AbstractWe find two different families of $$\mathbf{Sp}(4,\mathbb{R})$$
Sp
(
4
,
R
)
symmetric $$G_2$$
G
2
structures in seven dimensions. These are $$G_2$$
G
2
structures with $$G_2$$
G
2
being the split real form of the simple exceptional complex Lie group $$G_2$$
G
2
. The first family has $$\tau _2\equiv 0$$
τ
2
≡
0
, while the second family has $$\tau _1\equiv \tau _2\equiv 0$$
τ
1
≡
τ
2
≡
0
, where $$\tau _1$$
τ
1
, $$\tau _2$$
τ
2
are the celebrated $$G_2$$
G
2
-invariant parts of the intrinsic torsion of the $$G_2$$
G
2
structure. The families are different in the sense that the first one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$
Sp
(
4
,
R
)
/
SL
(
2
,
R
)
l
, and the second one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$
Sp
(
4
,
R
)
/
SL
(
2
,
R
)
s
. Here $$\mathbf{SL}(2,\mathbb{R})_l$$
SL
(
2
,
R
)
l
is an $$\mathbf{SL}(2,\mathbb{R})$$
SL
(
2
,
R
)
corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$
sl
(
2
,
R
)
related to the long roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$
sp
(
4
,
R
)
, and $$\mathbf{SL}(2,\mathbb{R})_s$$
SL
(
2
,
R
)
s
is an $$\mathbf{SL}(2,\mathbb{R})$$
SL
(
2
,
R
)
corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$
sl
(
2
,
R
)
related to the short roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$
sp
(
4
,
R
)
.
A decomposition formula for representations