(joint work with N.Wallach).
Let g be a complex simple Lie algebra and let G be
the adjoint group of g. Let h be the Coxeter number of
g. Some time ago I conjectured that if q= 2h+1 is a prime
power, then the finite simple group L2(q) embeds into G.
With the help of computers, in a number of cases, this has been
shown to be true. The most sophisticated case is when G = E 8.
Here q= 61. This embedding was first computer-established by
Cohen--Griess and later without computer by Serre. Griess--Ryba also
later (computer) proved that L2(49) and L2(41) embed into E8.
Write the three power primes 61, 49, 41 as qk where
k = 30, 24, 20 so that qk = 2 k +1.
In a 1959 paper I related,
for any simple g, the Coxeter element withthe principal
nilpotent element in g. Tony Springer, in a 1974 paper,
extending my result in the special case of E8, established a
similar connection, between three nilpotent elements,
ek in g,
and three (regular) elements of the Weyl group σk. The order
of σk is k.
Using some beautiful properties of σk
the main result in our presentation here at is the establishent of
a clear-cut connection between Springer's result, on the one hand,
with the Griess-Ryba embedding L2(qk) in E8 on the other.