  We construct modular categories from Hecke algebras at roots of unity. For a
special choice of the framing parameter, we recover the Reshetikhin-Turaev
invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N)
by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We
then discuss the choice of the framing parameter. This leads, for any rank N
and level K, to a modular category \tilde H^{N,K} and a reduced invariant
\tilde\tau_{N,K}. If N and K are coprime, then this invariant coincides with
the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show that the
reduced invariant admits spin or cohomological refinements, with a nice
decomposition formula which extends a theorem of H. Murakami.
