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Spectral curves, simple Lie algebras, and Prym-Tjurin varieties. (English) Zbl 0707.14041
Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, part 1, 627-645 (1989).
[For the entire collection see Zbl 0672.00003.]
The Lax equations \[ \frac{d}{dt}A(\xi,t)=[B(\xi,t),A(\xi,t)] \] where A and B are polynomials in \(\xi\), \(\xi^{-1}\) and take values in an irreducible representation of simple Lie algebra \({\mathfrak g}\subset gl_ n\) are considered. An accurate definition of the spectral curve C: \(\det(\eta I-A(\xi,t))=0\) is suggested. The symmetries of spectral curves for which A(\(\xi\),t) take values in simple Lie algebras of types \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) are investigated. It turns out that in case \({\mathfrak g}\subsetneqq sl_ n\) an endomorphism \(i: I(C)\to I(C)\) exists, which satisfies the equation \((i-1)(i+q-1)=0\) for some \(q\geq 2\). Here I(C) is a Jacobian of C. The number q depends on \({\mathfrak g}\) and on the representation \({\mathfrak g}\hookrightarrow gl_ n\). The abelian variety \(P(C,i)=(1-i)I(C)\) is called Prym-Tjurin variety. It is proved that if certain conditions on A are satisfied then, provided A and B take values in \({\mathfrak g}\subsetneqq sl_ n\), the flow generated by the dynamical system (*) can be linearized on the Prym-Tjurin variety \(P(C,i)\subset I(C)\). It is shown also that the isogeny class of Prym- Tjurin varieties obtained from spectral curves does not depend on the representation of \({\mathfrak g}\). In the construction above theta functions on Jacobi varieties correspond to \(A_ n\), theta functions on Prym varieties \((q=2)\) correspond to \(D_ n\), theta functions on Prym-Tjurin varieties with \(q=6\) (resp. \(q=12)\) correspond to \(E_ 6\) (resp. \(E_ 7)\).
Let the following data be given: (i) a complex, reductive Lie-algebra \({\mathfrak g}\) and a rational map \(f: {\mathbb{C}}\to {\mathfrak g}\) such that for almost all \(\xi\in {\mathbb{C}}\) the element \(f(\xi\)) is regular and semi- simple: (ii) a non-trivial, semi-simple, irreducible representation \(\rho: {\mathfrak g}\to gl(V)\), where \(\dim (V)=n\). Then, (i) and (ii) determine uniquely a non-singular, projective, possibly non-connected curve C, the so-called spectral curve, with a d-sheeted covering \(p: C\to {\mathbb{P}}^ 1. \) C can be considered as the desingularization of certain irreducible components of the closure (in \({\mathbb{P}}^ 2)\) of the plane curve \(X\subset {\mathbb{C}}^ 2\) determined by a completely integrable Hamiltonian system with Lax representation: \[ (1)\quad \frac{d}{dt}A(\xi,t)= [B(\xi,t),A(\xi,t)], \] where A and B are polynomials in \(\xi,\xi^{-1}\) and take values in \(gl_ n\). An interesting example of a spectral curve arises when \({\mathfrak g}\) is of type \(E_ 6\) and one imposes a suitable condition on the weights of \(\rho\). In this case C is a 27-sheeted covering of \({\mathbb{P}}^ 1\), and the general fibre can be identified with the 27 lines on a cubic surface.
An important role is played by the Prym-Tjurin varieties. Assume \({\mathfrak g}\subsetneqq sl_ n\), then one can construct a non-zero correspondence D on C that induces an endomorphism \(i: Jac(C)\to Jac(C)\), such that \((i- 1)(i+q-1)=0\) for some \(q\geq 2\). The integer q depends on \({\mathfrak g}\) and the representation \(\rho\). In the aforementioned example \(q=6\). For other root systems one can work with Del Pezzo surfaces and one finds e.g. for \(A_ 4, D_ 4, E_ 7, E_ 8\) the q-values 3, 4, 12 and 60, respectively. The abelian variety \(P(C,i)=(1-i)Jac(C)\) is called the Prym-Tjurin variety.
One has the following theorem: Let \({\mathfrak g}\) be a Lie-algebra of type \(A_ n, D_ n, E_ 6, E_ 7\) or \(E_ 8\), and let f be as in (i). Let \(\rho\) and \(\rho'\) be two representations as in (ii). Then the corresponding Prym-Tjurin varieties \(P(C,i)\) and \(P(C',i')\) are isogenous. If, furthermore, f is in ‘general position’, and \(\rho\), \(\rho'\) correspond to so-called minuscule weights, then there is a polarized isomorphism \((P(C,i),\Xi)\overset \sim \rightarrow (P(C',i'),\Xi')\). - Another result of the paper says that under suitable conditions on A and B the flow generated by the dynamical system (1) can be linearized on \(P(C,i)\). This result may be considered with respect to the well-known fact that for \({\mathfrak g}=so(2n)\) the flow determined by (1) can be linearized on a complex subtorus, the Prym variety, \(P(C,\sigma)=(1-\sigma)Jac(C)\), where \(\sigma\) is an involution on C. In this case the Prym-Tjurin variety coincides with the Prym variety.

14K30 Picard schemes, higher Jacobians
14H40 Jacobians, Prym varieties
14K25 Theta functions and abelian varieties
14K10 Algebraic moduli of abelian varieties, classification
17B20 Simple, semisimple, reductive (super)algebras
14K02 Isogeny