Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces

Let G/K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,G/K\,$$\end{document} be an irreducible non-compact Hermitian symmetric space and let D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,D\,$$\end{document} be a K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,K$$\end{document}-invariant domain in G/K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,G/K$$\end{document}. In this paper we characterize several classes of K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,K$$\end{document}-invariant plurisubharmonic functions on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,D\,$$\end{document} in terms of their restrictions to a slice intersecting all K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,K$$\end{document}-orbits. As applications we show that K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,K$$\end{document}-invariant plurisubharmonic functions on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,D\,$$\end{document} are necessarily continuous and we reproduce the classification of Stein K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,K$$\end{document}-invariant domains in G/K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,G/K\,$$\end{document} obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).


Introduction
Let G/K be an irreducible non-compact Hermitian symmetric space of rank r . By the polydisk theorem the space G/K contains a closed subspace r , biholomorphic to an rdimensional polydisk, with the property that G/K = K · r . If D is a K -invariant domain in G/K , then D = K · R, where R := D ∩ r is a Reinhardt domain in r . The polydisk r and R are invariant under the group T S r , generated by rotations and coordinate permutations.
As the Reinhardt domain R intersects all the K -orbits in D, it encodes all information on the K -invariant objects in D. In this paper we focus on the K -invariant plurisubhar-The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome "Tor Vergata", CUP E83C18000100006. This research was partially supported by GNSAGA-INDAM. B Laura Geatti geatti@mat.uniroma2.it Andrea Iannuzzi iannuzzi@mat.uniroma2.it 1 Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Rome, Italy monic functions. When D is Stein, we obtain the following characterization of the class P ∞,+ (D) K of smooth, K -invariant, strictly plurisubharmonic functions on D : f ∈ P ∞,+ (D) K if and only if f | R ∈ P ∞,+ (R) T S r , where f | R is the restriction of f to R. Such result is later extended to wider classes of plurisubharmonic functions as follows. Let P ∞ (D) K denote the class of smooth, Kinvariant, plurisubharmonic functions and P + (D) K (resp. P(D) K ) the class of K -invariant, strictly plurisubharmonic (resp. plurisubharmonic) functions on D. One has: Theorem 4. 13 The restriction map f → f | R is a bijection between (i) P ∞,+ (D) K and P ∞,+ (R) T S r , (ii) P(D) K and P (R) T S r , (iii) P ∞ (D) K and P ∞ (R) T S r , (iv) P + (D) K and P + (R) T S r .
As a by-product we reproduce the classification of Stein K -invariant domains in G/K obtained by Bedford and Dadok in some classical cases by direct computations [2] (see also [5] for related results).

(i) If G/K is of tube type, then D is Stein if and only if R is Stein and connected. (ii) If G/K is not of tube type, then D is Stein if and only if R is Stein and complete. In particular R contains the origin and it is connected.
The proof of our results is carried out as follows. Let g = k⊕p be a Cartan decomposition of the Lie algebra g of G, let a be a maximal abelian subspace of p, with Weyl group W , and let G = K exp a K be the corresponding decomposition of G. Every K -invariant domain D in G/K is uniquely determined by a W -invariant domain D a in a by (cf. [4,6]).
As a first step we explicitly express the Levi form of f in terms of the first and second derivatives of f . This is achieved in Proposition 3.1 by means of a fine decomposition of the tangent bundle of D, induced by the restricted root decomposition of g, and a simple pluripotential argument which enable us to maximally exploit the symmetries at hand.
The Levi form computation is a key ingredient for our results. It leads to the following characterization of smooth K -invariant strictly plurisubharmonic functions on a Stein Kinvariant domain D (Theorem 4.5): f ∈ P ∞,+ (D) K if and only if f ∈ LogConv ∞,+ (D a ) W , where the latter class consists of smooth W -invariant functions on D a satisfying the appropriate differential positivity condition. We also show thatf belongs to LogConv ∞,+ (D a ) W if and only if the corresponding T S r -invariant function on the associated Reinhardt domain R is smooth and strictly plurisubharmonic.
This fact, which may be of independent interest in the context of Reinhardt domains, implies (i) in the above theorem. When extending such characterization to the non-smooth setting (Theorem 4.12), it turns out that the K -invariant plurisubharmonic functions on D are necessarily continuous.
In the appendix we explicitly determine a K -invariant potential of the Killing metric on G/K in a Lie theoretical fashion (Proposition 5.1) and we observe that, up to an additive constant, it coincides with the logarithm of the Bergman kernel function.
Finally, we point out that our methods require no classification results, nor any distinction between classical and exceptional cases.
We wish to thank our colleague Stefano Trapani for several useful discussions and suggestions.

Preliminaries
Let g be a non-compact semisimple Lie algebra and let k be a maximal compact subalgebra of g . Let g = k⊕p be the Cartan decomposition of g with respect to k, with Cartan involution θ . Let a be a maximal abelian subspace in p. The dimension r of a is by definition the rank of G/K . Let g = m⊕a⊕ α∈ g α be the restricted root decomposition of g, where m is the centralizer of a in k, the joint eigenspace g α = {X ∈ g | [H , X ] = α(H )X , for all H ∈ a} is the α-restricted root space and the restricted root system consists of those α ∈ a * for which g α = {0}. Denote by B( · , · ) the Killing form of g, as well as its holomorphic extension to g C (which coincides with the Killing form of g C ).
For α ∈ , consider the θ -stable space g[α] := g α ⊕ g −α , and denote by k[α] and p[α] the projections of g[α] along p and k, respectively. Let + be a choice of positive roots in . Then are B-orthogonal decompositions of k and p, respectively.

Lemma 2.1
Every element X in p decomposes in a unique way as where X a ∈ a and P α ∈ p[α]. The vector P α can be written uniquely as Proof By the restricted root decomposition, every X ∈ g can be written as Then X ∈ p if and only if X m = 0 and θ(X α + X −α ) = −(X α + X −α ), for all α ∈ . In particular θ X α = −X −α , and the lemma follows.
The restricted root system of a simple Lie algebra g of Hermitian type is either of type C r (if G/K is of tube type) or of type BC r (if G/K is not of tube type), i.e. there exists a basis {e 1 , . . . , e r } of a * for which + = {2e j , 1 ≤ j ≤ r , e k ± e l , 1 ≤ k < l ≤ r }, for type C r , With such a choice of a positive system + , the roots 2e 1 , . . . , 2e r form a maximal set of long strongly orthogonal positive restricted roots, i.e. such that 2e k ± 2e l / ∈ , for k = l. For j = 1, . . . , r , the root spaces g 2e j are one-dimensional. Choose generators E j ∈ g 2e j such that the sl (2) Denote by I 0 the G-invariant complex structure of G/K . We also assume that I 0 (E j − θ E j ) = A j (see [7], Def. 2.1). By the strong orthogonality of 2e 1 , . . . , 2e r , the vectors A 1 , . . . , A r form a B-orthogonal basis of a , dual to the basis e 1 , . . . , e r of a * , and the associated sl(2)-triples pairwise commute. For j = 1, . . . , r , define Denote by W the Weyl group of a, i.e. the quotient of the normalizer over the centralizer of a in K . As g is of Hermitian type, W acts on a by signed permutations of the coordinates determined by A 1 , . . . , A r .
On p ∼ = T eK G/K the complex structure I 0 coincides with the adjoint action of the element Z 0 ∈ Z (k) given by for some element S 0 in a Cartan subalgebra s of m. In the tube case, one has S 0 = 0 (see [7], Lem. 2.2). The complex structure I 0 permutes the blocks of the decomposition (1) of p (cf. [11]), namely The next lemma gives a more detailed description of the complex structure I 0 on p. In order to state it, we need to recall a few more facts. Let g C = h C ⊕ μ∈ g μ be the root decomposition of g C with respect to the maximally split Cartan subalgebra h = s ⊕ a of g. Let σ be the conjugation of g C with respect to g. Let θ denote also the C-linear extension of θ to g C . One has θσ = σ θ. Write Z := σ Z , for Z ∈ g C . As σ and θ stabilize h, they induce actions on , defined byμ(H ) := μ(H ) and θμ(H ) := μ(θ(H )), for H ∈ h, respectively. Fix a positive root system + compatible with + , meaning that μ| a = Re(μ) ∈ + implies μ ∈ + . Then σ + = + .
Given a restricted root α ∈ , the corresponding restricted root space g α decomposes into the direct sum of ordinary root spaces with respect to the Cartan subalgebra h = s ⊕ a as follows where λ ∈ is possibly a root satisfying λ =λ and Re(λ) = α. Lemma 2.2 (a) For j = 1, . . . , r, let A j and P j be as in (2) and (3). One has I 0 P j = A j and I 0 A j = −P j .
(b) By (4), (5) and the fact that [S 0 , g μ ] ⊂ g μ , for every μ ∈ , the action of S 0 is necessarily trivial on p[e j + e l ]. Moreover, if X ∈ g e j +e l , then Denote by λ the root in with real part e j − e l and the same imaginary part as μ. By comparing terms in the same root spaces in (6), one obtains the relations By comparing terms in the same root spaces, one obtains the relations which imply As μ(S 0 ) =: iμ 0 ∈ iR, the above expression becomes From I 2 0 = −I d, one obtains μ 0 = ± 1 2 . Depending on the value μ 0 , the pairs of roots μ,μ can be relabelled so that I 0 P has the desired expression.

Remark 2.3
In view of Lemma 2.2, one can choose a I 0 -stable basis of p, compatible with the decomposition (1).
(a) As a basis of a ⊕ j p[2e j ], take pairs of elements A j , P j = −I 0 A j , for j = 1, . . . , r , normalized as in (2) and (3); (b) As a basis of p[e j + e l ] ⊕ p[e j − e l ], take 4-tuples of elements P, P , I 0 P, I 0 P , parametrized by the pairs of roots μ =μ ∈ + satisfying Re(μ) = e j + e l (with no repetition). More precisely, one has where For μ =μ, one may assume Z μ = Z μ and take the pair P, I 0 P. (c) As a basis of p[e j ] (non-tube case), take pairs of elements P, I 0 P, parametrized by the pairs of roots μ =μ ∈ + satisfying Re(μ) = e j (with no repetition). More precisely, one has P = X − θ X and with Z μ a root vector in g μ .

Lemma 2.4
Let μ ∈ + be a root satisfying Re(μ) = e j + e l and let Z μ be a root vector in g μ . Let X = Z μ + Z μ ∈ g e j +e l and Y = [K l , X ] ∈ g e j −e l . Then Let μ be a root in + , with Re(μ) = e j (non-tube case) and let Z μ be a root vector in (c) One has The first and the fourth terms of the above expression are both zero because otherwise there would exist a root in + with real part equal to 2e j and non-zero imaginary part. The second and the third term sum up to zero by the Jacobi identity and the fact that Arguing as in the previous case, the first and the fourth terms are equal to zero. The second and the third terms sum up to 2Im( By the Jacobi identity Observe that [Z μ , θ Z μ ] ∈ a⊕is. Since K l centralizes s, one has that It follows that the expression in (8) reduces to For X ∈ g, denote by X the vector field induced on G/K by the left G-action, namely The above identity was proved in [9], Lemma 7.1, for f strictly plurisubharmonic. However the same argument works for arbitrary smooth K -invariant functions. This result will be used to compute the Levi form of an arbitrary smooth K -invariant function on G/K .
When the function f is strictly plurisubharmonic, then −dd c f is a K -invariant Kähler form and the map μ : G/K → k * , defined by is a moment map. It is referred to as the moment map associated with f . We conclude the preliminaries with a lemma which is needed in the next section. Let be the unit disc in C. Consider the (T S 2 )-action on the bidisk 2 , where T = (S 1 ) 2 acts by rotations and S 2 by permutations of the coordinates. Let W R 2 = (Z 2 ) 2 S 2 be the group acting on R 2 by signed permutations of the coordinates.
(iii) If Gf extends continuously to a strictly positive function on R 2 , then r = s > 0 and, Proof (i) For a 1 > 0, write tanh a 1 = e s 1 , for some s 1 ∈ (−∞, 0). Since f is T -invariant and strictly plurisubharmonic, the function s 1 → f (e s 1 , tanh a 2 ) is strictly convex and the limit lim Hence the function is strictly increasing and its derivative e and tanh a 1 = e s 1 , the first part of statement (i) follows. The second one follows from the ( Since such quantity is assumed to be strictly positive and ∂ f ∂a 1 (a 1 , 0) > 0, then r > 0. By taking a 1 = 0 and a 2 > 0, one obtains that s > 0. .
Consequently, for a 1 converging to a fixed a 2 > 0, statements (i) and (ii) imply r s ≥ 1. An analogous argument, with 0 < a 1 < a 2 , implies r s ≤ 1. As a consequence, r s = 1.

The Levi form of a K -invariant function
Let G/K be an irreducible non-compact Hermitian symmetric space of rank r . From the Similarly, every K -invariant function f : D → R is uniquely determined by the W -invariant functionf : D a → R, given byf The goal of this section is to express the real symmetric I 0 -invariant bilinear form of a smooth K -invariant function f on a K -invariant domain D ⊂ G/K in terms of the first and second derivatives of the functionf on D a . This will enable us to characterize smooth K -invariant strictly plurisubharmonic functions on a Stein K -invariant domain D in G/K by an appropriate differential positivity condition on the corresponding functions on D a (see Theorem 4.5 and Corollary 4.6). As f is K -invariant, h f is K -invariant as well. Therefore it will be sufficient to carry out the computations along the slice exp D a K , which meets all the K -orbits in D.
For z = aK , with a = exp(H ) and H ∈ a, one has for all X ∈ g, where F a : g → p is the map given by F a := π # • Ad a −1 , and π # : g → p is the linear projection along k. One can verify that Denote by a 1 , . . . , a r the coordinates induced on a by the basis A 1 , . . . , A r of a (cf. Remark 2.3(a)).
Then, in the basis of p defined in Remark 2.3, the form h f at z = aK ∈ D is given as follows.
As the form h f is I 0 -invariant, by (5) it is determined by its restrictions to the blocks a * a, a * p[e j + e l ] and a * p[e j ]. The non-zero entries of h f on each of these blocks are given as follows.
In particular, with respect to the basis of a * p[e j ] defined in Remark 2.3 (c), the form h f is diagonal.
Proof We compute the form h f by exploiting relation (10). We begin by determining d c f ( X z ), for X ∈ k and z ∈ G/K . By the K -invariance of f and of I 0 one has for every z ∈ G/K and k ∈ K . Thus it is sufficient to take z = aK in exp D a K . We first assume that α(H ) = 0 for all α ∈ , and later obtain the complete result by passing to the limit for H approaching the hyperplanes {α = 0} in a.
On the blocks of decomposition (1) of k, one has Indeed, for M ∈ m, one has M z = 0 and therefore Then, by (15) and the K -invariance of f and of I 0 , one has (3)). One has Proof of statement (i). As a first step we show that a * p[α] and a * p[γ ] are h f -orthogonal for any distinct roots α ∈ + and γ ∈ {0} ∪ ( + \ {2e 1 , . . . , 2e r }), with the convention (5)). Then by (15) which, by (16), becomes The brackets (ii) The form h f on a * a.
Let A j , A l ∈ a. Since I 0 A l = −P l , one has The above expression is well defined also for those H = j a j A j with some zero coordinate. Assume for example a l = 0. As it is W -invariant, f is an even function of the coordinate a l . Consequently its derivative ∂ f ∂a l vanishes for a l = 0 and l smoothly extends to the hyperplane a l = 0. This concludes the proof of (ii).

(iii)
The form h f on a * p[e j + e l ].
Next, let P, P ∈ p[e j + e l ] and I 0 P, I 0 P ∈ p[e j − e l ] be elements of the basis of Remark 2.3 (b), arising from the same root μ ∈ + .
From (18) it follows that where K = X + θ X and C = Y + θ Y , for X and Y as in (7). By Lemma 2.4(a)(b) and (17), the above expression equals for some r , s ∈ R. In a similar way, one obtains and, from Lemma 2.4(c), h f (a * P, a * P ) = 0.
Also, by (i), one has h f (a * P, a * I 0 P) = h f (a * P, a * I 0 P ) = 0. For the strictly plurisubharmonic potential ρ of the Killing metric of G/K given in Proposition 5.1, the quantity in (19) smoothly extends to a strictly positive function on R 2 . Hence (iii) of Lemma 2.5 implies that r = s > 0. Finally, as h ρ (a * P, a * P) = B(P, P), a simple computation shows that r = B(P, P)/b. This concludes the proof of (iii).
(iv) The form h f on a * p[e j ].
Let P, Q ∈ p[e j ] be elements of the basis of Remark 2.3 (c), arising from roots μ, ν ∈ + , respectively, with ν = μ,μ. Then h f (a * P, a * Q) = 0, because [Z μ ± Z μ , Z ν ± Z ν ] = 0, for all Z μ ∈ g μ and Z ν ∈ g ν . In addition, by the I 0 -invariance of h f one has h f (a * P, a * I 0 P) = 0. In order to compute h f (a * P, a * P), write P = X − θ X and

.2 (c)). Then, from (18) it follows that
The above formula smoothly extends to D a , since ∂ f ∂a j is identically zero on the hyperplane a j = 0.
Finally, by computing the above quantity for the strictly plurisubharmonic potential ρ of the Killing metric of G/K given in Proposition 5.1, one obtains that t = B(P, P)/b . This completes the proof of statement (iv) and of the proposition.

Remark 3.2 The Levi form L C
f of f is given by

K -invariant psh functions vs. W-invariant logcvx functions
Let G/K be an irreducible non-compact Hermitian symmetric space of rank r and let D ⊂ G/K be a Stein, K -invariant domain. The goal of this section is to prove a characterization of various classes of K -invariant plurisubharmonic functions on D by appropriate conditions on the corresponding functions on D a (see (12) and (13)). In the smooth case we prove that a smooth K -invariant function f of D is strictly plurisubharmonic if and only if the associated functionf satisfies a positivity condition arising from Proposition 3.1(ii).
As an application, in Corollary 4.8, we reproduce the characterization of Stein K -invariant domains in G/K outlined in [2], Thm. 3 and Thm. 4.
Denote by r the orbit of the base point eK ∈ G/K under the product of the r commuting copies of SU (1, 1) determined by (2) and (3). One has r = T exp aK , where T ∼ = (S 1 ) r is the r -dimensional torus in K whose Lie algebra is generated by K 1 , . . . , K r . It is wellknown that r may be identified with the unit polydisk in C r (cf. [13], p.280), and under this identification exp(a 1 , . . . , a r )K = (tanh(a 1 ), . . . , tanh(a 1 )), for (a 1 , . . . , a r The polydisk r is a "thick slice" for the K -action in G/K , in the sense that G/K = K · r . If D is a K -invariant domain in G/K , then R := D ∩ r is by definition the Reinhardt domain associated to D and satisfies D = K · R . We will show that if D is Stein, then R is necessarily connected. It should be remarked that, despite its appellation, a Reinhardt domain is open in C r but need not be connected (in our context the quotient of R under the action induced by the Weyl group is always connected).
Denote by LogConv ∞,+ (D) (Z 2 ) r the class of smooth functions on D which are even in each variable and such that the form defined in (ii) of Proposition 3.1, i.e.
for j, l = 1, . . . , r, is strictly positive definite, for every H ∈ D. The next proposition characterizes T -invariant smooth strictly plurisubharmonic functions on R by elements in LogConv ∞,+ (D) (Z 2 ) r . It is an intermediate step in the proof of the main theorem in the smooth case, but it may be of independent interest in the context of Reinhardt domains.

Proposition 4.1 Let f be a smooth T -invariant function on a Reinhardt domain R in r . Then f is strictly plurisubharmonic if and only iff belongs to LogConv
Proof In polar coordinates (ρ j , θ j ) , with z j = ρ j e iθ j = 0, one has One easily sees that, for z j z l = 0, As it is T -invariant, f is an even function in each of the variables ρ 1 , . . . , ρ r . Consequently, the above quantity extends smoothly through the hyperplanes z j = 0 (and therefore to the whole domain) whenever j = l, while . . . , z r ) = 0, for j = l and z j z l = 0.
Then, for (z 1 , . . . , z r ) ∈ R, one has where C is the diagonal matrix with diagonal entries for z j = 0.
It follows that f is strictly plurisubharmonic if and only iff belongs to the class LogConv ∞,+ (D) (Z 2 ) r .
Let R be a Reinhardt domain in ( * ) r and let  (s 1 , . . . , s r ) .

Proposition 4.4 Let D and R be as above and let f : D → R be a smooth, K -invariant strictly plurisubharmonic exhaustion function of the Stein domain D. (i) If R contains the origin, then R is connected andf has a unique minimum point at the origin of D a . (ii) If R does not contain the origin, thenf has a unique minimum point on the intersection
D a ∩ {a 1 = · · · = a r > 0}. In particular R is connected. In this case G/K is necessarily of tube type.

Proof
The minimum set of a K -invariant exhaustion function f of D intersects R = T · exp D a K in a non-empty T -invariant set. Moreover, exp(H )K ∈ R is a minimum point of f | R , the restriction of f to R, if and only if H ∈ D a is a minimum point of f .
(i) As R intersects the coordinate hyperplanes, it is complete, as mentioned above. Assume thatf has a minimum point H = (a 1 , . . . , a r ) , different from the origin. Then f | R has a minimum point in P = exp(H )K . For ε > 0 small enough there is a holomorphic immersion where 1+ε denotes the disc of center 0 and radius 1 + ε in C. The pull-back f • ι of f via ι is a smooth strictly subharmonic S 1 -invariant function on 1+ε . It has a minimum point in 0 and, by construction, in 1. Then f • ι is necessarily constant, contradicting the fact that it is strictly subharmonic. (ii) Let H = (a 1 , . . . , a r ) , with a j ≥ 0, be a minimum point off . As R does not intersect the coordinate hyperplanes, all a j 's are different from 0 . As a consequence 2 coth(a j ) ∂ f ∂a j (H ) = 0 , for j = 1, . . . , r . In the non-tube case this contradicts the strict plurisubharmonicity of f by (iv) of Proposition 3.1, implying that the space G/K is necessarily of tube type. The strict plurisubharmonicity of f along with (iii) of Proposition 3.1, implies that a j = a k for every j, k = 1, . . . , r . Hence H lies on the positive diagonal of a. Consider the Weyl chamber a + = {a 1 ≥ a 2 ≥ · · · ≥ a r ≥ 0}. Since D a ∩ a + is connected by the connectedness of D and H belongs to the boundary of every Weyl chamber in {(a 1 , . . . a r ) ∈ a : a j ≥ 0, j = 1, . . . , r }, it follows that R ∩ (R >0 ) r is connected (as well as D log ). Hence the Reinhardt domain R = T · (R ∩ (R >0 ) r ) is connected. The uniqueness of the minimum point of f follows from standard arguments as in [1], or from the following direct argument.
The region D log is convex by the Steinness of D. By Remark 4.2, the associated function f has everywhere strictly positive definite Hessian. In particular its restriction to the diagonal D log ∩ {s 1 = · · · = s r } is a strictly convex exhaustion function. Consequently it has a unique minimum point, implying thatf has a unique minimum point on D a ∩{a 1 = · · · = a r > 0} .
Consider the following classes of functions: Since the K -action on D is proper and every K -orbit intersects the slice exp D a K in a W -orbit, the map f →f is a bijection from C 0 (D) K onto C 0 (D a ) W . By Theorem 4.1 in [6] (see also [4]) such a map is also a bijection from C ∞ (D) K  Our first result is the following theorem. Conversely, assume thatf ∈ LogConv ∞,+ (D a ) W . We need to show that the terms in (iii) and (iv) of Proposition 3.1 are strictly positive (the ones in (iii) occurring only if r > 1, the ones in (iv) occurring only in the non-tube case).
For the terms in (iii), without loss of generality, it is sufficient to consider the case r = 2, and H = (a 1 , a 2 ) ∈ a + , with a 1 ≥ a 2 ≥ 0. Assume first a 1 > a 2 > 0. Then  (tanh a 1 , tanh a 2 ) = (e s 1 , e s 2 ) ∈ R * , where R is the Reinhardt domain associated to D. Let d 0 < 0 and t 0 > 0 be real numbers defined by (s 1 , s 2 From now on, refer to the smooth functions with everywhere positive definite Hessian as SSC (smooth stably convex). By Remark 4.2, the functionf , which is invariant under coordinate permutations, is SSC. Therefore g(t) :=f (d 0 + t, d 0 − t) is even and SSC. Consequently, for t 0 as above, the inequality true. This, combined with formulas (23), implies giving the desired positivity.
Next consider H = (a, a) , with a = 0. Set tanh a = e d 0 and (tanh a 1 , tanh a 2 ) = (e d 0 +t , e d 0 −t ). Recall that g (0) = 0. Then the corresponding term in (iii) of Proposition 3.1 is the limit which is positive since c(a) is a positive real number and g (0) > 0 (g is even and SSC).
If H = (a 1 , 0) ∈ D a , with a 1 > 0, then the Reinhardt domain R associated to D is necessarily complete and the term to be evaluated reduces to a 1 , 0) . 0) is positive, as wished. Finally, for a 1 = a 2 = 0, the analytic extension of our term is given by which is strictly positive by assumption.
We are left to examine the terms in (iv), which only appear in the non-tube case. The arguments are similar to the ones used in the previous case. By is strictly positive as well, by assumption. This completes the proof of the theorem.
Consider the (T S r )-action on r , where S r denotes the group of coordinate permutations. From Proposition 4.1 one deduces the following corollary.

Corollary 4.6 Let D be a Stein K -invariant domain in an irreducible non-compact Hermitian symmetric space G/K and let R be the associated Reinhardt domain.
The map f → f | R is a bijection between P ∞,+ (D) K and P ∞,+ (R) T S r .

Remark 4.7
If R does not contain the origin, then, by Remark 4.2, the condition f ∈ P ∞,+ (D) K is also equivalent to requiring that the smooth invariant functionf has strictly positive definite Hessian on D log . Our next goal is to extend the characterization of smooth, K -invariant, strictly plurisubharmonic functions on D obtained in Theorem 4.5 to the following classes of K -invariant functions:

Corollary 4.8 (See [2, Thm.3 and Thm. 4]) Let D be a Stein K -invariant domain in an irreducible non-compact Hermitian symmetric space G/K and let R be the associated Reinhardt domain. Then (i) If G/K is of tube type, then D is Stein if and only if R is Stein and connected. (ii) If G/K is not of tube type, then D is Stein if and only if R is
-P(D) K : plurisubharmonic, K -invariant functions on D, -P ∞ (D) K : smooth, plurisubharmonic, K -invariant functions on D, -P + (D) K : functions which, on every relatively compact K -invariant domain C in D , are the sum g + h , of some g ∈ P(C) K and h ∈ P ∞,+ (C) K .
In order to do that we need to define the corresponding appropriate classes of functions on the associated domain  By proceeding inductively, one obtains the statement for r > 2.
(ii) By Theorem 4.5, to a decreasing sequencef n of functions in LogConv ∞,+ (D a ) W there corresponds a decreasing sequence f n in P ∞,+ (D) K , whose limit f necessarily belongs to P(D) K . The restriction f | R of f to R is a plurisubharmonic T -invariant function. By part (i), the function f | R is continuous. Consequently so is the correspondingf in LogConv(D a , [−∞, ∞)) W , which is the limit of thef n .
Summarizing, the following inclusions hold true Our complete result is stated in the next theorem. In particular, from the above inclusions, it follows that the K -invariant plurisubharmonic functions on D are continuous.
Proof (i) is the content of Theorem 4.5. By averaging over K , a K -invariant, plurisubharmonic function on D is the decreasing limit of smooth K -invariant, strictly plurisubharmonic functions (cf. [8, Sect. K]). Then (ii) follows from (i). As smooth K -invariant functions on D correspond to smooth W -invariant functions on D a , an analogous argument also proves statement (iii). Finally (iv) follows from the definitions of LogConv + (D) W and P + (D) K , by averaging the summands over W and K , respectively.
Let T S r act on r as in Corollary 4.6. The previous theorem can be reformulated as follows.

Theorem 4.13 Let D be a Stein K -invariant domain in an irreducible non-compact Hermitian symmetric space G/K and let R be the associated Reinhardt domain. The map f → f | R is a bijection between
(i) P ∞,+ (D) K and P ∞,+ (R) T S r , (ii) P (D) K and P (R) T S r , (iii) P ∞ (D) K and P ∞ (R) T S r , (iv) P + (D) K and P + (R) T S r .

Appendix: A K -invariant potential of the Killing metric.
Let G/K be an irreducible non-compact Hermitian symmetric space. The Killing form B of g, restricted to p, induces a G-invariant Kähler metric on G/K , which we refer to as the Killing metric. In this section we exhibit a K -invariant potential ρ of this metric in a Lie theoretical fashion. We also show that such a K -invariant potential coincides, up to an additive constant, with the logarithm of the Bergman kernel function (Remark 5.2 and Corollary 5.3).
In order to define ρ, according to the decomposition G = K exp a K , write an element of G/K as ka K , where k ∈ K and a = exp H , with H = j a j A j ∈ a.
The form h ρ on a * p[e j + e l ].
Here α = e j + e l and β = e j − e l . Then for P ∈ p[e j + e l ], one has h ρ (a * P, a * P) = Here α = β = e j . Then for P ∈ p[e j ], one has h ρ (a * P, a * P) = This concludes the proof of (i) and of the proposition.
The following remark shows that a K -invariant potential of the Killing metric is unique, up to an additive constant.
Proof As ρ 1 − ρ 2 is pluriharmonic and G/K is contractible, there exists a holomorphic function f : G/K → C, such that Re f = ρ 1 − ρ 2 , which is unique up to an imaginary constant (cf. [8,Sect. K]). By averaging f over K , its real part ρ 1 − ρ 2 does not change. Hence f itself is K -invariant. Moreover, being holomorphic, f is also invariant with respect to the induced local K C -action on G/K . Since K C acts locally transitively on an open subset of G/K (cf. [13]), the function f is constant and so is its real part ρ 1 − ρ 2 .
Since the logarithm of the Bergman kernel function is a K -invariant potential of the Killing metric (see [10], Vol. 2, Exa. 6.6, p. 162 and Thm. 9.6, p. 262), one can draw the following conclusion.

Corollary 5.3
Up to an addictive constant, the smooth K -invariant exhaustion function ρ coincides with the logarithm of the Bergman kernel function.

Example 5.4
As an example, consider the unit disc = G/K , where G = SU (1, 1) acts on by linear fractional transformations. Fix the basis of g, normalized as in (3): Then exp a 1 A 1 K = tanh a 1 = |z|. Take ρ(t) = − ln 1 cosh t+1 , which satisfies the differential equation ρ (t) = cosh t−1 sinh t . Since B(A 1 , A 1 ) = 8, then up to an addictive constant, the logarithm of the Bergman kernel function is given by Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.