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Abstract
A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with exponential map , Exp, from L G/L H to G/H, an action, : K → Autн(G) (where Autн (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, # , of K on L G/L H defined by g L (g), along with a quadruple (s, E, j, E#), where s is a # - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C∞ (Exp s), j Є C∞ (s), and E# is a test-function subspace of C∞ (s) which contains { j.f Exp: f Є E }.
Of interest is the question: Is the function : Φ Φυ, where υ: f j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s C, to F, with the usual convolution for its multiplication (where F is the space of all - invariant distributions of E and F# is the space of all # - invariant distributions of E#)?
For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be absorbed into the function e. Also, when K is compact, we can show that
∩ ker Φ = { f Є E : ∫k f (g) dg = 0}.
These results turn out to be very useful for calculations on s2 ≠ G/H, where G = SO(3) and H≤ SO(3) with H ≠ SO(2) with : h Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(ŗ,ŋ) = e(ŋ,ŗ).