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MATHEMATICS
T. S. Bhanumurthy
ASYMPTOTICS OF ZONAL SPHERICAL FUNCTIONS ON THE SIEGEL UPPER HALF-PLANE
(Presented by Academician I. G. Petrovsky, 4.VII 1960)
1. Let \(G\) be a connected noncompact semisimple Lie group; \(K\) its maximal compact subgroup; \(M = G/K\) the corresponding symmetric space; \(G = KA_{+}K\), where \(A_{+}\) is the Cartan subgroup of the space \(M\). Let \(R^{n}\) \((n=\operatorname{rank} M)\) be the Cartan algebra; \(W\) the Weyl group for the space \(M\); \(w\) the order of this finite group. Finally, let \(P_{+}\) be the system of all positive roots that do not vanish on \([R^{n}] = R^{n} + \sqrt{-1}\,R^{n}\). Put
\[ \rho = \frac{1}{2}\sum_{\alpha\in P_{+}}\alpha. \]
Denote by \(\varphi_{\lambda}\) the zonal spherical function of positive-definite type corresponding to the vector \(\lambda \in R^{n}\) \((^{1-3})\). For every function \(f\) constant on the double cosets of the group \(G\) with respect to the subgroup \(K\), put
\[ \widetilde f(\lambda)=\int_G f(x)\,\overline{\varphi_{\lambda}(x)}\,dx. \]
It is known that
\[ \int_G |f(x)|^{2}\,dx=\frac{1}{w}\int |\widetilde f(\lambda)|^{2}\,d\mu(\lambda); \]
the measure \(d\mu(\lambda)\) is the (uniquely determined) Plancherel measure for the space \(M\). Let \(d\lambda\) be Euclidean measure in \(R^{n}\); as shown in Harish-Chandra’s paper \((^{7})\),
\[ d\mu=\frac{1}{|c(\lambda)|^{2}}\,d\lambda. \]
The function \(c(\lambda)\) is connected with the asymptotic behavior of the zonal spherical function \(\varphi_{\lambda}\) in the following way (see \((^{7,5})\)). Let \(t=(t_{1},\ldots,t_{n})\) be the canonical coordinates of an element \(h\in A_{+}\). We shall say that \(h\to\infty\) on the Cartan group \(A_{+}\) if \((\alpha,t)\to+\infty\) for every \(\alpha\in P_{+}\). Then
\[ \varphi_{\lambda}(h)\underset{h\to\infty}{\sim} e^{-(\rho,t)} \sum_{s\in W} c(s\lambda)e^{i(\lambda,st)}, \]
\[ \lambda\in R^{n}+\sqrt{-1}R^{n},\quad |\operatorname{Im}\lambda|>\varepsilon,\quad a(\lambda)\ne 0\quad \text{for } \alpha\in P_{+}, \]
and moreover
\[ |c(s\lambda)|=|c(\lambda)|\qquad (s\in W). \]
In the present paper the Plancherel measure is computed when \(G\) is the real symplectic group.
2. Denote by \(\mathfrak S_{n}\) the space of all complex matrices \(Z=X+iY\) with positive-definite imaginary part: \(Y=\operatorname{Im} Z>0\).
This domain is called the Siegel upper half-plane and is a classical domain of type \((CI)\) in Cartan’s classification. The motion group of this domain is isomorphic to the real symplectic group \(\operatorname{Sp}(n;\mathbb R)\).
First we shall indicate the subgroups \(\mathcal K, \mathcal A_+, \mathcal N^+, \mathcal N^-\) of the group \(\operatorname{Sp}(n;\mathbb R)\) that we need—subgroups typical for arbitrary semisimple Lie groups. Let \(E_n\) be the identity matrix of order \(n\). Then, by definition,
\[ \operatorname{Sp}(n;\mathbb R)=\{g\in GL(2n;\mathbb R):gJg'=J\},\quad \text{where }\quad J=\begin{pmatrix}0&E_n\\-E_n&0\end{pmatrix}. \]
The subgroup \(\mathcal K\) consists of all symplectic orthogonal matrices. The subgroup \(\mathcal A_+\) consists of all symplectic diagonal matrices with positive entries:
\[ \mathcal A_+=\left\{h=\begin{pmatrix}\varepsilon&0\\0&\varepsilon^{-1}\end{pmatrix},\quad \varepsilon=\operatorname{Diag}(\varepsilon_1,\ldots,\varepsilon_n),\quad \varepsilon_1>0,\ldots,\varepsilon_n>0\right\}. \]
The subgroup \(\mathcal N^+\)
\[ \mathcal N^+=\left\{n^+=\begin{pmatrix}Y'^{-1}&BY\\0&Y\end{pmatrix},\quad B=B',\quad Y=\begin{pmatrix} 1&0\ldots 0\\ y_{21}&1\ldots 0\\ \vdots& & \\ y_{n1}&y_{n2}\ldots 1 \end{pmatrix}\right\}. \]
The subgroup \(\mathcal N^-\):
\[ \mathcal N^-=\left\{n^-=\begin{pmatrix}X&0\\SX&X'^{-1}\end{pmatrix},\quad S=S'=(\sigma_{ij}),\quad X=\begin{pmatrix} 1&0\ldots 0\\ x_{21}&1\ldots 0\\ \vdots& & \\ x_{n1}&x_{n2}\ldots 1 \end{pmatrix}\right\}. \]
Then \(\operatorname{Sp}(n;\mathbb R)=\mathcal K\mathcal A_+\mathcal N^+\), i.e. every element \(g\in\operatorname{Sp}(n;\mathbb R)\) is uniquely representable in the form \(g=khn^+\), where \(k\in\mathcal K,\ h\in\mathcal A_+,\ n^+\in\mathcal N^+\).
We shall call the eigenvalues of the matrix \(h\) the composite radius of the element \(g\). Let us find the composite radius (?) \((r_1,\ldots,r_n)\) of an element \(n^-\in\mathcal N^-\); for this purpose denote by \(\xi_1,\ldots,\xi_n\) the columns of the matrix \(X\), and put
\[ \widetilde D_p=\det\|(\xi_i,T\xi_j)\|_1^p \quad (p=1,2,\ldots,n-1), \]
\[ \Delta=\det T,\qquad T=E_n+S^2. \]
Then, as is easy to see,
\[ r_1^2=\widetilde D_1,\qquad r_1^2r_2^2=\widetilde D_2,\ldots,\qquad r_1^2\cdots r_{n-1}^2=\widetilde D_{n-1},\qquad r_1^2\cdots r_n^2=\Delta. \]
Taking into account that the Cartan algebra \(R^n\) consists of matrices of the form
\[ H=\operatorname{Diag}(h_1,\ldots,h_n,-h_1,\ldots,-h_n) \]
and
\[ \rho=\frac12\sum_{\alpha\in P_+}\alpha =ne_1+(n-1)e_1+\cdots+2e_{n-1}+e_n;\quad e_i(H)=h_i\ (i=1,\ldots,n), \]
we obtain
\[ r_1^{i\lambda_1+n}r_2^{i\lambda_2+n-1}\cdots r_{n-1}^{i\lambda_{n-1}+2}r_n^{i\lambda_n+1} = \widetilde D_1^{\,i\frac{\lambda_1-\lambda_2}{2}+\frac12} \cdots \widetilde D_{n-1}^{\,i\frac{\lambda_{n-1}-\lambda_n}{2}+\frac12} \cdot \Delta^{\,i\frac{\lambda_n}{2}+\frac12}. \]
The composite radius of the element \(hn^-h^{-1}\) is immediately obtained from the composite radius of the element \(n^-\), when in the determinants \(\widetilde D_p,\Delta\) we replace all \(x_{ij}\) by
\[ x_{ij}\frac{\varepsilon_i}{\varepsilon_j} \]
and all \(\sigma_{ij}\) by
\[ \frac{\sigma_{ij}}{\varepsilon_i\varepsilon_j}; \]
the determinants thus obtained will be denoted by \(\widetilde D_p(\varepsilon)\) and \(\Delta(\varepsilon)\).
Theorem 1. The zonal spherical functions on the Siegel upper half-plane are given by the integrals
\[ \varphi_\lambda(\varepsilon)= \frac{l}{\varepsilon_1^n\varepsilon_2^{\,n-1}\ldots \varepsilon_{n-1}^2\varepsilon_n} \times \]
\[ \times \iint_{(X,S)} \frac{ \widetilde D_1(\varepsilon)^{\,i\frac{\lambda_1-\lambda_2}{2}-\frac12} \ldots \widetilde D_{n-1}(\varepsilon)^{\,i\frac{\lambda_{n-1}-\lambda_n}{2}-\frac12} \cdot \Delta(\varepsilon)^{\,i\frac{\lambda_n}{2}-\frac12} }{ \widetilde D_1^{\,i\frac{\lambda_1-\lambda_2}{2}+\frac12} \ldots \widetilde D_{n-1}^{\,i\frac{\lambda_{n-1}-\lambda_n}{2}+\frac12} \cdot \Delta^{\,i\frac{\lambda_n}{2}+\frac12} } \,dX\,dS, \tag{1} \]
where
\[ dX=\prod_{i>j} dx_{ij},\qquad dS=\prod_{i\le j} d\sigma_{ij}. \]
Moreover, if \(\lambda\) is a real vector, i.e. if \(\lambda\in R^n\), then \(\varphi_\lambda\) is also positive definite. The constant \(l\) is determined by the condition \(\varphi_\lambda(1)=1\).
Applying Harish-Chandra’s results (⁷), we conclude
\[ \varphi_\lambda(\varepsilon)\cong \frac{1}{\varepsilon_1^n\varepsilon_2^{\,n-1}\ldots \varepsilon_{n-1}^2\varepsilon_n} \sum_{(k_1,\ldots,k_n)} c(\pm\lambda_{k_1},\ldots,\pm\lambda_{k_n}) \varepsilon_1^{\pm i\lambda_{k_1}}\ldots \varepsilon_n^{\pm i\lambda_{k_n}}, \]
where the sum runs over \(n!\) permutations \((k_1,\ldots,k_n)\) of \((1,2,\ldots,n)\). Further,
\[ |c(\pm\lambda_{k_1},\ldots,\pm\lambda_{k_n})| = |c(\lambda_1,\ldots,\lambda_n)|, \qquad \operatorname{Re}\lambda_1>\cdots>\operatorname{Re}\lambda_n>0, \]
and, if \(\lambda\in R^n,\ \lambda_1>\cdots>\lambda_n>0\),
\[ c(\lambda)=c(\lambda_1,\ldots,\lambda_n) = \lim_{\theta\to+0} c(\lambda_1-in\theta,\lambda_2-i(n-1)\theta,\ldots,\lambda_n-i\theta). \]
We introduce for consideration the integral
\[ C(\alpha)=C(\alpha_1,\ldots,\alpha_n) = \iint_{(X,S)} \frac{dX\,dS}{ \widetilde D_1^{\,\alpha_1-\alpha_2+\frac12} \ldots \widetilde D_{n-1}^{\,\alpha_{n-1}-\alpha_n+\frac12} \cdot \Delta^{\,\alpha_n+\frac12} }. \tag{2} \]
As follows from the paper (⁷),
\[ c(\lambda)= \lim_{\theta\to+0} C\left( \frac{i\lambda_1+n\theta}{2}, \frac{i\lambda_2+(n-1)\theta}{2}, \ldots, \frac{i\lambda_n+\theta}{2} \right). \tag{3} \]
We note in passing that the constant \(l\) in formula (1) is determined by the relation
\[ \frac{1}{l}=C\left(\frac n2,\frac{n-1}{2},\ldots,\frac12\right). \tag{4} \]
Theorem 2.
\[ C(\alpha)=a(\alpha)b(\alpha), \]
where
\[ a(\alpha)= \int_{(X)} \frac{dX}{ D_1^{\,\alpha_1-\alpha_2+\frac12} \ldots D_{n-1}^{\,\alpha_{n-1}-\alpha_n+\frac12} }, \]
\[ b(\alpha)= \int_{(S)} \frac{ \Delta_1^{\,\alpha_1-\alpha_2-\frac12} \ldots \Delta_{n-1}^{\,\alpha_{n-1}-\alpha_n-\frac12} }{ \Delta^{\,\alpha_1+\frac12} } \,dS, \]
\[ D_p=\det\|(\xi_i,\xi_j)\|_1^p,\qquad (p=1,\ldots,n-1). \]
Here
\[ \Delta=\det T,\qquad T=E_n+S^2=(t_{ij}),\qquad \Delta_p=\det\|t_{ij}\|_{p+1}^{n}\quad (p=1,\ldots,n-1). \]
The integral for \(a(\alpha)\) has already been computed by us in (8):
\[ a(\alpha)=\prod_{1\le i<j\le n}\mathrm B\left(\alpha_i-\alpha_j;\frac12\right); \tag{5} \]
where \(\mathrm B(x,y)\) is Euler’s beta function. The computation of the second integral will be carried out inductively.
More explicitly, put
\[ S=\begin{pmatrix}\sigma&v\\ v'&\widetilde S\end{pmatrix},\qquad E_{n-1}+\widetilde S^{\,2}=\widetilde T=\|\widetilde t_{ij}\|_2^n \]
(\(\widetilde S\) is a matrix of order \(n-1\)),
\[ \widetilde\Delta=\det\widetilde T,\qquad \widetilde\Delta_p=\det\|\widetilde t_{ij}\|_{p+1}^{n}\quad (p=2,\ldots,n-1), \]
and let
\[ \widetilde b(\alpha_2,\ldots,\alpha_n) = \int_{(\widetilde S)} \frac{ \widetilde\Delta_2^{\,\alpha_2-\alpha_3-\frac12}\cdots \widetilde\Delta_{n-1}^{\,\alpha_{n-1}-\alpha_n-\frac12} }{ \widetilde\Delta^{\,\alpha_2+\frac12} }\,d\widetilde S . \]
Theorem 3.
\[ b(\alpha)=\widetilde b(\alpha_2,\ldots,\alpha_n)\, \mathrm B\left(\alpha_1;\frac12\right) \mathrm B\left(\alpha_1+\alpha_2;\frac12\right)\cdots \mathrm B\left(\alpha_1+\alpha_n;\frac12\right). \]
It should be noted that analogous, but somewhat different, integrals, in which the argument ranges over various sets of matrices, were considered by Hua Loo-keng (see, for example, (6)).
From Theorems 2 and 3 our main assertion finally follows:
Theorem 4. The Plancherel measure \(d\mu\) in the case of spherical functions on the Siegel half-plane is given by the formula
\[ d\mu= \frac{1}{\pi^{n}l^{2}}\, \prod_{1\le p<q\le n} \frac{\lambda_p-\lambda_q}{2}\, \operatorname{th}\frac{\lambda_p-\lambda_q}{2}\pi \cdot \prod_{1\le p<q\le n} \frac{\lambda_p+\lambda_q}{2}\, \operatorname{th}\frac{\lambda_p+\lambda_q}{2}\pi \times \]
\[ \times \prod_{p=1}^{n} \frac{\lambda_p}{2}\, \operatorname{th}\frac{\lambda_p}{2}\pi \cdot d\lambda_1\cdots d\lambda_n, \]
where, according to (4),
\[ \frac{1}{l} \prod_{1\le p<q\le n} \mathrm B\left(\frac{q-p}{2};\frac12\right) \cdot \prod_{1\le p<q\le n} \mathrm B\left(n+1-\frac{p+q}{2};\frac12\right) \times \]
\[ \times \prod_{p=1}^{n} \mathrm B\left(\frac{n-p+1}{2};\frac12\right). \]
The author expresses his heartfelt gratitude to F. I. Karpelevich, who suggested that he consider the symplectic group.
Moscow State University
named after M. V. Lomonosov
Received
30 VI 1960
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