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Automorphic forms with singularities on Grassmannians. (English) Zbl 0919.11036
Let $$G(b^+, b^-)$$ 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.

##### 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
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