Some Results on Multiplicities for SL(n

We define the multiplicity and the global multiplicity of an L-packet of SL(n), unifying lack of multiplicity one and non-rigidity of L-packets. The first examples of these phenomena were given by Blasius. Giving a heuristic approach to its calculation, based on Langlands ’ Tannakian formalism, we conjecture that the global multiplicity is bounded in terms of n only. We justify the heuristics in a special case of L-packets attached to Hecke characters on an Abelian or p-extension. We then focus on L-packets lifted from endoscopic tori. A full description of their global multiplicities is given in the case where n is prime. 1