varies. (If $x \in X$, then $T_x \colon X \rightarrow X$ denotes translation by $x$.)

As an example, Pareschi-Popa used their index conditions to relate the cup-product map to the fiberwise evaluation map of the vector bundle $$p_{1 *}(m^* A^{ n} \otimes p_{2 }^* E)$$ on $X$; here $p_i$ denotes the projection of $X \times X$ onto the first and second factors respectively (for $i = 1,2$) and $m$ denotes multiplication in the group law. Pareschi-Popa then gave cohomological criteria for vector bundles on $X$ to be globally generated. That certain cup-product maps of the form are surjective then became a question of applying this criteria to the vector bundle $p_{1 *}(m^* A^{n} \otimes p_{2 }^* E)$.

Pareschi-Popa subsequently generalized their approach and went on to study families of cup-product maps having the form $${\mathrm{H}}^0(X, T_x^* E) \otimes {\mathrm{H}}^0(X, F) \xrightarrow{\cup} {\mathrm{H}}^0(X, T^*_x(E) \otimes F)$$ where $E$ and $F$ are semi-homogenous vector bundles on $X$ and $x \in X$ varies [@Par:Popa:III §7.3].

The purpose of this paper is to study matters related to cup-product maps arising from pairs of vector bundles on $X$ which satisfy the *pair index condition* and have *nonzero index*. The *pair index condition*, which we define 