The linear action of the group
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
on the vector space
C
n
×
k
{{\mathbf {C}}^{n \times k}}
extends to an action on the algebra of polynomials on
C
n
×
k
{{\mathbf {C}}^{n \times k}}
. The polynomials that are fixed under this action are called
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
-invariant. The
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
-harmonic polynomials are common solutions of the
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
-invariant differential operators. The ideal of all
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
-invariants without constant terms, the null cone of this ideal, and the orbits of
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
on this null cone are studied in great detail. All irreducible holomorphic representations of
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
are concretely realized on the space of
S
O
(
k
,
C
)
SO(k,{\mathbf {C}})
-harmonic polynomials.