# p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some exceptional ones (see e.g. wikipedia). In Paul Garrett's answer to this MSE question, after giving the non-compact forms of the infinite families of irreducible Riemannian symmetric spaces, he says

Note that in all cases the three $$\mathbb{R}$$-algebras $$\mathbb{R}$$,$$\mathbb{C}$$,$$\mathbb{H}$$ play roles. In fact, a completely parallel thing happens for classical groups over $$p$$-adic fields and in other cases, as in A. Weil's "Algebras with involutions and classical groups", Indian (not Indiana) J. Math. 1960.

I flipped through Weil's paper but wasn't enlightened. What I didn't find and am looking for in this question is an explicit list of p-adic homogeneous spaces $$G/K$$, where $$K$$ is presumably the fixed points of some involution, which is the analogue in whatever suitable sense of the irreducible Riemannian symmetric spaces; additionally I would like an explanation of what this 'suitable sense' is. Any help is appreciated!

• At least for the analogue of symmetric spaces of noncompact type, the answer is probably "Euclidean building", as developed by Bruhat and Tits in the late 60's. The problem with $G/K$ is that it's just a discrete space. But the point is that it's (close to be) the 1-skeleton of a nice complex.
– YCor
Commented Mar 20, 2020 at 15:36
• To complement Yve's comment: For compact symmetric spaces an analogue would be Tits buildings. Their classification (in the compact case) is well-known and can be found in any book on buildings, e.g. Brown or Ronan. Commented Mar 20, 2020 at 17:20
• @YCor: are there pairs $(G,K)$ associated to these Euclidean buildings or to compact buildings, or are they something more complicated? (sorry, I know almost nothing about buildings). The fact that $G/K$ is a discrete space isn't a problem for my purposes. I'm really after a list of pairs of groups $(G,K)$, as I'm coming from a random matrix theory viewpoint. Can such a list be extracted from the classification of buildings? It looked like Ronan's treatment of the classification is using an axiomatic approach to buildings and not referencing p-adic groups too much. Thanks! Commented Mar 20, 2020 at 18:43
• In the Riemannian case, the point is that you can define on something purely local (Riemannian symmetric space $X$ of non-positive sectional curvature) which essentially characterizes the pair $(G,K)$ (say, taking the isometry group of the universal covering and one point stabilizer). In the $p$-adic case, $G/K$ being discrete, one needs something new, and the new feature is the building structure. (Here I'm only addressing your sentence "the fact that $G/K$ is discrete isn't a problem for my purposes". I don't know what a good reference for Bruhat-Tits buildings is.)
– YCor
Commented Mar 20, 2020 at 19:26