Spectral multipliers for functions of fixed $K$-type on $L^p(SL(2,\mathbb{R}))$

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 $K$.