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The fixed-point spectrum of a locally compact second countable group G on lpis defined to be the set of real numbers p such that every action by affineisometries of G on lp admits a fixed-point. We show that this set is eitherempty, or is equal to a set of one of the following forms : [1,\pc[,[1,\pc[\{2} for some \pc<\infty or \pc=\infty, or [1,\pc], [1,\pc]\{2} for somepc<infty. This answers a question closely related to a conjecture of C. Drutuwhich asserts that the fixed-point spectrum is connected for isometric actionson Lp(0,1). We also study the topological properties of the fixed-pointspectrum on Lp(X,\mu) for general measure spaces (X,\mu), and show partialresults toward the conjecture for actions on Lp(0,1). In particular, we provethat the spectrum F_{L^{\infty}(X,\mu)(G,\pi) of actions with linear part \piis either empty, or an interval of the form [1,\pc] or [1,\infty[, whenever \piis an orthogonal representation associated to a measure-preserving ergodicaction on a finite measure space (X,\mu).

Fixed-point spectrum for group actions by affine isometries on Lp-spaces