  We establish precise regularity conditions for $L_p$-boundedness of Fourier
multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is
inspired by H\"ormander-Mikhlin criterion from classical harmonic analysis,
although it is substantially and necessarily different. Locally, we get sharp
growth rates of Lie derivatives around the singularity and nearly optimal
regularity order. The asymptotics also match Mikhlin formula for a
exponentially growing weight with respect to the word length. Additional decay
comes imposed by this growth and Mikhlin condition for high order terms.
Lafforgue/de la Salle's rigidity theorem fits here. The proof includes a new
relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By
transference, matters are reduced to a rather nontrivial $RC_p$-inequality for
$SL_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional
laplacians.
  Our second result gives a new and much stronger rigidity theorem for radial
multipliers in $SL_n(\mathbf{R})$. More precisely, additional regularity and
Mikhlin type conditions are proved to be necessary up to an order $\sim
|\frac12 - \frac1p| (n-1)$ for large enough $n$ in terms of $p$. Locally,
necessary and sufficient growth rates match up to that order. Asymptotically,
extra decay for the symbol and its derivatives imposes more accurate and
additional rigidity in a wider range of $L_p$-spaces. This rigidity increases
with the rank, so we can construct radial generating functions satisfying our
H\"ormander-Mikhlin sufficient conditions in rank $n$ and failing the rigidity
conditions for ranks $m >> n$. We also prove automatic regularity and rigidity
estimates for first and higher order derivatives of $\mathrm{K}$-biinvariant
multipliers in the rank 1 groups $SO(n,1)$.
