4 Chris Jantzen
We also remark that the modular function for P is given by
8(m) = |detAi| 2n+1 -* 1 IdetA^ 2 ^-* 1 ^ 1 -* 2 IdetA,.!!2^"*1—-fcl"2)+1-*1-1.
The parahoric subgroups may be constructed in a similar fashion. A stan-
dard Iwahori subgroup for Sp2n(F) may be constructed as follows: let ^ : K —*
Sp2n(Fq) be the reduction mod V homomorphism (where F
g
is the finite field O/V).
Then, take I ^ _ 1 (P
m
i
n
(Fg)) as the standard Iwahori subgroup. Let
so =
1
VD
' l
-zu-1
1
' l
so that {so, s i , . . . ,s
n
} generate the affine Weyl group. The standard parahoric
subgroups are in bijective correspondence with the subsets $ of {s0, « i , . . . , «sn}, the
correspondence being given by
B* = / , $ .
Note that if $ C { s i , . . . , s
n
} , then
B* = V-\Pz(Fq)).
These are the parahoric subgroups we shall be most interested in.
1.2. Induced representations and Jacquet modules
We now review the construction of induced representations and Jacquet
modules.
We start by reviewing the construction of induced representations. Let G
be a reductive p-adic group and P = MU a parabolic subgroup of G. Let (p,X) be
an admissible representation of M. Then p may be extended trivially to U to get a
representation of P , p (g 1. The induced representation Indp^® 1 acts on the space
V \f:G-+X
jf smooth 1
If (mug) = 8*{m)p(m)f(g) Vra G M, u G U, g G G J '
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