  In this paper, we study the $L^2$-minimal extension problem for polarized
variations of Hodge structures over Hermitian symmetric domains. We are able to
explicitly find the $L^2$-minimal extensions using a group-theoretic
construction. In particular, this gives a construction without using
$L^2$-estimates as in the Ohsawa-Takegoshi type extension theorems. The key
ingredient is the Harish-Chandra embedding of Hermitian symmetric domains.
  The construction of holomorphic sections might be of independent interest
since it gives a concrete description in the setting of Hermitian VHS.
