Relationships of divisibility between local L-functions associated to representations of complex reductive groups Martin Andler Massachussetts Institute of Technology CNRS-Universite Paris 7 I.Introduction Let C be a complex connected reductive algebraic Lie group, lYo its real Lie algebra (i.e. the underlying real algebra of the complex Lie algebra), the complexification of ' and Ka maximal compact subgroup of C. The Langlands classification asserts that the set of equivalence classes of irreducible representations of C (precise definitions will be given later! is parametrized by the set of (conjugacy classes of) morphisms of into the set of semi-simple elements of the connected component of the L-group LCD of C. If r is a finite representation of LCD, a morphism of into LCD defines a finite dimensional representation of and hence an L-function in the sense of Artin. We thus have a triangle whose vertices are : a. representations of C b. morphisms of into LCD c. L-functions. Unfortunately, little is known about links between representations and L- functions and £.) without going through £ .. The only case which is well understood is the case of and r the standard representation of LCD = C. The results there are due to Jacquet «J>, see also <C-J>, <J-L». As a motivation for the present paper, let us describe Jacquet's results. He gives a method for computing the L-function associated to a repre- sentation 11 (recalI that r is the trivial representation of LCD). It goes roughly in the following way. Let be the set of nxn matrices with complex coefficients, and the set of functions on M(n,[) of the form P(z) = P(Zij,Zij) exp(-211)' zi}ij)