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§2. Generalities on -Adic Representations 295

Proof. Let T be a lattice in V with generators x 1 ,..., x m , and consider for j =

1,..., m the function which assigns to each s in G the coset sx j + T in V/T . This
is a finite valued function with all sx j in −N T for some large N. The subgroup

H of all s in G with sx j ∈ T , for all j, sT ⊂ T is an open subgroup of finite

index in G. Let T be the submodule over Z generated by sT for sH ∈ G/H. Then
T ⊂ T ⊂ −N T and T is a lattice with sT ⊂ T for all s in G.Hence sT = T for

all s ∈ G. This proves the proposition.
In the previous proof, we used the usual convention that sx denotes ρ(s)x when
ρ is clear from the context.
(2.5) Remarks. If ρ : G → GL(V ) is a representation of a group on a finite di-
mensional vector space V over a field F, then the character χ ρ of ρ is the func-
tion χ ρ (s) = Tr(ρ(s)). A particular value of χ ρ is χ ρ (1) = dim F V .If F is of
characteristic zero and if G is finite, then we know that two representations ρ and

ρ : G → GL(V ) are isomorphic if and only if χ ρ = χ ρ .

There is a similar assertion for infinite groups and semisimple representations.
This can be viewed in terms of the group algebra A = F[G]of G over F consisting

of (finite) linear combinations a s s with scalars a s ∈ F and algebra structure
s∈G
given by

a s s b t t = (a s b t )st.
s t s,t
Then a representation ρ : G → GL(V ) defines an A = F[G] module struc-
ture on V by ( a s s)v = a s ρ(s)v and, conversely, A-modules restrict to G-
s s
representations. Two A-modules V and V are isomorphic if and only if the repre-

sentations are isomorphic. The A-module V is simple if and only if the representa-
tion V is irreducible or simple. In either case, the term semisimple means a direct
sum of simple modules or representations. The following proposition shows that χ ρ
determines ρ up to isomorphism in the case of finite dimension, semisimple repre-
sentations over a field F of characteristic 0.

(2.6) Proposition. Let F be a field of characteristic zero, let A be an algebra over
F, and let M and N be two semisimple A-modules which are finite dimensional over
F. If Tr M (a) = Tr N (a) for all a in A, then M and N are isomorphic.
Proof. Let I be the set of isomorphism classes of simple A-modules, and let E(i) be
m(i)
a representative module in the class i ∈ I. Then M is isomorphic to i E(i) and
n(i)
N to i E(i) for some m(i), n(i) ∈ N. For each a in A and j ∈ I there exist
by the density theorem (see Lang, Algebra) elements a( j) ∈ A with scalar action on
E(i) given by

0 for i = j,
a( j) E(i) =
a E( j) for i = j.