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.
Previous Page Next Page