# What are the characters of SL_2 and PSL_2?

How to calculate $X^*(SL_2) = \operatorname{Hom}(SL_2,\mathbb{G_m})$ and $X^*(PSL_2) = \operatorname{Hom}(PSL_2,\mathbb{G_m})$ ? ($SL_2$ and $PSL_2$ are viewed as algebraic groups over a field $K$)

For $SL_2$, I tried to do it with Hopf algebra, which leads to calculate $\operatorname{Hom}(K[X,X^{-1}],K[A,B,C,D]/(AD-BC-1))$, but I have difficulty calculating the group of invertible elements of $K[A,B,C,D]/(AD-BC-1)$.

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"The algebraic group $PSL_2$" doesn't actually exist. See mathoverflow.net/questions/16145/…. – David Loeffler Jan 27 '12 at 10:38

$SL_2$ is a simple algebraic group, and so admits no non-trivial characters.
$PSL_2$ is a quotient of $SL_2$, and so any character of $PSL_2$ would also be a character of $SL_2$; thus $PSL_2$ also admits no non-trivial characters.