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).