Automorphic forms with singularities on Grassmannians. 
29 September 1996, corrected 29 May
1997. 

Richard E. Borcherds,
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 

home page: http://www.dpmms.cam.ac.uk/~reb 



We construct some families of automorphic forms on 
Grassmannians which have singularities along
smaller sub Grassmannians, using Harvey and Moore's 
extension of the Howe (or theta)
correspondence to modular forms with poles at cusps. 
Some of the applications are as follows. We
construct families of holomorphic automorphic forms 
which can be written as infinite products, which
give many new examples of generalized Kac-Moody 
superalgebras. We extend the Shimura and
Maass-Gritsenko correspondences to modular forms 
with singularities. We prove some congruences
satisfied by the theta functions of positive definite 
lattices, and find a sufficient condition for a Lorentzian
lattice to have a reflection group with a finite 
volume fundamental domain. We give some examples
suggesting that these automorphic forms with 
singularities are related to Donaldson polynomials and to
mirror symmetry for K3 surfaces.