X$, then $T_x \colon X \rightarrow X$ denotes translation by $x$ in the group law. We define a cohomological condition which we place on a pair of vector bundles on $X$. To state this criterion we first make some auxiliary definitions.

Non-degenerate vector bundles and the index condition {#index:defn}
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We make definitions which apply to vector bundles of all ranks and extend the basic cohomological properties of non-degenerate line bundles. We discuss examples of vector bundles which satisfy these conditions in §\[eg:semi:homog:index\] and §\[real:multi:vect\].

[**Definitions.**]{} Let $E$ be a vector bundle on $X$.

-   [We say that $E$ is *non-degenerate* if $\chi(E) \not = 0$.]{}

-   [If $E$ admits exactly one nonzero cohomology group, then we say that $E$ satisfies the *index condition*.]{}

-   [If $E$ satisfies the index condition, then we let ${\operatorname{i}}(E)$ denote the unique integer $b$ for which ${\mathrm{H}}^b(X,E)$ is nonzero and say that ${\operatorname{i}}(E)$ is the *index* of $E$.]{}

Simple semi-homogeneous vector bundles and the index condition {#eg:semi:homog:index}
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Following Mukai, we say that a vector bundle $E$ on $X$ is *semi-homogeneous* if, for all $x \in X$, there exists a line bundle $L$ such that $T^*_x E \cong E \otimes L$, see [@Muk78 p. 