Ag.algebraische Geometrie – Analogues between representation theory of real groups and p-adic groups

Harish-Chandra uses the term "Lefschetz Principle" to describe a guiding principle: Everything that applies to reductive Lie groups should apply as well $ p $-adic reductive algebraic groups.

I wonder how general this analogue is. First, many basic notations are similar. Let us name a few examples:

  • Harish Chandra Limit Formula – Shalika Germs
  • Riemannian symmetrical room — Bruhat-Tits building
  • Classification of discrete series — Classification of regular supercuspidal presentations

Are there more (lesser known) examples? Is there a conceptual explanation to believe such analogues?

Finally, is there anything on the one side that we can not find on the other side right now?