Compatibility and full-intertwining beta extensions for inner forms of classical groups

Determine whether, for inner forms of p-adic classical groups with odd residual characteristic, there exist compatible families of beta extensions (as defined via the chamber-based compatibility condition in the building of the centralizer), and ascertain whether there always exist beta extensions with full intertwining for such groups.

Background

A family of beta extensions associated to an m-realization and a chamber in the building of the centralizer G_β is called compatible if the extensions are coherent under conjugation by elements of G_β across vertices (Definition 6.6). Proposition 6.7 shows that full intertwining implies compatibility.

For inner forms of general linear groups, compatible families (and in fact full-intertwining families) are obtainable by adapting the classical method for GL_n(F). The situation for inner forms of classical groups remains unsettled; resolving this would clarify the construction and comparison of types in these settings and strengthen the link with depth-zero theory.