on $Y$. Using Mumford’s index theorem [@Mum p. 145] applied to $L$, together with equation , we conclude that the roots of the polynomial $\chi(A^{\otimes N} \otimes E)$ are real and that ${\operatorname{i}}(E)$ equals the number of positive roots counted with multiplicity.

[**Remark.**]{} If $E$ is a non-degenerate simple semi-homogeneous vector bundle on $X$ and if $\chi(E)$ is not divisible by the characteristic of the ground field, then the cohomology group ${\mathrm{H}}^{{\operatorname{i}}(E)}(X, E)$ is the unique irreducible weight $1$ representation of the (non-degenerate) theta group ${\operatorname{G}}(E)$. This fact, which extends work of Mumford [@MumI §1], is proven in [@Grieve:PhD:Thesis §6]. The higher weight representation theory of the group ${\operatorname{G}}(E)$ is known by work of E.Z. Goren [@Goren:theta:preprint Theorem A1.4] and, independently, by [@Grieve:PhD:Thesis §6].

The pair index condition {#pair:index}
------------------------

We now define the *pair index condition*. This condition allows us to study families of cup-product maps and, in §\[families\], to relate these maps to the fiberwise evaluation map of suitably defined vector bundles.

[**Definition.**]{} A pair $(E,F)$ of vector bundles on $X$ satisfies the *pair index condition* if, for all $x \in X$, the vector bundles $\text{$T^*_xE$, $F$, and $T^*_x(E)\otimes F$}$ satisfy the index condition and ${\operatorname{i}}(T^*_x(E) \otimes F) 