Automorphic forms and Lie algebras. 30 Sept 1996 
(EXPOSITORY) 
Current Developments in mathematics 1996, International Press 1998. 

Richard E. Borcherds,
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 
home page: http://www.dpmms.cam.ac.uk/~reb 

0. Introduction. This paper is mainly an advertisement for one particular Lie 
algebra called the fake monster Lie algebra. The justification for looking at 
just one rather obscure object in what is supposed to be a general survey is 
that the fake monster Lie algebra has already led directly to the definition 
of vertex algebras, the definition of generalized Kac-Moody algebras, the 
proof of the moonshine conjectures, a new family of automorphic forms, and it 
is likely that there is still more to come from it. 

The reader may wonder why we do not start by looking at a simpler example 
than the fake monster Lie algebra. The reason is that the fake monster is the 
simplest known example in the theory of non-affine Lie algebras; the other 
examples are even worse. 

Here is a quick summary of the rest of this paper. The first sign of the 
existence of the fake monster came from Conway's discovery that the Dynkin 
diagram of the lattice II25,1 is essentially the Leech lattice. We explain 
this in section 1. For any Dynkin diagram we can construct a Kac-Moody 
algebra, and a first approximation to the fake monster Lie algebra is the 
Kac-Moody algebra with Dynkin diagram the Leech lattice. It turns out that 
to get a really good Lie algebra we have to add a little bit more; more 
precisely, we have to add some ``imaginary simple roots'', to get a 
generalized Kac-Moody algebra. Next we can look at the ``denominator 
function'' of this fake monster Lie algebra. For affine Lie algebras the 
denominator function is a Jacobi form which can be written as an infinite 
product [K, chapter 13]. For the fake monster Lie algebra the denominator 
function turns out to be an automorphic form for an orthogonal group, which 
can be written as an infinite product. There is an infinite family of such 
automorphic forms, all of which have explicitly known zeros. (In fact it 
seems possible that automorphic forms constructed in a similar way account 
for all automorphic forms whose zeros have a ``simple'' description.) Finally 
we briefly mention some connections with other areas of mathematics, such as 
reflection groups and moduli spaces of algebraic surfaces.