  We partially generalize the theory of semihomogeneous bundles on an abelian
variety $A$ developed by Mukai. This involves considering abelian subvarieties
$Y\subset X_A=A\times\hat{A}$ and studying coherent sheaves on $A$ invariant
under the action of $Y$. The natural condition to impose on $Y$ is that of
being Lagrangian with respect to a certain skew-symmetric biextension of
$X_A\times X_A$. We prove that in this case any $Y$-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 $D^b(A)\to
D^b(B)$ associated with kernels in $D^b(A\times B)$ that are invariant with
respect to some Lagrangian subvariety in $X_A\times X_B$. We calculate their
composition and prove that in characteristic zero it can be decomposed into a
direct sum of LI-functors. In the case $B=A$ this leads to an interesting
central extension of the group of symplectic automorphisms of $X_A$ in the
category of abelian varieties up to isogeny.
