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 '