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P. Broussous and S. Stevens studied maps between enlarged Bruhat-Titsbuildings to construct types for p-adic unitary groups. They needed maps whichrespect the Moy-Prasad filtrations. That property is called (CLF), i.e.compatibility with the Lie algebra filtrations. In the first part of thisthesis we generalise their results on such maps to the Quaternion-algebra case.Let k0 be a p-adic field of residue characteristic not two. We consider asemisimple k0-rational Lie algebra element beta of a unitary group G:=U(h)defined over k0 with a signed hermitian form h. Let H be the centraliser ofbeta in G. We prove the existence of an affine H(k0)-equivariant CLF-map j fromthe enlarged Bruhat-Tits building B^1(H,k0) to B^1(G,k0). As conjectured byBroussous the CLF-property determines j, if none of the factors of H isk0-isomorphic to the isotropic orthogonal group of k0-rank one and all factorsare unitary groups. Under the weaker assumption that the affine CLF-map j isonly equivariant under the center of H^0(k0) it is uniquely determined up to atranslation of B^1(H,k0). The second part is devoted to the decoding ofembedding types by the geometry of a CLF-map. Embedding types have been studiedby Broussous and M. Grabitz. We consider a division algebra D of finite indexwith a p-adic center F. The construction of simple types for GLn(D) in theBudhnell-Kutzko framework required an investigation of strata which had tofulfil a rigidity property. Giving a stratum especially means to fix a pair(E,a) consisting of a field extension E|F in Mn(D) and a hereditary order awhich is stable under conjugation by E^x, in other words we fix an embedding ofE^x into the normalizer of a. Broussous and Grabitz classified these pairs withinvariants. We describe and prove a way to decode these invariants using thegeometry of a CLF-map.

Embedding types and canonical affine maps between Bruhat-Tits buildings of classical groups (Thesis)