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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Let M be a manifold with a closed, integral (k+1)-form ω⁠, and let G be a Fréchet–Lie group acting on (M,ω)⁠. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of g by R⁠, indexed by Hk−1(M,R)∗⁠. We show that the image of Hk−1(M,Z) in Hk−1(M,R)∗ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of G by the circle group T⁠. The idea is to represent a class in Hk−1(M,Z) by a weighted submanifold (S,β)⁠, where β is a closed, integral form on S⁠. We use transgression of differential characters from S and M to the mapping space C∞(S,M) and apply the Kostant–Souriau construction on C∞(S,M)⁠.@en

We construct an L1-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.

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There are eight possible Pin groups that can be used to describe the transformation behavior of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group.@en

We present a geometric construction of central S 1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S 1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S1-extensions of Lie groups that act by exact strict contact transformations. @en

For an infinite dimensional Lie group G modelled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G♯ of G. (The main point is the smooth structure on G♯.) For infinite dimensional Lie groups G which are 1-connected, regular, and modelled on a barrelled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G♯, and the appropriate unitary representations of its Lie algebra g♯.@en

Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980).@en