We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component
X
¯
\overline X
of
O
¯
m
i
n
∩
n
+
\overline O_{\mathrm {min}}\cap \mathfrak n_+
(where
O
¯
m
i
n
\overline O_{\mathrm {min}}
is the (Zariski) closure of the minimal nilpotent orbit of
s
p
2
n
\mathfrak {sp}_{2n}
and
n
+
\mathfrak n_+
is the Borel subalgebra of
s
p
2
n
\mathfrak {sp}_{2n}
) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of
U
(
s
p
2
n
)
U(\mathfrak {sp}_{2n})
, which contains the maximal parabolic subalgebra
p
\mathfrak p
determining
O
¯
min
\overline O_{\min }
. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same family of subalgebras. Finally, we investigate this family of subalgebras from the representation-theoretical point of view and, among other things, rediscover in a different framework irreducible highest weight modules for the UEA of
s
p
2
n
\mathfrak {sp}_{2n}
.