  We analyse the noncommutative space underlying the quantum group SUq(2) from
the spectral point of view which is the basis of noncommutative geometry, and
show how the general theory developped in our joint work with H. Moscovici
applies to the specific spectral triple defined by Chakraborty and Pal. This
provides the pseudo-differential calculus, the Wodzciki-type residue, and the
local cyclic cocycle giving the index formula. The cochain whose coboundary is
the difference between the original Chern character and the local one is given
by the remainders in the rational approximation of the logarithmic derivative
of the Dedekind eta function. This specific example allows to illustrate the
general notion of locality in NCG. The formulas computing the residue are
"local". Locality by stripping all the expressions from irrelevant details
makes them computable. The key feature of this spectral triple is its
equivariance, i.e. the $SU_q(2)$-symmetry. We shall explain how this leads
naturally to the general concept of invariant cyclic cohomology in the
framework of quantum group symmetries.
