Every irreducible representation is either even or odd.

Question

Let $V$ be any $n$-dimensional complex vector space and $SL(2,\Bbb{Z})$ is special linear group. Let $\rho:SL(2,\Bbb{Z}) \rightarrow GL(V)$ be a representation. It is even if $\rho(-I)=\Bbb{id}_V$ and odd if $\rho(-I)=\Bbb{-id}_V$. Then show that any irreducible representation is either even or odd.

Answer 1

Hints:

1. Use Schur's lemma to see that $p(I)$ and $p(-I)$ must act by scalars.
2. Note that $p(-I)^2 = p((-I)^2)) = p(I)$.