Non-basic rigid packets for discrete L-parameters

This article introduces the theory of non-basic rigid inner forms over p-adic local fields, extending the basic theory developed by Kaletha. Motivated by the recent work of Bertoloni Meli--Oi on the $B(G)$-parametrization of the local Langlands conjectures, our main application is to extend the basic rigid refined local Langlands conjectures for a discrete L-parameter φ of a quasi-split connected reductive group G. The packets of our extended construction are Weyl orbits of representations of inner forms of twisted Levi subgroups N of G for which φ factors through a member of the canonical G-conjugacy class of embeddings LN± → 1mm LG constructed by Kaletha.