Applications of loop groups and standard modules to Jacobians and theta functions of isospectral curves

Willi Schwarz

Vol. 176 (1996), No. 2, 463–506

DOI: 10.2140/pjm.1996.176.463

Abstract

Let L(z) be an element of M_n(ℂ[z,z⁻¹]). In this work we study the structure of isospectral curves given by f(z,λ) = 0, f(z,λ) = det(L(z) − λ), their Jacobians and the relationship between standard modules and the corresponding theta functions. We assume that f(z,λ) is irreducible and nonsingular for f(z,λ) = 0 and z ∈ ℂ^∗.

The element L(z) will be called good, if the centralizers C_±(L) of L(z) in M_n(ℂ[z]) (resp. M_n(ℂ[z⁻¹])) are the integral closure of ℂ[z,z^pL] (resp. M_n(ℂ[z⁻¹,z^−qL])) in M_n(ℂ[z,z⁻¹]). The class of curves we analyze include nonsingular curves and the isospectral curve of the periodic Toda lattice. The latter curve is represented by a “tridiagonal” matrix L(z).

The Jacobian variety is expressed as a quotient of certain centralizers of L(z) which are computed in a completion M_n(A_w) of M_n(ℂ[z,z⁻¹]). If we assume further that L(z) is an element of SL_n(ℂ[z,z⁻¹]) then the basic module of the universal central extension SL_n(A_w) of SL_n(A_w) is employed to define a function Θ. This function Θ is defined in terms of representative functions on the “Lie theoretic” Jacobian and satisfies the functional equation of theta functions.

Mathematical Subject Classification 2000

Primary: 58F07

Secondary: 14H40, 22E67

Milestones

Received: 6 January 1994

Revised: 15 April 1994

Published: 1 December 1996