Homological Finiteness of Representations of Almost Linear Nash Groups

Let G be an almost linear Nash group, namely, a Nash group that admits a Nash homomorphism with finite kernel to some GL k (R).A smooth Frchet representation V with moderate growth of G is called homologically finite if the Schwartz homology H S i (G; V ) is finite dimensional for every i Z.We show that the space of Schwartz sections (X, E) of a tempered G -vector bundle (X, E) is homologically finite as a representation of G , under some mild assumptions.