Automorphic forms with singularities on Grassmannians.

*(English)*Zbl 0919.11036Let \(G(b^+, b^-)\) be the Grassmannian of \(B^+\)-dimensional positive definite subspaces of the inner product space \(\mathbb R^{b^+, b^-}\) of signature \((b^+, b^-)\). This paper concerns the construction of automorphic forms on \(G(b^+, b^-)\) which have singularities along smaller sub-Grassmannians. The main tool used in the paper is the extension of the usual theta correspondence to automorphic forms with singularities developed by J. Harvey and G. Moore [Nucl. Phys. B 463, 315–368 (1996; Zbl 0912.53056)]. It is used to construct families of holomorphic automorphic forms which can be written as infinite products. This extends the previous results for \(G(2, b^-)\) by the author [R. E. Borcherds, Invent. Math. 120, 161–213 (1995; Zbl 0932.11028)], and such automorphic forms provide many new examples of generalized Kac-Moody superalgebras.

The paper gives a common generalization of several well-known correspondences, including the Shimura and Maass-Gritsenko correspondences, to modular forms with poles at cusps. It also contains proofs of some congruences satisfied by the theta functions of positive definite lattices and provides a sufficient condition for a Lorentzian lattice to have a reflection group with a fundamental domain of finite volume. Finally, the paper discusses some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for \(K3\) surfaces.

The paper gives a common generalization of several well-known correspondences, including the Shimura and Maass-Gritsenko correspondences, to modular forms with poles at cusps. It also contains proofs of some congruences satisfied by the theta functions of positive definite lattices and provides a sufficient condition for a Lorentzian lattice to have a reflection group with a fundamental domain of finite volume. Finally, the paper discusses some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for \(K3\) surfaces.

Reviewer: Min Ho Lee (Cedar Falls)

##### MSC:

11F55 | Other groups and their modular and automorphic forms (several variables) |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

11F22 | Relationship to Lie algebras and finite simple groups |

11F32 | Modular correspondences, etc. |

14M15 | Grassmannians, Schubert varieties, flag manifolds |