In William Goldman's book Complex Hyperbolic Geometry, bisector hypersurfaces play an important role. Given two points $x,y$, the bisector is the set of points equidistant from $x$ and from $y$. Do they also play an important role in the study of higher rank symmetric spaces, and do they admit concrete descriptions, say in the Siegel upper half-plane? References are especially welcome.
Although I cannot address your specific question, of whether bisectors "play an important role in the study of higher rank symmetric spaces," I can say that bisectors are the essence of Voronoi diagrams. So the 2009 paper by Frank Nielsen and Richard Nock, entitled "Hyperbolic Voronoi diagrams made easy" (arXiv link), could well be relevant: