  In this paper we study the microscopic mechanism for the retarded decay of
K-isomers to lower K bands. We do angular momentum projection from suitable
intrinsic states. The retardation arises from poor overlap between the low K
and high K bands in the integral over Euler angles. Deformed HF and angular
momentum projection calculations are done for the decay of the $K=10^+$ isomer
band to the ground band of $^{182}W$. K-mixing is unimportant and the K quantum
numbers of the bands are quite good. There is significant difference in the
reduced matrix elements of transition operators in our formalism and that in
the rotational model. Angular momentum projection gives J-selection rule but
there is no K selection rule for reduced matrix elements of electromagnetic
multipole operators. Thus, E2 and M1 transitions from the K isomer to the
ground band of $^{182}W$ are finite but retarded in angular momentum projection
theory, as in experiments. This provides a theoretical basis for the study of
K-isomers and their decay modes. Quantitative results are presented. The
microscopic model gives J-selection rule and angular momentum conservation for
combined matter and radiation systems.
