Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $\mu \in \Lambda^{\vee}$, we can construct what seems like a fairly down to earth thing--a theta function for a shifted lattice:
$$\theta_{\Lambda}^{(\mu)}(q,w_{1},\ldots,w_{n}) = \sum_{v \in \Lambda+\mu} q^{\frac{1}{2} (v,v)} w_{1}^{v_{1}}\cdots w_{n}^{v_{n}}.$$
Here, we've chosen a base $\Delta$ and expressed $v$ in its (generally rational!) coordinates, and the $w_{i}$ are merely formal variables. If you look at Equation (3.2) of this paper and set their $m=1$, they say these shifted theta functions I write above are precisely "theta functions (of Kac-Peterson) for the integral representations of the affine Lie algebra".
I am hoping someone can help me wrap my head around this. I've tried, with not so much luck, to understand the original Kac-Peterson paper, and I think Equation (0.1) should be somehow equivalent to the shifted theta function above in some cases. Integrable representations of affine Lie algebras are beyond my knowledge, yet these shifted theta functions show up in my work all the time.
Here's a somewhat related question I'm interested in. If I make the following specialization in the above theta function: $$w_{i} = \exp\bigg( \frac{2 \pi i}{h^{\vee}+1} \bigg)$$ for all $i$, something very interesting happens. Here, $h^{\vee}$ is the dual Coxeter number. This shows up sort of unexplained for me, so I'm curious if this brings something to mind for anyone? In particular, the top of page 31 of this thesis involving Kac-Peterson modular forms has something tantalizingly close! Maybe someone can explain what role this plays in what they're doing, specialized to my simple ADE setting.
