Let G be a real noncompact semisimple connected Lie group and let
$$\rho : G \longrightarrow \text {SL}(V)$$
ρ
:
G
⟶
SL
(
V
)
be a faithful irreducible representation on a finite-dimensional vector space V over
$$\mathbb {R}$$
R
. We suppose that there exists a scalar product
$$\texttt {g}$$
g
on V such that
$$\rho (G)=K\exp ({\mathfrak {p}})$$
ρ
(
G
)
=
K
exp
(
p
)
, where
$$K=\text {SO}(V,\texttt {g})\cap \rho (G)$$
K
=
SO
(
V
,
g
)
∩
ρ
(
G
)
and
$${\mathfrak {p}}=\text {Sym}_o (V,\texttt {g})\cap (\text {d} \rho )_e ({\mathfrak {g}})$$
p
=
Sym
o
(
V
,
g
)
∩
(
d
ρ
)
e
(
g
)
. Here,
$${\mathfrak {g}}$$
g
denotes the Lie algebra of G,
$$\text {SO}(V,\texttt {g})$$
SO
(
V
,
g
)
denotes the connected component of the orthogonal group containing the identity element and
$$\text {Sym}_o (V,\texttt {g})$$
Sym
o
(
V
,
g
)
denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on
$${\mathbb {P}}(V)$$
P
(
V
)
arising from
$$\rho $$
ρ
. There is a corresponding G-gradient map
$$\mu _{\mathfrak {p}}:{\mathbb {P}}(V) \longrightarrow {\mathfrak {p}}$$
μ
p
:
P
(
V
)
⟶
p
. Using G-gradient map techniques, we prove that the unique compact G orbit
$${\mathcal {O}}$$
O
inside the unique compact
$$U^\mathbb {C}$$
U
C
orbit
$${\mathcal {O}}'$$
O
′
in
$${\mathbb {P}} (V^\mathbb {C})$$
P
(
V
C
)
, where U is the semisimple connected compact Lie group with Lie algebra
$${\mathfrak {k}} \oplus {\textbf {i}} {\mathfrak {p}}\subseteq \mathfrak {sl}(V^\mathbb {C})$$
k
⊕
i
p
⊆
sl
(
V
C
)
, is the set of fixed points of an anti-holomorphic involutive isometry of
$${\mathcal {O}}'$$
O
′
and so a totally geodesic Lagrangian submanifold of
$${\mathcal {O}}'$$
O
′
. Moreover,
$${\mathcal {O}}$$
O
is contained in
$${\mathbb {P}}(V)$$
P
(
V
)
. The restriction of the function
$$\mu _{\mathfrak {p}}^\beta (x):=\langle \mu _{\mathfrak {p}}(x),\beta \rangle $$
μ
p
β
(
x
)
:
=
⟨
μ
p
(
x
)
,
β
⟩
, where
$$\langle \cdot , \cdot \rangle $$
⟨
·
,
·
⟩
is an
$$\text {Ad}(K)$$
Ad
(
K
)
-invariant scalar product on
$${\mathfrak {p}}$$
p
, to
$${\mathcal {O}}$$
O
achieves the maximum on the unique compact orbit of a suitable parabolic subgroup and this orbit is connected. We also describe the irreducible representations of parabolic subgroups of G in terms of the facial structure of the convex body given by the convex envelope of the image
$$\mu _{\mathfrak {p}}({\mathbb {P}}(V))$$
μ
p
(
P
(
V
)
)
.
