Notes on Representations of GL(r) over a Finite Field

15 Pages·00.207 MB·English

by  Daniel Bump

NOTES ON REPRESENTATIONS
OF GL(r) OVER A FINITE FIELD
by Daniel Bump
1. Indu
ed representations of (cid:12)nite groups. Let G be a (cid:12)nite group, and
H a subgroup. Let V be a (cid:12)nite-dimensional H-module. The indu
ed module
G
V is by de(cid:12)nition the spa
e of all fun
tions F : G ! V whi
h have the prop-
erty that F(hg) = h:F(g) for all h 2 H. This is a G-module, with group a
tion
0 0
(gF)(g ) = F(g g). On the other hand, if U is a G-module, let UH denote the
H-module obtained by restri
ting the a
tion of G on U to the subgroup H. Thus
the underlying spa
e of UH is the same as that of U. If V is a G-module, we will
o

asionally denote by (cid:25)V : G ! GL(V) the representation of G asso
iated with
V. Thus if g 2 G, v 2 V, g:v and (cid:25)V(g)(v) are synonymous.
The Frobenius re
ipro
ity law amounts to a natural isomorphism
G (cid:24)
(1.1) HomG(U;V ) = HomH(UH;V):
G 0
This is the
orresponden
e between (cid:30) 2 HomG(U;V ) and (cid:30) 2 HomH(UH;V),
0 0
where (cid:30) (w) = (cid:30)(w)(1), and
onversely, (cid:30)(w)(g) = (cid:30) (gw). There is also a natural
isomorphism
G (cid:24)
(1.2) HomG(V ;U) = HomH(V;UH):
This may be des
ribed as follows. Given an element (cid:30) 2 HomH(V;UH), we may
0 G
asso
iate an element (cid:30) 2 HomG(V ;U), de(cid:12)ned by
X
0 (cid:0)1
(cid:30) (f) = (cid:13)(cid:30)(f((cid:13) )):
(cid:13)2G=H
Thus indu
tion and restri
tion are adjoint fun
tors between the
ategories of
H-modules and G-modules. It is also worth noting that they are exa
t fun
tors.
Another important property of indu
tion is transitivity. Thus if H (cid:18) K (cid:18) G,
where H and K are subgroups of G, and if V is an H-module, then
K G (cid:24) G
(1.3) (V ) = V :
K G K
The isomorphism is as follows. Suppose that F 2 (V ) . Thus F : G ! V .
G
We asso
iate with this the element f of V de(cid:12)ned by f(g) = F(g)(1). It is easily
K G G

he
ked that this F ! f is an isomorphism (V ) ! V .
The problem of
lassifying intertwining operators between indu
ed representa-
tions was
onsidered by Ma
key [Ma
℄ who proved the following Theorem.
Typeset by AMS-TEX
1
2 BY DANIEL BUMP
Theorem 1.1. Suppose that G is a (cid:12)nite group, that H1 and H2 are subgroups of
G G
G, and that V1, V2 are H1- and H2-modules, respe
tively. Then HomG(V1 ;V2 )
is naturally isomorphi
to the spa
e of all fun
tions (cid:1) : G ! HomC(V1;V2) whi
h
satisfy
(1.4) (cid:1)(h2gh1) = (cid:25)V2(h2)Æ(cid:1)(g)Æ(cid:25)V1(h1):
We may exhibit the
orresponden
e expli
itly as follows. Firstly, let us de(cid:12)ne a
G

olle
tion fg;v1 of elements of V1 indexed by g 2 G, v1 2 V1. Indeed, we de(cid:12)ne
(cid:26)
0 (cid:0)1 0 (cid:0)1
0 g g v1 if g g 2 H1;
fg;v1(g ) =
0 otherwise.
0
It is easily veri(cid:12)ed that if g, g 2 G, h1 2 H1, v1 2 V1, then
0
fh1g;h1v1 = fg;v1; g fgg0;v1 = fg;v1;
G
and if F 2 V1 ,
X
F = f(cid:13);F((cid:13)):
(cid:13)2H1nG
From these relations, the existen
e of a
orresponden
e as in the theorem is easily
G G
dedu
ed, where if L 2 HomG(V1 ;V2 )
orresponds to the fun
tion (cid:1) : V1 ! V2,
then
(cid:1)(g)v1 = (Lfg(cid:0)1;v1)(1);
G
and, for F 2 V1 ,
X
(cid:0)1
(1.5) (LF)(g) = (cid:1)((cid:13) )F((cid:13)g):
(cid:13)2H1nG
This
ompletes the proof of Theorem 1.1.
G G
An intertwining operator L : V1 ! V2 therefore determines a fun
tion (cid:1) on G.
We say that L is supported on a double
oset H2ng=H1 if the fun
tion (cid:1) vanishes
o(cid:11) this double
oset. The situation is parti
ularly simple if V1 and V2 are one
G
dimensional. If (cid:31) is a
hara
ter of a subgroup H of G, we will denote by (cid:31) the
G
representation V , where V is a one-dimensional representation of H a(cid:11)ording the
g g (cid:0)1

hara
ter (cid:31). Also, if g 2 G, we will denote by (cid:31) the
hara
ter (cid:31)(h) = (cid:31)(g hg)
(cid:0)1
of the group gHg .
Corollary 1.2. If (cid:31)1 and (cid:31)2 are
hara
ters of the subgroups H1 and H2 of G, the
G G
dimension of HomG((cid:31)1 ;(cid:31)2 ) is equal to the number of double
osets in H2ng=H1
G G
whi
h support intertwining operators (cid:31)1 ! (cid:31)2 . Moreover, a double
oset H2ng=H1
g
supports an intertwining operator if and only if the
hara
ters (cid:31)2 and (cid:31)1 agree on
(cid:0)1
the group H1 \g H2g.
The
omposition of intertwining operators
orresponds to the
onvolution of the
fun
tions (cid:1) satisfying (1.4). More pre
isely,
NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD 3
Corollary 1.3. Let V1, V2 and V3 be modules of the subgroups H1, H2 and H3
G G G G
respe
tively, and suppose that L1 2 HomG(V1 ;V2 ) and L2 2 HomG(V2 ;V3 )

orrespond by Theorem 1.1 to fun
tions (cid:1)1 : G ! HomC(V1;V2) and (cid:1)2 : G !
HomC(V2;V3). Then the
omposition L2L1
orresponds to the
onvolution
X
(cid:0)1
(1.6) (cid:1)(g) = (cid:1)2(g(cid:13) )Æ(cid:1)1((cid:13)):
(cid:13)2H2nG
This is easily
he
ked.
An important spe
ial
ase:
Corollary 1.4. If V is a module of the subgroup H of G, the endomorphism ring
G
EndG(V ) is isomorphi
to the
onvolution algebra of fun
tions (cid:1) : G ! EndC(V)
whi
h satisfy (cid:1)(h2gh1) = (cid:25)V(h2)Æ(cid:1)(g)Æ(cid:25)V(h1) for h1, h2 2 H.
2. The Bruhat De
omposition. Let F = Fq be a (cid:12)nite (cid:12)eld with q elements,
and let G = GL(r;F). By an ordered partition of r, we mean a sequen
e J =
fj1;::: ;jkg of positive integers whose sum is r. If J is su
h a ordered partition,
then P = PJ will denote the subgroup of elements of G of the form
0 1
G11 G12 (cid:1)(cid:1)(cid:1) G1k
B 0 G22 (cid:1)(cid:1)(cid:1) G2k C
B .. .. .. CA;
. . .
0 (cid:1)(cid:1)(cid:1) 0 Gkk
where Guv is a ju(cid:2)jv blo
k. The subgroups of the form PJ are
alled the standard
paraboli
subgroups of G. More generally, a paraboli
subgroup of G is any subgroup

onjugate to a standard paraboli
subgroup. The term maximal paraboli
subgroup
refers to a subgroup whi
h is maximal among the proper paraboli
subgroups of G.
Thus PJ is maximal if the
ardinality of J is two.
We have PJ = MJ NJ where M = MJ is the subgroup
hara
terized by Guv = 0
if u 6= v, and N = NJ is the subgroup
hara
terized by Guu = Iju (the ju (cid:2) ju
(cid:24)
identity matrix). Evidently M = GL(j1;F)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)GL(jk;F). The subgroup M is

alled the Levi fa
tor of P, and the subgroup N is
alled the unipotent radi
al.
We wish to extend the de(cid:12)nitions of paraboli
subgroups to subgroups of groups
(cid:24)
of the form G = G1 (cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Gk, where Gj = GL(jk;F). By a paraboli
subgroup of
su
h a group G, we mean a subgroup of the form P = P1 (cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Pk, where Pj is
a paraboli
subgroup of Gj. We will
all su
h a P a maximal paraboli
subgroup
if exa
tly one of the Pj is a proper subgroup, and that subgroup is a maximal
paraboli
subgroup. If Mj and Nj are the Levi fa
tors and unipotent radi
als of
the Pj, then M = M1 (cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Mk and N = N1 (cid:2)(cid:1)(cid:1)(cid:1)(cid:2)Nk will be
alled the Levi
fa
tor and unipotent radi
al respe
tively of P.
A permutation matrix is by de(cid:12)nition a square matrix whi
h has exa
tly one
nonzero entry in ea
h row and
olumn, ea
h nonzero entry being equal to one. We
will also use the term subpermutation matrix to denote a matrix, not ne
essarily
square, whi
h has at most one nonzero entry in ea
h row and
olumn, ea
h nonzero
entry being equal to one. Thus a permutation matrix is a subpermutation matrix,
and ea
h minorin a subpermutation matrixis equal to one. Let W be the subgroup
of G
onsisting of permutation matri
es. If M is the Levi fa
tor of a standard
paraboli
subgroup, let WM = W \M. If M = MJ, we will also denote WM = WJ.
If M is the Levi fa
tor of P, then in fa
t WM = W \P.
4 BY DANIEL BUMP
0
Proposition 2.1. Let M and M be the Levi fa
tors of standard paraboli
sub-
0
groups P and P , respe
tively, of G. The in
lusion of W in G indu
es a bije
tion
0
between the double
osets WM0nW=WM and P nG=P.
0
First let us prove that the natural map W ! P nG=P is surje
tive. Indeed, we will
0 0
show that if g = (guv) 2 G, then there exists b 2 B su
h that b g has the form wb,
0 0
for b 2 B. Sin
e B (cid:18) P, P this will show that w 2 P ng=P. We will re
ursively
de(cid:12)ne a sequen
e of integers (cid:21)(r), (cid:21)(r (cid:0)1);(cid:1)(cid:1)(cid:1) ;(cid:21)(1) as follows. (cid:21)(r) is to be the
(cid:12)rst positive integer su
h that gr;(cid:21)(r) 6= 0. Assuming (cid:21)(r), (cid:21)(r (cid:0)1);(cid:1)(cid:1)(cid:1) ;(cid:21)(u+1)
to be de(cid:12)ned, we will let (cid:21)(u) be the (cid:12)rst positive integer su
h that the minor
0 1
gu;(cid:21)(r) gu;(cid:21)(r(cid:0)1) (cid:1)(cid:1)(cid:1) gu;(cid:21)(u)
BBgu+1;(cid:21)(r) gu+1;(cid:21)(r(cid:0)1) (cid:1)(cid:1)(cid:1) gu+1;(cid:21)(u)CC
detB . . C 6= 0:
 . . A
. .
gr;(cid:21)(r) gr;(cid:21)(r(cid:0)1) (cid:1)(cid:1)(cid:1) gr;(cid:21)(u)
0 0
Now the
olumns of b are to be spe
i(cid:12)ed as follows. The last
olumn of b is to be
0
determined by the requirement that the (cid:21)(r)-th
olumn of b g is to have 1 in the
0
r; (cid:21)(r) position, and zeros above. Assuming that the last u(cid:0)1
olumns of b have
0
been spe
i(cid:12)ed, we spe
ify the u-th
olumn from the right of b by requiring that the
0
r(cid:0)u;(cid:21)(r(cid:0)u) entry of b g equal 1, while if v < r(cid:0)u, then the v;(cid:21)(r(cid:0)u) entry of
0
b g equals zero. Now let w be the permutation matrix with has ones in the u;(cid:21)(u)
(cid:0)1 0
positions, zeros elsewhere. Then b = w b g is upper triangular. This shows that
0
the natural map W ! P nG=P is surje
tive.
0
Now let us show that the indu
ed map WM0nW=WM ! P nG=P is inje
tive.
0 0 0 0
Suppose that P = PJ, P = PJ0, J = fj1;(cid:1)(cid:1)(cid:1) ;jkg, J = fj1;(cid:1)(cid:1)(cid:1) ;jlg. Suppose that
0 0 0 0 0 0
w, w 2 W, p, p 2 P su
h that p wp = w . We must show that w and w lie in the
same double
oset of WM0nW=WM. Let us write
0 1
0 1 0 0
W11 (cid:1)(cid:1)(cid:1) W1k W11 (cid:1)(cid:1)(cid:1) W1k
w =  ... ... A w0 = B ... ... CA;
0 0
Wl1 (cid:1)(cid:1)(cid:1) Wlk; Wl1 (cid:1)(cid:1)(cid:1) Wlk
0 1 0 0 0 0 1
P11 P12 (cid:1)(cid:1)(cid:1) P1l P11 P12 (cid:1)(cid:1)(cid:1) P1k
0 0
p = BB 0... P22 (cid:1).(cid:1).(cid:1). P...2lCCA; p0 = BB 0... P22 (cid:1).(cid:1).(cid:1). P...2k CCA;
0
0 0 (cid:1)(cid:1)(cid:1) Pll 0 0 (cid:1)(cid:1)(cid:1) Pkk
0 0 0
where ea
h matrix Wuv or Wuv is a ju (cid:2)jv blo
k, Puv is a ju (cid:2)jv blo
k, and Puv
0 0
is a ju (cid:2)jv blo
k. Let us also denote
0 1
0 1 0 0
Wt1 (cid:1)(cid:1)(cid:1) Wtv Wt1 (cid:1)(cid:1)(cid:1) Wtv
Wftv =  ... ... A; Wft0v = B ... ... CA;
0 0
Wlv (cid:1)(cid:1)(cid:1) Wlv Wlv (cid:1)(cid:1)(cid:1) Wlv
0 1
0 0 0 1
Ptt (cid:1)(cid:1)(cid:1) Ptl P11 (cid:1)(cid:1)(cid:1) P1v
Pet0;= B ... ... CA; Pet;=  ... ... A:
0
0 (cid:1)(cid:1)(cid:1) Pll 0 (cid:1)(cid:1)(cid:1) Pvv
NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD 5
Evidently Wfu0v = Peu0 WftvPev, so Wt0v and Wtv have the same rank. Now the rank of
a subpermutation matrix is simply equal to the number of nonzero entries, and so
rank(Wtv) = rank(Wftv)(cid:0)rank(Wft(cid:0)1;v)(cid:0)rank(Wft;v+1)+rank(Wft(cid:0)1;v+1):
Thus rank(Wfuv) = rank(Wfu0v). Sin
e this is true for every u, v it is apparent
that we
an (cid:12)nd elements multiply w on the left by elements of wM 2 WM and
0
wM0 2 WM0 su
h that wMwwM0 = w .
Corollary 2.2 (The Bruhat De
omposition). We have
[
G = BwB (disjoint):
w2W
This follows by taking P = B.
3. Paraboli
indu
tion and the \Philosophy of Cusp forms". Let J =
fj1;::: ;jkg be an ordered partition of r, and let Vu, u = 1;(cid:1)(cid:1)(cid:1) ;k be modules for
GL(ju;F). Then V = V1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vk is a module for MJ. We may extend the a
tion
of MJ on V to all of P, by allowing NJ to a
t trivially. Then let I(V) = IM;G(V)
denote the G-module obtained by indu
ing V from P. The module I(V) is be said
to be formed from V by paraboli
indu
tion.
In order to have an analog of Frobenius re
ipro
ity for paraboli
indu
tion, it
is ne
essary to de(cid:12)ne a fun
tor from the
ategory of G-modules to the
ategory
of M-modules, the so-
alled \Ja
quet fun
tor." Thus, let W be a G-module. We
de(cid:12)ne amoduleJ(W) = JG;M(W) to be the set of allelements usu
h that n:u = u
for all n 2 N. Sin
e N is normalized by M, J(W) is an M-submodule of W. It is

alled the Ja
quet module. Other names for the Ja
quet fun
tor from the
ategory
of G-modules to the
ategory of M-modules whi
h have o

urred in the literature
are \lo
alization fun
tor," \trun
ation," and \fun
tor of
oinvariants."
Lemma 3.1. Let U be a G-module, U0 the additive subgroup generated by all
elements of the form u (cid:0) nu with u 2 U, n 2 N. Then U0 is an M-submodule
of U and we have a dire
t sum de
omposition
U = J(U)(cid:8)U0:
Sin
eM normalizesN, U0 isanM-submoduleofU. Weshow(cid:12)rstthatJ(U)\U0 =
P
f0g. Indeed, suppose that u0 = i(ui (cid:0)niui) is an element of U0. If this element
is also in J(U), then
" #
X X X X
1 1
u0 = nu0 = nui (cid:0) nniui = 0:
jNj jNj
n2N i n2N n2N
On the other hand, to show U = J(U)+U0, let u 2 U. Then
X X
1 1
u = nu+ (u(cid:0)nu);
jNj jNj
n2N n2N
where the (cid:12)rst element on the right is in J(U), while the se
ond is in U0.
6 BY DANIEL BUMP
Proposition 3.2. Let U be a G-module, M and N the Levi fa
tor and unipotent
radi
al, respe
tively, of a proper standard paraboli
subgroup P of G. Then a ne
-
essary and suÆent
ondition for JG;M(U) 6= 0 is that there exists a nonzero linear
fun
tional T on U su
h that T(nu) = T(u) for all n 2 N, u 2 U.
Indeed, a ne
essary and suÆ
ient
ondition for a given linear fun
tional T to have
the property that T(nu) = T(u) for all n 2 N, u 2 U is that its kernel
ontain the
submodule U0 of Lemma 3.1. Thus there will exist a nonzero su
h fun
tional if and
only if U0 6= U, i.e. if and only if J(U) 6= 0.
Proposition 3.3. Suppose that V is an M-module and U a G-module. We have
a natural isomorphism
(cid:24)
(3.1) HomG(U;I(V)) = HomM(J(U);V):
Indeed, by Frobenius re
ipro
ity (1.1), we have an isomorphism
(cid:24)
HomG(U;I(V)) = HomP(UP;V):
Re
all that the a
tion of M on V is extended by de(cid:12)nition to an a
tion of P by
allowing N to a
t trivially. Then it is
lear that a given M-module homomor-
phism (cid:30) : U ! V is a P-module homomorphism if and only if (cid:30)(U0) = 0. Thus
(cid:24) (cid:24)
HomP(UP;V) = HomM(UM=U0;V). By Lemma 3.1, UM=U0 = J(U).
Proposition 3.3 shows that the Ja
quet
onstru
tion and paraboli
indu
tion are
adjoint fun
tors.
There is also a transitivity property of paraboli
indu
tion, analogous to (1.3).
Proposition 3.4. Let M is the Levi fa
tor of a paraboli
subgroup P of G, and let
Q be a paraboli
subgroup of M with Levi fa
tor M0. Then there exists a paraboli
subgroup P0 (cid:18) P of G su
h that the Levi fa
tor of P0 is also M0. Thus if V is an
M0-module, then both IM0;M(V) and IM;G(V) are de(cid:12)ned as M- and G-modules,
respe
tively. We have
(cid:24)
(3.2) IM;G(IM0;M(V)) = IM0;G(V):
We leave the proof to the reader.
It is also very easy to show that:
Proposition 3.5. The Ja
quet and paraboli
indu
tion fun
tors are exa
t.
An important strategy in
lassifying the irredu
ible representations of redu
tive
groups over (cid:12)nite or lo
al (cid:12)elds
onsists in trying to build up the representations
from lower rank groups by paraboli
indu
tion. This strategy was
alled the Phi-
losophy of Cusp Forms by Harish-Chandra, who found motivation in the work of
Selberg and Langlands on the spe
tral theory of redu
tive groups. An irredu
ible
representation whi
h does not o

ur in IM;G(V) for any representation V of the
Levi fa
tor of a proper paraboli
subgroup is
alled
uspidal.
Proposition 3.6. Any irredu
ible representation of G o

urs in the
omposition
series of some represention of the form IM;G(V), where V is a
uspidal represen-
tation of the Levi fa
tor M of a paraboli
subgroup P.
Indeed, let P be minimal among the paraboli
subgroups su
h that the given rep-
resentation of G o

urs as a
omposition fa
tor of IM;G(V) for some representation
NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD 7
V of the Levi fa
tor M of P. (There always exist su
h paraboli
s P, sin
e we
may take P to be G itself.) If V is not
uspidal, it o

urs in the
omposition se-
ries of IM0;M(V0) for some Levi fa
tor M0 of a proper paraboli
subgroup of M.
Now in Proposition 3.3 the given representation of G also o

urs in IM0;G(V0),

ontradi
ting the assumed minimality of P.
A

ording to the Philosophy of Cusp Forms, the
uspidal representations should
be regarded as the basi
building blo
ks from whi
h other representations are
on-
stru
ted by the pro
ess of paraboli
indu
tion. Proposition 3.6 shows that every
irredu
ible representation of G may be realized as a subrepresentation of IM;G(V)
where V is a
uspidal representation of the Levi
omponent P of a paraboli
sub-
group. Moreover, there is also a sense in whi
h this realization is unique: although
P is not unique, its Levi fa
tor M and the representation V are determined up to
isomorphism. More pre
isely, we have
0
Theorem 3.7. Let V and V be
uspidal representations of the Levi fa
tors M
0 0
and M of standard paraboli
subgroups P and P respe
tively of G. Then either
0
IM;G(V) and IM0;G(V ) have no
omposition fa
tor in
ommon, or there is a Weyl
(cid:0)1 0
group element w in G su
h that wMw = M , and a ve
tor spa
e isomorphism
0
(cid:30) : V ! V su
h that
(cid:0)1
(3.3) (cid:30)(mv) = wmw (cid:30)(v) for m 2 M, v 2 V.
0
In the latter
ase, the modules IM;G(V) and IM0;G(V ) have the same
omposition
fa
tors.
(cid:0)1 0 0
Remark. The signi(cid:12)
an
e of (3.3) is that if wMw = M , then M and M are
0
isomorphi
, and so V may be regarded as an M-module. Thus (3.3) shows that
0
(cid:30) is an isomorphism of V and V as M-modules. In other words, for the indu
ed
0
representations to have a
ommon
omposition fa
tor, not only do M and M have
0
to be
onjugate, but V and V must be isomorphi
as M-modules.
We will defer the proof of this Theorem until the next se
tion. Of
ourse the

omplete redu
ibility of representations of a (cid:12)nite group G implies that two G-
modules have the same
omposition fa
tors if and only if they are isomorphi
. We
have stated the theorem this way be
ause this is the
orre
t formulation over a
lo
al (cid:12)eld. Over a lo
al (cid:12)eld, one en
ounters indu
ed representations whi
h may
have the same
omposition fa
tors, but still fail to be isomorphi
.
There are two problems to be solved a

ording to the Philosophy of Cusp forms:
(cid:12)rstly, the
onstru
tion of the
uspidal representations; and se
ondly, the de
om-
position of the representations obtained by paraboli
indu
tion from the
uspidal
ones. We will examine the se
ond problem in the following se
tions.
It follows from transitivity of indu
tion that it is suÆ
ient for a given irredu
ible
representation to be
uspidal that the representation does not o

ur in IM;G(V)
for any maximal paraboli
subgroup P. Indeed, suppose that the representation is
not
uspidal. Then it o

urs in IM0;G(V) for some proper paraboli
subgroup P0
of G, and some representation V of the Levi fa
tor M0 of P0. If P is a maximal
paraboli
subgroup
ontaining P0, and if M is the Levi fa
tor of P, then by (3.2) it
also o

urs in IM;G(IM0;M(V)). Thus if an irredu
ible representation o

urs in a
8 BY DANIEL BUMP
representation indu
ed from a proper paraboli
subgroup, it may be assumed that
the paraboli
is maximal.
It follows from Proposition 3.3 that a a ne
essary and suÆ
ient
ondition for the
representation U to be
uspidal is that JG;M(U) = 0 for every Levi fa
tor M of a
maximal paraboli
subgroup.
4. Intertwining operators for indu
ed representations. In this se
tion we
shall analyze the intertwining operators between two indu
ed representations. We
shall also prove Theorem 3.3. First let us establish
0 0 0
Proposition 4.1. Let J = fj1;(cid:1)(cid:1)(cid:1) ;jkg, J = fj1;(cid:1)(cid:1)(cid:1) ;jlg be two ordered partitions
of r, and let P = PJ, P0 = PJ0. Let M (cid:24)= GL(j1;F)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)GL(jk;F) and M0 (cid:24)=
0 0 0
GL(j1;F)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)GL(jl;F) be their respe
tive Levi fa
tors. Let Vu, (resp. Vu) be
0
given
uspidal GL(ju;F)-modules (resp. GL(ju;F)-modules). Let V = V1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vk,
0 0 0
V = V1 (cid:10) (cid:1)(cid:1)(cid:1) (cid:10) Vl, and let d = dimC HomG(IM;G(V);IM0;G(V )). Then d = 0
unless k = l, in whi
h
ase, d is equal to the number of permutations (cid:27) of f1;(cid:1)(cid:1)(cid:1) ;kg
0 (cid:24) 0
su
h that j(cid:27)(u) = ju; and su
h that V(cid:27)(u) = Vu as GL(j(cid:27)(u);F)-modules for ea
h
u = 1;(cid:1)(cid:1)(cid:1) ;k.
To prove Proposition 4.1, assume (cid:12)rst that we are given nonzero intertwining
0
operator in HomG(IM;G(V);IM0;G(V )), whi
h is supported on a single double
0

oset of P nG=P. We will asso
iate with this intertwining operator a bije
tion
(cid:27) : f1;(cid:1)(cid:1)(cid:1) ;kg ! f1;(cid:1)(cid:1)(cid:1) ;lg whi
h has the required properties. Then we will show
that the
orresponden
e between double
osets whi
h support intertwining oper-
ators and su
h (cid:27) is a bije
tion, and that no
oset
an support more than one
intertwining operator. This will show that the dimension d is equal to the number
of su
h (cid:27).
0
Given the intertwining operator, let (cid:1) : G ! HomC(V;V ) be the fun
tion
asso
iated in Theorem 3.1. Thus
0 0 0 0
(4.1) (cid:1)(p gp):v = p (cid:1)(g)p:v for p 2 P, p 2 P , v 2 V.
0
Re
all that we are assuming that (cid:1) is supported on a single double
oset P wP,
where by the Bruhat de
omposition we may take the representative w 2 W. Let
0
(cid:30) = (cid:1)(w) : V ! V .
(cid:0)1 0
Now let us show that wMw = M .
0 (cid:0)1 0 (cid:0)1
First we show that M (cid:18) wMw . Suppose on the
ontrary that M 6(cid:18) wMw .
0 (cid:0)1
Let NP be the unipotent radi
al of P. Then Q = M \ wPw is a proper (not
0
ne
essarily standard) paraboli
subgroup of M , whose unipotent radi
al NQ =
0 (cid:0)1 (cid:0)1
M \ wNPw is
ontained in the unipotent radi
al of wPw . Now let v 2 V,
(cid:0)1 (cid:0)1 0
and n 2 NQ. Applying (4.1) with g = w, p = w n w, p = n, we see that
(cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1
n:(cid:30)(w n w:v) = (cid:30)(v). Now sin
e w n w is
ontained in the unipotent radi
al
(cid:0)1 (cid:0)1
of P, w n w:v = v, and so if n 2 NQ we have n:(cid:30)(v) = (cid:30)(v). Thus (cid:30)(v) = 0,
0
sin
e V is
uspidal. This shows that (cid:30) is the zero map, whi
h is a
ontradi
tion.
0 (cid:0)1 0 (cid:0)1
Therefore M (cid:18) wMw . The proof of the opposite inequality M (cid:19) wMw is
similar.
(cid:0)1 0 0
Now the isomorphism m ! wmw of M onto M makes V into an M-module.
(4.1) implies that if m 2 M, v 2 V, then
(cid:0)1
(4.2) wmw (cid:30)(v) = (cid:30)(mv):
NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD 9
0
This implies that if V, V are regarded as M-modules, they are isomorphi
, sin
e
(cid:30) is an isomorphism. By S
hur's Lemma, there
an be only one isomorphism
(up to
onstant multiple) between irredu
ible M-modules, and sin
e by (4.1) (cid:1) is
determined by (cid:30), it follows that there
an be at most one intertwining operator
0
supported on ea
h double
oset. On the other hand, if w is given su
h that M =
(cid:0)1 0 0
wMw , and ifthe indu
ed M-modulestru
ture onV makesV andV isomorphi
,
then denoting by (cid:30) su
h an isomorphism, so that (4.2) is satis(cid:12)ed, it is
lear that
0
(4.1) is also satis(cid:12)ed, sin
e the unipotent matri
es in both P and P a
t trivially
0
on V and V .
0
ThusbyProposition2.1,thedimensiondofthespa
eHomG(IM;G(V);IM0;G(V ))
of intertwining operators is exa
tly equal to the number of w 2 WM0nW=WM su
h
(cid:0)1 0 0
that wMw = M , and su
h that the indu
ed M-module stru
ture on V makes
(cid:24) 0
V = V . It is
lear that this is equal to the number of permutations (cid:27) of f1;(cid:1)(cid:1)(cid:1) ;kg
0 (cid:24) 0
su
h that j(cid:27)(i) = ji; and su
h that V(cid:27)(i) = Vi as GL(j(cid:27)(i);F)-modules for ea
h
i = 1;(cid:1)(cid:1)(cid:1) ;k.
Corollary 4.2. Let P (cid:18) G be the standard paraboli
subgroup PJ where J is the
(cid:24)
ordered partition fj1;(cid:1)(cid:1)(cid:1) ;jkg of r. Let M = GL(j1;F)(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)GL(jk;F) be the Levi
fa
tor of P, and let Vu be given
uspidal GL(ju;F)-modules. Let V = V1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vk.
Then I(V) is redu
ible if and only there exist distin
t u, v su
h that ju = jv, and
(cid:24)
Vu = Vv as GL(ju;F)-modules.
0
This follows from Proposition 4.1 by taking P = P.
We now give the proof of Theorem 3.7. The only part whi
h is not
ontained in
Proposition 4.1 is the (cid:12)nal assertion that if there exists a Weyl group element w in
(cid:0)1
G su
h that wMw , and (cid:30) su
h that (3.3) is satis(cid:12)ed, then the indu
ed modules
have the same
omposition fa
tors.
0 0 0
Theproblemmaybestatedasfollows. LetJ = fj1;(cid:1)(cid:1)(cid:1) ;jkgandJ = fj1;(cid:1)(cid:1)(cid:1) ;jkg
0
be two sets of positive integers whose sum is r, and assume that J is obtained
from J by permuting the indi
es j1. Thus there exists a bije
tion (cid:27) : f1;(cid:1)(cid:1)(cid:1) ;kg !
0
f1;(cid:1)(cid:1)(cid:1) ;kg su
h that j(cid:27)(i) = ji: Let Vi be a
uspidal GL(ji;F)-module for i =
0 0 0
1;(cid:1)(cid:1)(cid:1) ;k, and let Vi = V(cid:27)(i). Let P = PJ, P = PJ0, M = MP, and M = MP0.
0 0 0 0
Then V = V1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vk and V = V1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Vk are M- and M -modules respe
tively.
0
What is to be shown is that IM;G(V) and IM0;G(V ) have the same
omposition
fa
tors. Clearly it is suÆ
ient to show this when (cid:27) simply inter
hanges two adja-

ent
omponents. Moreover in that
ase, by transitivity and exa
tness of paraboli
indu
tion (Propositions 3.4 and 3.5) it is suÆ
ient to show this when k = 2. Now
(cid:24) 0 (cid:24) 0
if V = V this is obvious. On the other hand, if V 6= V then by Corollary 4.2,
0 0
IM0;G(V ) is irredu
ible, and a nonzero intertwining map IM;G(V) ! IM0;G(V )
exists by Proposition 4.1. Consequently IM;G(V) (cid:24)= IM0;G(V0). This
ompletes the
proof of Theorem 3.7.
5. The Kirillov Representation. Let G = Gr = GL(r;F) as before, and let
Pr be the subspa
e
onsisting of elements having bottom row (0;::: ;0;1). Let
N = Nr be the subgroup of unipotent upper triangular matri
es, and let Ur be the
subgroup
onsisting of matri
es whi
h have only zeros above the diagonal, ex
ept
(cid:24) r(cid:0)1
for the entries in the last
olumn. Thus Ur = F . If k (cid:20) r, we will denote by Gk
the subgroup of G, isomorphi
to GL(k;F),
onsisting of matri
es of the form
(cid:18) (cid:19)
(cid:3) 0
;
0 Ir(cid:0)k
10 BY DANIEL BUMP
where `(cid:3)' denotes an arbitrary k (cid:2) k blo
k, and Ir(cid:0)k denotes the r (cid:0) k (cid:2) r (cid:0) k
identity matrix. Identifying GL(k;F) with this subgroup of Gr, the subgroups Pk,
Nk and Uk are then
ontained as subgroups of Gr. Thus Pk = Gk(cid:0)1:Uk (semidire
t
produ
t).
Let = F be a (cid:12)xed nontrivial
hara
ter of the additive group of F. For k (cid:20) r,
(cid:2)
let (cid:18)k : Nk ! C be the
hara
ter of Nk de(cid:12)ned by
00 11
1 x12 x13 (cid:1)(cid:1)(cid:1) x1k
BB 1 x23 (cid:1)(cid:1)(cid:1) x2kCC
BB CC
BB .. .. CC
(cid:18)kBB 1 . . CC = (x12+x23 +(cid:1)(cid:1)(cid:1)+xk(cid:0)1;k):
BB CC
.
 . AA
.
1
We will denote (cid:18)r as simply (cid:18). Then let K = Kr be the module of Pr indu
ed from
the
hara
ter (cid:18) of Nr. By de(cid:12)nition K is a spa
e of fun
tions Pr ! C. However,
ea
h fun
tion in K is determined by its value on Gr(cid:0)1, and it is most
onvenient
to regard K as a spa
e of fun
tions on Gr(cid:0)1. Spe
i(cid:12)
ally,
K = ff : Gr(cid:0)1 ! Cjf(ng) = (cid:18)r(cid:0)1(ng)f(g) for n 2 Nr(cid:0)1g;
and the group a
tion is de(cid:12)ned by
(cid:18)(cid:18) (cid:19) (cid:19)
h u
f (g) = (cid:18)(gu)f(gh) for g, h 2 Gr(cid:0)1, u 2 Ur, f 2 K.
1
K is
alled the Kirillov representation of the group Pr.
Theorem 5.1. The Kirillov representation of Pr is irredu
ible.
To prove this, it is suÆ
ient to show that HomPr(K;K) is one dimensional. Let
there be given a double
oset in NrnPr=Nr whi
h supports an intertwining operator
K ! K. Let (cid:1) : Pr ! C be the fun
tion asso
iated with the given intertwining
operator. We may take a
oset representative h whi
h lies in Gr(cid:0)1. We will prove
that h = Ir(cid:0)1. Theorem 5.1 will then follow from Corollary 1.2.
Suppose by indu
tion that we have shown that h 2 Gk, where 1 (cid:20) k (cid:20) r(cid:0)1. We
will show then that h 2 Gk(cid:0)1. (If k = 1, this is to be interpreted as the assertion
that h = 1.) Let
(cid:18) (cid:19)
Ik(cid:0)1 u
2 Uk;
1
k(cid:0)1
where u is a
olumn ve
tor in F . We have
0 1 0 1(cid:0)1
(cid:18) (cid:19) Ik(cid:0)1 h:u (cid:18) (cid:19) Ir(cid:0)1 u
h h
=  1 A  1 A ;
Ir(cid:0)k Ir(cid:0)k
Ir(cid:0)k+1 Ir(cid:0)k+1
and the bi-invarian
e property (1.4) of (cid:1) implies that
00 11 00 11
Ik(cid:0)1 h:u Ik(cid:0)1 u
(cid:18)k 1 AA = (cid:18)k 1 AA:
Ir(cid:0)k Ir(cid:0)k
Sin
e this is true for all u, it follows that h 2 Gk(cid:0)1.