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. \(E_ 7)\).

A.Bobenko

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.

W.Hulsbergen

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. \(E_ 7)\).

A.Bobenko

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.

W.Hulsbergen

##### MSC:

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 |