Let Γ<G be a discrete subgroup of a locally compact unimodular group G. Let m∈C _b(G) be a p-multiplier on G with 1≤p<∞ and let T _m:L _p(G^)→L _p(G^) be the corresponding Fourier multiplier. Similarly, let Tm| _Γ:L _p(Γ^)→L _p(Γ^) be the Fourier multiplier associated to the restriction m| _Γ of m to Γ. We show that (Formula presented.) for a specific constant 0≤c(U)≤1 that is defined for every U⊆Γ. The function c quantifies the failure of G to admit small almost Γ-invariant neighbourhoods and can be determined explicitly in concrete cases. In particular, c(Γ)=1 when G has small almost Γ-invariant neighbourhoods. Our result thus extends the de Leeuw restriction theorem from Caspers et al. (Forum Math Sigma 3(e21):59, 2015) as well as de Leeuw’s classical theorem (Ann Math 81(2):364–379, 1965). For real reductive Lie groups G we provide an explicit lower bound for c in terms of the maximal dimension d of a nilpotent orbit in the adjoint representation. We show that c(B _ρ ^G)≥ρ ^-d/4 where B _ρ ^G is the ball of g∈G with ‖Ad _g‖<ρ. We further prove several results for multilinear Fourier multipliers. Most significantly, we prove a multilinear de Leeuw restriction theorem for pairs Γ<G with c(Γ)=1. We also obtain multilinear versions of the lattice approximation theorem, the compactification theorem and the periodization theorem. Consequently, we are able to provide the first examples of bilinear multipliers on nonabelian groups.

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The subject of this thesis is twofold: The first part is the study of local-to-global principles for the continuous Lie algebra (co-)homology of certain infinite-dimensional Lie algebras of geometric origin, specifically, Gelfand-Fuks cohomology and continuous cohomology of gauge algebras. It includes both an exposition to classical results of Gelfand and Fuks, and new methods to construct general local-to-global spectral sequences for the Lie algebra cohomology of section spaces of Lie algebroids. This includes a close functional-analytics study of the involved spaces and attention to complications within LF- and Fréchet spaces. The second part contains a study of two certain measure-theoretic problems on Lie groups. The first such problem is the study of Haar measures of certain identity-neighbourhoods relevant to de Leeuw inequalities in Harmonic Analysis. The second problem is the evaluation of expectation values of polynomials on compact Lie groups, motivated by the study of Weingarten functions and Wilson loops from lattice gauge theory.@en

Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perform a general comparison of these notions, give an interpretation of the lower orders of the invariant cohomology spaces and conclude as our main result that for proper group actions both spaces are isomorphic. As a corollary and a geometric interpretation, an invariant version of the Hochschild-Kostant-Rosenberg theorem is given, identifying the cohomology of invariant cochains with invariant multivector fields. Using this theorem, we shortly discuss the invariant Hochschild cohomology in the case of homogeneous spaces.

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