MATHEMATICS

F. A. BEREZIN and F. I. KARPELEVICH

ZONAL SPHERICAL FUNCTIONS AND LAPLACE OPERATORS ON SOME SYMMETRIC SPACES

(Presented by Academician P. S. Aleksandrov, 24 VI 1957)

Let (\mathfrak{M}=G/H) be a homogeneous space, whose stationary subgroup (H) we shall assume to be compact. Consider on (\mathfrak{M}) a differential operator (\Delta) that commutes with the translation operators (f(M)\to f(gM)). Such differential operators form a ring and are called Laplace operators on (\mathfrak{M}) ((^1)). Let (R) be the set of functions on (\mathfrak{M}) that are constant on the transitivity surfaces of the subgroup (H). (It is clear that (R) may be interpreted as the set of functions constant on the double cosets of the group (G) with respect to (H).) If (\Delta) is a Laplace operator and (f\in R), then (\Delta f\in R). Consequently, every Laplace operator induces a certain differential operator on (R), which we call the radial part of the operator (\Delta) and denote by (\overset{0}{\Delta}). In the present paper the radial parts of Laplace operators and the zonal spherical functions belonging to irreducible representations are computed for certain symmetric spaces, namely: for the space (\mathfrak{M}^{+}{n,k}) ((n\ge 2k))—the manifold of (k)-dimensional subspaces of an (n)-dimensional complex space (the so-called complex Grassmann spaces), and also for two other symmetric spaces (\mathfrak{M}^{-}}) and (\mathfrak{M}^{0{n,k}), which are closely connected with (\mathfrak{M}^{+}) and will be described below.

If the space (\mathfrak{M}) is symmetric, then the spherical functions (\varphi) belonging to irreducible representations of the group (G) satisfy the system of equations ((^1))

[
\Delta \varphi_i^n=\lambda(\Delta)\varphi,
\tag{1}
]

where (\Delta\to \lambda(\Delta)) is a homomorphism of the ring of Laplace operators into the field of complex numbers.

In the case when (\varphi) is a zonal spherical function, the system (1) can be rewritten in the form

[
\overset{0}{\Delta}\varphi=\lambda(\Delta)\varphi.
\tag{2}
]

Let us now note that if the group (G) of motions of the symmetric space (\mathfrak{M}) is semisimple, then every element (g) of (G) is representable in the form

[
g=h_1\varepsilon h_2,
\tag{3}
]

where (h_1) and (h_2) are in (H), and (\varepsilon) belongs to some commutative subgroup (E) ((^2)). The set of inner automorphisms (g\to hgh^{-1}) ((h\in H)) of the group (G) that carry (E) into itself induces on (E) some finite group (S) of automorphisms of (E). Formula (3) assigns to each element (g) of (G) a finite number of elements (\varepsilon) of (E). The elements (\varepsilon_1) and (\varepsilon_2) correspond, by formula (3), to one and the same element (g) if and only if their squares

can be transformed into one another by means of the group (S). Hence it is clear that every function from (R), i.e., a function constant on double cosets with respect to (H), may be regarded as a function of (\varepsilon^2), and moreover one such that (f(\varepsilon^2)=f(s\varepsilon^2)) for every (s) in (S). It also follows from what has been said that the operator (\overset{0}{\Delta}) may be interpreted as a differential operator on (E).

Let now (\mathfrak{M}^{+}{n,k}) be the complex Grassmann space. The group of motions of (\mathfrak{M}^{+}) is the group (SU(n)) of unimodular unitary transformations of the (n)-dimensional complex space; the stationary subgroup (H) is the subgroup of (SU(n)) leaving invariant a certain (k)-dimensional subspace. In the group (E) one can introduce canonical coordinates (t_1,\ldots,t_k) so that the group (S) will consist of all possible permutations of the variables (t_1,\ldots,t_k) and all possible changes of sign of them.*

In the coordinates (t_1,\ldots,t_k), the radial parts of the Laplace operators on (\mathfrak{M}^{+}_{n,k}) are written in the form

[
\overset{0}{\Delta}=\frac{1}{j(t)}\,P(L_1,\ldots,L_k)\,j(t),
\tag{4}
]

where

[
j(t)=\prod_{p<q}\left(\sin^2 t_p-\sin^2 t_q\right),
]

[
L_p=\frac14\,\frac{\partial^2}{\partial t_p^2}
+\frac12\left[\operatorname{ctg} 2t_p+(n-2k)\operatorname{ctg} t_p\right]\frac{\partial}{\partial t_p},
]

(P(x_1,\ldots,x_k)) is an arbitrary symmetric polynomial in (x_1,\ldots,x_k).

Solving system (2) and taking into account that the zonal spherical function (\varphi) must be a polynomial in (e^{2it_p}) and possess the symmetry properties indicated earlier, we obtain**

[
\varphi=\frac{d(r_1,\ldots,r_k)}{j(t)}\,\lambda(r_1,\ldots,r_k),
\tag{5}
]

where

[
d(r_1,\ldots,r_k)=\det\left|J^{m}_{r_j}(x_i)\right|;\qquad
m=n-2k;\qquad x_i=\sin^2 t_i;
]

[
J^{m}_{r}(x)=F(m+r+1,-r,m+1;x);
]

(F(\alpha,\beta,\gamma;x)) is the hypergeometric function;

[
\lambda(r_1,\ldots,r_k)=
\frac{1}{\displaystyle\prod_{i<j}(\rho_i-\rho_j)}
\prod_{s=1}^{k-1}s!\,(m+s)^{\,k-s};
\qquad
\rho_i=-r_i(m+1+r_i).
]

Let us note that the functions (J^{m}{r}(x)) coincide, up to a constant factor, with the Jacobi polynomials (P^{(m,0)}(\cos 2t)). The numbers (r_1,\ldots,r_k) are connected with the homomorphism (\Delta\to\lambda(\Delta)) and with the highest weight of the representation to which (\varphi) belongs, as follows: if (\Delta) is given by formula (4), then

[
\lambda(\Delta)=P(\rho_1,\ldots,\rho_k),\qquad r_i=p_i+k-i,
\tag{6}
]

where (p_1,\ldots,p_k) are the numbers defining the representation. (If (l_1\geq l_2\geq\cdots\geq l_n) are the coordinates of the highest weight of the representation according to Cartan, and (\sum l_i=0), then (p_i=l_i=-l_{n-i+1}), (i=1,\ldots,k), while the remaining (l_i) are equal to 0.) From formula (6)

* (e^{\pm it_1},\ldots,e^{\pm it_k}) are the characteristic roots of the transformation from (E) (the remaining roots are equal to 1).

** The function (\varphi) is normalized so that (\varphi(0)=1).

it is seen that the homomorphism (\Delta \to \lambda(\Delta)) determines the highest weight of the representation realized in spherical functions. Formula (5) for the function (\varphi) may also be obtained by orthogonalizing the sequence of symmetric polynomials in (\sin^2 t_1,\ldots,\sin^2 t_k) with weight equal to
[
j(t)^2 \prod |\sin 2t_i|\,\sin^2 t_i .
]
(If the group (G) is compact, then the zonal spherical functions belonging to irreducible representations form a complete orthogonal system of functions in (R) with the weight induced by the invariant measure in (\mathfrak M). For the space (\mathfrak M^+_{n,k}) this weight is equal to
[
j(t)^2 \prod |\sin 2t_i|\,\sin^2 t_i .
]
)

Other spaces for which the radial parts of the Laplace operators and the zonal spherical functions are computed are the space (\mathfrak M^-{n,k}), Cartan-dual to (\mathfrak M^+), which may be regarded as a limiting case of (\mathfrak M^+}), and the space (\mathfrak M^0_{n,k{n,k}) and (\mathfrak M^-).

The group (G) of motions of the space (\mathfrak M^-{n,k}) is the group of all unimodular complex matrices of order (n) that preserve the form
[
\sum_1^k z_i \bar z_i-\sum^n z_i \bar z_i .
]
The stationary subgroup (H=G\cap SU(n)) evidently coincides with the stationary subgroup of the space (\mathfrak M^+{n,k}). The formulas for the Laplace operators on (\mathfrak M^-\,t).}) and for the zonal spherical functions may be obtained from formulas (4) and (5), respectively, if in them (t) is replaced by (\sqrt{-1

The relation between the homomorphism (\Delta\to\lambda(\Delta)) and the numbers (r_1,\ldots,r_k) is still given by formula (6). From formula (6) it is seen that the homomorphism (\Delta\to\lambda(\Delta)) uniquely determines the system of numbers ((\rho_1,\ldots,\rho_k)), considered up to all possible permutations. In turn, the system ((\rho_1,\ldots,\rho_k)) uniquely determines, by formula (5), the function (\varphi), and hence also the representation realized in spherical functions. For arbitrary complex (\rho_1,\ldots,\rho_k) the corresponding representation is a representation by means of bounded operators in a Banach space. For real integral (r_1,\ldots,r_k) this representation is finite-dimensional. If the system of numbers ((\rho_1,\ldots,\rho_k)) coincides with the system of numbers ((\bar\rho_{i_1},\ldots,\bar\rho_{i_k})) ((i_1,\ldots,i_k) is some permutation of the indices (1,\ldots,k)), then the representation admits an invariant Hermitian form, perhaps not positive definite.

The space (\mathfrak M^0_{n,k}) is the space of all complex matrices with (k) rows and (n-k) columns. The motions in (\mathfrak M^0_{n,k}) are given by the formula (A\to U_k A U_{n-k}^{-1}+B), where (U_k) and (U_{n-k}) are arbitrary unitary matrices of orders (k) and (n-k), respectively, connected by the relation
[
\det U_k\cdot \det U_{n-k}=1,
]
and (A) and (B) are in (\mathfrak M^0_{n,k}). Although the group of motions of the space (\mathfrak M^0_{n,k}) is not semisimple, for (\mathfrak M^0_{n,k}) a number of assertions of the theory of symmetric spaces with a semisimple group of motions hold. In particular, the spherical functions on (\mathfrak M^0_{n,k}) are eigenfunctions for the Laplace operators. Every element of (\mathfrak M^0_{n,k}) can be represented in the form (U_k E U_{n-k}^{-1}), where (E) is a matrix in (\mathfrak M^0_{n,k}) of the form
[
E=
\begin{pmatrix}
t_1 & & 0\
& \cdot & \
0 & & \
& t_k\,0\ldots 0
\end{pmatrix},
]
where the numbers (t_1,\ldots,t_k) are real and are determined up to an arbitrary permutation and an arbitrary change of sign. Therefore,

the operators (\overset{0}{\Delta}) are operators in the space of symmetric even functions of (t_1,\ldots,t_k).

The radial part of the Laplace operators on (\mathfrak M^0_{n,k}) has the form

[
\overset{0}{\Delta}=\frac{1}{j(t)}\,P(L_1,\ldots,L_k)\,j(t),
\tag{7}
]

where

[
j(t)=\prod_{p