Abstract
Let ${\mathfrak g}$ be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with ${\mathfrak h}$ the Cartan subalgebra and $Q \subset {\mathfrak h}^{*}$ the root lattice. Denote by $\Theta_{Q}(q)$ the theta series of the root lattice $Q$ of ${\mathfrak g}$. We prove a curious "combinatorial" identity for the derivative of $\Theta_{Q}(q)$, i.e.\ for $q \frac{d}{dq} \Theta_{Q}(q)$, by using the representation theory of an affine Lie algebra.
Citation
Satoshi Naito. "A combinatorial identity for the derivative of a theta series of a finite type root lattice." Nagoya Math. J. 172 1 - 30, 2003.
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