We generalize Jacobi forms of an arbitrary degree and construct torus bundles over abelian schemes whose sections can be identified with such generalized Jacobi forms.

In the first section of this paper we give a characterization of those closed convex cones (wedges) $W$ in the Lie algebra $sl(2,\mathbf{R}{)}^n$ which are invariant under the maximal compact subgroup of the adjoint group and which are controllable in the associated simply connected Lie group $Sl(2,\mathbf{R}{)}^{n^{\sim }}$, i.e., for which the subsemigroup $S=(expW)$ generated by the exponential image of $W$ agrees with the whole group $G$ (Theorem 13). In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly...