S. G. Gindikin

Integral Formulas for Siegel Domains of the Second Kind

(Presented by Academician P. S. Novikov on 12 VI 1961)

In the present note we consider the Bergman and Cauchy–Szegő integral kernels for Siegel domains of the second kind \((^1)\). The explicit form of the kernels is found for affinely homogeneous domains associated with self-adjoint cones and, in particular, for all symmetric domains. Integral formulas for classical domains in another realization were obtained earlier by Hua Loo-keng \((^2)\), and for some homogeneous Siegel domains of the first kind by Bochner \((^3)\).

Let \(\mathfrak{D}\) be a Siegel domain of the second kind, i.e. the set of points
\[ \zeta=(z,u)\in C^{n+m},\quad z\in C^n,\quad u\in C^m, \]
such that
\[ \operatorname{Im} z-F(u,u)\in V, \]
where \(V\) is a convex cone in \(R^n\) containing no straight lines, and \(F(u,v)\) is a \(V\)-Hermitian form. The Bergman kernel of the domain \(\mathfrak{D}\) is the function \(B(\eta,\zeta)\) on \(\mathfrak{D}\times\mathfrak{D}\), analytic in \(\eta\) and antianalytic in \(\zeta\), for which
\[ \int_{\mathfrak{D}} B(\eta,\zeta) f(\zeta)\, d\zeta\, d\bar{\zeta}=f(\eta), \]
where \(f\) is an arbitrary bounded analytic function with integrable square of the modulus in \(\mathfrak{D}\), continuous in \(\overline{\mathfrak{D}}\). For simplicity we shall give the definition of the Cauchy–Szegő kernel only as applied to Siegel domains. The Cauchy–Szegő kernel is a function \(S(\eta,\zeta)\) on \(\mathfrak{D}\times\overline{\mathfrak{D}}\), possessing the same analyticity properties as \(B\), and such that
\[ \int_{\Omega_{\mathfrak{D}}} S(\eta,\zeta) f(\zeta)\, dx\, du\, d\bar{u}=f(\eta), \]
where
\[ \Omega_{\mathfrak{D}}=\{\zeta=(z,u): \operatorname{Im} z=F(u,u)\} \]
is the skeleton of the boundary of \(\mathfrak{D}\), \(x=\operatorname{Re} z\), and \(f\) satisfies the same conditions as above.

Analytic automorphisms of the domain lead to functional equations for the Bergman kernel. This is expressed by the fact that \(B(\zeta,\zeta)\) is the density of an invariant measure in \(\mathfrak{D}\). In the case of a homogeneous domain, the resulting equations together with the analyticity condition are sufficient to determine \(B\) up to a constant factor. The equations take an especially simple form for linear automorphisms. The use of automorphisms of the domain in finding the Cauchy–Szegő kernel causes certain difficulties. However, in the case of linear automorphisms these difficulties are absent, and one obtains very simple relations also for the kernel \(S\). Therefore it is convenient to carry out the computation of the kernels for homogeneous domains by realizing them as Siegel domains, since in this case there is a transitive group of linear transformations.

We now proceed to the direct computation of the kernels. First of all, note that it is sufficient to determine \(B(\eta,\zeta)\) and \(S(\eta,\zeta)\) for \(\eta=\zeta\), after which the kernels are uniquely found by means of the analyticity condition. If \(g\) is a linear transformation of \(\mathfrak{D}\), then
\[ B(\eta,\zeta)=B(g\eta,g\zeta)j^2(g), \]
where \(j(g)\) is the Jacobian of \(g\) (by linearity of \(g\) it does not depend on \(\zeta\));
\[ S(\eta,\zeta)=S(g\eta,g\zeta)\tilde{j}(g), \]
where
\[ \tilde{j}(g)=d(gx)\,d(gu)\,d(\overline{gu})/dx\,du\,d\bar{u}. \]
Consider, in particular, the nilpotent group of automorphisms
\[ N(\mathfrak{D}):(z,u)\to (z+a+2iF(u,c)+iF(c,c),\,u+c), \]
where \(a\in R^n,\ c\in C^m\). Then \(\tilde{j}=j=1\), and as a result it is enough to determine \(B\) and \(S\) for
\[ \zeta=\eta=(iy,0),\quad y\in V. \]
Let, for these values of the argument,
\[ B(\zeta,\zeta)=b(y),\qquad S(\zeta,\zeta)=s(y). \]
Then for

in order to obtain \(B\) and \(S\), in \(b\) and \(s\) one must replace the argument \(y\) by
\[ \rho(\eta,\xi)=(w-\bar z)/2i-F(v,u), \]
where \(\eta=(w,v)\), \(\xi=(z,u)\).

We now restrict ourselves to the case of an affinely homogeneous Siegel domain. In this case the cone \(V\) is also affinely homogeneous, and there exists a simply transitive group \(K(V)\) of linear transformations of \(V\) that extend to linear transformations of \(\mathfrak D\). Let \((\lambda,\mu)\) be some scalar product in \(R^n\), and let \(e\) be a fixed point in \(V\) such that \(\sigma(\lambda)=(e,\lambda)>0\) for all \(\lambda\in V\). Denote by \(T_\lambda\) the mapping from \(K(V)\) that carries \(e\) to \(\lambda\), by \(\widetilde T_\lambda\) the extension of \(T_\lambda\) to \(C^m\) \((T_\lambda F(u,u)=F(\widetilde T_\lambda u,\widetilde T_\lambda u))\), and by \(g_\lambda\) the resulting transformation of \(\mathfrak D\). Then
\[ b(y)=c_b j_z^{-2}(g_y)=c_b j_z^{-2}(y)j_u^{-2}(y);\qquad s(y)=c_s \widetilde j^{-1}(g_y)=c_s j_z^{-1}(y)j_u^{-2}(y), \]
where
\[ j_z(y)=d(T_y z)/dz,\qquad j_u(y)=d(\widetilde T_yu)/du. \]

It remains for us to determine the numerical factors \(c_b\) and \(c_s\). The following integral representations for the kernels hold:
\[ \int_V j_u^2(\lambda)\exp(i\sigma(\rho(g_\lambda\eta,g_\lambda\xi)))\,d\lambda_1\cdots d\lambda_n =\frac{\Gamma_s S(\eta,\xi)}{c_s}, \tag{1} \]
\[ \int_V j_u^2(\lambda)j_z(\lambda)\exp(i\sigma(\rho(g_\lambda\eta,g_\lambda\xi)))\,d\lambda_1\cdots d\lambda_n =\frac{\Gamma_b B(\eta,\xi)}{c_b}. \tag{2} \]

These representations are obtained analogously to the generalized Siegel integral \({}^{(6)}\), into which they pass for \(\eta=\xi,\ u=0\). In view of this, the computation of \(\Gamma_s\) and \(\Gamma_b\) reduces to the computation of Siegel integrals. Applying the operators (1) and (2) to the functions \(\exp(i\sigma(T_\lambda z/2))\), \(\lambda\in V\), we obtain that the expressions (1) and (2) are respectively equal to \(\alpha\beta S(\eta,\xi)\) and \(\alpha\beta\gamma B(\eta,\xi)\), where
\[ \int_{R^n}\exp\left(i\frac{\sigma(T_\lambda x-T_\mu x')}{2}\right)\,dx =\alpha\delta(\lambda-\mu),\qquad \beta=\int_{C^m}\exp(-\sigma(F(u,u)))\,du\,d\bar u, \]
\[ \gamma=\int_V \exp(-\sigma(y))\,dy. \]
The last integral is also a special case of the Siegel integral. Thus,
\[ c_s=\Gamma_s/\alpha\beta,\qquad c_b=\Gamma_b/\alpha\beta\gamma. \]

Let us now consider the case when the cone \(V\) is self-dual with respect to the chosen bilinear form \((\lambda,\mu)\). There exist four series of (classical) self-dual cones and one exceptional self-dual cone \({}^{(4,5)}\). In the course of the computations the explicit form of the Siegel integral is used, and examples of Siegel domains connected with self-dual cones are also considered. The results relevant here for the classical cones are contained essentially in \({}^{(6,7)}\), while for the exceptional cone they are new. We shall investigate each of the types of cones.

1. Let \(V\) be the cone of symmetric positive-definite matrices \(\lambda\) of order \(p\). Denote by \(\Delta_i(\lambda)\) the principal minor formed by the first \(i\) rows and columns. Put
   \[ (\lambda,\mu)=sp(\lambda\mu);\qquad e=E. \]
   In this case the space \(C^m\) decomposes into the direct sum of subspaces
   \[ C,\ldots,C^{s_p},\qquad s_1\ge \cdots \ge s_p, \]
   and in \(C^{s_i}\) is concentrated the component \(F_{ii}(u,v)\) of the vector-function \(F(u,v)\) (one may realize \(C^m\) as a collection of rectangular matrices of order \(p\times s_1\), in which in the \(i\)-th row only the last \(s_i\) elements can be different from zero), putting
   \[ F(u,v)=\frac12(uv^*+\overline{vu'}). \]
   We have
   \[ j_z(y)=(\Delta_p(y))^{(p+1)/2};\qquad j_u^2(y)=\prod_{i=1}^p(\Delta_i(y))^{s_i-s_{i+1}}; \]
   \[ \Gamma_s=\pi^{p(p-1)/4}\prod_{i=1}^p \Gamma\left(s_i+\frac{p-i}{2}+1\right);\qquad \Gamma_b=\pi^{p(p-1)/4}\prod_{i=1}^p \Gamma\left(s_i+p-\frac{i-3}{2}\right); \]
   \[ \alpha=2^{p(p+3)/2}\pi^{p(p+1)/2};\qquad \beta=\pi^m,\qquad m=\sum_{i=1}^p s_i;\qquad \gamma=\pi^{p(p-1)/4}\prod_{i=1}^p \Gamma\left(\frac{p-i}{2}+1\right). \]

1. For the cone of Hermitian positive-definite matrices of order \(p\), just as in the preceding case, \((\lambda,\mu)\), \(e\), \(\Delta_i(\lambda)\) are defined, and the space \(C^m\) is decomposed into a direct sum of subspaces \(C^{s_i}\). Examples of \(V\)-Hermitian forms associated with homogeneous Siegel domains may be furnished by forms defined on the space of pairs of matrices \(u^{(1)}, u^{(2)}\) of orders \(p \times s_1^{(1)}\) and \(p \times s_1^{(2)}\), respectively, for which in the \(i\)-th row only the last \(s_i^{(1)}\) \((s_i^{(2)})\) elements are nonzero; \(s_1^{(1)} \geq \cdots \geq s_p^{(1)}\), \(s_1^{(2)} \geq \cdots \geq s_p^{(2)}\); \(s_i=s_i^{(1)}+s_i^{(2)}\). The vector-function \(F(u,v)\) is given by the formula
   \[ F(u,v)=u^{(1)}v^{(1)*}+\bar v^{(2)}u^{(2)'}. \]
   In this way not all Siegel domains associated with this cone can be realized. One may, for example, also replace the space spanned by the first \(s_i^{(1)}-s_{i+1}^{(1)}\) elements of the \(i\)-th row of the matrix \(u^{(1)}\) and the first \(s_i^{(2)}-s_{i+1}^{(2)}\) elements of the \(i\)-th row of the matrix \(u^{(2)}\) by any of its subspaces. In the case under consideration we have:
   \[ j_z(y)=(\Delta_p(y))^p;\qquad j_u^2(y)=\prod_{i=1}^{p}(\Delta_i(y))^{s_i-s_{i+1}};\qquad \Gamma_s=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(s_i \]
   \[ -i+p+1);\qquad \Gamma_b=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(s_i-i+2p+1);\qquad \alpha=2^{p(p+1)/2}\pi^{p^2};\quad \beta=\pi^m, \]
   \[ m=\sum_{i=1}^{p}s_i;\qquad \gamma=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(p-i+1). \]

2. Let now \(V\) be the cone of Hermitian positive-definite quaternionic matrices of order \(p\), or, equivalently, of Hermitian positive-definite matrices \(\lambda\) of order \(2p\) for which
   \[ \lambda J=J\bar\lambda,\qquad J=\begin{pmatrix}j\cdot & 0\\ 0 & j\cdot\end{pmatrix},\qquad j=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}. \]
   Set
   \[ (\lambda,\mu)=\tfrac12\operatorname{sp}(\lambda\mu); \]
   by \(\Delta_i\) we shall denote the principal minor of order \(2i\). The space \(C^m\) decomposes into a direct sum of \(p\) subspaces \(C^{s_i}\) \((s_i \geq s_j\) for \(i<j)\), on which, respectively, the components \(F_{2i-1,\,2i-1}=F_{2i,\,2i}\) of the form \(F\) are concentrated. For example, as \(C^m\) one may take the space of rectangular matrices of order \(2p\times t_1\), in which in the \(i\)-th row only the last \(t_i\) elements are nonzero, \(t_{i+1}\leq t_i\); \(s_i=t_{2i-1}+t_{2i}\), and set
   \[ F(u,v)=\tfrac12(uv^*+\bar Jv(Ju)'). \]
   One may also replace each space spanned by the first \((t_{2i-1}-t_{2i+1})\) elements in the \((2i-1)\)-st and \((2i)\)-th rows by any of its subspaces. Finally:
   \[ j_z(y)=(\Delta_p(y))^{p-\frac12};\qquad j_u^2(y)=\prod_{i=1}^{p}(\Delta_i(y))^{(s_i-s_{i+1})/2} \]
   \[ \Gamma_s=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(s_i-2i+2p+1);\qquad \Gamma_b=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(s_i-2i+4p); \]
   \[ \alpha=2^{2p^2}\pi^{p(2p-1)};\qquad \beta=\pi^m,\quad m=\sum_{i=1}^{p}s_p;\qquad \gamma=\pi^{p(p-1)/2}\prod_{i=1}^{p}\Gamma(2p-2i+1). \]

3. We proceed to consider the cone in \(R^{n+2}\) consisting of points for which
   \[ |\lambda|=\lambda_1\lambda_2-\lambda_3^2-\cdots-\lambda_{n+1}^2>0;\qquad \lambda_1>0. \]
   Put
   \[ (\lambda,\mu)=\lambda_1\mu_1+\lambda_2\mu_2+2\sum_{3}^{n+2}\lambda_k\mu_k;\qquad e=(1,1,0,\ldots,0). \]
   The space \(C^m\) is decomposed into the direct sum of subspaces \(C^{s_1}\) and \(C^{s_2}\), on one of which the component \(F_1(u,v)\) is concentrated, and on the other \(F_2(u,v)\); \(s_1\geq s_2\). For example, one may consider the space consisting of triples \((u_0,u_1,u_2)\), where \(u_0\in C^t\), \(u_1\in C^N\), \(s_1=t+N\), and \(u_2\in C^{s_2}\) \((C^{s_2}\) is some fixed subspace in \(C^N)\). The vector-function \(F(u,v)\) is given by the formulas:
   \[ F_1(u,v)=(u_0,v_0)+(u_1,v_1);\qquad F_2(u,v)=(u_2,v_2);\qquad F_{k+2}(u,v)=\tfrac12((u_1,T_kv_2)+ \]

\[ +(T_k u_2,v_1)), \]
where \((u,v)\) is the ordinary scalar product, and \(T_i\) \((i=1,\ldots,n)\) are unitary matrices of order \(N\) satisfying the condition:
\[ T_kT_m^*+T_mT_k^*=0 \quad \text{for } m\ne k. \]
We have:
\[ j_z(y)=|y|^{n/2+1};\quad j_u^2(y)=|y|^{s_1}y_2^{-s_1+s_2}; \]
\[ \Gamma_s=\pi^{n/2}\Gamma(s_1+n/2+1)\Gamma(s_2+1); \]
\[ \Gamma_b=\pi^{n/2}\Gamma(s_1+n-2)\Gamma(s_2+n/2+2); \]
\[ \alpha=2^{n+4}\pi^{n+2};\quad \beta=\pi^m,\quad m=s+N+t;\quad \gamma=\pi^{n/2}\Gamma(n/2+1). \]

1. Finally, let us consider the special cone \(V\) of Hermitian positive-definite octavic matrices of the third order \(\lambda=\|\lambda_{ik}\|\)*. It consists of Hermitian octavic matrices representable in the form \(\lambda=T_\lambda^*T_\lambda\), where \(T_\lambda\) is an upper triangular octavic matrix of the 3rd order with positive real entries on the main diagonal. The matrices \(T_\lambda\) form a group which acts simply transitively on \(V\) by the formula
   \[ T_\mu\lambda=(T_\lambda T_\mu)^*(T_\lambda T_\mu). \]
   Let
   \[ T_\lambda=\|t_{ij}(\lambda)\|; \quad \Delta_1(\lambda)=\lambda_{11}=t_{11}^2(\lambda), \]
   \[ \Delta_2(\lambda)=t_{11}^2(\lambda)t_{22}^2(\lambda), \quad \Delta_3(\lambda)=t_{11}^2(\lambda)t_{22}^2(\lambda)t_{33}^2(\lambda). \]
   The invariant volume element in \(V\) has the form
   \[ dv=(\Delta_3(\lambda))^{-9}\,d\lambda, \]
   where \(d\lambda\) is the Euclidean volume element;
   \[ (\lambda,\mu)=\operatorname{sp}(\lambda\mu) =\sum_{i=1}^3\lambda_{ii}\mu_{ii} +2\operatorname{Re}\sum_{i<j}\lambda_{ij}'\mu_{ij}. \]
   Let
   \[ e_{\alpha_1,\alpha_2,\alpha_3} =\Delta_1^{\alpha_1}\Delta_2^{\alpha_2}\Delta_3^{\alpha_3}. \]
   The formula holds
   \[ \int_V e^{-(\lambda,\mu)}e_{\alpha_1,\alpha_2,\alpha_3}(\mu)\,dv(\mu) = G_{\alpha_1,\alpha_2,\alpha_3}e_{\alpha_1,\alpha_2,\alpha_3}(\lambda^{-1}), \]
   where
   \[ G_{\alpha_1,\alpha_2,\alpha_3} =\pi^{12}\Gamma(\alpha_1+\alpha_2+\alpha_3) \Gamma(\alpha_2+\alpha_3-4)\Gamma(\alpha_3-8). \]

The space \(C^m\) decomposes into the direct sum of three subspaces \(C^{s_1}, C^{s_2}, C^{s_3}\), \(s_1\ge s_2\ge s_3\), and \(F_{ii}(u,v)\) is concentrated on \(C^{s_i}\). Let us give examples. We shall call a complex octavic linear combination with complex coefficients of the basis elements of the octavic algebra
\[ a=\sum_{k=0}^7 a_ke_k,\quad a_k\in C. \]
Put:
\[ a'=a_0e_0-\sum_{k=1}^7a_ke_k, \quad \bar a=\sum_{k=0}^7\bar a_ke_k, \quad a^*=(\bar a)'. \]
As the space \(C^m\) we take the space of rectangular matrices of three rows whose elements are complex octaves, with all elements in the \(i\)-th row vanishing except the last \(t_i\), \(t_1\ge t_2\ge t_3\), \(8t_i=s_i\). The elements of the group \(T_\lambda\) act in this space by multiplication on the left. Further,
\[ F_{ij}(u,v)=\sum_{k=1}^{t_1}u_{ik}v_{kj}^*+\bar v_{ik}u_{kj}'. \]
Moreover, as in the preceding examples, one may replace the space generated by the first \(s_i-s_{i+1}\) elements of the \(i\)-th row by an arbitrary subspace. Finally we obtain:
\[ j_z(y)=(\Delta_3(y))^9; \quad j_u^2(y)=\prod_{i=1}^3(\Delta_i(y))^{s_i-s_{i+1}} \]
\[ \Gamma_s=\pi^{12}\Gamma(s_1+9)\Gamma(s_2+5)\Gamma(s_3+1); \]
\[ \Gamma_b=\pi^{12}\Gamma(s_1+18)\Gamma(s_2+14)\Gamma(s_3+10); \]
\[ \alpha=2^{30}\pi^{27}; \quad \beta=\pi^{(s_1+s_2+s_3)}; \quad \gamma=\pi^{12}\Gamma(9)\Gamma(5). \]

The author expresses gratitude to I. I. Pyatetskii-Shapiro for his attention to the work.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
8 VI 1961

CITED LITERATURE

1. I. I. Pyatetskii-Shapiro, DAN, 113, No. 5 (1957).
2. Hua Lo-keng, Harmonic Analysis of Functions of Several Complex Variables in Classical Domains, IL, 1959.
3. S. Bochner, Ann. of Math., 45, 686 (1944).
4. E. B. Vinberg, DAN, 133, No. 1 (1960).
5. M. Koecher, Math. Ann., 135, 3 (1958).
6. I. I. Pyatetskii-Shapiro, DAN, 116, No. 2 (1957).
7. I. I. Pyatetskii-Shapiro, Geometry of Classical Domains and Theory of Automorphic Functions, Moscow, 1961.
8. G. Freudenthal, Mathematica, 1, 1 (1957).

* For the definition of octaves (Cayley numbers), see, for example, in (8).