Abstract : Satake duality is an isomorphism between the algebra of characters of a simple Lie group G and the Hecke algebra of the affine Lie group corresponding to the Langlands dual to G. We will suggest a (conjectural) generalisation of this isomorphism replacing the characters by functions on local systems on surfaces and Hecke algebra by the algebra on cells on local systems with values in an affine Lie group.

Keywords : Satake correspondence; higher laminations; cluster varieties