|Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1999: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Let G be a semisimple Lie group with finite center, G=KAN an Iwasawa decomposition of G and P=MAN be a minimal parabolic subgroup. Let τ be an irreducible representation of the maximal compact subgroup K. Let EィイD2τィエD2 be the vector bundle associated to τ and let DィイD2τィエD2 be the algebra of the invariant differential operators on EィイD2τィエD2. Let σ be an irreducible representation of M which appears in the restriction τィイD2MィエD2 of τ to M. Take an linear form λ of the complexification of the Lie algebra α of A. Then for the triple (τ, σ, λ), a finite dimensional representation XィイD2τ,σ,λィエD2 of the algebra DィイD2τィエD2 is defined. We denote by εィイD1∞ィエD1(EィイD2τィエD2, X) the space of the functions (sections) of the vector bundle EィイD2τィエD2 which increase at most exponentilly. Let V be the representation space of τ, V(ィイD2σィエD2 the subspace of the vectors in V which transform according to σ and let FィイD2(σ),λィエD2 be the vector bundle associated to (VィイD2(σ),ィエD2λ) over G/P. Then we have a so-called Poisson tranform PィイD2τ,σ,λィエD2 from CィイD1∞ィエD1 (FィイD2(σ),λィエD2), the space of all CィイD1∞ィエD1 sections fo FィイD2(σ),λィエD2, into εィイD1∞ィエD1 (EィイD2τィエD2, X).
Now, we assume that for a fixed λ, all XィイD2τ,σ,λィエD2 (σ appearing in τィイD2MィエD2) are irreducible and inequivalent each other. Then, as is the case of the symmetric spaces, we have the injectivity and surjectivity of the map PィイD2τ,σ,λィエD2 from CィイD1∞ィエD1 (FィイD2(σ),λィエD2) to εィイD1∞ィエD1 (EィイD2τ,ィエD2 X) for almost all λ.
The above result is not satisfactory for us, since it does not treat all of the eigensections. But it has a meaning that the result is described only by τ, σ, λ instead of the center of the universal enveloping algebra.