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We partially generalize the theory of semihomogeneous bundles on an abelian variety developed by Mukai, Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ. 18 (1978), 239–272. This involves considering abelian subvarieties and studying coherent sheaves on invariant under the action of . The natural condition to impose on is that of being Lagrangian with respect to a certain skew-symmetric biextension of by . We prove that in this case any -invariant sheaf is a direct sum of several copies of a single coherent sheaf. We call such sheaves Lagrangian-invariant (or LI-sheaves). We also study LI-functors associated with kernels in that are invariant with respect to some Lagrangian subvariety in . We calculate their composition and prove that in characteristic zero it can be decomposed into a direct sum of LI-functors. In the case this leads to an interesting central extension of the group of symplectic automorphisms of in the category of abelian varieties up to isogeny.