  The Wigner crystal of composite fermions is a strongly correlated state of
complex emergent particles, and therefore its unambiguous detection would be of
significant importance. Recent observation of optical resonances in the
vicinity of filling factor {\nu} = 1/3 has been interpreted as evidence for a
pinned Wigner crystal of composite fermions [Zhu et al., Phys. Rev. Lett. 105,
126803 (2010)]. We evaluate in a microscopic theory the shear modulus and the
magnetophonon and magnetoplasmon dispersions of the composite fermion Wigner
crystal in the vicinity of filling factors 1/3, 2/5, and 3/7. We determine the
region of stability of the crystal phase, and also relate the frequency of its
pinning mode to that of the corresponding electron crystal near integer
fillings. These results are in good semiquantitative agreement with experiment,
and therefore support the identification of the optical resonance as the
pinning mode of the composite fermions Wigner crystal. Our calculations also
bring out certain puzzling features, such as a relatively small melting
temperature for the composite fermion Wigner crystal, and also suggest a higher
asymmetry between Wigner crystals of composite fermion particles and holes than
that observed experimentally.
