From: Allan Adler <ara@zurich.ai.mit.edu>
Subject: projective representations
X-Submissions-To: sci-math-research@uiuc.edu
Organization: M.I.T. Artificial Intelligence Lab.
Date: Tue, 7 Apr 1992 20:48:28 GMT
X-Administrivia-To: sci-math-research-request@uiuc.edu


I'm confused by Michael Larsen's posting. It looks ok. On the other hand,
it seems quite general, giving rise to a complex projective representation of
the semidirect product Sp(2n,2).B, where B is elementary abelian of order 
2^(2n), of degree 2^n. For reference, Michael's posting is quoted at the
end of this one.

What bothers me is the following theorem of Feit and Tits (Can.J.Math. 30 
(1078) 1092-1102):

Theorem. Let F be an algebraically closed field of characteristic not equal
to 2. For every positive integer n there exists an exact sequence
   1 --> (Z/2Z)^(2n) --> An ---> Sp(2n,2) --> 1
(denote the mapping An-->Sp(2n,2) by alpha_n) and a projective representation
rho_n:An-->PGL(2^n,F) such that alpha_n does not map the kernel of rho_n
onto Sp(2n,2). If n>3 then this property characterizes the exact sequence
uniquely and the pair {rho_n,rho_n^*}, where rho_n^* is the 
contragredient of rho_n. Furthermore (An,alpha_n,rho_n) is minimal with
respect to Sp(2n,2) and F. In particular, rho_n is irreducible and
the sequence does not split.

To say that (An,alpha_n,rho_n) is minimal says that no proper subgroup of
An maps onto Sp(2n,2) and that An has no nontrivial projective representation
of degree < 2^n.

So what I am confused about is why Michael's apparently general argument
does not contradict this theorem. I understand he is looking at n=3, while
the theorem assumes n>3, but he does seem to be making general arguments that 
don't depend on this.

Allan Adler
ara@altdorf.ai.mit.edu

=========================================================================


>
The short answer is yes.  The point, as the original poster probably
surmised, is to use a characteristic 2 version of the Weil representation.

Explicitly, let V denote a vector space of dimension 3 over the field
with 2 elements.  Let C[V] denote the complex group ring of V.  We can
view the dual space $V^*$ as a space of complex functions on V via
the sign character on Z/2Z.  Therefore V^* acts on C[V] by multiplication,
while V acts on C[V] by translation.  The two actions don't quite commute,
but they do give rise to a projective (8-dimensional) complex representation
of V x V^*.  This representation lifts to an actual representation
on the Heisenberg group H (central extension of V x V^* by Z/2Z with
2-cocycle given by the natural pairing (a,a*) x (b,b*) --> a*(b) + b*(a).)
The character of the representation rho is easy to calculate; it is zero
on non-zentral elements, 8 on the identity, and -8 on the other
central element.  Since the order of the group is 128, rho 
is irreducible.  On the other hand, any automorphism of the Z/2Z-vector
space W = V x V^* which preserves the symplectic form on W gives
rise to an automorphism of H, and conjugating by
this automorphism does not change the character of rho.
Thus the elements of Sp(W) intertwine rho with itself.  By Schur's lemma,
the semidirect product of Sp(W) by H admits a projective 8 dimensional
representation (which in fact is an actual representation if you replace
Sp(W) by Mp(W).)  The restriction of the representation to Sp(W) gives
the Steinberg representation.  

To solve the given problem, we realize the 168-group as the Levi component of 
a maximal parabolic subgroup of Sp(W) and restrict.

						-Michael Larsen

P.S.  Has anyone worked out a theory of dual reductive pairs over finite
fields?