The category of projective functors on a block of the category
O
(
g
)
\mathcal O(\mathfrak g)
of Bernstein, Gelfand and Gelfand, over a complex semisimple Lie algebra
g
\mathfrak g
, embeds to a corresponding block of the category
O
(
g
×
g
)
\mathcal O(\mathfrak g \times \mathfrak g)
. In this paper we give a nice description of the object
V
V
in
O
(
g
×
g
)
\mathcal O(\mathfrak g \times \mathfrak g)
corresponding to the identity functor; we show that
V
V
is isomorphic to the module of invariants, under the diagonal action of the center
Z
\mathcal Z
of the universal enveloping algebra of
g
\mathfrak g
, in the so-called anti-dominant projective. As an application we use Soergel’s theory about modules over the coinvariant algebra
C
C
, of the Weyl group, to describe the space of homomorphisms of two projective functors
T
T
and
T
′
T’
. We show that there exists a natural
C
C
-bimodule structure on
Hom
{
Functors
}
(
T
,
T
′
)
\operatorname {Hom}_{\{\operatorname {Functors}\}}(T, T’)
such that this space becomes free as a left (and right)
C
C
-module and that evaluation induces a canonical isomorphism
k
⊗
C
Hom
{
Functors
}
(
T
,
T
′
)
≅
Hom
O
(
g
)
(
T
(
M
e
)
,
T
′
(
M
e
)
)
k \otimes _C \operatorname {Hom}_{\{\operatorname {Functors}\}} (T, T’) \cong \operatorname {Hom}_{\mathcal O(\mathfrak g)}(T(M_e), T’(M_e))
, where
M
e
M_e
denotes the dominant Verma module in the block and
k
k
is the complex numbers.