As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of the $Pic(X)$ and $K_X$ is the canonical bundle of $X.$ Hence $K_X=rL$, where $r$ is a negative integer.
My question is what is $r$ in each case of Hermitian symmetric spaces (i.e. for Grassmannians, orthogonal Grassmannians, symplectic Grassmanians, quadrics and two exceptional classes).
You can get the numbers "r" from Table 1 page 612 of the following paper:
http://projecteuclid.org/download/pdf_1/euclid.tmj/1178228413
If you are not used with the notation you can find useful Remark (1.6) to distinguish the Hermitian symmetric spaces. For example, the number "r" for the two exceptional ones are : -12 for the 16 dimensional and -18 for the 27 dimensional one.
Table 1 comes from a paper by Borel&Hirzebruch. So perhaps it is a good idea to have a look to this original paper.
