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?