old forms on gln

Old Forms on GLnAuthor(s): Mark ReederSource: American Journal of Mathematics, Vol. 113, No. 5 (Oct., 1991), pp. 911-930Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2374790 .

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OLD FORMS ON GLn

By MARK REEDER

0. Introduction. This note presents several quantitative results about vectors in a generic representation of GLn over a p-adic field which are invariant under certain compact open subgroups. An appli- cation to the cuspidal cohomology of arithmetic groups is also given. This work is based to a large extent on the ideas in [J-P-S].

Let F be a non archimedian local field and ar an irreducible admissible representation of Gn = GLn(F). Let L(s, 7r) and E(s, rr) be as in [G-J]. In particular, E(s, 'r) = MYC, where M E C', y = q-S and c = c(7r) is a nonnegative integer. It was shown in [J-P-S] that there is a decreasing family of compact open subgroups K(m) of Gn (defined below) such that if ar is generic (i.e., if ar has a Whittaker model) then the space rK(c) of K(c) invariant vectors is one dimensional and IK(c-1)

= 0. There is a canonical generator of rK(c), called a "new vector." Our new results are as follows.

Section 2. We give a basis of 11K(c+i), i > 0 in terms of convolution by the Hecke algebra 1n-, of spherical functions on Gn l. These convolution operators are generalizations of Atkin-Lehner operators for GL2. (See [Cl], [D].) In terms of automorphic forms, our result gives a basis of the space of old forms at a given level, once the new forms at lower levels are known.

Section 3. Let I(x) be a (possibly reducible) unramified principal series representation of G, where X is a character of the diagonal torus. Then I(x) has a unique generic constituent u(x). Conjugating X by the Weyl group, we can arrange that o(X) is the irreducible submodule of I(X)

We compute the L and E-factors of u(x) in terms of the coroots &t such that Xa& is the modulus of F. We then identify the new vector in

911

Manuscript received 24 January 1990. Research partially supported by NSA grant MDA904-89-H-2027. American Journal of Mathematics 113 (1991), 911-930.



912 MARK REEDER

a(x) as a function in I(x). Again the Hecke algebra from Gn-1 plays a role, this time acting on I(x). The new vector is given by the convolution 'p * uP, where 'p E bn-1 is described in terms of the L-functions of I(x) and u(X), and uP, is the spherical vector in I(x), suitably normalized.

Section 4. Now let F be an algebraic number field, with adele ring A. Write A., Af for the subrings of infinite and finite adeles, respectively. Recall that if F is an arithmetic subgroup of SLn(F) then we have a canonical isomorphism

Hc*usp(, C) & H*(g, K., a.) 0 flT.

Here g and K. are the Lie algebra and a maximal compact subgroup of SLn(A.), and 7r = 7Fa. 0 ITf runs over the irreducible cuspidal automorphic representations of SLn(A). Each ar is a direct summand of the restriction to SLn of a cuspidal automorphic representation -r of GLn. If the local component fr, of fr remains irreducible under SLn(Fp), we can apply the generalized Atkin-Lehner operators to get lower bounds on the dimension of cuspidal cohomology for certain F's. We carry this out when n = 3, F = Q, and F is the congruence group Fo(pm), p a prime.

I thank Professor Ash for suggesting the topic, and Professor Rallis for a helpful conversation.

1. Notation and preliminaries. Let F be a non archimedian local field with ring of integers o, w a generator of the maximal ideal of o, and q the cardinality of residue field. Also let Gn = GLn(F), Kn = GL4(0), Nn = upper triangular matrices in Gn, An = diagonal matrices in Gn, Bn = NnAn. We view Gn < Gn+j and Kn < K,+1 embedded in the upper left corner. The Haar measure on Gn is chosen so that the volume of Kn is 1.

Let T be a character of the additive group of F with T trivial on o, T(7-1) # 1. Let On be the character of Nn, defined for x = (xi,) E Nn, by 0(x) = Tll-i-n1 (Xi,i+l)

If X is a (quasi)character of Bn, the induced representation

I(X) = IndBn(X)



OLD FORMS ON GLn 913

consists of those locally constant C-valued functions f on Gn such that

f(bg) = 8(b)"2x(b)f(g) for b E Bn, g E Gn,

where 8 is the modulus of Bn. The induced representation

Ion = IndNn(0n)

is defined similarly except the modulus is trivial. We will be considering admissible representations (7r, V) which admit a nontrivial Gn-equivari- ant homomorphism 0W: V -> Ion. If such a ar is irreducible, it is called "generic."

For example, let rr = '(x). Then (cf. [C-S]) there is (up to scalar) a unique W,: '(x) I-> On such that

VW,(f)(e) = IN f(wou)du

for all f E I(x) with support in the big cell BnwoBn. Let fn be the convolution algebra of Kn bi-invariant functions on

Gn with compact support. We have the Satake isomorphism

s: Sn '-> Iin

where Sn = C[T1, . . ., Tn, Tn-'] and T, is the ith elementary symmetric polynomial in variables X1, . . . , Xn. If x = (x1, . . . , xn) E (Cx)n, then we get an unramified character X of Bn via diag (a1, ..., an) ->

4i(a), extended to Bn. The space of Kn-fixed vectors in I(x) is one dimensional and affords an 1n eigencharacter Ax. In fact s is determined by the equation AX(S(P)) = P(x).

Let Ai(n) = {f = (fl, . . ., fn) E Znf12 fn}, A(n, i) =

{f E Ai(n) I E fj = i}. For f E Zn, let wf = diag(fl, ... , wifn) E An. For x E (CX)n, let W(g) = W(g, x1, . . ., xn, T) be the unique Kn-

fixed vector in Oen which transforms by Ax under jn, such that W(e) = 1. We have the following formula for W(g) ([S], [C-S]). Let n E Nn, k E Kn, f E Zn. Then

W(nwfk) = 0(n)8(wf) /2pf(X),



914 MARK REEDER

where pf(x) = 0 unless f E Ai(r), in which case pf(x) is the character of the finite dimensional GLn(C) representation with highest weight (with respect to Bn) diag(a, . . . , an) ->H a.ii

Let ((X) = lij<(qxi - x,) and let uPx E I(x) be defined by p(bk) = X81/2(b) for b E B, k E Kn. It is shown in [C-S] that for an appropriate choice of Haar measure on N, we have 0W(up) = t(X)W

By replacing the xi's with indeterminants Xi we can view W as a right K&-invariant function

W G-> Sn.

Notice that if P E Sn then

(;(P) * W)(g) fG (P)(h)W(gh)dh = PW(g).

Now let (7r, V) be any admissible representation of G+ 1. We further suppose we have a Whittaker map 'W E HomG,+1(V, Ien+1).

Denote the ring of formal Laurent series in an indeterminant Y with coefficients in S,, by S,(Y). The subring of polynomials is denoted by S4[Y].

Define a linear map

W V - Sn,(Y)

by I(v) = E ai(v)YL, where

ai(v) = i/2 W(V)(( ( ))W(g, X1, * * , X,,, T1)dg. Idet gl=q-i

It is shown in [J-P-S, Section 3] that the integrands have compact support so ai(v) indeed belongs to Sn, and moreover, that t(v) = 0 implies OW(V) I G,

= 0 for v E VKn.

Define an action of in on V by

'P * V = f 'P(h)IT(( ))v|det h| 112dh,

for ' E jn, v E V. This action preserves VKn.



OLD FORMS ON GL. 915

(1.1) LEMMA. If P E Sn, and v E V, then

r(s(P) * v) = P(Y X, . . . , Y Xn)I(v)

Proof. Let P E Sn be homogeneous of degree d in the Xi's, so that s(P) is supported on {g E Gn I Idet gl = q-d}. Using this, we compute, for v E V,

ai(s(P) *v) =qi/ VfOlG W(P) * v)( 0) tNn\G, (0 1)

Idet gl =q-'

* W(g, Xi, . .. , Xn, T-)dg

= JN . 1 ?lW?(V) (( g

)> (P)(h) det g lh 1/2 Nnl\Gn G,,0 Idet gl =q-'

*W(g, Xi, ... , X,, T-l)dg

fNn\G, GW(V)(( ))d (P)(h) Idet gl -1/2

Idet gI =q-t+d

Idet gl =q-'+d

s(P) * W(g, Xi, . . , X,, T-1)dg

= P(X,. . . , Xn)ai-d(V)-

The lemma follows readily. U

2. The basis of old vectors. Assume now that (7r, V) is an (irreducible) generic representation of Gn+,. Let 'P1(Y) E C[Y] be the polynomial such that L(rr, s) = 9P(q-s)-l. Then ([J-P-S, Section 4])



916 MARK REEDER

n H 9?,(Y X,) -W(v) E Sn[Y, Y]

j= 1

for every v E V. Thus, evaluation of Hlj=1 9?,(Y Xj) -(v) at Y = 1 defines a linear map V -> Sn and by restriction a map VK-

Sn.

The following is essentially a reformulation of the first main theorem of [J-P-S].

(2.1) THEOREM [J-P-S]. If (wr, V) is generic, then - : VKN 5, is an isomorphism of Sn modules, where Sn acts on VKn as above and on itself by multiplication.

In particular there is a "new vector" vo E VKn such that *(vo) = 1. This means (see Section 1)

/s f O\ n - W(vo) 8-1"2(Wf)pf = H 9P,(Xj)'. fEA(n) \0 1/

as formal power series in T1, . T. , Tn. The inverse of - is given by

(p)= (P) * vo.

For nonnegative integers m and k, let K(m, k) be the subgroup of Kn,+1 consisting of those matrices of the form

g x

where g E Kn, y E (Wmo)n x E (11ko)n, Z - 1 E 1k0o nw mO. Set K(m) = K(m, 0), K(??) = nm?l K(m). Note that

-W(,rr)K.= U 0W1('r)K(m)k)

m,k

Correspondingly, we define Sn(m, k) to be those elements in Sn = C[Ti, ... , Tn, Tn-'] whose degree in the Ti's is ?m and whose degree in Tn is ? - k. Also let Sn(m) = Sn(m, 0). The dimension of Sn(M, k) iS (m+k+n) 1 m+kJ



OLD FORMS ON GL, 917

The conductor c = cond (ar) of a generic representation (ar, V) is the smallest integer such that v0 E VK(c).

(2.2) THEOREM 1. Let (rr, V) be a generic representation of GLn+l(F) with conductor c and new vector vo. Then

-(VK(m+c,k)) - Sn(m, k)

for m 2 0, k. 0 Consequently, the map S(m, k) -> V defined by P h-4

s(P) * vo gives a linear isomorphism

Sn(m, k) - VK(m+c,k)

We can check the theorem directly in the simple case when c =

k = 0, m = 1. Here wr is spherical and generic, so wr must be an '(x) for some X. By counting double cosets, one checks that dim NK(1) = -

+ 1. On the other hand, Sn(1) is the space of homogeneous polynomials with n variables and degree <1 so dim S(1, 0) = n + 1. Curiously, these dimension counts are unchanged if I(x) is reducible, but the theorem is false for any v0 E I(x) unless I(x) itself has a Whittaker model. We discuss this further in the next section.

Proof of Theorem 1. First suppose k = 0. We first show that

X-'S (m) C VK(m+c).

Let 4ij = q(Ti) where s: Sn --> l is the Satake isomorphism. In fact, 4i

equals qi(i-n)/2 times the characteristic function of

KnwJf(i)Kn

where f(i) = (1, 1, . . , 1, O, . .. , 0) E A (n, i). Hence if v E VKn,

we have

1 * v = q i(i - -1)/2 f'Tr~.,( vdh. qJi V q J~Knllf(')Kn

v( 0 1)V

Let

U(d) = {K ]: u E (d o)n},



918 MARK REEDER

and observe that if h E Kn,Jf(i)Kn then

hU(d + 1)h- C U(d)

and

h(U(O)')h-1 C U(O)'.

Hence if v is invariant under both U(d) and U(O)' then ' * v is invariant under U(d + 1) and U(O)'. Thus

p * VK(d) C VK(d+1)

By induction on m, if P C Sn(m) then

( -) P((p) , pPr) * v0 E VK(m + c)

To get the other containment, we must show that if up E W(V)K(m+c) then Ep) C C[T1,. . , Tn] and the degree of any monomial appearing in -p) is at most m.

Since up is invariant under K(oo), we must have

(& 0)

to 1J

if fn < 0 ([J-P-S], 3.2). Moreover, W( X1, . Xn) E C[T1, .

T,] if fn ? 0. It follows, writing E fj = If, that

/f & (Qp) = H QsP9(X1) z qI K 1) W(fT, X1l . . . , Xn) fEHEA(n) \0 1/

belongs to C[T1, ... , Tn]. For the second assertion we use ([J-P-S, (5.3)]), where it is shown

that there is a Q E C[T1, . . ., Tn] C C[X1, . . . , Xn] such that

Tnm+cQ(q-lX1... , q-1X-1) = Tc(p).




Let T = Ti(q1 X1, , q1X-1) = q-'Tn i7T, and suppose

Q =, EAOT .. TOni

where In = (1, . . , 8n) E Nn and A E C. Set 18 = E i, (i) =

pi + 202 + + nOn. Then

-(p) = TnmzAcTfi 1 *Ti

q z q-(P)AOTn--

... T.. . T-pl.

The degree of the th monomial is m - ?n- m. This completes the

proof in case n = 0. To finish, note that

WV e )K(m+c,k) < 1k * E W(T)K(m+k+c)

(Jn* (p) = Tnp)E Sn(m + k)

< =((p) E TnkSn(M + k) = Sn(M, k). U

(2.3) Example. Let (i,u W) be a cuspidal representation of H =

GLn+ 1(Fq), pulled back to Kn+1 and extended to ZKn+1, where Z is the center of Gn+ 1. Consider the representation (7r, V) where V =

Indz+ jQfi. Here we are using compact induction, and V is an irreducible supercuspidal representation. All supercuspidal representations of Gn+1

are generic. It is known (c.f. [B-F]) that cond (7r) = n + 1. We will write down an explicit basis of VK(,) which is different from the one given in Theorem 1, but which verifies the equation dim VK(n+l+i) =

dim Sn(i). The restriction of W to H is the "Gelfand-Graev" representation

of H, and therefore contains a unique vector wo invariant under the lower triangular Borel subgroup B of H, embedded as always in the upper left corner of GLn+j (Fq). Let f = (fl, . . f,) E Zn be such

that fi > f2> ... fn > 01 and set w3jf = diag(w&1,. . ., IwfnI 1) E Gn+l.



920 MARK REEDER

Then if N 2 n + 1, one checks that wfK(N)w-f n Kn+, maps into B upon reduction modulo wo. We can therefore define

yf ZKn- ifK(N) -> W

by pf(kfTh) = 4(k)wo, for k E ZKn+1, h E K(N). The ypfs are linearly independent by the Cartan decomposition.

Then the new vector is (up to scalar) wpf, where fo = (n, n - 1, . 1). More generally, VK(N) is spanned by {pf: fi < N} and the

number of such f's equals the dimension of Sn(N - n - 1).

3. The unramified principal series. In this section we consider representations occurring in the unramified principal series of Gn+

Let D, D+ i A be the roots, positive roots and simple roots of Gn+1 with respect to Bn+1, and let W be the Weyl group. For ao E D, we set

w= &() where & is the coroot of ao. Let X be an unramified character of An+1 corresponding to (x1, . . . , Xn+1) E (CxX)n+. '(X) now stands for Ind'++,(x) and u(x) is the unique generic constituent of '(x). Also, (Px is the Kn+i invariant function in I(x) such that qp(e) = 1. We set

SX = {a I X(a.) = q-l}

This is called S in [R]. From [B-Z] we know that I(x) is irreducible if and only if Sx = 0. Furthermore, if 'W: (x) -> n+ is the (unique up to scalar) Gn+i homomorphism (see Section 1), then 'W is injective if and only if Sx 5 ?+.

In order to explicitly determine the K(oo) invariants in u(X), we need a formula for the conductor. We may assume Sx C ?+.

We set

Cx = {i I xixj = q for some j > i}.

In particular, I Cx equals the number of simple roots beginning a root in Sx. Also let DX be the set of j such that no root of the form aoi + * + aoj, i < j, belongs to Sx.




(3.1) PROPOSITION. When S, S D', we have

L(s, o(x)) = I (1 - xjq-s), jEDx

E(S, a(X)) = (-q)lcxl(s) II xi. iECx

In particular, the conductor of u(x) is ICxI.

Proof. Let Y = q-S. From [G-J, Chapter 3] we have the relation

ESa)L (1 s, o(X) E E(s, '(X)) L (1 - s, I(X) E(s, ) L( -s, o(X)) L(s, I(x))

where E(s, u) = MYC, c = cond (u(x)) and M E CX. Moreover, E(S, I(x)) = 1.

The right hand side is

(-qY)f+l nj1 xi(1 - Yxi) i =1 (1 - q Yxi)'

The ith pole cancels iff qxi = xj for some j. In this case, we have j > i since Sx C D+. The n + 1th pole never cancels.

On the other hand, there are subsets T, T C {1, . . ., n + 1} such that

L(s, a(x)) = H (1 - Yxj)-1 and L(s, (X-1)) = H (1 - Yx-1)-1. jET iET

It follows that the left side is

M(-qY)ItlYc n xi HET (1 - YX)

iE=-t fliEET (1 - q Yxi)'

Comparing the order of vanishing at Y = 0, we get c + =

n + 1. Moreover, if we set

I = {|Xj qxi for any i < j} U {1},

I = {i xj qxi for any j> i} U {n},



922 MARK REEDER

then

Ie, ( 1- Yxj) -h (1 - Yxi) fHJE T (1 - Yx) HiEi (1 qYxi) E=l (1 qYxi) H'iet (1 - qYxi)

Up to now, everything we have said holds for any subquotient of I(x). However, because u(x) is generic, we know ([J-P-S, (2.1)]) that the numerator and denominator of the last quotient are relatively prime in C[Y]. Comparing zeros and poles, we get I = T, I = T. Note that DX = T and Cx = {1, . . ., n + 1} - T.

Finally, we have also shown that

n+1

(.q)fn+1 fl xi = M(-q)tl Hl xi, i=1 jET

and this leads to the formula for E(S, U(X)). U

(3.2) Remarks.

(1) We may replace X by a conjugate under the Weyl group to arrange that Sx C A, and in this case, cond (u(X)) = SxI. This result combined with Theorem 1 gives the dimension of the space of K(m)-invariant vectors in u(x) for any m.

(2) If ar is any generic representation of Gn with unramified central quasicharacter and cond (7r) = 1, then ar contains a nonzero fixed vector under an Iwahori subgroup, so by [C, (2.6)] we have r u a(X) for some X. By (3.1) we must have Sx = {al} for some ao C (D.

Let us consider the extent to which Theorem 1 holds for reducible I(x). It is no great loss to consider Sn+ = C[T1, .. ., Tn], and this will simplify the notation. We define a map

Ql: Sn _> I(x)K(-)

by fl(P) = s(P) * qPx. This is the same as - when I(x) is irreducible. We have the filtrations

* I(Sn(m) C Sn((m + 1) C ... Sn

***I(x)K(-) C I(X)K( +l1) C ... I(X)K(-)



OLD FORMS ON GL,, 923

From the proof of Theorem 1, we know that Q preserves these filtrations. What is more, the corresponding terms have the same dimension. To see this, we observe that a set of double coset represen- tatives for

Bn +l\ Gn +1 IK(m)

is given by

1 0 0 ..O \

0 1 0 *.. O

O ... 0 1 0

. mI . m2 ... Mn 1

where m1, ..., Mn are integers such that m ? i1 m M2 M I in 2

0. To such a sequence of mi's we associate the polynomial

T MI-m2T m2-M3 ... TMn l-mnT Mn

This gives a bijection between Bn+ \Gn+1lK(m) and a basis of Sn((m). It is not the case, however, that Q is always an isomorphism. For

example, it might happen that qP, belongs to a proper irreducible submodule of I(x), so that im Q misses the K(oo)-invariants in u(X). The main point in the following result is that Q is an isomorphism for reducible I(x) precisely when the Whittaker map 0W on I(x) is injective.

We will assume that X is regular, i.e., the stabilizer in W of X is trivial. Then the composition factors of I(x) are parametrized by |Sxi- tuples of + l's: e = (e,)esx. The corresponding composition factor F,e

can be characterized in one of two ways as follows ([B-Z], [R]). Let We = {w E W I eatw-'ao > 0 Vao E Sxj.

(1) ale is isomorphic to the irreducible submodule of I(wX) if and only if w E We.

(2) The Jacquet module (fFe)N iS isomorphic to 1wEDwEe(WX)812*

Moreover, u(X) is the se for which eat = + 1 < a > 0.



924 MARK REEDER

(3.3) THEOREM 2. Assume that X is regular. Then the following are equivalent:

(1) W((PX) # 0 (2) SX C D+

(3) qPX generates '(X)- (4) eWW '(x) -> I,,+ is injective. (5) Q Sn+ I(X)K(-) is injective. (6) Q is a filtration-preserving isomorphism of Sn+ modules.

Remark. One reason for arranging the statement this way is to point out the equivalence of the first four statements for any split reductive group. This implies in particular a converse to [C, (3.6)]. Note that for GLn, the equivalence of (2) and (4) is a special case of [B-Z, 4.11].

Proof. We use the notation of [C-S]. In particular, define

cw(x) = H CA(x), woz<O

Ca(X) = 1 -q X(a.)

Note that c,(X) = 0 < - a C S,. For w C W, Tw is a generator of

HomG,+#(I(X), I(w)) C.

Finally, fw E I(X) is the unique (up to scalar) function invariant under the upper triangular Iwahori subgroup whose image in '(X) transforms by (wX)8"2 under An+l.

(1) X* (2). From [C-S], we know that 0W(up) = 0 if and only if cw0 = 0 iff Hli<j (xi - qxj) = 0, and this occurs precisely when S, contains a negative root.

(2) <* (3). Suppose qP, generates '(X) and - a E S, for some positive root ao. Choose w C W such that wao < 0. Then cw(x) = 0, and by [C, (3.1)], we must have Tw 0, which is not the case.

For the other direction, we observe that Sx -i = - Sx,, so we may as well prove the dual version: Sx C - D+ ' px belongs to the irreducible submodule of '(X).




By [C, (3.8)], we may write

x= E cw(x)fw. wsx<o

Let ar be the spherical composition factor of I(x). The Jacquet module 7rN is a direct sum of characters of the form (wx)8112 for certain w E W. Since X is regular, these characters are distinct. Now the projection of each fw in 1N transforms by (wx)81'2 and qpx projects nontrivially into IN

by [C, (2.4)]. It follows that (wx)8112 occurs in 1N for each w with wSx < 0.

On the other hand, the main theorem in [R] says^the Jacquet module of the irreducible submodule of I(x) affords the characters (wx)&'2 for w E We where ea = + 1 for all ao E Sx. It is easy to see that w E W, X

wSx < 0. Since distinct constituents of I(x) have no common characters in their Jacquet modules, rr must be the irreducible submodule of I(x).

(3) => (4). Recall that if Sx C ?+ then the irreducible submodule of I(x) is also the generic constituent u(x). If 'W were not injective, its kernel would contain u(x) and we would also have a nonzero irreducible submodule of im 'W. This would imply that I(x) had at least two generic constituents, which is false.

(4) => (5). Let P E S+ and suppose s(P) *Px = 0. Then by (1.1),

0 = *('W(s(P) * q,j) = P(Y Xi, . . . , Y Xn)I('W(.pX)),

so 'IW((Px)) = 0. Now [J-P-S (3.5)] implies W(upx)|Gfl = 0, and one sees from the formula for W(qPx) that we have in fact W(upx) = 0.

(5) => (6). That Q is filtration preserving follows from the proof of Theorem 1, and as remarked above, the terms in the filtrations S+ = US.(m), I(X)K(w) = UI(X)K(m) have the same (finite) dimension.

(6) => (1). Suppose W(upx) = 0. Then the image of Q is contained in ker 'W. Also, I(X)/ker 'W has an irreducible generic submodule which by [J-P-S] contains K(oo) invariants. Thus Q is not surjective. U

Remark. It is not hard to see that the image of Q is the space of Kn-invariants in the submodule of I(x) generated by px.

Suppose now that Sx = {aoil, . . ., oaid -C A. We know from Prop- osition 2 that the conductor of u(x) is ISxI = c. By the implications (4) => (5) => (6) of Theorem 2 (which did not require that X be regular), there must be a P0 E S+ (c) such that s(P0) * qPx spans U(X)K(c). We will



926 MARK REEDER

show that P0 is related to the L-functions of '(x) and u(x) and thereby give a formula for the new vector of a(X). Recall that ((X) # 0 because Sx c (p+.

(3.4) PROPOSrMON. Let L(s, I(x)) = gPj(q-S)-l, L(s, u(X)) =

gl,(q-s)-l* Then the new vector in a(X)K(c) is

((X) s(00 * Wx)

where

PO rl fl (I-xY)T1 i = 1 D (Xi) je(DT

and Dx = {i, + 1, ic + 1} = {1, . n + 1}-Dx.

Proof. We first note that the identity

n R((X) = ((X) ri 9P1YX')-' i=l

is a collection of identities between holomorphic functions of X, which holds when '(X) is irreducible (Sx = 0) and hence for all X. From (1.1), we get, for Q E Sn,

''(s(Q) * uPx) = t(x)Q(YXI, , YXn) n ?(YXi)1.

On the other hand, if s(Q) * uPx is the new vector in u(X), we have

n q(s(Q) * P= II P(YXi)'.

The first equality is now immediate. For the second, recall from (3.1) that gPa(Y) = fIjEDX (1 - Yxj), so we have

Po = LI I2I (1 - xjXj) = LI H T.(-xjXi,..., -xjXn) jEDx i1 jEDX i1

= I (-XTi.



OLD FORMS ON GLn 927

4. Cuspidal cohomology. We now apply the previous results to the cuspidal cohomology of arithmetic subgroups of G = SL3, viewed as an algebraic group over Q. Let A and Af be the rings of adeles and finite adeles, respectively, of Q.

If N is a positive integer, we let ro(N) denote the subgroup of matrices in SL3(Z) whose bottom row is congruent to (0, 0, *) modulo N. Let Ko(N) be the closure of F0(N) in SL3(Af), where Fo(N) is embedded diagonally.

Let a = wrr0. 0 uf be an irreducible cuspidal automorphic representation of G(A) = SL3(A). We assume that wr contains a nonzero vector invariant under Ko(p), and that

H* (!G (R), S03(R); 7r) 7& O.

As with all cuspidal representations of G(A), a belongs to an L-packet of representations whose members are the constituents of an irreducible cuspidal automorphic representation fTr of G(A) = GL3(A) ([L-S, (3.5)]) restricted to G(A). The same is true of the local components N, of 'F

(v = any place of Q) which are representations of Gp = SL3(Qp).

(4.1) PROPOSMTON. Let p be a prime. Suppose *f?o(P) # 0 and

H*(!f3(R), S03(R); w.) # 0.

Then

*p- 0 u(x),

where , is a character of the determinant, and X is unramified. Moreover, *fp is irreducible for Gp = SL3(Qp).

Proof. The p-component of Ko(p) contains an Iwahori subgroup of Gp, so ([C, (2.6)]) 1Tp is a constituent of some unramified principal series I(Xi) for Gp. Extend X1 to an unramified character X of the diagonal matrices in Gp = GL3(Qp). Then restriction of functions is a Gp equivalence I(X) G I(Xi). Let p be the Gp-irreducible subquotient of '(x) whose restriction to Gp contains 1Tp. Then p and 7rp are irreducible admissible representations of Gp containing a common Gp constituent, namely 7rp. By [L-S, (3.3)], we have Pre , j 0 p for some character ,u of the determinant. Since 'fp is the local component of a cuspidal rep-



928 MARK REEDER

resentation of GL3, both ft, and p are generic. Hence p a-(x) and the first assertion follows.

To check irreducibility, we let Z be the center of Gp and use [G-K, (3.2)] to see that

dim EndGp(*p) = #{v E (GIpZGp)A I*p 0 v lTp}.

Since a(Iix) 0 v oa(IXv) and a(Xv) a(X) if and only if xv = wx for some w E W, this last set is, with the obvious abuse of notation,

{v E QpX I (v o det)X = wx, some w E W}.

It is easy to verify that if v is a nontrivial character and v o det = wX for some w, then we must have v3 = 1 and, up to a permutation, X A 0 kv 0 Xv2 for some quasi-character A of QpX.

I claim that, with our hypotheses, X cannot be of this form. Suppose it is. Then since v is unitary, we have a(x) = I(x) and I(x) decomposes under Gp into three distinct constituents, one of which is spherical. These constituents are permuted transitively by the subgroup of GL3(Q) generated by y = diag(p, 1, 1). Consider the whole automorphic representation al and view y as diagonally embedded in GL3(Q)SL3(A). Re- call [L-S, (3.5)] that this group acts on the space of cusp forms on SL3 by

(hg.4)(x) = 4(h-lxhg),

where

h E GL3(Q), g E SL3(A), 4 E LC2Up(SL3(Q)\SL3(A)).

This action sends the space of -n to a space of functions transforming according to ih = 0@ iih, where wh(g) = I(h-lgh). We take h = a, and observe that for a place v different from p, we have r-n -niv. At v = 0o, this is because -h. has nonzero (g, K)-cohomology and since n = 3, the L-packet of -z. has only one element (see, eg. [(L-S), (5.3)]). For the finite places v # p, -rv is spherical, and we may apply the first part of this proof. We either have uv = lfvr, where our assertion is clear, or fv I( 0 t-q 0 tq2) where -q is a character of QVX of order three




and t is some other character. In the latter case, y sends the spherical vector in r, to a multiple of itself, and this implies 7-n 7.

Consequently, if we conjugate al by an appropriate power of y, we can make the p-component spherical without changing the other components. This yields a cuspidal automorphic representation 'a' of SL3 which is spherical at every finite place and for which

H*(4f3(R), S03(R); 'rr ) #A 0.

However it is well-known (cf. [A-G-G, Section 5]) that

H*USp(SL3(Z), C) = O,

so we have a contradiction, and this completes the proof. U

Write the character pt in (4.1) as p,u [ * Is, with s E C, and p,u a character of Zp extended by the rule ,t'(p) = 1. By strong approximation, we can extend ptj to a grossencharacter of Qx \AX and then twist fr by the resulting character of the determinant of GL3(A). We get another cuspidal representation of GL3 containing 7i. Thus we may assume frp a(X) with X unramified.

This implies 7rf(o(Pm) jp(m) for every m - 1 (notation of Section 2). As shown in the proof of (4.1), a(x) cannot be spherical. Also, a special representation contains no Fo(p)-invariants. It then follows from (3.1) that cond (fp) = 1. Theorem 1 now implies that *p(m) S2(M 1), for every m - 1.

(4.2) Example. The computations in [A-G-G] show that

dim H3USP(Fo(p), C) = 2

for p = 53, 61, 79, 89. From our remarks above, we conclude that these classes are coming from two distinct cuspidal automorphic representations and that

dim H3usp(ro(p?), C) ? m(m + 1)

for the above primes.

UNIVERSITY OF OKLAHOMA



930 MARK REEDER

REFERENCES

[A-G-G] A. Ash, D. Grayson and P. Green, Computations of cuspidal cohomology for congruence subgroups of SL3(Z), Jn. Numb. Thy., 19 (1984), 412-436.

[B-F] C. Bushnell and A. Frolich, Non-abelian congruence Gauss sums and p-adic simple algebras, Proc. London Math. Soc., 50 (1985), 207-264.

[B-Z] I. N. Bernstein and A. V. Zelevinskii, Induced representations of reductive p-adic groups I, Ann. Scient. Ec. Norm. Sup., 4 (1977), 441-472.

[C] W. Casselman, The unramified principal series of p-adic groups I, Comp. Math., 40 (1980), 387-406.

[C1] , On some results of Atkin and Lehner, Math. Ann., 206 (1973), 311-318. [C-S] and J. Shalika, The unramified principal series of p-adic groups II, Comp. Math.,

41 (1980), 207-231. [D] P. Deligne, Formes modulaires et representations de GL(2), in Modular Functions of One

Variable II, Springer-Verlag, Lecture Notes No. 349, 1973. [G-J] R. Godement and H. Jacquet, Zeta Functions and Simple Algebras, Lecture Notes,

Springer-Verlag, No. 260, 1972. [G-K] S. Gelbart and A. Knapp, Irreducible constituents of principal series of SL(k), Duke

Jn., 48 (1981), 313-327. [J-P-S] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Conducteur des representations du group

lineaire, Math. Ann., 256 (1981), 199-214. [K-M] P. Kutzko and D. Manderscheid, On the supercuspidal representations of GL4, Duke

Jn., 52 (1985), 841-867. [L-S] J.-P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups,

Inv. Math., 83 (1986), 383-401. [R] F. Rodier, Decomposition de la serie principale, in Non-Commutative Harmonic Analysis,

Lecture Notes, Springer-Verlag, No. 880, Berlin, 1980. [S] T. Shintani, On an explicit formula for class-1 Whittaker functions, Proc. Japan Acad., 52

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