10 DANIEL K. NAKANO
Let {L(X) : X £ T} be a complete set of irreducible U{GQ) modules. From Proposition 1.2.3 for
each A E T, L(X) 8t/-(j3±) U(g) has unique head in the categories C (resp. Cgr). These irreducible
modules will be denoted by £(X) (resp. £gr(X)). Let £p(A) = £(X) if £g = C# and £-p(A) = Cgr(X)
if VQ = CgrQ.
Proposition 1.2.4. The set {£x(X) : X £ T} is a complete set of pairwise non-isomorphic
irreducible modules for VQ.
Proof. Let M be an irreducible module of U(#). For some A £ T there exists a monomorphism
« : L(X) i—• .M |t/(B±)
with X(A) an irreducible module for
U(B±).
It follows that K will extend to a U(Q) module
homomorphism from L(X)
®u(B±)
U(&)
t o
-M. Therefore, M = £v(X).
Now substitute M = L(X) in (*) (Prop. 1.2.3). If R = rad U(Q) then
[V&W/V&MR : £(X)} = 4imFHomu{9)(y£rW,£(\)).
Suppose there exists f: V^:r(/i) i—• £(X). Then we have the following sequence of maps:
V&Ox) l u ( B * ) - - £(A) |t,
( f l
T).— 1(A).
where X(A) is an irreducible U(B^) module. The argument in the last proposition shows that the
composition of these maps is 0 unless A = //. Since image j is 0 or £{X) it follows that / = 0
unless X = fi. This shows that {£p(A) : A £ T} are pairwise non-isomorphic modules in V& because
{X(A) : A £ T } are pairwise non-isomorphic U{GQ) modules.•
Let P(X) be the projective cover of the irreducible U(GQ) module L(X) and let V(X) denote the
projective cover of the module £(X) in C$. Set
/(A) = P(\) ®u{Go) U(g).
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