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We show that a system L of real vector fields on a general compact nilmanifold Γ{minus 45 degree rule};N induced by the Lie algebra N of N is globally hypoelliptic (GH) iff (1{ring operator}) The symbols of the vector fields of L projected onto the associated torus T = Γ[N, N]{minus 45 degree rule}N as functions on the integral lattice T ̂ collectively decrease at infinity not faster than a reciprocal of a polynomial and (2{ring operator}) the Lie subalgebra of N that L generates is not annihilated by any non-zero integral linear functional on any Nj Nj + 1, j=0, 1,..., (Nj + 1 = [ N, Nj], No = N). It follows that (GH) is equivalent to injectivity of the system L on the dual on the space of C∞-vectors of all the non-trivial representations in the spectrum of Γ{minus 45 degree rule}N (a "Rockland type" condition) plus a number-theoretic condition on L on the associated torus (to avoid "small divisors"). © 1988.

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Journal of Functional Analysis

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