Degenerate Principal Series 3
1.
I
1
Sn1 =
1.
' I
V
/
V
/
We now review the parabolic and parahoric subgroups for Sp2n(F). A standard
minimal parabolic subgroup for Sp2n(P) is the following:
X
TX1
£ Sp2n(F)\X € GLn(F) is upper triangular
Note that TX~X will be lower triangular so that Pmin is not the upper triangular
matrices in Sp2n(F) (we could arrange for Pmin to be upper triangular had we chosen
a different J in defining Sp2n(F)). Let 3 C {.Si,... , s
n
} . The standard parabolic
subgroups of Sp2n(F) are parameterized by such subsets of the simple reflections.
Associated to 3 is the parabolic subgroup P$ = Pmin, $ •
If P = MN is the Levi factorization of P then
M * GLkl(F) x GLk2(F) x • • • x GLkl^(F) x Sp2h(F),
with fci + ^2 + • • • + &i = "• This will be embedded in Sp2n(F) as
M = \
(
A,
l \
C
\
5
t^Ar1
D
J
Ai G GZt..(F)
and
C Z?
G ^P2.,(^)
r
We note that in terms of the parameterization of parabolic subgroups by subsets of
{si,.. . , 5
n
}, this parabolic subgroup corresponds to
$ = { S i , . . . , 3
n
} \ {5fc
1
,6jfc
1 +
fc
2
,...,5j
b l +
...ib

_
1
,}.