Abstract Generated abstract
This paper computes the Plancherel measure for the Riemannian symmetric space \(SL(n,\mathbb R)/SO(n,\mathbb R)\). It first gives an integral representation of the positive definite zonal spherical functions in terms of lower triangular unipotent matrices and Gram determinants, normalized by an explicit beta function constant. Using Harish-Chandra’s relation between the Plancherel measure and the asymptotic coefficient of spherical functions, the paper reduces the calculation to an integral \(C_n(\lambda)\), evaluates it inductively through beta functions, and obtains an explicit product formula for the measure in terms of the spectral parameters \(\lambda_1,\ldots,\lambda_n\) subject to \(\sum \lambda_i=0\).
Full Text
T. S. Bhanumurthy
THE PLANCHEREL MEASURE FOR THE QUOTIENT SPACE
\(SL(n; R)/SO(n; R)\)
(Presented by Academician I. G. Petrovskii, 22 III 1960)
1. Let \(G\) be a connected real semisimple Lie group with finite center, and let \(K\) be its maximal compact subgroup (we shall denote elements of the group \(G\) by \(g\), and of the group \(K\) by \(k\)). Further, let \(T_g^\rho\) be an irreducible unitary representation of the group \(G\), acting in a Hilbert space \(H\), where \(\rho\) is the parameter determining this representation. The representation \(T_g^\rho\) is called a representation of class I if in \(H\) there exists a vector \(\xi_0\) satisfying the condition \(T_k \xi_0=\xi_0\).* Let \(T_g^\rho\) be a representation of class I, and let \(\xi_0\in H\) be a vector of unit length satisfying the condition \(T_g^\rho \xi_0=\xi_0\) when \(g\in K\). Denote by \(f_\rho(g)\) the function
\[ f_\rho(g)=(\xi_0,T_g^\rho \xi_0). \tag{1} \]
The function \(f_\rho(g)\) is called a zonal spherical function \((^{1-3})\) belonging to the representation \(T_g^\rho\). We note that
\[ f_\rho(k_1 g k_2)=f_\rho(g). \]
It is known that every function having this property can be expanded into an integral over the functions (1)
\[ f(g)=\int a(\rho)f_\rho(g)\,d\mu(\rho), \]
where
\[ \int |f(g)|^2\,dg=\int |a(\rho)|^2\,d\mu(\rho). \]
The aim of the present work is to compute the Plancherel measure \(d\mu(\rho)\) for the case when \(G\) is the group of all real unimodular transformations of \(n\)-dimensional space.
2. We shall describe the integral representation, needed in what follows, of these zonal spherical functions \((^{1,5})\).** Denote by \(X\) the group of lower triangular matrices with ones on the diagonal
\[ x= \left\| \begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0\\ x_{21} & 1 & 0 & \cdots & 0\\ \vdots & & & & \\ x_{n1} & x_{n2} & x_{n3} & \cdots & 1 \end{array} \right\| \qquad (x\in X). \]
Let \(x_1,x_2,\ldots,x_n\) be its column vectors, and let \(D_p(x)\) be the Gram determinant of the first \(p\) columns of the matrix \(x\).
* It is known that, up to a scalar factor, the vector \(\xi_0\) is unique.
** These functions have been computed explicitly on the manifold \(G/K\) when \(G\) is a complex semisimple group \((^1)\) and on Grassmann manifolds \((^4)\).
Spherical functions are functions on the group of diagonal matrices with positive entries:
\[ \varepsilon = \left\| \begin{array}{ccccc} \varepsilon_1 & 0 & 0 & \cdots & 0\\ 0 & \varepsilon_2 & 0 & \cdots & 0\\ \vdots & & & & \\ 0 & 0 & 0 & \cdots & \varepsilon_n \end{array} \right\|, \qquad \varepsilon_1>0,\ldots,\varepsilon_n>0; \qquad \varepsilon_1\varepsilon_2\cdots\varepsilon_n=1. \]
Consider the matrix \(\varepsilon x\varepsilon^{-1}\), equal to
\[ \xi = \left\| \begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0\\ \xi_{21} & 1 & 0 & \cdots & 0\\ \vdots & & & & \\ \xi_{n1} & \xi_{n2} & \xi_{n3} & \cdots & 1 \end{array} \right\|, \qquad \text{where}\quad \xi_{ij}=x_{ij}\frac{\varepsilon_j}{\varepsilon_i}. \]
Let \(\vec{\xi}_1,\vec{\xi}_2,\ldots,\vec{\xi}_n\) be the columns of the matrix \(\varepsilon x\varepsilon^{-1}\), and let \(\Delta_p(\xi)\) be the determinant of the Gram matrix of the first \(p\) columns of the matrix \(\varepsilon x\varepsilon^{-1}\). Thus,
\[ D_p(x)= \left| \begin{array}{ccc} (x_1x_1) & \cdots & (x_1x_p)\\ \vdots & & \\ (x_px_1) & \cdots & (x_px_p) \end{array} \right|, \]
\[ \Delta_p(\xi)= \left| \begin{array}{ccc} (\vec{\xi}_1\vec{\xi}_1) & \cdots & (\vec{\xi}_1\vec{\xi}_p)\\ \vdots & & \\ (\vec{\xi}_p\vec{\xi}_1) & \cdots & (\vec{\xi}_p\vec{\xi}_p) \end{array} \right| \qquad (p=1,\ldots,n). \]
It is easy to see that
\[ D_1\geq 1,\ldots,D_{n-1}\geq 1,\qquad D_n=1. \]
Denote by \(dx\) the product of all differentials \(dx_{ij}\):
\[ dx=\prod_{i>j} dx_{ij}. \]
Theorem 1. The zonal spherical functions of positive-definite type on the symmetric space \(SL(n;\mathbb R)/SO(n;\mathbb R)\) are given by the integrals
\[ \Phi_\lambda(\varepsilon) = \varepsilon_1^{\,i\lambda_1-\frac{n-1}{2}} \varepsilon_2^{\,i\lambda_2-\frac{n-2}{2}} \cdots \varepsilon_{n-1}^{\,i\lambda_{n-1}+\frac{n-2}{2}} \varepsilon_n^{\,i\lambda_n+\frac{n-1}{2}} \times \]
\[ \times \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\cdots \int_{-\infty}^{+\infty} \frac{ \Delta_1^{\,i\frac{\lambda_1-\lambda_2}{2}-\frac12} \cdots \Delta_{n-1}^{\,i\frac{\lambda_{n-1}-\lambda_n}{2}-\frac12} }{ D_1^{\,i\frac{\lambda_1-\lambda_2}{2}+\frac12} \cdots D_{n-1}^{\,i\frac{\lambda_{n-1}-\lambda_n}{2}+\frac12} } \,[dx], \tag{2} \]
where
\[ [dx]=\frac{1}{c}\,dx,\qquad c=\prod_{p=1}^{n-1}\left[B\left(\frac p2;\frac12\right)\right]^{n-p}, \tag{3} \]
\(\lambda_1,\ldots,\lambda_n\) are real numbers.
The constant \(c\) is chosen so that
\[ \int_X \frac{[dx]}{D_1D_2\cdots D_{n-1}}=1. \]
Thus,
\[ c=\int_X \frac{dx}{D_1D_2\cdots D_{n-1}}. \]
Zonal spherical functions in integral form are known from the works of I. M. Gelfand and M. A. Naimark \((^{1,2})\). The proof of Theorem 1 is based on one result of Harish-Chandra (see \((^5)\), No. 2, Corollary 1 to Lemma 44 (p. 288)).
- In the work \((^5)\) it is proved that the Plancherel measure on the symmetric space \(G/K\) is closely connected with the asymptotic behavior of zonal spherical functions on this space. Namely, the zonal spherical function \(f_\rho(h)\), as \(h \to \infty\) on the Cartan subgroup, has the asymptotic form
\[ f_\rho(h)\simeq c(\rho)e^{i(\rho,t)} \]
and the Plancherel measure \(d\mu\) for the space \(G/K\) is given by the formula
\[ d\mu=\frac{1}{|c(\lambda)|^2}\,d\lambda, \]
where \(d\lambda\) denotes Euclidean measure on the \(l\)-dimensional real Euclidean space which parametrizes the family of zonal spherical functions of positive-definite type on the symmetric space \(G/K\) (\(l=\operatorname{rank} G/K\)). Everything reduces to finding the function \(c(\lambda)\).
In the case under consideration, \(\varepsilon_j/\varepsilon_i\to 0\) for \(i>j\), and, therefore, \(\xi_{ij}\to 0\). Hence it follows:
Theorem 2.
\[ c(\lambda)=\frac{1}{c}C_n(\lambda), \]
where
\[ C_n(\lambda)= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \cdots \int_{-\infty}^{+\infty} D_1^{-i\frac{\lambda_1-\lambda_2}{2}-\frac12} \cdots D_{n-1}^{-i\frac{\lambda_{n-1}-\lambda_n}{2}-\frac12}\,dx, \tag{4} \]
\[ \lambda=(\lambda_1,\ldots,\lambda_n). \]
The coefficient \(c\) is determined by formula (3).
The evaluation of the integral (4) will be carried out inductively. Indeed, the following theorem holds:
Theorem 3. The coefficients \(C_n(\lambda_1,\ldots,\lambda_n)\) and \(C_{n-1}(\lambda_1,\ldots,\lambda_{n-1})\), corresponding to the groups \(SL(n;\mathbb R)\) and \(SL(n-1;\mathbb R)\), respectively, are connected by the relation
\[ C_n(\lambda)= B\left(i\frac{\lambda_1-\lambda_n}{2};\frac12\right) B\left(i\frac{\lambda_2-\lambda_n}{2};\frac12\right)\cdots \]
\[ \cdots B\left(i\frac{\lambda_{n-1}-\lambda_n}{2};\frac12\right) C_{n-1}(\lambda_1,\ldots,\lambda_{n-1}). \]
Let us note that, if \(r\) takes real values,
\[ \left|B\left(ir;\frac12\right)\right|^2 = \frac{|\Gamma(1/2)|^2|\Gamma(ir)|^2}{|\Gamma(ir+1/2)|^2} = \pi\,\frac{1}{r\,\operatorname{th}(r\pi)}. \]
From Theorem 3 the final result immediately follows:
Theorem 4. The Plancherel measure in the case of the symmetric space \(SL(n;\mathbb R)/SO(n;\mathbb R)\) is given by the formula
\[ d\mu= \frac{c^2}{\pi^{\frac{n(n-1)}{2}}} \prod_{1\le p\le q\le n} \frac{\lambda_p-\lambda_q}{2}\, \operatorname{th}\frac{\lambda_p-\lambda_q}{2}\,\pi\, d\lambda_1\cdots d\lambda_{n-1} \]
\[ (\lambda_1+\cdots+\lambda_n=0). \]
The coefficient \(c\) is determined by formula (3).
The asymptotics of zonal spherical functions on \(SL(n; R)/SO(n; R)\), for \(n=3\), was previously obtained by F. I. Karpelevich by another method.
In conclusion I take this opportunity to express my sincere gratitude to F. I. Karpelevich for his constant assistance, valuable guidance, and fruitful discussions.
Received
22 III 1960
REFERENCES
\({}^{1}\) I. M. Gelfand, M. A. Naimark, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 36 (1950).
\({}^{2}\) I. M. Gelfand, M. A. Naimark, Tr. Mosk. matem. obshch., 1, 423 (1952).
\({}^{3}\) F. A. Berezin, I. M. Gelfand, Tr. Mosk. matem. obshch., 5, 311 (1956).
\({}^{4}\) F. A. Berezin, F. I. Karpelevich, DAN, 118, No. 1 (1958).
\({}^{5}\) Harish-Chandra, Am. J. Math., 80, No. 2, No. 3 (1958).