# Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups.

I have been looking for a good proof for this claim for a little while now.

I am new here, so I could not comment, whence I made a new post.

It seems to me that the claim verified by the 'known theorem' is false. It is known that the map should be an injective Lie Homomorphism (hence an immersion everywhere by a simple homogeneity argument). But we cannot guarantee that the inverse of this map is continuous (or that the image is closed, in which case it also follows by a known theorem that it is an embedded submanifold).

The wikipedia page on Lorentz Group Representations also gives the same argument, and it cites a physics paper (and a theorem in Hall's book which does not suffice) without further context.

It is true that the group generated by the exponentials of a Lie Subalgebra has a unique structure of Lie Subgroup, but this need not be an embedded submanifold, it is only immersed (cf. dense curves on tori et cetera). I know this is not a counterexample to the claim, because the real line is not simple. But at least it seems a bit more should be said here. If we only know that the image is immersed in a compact Lie Group, the remainder of the argument fails.

My thesis advisor briefly sketched a different argument involving negative definiteness of the killing form on the Lie Algebra of $SU(n)$. If someone could give me details on this, that would be greatly appreciated as well.

• Related to the last subquestion: physics.stackexchange.com/q/364185/2451 Commented Feb 28, 2018 at 11:25
• Thanks! As I expected, the argument is a bit more subtle. At least I roughly understand the steps now. Do you think the original argument can be adapted (i.e. by using more explicitly that the algebra is simple) into a correct one? Commented Feb 28, 2018 at 12:43
• Please see proposition 6.15 in Mitsuo Sugiura's book books.google.co.il/books/about/…, it refers to the goup $SU(1,1)$, however I think it can be easily generalized to any simple group. Commented Feb 28, 2018 at 14:20