to
a smooth projective unitary representation
$\overline{\rho}_{\lambda}\colon\mathop{\rm U{}}\nolimits_{2}({\sf
H})\to\mathop{\rm PU{}}\nolimits(\mathcal{H}_{\lambda})$ (see [JN15] for the
concept of a smooth projective unitary representation).

If $g\in\mathop{\rm U{}}\nolimits({\sf H})_{\lambda}$, then the density of
$\mathop{\rm U{}}\nolimits_{1}({\sf H})$ in $\mathop{\rm
U{}}\nolimits_{2}({\sf H})$ implies that the projective unitary representation
$\overline{\rho}_{\lambda}\circ\alpha_{g}$ is equivalent to
$\overline{\rho}_{\lambda}$, and this implies in particular that the
corresponding central extensions are equivalent. On the Lie algebra level a
cocycle of the central extension is given by

$\omega_{\lambda}(x,y)=i\lambda([x,y])=\mathop{{\rm
tr}}\nolimits(\mathbf{d}[x,y])\quad\mbox{ for }\quad
x,y\in{\mathfrak{u}}_{2}({\sf H})$

(see (39) in Subsection 5.3). We thus obtain the necessary condition

$g.[\omega_{\lambda}]=[\omega_{\lambda}]\quad\mbox{ in }\quad
H^{2}({\mathfrak{u}}_{2}({\sf H}),{\mathbb{R}}).$

From

$(g.\omega_{\lambda})(x,y)=\omega_{\lambda}(\alpha_{g}^{-1}x,\alpha_{g}^{-1}y)=\mathop{{\rm
tr}}\nolimits(\mathbf{d}\alpha_{g}^{-1}[x,y])=\mathop{{\rm
tr}}\nolimits(\alpha_{g}(\mathbf{d})[x,y])$

we derive that the operator $g\mathbf{d}g^{-1}-\mathbf{d}$ has to represent a
trivial cocycle, which means that

$g\mathbf{d}g^{-1}-\mathbf{d}\in{\mathbb{R}}i\mathbf{1}+{\mathfrak{u}}_{2}({\sf
H})$

([Ne02c, Ex. 2.18] and Subsection 4.2). We consider the $1$-cocycle

$\beta\colon\mathop{\rm U{}}\nolimits({\sf H})\to{\mathfrak{u}}({\sf
H}),\quad\beta(g):=g\mathbf{d}g^{-1}-\mathbf{d}.$

Suppose that $\beta(g)-\mu i\mathbf{1}\in{\mathfrak{u}}_{2}({\sf H}).$ Let
$\overline{\beta}\colon\mathop{\rm U{}}\nolimits({\sf
H})\to{\mathfrak{u}}({\sf H})/{\mathfrak{u}}_{\infty}({\sf H})$ be the
corresponding cocycle modulo the subspace ${\mathfrak{u}}_{\infty}({\sf H})$
of compact skew-hermitian operators. Then
$\overline{\beta}(g)=\mu[i\mathbf{1}]$, and since $[i\mathbf{1}]$ is fixed, we
obtain

$\overline{\beta}(g^{n})=n\mu[i\mathbf{1}]\quad\mbox{ for }\quad
n\in{\mathbb{Z}}.$

As $\overline{\beta}$ is bounded, it follows that $\mu=0$, hence that
$\beta(g)$ is Hilbert–Schmidt, which is equivalent to
$g\mathbf{d}-\mathbf{d}g=(g\mathbf{d}g^{-1}-\mathbf{d})g\in B_{2}({\sf H})$,
i.e., to $g\in\mathop{\rm U{}}\nolimits_{\rm res}({\sf H},\mathbf{d})$. ∎

Note that the group $\mathop{\rm U{}}\nolimits_{\rm res}({\sf H},\mathbf{d})$
does not change in the theorem above, if we replace $\mathbf{d}$ by
$\mathbf{d}+\mathop{{\rm ad}}\nolimits x$ for some
$x\in{\mathfrak{u}}_{2}({\sf H})$ (see Corollary 45).

######  Remark 518.

Assume that $\mathbf{d}$ is diagonalizable with finitely many eigenvalues and
eigenspaces ${\sf H}_{j}:=\ker(\mathbf{d}-id_{j}\mathbf{1})$, $j=1,\ldots,r$.
Accordingly, we write $g\in\mathop{\rm U{}}\nolimits({\sf H})$ as a matrix
$g=(g_{ij})_{1\leq i,j\leq r}$ with $g_{ij}\in B({\sf H}_{j},{\sf H}_{i})$.
The relation $g\mathbf{d}-\mathbf{d}g\in B_{2}({\sf H})$ means that, for
$i\not=j$, the corresponding matrix entry $(d_{j}-d_{i})g_{ij}$ is
Hilbert–Schmidt, and this is equivalent to $g_{ij}$ being Hilbert–Schmidt. In
this sense we have

$\mathop{\rm U{}}\nolimits_{\rm res}({\sf
H},\mathbf{d})=\\{g=(g_{ij})\colon(\forall i\not=j)\,g_{ij}\in B_{2}({\sf
H}_{j},{\sf H}_{i})\\}.$

##  6 Perspectives

Although we formulated several challenging problems in the main text, we
briefly discuss in this short final section some directions of research that
target important tools to analyze unitary representations of infinite-
dimensional Lie groups.

###  6.1 Representations of abelian Hilbert–Lie groups

######  Definition 61.

A toroidal Hilbert–Lie group is a group of the form $T={\mathfrak{t}}/\Gamma$,
where $\Gamma$ is a discrete subgroup of the real Hilbert space
${\mathfrak{t}}$, considered as an abelian Hilbert–Lie algebra ([Ba91]), and
$\Gamma$ spans a dense subspace of ${\mathfrak{t}}$. Assigning to
$\alpha\in{\mathfrak{t}}^{\prime}$ with $\alpha(\Gamma)\subseteq{\mathbb{Z}}$
the character $\chi_{\alpha}(x+\Gamma):=e^{2\pi i\alpha(x)}$, we then obtain
an isomorphism of groups

$\widehat{T}\cong\\{\alpha\in{\mathfrak{t}}^{\prime}\colon\alpha(\Gamma)\subseteq{\mathbb{Z}}\\}\hookrightarrow\mathop{{\rm
Hom}}\nolimits(\Gamma,{\mathbb{Z}}).$

Any toroidal Lie group contains the torus groups $T_{F}:=(\mathop{{\rm
span}}\nolimits_{\mathbb{R}}\Gamma_{F})/\Gamma_{F}$,
$\Gamma_{F}\subseteq\Gamma$ finitely generated, and the directed union $T_{0}$
of these subgroups is dense in $T$. Assume that $\Gamma$ is countable. Then
$T_{0}$ carries an intrinsic Lie group structure whose underlying topology is
the direct limit topology ([Gl05]). We identify the character group
$\widehat{T}$ with a subgroup of $\widehat{T_{0}}\cong\mathop{{\rm
Hom}}\nolimits(\Gamma,{\mathbb{Z}})$, consisting of those characters of
$T_{0}$ which extend continuously to $T$.

An important example is the group $T:=\ell^{2}/{\mathbb{Z}}^{({\mathbb{N}})}$
with the dense subgroup
$T_{0}:={\mathbb{R}}^{({\mathbb{N}})}/{\mathbb{Z}}^{({\mathbb{N}})}$ and
$\widehat{T_{0}}\cong{\mathbb{Z}}^{\mathbb{N}}\supseteq\widehat{T}\cong{\mathbb{Z}}^{({\mathbb{N}})}$.
That bounded representations of $T$ are direct sums of eigenspaces follows
from Theorem 36 because $\widehat{T}$ is countable.

It is an interesting problem to understand the continuous unitary
representations of $T$ in terms of their restrictions to $T_{0}$. This problem
immediately reduces to cyclic representations, which are given by positive
definite functions. As $T_{0}$ is a nuclear group, positive definite functions
are Fourier transforms of measures $\mu$ on $\widehat{T_{0}}$ ([Ba91]).

######  Problem 62.

Characterize those Borel measures $\mu$ on $\widehat{T_{0}}\cong\mathop{{\rm
Hom}}\nolimits(\Gamma,{\mathbb{Z}})$ whose Fourier transform $\widehat{\mu}$
extends to a continuous function on $T$. This is equivalent to the function
$\widehat{\mu}$ to extend from $\mathop{{\rm span}}\nolimits\Gamma$ to a
continuous function on ${\mathfrak{t}}$. As $\widehat{T}$ is a countable
subset of $\widehat{T_{0}}$ and $\mu$ is atomic on $\widehat{T}$, it suffices
to consider measures on the complement $\widehat{T_{0}}\setminus\widehat{T}$.

We expect that this problem can be addressed by using the Sazonov topology on
Hilbert spaces; see [Ya85] and [Xia72] for tools on measures on infinite-
dimensional spaces.

###  6.2 Representations of non-Lorentzian double extensions

The construction in Section 5.4 produces in particular projective semibounded
representations of groups $G\rtimes_{\alpha}{\mathbb{R}}$, $G=\mathop{\rm
U{}}\nolimits_{2}({\sf H})$, where $\alpha$ is defined by a bounded diagonal
operator $\mathbf{d}$ on ${\sf H}$ with integral eigenvalues and the cocycle
$\omega$ of the corresponding central Lie algebra extension
${\mathfrak{g}}^{\sharp}={\mathbb{R}}\oplus_{\omega}{\mathfrak{g}}$ is also
given by $\mathbf{d}$.

In general, one may consider central extensions whose cocycle is given by a
bounded operator $\mathbf{d}_{0}$, and $\alpha$ is generated by a, possibly
unbounded diagonal operator $\mathbf{d}$. It would be interesting to
understand the precise relations on the pair $(\mathbf{d}_{0},\mathbf{d})$
that characterizes the existence of projective semibounded representations of
$G\rtimes_{\alpha}{\mathbb{R}}$.

For extremal weight representations $(\pi_{\lambda},\mathcal{H}_{\lambda})$ as
in Theorem 513, this condition only depends on the class of $\mathbf{d}$
modulo $\ell^{1}$ if the weight $\lambda$ is bounded; see [MN16]. If the
spectrum of $\mathbf{d}$ is bounded from below, a suitable diagonal
modification $\widetilde{\mathbf{d}}$ of $\mathbf{d}$ should have a non-zero
minimal eigenspace.

###  6.3 Restrictions imposed by spectral conditions

Suppose that $(U,\mathcal{H})$ is a unitary representation of $G=\mathop{\rm
U{}}\nolimits_{2}({\sf H})$ and
$\alpha_{t}(g)=e^{t\mathbf{d}}ge^{-t\mathbf{d}}$ is a one-parameter group of
automorphisms for which $U$ extends to a unitary representation $\widehat{U}$
of $G\rtimes_{\alpha}{\mathbb{R}}$ for which the one-parameter group
$V_{t}=U(e,t)$ has semibounded spectrum. In view of Corollary 59, we only
expect this to happen if $i\mathbf{d}$ is semibounded.

######  Problem 63.

Determine the restrictions on the representation $(U,\mathcal{H})$ of
$G=\mathop{\rm U{}}\nolimits_{2}({\sf H})$ imposed by the semiboundedness of
$V$.

In view of Problem 421, we do not expect any conclusive restrictions without
semiboundedness of $V$.

##  Appendix A Invariant convex subsets of Hilbert–Lie algebras

######  Lemma A1.

([Ne20, Lemma A.4]) Let $(V,\|\cdot\|)$ be a normed space and $U\subseteq V$
be a proper open convex subset with complement $U^{c}=V\setminus U$. Then the
distance function

$d_{U}(x):=\mathop{{\rm dist}}\nolimits(x,U^{c})=\inf\\{\|x-y\|\colon y\in
U^{c}\\}$

is continuous, concave and $d_{U}^{-1}\colon U\to{\mathbb{R}}$ is a continuous
convex function on $U$ with

$\lim_{x_{n}\to x\in\partial U}d_{U}^{-1}(x_{n})=\infty.$

In particular, all the sets

$U_{c}:=\\{x\in U\colon d_{U}(x)\geq c\\},\quad c>0,$

are closed subsets of $V$.

######  Proposition A2.

If $G$ is a connected Hilbert–Lie group, then each non-empty open
$\mathop{{\rm Ad}}\nolimits(G)$-invariant convex subset
$U\subseteq{\mathfrak{g}}$ intersects the center.

###### Proof.

If $U={\mathfrak{g}}$, there is nothing to show because $0\in U$. We may
therefore assume that $U\not={\mathfrak{g}}$. Then $U$ is a proper open convex
subset and the group $\mathop{{\rm Ad}}\nolimits(G)$ acts isometrically on
${\mathfrak{g}}$ preserving $U$. Let $d_{U}\colon V\to{\mathbb{R}}$ denote the
distance from $\partial U$. Then Lemma A1 shows that, for every $c>0$,

$U_{c}:=\\{x\in U\colon d_{U}(x)\geq c\\}$

is a closed convex subset of ${\mathfrak{g}}$. Let $c>0$ be such that
$U_{c}\not=\emptyset$. All these sets are $\mathop{{\rm
Ad}}\nolimits(G)$-invariant. Now $U_{c}$ is a Bruhat–Tits space with respect
to the induced Hilbert metric from ${\mathfrak{g}}$ and since $\mathop{{\rm
Ad}}\nolimits(G)$ acts on this space isometrically with bounded orbits, the
Bruhat–Tits Fixed Point Theorem implies the existence of a fixed point $z\in
U_{c}\subseteq U$ ([La99]). Now it only remain to observe that
${\mathfrak{z}}({\mathfrak{g}})$ is the set of fixed points for the action of
$\mathop{{\rm Ad}}\nolimits(G)$ on ${\mathfrak{g}}$. ∎

##  Appendix B Central extensions and convexity

######  Definition B1.

(a) Let $V$ be a locally convex space. We consider the affine group
$\mathop{{\rm Aff}}\nolimits(V)\cong V\rtimes\mathop{{\rm GL}}\nolimits(V)$
which acts on $V$ by $(x,g).v=g.v+x$. On the space
$\widetilde{V}:=V\times{\mathbb{R}}$ the group $\mathop{{\rm
Aff}}\nolimits(V)$ acts by linear maps

$(x,g).(v,z):=(g.v+zx,z),$

and we thus obtain a realization of $\mathop{{\rm Aff}}\nolimits(V)$ as a
subgroup of $\mathop{{\rm GL}}\nolimits(\widetilde{V})$. The corresponding Lie
algebra is $\mathop{{\mathfrak{aff}}}\nolimits(V)\cong
V\rtimes\mathop{{\mathfrak{gl}}}\nolimits(V)$ with the bracket

$[(v,A),(v^{\prime},A^{\prime})]=(Av^{\prime}-A^{\prime}v,[A,A^{\prime}]).$

(b) A homomorphism
$\rho\colon{\mathfrak{g}}\to\mathop{{\mathfrak{aff}}}\nolimits(V)$ is
therefore given by a pair $(\rho_{l},\theta)$, consisting of a linear
representation
$\rho_{l}\colon{\mathfrak{g}}\to\mathop{{\mathfrak{gl}}}\nolimits(V)$ and map
$\theta\colon{\mathfrak{g}}\to V$ satisfying

$\theta([x,y])=\rho_{l}(x)\theta(y)-\rho_{l}(y)\theta(x)\quad\hbox{ for }\quad
x,y\in{\mathfrak{g}}.$ (42)

A linear map $\theta\colon{\mathfrak{g}}\to V$ satisfying (42) is called a
$1$-cocycle with values in the representation $(\rho_{l},V)$ of
${\mathfrak{g}}$. We write $Z^{1}({\mathfrak{g}},V)_{\rho_{l}}$ for the set of
all such cocycles.

(c) On the group level, a homomorphism $\rho\colon G\to\mathop{{\rm
Aff}}\nolimits(V)$ is given by a pair $(\rho_{l},\gamma)$ of a linear
representation $\rho_{l}\colon G\to\mathop{{\rm GL}}\nolimits(V)$ and map
$\gamma\colon G\to V$ satisfying

$\gamma(g_{1}g_{2})=\gamma(g_{1})+\rho_{l}(g_{1})\gamma(g_{2})\quad\hbox{ for
}\quad g_{1},g_{2}\in G.$ (43)

A map $\gamma\colon G\to V$ satisfying (43) is called a $1$-cocycle with
values in the representation $(\rho_{l},V)$ of $G$. We write
$Z^{1}(G,V)_{\rho_{l}}$ for the set of all such smooth cocycles. Typical
examples of $1$-cocycles are the coboundaries $\gamma(g):=\rho_{l}(g)v-v$ for
some $v\in V.$

######  Proposition B2.

Let $G$ be a connected Lie group with Lie algebra ${\mathfrak{g}}$ and
$\omega\in Z^{2}({\mathfrak{g}},{\mathbb{R}})$ be a continuous $2$-cocycle.
Then the following assertions hold

  * (a)

$\theta_{\omega}(x)(y):=\omega(x,y)$ is a $1$-cocycle on ${\mathfrak{g}}$ with
values in the coadjoint representation $(\mathop{{\rm
ad}}\nolimits^{*},{\mathfrak{g}}^{\prime})$, where $\mathop{{\rm
ad}}\nolimits^{*}(x)\beta=-\beta\circ\mathop{{\rm ad}}\nolimits x$.

  * (b)

If $G$ is simply connected and
$\widehat{\mathfrak{g}}={\mathbb{R}}\oplus_{\omega}{\mathfrak{g}}$ the central
extension defined by $\omega$, then there exists a unique smooth
representation

$\mathop{{\rm Ad}}\nolimits_{\widehat{\mathfrak{g}}}\colon G\to\mathop{{\rm
Aut}}\nolimits(\widehat{\mathfrak{g}}),\quad\mathop{{\rm
Ad}}\nolimits_{\widehat{\mathfrak{g}}}(g)(z,y)=(z+\Theta_{\omega}(g^{-1})(y),\mathop{{\rm
Ad}}\nolimits(g)y)$ (44)

whose derived representation is given by

$\mathop{{\rm
ad}}\nolimits_{\widehat{\mathfrak{g}}}(x)(z,y)=(\omega(x,y),[x,y])=[(0,x),(z,y)],$

so that $T_{e}(\Theta_{\omega})(x)=-\theta_{\omega}(x)=\omega(\cdot,x)$. The
corresponding dual representation of $G$ on
$\widehat{\mathfrak{g}}^{\prime}\cong{\mathbb{R}}\times{\mathfrak{g}}^{\prime}$
is given by

$\mathop{{\rm
Ad}}\nolimits^{*}_{\widehat{\mathfrak{g}}}(g)(z,\alpha)=(z,\mathop{{\rm
Ad}}\nolimits^{*}(g)\alpha+z\Theta_{\omega}(g)).$

  * (c)

Identifying ${\mathfrak{g}}^{\prime}$ with the hyperplane
$\\{1\\}\times{\mathfrak{g}}^{\prime}$ in $\widehat{\mathfrak{g}}^{\prime}$,
we thus obtain an affine action

$\mathop{{\rm Ad}}\nolimits^{*}_{\omega}(g)\alpha=\mathop{{\rm
Ad}}\nolimits^{*}(g)\alpha+\Theta_{\omega}(g)$

of $G$ on ${\mathfrak{g}}^{\prime}$.

###### Proof.

(a) follows from an easy calculation ([Ne04, Lemma VI.4]).

(b) The uniqueness of the affine representation follows from the general fact
that for a connected Lie group, smooth representations are uniquely determined
by their derived representations ([Ne06, Rem. II.3.7]). The existence follows
form [Ne02, Prop. 7.6] because ${\mathbb{R}}$ is complete. In loc. cit. one
finds more details on how to obtain the corresponding cocycle: For each
$x\in{\mathfrak{g}}$, let $f_{x}\in C^{\infty}(G,{\mathbb{R}})$ be the unique
function with ${\tt d}f_{x}=i_{x_{r}}\Omega$ and $f_{x}(e)=0$, where
$\Omega\in\Omega^{2}(G,{\mathbb{R}})$ is the left invariant $2$-form with
$\Omega_{e}=\omega$ and $x_{r}(g)=x.g$ is the right invariant vector field
with $x_{r}(e)=x$. Then $\Theta_{\omega}(g)(y):=f_{y}(g)$ defines a smooth
function $G\times{\mathfrak{g}}\to{\mathbb{R}}$:

$\Theta_{\omega}(g)(y)=\int_{0}^{1}\Omega_{\gamma_{g}(t)}\big{(}y_{r}(\gamma_{g}(t)),\gamma_{g}^{\prime}(t)\big{)}\,dt=\int_{0}^{1}\omega\big{(}\mathop{{\rm
Ad}}\nolimits(\gamma_{g}(t))^{-1}y,\gamma_{g}^{\prime}(t)\big{)}\,dt,$

where $\gamma_{g}$ is any piecewise smooth path in $G$ from $e$ to $g$. Note
that

$\big{(}T_{e}(\Theta_{\omega})(x)\big{)}(y)=(i_{y_{r}}\Omega)_{e}(x)=\omega(y,x)=-\theta_{\omega}(x)(y).$

(c) is an immediate consequence of (b). ∎

######  Definition B3.

We call a Banach–Lie algebra elliptic if the adjoint group $\mathop{{\rm
Inn}}\nolimits({\mathfrak{g}}):=\langle e^{\mathop{{\rm
ad}}\nolimits{\mathfrak{g}}}\rangle$ is bounded.

######  Corollary B4.

Suppose that ${\mathfrak{g}}$ is a Banach–Lie algebra and
$\widehat{\mathfrak{g}}={\mathbb{R}}\oplus_{\omega}{\mathfrak{g}}$ is the
central extension defined by $\omega\in Z^{2}({\mathfrak{g}},{\mathbb{R}})$.
Then $\widehat{\mathfrak{g}}$ is elliptic if and only if ${\mathfrak{g}}$ is
elliptic and $\Theta_{\omega}$ is bounded.

###### Proof.

From (44) is follows that the adjoint group $\langle e^{\mathop{{\rm
ad}}\nolimits\widehat{\mathfrak{g}}}\rangle=\langle e^{\mathop{{\rm
ad}}\nolimits_{\widehat{\mathfrak{g}}}{\mathfrak{g}}}\rangle$ of
$\widehat{\mathfrak{g}}$ is bounded if and only if the adjoint group of
${\mathfrak{g}}$ is bounded and $\Theta_{\omega}(G)$ is a bounded subset of
${\mathfrak{g}}^{\prime}$. ∎

The following theorem extends some techniques developed in [NZ13, §3] for
abelian Hilbert–Lie algebras (for which the double extensions are oscillator
algebras) to the case of simple Hilbert–Lie algebras.

######  Definition B5.

Let $V$ be a locally convex space. We call a subset $X\subseteq V^{\prime}$
semi-equicontinuous if its support functional

$s_{X}\colon V\to{\mathbb{R}}\cup\\{\infty\\},\quad s_{X}(v)=-\inf\langle
X,v\rangle=\sup\langle X,-v\rangle$

is bounded on some non-empty open subset of $V$.

######  Theorem B6.

Let ${\mathfrak{g}}$ be a Hilbert–Lie algebra, $G$ a corresponding simply
connected Lie group, $\omega\in Z^{2}({\mathfrak{g}},{\mathbb{R}})$ be a
continuous cocycle, and $\Theta_{\omega}\colon G\to{\mathfrak{g}}^{\prime}$
the corresponding $1$-cocycle with $T_{e}(\Theta_{\omega})x=\omega(\cdot,x)$
for $x\in{\mathfrak{g}}$. On ${\mathfrak{g}}^{\prime}$ we consider the affine
action $g*\nu:=\mathop{{\rm Ad}}\nolimits^{*}(g)\nu+\Theta_{\omega}(g)$. Then
the following are equivalent:

  * (a)

$\omega$ is a coboundary.

  * (b)

$\Theta_{\omega}$ is bounded.

  * (c)

$\Theta_{\omega}(G)$ is semi-equicontinuous.

  * (d)

One orbit of the affine action of $G$ on ${\mathfrak{g}}^{\prime}$ is semi-
equicontinuous.

  * (e)

All orbits of the affine action of $G$ on ${\mathfrak{g}}^{\prime}$ are
bounded.

###### Proof.

(a) $\Rightarrow$ (b): If $\omega$ is a coboundary, then there exists an
$\alpha\in{\mathfrak{g}}^{\prime}$ with $\omega(x,y)=\alpha([y,x])$ for
$x,y\in{\mathfrak{g}}$ and $\Theta_{\omega}=\alpha-\mathop{{\rm
Ad}}\nolimits^{*}(g)\alpha$ follows from the equality of derivatives

$(\mathop{{\rm
ad}}\nolimits^{*}x)\alpha=-\alpha([x,\cdot])=\omega(x,\cdot)=-T_{e}(\Theta_{\omega})(x).$

Since the coadjoint action preserves the norm on ${\mathfrak{g}}^{\prime}$,
the boundedness of $\Theta_{\omega}$ follows.

(b) $\Rightarrow$ (c) is trivial.

(c) $\Leftrightarrow$ (d) follows from the fact that all orbits of the
coadjoint action of $G$ on ${\mathfrak{g}}^{\prime}$ are bounded.

(b) $\Leftrightarrow$ (e) follows with the same argument.

(c) $\Rightarrow$ (b): Since $Z(G)_{0}$ acts trivially on ${\mathfrak{g}}$, we
obtain for $z\in{\mathfrak{z}}:={\mathfrak{z}}({\mathfrak{g}})$ the relation

$\Theta_{\omega}(\exp z)=\omega(\cdot,z).$

Therefore $\Theta_{\omega}(\exp{\mathfrak{z}})$ is a linear space. As it is
semi-equicontinuous it is trivial. This means that
${\omega({\mathfrak{z}},{\mathfrak{g}})=\\{0\\}}$. Accordingly,
$\Theta_{\omega}(G)\subseteq{\mathfrak{z}}^{\bot}\cong[{\mathfrak{g}},{\mathfrak{g}}]^{\prime}$.

The set $W$ of all points $x_{0}\in{\mathfrak{g}}$ for which the support
functional $s_{\Theta_{\omega}(G)}(x):=\sup\Theta_{\omega}(G)(-x)$ is bounded
in a neighborhood of $x_{0}$ is a non-empty open invariant convex cone. We
have seen in the preceding paragraph that $W+{\mathfrak{z}}=W$. Hence
$W\cap[{\mathfrak{g}},{\mathfrak{g}}]\not=\emptyset$, and therefore
$[{\mathfrak{g}},{\mathfrak{g}}]\subseteq W$ follows from Proposition A2.
Therefore $W={\mathfrak{g}}$, which implies that $\Theta_{\omega}(G)$ is
equicontinuous, i.e., that $\Theta_{\omega}$ is bounded.

(b) $\Rightarrow$ (a): If $\Theta_{\omega}$ is bounded, then the orbit of
$0\in{\mathfrak{g}}^{\prime}$ under the affine action defined by
$\Theta_{\omega}$ is bounded. Hence the Bruhat–Tits Theorem ([La99]) implies
the existence of a fixed point $\alpha$ of the corresponding affine isometric
action on ${\mathfrak{g}}^{\prime}$. This means that
$\Theta_{\omega}(g)=\alpha-\mathop{{\rm Ad}}\nolimits^{*}(g)\alpha$ for each
$g\in G$, and by taking derivatives in $e$, we see that
$\omega(y,x)=\alpha([x,y])$ is a coboundary. ∎

######  Remark B7.

Let ${\mathfrak{g}}^{\sharp}={\mathbb{R}}\oplus_{\omega}{\mathfrak{g}}$ be the
central Lie algebra extension corresponding to the cocycle $\omega\in
Z^{2}({\mathfrak{g}},{\mathbb{R}})$. Then we have a natural affine embedding
${\mathfrak{g}}^{\prime}\hookrightarrow({\mathfrak{g}}^{\sharp})^{\prime},\alpha\mapsto(1,\alpha)$,
and this embedding intertwines the coadjoint action of the group $G$ in
Theorem B6 with the affine action on ${\mathfrak{g}}^{\prime}$, specified by
$(g,\alpha)\mapsto g*\alpha$. Therefore coadjoint orbits of
${\mathfrak{g}}^{\sharp}$ not vanishing on the central element $(1,0)$
correspond to affine orbits in ${\mathfrak{g}}^{\prime}$.

######  Example B8.

Let ${\sf H}$ be a complex Hilbert space and
${\mathfrak{g}}={\mathfrak{u}}_{2}({\sf H})$.

(a) As we have seen in Subsection 4.2, each cocycle $\omega\in
Z^{2}({\mathfrak{g}},{\mathbb{R}})$ can be written as

$\omega(x,y)=\mathop{{\rm tr}}\nolimits([\mathbf{d},x]y)=\mathop{{\rm
tr}}\nolimits(\mathbf{d}[x,y])$

for some bounded skew hermitian operator $\mathbf{d}\in{\mathfrak{u}}({\sf
H})$ ([Ne03, Prop.III.19 & proof]). Then $\theta_{\omega}(x)=[\mathbf{d},x]$
if we identify ${\mathfrak{g}}$ and ${\mathfrak{g}}^{\prime}$ via the
invariant scalar product defined by $(x,y):=-\mathop{{\rm
tr}}\nolimits(xy)=\mathop{{\rm tr}}\nolimits(xy^{*})$. From that we easily
derive that $\Theta_{\omega}\colon G=\mathop{\rm U{}}\nolimits_{2}({\sf
H})\to{\mathfrak{g}}$ is given by

$\Theta_{\omega}(g)=g\mathbf{d}g^{-1}-\mathbf{d}.$

This is a cocycle for the adjoint action, which is an orthogonal
representation of $G$ on the real Hilbert space ${\mathfrak{g}}$. In view of
Theorem B6, $\Theta_{\omega}$ is bounded if and only if $\omega$ is a
coboundary, which is equivalent to
$\mathbf{d}\in{\mathbb{R}}i\mathbf{1}+{\mathfrak{u}}_{2}({\sf H})$. In
particular, the central extension defined by a non-trivial $2$-cocycle
$\omega$ is not an elliptic Banach–Lie algebra, hence in particular not
Hilbert (Corollary B4).

(b) The existence of unbounded cocycles $\theta$ for the adjoint action of
$\mathop{\rm U{}}\nolimits_{2}({\sf H})$ also implies that we obtain
interesting representations of this group by composing the Fock
representations of the affine group ${\rm Mot}({\mathfrak{u}}_{2}({\sf
H}))\cong{\mathfrak{u}}_{2}({\sf H})\rtimes\mathop{\rm
O{}}\nolimits({\mathfrak{u}}_{2}({\sf H}))$ with the homomorphism
$(\theta,\mathop{{\rm id}}\nolimits)\colon\mathop{\rm U{}}\nolimits_{2}({\sf
H})\to{\rm Mot}({\mathfrak{u}}_{2}({\sf H}))$. For details we refer to [Gu72].

######  Remark B9.

We may now ask for the convexity properties of the affine orbit ${\cal
O}_{\omega}=\Theta_{\omega}(G)$, resp., the coadjoint orbit of $(1,0)$ in
$\widehat{\mathfrak{g}}^{\prime}$. Clearly, the convexity properties of this
orbit will depend on the cocycle $\omega$ and not only on its cohomology
class. F.i. if the central extension is trivial, then the affine action on
${\mathfrak{g}}^{\prime}$ is equivalent to the coadjoint action of
${\mathfrak{g}}^{\prime}$ and different orbits have very different convexity
properties.

(a) In view of

$\langle{\cal O}_{(1,0)},(t,x)\rangle=t+\Theta_{\omega}(G)(x),\quad\mbox{ we
have }\quad B({\cal O}_{(1,0)})={\mathbb{R}}\times
B(\Theta_{\omega}(G))\subseteq\widehat{\mathfrak{g}},$

where, for a subset $C\subseteq{\mathfrak{g}}^{\prime}$, we put

$B(C):=\\{x\in{\mathfrak{g}}\colon\inf\langle C,x\rangle>-\infty\\}.$

Therefore ${\cal O}_{(1,0)}$ is (semi-)equicontinuous in
$\widehat{\mathfrak{g}}^{\prime}$ if and only if
$\Theta_{\omega}(G)\subseteq{\mathfrak{g}}^{\prime}$ has the corresponding
property.

(b) The subspace

${\cal
O}_{(1,0)}^{\bot}=\\{(t,x)\in\widehat{\mathfrak{g}}\colon\Theta_{\omega}(G)(x)=\\{-t\\}\\}$

is an invariant ideal of $\widehat{\mathfrak{g}}$ which intersects the the
central line ${\mathbb{R}}\times\\{0\\}$ trivially. Its image in
${\mathfrak{g}}$ consists of all those elements $x\in{\mathfrak{g}}$ for which
the evaluation function

$\mathop{{\rm
ev}}\nolimits_{x}\colon{\mathfrak{g}}^{\prime}\to{\mathbb{R}},\quad\mathop{{\rm
ev}}\nolimits_{x}(\alpha)=\alpha(x)$

is constant on $\Theta_{\omega}(G)$. Since $\Theta_{\omega}$ is a $1$-cocycle
with values in ${\mathfrak{g}}^{\prime}$, we have for any $g\in G$:

$\displaystyle T_{g}(\Theta_{\omega})(g\cdot y)(x)$
$\displaystyle=\big{(}\mathop{{\rm
Ad}}\nolimits^{*}(g)T_{e}(\Theta_{\omega})(y)\big{)}(x)=T_{e}(\Theta_{\omega})(y)(\mathop{{\rm
Ad}}\nolimits(g)^{-1}x)=\omega\big{(}\mathop{{\rm
Ad}}\nolimits(g)^{-1}x,y\big{)}.$

All these expressions vanish if and only if $\mathop{{\rm Ad}}\nolimits(G)x$
is contained in the radical $\mathop{{\rm
rad}}\nolimits(\omega)=\\{y\in{\mathfrak{g}}\colon i_{y}\omega=0\\}$.
Therefore the image of ${\cal O}_{(1,0)}^{\bot}$ in ${\mathfrak{g}}$ is the
largest $\mathop{{\rm Ad}}\nolimits(G)$-invariant ideal of ${\mathfrak{g}}$
contained in $\mathop{{\rm rad}}\nolimits(\omega)$. The central extension
splits on this ideal.

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