By Tammo tom Dieck

This booklet is a jewel– it explains vital, beneficial and deep subject matters in Algebraic Topology that you simply won`t locate in other places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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F (exp txixi)] (a:n) o:. f (exp txixi)] (a:n) o:. f ((exp txixi) o)] (0) (0) (0) = (Do:)o(f). Therefore, we have fl(X1 o:1 • ... 2) is identical with the linear isomorphism of S(mf onto g;(G/K) obtained from the local coordinate system (x 1 , ••• , xn) in §2. (2) Making the same argument as above at the point x 0 o, we obtain (2). EXAMPLE 2. Let B be a K-invariant nondegenerate symmetric bilinear form on m. Let { X 1 , ••• , Xn} be a basis of m, and let (bi\~i ,J~n be the inverse of the matrix of B with respect to this basis.

We call Ge and (G / Kf the complexifications of G and of G / K , respectively (for these refer to Chevalley [3] and lwahori-Sugiura [15]). We shall denote by C( G, K) the space consisting of functions f E C( G/ K) which satisfy Lkf = f for any k E K . Then C (G, K) is a closed subalgebra of the Banach algebra C( G/ K) . Similarly, we shall denote by L 2 ( G, K) the space of functions f E L 2 ( G/ K) which satisfy Lkf = f for any k E K . 20 I. SPHERICAL FUNCTIONS Then L 2 ( G, K) is a closed subspace of the Hilbert space L 2 ( G / K) , and C( G , K) is a dense subspace of L 2 ( G , K) .

DIFFERENTIAL OPERATORS 31 isomorphism is given as follows: take a basis {u 1 , ••• , un} of V ; given an h-tuple (i 1 , ... , ih) of integers with 1 ~ i 1 , ••• , ih ~ n, we shall write ' } -_ { lQl , ••• , { i·1 , ••• , ih nan} if the h-tuple contains 1 a 1 times, ... 3) defines a graded algebra isomorphism of S(V) onto F[X1 , ••• , Xn], where the sum of left-hand side runs through all h-tuples (i 1 , ••• , ih) with {ii ' • • • ' ih} = { 1QI ' • • • ' nan } • Let T(M) be the tangent bundle of M, and let T(Mf be its complexification.

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