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In almost every introductory notes on Tits buildings these are motivated as structures capturing/ sharing several features of symmetric spaces. 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?

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    $\begingroup$ Just a quibble: one can take the viewpoint that the building itself is the primary thing, constructed in a more top-down way (e.g., classically by suitable "flags"), and the Coxeter groups arise just as automorphisms of the apartments. (J. Tits proved that the automorphisms of apartments in thick buildings are Coxeter groups... ) Also, the same Weyl group can easily occur for a variety of distinct buildings (e.g., the spherical building for $GL_n(k)$ for various fields $k$). $\endgroup$
    – paul garrett
    Commented Jun 17, 2023 at 17:19
  • $\begingroup$ @paulgarrett: what can one generally say about the group of isometries of a symmetric space $X$? By definition there is at every point $p \in X$ an involution given fixing $p$. Can one say something interesting about relations between the whole group of isometries of an symmetric space and the subgroup generated by the involutions? (eg much "bigger" whole group of isometries?) The latter can be naturally regarded as the pendant to Weyl group of buildings. $\endgroup$
    – user267839
    Commented Jun 19, 2023 at 14:07
  • $\begingroup$ I seem to recall that E. Cartan or someone subsequently did prove that, first, the whole group of isometries (or, anyway, the connected component of the identity...) is the expected semi-simple group $G$, so the whole thing is $G/K$. More subtly, perhaps at the same time, I think someone proved that the group generated by all the involutions (or closure?) is the whole $G$... (up to connected components?) $\endgroup$
    – paul garrett
    Commented Jun 19, 2023 at 18:50

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(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.

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, Invent. Math. 105, No. 1, 47-122 (1991). ZBL0751.14016.

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

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