Embeddings of L-Groups

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Abstract
To a real reductive group G there is attached a family of (real) groups, each of lower dimension but sharing Cartan subgroups with G (cf. [8]). The purpose of these groups is to provide “building blocks” (in a specific sense (cf. [11])) for analysis on G. Their définition is via an L-group construction; the connected component of the identity, LH0, in the L-group of such a group H is naturally a subgroup of LG0 but the requirement that H “share” Cartan subgroups with G precludes defining LH the full L-group of H, as a subgroup of LG. Nevertheless, the principle of functoriality in the L-group suggests that the embeddings of LH in LG will play a role in analysis. In this paper, we study the embeddings of LH in LG in order to resolve a problem about the normalization of orbital integrals.

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