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Root systems and elliptic curves. (English) Zbl 0358.17016
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. I. G. Macdonald, Invent. Math. 15, 91–143 (1972; Zbl 0244.17005) and also 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 \(\mathbb C/\mathbb Z \oplus \tau\mathbb Z\) \((\operatorname{Im}(\tau)>0)\). 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. B. Kostant [Adv. Math. 20, 179–212 (1976; Zbl 0339.10019)]).
Reviewer: Alain Robert

MSC:
17B22 Root systems
11F12 Automorphic forms, one variable
14K25 Theta functions and abelian varieties
14H45 Special algebraic curves and curves of low genus
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References:
[1] Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4, 5 et 6. Paris: Hermann 1969 · Zbl 0205.06001
[2] Freudenthal, H., de Vries, H.: Linear Lie groups. New York: Academic Press 1969 · Zbl 0377.22001
[3] Gunning, R. C.: Lectures on modular forms. Princeton, University Press 1962 · Zbl 0178.42901
[4] Lang, S.: Elliptic functions. Reading (Mass.): Addison Wesley 1973 · Zbl 0316.14001
[5] Macdonald, I. G.: Affine root systems and Dedekind’s ?-function. Inventiones math.15, 91-143 (1972) · Zbl 0244.17005
[6] Mumford, D.: Abelian varieties. Oxford: University Press 1970 · Zbl 0223.14022
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