POLYHEDRAL COMPACTIFICATIONS OF BRUHAT-TITS BUILDINGS OF QUASI-REDUCTIVE GROUPS

Abstract

Given a quasi-reductive group G over a local field k , using Berkovich geometry, we exhibit a family of

-equivariant compactifications of the Bruhat-Tits building

B ( G , k ) $\mathcal B(G, k)$ script upper B left parenthesis upper G comma k right parenthesis

, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists of mapping the building into the analytification

G an $G^{\operatorname {\mathrm {an}}}$ upper G Superscript an

of G , then composing this map with the projections from

G an $G^{\operatorname {\mathrm {an}}}$ upper G Superscript an

to its (in general non-compact) pseudo-flag varieties

( G / P ) an $(G/P)^{\operatorname {\mathrm {an}}}$ left parenthesis upper G divided by upper P right parenthesis Superscript an

, for P ranging among the pseudo-parabolic subgroups of G . This generalises previous constructions of Berkovich, then Rémy, Thuillier and Werner.

To define the embedding, we are led to giving a partial extension to the quasi-reductive context of results due to Rousseau on the functoriality of Bruhat-Tits buildings with respect to field extensions, which are of independent interest.

Finally, we conclude by investigating the geometry at infinity of these compactifications. The boundaries are shown to be stratified, each stratum being equivariantly homeomorphic to the Bruhat-Tits building of the maximal quasi-reductive quotient of a pseudo-parabolic subgroup.