  This paper is devoted to the study of generalized subdifferentials of
spectral functions over Euclidean Jordan algebras. Spectral functions appear
often in optimization problems playing the role of "regularizer", "barrier",
"penalty function" and many others. We provide formulae for the regular,
approximate and horizon subdifferentials of spectral functions. In addition,
under local lower semicontinuity, we also furnish a formula for the Clarke
subdifferential, thus extending an earlier result by Baes. As application, we
compute the generalized subdifferentials of the function that maps an element
to its k-th largest eigenvalue. Furthermore, in connection with recent
approaches for nonsmooth optimization, we present a study of the
Kurdyka-Lojasiewicz (KL) property for spectral functions and prove a transfer
principle for the KL-exponent. In our proofs, we make extensive use of recent
tools such as the commutation principle of Ram\'irez, Seeger and Sossa and
majorization principles developed by Gowda.
