Supercuspidal L-packets of positive depth and twisted Coxeter elements

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a p-adic field k and certain homomorphisms (Langlands parameters) from the Galois (or WeilDeligne group) of k into a complex Lie group Gwhich is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks on the local Langlands correspondence can be found in [18]. When G = GL1 this correspondence should reduce to local abelian class field theory. For G = GLn, the Langlands correspondence is known to exist [19], [22] and is uniquely determined by local ε factors [21]. So far this correspondence is not completely explicit, but much progress has been made in this direction; see [9], [10], for example. For groups other than GLn or PGLn, the theory is much less advanced; new phenomena appear, arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of Langlands parameters. This means that a single Langlands parameter φ should