S. G. Gindikin and F. I. Karpelevich

THE PLANCHEREL MEASURE FOR RIEMANNIAN SYMMETRIC SPACES OF NONPOSITIVE CURVATURE

(Presented by Academician P. S. Aleksandrov, February 21, 1962)

Let \(G\) be a connected semisimple Lie group with finite center, and let \(U\) be its maximal compact subgroup. Denote by \(\mathscr E\) the homogeneous space \(G/U\). As is known, \(\mathscr E\) is a Riemannian symmetric space of nonpositive curvature. In the present paper an explicit form of the Plancherel measure for such spaces is found.

Harish-Chandra showed \((^1)\) that the Plancherel measure is closely related to the asymptotics of zonal spherical functions on \(\mathscr E\), and found an integral representation for the density of this measure. T. S. Bhanu-Murti \((^2)\) computed the explicit form of the density of the Plancherel measure for all (except one) classical symmetric spaces of maximal rank. The formulas obtained in \((^2)\) made it possible, to a certain extent, to predict the structure of the density of the Plancherel measure for arbitrary symmetric spaces of the type under consideration.

Let \(\mathfrak h\) be a Cartan subalgebra* of the space \(\mathscr E\); let \(\Sigma\) be the system of its roots relative to \(\mathfrak h\); let \(\Sigma^+\) \((\Sigma^-)\) be the set of positive (negative) roots for some ordering. Denote by \(\mathfrak z^+\) \((\mathfrak z^-)\) the subalgebras spanned by the root vectors corresponding to the positive (negative) roots. By the Iwasawa theorem, every element \(g\) of \(G\) can be represented in the form \(g=z^-hu\) \((z^-\in Z^-,\, h\in H,\, u\in U)\). The element \(h\) occurring in this decomposition will be denoted by \(h(g)\). For every complex-valued linear form \(\chi\) on \(\mathfrak h\) set \(h^\chi=\exp(\chi(\xi))\), where \(\xi=\ln h\) is the preimage of the element \(h\) under the canonical mapping of the algebra \(\mathfrak h\) onto the group \(H\). Using the Cartan scalar product \((\xi_1,\xi_2)\), which is positive definite on \(\mathfrak h\), we shall identify the form \(\chi\) with the vector \(\chi=\xi+i\eta\) \((\xi,\eta\in\mathfrak h)\).

Let \(R\) be an arbitrary open half-space in \(\mathfrak h\), bounded by a hyperplane passing through the origin of the coordinates. Put
\[ \Sigma_R=\Sigma\cap R,\qquad \Sigma_R^+=\Sigma^+\cap R,\qquad \Sigma_R^-=\Sigma^-\cap R . \]
Denote by \(\mathfrak z_R,\mathfrak z_R^+,\mathfrak z_R^-\) the subalgebras spanned by the root vectors corresponding to the roots from \(\Sigma_R,\Sigma_R^+,\Sigma_R^-\), respectively. Consider the integral**
\[ I(\mathscr E,R,\nu)=\int h^\chi(z_R^+)\,dz_R^+, \]
where \(\chi=-(\rho+\nu)\),
\[ \rho=\frac12\sum_{\alpha\in\Sigma^+}\alpha \]
(in this sum each root occurs as many times as its multiplicity). If \(\Sigma_R^+\) is empty, then we put \(I(\mathscr E,R,\nu)=1\). Denote by \(\Delta_R\) the set of vectors \(\xi\in\mathfrak h\) such that \((\xi,\alpha)>0\) for all \(\alpha\in\Sigma_R^+\). It will be shown below that the integral \(I(\mathscr E,R,\nu)\) converges absolutely if \(\operatorname{Re}\nu\in\Delta_R\). In the case when \(\Sigma_R^+=\Sigma^+\), and hence \(\mathfrak z_R^+=\mathfrak z^+\), we shall denote the integral \(I(\mathscr E,R,\nu)\) simply by \(I(\mathscr E,\nu)\), and the set \(\Delta_R\) by \(\Delta\). The convergence of the integral

* We agree to denote Lie groups by capital Latin letters, and their Lie algebras by the corresponding lowercase Gothic letters.

** Everywhere in this article, \(\int f(p)\,dp\) denotes the integral over the whole group \(P\), where \(dp\) is the invariant measure on \(P\).

\(I(\mathscr E,\nu)\) for \(\operatorname{Re}\nu\in\Delta\) is proved in \((1^a)\). We note that if the rank of the space \(\mathscr E\) is equal to 1 and \(\Sigma_R^+\) is nonempty, then \(\Sigma_R^+=\Sigma^+\), and hence \(I(\mathscr E,R,\nu)=I(\mathscr E,\nu)\).

To each vector \(\lambda\in\Delta\) there corresponds a certain unitary representation of the group \(G\), realized in spherical functions on the space \(\mathscr E\). The regular representation of the group \(G\) in \(L_2\) on \(\mathscr E\) decomposes into a continuous direct sum of the indicated representations. In this case the Plancherel measure has the form \(\mu(d\lambda)=|c(\mathscr E,\lambda)|^{-2}d\lambda\), where \(d\lambda\) is Euclidean measure in \(\Delta\) and, as follows from (1), \(c(\mathscr E,\lambda)=I(\mathscr E,i\lambda)(I(\mathscr E,\rho))^{-1}\). The integral \(I(\mathscr E,i\lambda)\) diverges. It must be understood as the analytic continuation of \(I(\mathscr E,\nu)\) in \(\nu\). Thus the problem of computing the Plancherel measure reduces to computing the integral \(I(\mathscr E,\nu)\). We compute an even more general integral \(I(\mathscr E,R,\nu)\). It turns out that this integral is equal to the product of analogous integrals for certain symmetric spaces of rank 1 associated with the space \(\mathscr E\). Namely, denote by \(\Sigma_0^+\) the subset of positive roots which are not integral multiples of other positive roots. With each root \(\alpha\in\Sigma_0^+\) we associate a symmetric space \(\mathscr E_\alpha\) in the following way. Denote by \(\mathfrak G_\alpha\) the subalgebra generated by all root vectors corresponding to the roots \(\alpha\) and \(-\alpha\). The subalgebra \(\mathfrak G_\alpha\) is semisimple. It is invariant under the involutive automorphism which singles out the subalgebra \(\mathfrak U\) in \(\mathfrak G\). The subalgebra \(\mathfrak U_\alpha=\mathfrak U\cap\mathfrak G_\alpha\) is a maximal compact subalgebra in \(\mathfrak G_\alpha\). The symmetric space \(\mathscr E_\alpha=G_\alpha/U_\alpha\) has rank 1. As a Cartan subalgebra of the space \(\mathscr E_\alpha\) one may take the line \(\mathfrak H_\alpha\) in \(\mathfrak H\) on which the root \(\alpha\) is situated.*

The principal result of the present paper is the following theorem.

Theorem 1. The integral \(I(\mathscr E,R,\nu)\) converges absolutely if and only if \(\operatorname{Re}\nu\in\Delta_R\), and, with a suitable normalization of measures,

\[ I(\mathscr E,R,\nu)=\prod_{\alpha\in\Sigma_0^+\cap R} I(\mathscr E_\alpha,\nu_\alpha), \tag{1} \]

where \(\nu_\alpha\) is the restriction of the form \((\nu,\mathfrak H)\) to the line \(\mathfrak H_\alpha\).

From Theorem 1 it follows immediately that

Corollary. The formula
\[ c(\mathscr E,\lambda)=\prod_{\alpha\in\Sigma_0^+} c(\mathscr E_\alpha,\lambda_\alpha) \]
is valid.

For all spaces \(\mathscr E\) of rank 1 the integral \(I(\mathscr E,\nu)\) is easily computed directly.** After substituting the resulting expressions for \(I(\mathscr E_\alpha,\nu_\alpha)\) in (1), we obtain Theorem 2.

Theorem 2. With a suitable normalization of the measure \(dz_R^+\) on the group \(Z_R^+\), we have:

\[ I(\mathscr E,R,\nu)=\prod_{\alpha\in\Sigma_R^+} \mathrm B\left(\frac{p_\alpha}{2},\frac{p_{\alpha/2}}{4}+\frac{(\nu,\alpha)}{(\alpha,\alpha)}\right), \]

where \(p_\alpha\) is the multiplicity of the root \(\alpha\), and \(\mathrm B(x,y)\) is the beta function.

We shall precede the proof of Theorem 1 by two lemmas.

Lemma 1. Let a finite-dimensional algebra \(\mathfrak R\) be decomposed into the direct sum of two subspaces \(\mathfrak A\) and \(\mathfrak B\). Suppose that in \(\mathfrak R\) there exists a system of subspaces \(I_0,\ldots,I_{l+1}\) such that \(\mathfrak R=I_0\supset I_1\supset\cdots\supset I_{l+1}=\{0\}\), \([\mathfrak R,I_k]\subset I_{k+1}\), and \(I_k=\mathfrak A_k+\mathfrak B_k\), where \(\mathfrak A_k=\mathfrak A\cap I_k,\ \mathfrak B_k=\mathfrak B\cap I_k\). Choose a basis in the space \(\mathfrak A_l\) \((\mathfrak B_l)\), extend it to a basis in \(\mathfrak A_{l-1}\) \((\mathfrak B_{l-1})\), and so on. The aggregate of the coordinates of a vector \(\mathfrak n\in\mathfrak R\) with respect to the basis vectors,

* We note that if \(\mathscr E\) is a space of maximal rank, then the subalgebra \(\mathfrak G_\alpha\) is a three-dimensional subalgebra associated with the root \(\alpha\).

** See also \((1^a)\). We note that the final formula for spaces of rank 1, given in \((1^a)\) (no. 13, p. 303), contains an inaccuracy.

lying in $\mathfrak A_k(\mathfrak B_k)$, but not lying in $\mathfrak A_{k+1}(\mathfrak B_{k+1})$, by $a_k(n)$, $(b_k(n))$. Then
\[ \exp(n)=\exp(a_0)\ldots \exp(a_l)\exp(b_l)\ldots \exp(b_0)^*, \]
where $a_k\in\mathfrak A_k$, $b_k\in\mathfrak B_k$ and the coordinates $a_i(a_k)$ $(b_i(b_k))$ for $i\ne k$ are equal to zero. Moreover, if
\[ a_k(n)=a_k,\quad b_k(n)=b_k,\quad a_k(\hat a_k)=a'_k,\quad b_k(\hat b_k)=b'_k, \]
then
\[ a'_k=a_k+f_k(a_0,\ldots,a_{k-1};b_0,\ldots,b_{k-1}),\quad b'_k=b_k+\varphi_k(a_0,\ldots,a_{k-1},b_0,\ldots,b_{k-1}). \tag{2} \]

The proof of Lemma 1 is easily obtained, for example, by means of the Campbell–Hausdorff formula.

Remark. Let us note that the conditions of Lemma 1 are satisfied if $\mathfrak A$ is an ideal in $\mathfrak N$, and $\mathfrak B$ is an arbitrary complementary subspace. In this case, as the system of subspaces $I_k$ one may take the subspaces
\[ \mathfrak N=I_0\supset \mathfrak A\supset[\mathfrak N,\mathfrak A]\supset[\mathfrak N,[\mathfrak N,\mathfrak A]]\supset\cdots . \]
Moreover, $\mathfrak B_k=\{0\}$ for $k\ge1$.

Lemma 2. Let some set of roots $T\subset\Sigma$ of the space $\mathcal E$ be invariant with respect to some root $\alpha$ (i.e. if $\beta\in T$ and $\beta+k\alpha\in\Sigma$, then $\beta+k\alpha\in T$), and let $\delta=\sum_{\beta\in T}\beta$. Then $(\alpha,\delta)=0$.

Proof. Let $S_\alpha$ be the reflection in the plane perpendicular to the root $\alpha$. As is known, $S_\alpha$ is an element of the Weyl group of the space $\mathcal E$. By virtue of the invariance of $T$ with respect to $\alpha$, we have $S_\alpha T=T$, since
\[ S_\alpha\beta=\beta-2(\alpha,\beta)(\alpha,\alpha)^{-1}\cdot\alpha . \]
Consequently, $S_\alpha\delta=\delta$. On the other hand,
\[ S_\alpha\delta=\delta-2(\alpha,\delta)(\alpha,\alpha)^{-1}\cdot\alpha, \]
whence $(\alpha,\delta)=0$.

We now pass to the proof of Theorem 1. First note that formula (1) reduces the assertion on the convergence of $I(\mathcal E,R,\nu)$ to the corresponding assertion for a space of rank 1, whose validity is easily established directly. Thus it remains only to prove formula (1). We shall carry out the proof by induction on the number $k(R)$ of vectors contained in $\Sigma_0^+\cap R$. For $k(R)=0$ Theorem 1 is obvious. Suppose now that Theorem 1 is valid for all half-spaces $Q$ for which $k(Q)<k(R)$. For the half-space $R$ one can construct such an open half-space $Q$, bounded by a hyperplane $L$ passing through the origin, that
\[ \Sigma_Q^+\subset\Sigma_R^+,\quad \Sigma_R^+\setminus\Sigma_Q^+\subset L \]
and $\Sigma_0^+\cap L$ consists of one root $\alpha$. With this root $\alpha$ there is associated, in the manner indicated above, the symmetric space $\mathcal E_\alpha=G_\alpha/U_\alpha$ of rank 1 with Cartan subalgebra $\mathfrak h_\alpha$. In $\mathfrak G_\alpha$ choose subalgebras $\mathfrak Z_\alpha^+$ and $\mathfrak Z_\alpha^-$, analogous to the subalgebras $\mathfrak Z^+$ and $\mathfrak Z^-$ of the algebra $\mathfrak G$. It is easy to see that
\[ \mathfrak Z_R^+=\mathfrak Z_Q^++\mathfrak Z_\alpha^+ \]
and $\mathfrak Z_Q^-$ is an ideal in $\mathfrak Z_R^-$. On the basis of the remark to Lemma 1, every element $z_R^+\in Z_R^+$ can be represented in the form $z_R^+=z_Q^+z_\alpha^+$. In canonical coordinates the invariant measure on the nilpotent Lie group is Euclidean measure. From (2) it follows that
\[ dz_R^+=dz_Q^+dz_\alpha^+. \]
Therefore
\[ I(\mathcal E,R,\nu)=\int h^\chi(z_Q^+z_\alpha^+)\,dz_Q^+dz_\alpha^+. \]
The element $z_\alpha^+$ belongs to the half-simple group $G_\alpha$ and in it decomposes as:
\[ z_\alpha^+=z_\alpha h_\alpha(z_\alpha^+)u_\alpha . \]
Let us now observe that $[\mathfrak G_\alpha,\mathfrak Z_Q]\subset\mathfrak Z_Q$. Therefore
\[ (z_\alpha^-)^{-1}z_Q^+z_\alpha^-=z_Q(z_Q^+,z_\alpha^+)=z_Q\in Z_Q. \]
Obviously,
\[ h^\chi(z^-gu)=h^\chi(g), \]
where $z^-\in Z^-$ and $u\in U$. Hence
\[ I(\mathcal E,R,\nu)=\int h^\chi(z_Qh_\alpha(z_\alpha^+))\,dz_Q^+dz_\alpha^+. \]

For every closed half-space $P\subset Q$ put $\Xi_P=P\cap\Sigma$. There is only a finite number of different nonempty systems of roots $\Xi_P$. Number them in decreasing order so that
\[ \Sigma_Q=\Xi_0\supset\Xi_1\supset\ldots\supset\Xi_l . \]
Let $I_k$ be the subalgebra in $\mathfrak Z_Q$ spanned by the root vectors corresponding to roots from $\Xi_k$. We have
\[ \mathfrak Z_Q=I_0\supset I_1\supset\ldots\supset I_{l+1}=\{0\}. \]
It is obvious that if $I_i$ corresponds to $\Xi_i=\Xi_{P_i}$, and $I_j$ corresponds to $\Xi_j=\Xi_{P_j}$, and if the half-space $P_k$ is the arithmetic sum of the half-spaces $P_i$

* By $\exp(n)$ we denote the canonical mapping of the algebra $\mathfrak N$ onto the corresponding simply connected group $N$.

and \(P_l\), then \([I_i,I_j]\subset I_k\), where \(I_k\) corresponds to \(\Xi_k=\Xi_{P_k}\). Hence it follows that \([\mathfrak Z_Q,I_k]\subset I_{k+1}\). Put \(\mathfrak A=\mathfrak Z_Q^{-}\) and \(\mathfrak B=\mathfrak Z_Q^{+}\). The subalgebra \(\mathfrak Z_Q\) is nilpotent. The subspaces \(\mathfrak A,\mathfrak B\), and \(I_k\) satisfy the conditions of Lemma 1; moreover, as the basis appearing in its formulation we may choose root vectors. Consequently,

\[ z_Q=z_Q\left(z_Q^{-},z_\alpha^{+}\right)=z_Q^{-}\widetilde z_Q^{+}, \]

where \(z_Q^{-}\in Z_Q^{-}\), \(\widetilde z_Q^{+}\in Z_Q^{+}\). Let us now note that \([\mathfrak G_\alpha,I_k]\subset I_k\). Hence it follows that the canonical coordinates \(a_k(z_Q)=a_k(\ln z_Q)\) and \(b_k(z_Q)=b_k(\ln z_Q)\) are expressed linearly only in terms of the coordinates \(b_i(\widetilde z_Q^{+})=b_i(\ln z_Q^{+})\), where \(i\leq k\), and, by virtue of the nilpotency of the group \(Z_\alpha^{-}\), the transition matrix from \(b_i(z_Q^{+})\) to \(b_k(z_Q)\) is nilpotent. Therefore (see (2))

\[ b_i(\widetilde z_Q^{+})=b_k+\psi_k(b_0,\ldots,b_{k-1}), \]

where \(b_i=b_i(z_Q^{+})\). Thus, \(dz_Q^{+}=d\widetilde z_Q^{+}\). As a result we obtain

\[ I(\mathcal E,R,\nu)=\int h^\chi\!\left(z_Q^{+}h_\alpha(z_\alpha^{+})\right)\,dz_Q^{+}\,dz_\alpha^{+}. \]

Next,

\[ z_Q^{+}h_\alpha(z_\alpha^{+})=h_\alpha(z_\alpha^{+})\,\widehat z_Q^{+}, \]

where

\[ \widehat z_Q^{+}=h_\alpha^{-1}(z_\alpha^{+})\,z_Q^{+}h_\alpha(z_\alpha^{+})\in Z_Q^{+}. \]

The measures \(dz_Q^{+}\) and \(d\widehat z_Q^{+}\) are connected by the relation

\[ dz_Q^{+}=h_\alpha^{\vartheta}(z_\alpha^{+})\,d\widehat z_Q^{+}, \]

where \(\vartheta=\sum_{\beta\in\Sigma_Q^{+}}\beta\). Let us now note that, if \(h_0\in H\), then

\[ h^\chi(h_0g)=h_0^\chi\cdot h^\chi(g). \]

Consequently,

\[ I(\mathcal E,R,\nu) = \int h_\alpha^{\chi+\vartheta}(z_\alpha^{+})\,dz_\alpha^{+}\cdot \int h^\chi(\widehat z_Q^{+})\,d\widehat z_Q^{+}. \]

It is obvious that

\[ h_\alpha^{\chi+\vartheta}(z_\alpha^{+}) = h_\alpha^{\chi'+\vartheta'}(z_\alpha^{+}), \]

where \(\chi',\vartheta'\) are the projections of the vectors \(\chi\) and \(\vartheta\) onto the line \(\mathfrak H_\alpha\). We have:

\[ -\rho+\vartheta=-\rho_\alpha+\frac12\sum_{\beta\in\Sigma_Q}\beta, \]

where

\[ \rho_\alpha=\frac12\sum_{\beta\in\Sigma^{+}\cap L}\beta, \]

i.e. \(\rho_\alpha\) is the vector appearing in the definition of the integral \(I(\mathcal E_\alpha,\nu_\alpha)\). The set of roots \(\Sigma_Q\) is invariant with respect to the root \(\alpha\). Therefore, by Lemma 2, the projection of the vector \(-\rho+\vartheta\) onto the line \(\mathfrak H_\alpha\) coincides with the vector \(-\rho_\alpha\). The projections of the vectors onto the line \(\mathfrak H_\alpha\) correspond to the restrictions of the corresponding forms to this line. Thus,

\[ I(\mathcal E,R,\nu)=I(\mathcal E_\alpha,\nu_\alpha)\,I(\mathcal E,Q,\nu), \]

and the validity of Theorem 1 for the integral \(I(\mathcal E,R,\nu)\) follows directly from the induction hypothesis.

Received
14 II 1962

REFERENCES

1. Harish-Chandra, a) Am. J. Math., 80, No. 2 (1958); b) 80, No. 3 (1958).
2. T. S. Bhanu-Murti, a) DAN, 133, No. 3 (1960); b) 135, No. 5 (1960); c) Dissertation, Moscow State University, 1961.