In almost every introductory notes on [Tits buildings](https://en.wikipedia.org/wiki/Building_(mathematics)) these are motivated
as structures capturing/ sharing several features of [symmetric spaces](https://en.wikipedia.org/wiki/Symmetric_space). Could somebody elaborate what are precisely the main structural similarities between buildings and
symmetric spaces motivating the vievpoint that buildings can
be regarded in some sense as generalizations of symmetric spaces?


(so is there something more 'substantial' beyond the sloppy slogan
that both objects carry 'high degree
of symmetry'; so what are the 'characteristic' features buildings and
symmetric spaces have in common in deeper sense?)


The central construction in the theory of buildings is certain simplicial complex $\Sigma= \Sigma(W,S)$ for appropriate Coxeter group $W$ ( the Weyl group of the buildings) with generators S gouverning the symmetry of the complex. Is there a kind of 'Weyl group' which naturally reflects the symmetry of symmetric spaces?



Buildings as generalizations of symmetric spaces
(Wanted to post as a comment but didn't have enough reputation)


I have a partial answer only for your first question.


Let $F$ be a non-Archimedean field. For p-adic(or $\pi-$adic) symmetric space $\Omega^r \subset \mathbb{P}^{r-1}\_F$(now also called Drinfeld period domain), Drinfeld found a very clear relation between the symmetric space and the Bruhat-Tits building for $GL\_r(F)$, $\mathcal{BT}^\bullet$(Tits building being its boundary) via a certain *reduction map* $$\rho:\Omega^r \rightarrow |\mathcal{BT}^\bullet|$$ (|.| means the geometric realization of the simplicial complex) and he used this map to construct an admissible covering of the symmetric space s.t. the nerve of the covering is the barycentric subdivision of the $\mathcal{BT}^\bullet$. You can see this in Drinfeld's seminal paper on "Elliptic modules"(Section 6) or also in Chapter 3 of


*Deligne, Pierre; Husemoller, Dale H.*, Survey of Drinfel’d modules, Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 25-91 (1987). [ZBL0627.14026](https://zbmath.org/?q=an:0627.14026).


If you want to know a specific relationship of the symmetric space with the Tits building, then one such is contained in the work of P. Schneider and U. Stuehler(in section 3)


*Schneider, P.; Stuhler, U.*, [**The cohomology of (p)-adic symmetric spaces**](https://doi.org/10.1007/BF01232257), Invent. Math. 105, No. 1, 47-122 (1991). [ZBL0751.14016](https://zbmath.org/?q=an:0751.14016).


contains some results relating the cohomology of symmetric space and certain generalized Tits building .


