3

The context for the question is due to a comment in Milne’s Introduction to Shimura Varieties. I have limited background in algebraic groups (i.e. just enough to know what it is when it is used).

Besides the point, how does one determine if an algebraic group have $PSL_2$ as a quotient? Say, $U(2,1)$ for example. Certainly dimension is a nice enough way to check so anything dimension $<3$ cannot have it as a quotient.

Is there anything else one can do? Allegedly, this is doable via Dynkin diagrams. See "Edit II" and an answer describing this story is perfectly acceptable as well.

Edit: I realize it isn’t $PSL_2$ that I care about, but $PGL_2$. In either case, there’s many ways to generalize the above question so let me keep it to $PSL_2$. For the $PGL_2$ case, one can use the fact $PGL_2$ is noncompact so anything that is compact, such as $U(2,1)$, cannot have it as a quotient.

Edit II: According to @jackson, there should be a component of the Dynkin diagram that is of type $A_1$ and this should follow from the classification of split connected reductive groups.

  • Do you want to restrict to reductive groups? If so, there’s not really so many. I’m a bit spotty on the non-split cases here but for split groups, you would need a component of the Dynkin diagram to be type $A_1$, and I think that’s sufficient – jackson Sep 12 '23 at 05:02
  • @jackson Yes, I want to restrict to reductive groups since that's what Shimura varieties work best for. Can you explain why the condition you say is sufficient –  Sep 12 '23 at 23:21
  • The Dynkin diagram is a way of describing the root system of a Lie/reductive group. Basically I think the classification of split connected reductive groups (which is really similar to the classification of semisimple Lie groups) basically answers your question, but it’s kind of tough to describe here – jackson Sep 12 '23 at 23:53

0 Answers0