I'm reading the book 'A course in Modern Mathematical Physics' by
'Szekeres' and encountered a problem in interpreting the proof of the
following corollary of Schur's lemma.

The corollary and the proof in the book is as follows (I mark my problem
area's in the proof with (<1>) and (<2>) :

Corollary 4.2. Let $T : G \rightarrow {GL} (V)$ be a irreducible 
representation of a finite (<1>) group G on a complex vector space $V$ and $A
: V \rightarrow V$ an operator such that $\forall g \in G$ we have $T (g)
\circ A = A \circ T (g)$ then $A = \alpha {id}_V$ for some complex scalar
$\alpha$.

Proof: As $\alpha {id}_V$ commutes with $T (g)$ for every $g$ and every complex $\alpha$
 we have $\forall g \in G$ that $T (g) \circ (A - \alpha
{id}_V) = (A - \alpha {id}_V) \circ T (g)$ and from Shur's lemma it
follows then that either $A - \alpha {id}_V = 0$ or $A - \alpha
{id}_V$ is a isomorphism (and thus invertible). Now if $V$ is complex
then $A$ must have a eigenvalue $\alpha$ (<2>) so that $A - \alpha
{id}_V$ is not invertible and thus we must have $A = \alpha {id}_V$



The problems I have with this proof are as follows :

1. For (<2>) to be valid (that $A$ has a eigenvalue) must we not have that $V$
is finite dimensional. The corollary tells nothing about finite dimensionality
of $V$.

2. For what do we need (<1>) ($G$ is finite). In the same book at the
formulation of Shur's lemma $G$ does not need to be finite, so I think the
fact that $G$ is finite should be used somewhere in the proof of the corollary
but I fail to see where (unless from $G$ finite it follows that $V$ is finite
which would solve my previous problem but I fail to see how this could be
true). This is my biggest problem, the previous problem is in my opinion a typo in the text of the book, but here I have the impression that I'm missing something in the proof, in my opinion none of the steps in the proof do not need finiteness of G so can we omit the requirement that G must be finite in the corollary.



Thanks a lot in advance



Marc Mertens