By Euler L.

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Pr2 : Fix0 (φ ) → pr2 (M ) is a C ∞ submersion. Proof. Since the leaves of F are contained in the ﬁbers of pr2 , the tangent map pr2∗ induces a homomorphism pr2∗ : T (M ×G)/T F → T G. Take any (x, g) ∈ Fix0 (φ ). 1, the restrictions T(x,g) Fix0 (φ ) −−−−→ T(x,g) M /T(x,g)F ←−−−− V (x,g) . of the quotient map T (M × G) → T (M × G)/T F are isomorphisms. Moreover pr2∗ corresponds to pr2∗ by these isomorphisms. 1. 8, the measure given by Λ on Fix0 (φ ) is denoted by ΛFix0 (φ ) . The direct image pr2∗ ( · ΛFix0 (φ ) ) is supported in pr2 (M ) ∩ O.

Kordyukov [20] B. M¨ umken. On tangential cohomology of Riemannian foliations. Amer. J. Math. 128 (2006), 1391–1408. [21] A. Nestke and P. Zuckermann. The index of transversally elliptic complexes. Rend. Circ. Mat. Palermo 34, Suppl. 9 (1985), 165–175. [22] K. Nomizu. On the cohomology of compact homogeneous spaces of nilpotent lie groups. Ann. of Math. 59 (1954), 531–538. L. Reinhart. Foliated manifolds with bundle-like metrics. Ann. of Math. 69 (1959), 119–132. F. Rich. A Lefschetz theorem for foliated manifolds.

11]). In this case, we have Tridis (F ) = βΛi (F ) · δe , but Tridis (F ) − βΛi (F ) · δe is C ∞ . 3. Bundles over homogeneous spaces and the Selberg trace formula Let G be a simply connected Lie group, Γ a discrete cocompact subgroup in G, and α an injective homomorphism of Γ to the diﬀeomorphism group Diﬀ(X) of some compact connected C ∞ manifold X. Consider a left action of Γ on M = G × X given by γ · (a, x) = (γa, α(γ)(x)) . Let M = Γ\(G × X), and let [a, x] be the element of M represented by any (a, x) ∈ M .

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