if $E$ is a vector bundle on $X$, then $\det E \cong L(H,\chi)$ where $L(H,\chi)$ is a line bundle on $X$ determined by Appell-Humbert data. We then have that $\operatorname{slope}(E) := \frac{[\det E ]}{\operatorname{rank} E} \in \operatorname{NS}(X)_\QQ$ is identified with $\frac{1}{\operatorname{rank} E} H$. (Here $[\det E]$ denotes the class of $\det E$ in $\operatorname{NS}(X) := {\operatorname{Pic}}(X) / {\operatorname{Pic}}^0(X)$.)

We prove

\[Pair:index:theorem\] Let $(X, H, \iota)$ be a polarized abelian variety, of dimension $g$, with endomorphism structure $(K, \operatorname{id}_K, \rho)$. Then $X$ admits, for all $e {\geqslant}\ell {\geqslant}0$, $e {\geqslant}k {\geqslant}0$, classes $a, b \in \operatorname{NS}(X)_\QQ$, and non-degenerate simple semi-homogeneous vector bundles $E$ and $F$ with the properties that $\operatorname{slope}(E) = a$, $\operatorname{slope}(F) = b$, ${\operatorname{i}}(E) = m \ell$, ${\operatorname{i}}(F) = m k$, $E \otimes F$ is non-degenerate and ${\operatorname{i}}(E \otimes F) = m(\ell + k)$.

Before proving Theorem \[Pair:index:theorem\] we remark that if $\mathcal{M} \subseteq K^{\oplus 2m}$ is a free abelian group of rank $2 e$ and $Z = (Z_1,\dots, Z_e) \in \mathfrak{h}^{\oplus e}_m$, then the image of $\mathcal{M}$ in $\CC^g$ under the map $J_Z \colon K^{\oplus 2 m} \rightarrow \CC^g $ defined by sending $(\eta_1 , \dots, \eta_{2 m} ) $ to $${\operatorname{diag}}((Z_1 , \mathbf{1}_m), \dots, (Z_e , \mathbf{1}_m)) 