Degenerate Principal Series

5

where 6 denotes the modular function for P. The action of G on this space is

by right translation. We shall give some more convenient notation for induced

representation after we discuss Jacquet modules.

We now describe the construction of Jacquet modules. Again, suppose G

is a reductive p-adic group and P = MU a parabolic subgroup of G. Let (7r, V) be

a representation of G. The Jacquet module of 7r with respect to P , denoted 7T£/, is

a representation of M on a space which is denoted by VJJ. The space is

Vu = V/V(U),

where V(U) = span{7r(u)v — v\v G V , u E U}. The action of M is given by

7ru(m)(v + V(U)) = 6-±(m)ir(m)v + V(U)

(one checks that this defines a representation).

We shall frequently use the notation of Bernstein-Zelevinsky [B-Z] for in-

duced representations and Jacquet modules. If P = MU is a parabolic subgroup of

(7, (p, X) an admissible representation of M, (7r, V) an admissible representation of

(7, then set

icM(p) = Indfp (g) 1

and

rMG{n) = KU-

The following notation for induced representations in 5j2n(P)will also be

convenient. It is just an extension (cf. [S-T]) of the shorthand notation of Bernstein-

Zelevinsky for induced representation in GLn(F). Suppose P = MU is a standard

parabolic subgroup of Sp2n(P) with

M * GLkl(F) x ••• x GL^iF) x Sp2kl(F).

Let p i , . . . , pi_i be admissible representations of G L ^ ( P ) , . . . GL^^F), and r an

admissible representation of Sp2kt(F). Then, let

px x . . . x p/_i (xr = IGMPI 8 • • • ® pi-i (8) r.