  We prove an $L^p$ spectral multiplier theorem for functions of the
$K$-invariant sublaplacian $L$ acting on the space of functions of fixed
$K$-type on the group $SL(2,\mathbb{R}).$ As an application we compute the
joint $L^p(SL(2,\mathbb{R}))$ spectrum of $L$ and the derivative along $K$.
