AbstractGiven a regular supercuspidal representation $$\rho $$
ρ
of the Levi subgroup M of a standard parabolic subgroup $$P=MN$$
P
=
M
N
in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $$JH(Ind^G_P(\rho ))$$
J
H
(
I
n
d
P
G
(
ρ
)
)
of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation $$Ind^G_P(\rho )$$
I
n
d
P
G
(
ρ
)
, vastly generalizing Rodier structure theorem for $$P=B=TU$$
P
=
B
=
T
U
Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $$W_M=N_G(M)/M$$
W
M
=
N
G
(
M
)
/
M
is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $$W_T=N_G(T)/T$$
W
T
=
N
G
(
T
)
/
T
is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split $$(G)Sp_{2n}$$
(
G
)
S
p
2
n
and $$SO_{2n+1}$$
S
O
2
n
+
1
, and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.
MORITA'S THEORY FOR THE SYMPLECTIC GROUPS
