Root systems and elliptic curves
The Macdonald-Weyl formula \[ \prod_{w\in W_{\text{aff}}} \det(w)e^{2\pi is(w)(z)\tau}=C \prod_{a\in R^{+}_{\text{aff}}}(1-e^{2\pi ia(z)\tau}) \tag{*} \] where \(C = \prod\limits_{n=1}^\infty (1-e^{2\pi in \tau})\) is reproved here [cf. \textit{I. G. Macdonald}, Invent. Math. 15, 91--143 (1972; Zbl 0244.17005) and also \textit{M. Demazure}, Sémin. Bourbaki 1975/76, Lect. Notes Math. 567, Exp. No. 483 (1977; Zb1 0345.17003)]. The method consists in comparing two theta functions over the family of abelian varieties \(A_\tau = Q^\vee \otimes E_\tau\) where \(R\) is a root system, \(Q^\vee\) the lattice generated by the coroots and \(E_\tau\) is the elliptic curve \(\mathbb C/\mathbb Z \oplus \tau\mathbb Z\) \((\Im(\tau)>0)\). As the Weyl group \(W\) acts on \(Q^\vee\), it also acts on \(A_\tau\) and there is a basic \(W\)-antiinvariant theta function \(\theta_A\), relative to \(A_\tau\) with divisor \(\Delta = \sum\limits_{\alpha \in R_+} \text{Ker}^\alpha\) (each root \(\alpha\) determines a homomorphism \(A_\tau \to E_\tau\) hence \(|\Delta|\) is a union of reflection hypertori of \(A_\tau\)). The modular behavior \((\tau \mapsto \tau + 1, \tau \mapsto -1/\tau)\) of \(\theta_A\) leads to a uniqueness result and and eventually to the identity (*). The construction of this paper seems quite new and interesting: it throws some new light on the subject (for another point of view connected with group representation, cf. \textit{B. Kostant} [Adv. Math. 20, 179--212 (1976; Zbl 0339.10019)]).
scientific article; zbMATH DE number 3558015