Semisimplicity of the Lyapunov spectrum for irreducible cocycles
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Let G be a semisimple Lie group acting on a space X, let μ be a symmetric compactly supported measure on G, and let A be a strongly irreducible linear cocycle over the action of G. We then have a random walk on X, and let T be the associated shift map. We show that, under certain assumptions, the cocycle A over the action of T is conjugate to a block conformal cocycle.
This statement is used in the recent paper by Eskin–Mirzakhani on the classification of invariant measures for the SL(2, ℝ) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.
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