The column space of a real
n
×
k
n\times k
matrix
x
x
of rank
k
k
is a
k
k
-plane. Thus we get a map from the space
X
X
of such matrices to the Grassmannian
G
\mathbb {G}
of
k
k
-planes in
R
n
\mathbb {R}^{n}
, and hence a
G
L
n
GL_{n}
-equivariant isomorphism
\[
C
∞
(
G
)
≈
C
∞
(
X
)
G
L
k
.
C^{\infty }\left ( \mathbb {G}\right ) \approx C^{\infty }\left ( X\right ) ^{GL_{k}}\text {.}
\]
We consider the
O
n
×
G
L
k
O_{n}\times GL_{k}
-invariant differential operator
C
C
on
X
X
given by
\[
C
=
det
(
x
t
x
)
det
(
∂
t
∂
)
,
where
x
=
(
x
i
j
)
,
∂
=
(
∂
∂
x
i
j
)
.
C=\det \left ( x^{t}x\right ) \det \left ( \partial ^{t}\partial \right ),\quad \text {where }x=\left ( x_{ij}\right ),\text { }\partial =\left ( \frac {\partial }{\partial x_{ij}}\right ).
\]
By the above isomorphism,
C
C
defines an
O
n
O_{n}
-invariant operator on
G
\mathbb {G}
.
Since
G
\mathbb {G}
is a symmetric space for
O
n
O_{n}
, the irreducible
O
n
O_{n}
-submodules of
C
∞
(
G
)
C^{\infty }\left ( \mathbb {G}\right )
have multiplicity 1; thus,
O
n
O_{n}
-invariant operators act by scalars on these submodules. Our main result determines these scalars for a general class of such operators including
C
C
. This answers a question raised by Howe and Lee and also gives new Capelli-type identities for the orthogonal Lie algebra.