NEW VECTORS FOR GSp(4): A CONJECTURE AND SOMEEVIDENCE
BROOKS ROBERTS AND RALF SCHMIDTIn this paper we present and state evidence for a conjecture on the existence
and properties of new vectors for generic irreducible admissible representations ofGSp(4, F ) with trivial central character for F a nonarchimedean field of characteristic zero. To summarize the conjecture, let O be the ring of integers of F and let
P be the prime ideal of O. We define, by a simple formula, a sequence of compact
open subgroups K(P n ) of GSp(4, F ) indexed by nonnegative integers n. The firstgroup K(O) is GSp(4, O). The second group K(P) is the other maximal compact
subgroup of GSp(4, F ), up to conjugacy, and is called the paramodular group. Automorphic forms for the global version of this group have been considered by T.
Ibukiyama and his collaborators in a number of papers dealing with a genus two
version of Eichler’s correspondence and old and new forms. In general, we refer toK(P n ) as the paramodular group of level P n . Given a generic irreducible admissible
representation π of GSp(4, F ) with trivial central character, we consider the spaceof vectors fixed by each K(P n ). The conjecture for π makes three assertions. First,
for some nonnegative n, the space of K(P n ) fixed vectors is nonzero; second, if Nπ
is the smallest such n, then the space of K(P Nπ ) fixed vectors is one dimensional;
and third, this one dimensional space contains a vector Wπ whose Novodvorsky
zeta integral gives the Novodvorsky L-factor of the representation:Z(s, Wπ ) = L(s, π).
We call Wπ the new vector of π. Zeta integrals depend on a choice of Whittakermodel, which depends on a choice of nondegenerate character: we make a choice
independent of π.
Evidently, the conjecture is similar to the theory of new vectors for generic
irreducible admissible representations of GL(2, F ) with trivial central character.
Just as for GL(2, F ), there is a simple relation between new vectors and -factors.
Assume the conjecture holds for π. There exists an Atkin-Lehner type element uNπin GSp(4, F ) which normalizes K(P Nπ ) and whose square is in the center. Thus,
π(uNπ )Wπ = π Wπ for some π = ±1. Moreover, it is easy to show that
(s, π) = π q −Nπ (s−1/2)
so that (1/2, π) = π . Here, q is the order of O/P, and we use the mentioned
nondegenerate character in the definition of the -factor.
We state three pieces of evidence for the conjecture. First, the first two parts ofthe conjecture are true for all π containing a nonzero vector fixed by the Iwahori
subgroup. As evidence for the third part of the conjecture for such π one also has(s, ϕπ , ψ, dxψ ) = π q −Nπ (s−1/2)
This paper appeared in Surikaisekikenkyusho Kokyuroku no. 1338 (2003), 107–121.
The first named author was partially supported by a NSA Young Investigators Grant.
1

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BROOKS ROBERTS AND RALF SCHMIDT

where ϕπ is the L-parameter assigned to π by [KL]. Second, the first two parts ofthe conjecture are true for many π induced from the Siegel or Klingen parabolic
subgroups, and for these π, the level P Nπ is as expected. Finally, in proving theanalogue for GSp(4) of the dihedral case of the global Langlands-Tunnell theorem,[R1] defined certain local L-packets Π(τ ) and L-parameters ϕ(τ ) for GSp(4, F )which depend on a generic tempered irreducible admissible representation τ ofGL(2, E) with trivial central character, where E is either a quadratic extension of
F , or F × F . The work [R1] gave strong global evidence that Π(τ ) is the L-packetof ϕ(τ ). Assuming q is odd, we show that if E/F is unramified or E = F × F , thenthe generic element π of Π(τ ) contains a nonzero vector W fixed by K(P N ), where
N is defined by (s, ϕ(τ ), ψ, dxψ ) = cq −N (s−1/2) , and c is a constant. Moreover,Z(s, W ) = L(s, π).
To end this introduction, we emphasis that our conjecture is for generic irreducible admissible representations of GSp(4, F ) with trivial central character. In
gathering evidence we have encountered various related cases and questions, asmentioned below. But, for example, currently we are not in a position to state aconjecture for the case of nontrivial central character.
Notation. In this paper GSp(4, F ) is the group of g in GL(4, F ) such that




0
12
0
12
t
g
g = λ(g)
−12 0
−12 0
for some λ(g) in F × . Fix a continuous character ψ of F with conductor O and agenerator $ for P. Let | · | be the valuation on F such that if µ is a Haar measure
on F , then µ(xA) = |x|µ(A) for x in F and measurable sets A in F . If π is anirreducible admissible representation of a group of td-type [Car], let ωπ denote thecentral character of π. Let LF = WF × SU(2, R) be the Langlands group of F ,where WF is the Weil group of F . A GSp(4) L-parameter over F is a continuous
homomorphism ϕ : LF → GSp(4, C) such that ϕ(x) is semisimple for all x ∈ WF
and ϕ|1×SU(2,R) is a smooth representation. We denote the -factor of ϕ with respectto ψ and the Haar measure dxψ self-dual with respect to ψ by (s, ϕ, ψ, dxψ ). One
has (s, ϕ, ψ, dxψ ) = cq −N (s−1/2) for some nonnegative integer N and constant c.
1. The conjecture
To state the conjecture we need some definitions and results. First, we recall thefundamentals of the theory of Novodvorsky zeta integrals for GSp(4, F ), as provenin [T-B]. Fix c1 , c2 ∈ F × . Let π be an irreducible admissible representation ofGSp(4, F ). We say that π is generic if HomU (π, ψc1 ,c2 ) 6= 0, where U is the groupof all elements



1 u1
0
0
1 0 ∗ ∗
 0 1


0
0 
  0 1 ∗ u2  ,
u=
 0 0


1
0
0 0 1 0 
0 0 −u1 10 0 0 1and ψc1 ,c2 (u) = ψ(c1 u1 + c2 u2 ). Whether π is generic does not depend on the
choice of c1 and c2 . Assume π is generic. Consider the space of functions W :GSp(4, F ) → C such that W (ug) = ψc1 ,c2 (u)W (g) for u in U and g in GSp(4, F ),and W is right invariant under some compact open subgroup of GSp(4, F ). Thereexists a unique GSp(4, F ) subspace W (π, ψc1 ,c2 ) of this space which is isomorphic

NEW VECTORS

3

to π [Rod]. This subspace is called the Whittaker model of π with respect toψc1 ,c2 . Fix Haar measures on F × and F . Let µ : F × → C× be a continuous quasicharacter. If W is in W (π, ψc1 ,c2 ), the Novodvorsky zeta integral associated to
W and µ is
y 0 0 0Z Z
 0 y 0 0 
s−3/2
dx d× y.
Z(s, W, µ) =W ( 0 0 1 0 )µ(y)|y|×FF0 x 0 1The Z(s, W, µ) for W in W (π, ψc1 ,c2 ) converge absolutely in some right half plane
and are elements of C(q −s ). In C(q −s ) there is an element γ(s, π, µ, ψc1 ,c2 ) suchthat the following functional equation
0 J
Z(1 − s, π()W, (ωπ µ)−1 ) = γ(s, π, µ, ψc1 ,c2 )Z(s, W, µ)−J 0holds for W in W (π, ψc1 ,c2 ). This γ-factor does not depend on the choices of Haar
measure on F and F × . Here,


0 1
J=
.
−1 0
The C[q s , q −s ] module generated by the Z(s, W, µ) for W in W (π, ψc1 ,c2 ) is a fractional ideal of C(q −s ) with generator of the form 1/Q(q −s ) with Q(0) = 1, where
Q(X) is in C[X]. We defineL(s, π, µ) = 1/Q(q −s ).
This L-factor does not depend on the choices of Haar measures or c1 and c2 . We
also defineL(s, π, µ).
(s, π, µ, ψc1 ,c2 ) = γ(s, π, µ, ψc1 ,c2 )L(1 − s, π, (ωπ µ)−1 )The function (s, π, µ, ψc1 ,c2 ) is a nonzero monomial in q −s (e.g., see the top of p.
65 of [J]). The work [R2] verifies that L(s, π, µ) = L(s, ϕ, µ), and (s, π, µ, ψ1,−1 ) =(s, ϕ, µ, ψ, dxψ ) for the generic element π in Π(χ, τ ) and ϕ = ϕ(χ, τ ), where Π(χ, τ )and ϕ(χ, τ ) are the local L-packets and parameters defined in [R1]. We take c1 = 1
and c2 = −1 in the remainder of this paper, and write W (π) = W (π, ψ1,−1 ),γ(s, π, µ) = γ(s, π, µ, ψ1,−1 ) and (s, π, µ) = (s, π, µ, ψ1,−1 ). If µ = 1 we drop µ
from our notation.
Next, we define the paramodular group of level P n . This requires that we first
define the Klingen congruence subgroup of level P n . Let n be a nonnegative integer. The Klingen congruence subgroup Kl(P n ) of level P n is the subgroup ofGSp(4, F ) of all elements k such that λ(k) is in O× and


O
O O O
 Pn O O O k∈ Pn Pn O Pn  .
Pn O O O
Define the Atkin-Lehner element of level P n in GSp(4, F ) to be


0
J
un =
.
−$n J 0

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BROOKS ROBERTS AND RALF SCHMIDT

Evidently, u2n = $n is in the center of GSp(4, F ). We now define the paramodulargroup K(P n ) of level P n to be the subgroup of GSp(4, F ) generated by Kl(P n ) and−1nn
un Kl(P n )u−1
n = un Kl(P )un . Equivalently, K(P ) is the subgroup of GSp(4, F )×of all elements k such that λ(k) is in O andOO P −n O
 Pn OOO .
k∈ Pn Pn
O
Pn 
Pn O
O
O
1.1. Conjecture. Let π be a generic irreducible admissible representation ofGSp(4, F ) with trivial central character. For each nonnegative integer n, let π(P n )be the subspace of π of vectors fixed by K(P n ).
(1) For some nonnegative integer n the space π(P n ) is nonzero.
(2) If Nπ is the smallest n such that π(P n ) is nonzero, thendim π(P Nπ ) = 1.
(3) There exists Wπ in π(P Nπ ) such that
Z(s, Wπ ) = L(s, π).
In (3) of the conjecture we use the Whittaker model W (π) for π as defined above.
If the conjecture holds for π, we call P Nπ the level of π and Wπ the new vectorof π.
The reader will note that while the conjecture is quite similar to the theoryof new vectors for generic irreducible admissible representations of GL(2, F ) withtrivial central character, there is a significant difference: K(P n ) is not containedin K(P n+1 )! Thus, the theory of old vectors for GSp(4, F ) will not be strictly
analogous to that for GL(2, F ). Nevertheless, we have some evidence, which wewill not discuss here, that a coherent theory of old vectors for GSp(4, F ) does exist.
2. A formal heuristic
Before stating implications for -factors and our evidence, we will give some formal
motivation for the conjecture. As far as we know, there does not exist a conjecturalconceptual theory of new vectors for representations of the F points of an arbitrary
reductive algebraic group defined over F . The situation seems to be that, given a
particular group like GSp(4), a theory of new vectors would be useful, but one has
no reason to believe it exists. Groups for which new vectors have been considered
include GL(n) (see [Cas], [D], [J-PS-S]) and SL(2) (see [LR]); for GSp(4) see also [S]for the case of square-free level. In our considerations we mostly have been guided
by empirical facts. Still, for GSp(4) we can offer the following formal motivation.
Suppose one wants to derive the statement for a conjectural simple theory of
new vectors for generic irreducible admissible representations of GSp(4) with trivial
central character, and let π be one such representation. In π one might consider
the space of Klingen vectors of level P n , i.e., the subspace πKl (P n ) of vectors fixedby Kl(P n ). Alternatively, one might consider vectors fixed by Γ0 (P n ), the Siegel
congruence subgroup of level P n . However, without going into details, examples
show that these vectors will not give a simple theory. One might hope, then, that
Klingen vectors work, so that if N is the smallest n such that πKl (P n ) is nonzero,then dim πKl (P N ) = 1, and there exists a W in πKl (P N ) such that Z(s, W ) =L(s, π). One might also hope, as a consequence, that (s, π) = cq −N (s−1/2) forNEW VECTORS

5

some constant c. Examples show, however, for the smallest n such that πKl (P n )is nonzero one can have dim πKl (P n ) > 1: being a Klingen vector at the smallest
nontrivial level is not enough to give uniqueness. It seems an enlargement of theKlingen congruence subgroup is required.
How can one arrive at such an enlargement? One might start with a Klingen
vector W of level P N for which Z(s, W ) = L(s, π) and (s, π) = cq −N (s−1/2) and
see if W reasonably might be fixed by a natural larger compact open subgroup.
Using Z(s, W ) = L(s, π), the functional equation gives
0 Jγ(s, π)L(s, π) = Z(1 − s, π(
)W ).
−J 0
Dividing by L(1 − s, π), one obtains the -factor:
0 J(s, π) = Z(1 − s, π()W )/L(1 − s, π).
−J 0
Now (s, π) = cq −N (s−1/2) ; how can one make the right hand side look like this? A
bit of algebra yields
(s, π) =Z(1 − s, π(uN )W ) −N (s−1/2)·q.
L(1 − s, π)

It follows that Z(s, π(uN )W ) is a constant multiple of L(s, π), or equivalently,
Z(s, π(uN )W ) is a constant multiple of Z(s, W ). What condition on W can guarantee this? It would hold if π(uN )W is a constant multiple of W ; and if π(uN )W
is a constant multiple of W , then π(uN )W is fixed by Kl(P N ). Thus, one might
consider, for nonnegative integers n, vectors W such that W and un W are both
fixed by Kl(P n ), or equivalently, vectors fixed by K(P n ). Note that if W is fixedby Kl(P n ) then one has no reason to expect π(un )W to also be fixed by Kl(P n ),as un does not normalize Kl(P n ). On the other hand, un does normalize the Borel
congruence subgroup B(P n ) = Kl(P n ) ∩ Γ0 (P n ) of level P n , so if W is fixed byKl(P n ), then at least π(un )W will be fixed by B(P n ).
3. The connection to -factors
As mentioned in the introduction, the new vector and level of a representation
satisfying the conjecture are closely connected to its -factor. This is useful inproviding evidence for the conjecture.
3.1. Proposition. Let π be a generic irreducible admissible representation ofGSp(4, F ) with trivial central character. Assume (1) and (2) of the conjecturefor π hold. Then Wπ is an eigenvector for π(uNπ ) with eigenvalue π = ±1:π(uNπ )Wπ = π Wπ .
Assume (3) of the conjecture for π also holds. Then(s, π) = π q −Nπ (s−1/2) ,so that π = (1/2, π).
Proof. Assume (1) and (2) of the conjecture for π hold. A computation shows uNπnormalizes K(P Nπ ). This implies that π(uNπ )Wπ is in π(P Nπ ); since this space is
one dimensional, π(uNπ )Wπ = π Wπ for some π ∈ C× . As u2Nπ = $Nπ , and π has

6

BROOKS ROBERTS AND RALF SCHMIDT

trivial central character, we have π(uNπ )2 = 1, so that 2π = 1. Next, assume (3) of
the conjecture for π also holds. Applying the functional equation to Wπ , we obtain
0 J
Z(1 − s, π()Wπ ) = γ(s, π)Z(s, Wπ ).
−J 0
The definitions of the zeta integral and uNπ imply
0 J
Z(1 − s, π()Wπ ) = π q −Nπ (s−1/2) Z(1 − s, Wπ ).
−J 0Substituting this into the functional equation and using Z(s, Wπ ) = L(s, π), we
obtainπ q −Nπ (s−1/2) L(1 − s, π) = γ(s, π)L(s, π),so that (s, π) = π q −Nπ (s−1/2) .

This proposition can be used to supply evidence for the conjecture. For example,
suppose π is a generic irreducible admissible representation of GSp(4, F ) with trivial
central character, and parts (1) and (2) of the conjecture for π are known. To
obtain evidence for (3) of the conjecture for π we may proceed as follows. Supposethat it is believed that a certain L-parameter ϕ is the L-parameter associated toπ via the conjectural local Langlands correspondence, so that it is believed that
(s, ϕ, ψ, dxψ ) = (s, π) (or even suppose this equality is known). Then, in light of
Proposition 3.1, verifying
(1/2, ϕ, ψ, dxψ ) = π q −Nπ (s−1/2)
gives evidence that (3) of the conjecture for π holds.
4. Evidence
We currently have three different pieces of evidence for the conjecture. Our evidence considers a wide variety of generic irreducible admissible representations ofGSp(4, F ) with trivial central character, and includes all representations of lower
level and several families of induced and supercuspidal representations.
To state the first piece of evidence, define the Iwahori subgroup I of GSp(4, F )to be the subgroup of all k in GSp(4, F ) with λ(k) in O× andO O O O P O O O k∈ P P O P .
P P O O
Then we have the following theorem. The number π is defined in Proposition 3.1.
4.1. Theorem. Parts (1) and (2) of the conjecture are true for all generic irreducible admissible representations of GSp(4, F ) with trivial central character whichcontain a nonzero vector fixed by the Iwahori subgroup. Moreover, suppose π is a
generic irreducible admissible representation of GSp(4, F ) with trivial central character which contains a nonzero vector fixed by the Iwahori subgroup, and let ϕ be
the L-parameter associated to π by [KL]. Then(1/2, ϕ, ψ, dxψ ) = π q −Nπ (s−1/2) ,which gives evidence that (3) of the conjecture for π holds, as explained in section
3.

NEW VECTORS

7

In fact, we have computed the spaces of vectors fixed by K(P 0 ), K(P 1 ), K(P 2 )and K(P 3 ) in all the, possibly nongeneric, irreducible admissible representations ofGSp(4, F ) with trivial central character which contain a nonzero vector fixed by the
Iwahori subgroup. This information is displayed in the table in the next section,
which also includes information on how to understand the table. It is interesting
to observe that (1) and (2) of the conjecture and (1/2, ϕ, ψ, dxψ ) = π q −Nπ (s−1/2)
hold, with one exception, for all irreducible admissible representations of GSp(4, F )with trivial central character which contain a nonzero vector fixed by the Iwahori
subgroup. This exception is the representation VIb, which does not admit a nonzero
vector fixed by K(P 0 ), K(P 1 ), K(P 2 ) or K(P 3 ); we would expect a nonzero vector
fixed by K(P 2 ). However, the representations VIa and VIb form an L-packet, andthe conjecture holds for the representation VIa. This suggests that (1) and (2) of
the conjecture and the equality (1/2, ϕ, ψ, dxψ ) = π q −Nπ (s−1/2) may be true forall irreducible admissible representations of GSp(4, F ) with trivial central characterat the level of L-packets.
Our second parcel of evidence concerns certain induced representations. For
the representations considered in the following theorem there is a naturally associated L-parameter ϕ, which should be the L-parameter associated to π by
the conjectural local Langlands conjecture; define the nonnegative integer N by(s, ϕ, ψ, dxψ ) = cq −N (s−1/2) , where c is a constant. We use the notation of [ST]for induced representations.
4.2. Theorem. Let τ be a generic irreducible admissible representation of GL(2, F ).
Assume ωτ is unramified.
(1) (Siegel parabolic) Let σ be an unramified quasi-character of F × such thatωτ σ 2 = 1. Assume
π =τ oσis irreducible. Then π is a generic irreducible admissible representation ofGSp(4, F ) with trivial central character, and (1) and (2) of the conjecturefor π are true. Moreover, Nπ = N .
(2) (Klingen parabolic) Assume
π = ωτ−1 o τ
is irreducible. Then π is a generic irreducible admissible representation ofGSp(4, F ) with trivial central character, and (1) and (2) of the conjecturefor π are true. Moreover, Nπ = N .
Our final piece of evidence considers a broad distribution of representationsof GSp(4, F ), including supercuspidals. Recall that [R1] proved an analogue forGSp(4) of the global Langlands-Tunnell theorem. In doing so, [R1] defined certain
local L-packets of representations of GSp(4, F ). Let Π(τ ) = Π(1, τ ) be such a local
L-packet which happens to occur in a global situation as in Theorem 8.6 of [R1].
Thus, in particular, τ is a tempered generic irreducible admissible representation ofGL(2, E) with trivial central character, where E is either a quadratic extension of
F , or E = F × F . The packet Π(τ ) has one or two elements, and all elements aretempered irreducible admissible representations of GSp(4, F ) with trivial central
character. In [R2] it is shown that exactly one element π of Π(τ ) is generic. The
paper [R1] also associates to τ an L-parameter ϕ(τ ) = ϕ(1, τ ), and Theorem 8.6 of

8BROOKS ROBERTS AND RALF SCHMIDT[R1] provides evidence that Π(τ ) is the L-packet associated to ϕ(τ ) by the conjectural local Langlands correspondence for GSp(4, F ). Again, define the nonnegative
integer N by (s, ϕ(τ ), ψ, dxψ ) = cq −N (s−1/2) , where c is a constant.
4.3. Theorem. Assume q is odd. If E is unramified or E = F × F , then π contains
a vector W fixed by K(P N ) such that Z(s, W ) = L(s, π).
In writing Z(s, W ) = L(s, π) we are, as in the conjecture, using the Whittakermodel W (π) defined in section 1.
5. The table
The table gives information relevant to the conjecture about all the irreducible
admissible representations of GSp(4, F ) with trivial central character which containa nonzero vector fixed by the Iwahori subgroup.
The first column. By [Bo], an irreducible admissible representation of GSp(4, F )with trivial central character contains a nonzero vector fixed by I if and only if it
is an irreducible subquotient of a representation of GSp(4, F ) with trivial central
character induced from an unramified quasi-character of the Borel subgroup. The
basic reference on representations of GSp(4, F ) induced from a quasi-character ofthe Borel subgroup is section 3 of [ST], and we will use the notation of that paper.
Thus, St is the Steinberg representation, 1 is the trivial representation, and ν = | · |.
The reader will have to consult [ST] for more details. It is also useful to consult
section 4.1 of [T-B]. Let χ1 , χ2 and σ be unramified quasi-characters of F × with
χ1 χ2 σ 2 = 1, so that the representation χ1 × χ2 o σ of GSp(4, F ) induced from thequasi-character χ1 ⊗ χ2 ⊗ σ has trivial central character. Of course, χ1 × χ2 o σ
may be reducible. It turns out that by section 3 of [ST], there are six types ofχ1 × χ2 o σ such that every irreducible admissible representation of GSp(4, F )with trivial central character which contains a nonzero vector fixed by I is anirreducible subquotient of a representative of one of these six types, and that no two
representatives of two different types share a common irreducible subquotient. The
first column gives the name of the type. In the table we choose a representative fora type with the notation as below, and in subsequent columns we give information
about the irreducible subquotients of that representative. The types are describedas follows:
Type I. These are the χ1 ×χ2 oσ where χ1 , χ2 and σ are unramified quasi-charactersof F × such that χ1 χ2 σ 2 = 1 and χ1 × χ2 o σ is irreducible. See Lemma 3.2 of [ST].
Type II. These are the ν 1/2 χ × ν −1/2 χ o σ where χ and σ are unramified quasicharacters of F × such that χ2 σ 2 = 1. See Lemmas 3.3 and 3.7 of [ST].
Type III. These are the χ×νoν −1/2 σ where χ and σ are unramified quasi-charactersof F × such that χσ 2 = 1. See Lemmas 3.4 and 3.9 of [ST].
Type IV. These are the ν 2 × ν o ν −3/2 σ where σ is an unramified quasi-characterof F × such that σ 2 = 1. See Lemma 3.5 of [ST].
Type V. These are the νξ0 × ξ0 o ν −1/2 σ where ξ0 and σ are unramified quasicharacters of F × such that ξ0 has order two and σ 2 = 1. See Lemma 3.6 of [ST].
Type VI. These are the ν × 1 o ν −1/2 σ where σ is an unramified quasi-character ofF × such that σ 2 = 1. See Lemma 3.8 of [ST].

NEW VECTORS

9The second column. Choose a type as in the first column, and choose a representative χ1 × χ2 o σ of that type. Then χ1 × χ2 o σ admits a finite number ofirreducible subquotients, and this number depends only on the type of χ1 × χ2 o σ.
We index the irreducible subquotients by lower case Roman letters. The letter “a”
is reserved for the generic irreducible subquotient.
The third column. This column lists the irreducible subquotients of the representative of the type of the first column. We use the specific notation as in thediscussion of the first column.
The fourth column. Suppose π is an entry of the third column, and let ϕ be theL-parameter associated to π by [KL]. We define N by the equation
(s, ϕ, ψ, dxψ ) = cq −N (s−1/2) ,where c is a constant.
The fifth column. Using the notation of the explanation of the fourth column,
this is  = c = (1/2, ϕ, ψ, dxψ ).
The sixth, seventh, eighth and ninth columns. The numbers in the columns
give the dimensions of the K(P n ) fixed vectors for the representations in the thirdcolumn for n = 0, 1, 2 and 3. Note that to save space we have abbreviated K(P n )by K(n). The signs under the numbers indicate how these spaces of K(P n ) fixed
vectors split under the action of the Atkin-Lehner operator π(un ). The signs arecorrect if in the type II case, where the central character of π is χ2 σ 2 , the characterχσ is trivial, and in the type IV, V, and IV cases, where the central character of
π is σ 2 , the character σ is trivial. If these assumptions are not met, then the plus
and minus signs must be interchanged to obtain the correct signs.

10

BROOKS ROBERTS AND RALF SCHMIDT

representationNχ1 × χ2 o σ (irred.)01

χStGL(2) o σ1b χ1GL(2) o σ

0aχ o σStGL(2)2b χ o σ1GL(2)

0

a

3

I
a

−σχ($)

K(0) K(1) K(2) K(3)
1

2

4

6

+

+−

+++−

+++
−−−

0

1

2

4

−

+−

++−−

II
1
1

1

1

2

2

+

+

++

++

0

0

1

2

+

+−

III

σStGSp(4)2b L(ν , νIV
c

−1

σStGL(2) )3
2

L(ν StGL(2) , ν

− 23

σ)

δ([ξ0 , νξ0 ], ν

− 12

σ)
1

1

1

c L(ν 2 ξ0 StGL(2) , ξ0 ν 2 σ)
1d L(νξ0 , ξ0 o ν − 2 σ)1τ (S, ν − 2 σ)b τ (T, ν − 2 σ)c

1
2

L(ν StGL(2) , ν

1
1
0

2

3

4

+

+−

++−

++−−

0

0

0

1

− 21
1

σ)

d L(ν, 1F × o ν − 2 σ)

1
−σ($)
1
−1
σ($)
−σ($)
1

0
0

0

1

1

1

+

+

1

2

3

−

+−

+−−

1

1

1

1

+

+

+

+

0

0

1

2

−

+−

0
0
1

1

1

2

+

+

++

1

1

2

−

+

−−

0

1

0

+

2

1

VI

1

2

1

a

2

0

b L(ν 2 ξ0 StGL(2) , ν − 2 σ)
V

−σ($)

1

−

d σ1GSp(4)
a

1

0

+

0

1

2

+

+−

0

2

1

0

0

0

1

−σ($)

0

1

1

2

−

−

−−

0

1

1

1

2

2

+

+

++

++

Table 1. Representations containing a nonzero vector fixed by
the Iwahori subgroup. Consult section 5 for definitions and comments.

References
[Bo]
[Car]

[Cas]

A. Borel, Admissible representations of a semi-simple group over a local field with
vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259.
P. Cartier, Representations of p-adic groups: a survey, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, 111–155, Proc. Sympos. Pure Math., XXXIII, Amer. Math.
Soc., Providence, R.I., 1979.
W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–
314.

NEW VECTORS

[D]

[J]

[J-PS-S]
[KL]
[LR]
[R1]
[R2]
[Rod]

[S]

[ST]
[T-B]

11

P. Deligne, Formes modulaires et repŕesentations de GL(2), in Modular Functions of
One Variable, II, (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972),
55–105. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973.
H. Jacquet, Principal L-functions of the linear group, in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, 63–86, Proc. Sympos. Pure Math., XXXIII, Amer. Math.
Soc., Providence, R.I., 1979.
H. Jacquet, I.I. Piatetski-Shapiro, and J. Shalika, Conducteur des représentations
du groupe linéaire, Math. Ann. 256 (1981), 199–214.
D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke
algebras, Invent. Math. 87 (1987), 153–215.
J. Lansky and A. Raghuram, On conductors and new forms for SL(2) and U(1, 1),preprint.
B. Roberts, Global L-packets for GSp(2) and theta lifts, Doc. Math. 6 (2001), 247–
314.
B. Roberts, Epsilon factors for some representations of GSp(2) and Bessel coefficients, in preparation.
F. Rodier, Whittaker models for admissible representations of reductive p-adic split
groups, in Harmonic analysis on homogeneous spaces, (Proc. Sympos. Pure Math.,
Williams Coll., Williamstown, Mass., 1972), 425–430, Proc. Sympos. Pure Math.,Vol. XXVI, Amer. Math. Soc., Providence, R.I., 1973.
R. Schmidt, On Siegel modular forms of degree 2 with square free level, To appear
in the proceedings of the conference Automorphic forms and representations of algebraic groups, January 20-24, 2003, at RIMS, published by SūrikaisekikenkyūshoKōkyūroku, Research Institute for Mathematical Sciences, Kyoto University.
P. Sally and M. Tadić, Induced representations and classifications for GSp(2, F ) andSp(2, F ), Mem. Soc. Math. France (N.S.) No. 52 (1993), 75–133.
R. Takloo-Bighash, L-functions for the p-adic group GSp(4), Amer. J. Math. 122(2000), 1085–1120.

Brooks Roberts, University of Idaho, Department of Mathematics, PO Box 441103,
University of Idaho, Moscow ID 83844-1103, USA
E-mail address: brooksr@uidaho.edu, http://www.uidaho.edu/~brooksr/
Ralf Schmidt, Universität des Saarlandes, Fachrichtung 6.1 Mathematik, Postfach
15 11 50, 66041 Saarbrücken, Germany
E-mail address: rschmidt@math.uni-sb.de