Existence of a family of beta extensions with full intertwining

Establish the existence of a family {κ_j} of beta extensions attached to the vertices of a chamber in the strong simplicial structure of the Bruhat–Tits building of the centralizer G_β = C_G(β), arising from an m-realization θ of an endo-parameter t for G, such that for all i and j the intertwining subgroup IG(κ_i, κ_j) contains G_β (i.e., the family has full intertwining).

Background

Beta extensions are specific extensions of Heisenberg representations associated to semisimple characters determined by data ((V,h), φ, Λ, β), where β is a full semisimple element and G_β denotes its centralizer in G. For applications to constructing types and comparing them across facets, one seeks families of such beta extensions indexed by the vertices of a chamber in the building of G_β that are compatible and have strong intertwining properties.

Full intertwining means that any two members κi and κ_j of the family are intertwined by every element of Gβ. Proposition 6.7 shows that full intertwining implies the family is compatible in the sense of Definition 6.6, making this conjecture a strong formulation sufficient for the intended applications to types and block decompositions.