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2TTix 2TTiy 2TTiz Abelian Harmonic Analysis Algebraic analytic automorphism Brezin bundle ch(A Chapter character classical result Cohomologie computation shows converges Corollary decomposition define denote dense in H direct sum distinguished subspaces dx dy dz easily verified Edited elements equivalent Fejer kernel fiber follows formula Fourier analysis Fourier series function of period functional equations given Haar measure Hence Hilbert space implies integers intertwining operator invariant irreducible subspaces isomorphism Jacobi theta functions let F Let G linear term mapping multiplicity nilmanifold norm notation orbits order theta functions orthonormal basis Plancherel theorem proof of Theorem R-invariant and irreducible R-invariant subspace satisfy the functional space H stability group statement subspaces of H Theorem C.lO Theorem II.4 tion TTiTny unique unitary operator unitary representation vanishes vector space Vlll write F Z/nZ zeros