Summary: Complex and modular representations of the group Sp(4, q)
Conference ``Analyse harmonique sur le groupe Sp(4)''
organized by Paul Sally and MarieFrance Vign’eras
(Luminy, France, June 15--19 1998)
Let q be a power of a prime number p, and F q be a finite field of q elements. We denote by F q an
algebraic closure of F q . Let G = Sp 4 (F q ) be the group preserving the symplectic form given by # J 0
where J = # 0 1
-1 0 # . The group G is a connected reductive algebraic group defined over F q . The group
G = Sp(4, q) = Sp(4, F q ) is the finite group of the F --fixed points of G under a Frobenius endomorphism
F : G # G.
The order of G is |G| = q 4 (q 2 + 1)(q + 1) 2 (q - 1) 2 . Hence any prime number # #= p which divides |G|
either is equal to 2, either divides exactly one of the terms q 2 + 1, q + 1, q - 1.
In the first section we give an account of the theory of complex irreducible characters of Sp(4, q). In
the second section we consider the case of #modular representations, where # ia a prime not dividing q.
The concept of the decomposition matrix provides a link between modular and complex representations.
Characters of Sp(4, q)
We will assume that q is odd (for q even, the character table of Sp(4, q) has been determined by