  We explain the construction of minimal tilting complexes for objects of
highest weight categories and we study in detail the minimal tilting complexes
for standard objects and simple objects. For certain categories of
representations of complex simple Lie algebras, affine Kac-Moody algebras and
quantum groups at roots of unity, we relate the multiplicities of
indecomposable tilting objects appearing in the terms of these complexes to the
coefficients of Kazhdan-Lusztig polynomials.
