1 Introduction
By the results of Gindikin, Pijatetcki-Shapiro, and Vinberg (see [Reference Gindikin, Pyatetskii-Shapiro and Vinberg4], [Reference Pyatetskii-Shapiro8]), every bounded homogeneous domain
$\mathbf {D}$
in
${\mathbb {C}}^n$
admits a realization as a Siegel domain. Such a realization relies on the existence of a simply transitive real split solvable group S of holomorphic automorphisms of
$\mathbf {D}$
. In the symmetric case, the group
$G= Aut(\mathbf {D}) $
is semisimple and
$S=AN $
, where A and N are the abelian and the unipotent subgroups arising from an Iwasawa decomposition of G.
In [Reference Geatti and Iannuzzi3], the N-invariant Stein domains in irreducible symmetric Siegel domains were characterized. The goal of this note is to prove a similar characterization for N-invariant Stein domains in arbitrary homogeneous Siegel domains, which form a much wider class of domains containing the symmetric ones as special cases.
As in the symmetric case, to an N-invariant domain D in
$\mathbf {D}$
, we associate an r-dimensional tube domain in
${\mathbb {H}}^r$
, the product of r copies of the upper half plane
${\mathbb {H}}$
in
${\mathbb {C}}$
(here r is the rank of
$\mathbf {D}$
). Then we prove that D is Stein if and only if the base of the associated tube is convex and satisfies an additional geometric condition (see Theorem 3.4). In the symmetric case, such condition only depends on whether
$\mathbf {D}$
is of tube type or of non-tube type, while in the general case it depends on the specific root decomposition of the normal J-algebra
${\mathfrak {s}}=Lie(S)$
of
$\mathbf {D}$
.
The univalence of holomorphically separable,
$\,N$
-equivariant, Riemann domains over
$\mathbf {D}$
continues to hold true in this more general context, yielding a precise description of the envelope of holomorphy (cf. Reference Rossi[9]) of an arbitrary N-invariant domain in
$\mathbf {D}$
(see Corollary 3.5).
Finally, we exhibit an N-invariant potential of the Bergman metric of
$\mathbf {D}$
, expressed in a Lie theoretical fashion and obtained via an explicit N-moment map with respect to the Bergman Kähler structure of
$\mathbf {D}$
(see Proposition 4.3).
2 Preliminaries
Every bounded homogeneous domain
$\mathbf {D}$
in
${\mathbb {C}}^n$
admits a real split solvable group S of holomorphic automorphisms acting simply transitively on
$\mathbf {D}$
. The Lie algebra
${\mathfrak {s}}$
of S has the structure of a normal J-algebra, with the complex structure J inherited from
$\mathbf {D}$
and the linear form
$-f_0\in {\mathfrak {s}}^{*}$
inducing the Bergman metric (cf. [Reference Koszul7]). This means in particular that
${\omega (X,Y):=-f_0([X,Y])}$
is a nondegenerate skew-symmetric J-invariant bilinear form on
${\mathfrak {s}}$
and
${ \langle X,Y\rangle :=-f_0([JX,Y])}$
is a J-invariant positive definite inner product on
${\mathfrak {s}}$
.
2.1 The normal J-algebra of a bounded domain
For the structure of normal J-algebras, we mainly refer to [Reference Rossi and Vergne10, §5A]. Further details and comments can be found in [Reference Geatti and Iannuzzi3]. Denote by
${\mathfrak {n}}:=[{\mathfrak {s}},{\mathfrak {s}}]$
the nilradical of
${\mathfrak {s}}$
, and let
${\mathfrak {a}}$
be the orthogonal complement of
${\mathfrak {n}}$
in
${\mathfrak {s}}$
, with respect to the inner product
$ \langle \cdot ,\cdot \rangle $
. Then
${\mathfrak {a}}$
is an abelian subalgebra, whose dimension r is by definition the rank of D. The adjoint action of
${\mathfrak {a}}$
on
${\mathfrak {s}}$
is symmetric with respect to
$ \langle \cdot ,\cdot \rangle $
and decomposes
${\mathfrak {s}}$
into the orthogonal direct sum of root spaces
${\mathfrak {s}}^\alpha =\{X\in {\mathfrak {s}}~|~[H,X]=\alpha (H)X,~\forall H\in {\mathfrak {a}}\}$
. There exist
$e_1,\ldots ,e_r\in {\mathfrak {s}}^{*}$
such that the roots
$\alpha $
are of the form
$$ \begin{align*}e_j-e_l, ~~ \ \ e_j+e_l,\quad 1\le j<l\le r,\qquad 2e_j,~~\ e_j,\quad 1\le j \le r .\end{align*} $$
In the nonsymmetric case, not all possibilities need occur. Here, the roots are normalized so that, in the symmetric case, they coincide with the restricted roots. The complex structure J permutes the root spaces as follows:
$$ \begin{align*}J{\mathfrak{a}}=\bigoplus_j{\mathfrak{s}}^{2e_j},\qquad J{\mathfrak{s}}^{e_j-e_l}={\mathfrak{s}}^{e_j+e_l},\qquad J{\mathfrak{s}}^{e_j}={\mathfrak{s}}^{e_j}.\end{align*} $$
Let
$H_1,\ldots ,H_r$
be the basis of
${\mathfrak {a}}$
dual to
$e_1,\ldots , e_r\in {\mathfrak {a}}^{*}$
. As
$\dim {\mathfrak {s}}^{2e_j}=1 $
, for
$j=1,\ldots , r$
, one can fix generators
$E^j\in {\mathfrak {s}}^{2e_j}$
such that the pairs
$\{ H_j, \,E^j \}$
satisfy
$$ \begin{align*}\textstyle [H_j,E^l]=\delta_{jl} 2E^l, \quad JE^j=\frac{1}{2} H_j,\qquad \text{for } j,l=1,\ldots, r.\end{align*} $$
For
$j=1,\ldots , r$
, the real split solvable subalgebras generated by
$\{ H_j, \,E^j \}$
pairwise commute and are isomorphic to the
${\mathfrak {a}} \oplus {\mathfrak {n}}$
-component of an Iwasawa decomposition of
${\mathfrak {s}}{\mathfrak {l}}(2,{\mathbb {R}})$
.
Set
$\,H_0:=\frac {1}{2}\sum _j H_j\in {\mathfrak {a}}\,$
. The adjoint action of
$H_0$
decomposes
${\mathfrak {s}}$
and
${\mathfrak {n}}$
as
$$ \begin{align*}{\mathfrak{s}}={\mathfrak{s}}_{0}\oplus {\mathfrak{s}}_{1/2}\oplus {\mathfrak{s}}_1,\qquad {\mathfrak{n}}={\mathfrak{n}}_{0}\oplus {\mathfrak{n}}_{1/2}\oplus {\mathfrak{n}}_1\,,\end{align*} $$
where
$\,{\mathfrak {n}}_j={\mathfrak {n}}\cap {\mathfrak {s}}_j\,$
and
$$ \begin{align*}{\mathfrak{s}}_0={\mathfrak{a}}\oplus\bigoplus_{\scriptstyle1\le j<l\le r} {\mathfrak{s}}^{e_j-e_l},\quad {\mathfrak{s}}_{1/2}=\bigoplus_{\scriptstyle 1\le j\le r}{\mathfrak{s}}^{e_j}, \quad {\mathfrak{s}}_1=\bigoplus_{\scriptstyle 1\le j\le r}{\mathfrak{s}}^{2e_j}\oplus\bigoplus_{\scriptstyle 1\le j<l\le r}{\mathfrak{s}}^{e_j+e_l}. \end{align*} $$
If
${\mathfrak {s}}_{1/2}=\{0\}$
, then the domain
$\mathbf {D}$
is of tube type, otherwise it is of non-tube type.
Set
$E_0:= \sum_j E^j$
. The complex structure on
$ {\mathfrak {s}}_0$
is given by
$JX=[E_0,X]$
, for all
$X\in {\mathfrak {s}}_0$
. The orbit
$$ \begin{align*}V:=Ad_{\exp{\mathfrak{s}}_0}E_0\end{align*} $$
is a sharp convex cone in
${\mathfrak {s}}_1$
and
$$ \begin{align*}\textstyle F\colon {\mathfrak{s}}_{1/2}\times{\mathfrak{s}}_{1/2}\to {\mathfrak{s}}_1^{\mathbb{C}},\qquad F(W,W'):=\frac{1}{4}([JW',W]-i[W',W]),\end{align*} $$
is a
$ V$
-valued Hermitian form, that is, it is sesquilinear and
$F(W,W)\in \overline V$
(the topological closure of V), for all
$W \in {\mathfrak {s}}_{1/2} $
. The group S acts on
${\mathfrak {s}}_1^{\mathbb {C}}\oplus {\mathfrak {s}}_{1/2}$
by affine transformations, given by
$$ \begin{align} s\cdot(Z,W)= (Ad_{\exp \gamma} Z+\xi+2iF(Ad_{\exp \gamma}W,\zeta)+iF(\zeta,\zeta),Ad_{\exp \gamma}W+\zeta), \end{align} $$
where
$\, s=\exp \zeta \exp \xi \, \exp \gamma $
, with
$\, \zeta \in {\mathfrak {s}}_{1/2} $
,
$\xi \in {\mathfrak {s}}_1$
,
$\,\gamma \in {\mathfrak {s}}_0$
. If we fix the base point
${p_0:=(iE_0,0)\in {\mathfrak {s}}_1^{\mathbb {C}}\oplus {\mathfrak {s}}_{1/2}}$
, then the map
$$ \begin{align} \mathcal L\,\colon \, S \mapsto D(V,F),\qquad s \mapsto s\cdot p_0\, \end{align} $$
defines a biholomorphism between
$\mathbf {D}\cong S$
and the Siegel domain
$$ \begin{align*} D(V,F)=\{(Z,W)\in {\mathfrak{s}}_1^{\mathbb{C}}\oplus {\mathfrak{s}}_{1/2}~|~Im(Z)-F(W,W)\in V\}\end{align*} $$
(cf. [Reference Rossi and Vergne10, Lem. 5.2, p. 330]). Denote by
$$ \begin{align*}(E^1)^{*}, \ldots,(E^r)^{*}\end{align*} $$
the elements in the dual
${\mathfrak {n}}^{*}$
of
${\mathfrak {n}}$
, with the property that
$(E^j)^{*}(E^l)=\delta _{jl}$
and
$(E^j)^{*}(X)=0$
, for all
$X \in {\mathfrak {s}}^\alpha $
, with
$\alpha \notin \{2e_1,\ldots ,2e_r\}$
.
Lemma 2.1.
$\mathrm{(a)}$
The form
$\,-f_0\colon {\mathfrak {s}}\to {\mathbb {R}}\,$
is given by
$\,-f_0=\sum _kc_k(E^k)^{*} $
, for some
${c_k\in {\mathbb {R}}^{>0}}$
.
$\mathrm{(b)}$
Let
$\,X \in {\mathfrak {s}}^{e_j-e_l}\setminus \{0\}\,$
. Then
$\,[JX,X] =s E^j $
, for some
$\,s\in {\mathbb {R}}^{>0}$
. Let
$X,Y\in {\mathfrak {s}}^{e_j-e_l}\setminus \{0\}\,$
, satisfying
$\langle X,Y\rangle =0$
. Then
$[JX,Y]=0$
.
$\mathrm{(c)}$
Let
$\,X \in {\mathfrak {s}}^{e_j}\setminus \{0\}.$
Then
$\,[JX,X] =tE^j$
, for some
$\,t\in {\mathbb {R}}^{>0}$
. Let
$X,Y\in {\mathfrak {s}}^{e_j}\setminus \{0\}\,$
, satisfying
$\langle X,Y\rangle =0$
. Then
$[JX,Y]=0$
.
Proof. The proof of statement (a) is contained in [Reference Rossi and Vergne10]. For the sake of completeness, we recall the main arguments. Let
$f_0$
also denote the
$\,{\mathbb {C}}$
-linear extension of
$\,f_0$
to
${\mathfrak {s}}^{\mathbb {C}}$
. From the integrability of J, one has that
$f_0([X+iJX,Y+iJY])=0$
, for all
$X,Y\in {\mathfrak {s}}$
. This implies that
$f_0([H,X])=f_0(J[H,X])=0$
, for all
$H\in {\mathfrak {a}}$
and
$X\in {\mathfrak {q}}:={\mathfrak {s}}_{1/2}\oplus \bigoplus _{j<l}{\mathfrak {s}}^{e_j-e_l}$
. Since
$[{\mathfrak {a}},{\mathfrak {q}}]={\mathfrak {q}}$
and
$J{\mathfrak {q}}={\mathfrak {s}}_{1/2}\oplus \bigoplus _{j<l}{\mathfrak {s}}^{e_j+e_l}$
, the form
$f_0$
identically vanishes on
${\mathfrak {q}}$
and
$-f_0=\sum _jc_j (E^j)^{*}$
, for some
$c_j\in {\mathbb {R}}$
. The identity
$c_j=-f_0(E^j)=-\frac {1}{2}f_0([H_j,E^j])=-f_0([JE^j,E^j])= \langle E^j,E^j\rangle>0 $
concludes the proof.
(b) Let
$X\in {\mathfrak {s}}^{e_j-e_l} \setminus \{0\}$
. Then
$\,JX=[E^l,X]\in {\mathfrak {s}}^{e_j+e_l}$
. Since
${\mathfrak {s}}^{2e_j}$
is one-dimensional,
$ [JX,X]=sE^j$
, for some
$s\in {\mathbb {R}}$
. By applying
$-f_0$
to both terms, one obtains
$-f_0([JX,X])=\langle X,X\rangle =c_j s>0$
. Since
$c_j>0$
, also
$s>0$
. For the second part of the statement, write
$[JX,Y]=s E^j$
, for some
$s\in {\mathbb {R}}$
. Then, from
$$ \begin{align*} 0= \langle X,Y\rangle =-f_0([JX,Y])= -\sum_kc_k(E^k)^{*}( s E^j) = c_js , \end{align*} $$
one obtains
$s=0$
and therefore
$[JX,Y]=0$
, as desired.
As
${\mathfrak {s}}^{e_j}$
is J-invariant, statement (c) follows in a similar way.
Remark 2.2. The forms
$\sum _jc_j(E^j)^{*}$
, where the
$c_j$
’s vary in
${\mathbb {R}}^{>0}$
for
$j=1,\ldots ,r$
, determine all S-homogeneous Kähler metrics on
$D(V,F)$
(cf. [Reference Dorfmeister2, Th. 1, p. 304]). By [Reference D’Atri1, Th. 4], one such metric is Kähler–Einstein if and only if the quantity
$\textstyle \frac {1}{c_j}(1+\frac {1}{4} \dim {\mathfrak {s}}^{e_j}+\frac {1}{2}\sum _{j<l}{\mathfrak {s}}^{e_j+e_l})$
is a constant independent of
$j=1,\ldots ,r$
.
2.2
N-invariant domains in
$D(V,F)$
and tube domains in
${\mathbb {H}}^r$
In
$S=NA$
, consider the unipotent abelian subgroup
$\,R:= \exp J{\mathfrak {a}}$
, isomorphic to
${\mathbb {R}}^r$
. The
$\,R$
-invariant set
$$ \begin{align*} \,R \exp ({\mathfrak{a}}) \cdot p_0 \end{align*} $$
is an r-dimensional closed complex submanifold of
$\,D(V,F),\,$
intersecting all N-orbits in
$D(V,F)$
. Define the positive octant in
$J{\mathfrak {a}}$
$$ \begin{align*} \textstyle J{\mathfrak{a}}^+:=\{\sum y_kE^k~:~y_k>0, \ \mathrm{for} \ k=1,\ldots,r\}. \end{align*} $$
Then the map
$\mathcal L$
defined in (2.1) and (2.2) restricts to a biholomorphism
$$ \begin{align*} \,R \exp ({\mathfrak{a}}) \to J{\mathfrak{a}}\oplus iJ{\mathfrak{a}}^+, \end{align*} $$
given by
$$ \begin{align} \textstyle \exp(\sum_je_jE^j) \exp( \sum_k h_kH_k) \mapsto \sum_je_jE^j+ i Ad_{\exp(\sum_k h_kH_k)}E_0. \end{align} $$
In particular,
$\mathcal L|_{\exp ({\mathfrak {a}})}$
defines a diffeomorphism
$\ L\colon {\mathfrak {a}}\to J{\mathfrak {a}}^+ $
given by
$$ \begin{align} \sum_kh_kH_k\mapsto Ad_{\exp(\sum_kh_kH_k)}E_0=\sum_je^{2h_j}E^j. \end{align} $$
Write an N-invariant domain in a rank-r homogeneous Siegel domain
$D(V,F)$
as
${D=N\exp {\mathcal D}\cdot p_0}$
, for some domain
${\mathcal D}\subset {\mathfrak {a}}$
. Then, as in the symmetric case (see [Reference Geatti and Iannuzzi3, §3]), one can associate with D an r-dimensional tube domain.
Definition 2.3. The r-dimensional tube domain associated with an N-invariant domain D in
$D(V,F)$
is the image of the set
$\,R\exp (\mathcal D) \,$
under
$\mathcal L$
, namely,
$$ \begin{align*} D\cap (J{\mathfrak{a}}\oplus iJ{\mathfrak{a}}^+)=J{\mathfrak{a}} +i \Omega ,\quad \text{ where } \,\Omega:= L(\mathcal D). \end{align*} $$
3 N-invariant Stein domains in a homogeneous Siegel domain
Let
$D(V,F)$
be a homogeneous bounded domain. In this section, we give a characterization of the N-invariant Stein domains D in
$D( V,F)$
in terms of the associated tube domain. If D is Stein, then such tube domain is Stein and its base
$\Omega $
is an open convex set in
$J{\mathfrak {a}}^+$
. On the other hand, we will see that
$\Omega $
must satisfy some further geometric conditions which depend on the specific root decomposition of the normal J-algebra of
$D(V,F)$
.
Let D be an N-invariant domain in
$D(V,F)$
. Then
$$ \begin{align*} D=\{(Z,W)\in D( V,F)~|~ Im(Z)-F(W,W)\in \boldsymbol{\Omega} \}, \end{align*} $$
where
$ \boldsymbol {\Omega }$
is the
$Ad_{\exp {\mathfrak {n}}_0}$
-invariant open subset in
$ V$
determined by
$$ \begin{align*} i \boldsymbol{\Omega}:=D\cap i V. \end{align*} $$
By (2.2)–(2.4), the base of the associated tube is
$$ \begin{align*} \Omega = \boldsymbol{\Omega}\cap J{\mathfrak{a}}^+. \end{align*} $$
Note that, since
$Ad_AE_0=J{\mathfrak {a}}^+$
, the set
$i\Omega $
is a slice both for the
$Ad_{\exp {\mathfrak {n}}_0}$
-action on
$i\boldsymbol {\Omega }$
and for the
$ N$
-action on D.
For
$D(V,F)$
irreducible, define a cone in
$J{\mathfrak {a}}^+ $
as follows:
$$ \begin{align*} \textstyle C:=\begin{cases} {\mathcal C}_t, \text{in the tube case,}\\ {\mathcal C}_{nt}, \text{in the non-tube case,} \end{cases} \end{align*} $$
where
$ {\mathcal C}_t:=cone\{E^j\}_j $
, with
$j\in \{1,\ldots ,r-1\}$
such that
${\mathfrak {s}}^{e_j-e_l}\not =\{0\}$
for some
$l>j$
, and
$ {\mathcal C}_{nt}=cone\{E^j\}_j$
, with
$j\in \{1,\ldots ,r\}$
such that either
${\mathfrak {s}}^{e_j-e_l}\not =\{0\}$
for some
$l>j$
, or
${\mathfrak {s}}^{e_j}\not =\{0\}$
. (Here, given nonzero vectors
$\mathbf {v}_1,\ldots ,\mathbf {v}_k $
, we set
$cone\{\mathbf {v}_1,\ldots ,\mathbf {v}_k\}:=\{\sum _jt_j\mathbf {v}_j,~t_j> 0 \} $
.)
In the reducible case, if
$D(V,F)$
decomposes in the product of irreducible domains as
$D(V^{(1)},F^{(1)})\times \cdots \times D(V^{(m)},F^{(m)})$
, then the normal J-algebra
${\mathfrak {s}}$
and all its related objects decompose accordingly. In particular, the cone decomposes as
$C=C^{(1)}\times \cdots \times C^{(m)}$
, where
$C^{(i)}$
is the cone associated with the i-th irreducible component of
$D(V,F)$
.
Example 3.1. (a) If
$D(V,F)$
is irreducible symmetric, then
${\mathcal C}_t=cone\{E^1,\ldots ,E^{r-1}\}$
and
${\mathcal C}_{nt}=cone\{E^1,\ldots ,E^{r}\}$
(see (9) in [Reference Geatti and Iannuzzi3]).
(b) Let
$D(V)$
be the tube domain over the five-dimensional Vinberg cone
$$ \begin{align*} D(V)=\left\{\begin{pmatrix}z_{11}&0&z_{13}\\ 0&z_{22}&z_{23}\\ z_{13}& z_{23}&z_{33}\end{pmatrix}~|~z_{ij}=x_{ij}+iy_{ij}\in{\mathbb{C}}, ~\begin{cases} y_{11} y_{33}-y_{13}^2>0 \\ y_{22} y_{33}-y_{23}^2>0\end{cases} \mathrm{and}\quad y_{33}>0\right\}. \end{align*} $$
Then
$$ \begin{align*} {\mathfrak{s}}={\mathfrak{a}}\oplus J{\mathfrak{a}}\oplus {\mathfrak{s}}^{e_1\pm e_3}\oplus {\mathfrak{s}}^{e_2\pm e_3},\qquad \dim{\mathfrak{a}}=3,~\dim {\mathfrak{s}}^{e_j\pm e_l}=1 \end{align*} $$
and
${\mathcal C}_t=cone\{E^1,E^2\}$
.
(b) Let
$D(V,F)$
be the four-dimensional nonsymmetric domain
$$ \begin{align*}D(V,F)=\left\{\left(\begin{pmatrix}z_{11}&z_{12}\\ z_{12}&z_{22}\end{pmatrix},w\right)~|~z_{ij}=x_{ij}+iy_{ij},~w\in{\mathbb{C}}, ~\begin{cases} (y_{11}-|w|^2)y_{22}-y_{12}^2>0\\ y_{22} >0\end{cases}\!\!\!\!\!\!\!\!\!\right\}.\end{align*} $$
Then
$$ \begin{align*}{\mathfrak{s}}={\mathfrak{a}}\oplus J{\mathfrak{a}}\oplus {\mathfrak{s}}^{e_1\pm e_2}\oplus {\mathfrak{s}}^{e_1},\qquad \dim{\mathfrak{a}}=2,~\dim s^{e_1\pm e_2}=1,~\dim {\mathfrak{s}}^{e_1}=2 \end{align*} $$
and
${\mathcal C}_{nt}=cone\{E^1\}$
.
Definition 3.2. A domain
$\,\Omega \subset J{\mathfrak {a}}^+\,$
is C-invariant if
$\,E\in \Omega \, $
implies
$\,E+C \subset \Omega $
or, equivalently, if
$\,E\in \Omega \, $
implies
$\,E+\overline C \subset \Omega $
.
Denote by
$$ \begin{align*}p\colon i{\mathfrak{s}}_1\to iJ{\mathfrak{a}}\end{align*} $$
the projection onto
$iJ{\mathfrak {a}}$
, parallel to
$i( \oplus \, {\mathfrak {s}}^{e_j+e_l})$
and by
$$ \begin{align*}\, \widetilde p\colon {\mathfrak{s}}_1^{\mathbb{C}}\oplus{\mathfrak{s}}_{1/2} \to iJ{\mathfrak{a}}\,\end{align*} $$
the projection onto
$\, iJ{\mathfrak {a}}$
parallel to
${\mathfrak {s}}_1\oplus i(\oplus \,{\mathfrak {s}}^{e_j+e_l})\oplus {\mathfrak {s}}_{1/2}$
.
For simplicity, the next lemma is formulated in the irreducible case. In the reducible case, it applies to each irreducible component.
Lemma 3.3. The following statements hold true.
-
(i) Assume that
${\mathfrak {s}}^{e_j-e_l}\not =\{0\}$
, for some
$l>j$
, and let
$X\in {\mathfrak {s}}^{e_j-e_l}$
be a nonzero element. Then
$[[E^l,X],X]=s E^j$
, for some
$s\in {\mathbb {R}}^{> 0}$
. -
(ii) Let
$E=\sum y_kE^k \in J{\mathfrak {a}}^+$
. Then
$\,p(iAd_{\exp {\mathfrak {n}}_0} E)=i(E+\overline {\mathcal C}_{t}).$
-
(iii) Let
$E\in J{\mathfrak {a}}^+$
. Then
$\widetilde p(N\cdot (iE,0))=i(E+ \overline {\mathcal C}_{nt})$
.
Proof. (i) Since
$[[E^l,X],X]=[JX,X]$
, then the statement follows from Lemma 2.1(b).
(ii) Fix
$1\le j\le r-1$
and define
${\mathcal L}_j:=\oplus _{l>j}{\mathfrak {s}}^{e_j-e_l}$
. In each root space
${\mathfrak {s}}^{e_j-e_l}\not =\{0\}$
in
$ {\mathcal L}_j$
, there exists an orthogonal basis
$\{E^p_{jl}\}_p$
such that for
$X=\sum _{ l>j,\, p}x^p_{jl}E^p_{jl}\in {\mathcal L}_j$
, one has
$$ \begin{align*} \textstyle (E^j)^{*}(Ad_{\exp X} E) = y_j (1+\sum_{l>j,\, p} (x_{jl}^p)^2)\qquad\text{and}\qquad (E^r)^{*}(Ad_{\exp X} E) =y_r \end{align*} $$
(cf. Lemma 2.1(b) and (c)). Moreover, from a discussion similar to the one in [Reference Rossi and Vergne10, p. 363], one obtains
$$ \begin{align*} (E^j)^{*}(Ad_{\exp {\mathfrak{n}}_0}E)= (E^j)^{*}Ad_{\exp {\mathcal L}_j}E \end{align*} $$
[Reference Rossi and Vergne10, Th. 4.10 and (4.13)]. Hence,
$p( iAd_{\exp {{\mathfrak {n}}_0}} E)= i(E+\overline {\mathcal C}_{t})$
, as claimed.
(iii) The N-orbit of the point
$(iE,0)\in {\mathfrak {s}}_1^{\mathbb {C}}\oplus {\mathfrak {s}}_{1/2}$
is given by
$$ \begin{align} \{(\xi+i(Ad_{\exp{\mathfrak{n}}_0}E+F(\zeta,\zeta)),\zeta)~:~\xi\in{\mathfrak{s}}_1,~\zeta\in{\mathfrak{s}}_{1/2} \}. \end{align} $$
By (3.1) and Lemma 3.3(ii), one has
$$ \begin{align*}\widetilde p(N\cdot (iE,0))=i(E+\overline{\mathcal C}_{t})+ \{\tilde p(iF(\zeta,\zeta)) \ : \ \zeta\in{\mathfrak{s}}_{1/2}\}\,.\end{align*} $$
If
$ {\mathfrak {s}}^{e_j}\not =\{0\}$
and
$\zeta \not = 0$
in
$ {\mathfrak {s}}^{e_j} $
, then by Lemma 2.1(c), the element
$\,F(\zeta ,\zeta ) =\frac {1}{4} [J\zeta ,\zeta ] \,$
is a positive multiple of
$E^j$
. Therefore,
$\tilde p(N\cdot (iE,0))=i(E+\overline {\mathcal C}_{nt})$
, as claimed.
Theorem 3.4. Let
$\,D(V,F)\,$
be a homogeneous Siegel domain of rank r. Let D be an
$\,N$
-invariant domain in
$\,D(V,F)\,$
, and let
$ \Omega $
be the base of the associated tube domain. Then
$\,D\,$
is Stein if and only if
$\,\Omega \,$
is convex and C-invariant.
Proof of Theorem 3.4
We first prove that D Stein implies
$\Omega $
convex and C-invariant. Then we show that
$\Omega $
convex and C-invariant implies D convex and therefore Stein (cf. [Reference Gunning5, Vol. 1, Th. 10, p. 67]). In particular, if
$D $
is Stein, then it is necessarily convex. An essential fact is that the N-action on D is affine and every affine map commutes with taking convex hulls.
3.1
We begin with the tube case. An N-invariant domain D in a homogeneous tube domain
$D(V)$
is itself a tube domain with base the
$Ad_{\exp {\mathfrak {n}}_0}$
-invariant set
$ \boldsymbol {\Omega }$
. Since D is Stein if and only if its base is convex, all we have to show is that
$\Omega $
convex and
${\mathcal C}_t$
-invariant is equivalent to
$ \boldsymbol {\Omega }$
being convex.
If
$ \boldsymbol {\Omega }$
is convex, then
$\Omega $
is clearly convex. In order to prove that
$\Omega $
is
${\mathcal C}_t$
-invariant, let
$E=\sum _ky_kE^k\in \Omega $
, with
$y_k>0$
, for
$k=1,\ldots ,r$
. If the root space
${\mathfrak {s}}^{e_j-e_l} \not =\{0\}$
, let
$X $
be a nonzero element therein. Since
$ad_X$
is two-step nilpotent, for every
$t\in {\mathbb {R}}$
,
$$ \begin{align*}\textstyle Ad_{\exp tX}E=E+ty_l [X,E^l] +\frac{1}{2}t^2 y_l[X,[X,E^l]]\end{align*} $$
is an element of
$ \boldsymbol {\Omega }$
. As
$ \boldsymbol {\Omega }$
is convex, by replacing t with
$-t$
, one finds that also the midpoint
$E+ \frac {1}{2} t^2y_l [X,[X,E^l]]$
lies in
$\boldsymbol {\Omega }$
. This says that
$E+\lambda E^j $
lies in
$\boldsymbol {\Omega }$
, for all
$\lambda \ge 0$
. The same argument applied to all
$j\in \{1,\ldots ,r-1\}$
for which
${\mathfrak {s}}^{e_j-e_l} \not =\{0\}$
, for some
$l>j$
, and the convexity of
$\Omega $
imply that
$\Omega +{\mathcal C}_{t}\subset \Omega $
, as desired.
Conversely, assume that
$\Omega $
is convex and
${\mathcal C}_t$
-invariant. We are going to prove that
$conv( \boldsymbol {\Omega })\subset \boldsymbol {\Omega }$
. Since
$\, \boldsymbol {\Omega } =Ad_{\exp {\mathfrak {n}}_0}\Omega $
, from Lemma 3.3(ii) and the
${\mathcal C}_t$
-invariance of
$\Omega $
, one has
$$ \begin{align*} \textstyle p(i \boldsymbol{\Omega})=p(iAd_{\exp{\mathfrak{n}}_0}\Omega)=i(\Omega+\overline{\mathcal C}_{t}) \subset i\Omega.\end{align*} $$
From the above inclusion and the convexity of
$\Omega $
, one has
$$ \begin{align*}\textstyle conv( i\boldsymbol{\Omega})\cap iJ{\mathfrak{a}}\subset p( conv( i\boldsymbol{\Omega}))= conv(p( i\boldsymbol{\Omega}))\subset i\Omega. \end{align*} $$
Finally, from the
$Ad_{\exp {\mathfrak {n}}_0}$
-invariance of
$conv( i\boldsymbol {\Omega })$
, it follows that
$$ \begin{align*} \textstyle conv(i \boldsymbol{\Omega})=Ad_{\exp{\mathfrak{n}}_0} (conv( i\boldsymbol{\Omega})\cap iJ{\mathfrak{a}})\subset Ad_{\exp{\mathfrak{n}}_0} i\Omega= i\boldsymbol{\Omega}. \end{align*} $$
This completes the proof of the theorem in the tube case.
3.2
Next we deal with the non-tube case. Let D be an N-invariant domain in a homogeneous Siegel domain
$D( V,F)$
. Denote by
$conv(D)$
the convex hull of D in
${\mathfrak {s}}_1^{\mathbb {C}}\oplus {\mathfrak {s}}_{1/2}$
, which is N-invariant as well.
If D is Stein, then
$D\cap ({\mathfrak {s}}_1^{\mathbb {C}}\times \{0\})=\{(Z,0)\in {\mathfrak {s}}_1^{\mathbb {C}}\oplus {\mathfrak {s}}_{1/2}~| Im(Z)\in \boldsymbol {\Omega }\}$
is biholomorphic to a Stein tube domain in
${\mathfrak {s}}_1^{\mathbb {C}}$
, invariant under
$\exp ({\mathfrak {n}}_0\oplus {\mathfrak {n}}_1)$
. Hence, by Theorem 3.4 in the tube case, the set
$\Omega $
is convex and
$\Omega +\overline {\mathcal C}_{t}\subset \Omega $
. The fact that
$\Omega +\overline {\mathcal C}_{nt}\subset \Omega $
follows from (3.1) and the fact that
$F(\zeta ,\zeta )$
is an arbitrary positive multiple of
$E^j$
, when
$\zeta $
varies in
${\mathfrak {s}}^{e_j}\setminus \{0\}$
.
Conversely, assume that
$\Omega $
is convex and
${\mathcal C}_{nt} $
-invariant. By Lemma 3.3(iii), one has
$$ \begin{align*} \widetilde p(D)=\widetilde p(N \cdot i\Omega)=i(\Omega+\overline {\mathcal C}_{nt})\subset i\Omega. \end{align*} $$
Moreover,
$$ \begin{align*} conv(D)\cap iJ{\mathfrak{a}}\subset \widetilde p(conv(D))=conv(\widetilde p(D))\subset i\Omega. \end{align*} $$
By the N-invariance of
$conv(D)$
, one obtains
$$ \begin{align*} conv(D)=N\cdot (conv(D)\cap i J{\mathfrak{a}})\subset N\cdot i\Omega=D. \end{align*} $$
Hence, D is convex and therefore Stein (cf. [Reference Gunning5, Vol. 1, Th. 10, p. 67]). This concludes the proof of the theorem.
We conclude this section by observing that holomorphically separable,
$\,N$
-equivariant, Riemann domains over a bounded homogeneous domain
$\mathbf {D}$
are univalent: the same proof as in the symmetric case works in this more general case (see [Reference Geatti and Iannuzzi3, Prop. 3.7]). As a consequence, one obtains the following corollary.
Corollary 3.5. The envelope of holomorphy
$\widehat D$
of an
$\,N$
-invariant domain
$\,D\,$
in
$\,\mathbf {D}\,$
is the smallest Stein domain in
$\,\mathbf {D}\,$
containing D. Namely,
$\widehat D$
is the N-invariant domain such that the base
$\widehat \Omega $
of the associated tube is the convex C-invariant hull of
$\,\Omega $
.
4 An N-invariant potential of the Bergman metric
Let
$D(V,F)$
be a homogeneous Siegel domain, and let
$({\mathfrak {s}},J,-f_0)$
be the associated normal J-algebra, where
$-f_0\in {\mathfrak {s}}^{*}$
is the form inducing the Bergman metric g on
$D(V,F)$
. In this section, we exhibit an N-invariant potential of g, expressed in a Lie theoretical fashion. In order to do this, we determine an explicit formula for the N-moment map associated with g.
For
$X\in {\mathfrak {s}}$
, denote by
$\widetilde X$
the vector field on
$D(V,F)$
induced by the left S-action. Its value at
$z=s\cdot p_0 $
is given by
$\widetilde X_z=\frac {d}{dt}{\big |_{t=0}} \exp tX\cdot z$
. If
$z=a\cdot p_0$
, with
$a=\exp H$
and
$H\in {\mathfrak {a}}$
, and
$X\in {\mathfrak {s}}^\alpha $
, then
$\widetilde X_z=e^{-\alpha (H)}a_* X$
.
Lemma 4.1.
(a) The map
$\mu _S\colon D(V,F)\to {\mathfrak {s}}^{*},$
defined by
$$ \begin{align*} \mu_S(z)(X):=-f_0(Ad_{s^{-1}} X),\qquad z=s\cdot p_0,~X\in{\mathfrak{s}}, \end{align*} $$
is an S-moment map with respect to g.
(b) The map
$\mu _N\colon D(V,F)\to {\mathfrak {n}}^{*},$
defined by
$$ \begin{align*} \mu_N(z)(X):=-(Ad_a^{*}f_0)(Ad_{n^{-1}} X),\qquad z=na\cdot p_0,~X\in{\mathfrak{n}}, \end{align*} $$
is an N-moment map with respect to g.
Proof. (a) By definition, the map
$\mu _S$
is S-equivariant and satisfies
$\mu _S(p_0)=-f_0$
.
We identify
$D(V,F)$
with the group S by the map (2.1), and prove that
$$ \begin{align} d\mu^X_S (s)(Z) =\omega_s(\widetilde X_s, Z),\qquad Z\in T_sS, ~X\in{\mathfrak{s}}. \end{align} $$
Let
$W\in T_eS\cong {\mathfrak {s}}$
. Then
$$ \begin{align*} \textstyle d\mu^X_S (W)=\frac{d}{dt}{\big |_{t=0}} \mu_S^X(\exp tW )=\frac{d}{dt}{\big |_{t=0}} -f_0 (Ad_{ \exp(-tW)} X)=-f_0(\frac{d}{dt}{\big |_{t=0}} e^{ad_{-tW}}X) \end{align*} $$
$$ \begin{align*} \textstyle=-f_0(-[W,X])=-f_0([X,W])=\omega(X,W). \end{align*} $$
Now take
$s\in S$
and let
$s_*W\in T_{s}S\cong s_*{\mathfrak {s}}$
. On the left-hand side of (4.1), we find
$$ \begin{align*}\textstyle (d\mu^X_S)(s_*W)= \frac{d}{dt}{\big |_{t=0}} \, \mu^X_S (s \exp tW) =\frac{d}{dt}{\big |_{t=0}} -f_0(Ad_{\exp -tW} Ad_{s^{-1}} X) \end{align*} $$
$$ \begin{align*}=-f_0(-[W,Ad_{s^{-1}} X])=-f_0([ Ad_{s^{-1}} X,W]). \end{align*} $$
Since also the right-hand side of (4.1) is given by
$$ \begin{align*} \omega_s\Bigg(\frac{d}{dt}{\big |_{t=0}} \exp tX\cdot s,s_*W\Bigg)=\omega_s(s_* Ad_{s^{-1}}X,s_*W)=\omega(Ad_{s^{-1}}X,W)=-f_0([ Ad_{s^{-1}} X,W]), \end{align*} $$
the proof of (a) is complete.
(b) The restriction of
$\mu _S$
to
${\mathfrak {n}}$
defines an N-moment map
$\mu _N$
on
$D(V,F)$
. Since
$\mu _N$
is N-equivariant, it is uniquely determined by
$\mu _N(a\cdot p_0)(X) =-Ad_a^{*}f_0( X)$
, for
$X\in {\mathfrak {n}}$
. It follows that
$\mu _N(z)(X)=-(Ad_a^{*}f_0)(Ad_{n^{-1}} X)$
, is an
$N$
-invariant moment map with respect to
$g$
as claimed.
The moment map
$\,\mu _S\,$
defined in Lemma 4.1 is an embedding of
$\,D(V,F)\,$
in
$\,{\mathfrak {s}}^{*}$
as the coadjoint orbit of
$\,-f_0\,$
, with trivial isotropy subgroup. The image
$\,\mu _S(D(V,F))\,$
is the convex domain
$\,{\mathfrak {s}}_0^{*}+{\mathfrak {s}}_{1/2}^{*}+V^{*}\,$
in
$\,{\mathfrak {s}}^{*}$
, where
$V^{*}:=\{\phi \in {\mathfrak {s}}_1^{*}~|~\phi (X)>0, ~\forall X\in \overline V\setminus \{0\}\} $
is the dual cone of V in
${\mathfrak {s}}_1$
(cf. [Reference Rossi and Vergne10, Lem. 3.5, p. 350]). Similarly, the image
$\,\mu _N(D(V,F))\,$
is the convex domain
$\,{\mathfrak {n}}_0^{*}+{\mathfrak {s}}_{1/2}^{*}+V^{*}\,$
in
$\,{\mathfrak {n}}^{*}$
.
Convexity properties of the moment map have been studied in several settings (see [Reference Hilgert, Neeb and Plank6] and the references therein). Here, we show that the image under
$\,\mu _N\,$
of a Stein
$\,N$
-invariant domain in
$D(V,F)$
is not necessarily convex.
Let
$D(V,F)$
be a homogeneous Siegel domain, and let
$D=N\cdot i\Omega $
be an N-invariant Stein domain therein. One has
$$ \begin{align*}(J{\mathfrak{a}})^{*} \cap \mu_N(D(V,F))= (J{\mathfrak{a}})^{*} \cap V^{*} = (J{\mathfrak{a}}^+)^{*},\end{align*} $$
and one can easily verify that
$\mu _N$
maps
$A\cdot p_0=iJ{\mathfrak {a}}^+$
bijectively onto
$(J{\mathfrak {a}}^+)^{*}$
.
Therefore,
$\, \mu _N(i\Omega )=(J{\mathfrak {a}})^{*}\cap \mu _N(D)$
. Consequently, if
$\mu _N(i\Omega )$
is not convex, then
$\,\mu _N(D)$
is not convex either.
Example 4.2. Let
${\mathcal P}=\left \{ Z=X+iY~|~Z^t=Z,\, Y\gg 0\right \}$
be the Siegel upper half-plane of rank 2. Then
$$ \begin{align*} A\cdot p_0=\left\{ i\begin{pmatrix}y_1&0\\ 0&y_2\end{pmatrix}~|~y_1,\,y_2>0\right\} \end{align*} $$
and
$$ \begin{align*} \textstyle \mu_{S}\Big ( i\begin{pmatrix}y_1&0\\ 0&y_2\end{pmatrix} \Big )=-3(\frac{1}{y_1}(E^1)^{*}+\frac{1}{y_2}(E^2)^{*}). \end{align*} $$
Let
$\,D:= N\cdot i\Omega \,$
be the Stein N-invariant domain in
$ {\mathcal P}$
associated with the convex,
$C_t$
-invariant domain
$$ \begin{align*} \Omega : =\left\{ i\begin{pmatrix}y_1&0\\ 0&y_2\end{pmatrix}~|~y_1y_2>1\right\}. \end{align*} $$
The image
$$ \begin{align*} \mu_N(i\Omega) =\{\eta_1(E^1)^{*}+\eta_2(E^2)^{*}~|~ \eta_1,\,\eta_2<0,\, \eta_1\eta_2<9\} \end{align*} $$
is clearly not convex. Therefore,
$\mu _N(D)$
is not convex either.
As the domain
$\,\Omega \subset J{\mathfrak {a}}^+\,$
is convex and also
$\,{\mathcal C}_{nt}$
-invariant, a similar construction provides examples of N-invariant Stein domains with non-convex moment image in all rank-2 symmetric Siegel domains, both of tube type and of non-tube type.
Proposition 4.3. The N-invariant function
$\rho \colon D(V,F)\to {\mathbb {R}}$
, given by
$$ \begin{align*} \textstyle \rho(na\cdot p_0):=2\sum_k c_kh_k, \end{align*} $$
where
$a=\exp H$
, for
$H=\sum _k h_kH_k\in {\mathfrak {a}}\,$
, and
$-f_0=\sum _kc_k(E^k)^{*}$
, is a potential for the Kähler metric induced by
$-f_0$
.
Proof. As in the previous lemma, we identify
$\,D(V,F)\,$
with S. In order to check that
$-dd^c\rho =\omega $
, we need to show that
$\,d^c\rho (\widetilde X_s)= \,\mu _N^X(s)$
, for all
$\,s \in S$
. By the
$\,N$
-invariance of
$\,\rho \,$
and of
$\,J$
, one has
$$ \begin{align*} \textstyle \, d^c \rho(\widetilde X_{na})=d^c\rho( \widetilde {\mathrm{Ad}_{n^{-1}}X}_{a})\,, \end{align*} $$
for every
$\,na \in S$
. Then, as
$\,\mu _N\,$
is
$\,N$
-equivariant, it is enough to show that
$$ \begin{align} \,d^c\rho(\widetilde X_a)= \,\mu_N^X(a),\,\end{align} $$
for all
$\,a \in A\,$
and
$\,X\in {\mathfrak {n}}$
. If
$X=E^j$
, then
$$ \begin{align*}\textstyle d^c \rho((\widetilde {E^j})_a)= e^{-2e_j(H)}d \rho(a_* JE^j) =\textstyle \frac{1}{2} e^{-2h_j} \frac{d}{ds}{\big |_{s=0}} \rho(\exp(H+ sH_j)) \end{align*} $$
$$ \begin{align*}=e^{-2h_j} c_j = -f_0(Ad_{a^{-1}}E^j) = \mu_N^{E^j}(a). \end{align*} $$
If
$X\in {\mathfrak {s}}^\alpha $
, with
$0 \not = \alpha \notin \{2e_1,\dots ,2e_r\}$
, then
$JX\in {\mathfrak {s}}^\beta $
, with
$0\not = \beta \notin \{2e_1,\dots ,2e_r\}$
. By the N-invariance of
$\,\rho $
, one obtains
$$ \begin{align*} \textstyle d^c\rho(\widetilde X_a)= e^{-\alpha(H)} d\rho (a_* JX)= e^{-\alpha(H)+\beta(H)} \frac{d}{ds}{\big |_{s=0}} \rho(\exp(sJX)a)=0. \end{align*} $$
Since
$$ \begin{align*} \mu_N^{X}(a)=-f_0(Ad_{a^{-1}} X) = -e^{-\alpha(H)}f_0(X) =0, \end{align*} $$
equation (4.2) holds true and the proposition follows.
Remark 4.4. The above computation produces an N-invariant potential and an associated N-moment map, for any S-invariant Kähler metric on
$\mathbf {D}$
induced by an element
$\sum _jd_j(E^j)^{*}\in {\mathfrak {s}}^{*}$
, with
$d_j\in {\mathbb {R}}^{>0}$
, for
$j=1,\ldots ,r$
(cf. Remark 2.2).
Acknowledgements
The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome “Tor Vergata,” CUP E83C23000330006. This research was partially supported by GNSAGA-INDAM.
