I’m in California for 11 days, for (mostly) math. Here’s something fun I just learned from Dick Gross. Let be a split simple algebraic group over , and let be (a -lattice in) an irreducible algebraic representation of . It’s not hard to check that if is minuscule, the representation is an irreducible -representation for every prime . Are there any other examples of with this property? It turns out there is exactly one more example: the adjoint representation of . When I said it was surprising to me that another such example existed, Dick explained that I shouldn’t be surprised, because after all, “ is the greatest group that God ever created, so it’s not surprising that it has amazing properties like this”.

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