Over a
p
p
-adic local field
F
F
of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group
G
=
G
m
×
S
p
2
n
G={\mathbb {G}}_m\times {\mathrm {Sp}}_{2n}
. It is associated to the Langlands
γ
\gamma
-functions attached to any irreducible admissible representations
χ
⊗
π
\chi \otimes \pi
of
G
(
F
)
G(F)
and the standard representation
ρ
\rho
of the dual group
G
∨
(
C
)
G^\vee ({\mathbb {C}})
, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal (Braverman and Kazhdan, 2000) for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on
G
L
1
(
F
)
{\mathrm {GL}}_1(F)
, which is associated to a
γ
\gamma
-function
β
ψ
(
χ
s
)
\beta _\psi (\chi _s)
(a product of
n
+
1
n+1
certain abelian
γ
\gamma
-functions). Our work on
G
L
1
(
F
)
{\mathrm {GL}}_1(F)
plays an indispensable role in the development of our work on
G
(
F
)
G(F)
. These two types of harmonic analyses both specialize to the well-known local theory developed in Tate’s thesis (Tate, 1950) when
n
=
0
n=0
. The approach is to use the compactification of
S
p
2
n
{\mathrm {Sp}}_{2n}
in the Grassmannian variety of
S
p
4
n
{\mathrm {Sp}}_{4n}
, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis (1986) and many other works) on the doubling local zeta integrals for the standard
L
L
-functions of
S
p
2
n
{\mathrm {Sp}}_{2n}
.
The method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard
L
L
-function of
G
L
n
{\mathrm {GL}}_n
and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.