A Decomposition of the Set of Enhanced Langlands Parameters for a p-adic Reductive Group

Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will first introduce a notion
of cuspidality for these pairs. The cuspidal pairs are expected to correspond to the supercuspidal irreducible representations of G via the local Langlands correspondence.
We will next describe a construction of  a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig.
Finally, we will use this map to decompose  the set of enhanced Langlands parameters into Bernstein series.
This is joint work with Ahmed Moussaoui and Maarten Solleveld.