LOCAL TRANSFER AND REDUCIBILITY OF INDUCEDREPRESENTATIONS OF p-ADIC GROUPS OF CLASSICAL TYPE
MAHDI ASGARI, JAMES W. COGDELL, AND FREYDOON SHAHIDI

Abstract. We analyze reducibility points of representations of p-adic groups of classicaltype, induced from generic supercuspidal representations of maximal Levi subgroups,
both on and off the unitary axis. We are able to give general, uniform results in terms of
local functorial transfers of the generic representations of the groups we consider. The
existence of the local transfers follows from global generic transfers that were establishedearlier.

1. Introduction
In this paper we prove some general, uniform results on reducibility of representationinduced from irreducible, generic, supercuspidal representations of the Levi subgroups of the
maximal parabolics of p-adic groups. Some special cases of these results have been known
for some time. Our main contribution in this work is to cast these results in a general setup
in the framework of local Langlands Funcotriality from groups of classical type (cf. Section
2.1) to the general linear groups. This allows us to prove quite general, uniform results
using information about poles of local L-functions and image of local functorial transfers.
Moreover, this agrees with the conjecture on arithmetic R-groups, defined by Langlands andArthur [1, §7], through Local Langlands Correspondence (cf. [8]).
Let F denote a p-adic field of characteristic zero. Consider a maximal Levi subgroup of
the form M = GL(m)×G(n) in a larger group G(m+n), a connected linear algebraic groupover F, of the same type as G. We take the group G(n) to be any of the split semi-simple
groups SO(2n + 1), Sp(2n), SO(2n), the split reductive groups GSpin(2n + 1), GSpin(2n),or the non-split quasi-split groups SOE/F (2n), UE/F (2n) or UE/F (2n + 1). Here E/F isa quadratic extension over which our quasi-split group splits. These groups all have theproperty that their connected L-group has a classical derived group.
Given a connected reductive group H over F, let H = H(F ). In particular, we let G =G(F ) and M = M(F ), where G and M are as in the previous paragraph. Let B = TU
be a Borel subgroup of G with M ⊃ T. Denote by P = MN the parabolic subgroup of G,
standard via N ⊂ U or P ⊃ B. Let A0 ⊂ T be the maximal split subtorus of T and let
A ⊂ A0 be the split component of M.
If α is the unique simple root of A0 in Lie(N), we set α
e = hρ, α∨ i−1 ρ, where ρ is half the∨sum of the roots of A0 in Lie(N) as in [44, §1.2], α is the coroot of the root α, and h·, ·idenotes the pairing between roots and coroots. Then, αe ∈ a∗ , where a is the real Lie algebra∗of A and a is its dual (cf. [44]). Let s be a complex number. Then seα ∈ a∗ ⊗R C. Now let
τ be an irreducible supercuspidal representation of M. We are interested in understanding
M.A. was partially supported by Collaboration Grant # 245422 from the Simons Foundation.
J.W.C. was partially supported by NSF grant DMS–0968505.
F.S. was partially supported by NSF grant DMS–1162299.
1

2

M. ASGARI, J. COGDELL, AND F. SHAHIDI

reducibility of the normalized induced representation

hse
α,HM (·)iI(seα, τ ) = IndG
⊗1 ,MN τ ⊗ q(1.1)

where HM : M −→ a = Hom (X(M)F , R) is defined as in [44, §3.3] byq hχ,HM (m)i = |χ(m)|F ,∀m ∈ M,(1.2)

where X(M)F is the group of the F -rational characters of M, χ ∈ X(M)F and h·, ·i is thepairing between X(M)F and a.
The aim of this paper is to determine the reducibility points for I(seα, τ ) for all s ∈ C,
whenever τ is generic, i.e., it has a Whittaker model, in the setting of a pair (G, M) of
classical type, in terms of functoriality as we now explain.
By assumption M = GL(m) × G(n) and in each case L G embeds into GL(N, C) × WF
for a minimal N, with an image with a classical derived group (cf. (2.3) for more detail).
Write τ = σ ⊗ π. By local transfer, π transfers to Π = Π1  · · ·  Πd on GL(N, F ) (Theorem
3.2 here, [7, 12, 13, 30, 31]).
Our main tool is to consider the poles of the intertwining operators (thus zeros ofPlancherel measures), as proposed by Harish-Chandra [19, 46], which we determine throughpoles of certain L-functions [39] (Theorem 2.13 and Corollary 2.14 here). Local transfer
allows us to show that these poles exist only when σ is quasi-self-dual (conjugate-self-dual
when G(n) is unitary) and σ is of the opposite type to the L-group of G (i.e., orthogonal
versus symplectic, see Section 4) or σ is among the Πi when it is of the same type as the
L-group of G.
This provides us with complete information about reducibility points on the unitary axis
for all groups of classical type. The reducibility off the unitary axis follows from [39, Theorem
8.1] (Theorem 4.24 here). These results are stated as Theorem 4.12 and summarized forindividual groups as Propositions 4.26 – 4.31.
The case of induction from other discrete series representation of M must also be addressed and it is left for the future. One should also verify the equality of the arithmetic
R-groups, defined by Langlands and Arthur [1, §7], with the analytic R-groups, defined by
Knapp and Stein [8], as conjectured by Langlands and Arthur.
The poles of intertwining operators can also be determined by direct calculations andthere is a large body of work on this topic, starting with [42, 43, 16, 17, 18] and ending withsome recent work [45, 48, 34, 10], where the connection to functoriality is fully establishedin some rather general cases (SO(2n + 1, F )).
The theory developed in [39] (Theorem 4.24 here) applies to any quasi-split group and in[33], Jing Feng Lau has determined the complete reducibility results for exceptional groups
E6 , E7 , and E8 , where Mder is a product of three SL-groups, using poles of triple productL-functions which are of Artin type [32]. The case of exceptional group G2 was fully treatedin [39]This paper is organized as follows. In Section 2 we introduce our notation and review the
local L-functions from the Langlands-Shahidi method and their connection to reducibilityof induced representations of p-adic groups of classical type that we consider. In Section 3
we give a proof of the generic local transfer of supercuspidal representations of the p-adic
groups of classical type. Our main uniform result on reducibility is given as Theorem 4.12.
The purpose of Section 4.3 is then simply to summarize all the information we have, both on
and off the unitary axis, for each individual group in the hopes that it helps the interested
reader see what our results give for each individual group and, at the same time, it indicatesthe scope of these results covering all groups of classical type.

LOCAL TRANSFER AND REDUCIBILITY

3

Dedication. The first and third named authors would like to dedicate their contributions
to this paper to their coauthor, Jim Cogdell. We are very fortunate to have Jim as a friendand collaborator and have very much benefited from his kindness and generosity and it isour pleasure to submit this paper to this volume in his honor.

Contents
1. Introduction
2. Local Representations and L-functions
2.1. Notation
2.2. The Langlands-Shahidi Local L-functions
3. Generic Local Transfers - Supercuspidal Case
4. Reducibility of Local Representations
4.1. Reducibility on the Unitary Axis
4.2. Reducibility off the Unitary Axis
4.3. Reducibility for Groups of Classical Type
References

1
3
3
4
5
11
11
15
16
18

2. Local Representations and L-functions
2.1. Notation. Let F be a non-archimedean local field of characteristic zero, with | · |Fdenoting its p-adic absolute value, normalized so that |$|F = 1/q, where $ is a fixed
uniformizer of F and q is the the cardinality of the residue field of F. For later use, let us
also fix a quadratic extension E/F. Let δE/F denote the non-trivial quadratic character of
F × associated with E/F via Class Field Theory. We denote the Weil group of F and E by
WF and WE , respectively. Also, we let x 7→ x̄ denote the non-trivial element of Gal(E/F ).
Let G denote a connected, reductive, quasi-split, linear, algebraic group over F . We fix asplitting (B, T, {X}) for G, where B is a Borel subgroup of G, T is a maximal torus in B,
and {X} is a collection of root vectors, one for each simple root of T in B. Recall that G is
quasi-split over F if and only if it has an F -splitting, i.e., one preserved under Gal(F /F ).
We will assume G = G(n) to be one of the following groups: the split groups SO(2n + 1),Sp(2n), SO(2n), GSpin(2n + 1), GSpin(2n), or the quasi-split non-split groups UE/F (2n),UE/F (2n + 1), SOE/F (2n), or GSpinE/F (2n), where E/F is a quadratic extension. We
refer to these groups as groups of classical type, i.e., those whose connected L-groups haveclassical derived groups. The groups UE/F (2n) and UE/F (2n + 1) are of type 2 An and
SOE/F (2n) and GSpinE/F (2n) are of type 2 Dn .
We write B = TU, where U is the unipotent radical of B. Unless stated otherwise, all
the parabolic subgroups we encounter will be assumed to be standard, i.e., they contain B.
Any standard, maximal, parabolic subgroups P of G has a Levi decomposition P = MN
with M ∼= GL(m) × G(n − m), if G is orthogonal or symplectic, or M ∼= ResE/F GL(m) ×G(n − m), if G is unitary.
For later use we define the positive integer N = NG to be the dimension of the firstbfundamental representation, or the standard representation, of L G0 = G(C),the connected

4

M. ASGARI, J. COGDELL, AND F. SHAHIDI

component of L-group of G. To be more explicit, for G = G(n) as above, we have2n
if G = UE/F (2n),





2n + 1 if G = UE/F (2n + 1),



2n
if G = SO(2n + 1), GSpin(2n + 1),
N = NG =
2n + 1 if G = Sp(2n),





2n
if G = SO(2n), GSpin(2n),2nif G = SOE/F (2n), GSpinE/F (2n).

(2.1)

In each case the standard representation is a representation of L G0 on CN and there is anassociated representation of L G on CN , or CN × CN in the unitary group cases, giving rise
to a natural L-homomorphism
ι : L G −→ L H(N ),
(2.2)
where
(
H(N ) =

GL(N )
ResE/F GL(N )if G is orthogonal or symplectic,
if G is unitary.

(2.3)

We refer to [14, §1] for a detailed description of ι.
Let A0 denote the maximal split torus in T and denote by Φ = Φ(A0 , G) the restrictedroots of A0 in G containing positive roots Φ+ . Also, let ∆ ⊂ Φ+ denote the set of simple
roots. Given a standard maximal parabolic P there exists a unique α ∈ ∆ such that P = Pθ−1
is determined by the subset θ = ∆ \ {α} of ∆. Let w0 = wG wM
denote the longest elementof the Weyl group of G modulo that of M. By abuse of notation, we employ the same
symbol for a Weyl group element and its representative in the quotient group. We thenhave w0 (θ) ⊂ ∆ and w0 (α) < 0.
A maximal standard parabolic P = Pθ is called self-associate if w0 (θ) = θ.
Remark 2.4. The only non-self-associate case among those we consider above is the following(cf. [27, §4]):• Dn with n odd and θ = ∆ − {αn }. This corresponds to the Levi subgroup GL(n)in SO(2n) or GL(n) × GL(1) in GSpin(2n) with n odd.
2.2. The Langlands-Shahidi Local L-functions. Let P = Pθ be a maximal parabolicin G as above and let ρ = ρP denote half of the sum of positive roots in N. Also, let
αe = hρ, α∨ i−1 ρ.

(2.5)

We have L P = L ML N and we let r denote the adjoint action of L M on the Lie algebraL
n = Lie(L N). Let Vi be the subspace of L n spanned by the root vectors Xβ ∨ satisfying
heα, β ∨ i = i. Then we have an irreducible decomposition
M
r=
ri ,
(2.6)
i

where ri denotes the restriction of r to Vi .
We fix a non-trivial additive character ψ of F throughout. We can use ψ to define a
multiplicative character of U(F ), denoted again by ψ. Let τ be an irreducible ψ-genericrepresentation of M(F ) and let s ∈ C. Having fixed ψ, we often simply say generic to mean
ψ-generic in the remainder. Let
HM : M(F ) −→ a = Hom(X(M)F , R)

(2.7)

LOCAL TRANSFER AND REDUCIBILITY

5

denote the Harish-Chandra homomorphism defined via
q hχ,HM (m)i = |χ(m)|F ,m ∈ M(F ), χ ∈ X(M)F .

(2.8)

If τ is unramified, then it is given by a semisimple conjugacy class {Aτ } in L M and we set
−1L(s, τ, ri ) = det IVi − ri ({Aτ })q −s.
(2.9)
2.2.1. Intertwining Operators. Let W = W (A0 ) = NG (A0 )/T and denote the longest
element of W by w` . Also, let w`M ∈ WM (A0 ). Then w0 = w` w`M . Set N0 = w0 N− w0−1 .
We define the induced representation


G(F )
α,HM (·)iI(s, τ ) = IndM(F )N(F ) τ ⊗ q hse⊗1 ,(2.10)
where ρ denotes half of the sum of positive roots in N and αe = hρ, α∨ i−1 ρ. Here, h·, ·idenotes the Z-pairing between characters and cocharacters of (G, T). We also set
I(τ ) = I(0, τ ).
Define the intertwining operatorZ
A(s, τ, w0 ) =f (w0−1 n0 g)dn0 : I(s, τ ) −→ I (w0 (s), w0 (τ )) .

(2.11)(2.12)

N0 (F )

If τ is generic, then, via the Langlands-Shahidi method, the L-functions L(s, τ, ri ) are always
defined, whether τ is unramified or not, and agree with the definition in the unramified casegiven above.
The following two results are well-known (cf. [39, §7]).
Theorem 2.13. Assume that P is a self-associate maximal parabolic and let τ be generic,
unitary, supercuspidal. Then
L(s, τ, re1 )−1 L(2s, τ, re2 )−1 A(s, τ, w0 )is a holomorphic, non-vanishing operator on all of C.
A consequence of this theorem is the following, which is what we will use later.
Corollary 2.14. Suppose that τ is generic, sucpercuspidal and unitary.
(a) If w0 (τ ) ∼6 τ , then I(τ ) is irreducible. (In particular, induction from a non-self=
associate parabolic is always irreducible.)(b) If P is self-associate and w0 (τ ) ∼= τ , then I(τ ) is irreducible if and only if exactly
one of L(s, τ, re1 ) or L(s, τ, re2 ) has a pole at s = 0.
(We allow for the second L-function not to appear at all. This does occasionally occur inthe case of Siegel Levi subgroups, as we will explain later.)
3. Generic Local Transfers - Supercuspidal Case
One consequence of the generic global functoriality is that we can draw conclusions abouttransfer of local representations, once it is known that the image of the global functorialtransfer is an isobaric sum of unitary cuspidal representations. Given that the local transfers
are completely understood at the archimedean places, we will focus on the non-archimedean
local transfers.

6

M. ASGARI, J. COGDELL, AND F. SHAHIDI

Definition 3.1. Let F, G and N = NG be as before. Let π be an irreducible genericrepresentation of G(F ). We say an irreducible representation Π of GL(N, F ) is a local
transfer of π if
L(s, π × ρ) = L(s, Π × ρ) and (s, π × ρ, ψ) = (s, Π × ρ, ψ)or equivalently
L(s, π × ρ) = L(s, Π × ρ) and γ(s, π × ρ, ψ) = γ(s, Π × ρ, ψ)
for all irreducible, unitary, supercuspidal representations ρ of GL(m, F ), 1 ≤ m ≤ N − 1.
The L-, -, and γ-factors on the left hand side are those of the Langlands-Shahidi method
while those on the right hand side are defined via parameters of the Local Langlands Correspondence.
We recall that the GL × GL factors on the right hand side are known to equal thosedefined via the Rankin-Selberg or the Langlands-Shahidi methods.
We describe the local transfer for irreducible generic supercuspidal representations in thetheorem below. This is what we need for our results on reducibility of local representationin Section 4. This result is a consequence of the global generic functoriality and was provedin many of the cases we cover below along with the global results. We give the details in
the proof below.
Theorem 3.2. Let G = G(n) and E/F be as before. Let π be an irreducible, generic,supercuspidal, representations of G(F ). Then π has a unique local transfer Π to GL(N, F )if G is symplectic or orthogonal, or to GL(N, E) if G is unitary. Moreover, it is of the form
Π = Π1  · · ·  Πd = Ind (Π1 ⊗ · · · ⊗ Πd ) ,where each Πi is an irreducible, unitary, supercuspidal representation of GL(Ni , F ) orGL(Ni , E), as appropriate, and the induction is from the standard parabolic subgroup ofGL(N ) with Levi component of type (N1 , . . . , Nd ) with N1 + · · · + Nd = N. Furthermore,∼Πe i , L(s, Πi , ∧2 ) has• if G = SO(2n + 1), then each Ni is even, each Πi satisfies Πi =a pole at s = 0, and Πi =6∼ Πj for i 6= j.
e i,• if G = SO(2n) or SOE/F (2n), n ≥ 2, or G = Sp(2n), then each Πi satisfies Πi ∼=Π
2∼
L(s, Πi , Sym ) has a pole at s = 0, and Πi =6 Πj for i 6= j.
e i ⊗ ω, where• if G = GSpin(2n + 1), then each Ni is even, each Πi satisfies Πi ∼=Πω = ωπ is the central character of π, L(s, Πi , ∧2 ⊗ ω −1 ) has a pole at s = 0, andΠi 6∼= Πj for i 6= j.
e i ⊗ ω,• if G = GSpin(2n) or GSpinE/F (2n), n ≥ 2, then each Πi satisfies Πi ∼= Π
2−1
where ω = ωπ is again the central character of π, L(s, Πi , Sym ⊗ ω ) has a poleat s = 0, and Πi 6∼= Πj for i 6= j.
e i , the local Asai L-function• if G = UE/F (2n + 1), then each Πi satisfies Πi ∼= Π∼L(s, Πi , rA ) has a pole at s = 0, and Πi =6 Πj for i 6= j.
e i , the local twisted Asai L-function• if G = UE/F (2n), then each Πi satisfies Πi ∼=ΠL(s, Πi , rA ⊗ δE/F ), has a pole at s = 0, and Πi ∼6 Πj for i 6= j.
=e i denotes the contragredient of Πi and Πi denotes the Gal(E/F )-action on theHere, Πrepresentation Π, i.e., Π(g) = Π(ḡ).
Proof. For G = SO(2n+1), Sp(2n), or SO(2n), this is [13, Theorem 7.3]. For G = UE/F (2n)this is [30, Proposition 8.4] and for G = UE/F (2n + 1) it is [31, Proposition 4]. For
G = GSpin(2n + 1) or GSpin(2n) this is [20, Theorem 4.7]. All of these results are based

LOCAL TRANSFER AND REDUCIBILITY

7

on a local-global argument, using the fact that one can embed the generic supercuspidal
representation π as the local component of a global generic representation to which one canapply the global generic transfer, possibly several times, to obtain the result. Let us givesome of the details now to indicate that a similar argument works for all the groups we areconsidering.
We first show the existence of one local transfer Π. If π is unramified, then the choiceof Π is clear; we simply take the irreducible, unramified representation determined by thesemi-simple conjugacy class in GL(N, C) generated by the image of the class of π under thenatural embedding ι as in (2.2). We then know, as can be seen directly and is verified in
the proof of the global generic transfer in the cases we are considering, that we have therequirement of Definition 3.1, i.e., that the local L- and -factors of π and Π twisted byirreducible, unitary, supercuspidal representations ρ of GL(m, F ) for m up to N − 1 areequal.
Next, assume that π is a general (not necessarily unramified) generic supercuspidal representation. Since π is generic and supercuspidal, by [39, Proposition 5.1], there exist a
number field k, a non-archimedean place v0 of k, and a globally generic cuspidal automorphic representation σ of G(Ak ) such that kv0 = F, and σv0 = π, and for all non-archimedean
places v 6= v0 of k the local representations σv is unramified. Here, σ is generic with respect
to a global generic character Ψ whose local component at v0 is out fixed ψ. (In the non-split
quasi-split cases, we have a quadratic extension K/k of number fields, and a place w of K
lying above the place v of k such that Kw = E.)
We recall the globally generic automorphic representation σ of G(Ak ) is known to havea transfer Σ to GL(N, Ak ) for each of the groups we are considering, as proved in [5, 7, 12,
13, 30, 31]. To be more precise, Σv is the transfer of σv as above for v outside a finite setof places and σv unramified. In particular the twisted L- and -factors are equal for Σv andσv for such v. Now, just take Π to be the local component of Σ at v0 .
To show this Π satisfies the requirements of Definition 3.1, we again note that if ρ is anirreducible, unitary, supercuspidal representation of GL(m, F ), 1 ≤ m ≤ N − 1, then we
may again use [39, Proposition 5.1] to embed ρ in a global cuspidal representation τ 0 ofGL(m, Ak ) such that τv0 0 = ρ and τv0 is unramified for all non-archimedean v 6= v0 .
Let S be a finite set of non-archimedean places of k such that σv is unramified for v 6∈ S
and let S 0 = S − {v0 }. Choose an idele class character η such that ηv0 is trivial and ηv ishighly ramified at v ∈ S 0 . By a general result, usually referred to as stability of γ-factors,
and used in the establishing the global generic transfer results for each of the groups above,for v ∈ S 0 we haveγ(s, σv × (τv0 ⊗ ηv ), Ψv ) = γ(s, Σv × (τv0 ⊗ ηv ), Ψv ).

(3.3)

Let τ = τ 0 ⊗ η. Since ηv0 is trivial, we have τv0 = τv0 0 = ρ. On the other hand, we havethe global functional equations
L(s, σ × τ ) = (s, σ × τ )L(1 − s, σ
e × τe)(3.4)

e × τe),L(s, Σ × τ ) = (s, Σ × τ )L(1 − s, Σ

(3.5)

and

which we can rewrite as!
γ(s, π × ρ, ψ) =

Y
v∈S 0

γ(s, σv × τv , Ψv )−1

LS (s, σ × τ )S (s, σ × τ, Ψ)LS (1 − s, σ
e × τe)(3.6)

8

M. ASGARI, J. COGDELL, AND F. SHAHIDI

and!
γ(s, Π × ρ, ψ) =Yγ(s, Σv × τv , Ψv )−1v∈S 0

LS (s, Σ × τ )e × τe)S (s, Σ × τ, Ψ)LS (1 − s, Σ

.

(3.7)

Now, the fractions on the right hand sides of (3.6) and (3.7) are equal by the unramified
case mentioned above and the two products on the right hand sides are also equal as in(highly-ram) above. Hence, the left hand sides of (3.6) and (3.7) must be equal, whichmeans that Π is indeed a local transfer.
The uniqueness of Π follows from the “local converse theorem for GL(N )” which meansthat an irreducible, generic representation of GL(N, F ) (or GL(N, E) as the case may be) is
uniquely determined by its γ-factors twisted by supercuspidal representations of all smaller
rank general linear groups (cf. Remark after the Corollary of [23, Theorem 1.1]).
It remains to show that Π is of the form stated in the theorem. Being a local componentof a globally generic automorphic representation, Π is a generic, unitary irreducible representation of GL(N, F ) or GL(N, E) as the case may be. By classification of unitary genericrepresentations of the general linear groups [49] we have
Π = Ind δ1 ν r1 ⊗ · · · δk ν rk ⊗ Π1 ⊗ · · · ⊗ Πd ⊗ δ1 ν −rk ⊗ · · · δk ν −r1 ,(3.8)
where each δj and each Πi is a discrete series representation and 0 < rk ≤ · · · ≤ r1 < 12 .
Here, ν(·) = | det(·)|.
Recall that
γ(s, π × ρ, ψ) = γ(s, Π × ρ, ψ)(3.9)

for any unitary, supercuspidal representation ρ of GL(m) for m up to N − 1. In fact,
multiplicativity of γ-factors implies that (3.9) holds for ρ discrete series as well. To see this,note that if ρ is discrete series, then it can be realized as the irreducible quotient of an
induced representation


t−1
t−1
Ind ρ0 ν − 2 ⊗ · · · ⊗ ρ0 ν 2 ,(3.10)
where ρ0 is unitary supercuspidal and t is a positive integer. Then,γ(s, π × ρ, ψ)

=

t−1
Y

γ(s +

t−1
− j, π × ρ0 , ψ)
2

γ(s +

t−1
− j, Π × ρ0 , ψ)
2

j=1

=

t−1
Y
j=1

= γ(s, Π × ρ, ψ),i.e., (3.9) holds with ρ any discrete series representation of GL(m) for m up to N − 1.
Now, up to a monomial factor coming from the -factors, the numerator of γ(s, Π × ρ, ψ)
is given by

−1
k
d
Y
Y

L(s + rj , δj × ρ)L(s − rj , δj × ρ)L(s, Πi × ρ) .
(3.11)
j=1

i=1

Since neither of L(s, δj × ρ) or L(s, Πi × ρ) has a pole in <(s) > 0 this numerator can havezeros only in <(s) < 21 .

LOCAL TRANSFER AND REDUCIBILITY

9

Similarly, the denominator of γ(s, Π × ρ, ψ) is the polynomial

−1
k
d
Y
Y
e i × ρe) ,
L(1 − s − rj , δej × ρe)L(1 − s + rj , δej × ρe)
L(1 − s, Π
j=1

(3.12)

i=1

which can only have zeros in the region <(s) > 12 .
Hence, the numerator and denominator in the factorization coming from the multiplicativity of the γ-factor have not common zeros and, consequently, we conclude from theequality of the γ-factors that
L(s, π × ρ) =kY

L(s + rj , δj × ρ)L(s − rj , δj × ρ)

j=1

d
Y

L(s, Πi × ρ).

(3.13)

i=1

On the other hand, we know [26] that the same expression gives L(s, Π × ρ). Therefore,L(s, π × ρ) = L(s, Π × ρ),(3.14)

with ρ discrete series.
Fix 1 ≤ i ≤ k. We apply (3.14) with ρ = δej . Since δj and π are both tempered we knowthat L(s, π × δej ) is holomorphic for <(s) > 0. In general, this is the third author’s Tempered
L-function Conjecture [39, Conj. 7.1]. Many cases of this conjecture were proved by several
authors [3, 11, 22, 28, 29, 36] and a proof in the general case has now appeared in [22]. On
the other hand, we haveL(s, Π × δej ) =kYL(s + rj , δj × δei )L(s − rj , δj × δei )

j=1

d
Y

L(s, Πj × δei ).

(3.15)

j=1

The term L(s − ri , δi × δei ) produces a pole at s = ri which results in a pole of L(s, Π × δei )at s = ri > 0 as the local L-factors are never zero. This is a contradiction unless k = 0, i.e.,there are no δi ’s in (3.8). Hence,Π = Ind (Π1 ⊗ · · · Πd )(3.16)

is a full induced representation from unitary discrete series representations Πi . In particular,Π is tempered.
In fact, we claim that each Πi is unitary supercuspidal. To see this, we can again realize
the discrete series representation Πi as the irreducible quotient of the induced representation


ti −1
ti −1
(3.17)Ind Π0i ν − 2 ⊗ · · · ⊗ Π0i ν 2
ti −1

ti −1

associated with the segment [Π0i ν − 2 , Π0i ν 2 ] where Π0i is unitary supercuspidal and ti
e i we haveis a positive integer [9, 50]. Applying (3.14) again with ρ = Πe i ) = L(s, Π × Πe i ).
L(s, π × Π(3.18)Let us calculate both sides of this equality. On the right hand side we havee i) =L(s, Π × Π

d
Y

e i)
L(s, Πj × Π(3.19)

e 0i ).
L(s + k, Π0i × Π(3.20)j=1

ande i) =L(s, Πi × Π

tY
i −1
k=0

10

M. ASGARI, J. COGDELL, AND F. SHAHIDI

e 0)(The last equation is verified, for example, in [28, p. 575].) The local L-function L(s, Π0i ×Π
i00e ) has a pole at s = −(ti − 1). Since local
has a pole at s = 0 so that L(s + ti − 1, Πi × Πie i ) at s = 1 − ti .
L-functions are never zero, this pole persists to give a pole of L(s, Π × Π
As for the left hand side, from [39] we havee i ) = L(s +L(s, π × Πti − 1e i),π × Π
2(3.21)e i ) can have poles only for <(s) = 0, we see thatsince π is supercuspidal. Since L(s, π × ΠeL(s, π × Πi ) can only have poles on the line <(s) = −(ti − 1)/2. These locations of poles areinconsistent unless ti = 1, i.e., Πi = Π0i is supercuspidal, as we desire.
Finally, we show that the conditions in terms of the L-functions in the statement of thetheorem hold. Consider the case of the general spin groups first. Let ω = ωπ denote thecentral character of π. Consider the equalitye i ) = L(s, Π × Πe i ).
L(s, π × Π(3.22)The right hand side has a pole at s = 0 as before. For the left hand side to have a pole ats = 0, from the general properties of these local L-functions (cf. [39, Cor. 7.6], for example)
we must have Π ⊗ π ∼= w0 (Π ⊗ π) as representations of GL ×G. By Lemma 4.17 below wethis implies thate i ⊗ ω.
Πi ∼(3.23)=ΠMoreover, the order of pole at s = 0 on the left hand side of (3.22) is one while the orderof the pole on the right hand side is the number of j such that Πj ∼= Πi . Hence, Πi 6∼= Πj if
i 6= j.
Furthermore, assuming that we are in the odd general spin group case, [39] implies thatthe product
L(s, π × Πi )L(s, Πi , Sym2 ⊗ ω −1 )(3.24)
has a simple pole at s = 0 in this situation. This pole is already accounted for by the poleat s = 0 of L(s, π × Πi ). Therefore, L(s, Πi , Sym2 ⊗ ω −1 ) has no pole at s = 0. However,e i ) = L(s, Πi × Πi ⊗ ω −1 ) = L(s, Πi , Sym2 ⊗ ω −1 )L(s, Πi , ∧2 ⊗ ω −1 ),L(s, Πi × Π(3.25)

which implies that L(s, Πi , ∧2 ⊗ ω −1 ) has a pole at s = 0 (which can only happen if Ni iseven). If we are in the even general spin groups, the same argument works with the roles ofSym2 and ∧2 switched.
We end the proof by mentioning that a similar argument, with minor modifications
replacing ω-self-dual with self-dual or conjugate-self-dual as appropriate, establishes the Lfunction condition for the remaining groups in the statement of the theorem. We will not
repeat those arguments as they are similar and have already appeared in the literature. For
orthogonal and symplectic groups, this is done in [13, p. 203]. For unitary groups, it isverified in [30] and [31].
We should note here that the conditions that the transfers Π need to satisfy in the theorem above in fact determine the image of the transfer. In other words, every Π satisfying
these conditions is the transfer of an irreducible, generic, supercuspidal π from the appropriate group G to GL(N ). For this one needs the “local descent” for all the groups we areconsidering. For symplectic and orthogonal groups, as well as the unitary groups, thesefacts have already been established [47, 15] and for the general spin groups this is a work inprogress of Jing Feng Lau.

LOCAL TRANSFER AND REDUCIBILITY

11

4. Reducibility of Local Representations
As an application of our results on the generic local transfer in Section 3, we now givesome uniform results on reducibility of local induced representations.
4.1. Reducibility on the Unitary Axis. To state our main uniform results on irreducibility, we first define the orthogonal/symplectic representations of general linear groups.
Let F and ψ be as before. Let η be a character of F × and let σ be an irreduciblesupercuspidal representation of GL(m, F ). Let
φ : WF −→ GL(m, C)(4.1)

be the parameter of σ and set
L(s, σ, Sym2 ⊗ η)
2

L(s, σ, ∧ ⊗ η):= L(s, Sym2 φ · η),

(4.2)

2

(4.3)

:= L(s, ∧ φ · η).

When η = 1, these reduce to the usual untwisted L-factors.
Similarly, with E/F as before, let σ be an irreducible supercuspidal representation ofGL(m, E). Let
φ : WE −→ GL(m, C)(4.4)
be the parameter of σ and set
L(s, σ, rA )L(s, σ, rA ⊗ δE/F ):= L(s, rA ◦ φ),(4.5):= L(s, rA ◦ φ · δE/F ).

(4.6)

Here, rA is denote the Asai representation. For details about the Asai representation andthe corresponding L-function we refer to [30, §5 and §8].
The representation σ of GL(m, F ) is said to be η-self-dual if it satisfiesσe∼= σ ⊗ η.

(4.7)

If σ is η-self-dual, then we haveL(s, σ × σe)=L(s, σ × σ ⊗ η)=L(s, σ, ∧2 ⊗ η) · L(s, σ, Sym2 ⊗ η)

(4.8)

and exactly one of the two local L-function on the right hand side of (4.8) has a pole ats = 0. Conversely, if one of the L-functions on the right hand side of (4.8) has a pole ats = 0, then σ is η-self-dual. Again, when η = 1 the representation σ is said to be self-dual
and the (untwisted) exterior/symmetric square L-functions replace the twisted ones above.
Analogously, a representation σ of GL(m, E) is said to be conjugate-self-dual if it satisfies
σ
e∼
= σ.

(4.9)

We recall that x 7→ x denotes the action of Gal(E/F ) on the representation σ of GL(m, E)
and σ denotes the corresponding action on σ. For σ conjugate-self-dual, we haveL(s, σ × σe)= L(s, σ × σ)
= L(s, σ, rA ) · L(s, σ, rA ⊗ δE/F ),(4.10)

where the local L-functions on the right hand side are, as before, the Asai L-function andits twist by the quadratic character δE/F . Again, the L-function on the left hand side of(4.10) has a pole at s = 0 which implies that exactly one of those on the right hand side of(4.10) has a pole at s = 0.

12

M. ASGARI, J. COGDELL, AND F. SHAHIDI

Definition 4.11. An irreducible, unitary, supercuspidal representation σ of GL(m, F ) is
said to be η-symplectic if L(s, σ, ∧2 ⊗ η) has a pole at s = 0, and it is said to be η-orthogonal
if L(s, σ, Sym2 ⊗ η) has a pole at s = 0. When η = 1, then σ is simply called symplectic or
orthogonal.
Similarly, an irreducible, unitary, supercuspidal representation σ of GL(m, E) is said tobe conjugate-orthogonal if L(s, σ, rA ) has a pole at s = 0, and it is said to be conjugatesymplectic if L(s, σ, rA ⊗ δE/F ) has a pole at s = 0.
Note that if a representation σ of GL(m, F ) is either η-symplectic or η-orthogonal, then
(4.8) implies that σ is η-self-dual. Moreover, in the η-symplectic case, m must be even. In
the following theorem we use these notions with η = ω −1 , where ω = ωπ denotes the central
character of a representation of G(F ). Similarly, if a representation σ of GL(m, E) is either
conjugate-symplectic or conjugate-orthogonal, then (4.10) implies that σ is conjugate-selfdual.
Let G = G(n) be as in Section 2.1, i.e., G is one of the groups SO(2n+1), Sp(2n), SO(2n),GSpin(2n+1), GSpin(2n), or SOE/F (2n), GSpinE/F (2n), or UE/F (2n+1), UE/F (2n), whereb denote the connected component of its Langlands LE/F is a quadratic extension. Let G
L
b as follows:group G. We define the type of G
G

b
G

b
type of G

SO(2n + 1)Sp(2n, C)symplectic

Sp(2n)

SO(2n + 1, C)

orthogonal

SO(2n)

SO(2n, C)orthogonal

GSpin(2n + 1)

GSp(2n, C)

symplectic

GSpin(2n)GSO(2n, C)

orthogonal

SOE/F (2n)SO(2n, C)orthogonal

GSpinE/F (2n)GSO(2n, C)

orthogonal

UE/F (2n)GL(2n, C)(conjugate) symplectic

UE/F (2n + 1)

GL(2n + 1, C) (conjugate) orthogonal

Theorem 4.12. Let n ≥ 0 and m ≥ 1 be integers. Let G = G(n) and E/F be as before. Let
σ be an irreducible, unitary, supercuspidal representation of GL(m, F ) if G is orthogonalor symplectic type, or of GL(m, E) when G is unitary. Let π be an irreducible, generic,
unitary, supercuspidal representation of G(n, F ) with central character ω = ωπ . Considerτ = σ ⊗ π, an irreducible, generic, unitary, supercuspidal representation of M(F ), where
M = GL(m) × G(n) or M = ResE/F GL(m) × G(n), as appropriate, is a standard Levi
subgroup of a maximal parabolic subgroup P in the larger quasi-split group G(m + n). Let
I(τ ) = I(σ ⊗ π) be the induced representation of G(m + n, F ) as in (2.11).
If P is not self-associate (cf. Remark 2.4), then I(τ ) is irreducible .
If P is self-associate, then for G orthogonal, symplectic or a general spin group, we havethe following statements:(a) If σ is not ω −1 -self-dual, then I(σ ⊗ π) is irreducible.
b then I(σ ⊗ π) is irreducible.
(b) If σ is ω −1 -self-dual and of the type opposite to G,

LOCAL TRANSFER AND REDUCIBILITY

13

b then I(σ ⊗ π) is irreducible if and(c) If σ is ω −1 -self-dual and of the same type as G,
only if σ appears as a component of the transfer of π to GL(N, F ) as in Theorem
3.2. In particular, if m > N, then I(σ ⊗ π) is always reducible.
Analogously, for G unitary, we have the following statements:(d) If σ is not conjugate-self-dual, then I(σ ⊗ π) is irreducible.
b then I(σ ⊗ π) is irreducible.
(e) If σ is conjugate-self-dual and of the type opposite to G,b(f) If σ is conjugate-self-dual and of the same type as G, then I(σ ⊗ π) is irreducible
if and only if σ appears as a component of the transfer of π to GL(N, E) as in
Theorem 3.2. In particular, if m > N, then I(σ ⊗ π) is always reducible.
Proof. The theorem essentially follows from Corollary 2.14 combined with the local transfer
results Theorem 3.2 as we now explain.
If P is not self-associate, then clearly w0 (τ ) 6∼
= τ and I(τ ) is irreducible by Corollary 2.14.
Next, assume that P is self-associate. We then havew0 (σ ⊗ π) ∼e · ωπ ⊗ π.
(4.13)=σwhen G is orthogonal, symplectic, or a general spin group. Similarly,e ⊗ π.
w0 (σ ⊗ π) ∼=σ

(4.14)

when G is unitary (cf. Lemma 4.17).
Therefore, if G is orthogonal or symplectic type and σ is not ω −1 -self-dual, or if G is
unitary and σ is not conjugate-self-dual, then w0 (σ ⊗ π) ∼6 σ ⊗ π, and part (a) of Corollary=
2.14 implies that I(σ ⊗ π) is irreducible. This proves statements (a) and (d).
Now, let G = G(n) = SO(2n + 1) or GSpin(2n + 1) and let M = GL(m) × G(n) as a Levib is of symplectic type. Consider the case of n ≥ 1 (non-Siegel Levi)in G(m + n). Then G
first. Assume that σ is ω −1 -self-dual. Then σ is either ω −1 -orthogonal or ω −1 -symplecticb then σ is ω −1 -orthogonal, which(cf. Definition 4.11). If σ is of the type opposite to G,
2−1means that the local L-function L(s, σ, Sym ⊗ ω ) has a pole at s = 0. This local Lfunction is the second L-function appearing in Theorem 2.13 or part (b) of Corollary 2.14.
b then it is ω −1 -symplectic and, hence,
On the other hand, if σ is of the same type as G,
2−1
L(s, σ, ∧ ⊗ ω ) has a pole at s = 0 and L(s, σ, Sym2 ⊗ ω −1 ) does not. Now, the other(first) L-function in part (b) of Corollary 2.14 would have a pole at s = 0 if and only if σappears as a component in the transfer Π of π. To see this, note that the other L-functionisdYL(s, σe × π) = L(s, σe × Π) =L(s, σe × Πi ),(4.15)
i=1

where Πi ’s are the components of the transfer Π of π as in Theorem 3.2.
If n = 0, the group G(0) is either trivial in which case π is trivial, or isomorphic to GL(1)in which case π = ωπ is just a character. This is the Siegel Levi case and in this case only one
L-function, L(s, σ, Sym2 ⊗ ω −1 ), appears in Corollary 2.14. The above argument still holds
in the following sense. If σ is ω −1 -orthogonal, then the first (and only) local L-functionin Corollary 2.14 has a pole at s = 0 and I(σ ⊗ π) is irreducible. If σ is ω −1 -symplectic,
then I(σ ⊗ π) is reducible. Neither does any L-function in Corollary 2.14 have a pole ats = 0, nor does σ appear as a component of transfer Π of π. This proves (b) and (c) for
G = G(n) = SO(2n + 1) or GSpin(2n + 1).
Next, let G = G(n) = SO(2n), GSpin(2n) or their quasi-split forms. A similar argumentb is now of orthogonal type and if σ is of the type opposite
as above again holds, except that G−1b
to G, then it is ω -symplectic, which means that the local L-function L(s, σ, ∧2 ⊗ ω −1 ) has

14

M. ASGARI, J. COGDELL, AND F. SHAHIDI

a pole at s = 0. Now, this is the second L-function appearing in part (b) of Corollary 2.14.
b then it is σ −1 -orthogonal and, hence, L(s, σ, Sym2 ⊗ ω −1 )And if σ is of the same type at G,
has a pole at s = 0 and L(s, σ, ∧2 ⊗ ω −1 ) does not. Now, in a similar way, the other Lfunction in part (b) of Corollary 2.14 would have a pole at s = 0 if and only if σ appears
as a component of the transfer Π of π. When n = 0 a similar situation occurs with one one
local L-function appearing again.
When G = G(n) = Sp(2n), the above paragraph holds again. The difference is just thatthe transfer Π is a representation of GL(2n + 1, F ). When n ≥ 1 there are two L-functions,
namely, L(s, σ × π) and L(s, σ, ∧2 ). When n = 0, there are actually again two L-functions
appearing, namely, L(s, σ) (the standard L-function) which does not produce any poles at
s = 0, and L(s, σ, ∧2 ) which behaves the same way as above.
Hence, we have proved parts (b) and (c) for G = Sp(2n), SO(2n), GSpin(2n) and their
quasi-split forms.
Finally, let G = G(n) = UE/F (2n) or UE/F (2n + 1) and assume that the representation
σ of GL(m, E) is conjugate-self-dual. Then σ is either conjugate-orthogonal or conjugatesymplectic (cf. Definition 4.11).
b is (conjugate) symplectic. If σ is of type opposite
Consider G = UE/F (2n) first. Now, Gb
to G, then it is conjugate-orthogonal, which means that the local Asai L-function L(s, σ, rA )has a pole at s = 0. This L-function is the second L-function appearing in Theorem 2.13
b then itor part (b) of Corollary 2.14. On the other hand, if σ is of the same type as G,
is conjugate-symplectic and, hence, L(s, σ, rA ⊗ δE/F ) has a pole at s = 0 and L(s, σ, rA )does not. Now, in a similar way as above, the other L-function in part (b) of Corollary 2.14
would have a pole at s = 0 if and only if σ appears as a component of the transfer Π of π.
The argument for G = UE/F (2n + 1) is exactly the same with the words (conjugate)
symplectic and (conjugate) orthogonal switched.
Therefore, we have also proved (e) and (f) for G = UE/F (2n) and UE/F (2n + 1), whichfinishes the proof of the theorem.
Remark 4.16. It is worth pointing out that in the case of G = GSp(2n), if the representation
σ is ω −1 -self-dual, then ω = 1 (cf. [41, p. 286]). The case of non-trivial ω may occur for
general spin groups.
Lemma 4.17. Let m and n be non-negative integers and let G = G(m + n) and E/F
be as before. Let θ = ∆ − {α}, where ∆ denotes the set of simple roots of G and α is a
fixed simple root. Consider the standard maximal parabolic subgroup P = Pθ = MN with
the Levi M ∼= GL(m) × G(n) if G is one of the non-unitary groups we are considering, orM ∼= ResE/F GL(m) × G(n) if G is unitary. Let w0 be the unique element in the Weylgroup of G such that w0 (θ) ⊂ ∆ and w0 (α) < 0. We assume that P is self-associate, i.e.,
w0 (θ) = θ (cf. Remark 2.4).
Let σ be a representation of GL(m, F ), or of GL(m, E) when G is unitary, and let π bea representation of G(n, F ).
Then,w0 (σ ⊗ π) ∼e ⊗ π.
=σwhen G = SO(2n + 1), SO(2n), or SOE/F (2n), and∼σ
w0 (σ ⊗ π) =e · ωπ ⊗ π.
when G = GSpin(2n + 1), GSpin(2n), or GSpinE/F (2n).
For G = UE/F (2n) or UE/F (2n + 1), we havee ⊗ π.
w0 (σ ⊗ π) ∼=σLOCAL TRANSFER AND REDUCIBILITY

15

Proof. One verifies this lemma by considering the effect of conjugation by the Weyl groupelement w0 on an element of the Levi M. For special orthogonal, symplectic, or unitarygroups, we can do this by a standard matrix calculation, noting that the action of w0 is tosimply switch the upper left m × m block with the lower right block of the same size in theusual matrix representation of these groups.
For the general spin groups essentially the same observation works, except that one
expresses it in terms of root data due to lack of a convenient matrix realization and follows
the action of the Weyl group element w0 . Let us give some details for this case.
Consider G = G(m + n) = GSpin(2m + 2n + 1). Using the Bourbaki notation, a detailed
description of the root data for G is given in [7, §1] which we use below. Let
X = Zhe0 , e1 , . . . , em+n i(4.18)X ∨ = Zhe∗0 , e∗1 , . . . , e∗m+n i(4.19)

anddenote the character and cocharacter lattices of G, respectively, with the standard Z-pairing.
∨}With the simple roots ∆ = {α1 , . . . , αm+n } and the simple coroots ∆∨ = {α1∨ , . . . , αm+ndefined as in [7, §1], we have θ = ∆ − {αm } with w0 as in Section 2.1. Let M = Mθ be themaximal standard Levi subgroup corresponding to θ.Then
M∼= GL(m) × G(n)(4.20)

with X = X1 ⊕ X2 and X ∨ = X1∨ ⊕ X2∨ , where
X1 = Zhe1 , . . . , em i X2 = Zhe0 , em+1 , . . . , em+n i(4.21)∨∨∨ ∨∨X1∨ = Zhe∨
1 , . . . , em i X2 = Zhe0 , em+1 , . . . , em+n i.

(4.22)

andNow if we translate the action of w0 on the root data from M to GL(m) × G(n) via theisomorphism (4.20), we can conclude that for m = (A, g) with A ∈ GL(m) and g ∈ G(n)we havew0 (m) = (µ · t A−1 , g),(4.23)
where µ = e0 (g) is the “similitude character”. This proves the statement of the lemma inthis case.
The case of even general spin groups is similar. However, in the even case (4.23) holds
provided that we are in the self-associate case (cf. Remark 2.4). This proves the lemma. 
4.2. Reducibility off the Unitary Axis. Theorem 4.12 determines the reducibility ofrepresentations of classical groups induced from unitary, generic, supercuspidal representation of a maximal Levi in a satisfactory way. The analogous question for when the inducing
representation is non-unitary is fortunately reduced to the unitary case thanks to the following rather general theorem, a well-known result in the Langlans-Shahidi method (cf. [39,Theorem 8.1] or [41, Theorem 5.1], for example.)
Theorem 4.24. Let G, P = MN, τ = σ ⊗ π and w0 be as before. Assume that w0 (τ ) ∼=τand that I(τ ) is irreducible. Let i = 1 or 2 be the unique index such that L(s, τ, rei ) has apole at s = 0 as in Corollary 2.14. Then, the induced representation I(s, τ ) of (2.10) is(a) irreducible for 0 < s < 1/i.
(b) reducible for s = 1/i.
(c) irreducible for s > 1/i.
If w0 (τ ) ∼= τ and I(τ ) reduces, then I(s, τ ) is irreducible for s > 0.

16

M. ASGARI, J. COGDELL, AND F. SHAHIDI

Recall that for the groups we are considering, we always have i = 1 or i = 2. Hence, thereducibility point of the induced representation I(s, τ ) is always at either s = 1/2 or s = 1,
if any, in the region s > 0. In Section 4.3 we specify these reducibility points for each groupindividually.
Moreover, we also recall, as one checks easily using the roots of G in each case, that thefollowing equalities are immediate from (2.10):s
I (ν σ ⊗ π)I (ν s σ ⊗ π)I(s, τ ) =I ν s/2 σ ⊗ πI (ν s σ ⊗ π)if G(n) is of type A (unitary),if G(n) is of type B or D and n ≥ 1,
if G(n) is of type B or D and n = 0,
if G(n) is of type C.

(4.25)

Here, ν = |det| denotes the p-adic absolute value of the determinant character on GL(m, F )(or GL(m, E) as the case may be). When we summarize our reducibility results for eachindividual group in Section 4.3, we will state them in terms of det rather than αe in (2.10).
4.3. Reducibility for Groups of Classical Type. We now summarize our results on
reducibility points of the induced representations for each of the groups we consider in thisarticle. Below F continues to denote a non-archimedean local field of characteristic zeroand, when appropriate, E/F denotes a quadratic extension, as before.
Proposition 4.26. Reducibility for SO(2n + 1).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, F ).
Let n ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation ofSO(2n + 1, F ). Let I(s) = Ind (| det |s σ ⊗ π) denote the parabolically induced representationof SO(2m + 2n + 1, F ). The following hold:• If σ is not self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is self-dual and L(s, σ, Sym2 ) has a pole at s = 0, then I(s) is irreducible for0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (The reducibility
point is the same whether n = 0 or n ≥ 1.)• If σ is self-dual, L(s, σ, ∧2 ) has a pole at s = 0 and n = 0, then I(s) is reducible fors = 0 and irreducible for s > 0.
• If σ is self-dual, L(s, σ, ∧2 ) has a pole at s = 0, n ≥ 1, and σ appears as a componentof the transfer of π to GL(2n, F ), then I(s) is irreducible for 0 ≤ s < 1, reducible
for s = 1, and irreducible for s > 1.
• If σ is self-dual, L(s, σ, ∧2 ) has a pole at s = 0, n ≥ 1 and σ does not appear as acomponent of the transfer of π, then I(s) is reducible for s = 0 and irreducible fors > 0.
Proposition 4.27. Reducibility for GSpin(2n + 1).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, F ).
Let n ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation of
GSpin(2n + 1, F ) with central character ω = ωπ . Let I(s) = Ind (| det |s σ ⊗ π) denote theparabolically induced representation of GSpin(2m + 2n + 1, F ). The following hold:• If σ is not ω −1 -self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is ω −1 -self-dual and L(s, σ, Sym2 ⊗ ω −1 ) has a pole at s = 0, then I(s) is
irreducible for 0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (Thereducibility point is the same whether n = 0 or n ≥ 1.)• If σ is ω −1 -self-dual, L(s, σ, ∧2 ⊗ ω −1 ) has a pole at s = 0 and n = 0, then I(s) is
reducible for s = 0 and irreducible for s > 0.

LOCAL TRANSFER AND REDUCIBILITY

17• If σ is ω −1 -self-dual, L(s, σ, ∧2 ⊗ω −1 ) has a pole at s = 0, n ≥ 1, and σ appears as acomponent of the transfer of π to GL(2n, F ), then I(s) is irreducible for 0 ≤ s < 1,
reducible for s = 1, and irreducible for s > 1.
• If σ is ω −1 -self-dual, L(s, σ, ∧2 ⊗ ω −1 ) has a pole at s = 0, n ≥ 1 and σ does notappear as a component of the transfer of π, then I(s) is reducible for s = 0 andirreducible for s > 0.
Proposition 4.28. Reducibility for Sp(2n).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, F ).
Let n ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation ofSp(2n, F ). Let I(s) = Ind (| det |s σ ⊗ π) denote the parabolically induced representation ofSp(2m + 2n, F ). The following hold:• If σ is not self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is self-dual and L(s, σ, ∧2 ) has a pole at s = 0, then I(s) is irreducible for0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (The reducibility
point is the same whether n = 0 or n ≥ 1.)• If σ is self-dual, L(s, σ, Sym2 ) has a pole at s = 0 and n = 0, then I(s) is reducible
for s = 0 and irreducible for s > 0.
• If σ is self-dual, L(s, σ, ∧2 ) has a pole at s = 0, n ≥ 1, and σ appears as a componentof the transfer of π to GL(2n, F ), then I(s) is irreducible for 0 ≤ s < 1, reducible
for s = 1, and irreducible for s > 1.
• If σ is self-dual, L(s, σ, ∧2 ) has a pole at s = 0, n ≥ 1 and σ does not appear as acomponent of the transfer of π, then I(s) is reducible for s = 0 and irreducible fors > 0.
Proposition 4.29. Reducibility for G(n) = SO(2n) or SOE/F (2n).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, F ). Letn ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation of G(n, F ).
Let I(s) = Ind (| det |s σ ⊗ π) denote the parabolically induced representation of G(m + n, F ).
If n = 0 and m is odd, i.e., non-self-associate parabolic (cf. Remark 2.4), then I(s) isalways irreducible. Otherwise, the following hold:• If σ is not self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is self-dual and L(s, σ, ∧2 ) has a pole at s = 0, then I(s) is irreducible for0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (The reducibility
point is the same whether n = 0 or n ≥ 1.)• If σ is self-dual, L(s, σ, Sym2 ) has a pole at s = 0 and n = 0, then I(s) is reducible
for s = 0 and irreducible for s > 0.
• If σ is self-dual, L(s, σ, Sym2 ) has a pole at s = 0, n ≥ 1, and σ appears as acomponent of the transfer of π to GL(2n, F ), then I(s) is irreducible for 0 ≤ s < 1,
reducible for s = 1, and irreducible for s > 1.
• If σ is self-dual, L(s, σ, Sym2 ) has a pole at s = 0, n ≥ 1 and σ does not appear as
a component of the transfer of π, then I(s) is reducible for s = 0 and irreducible fors > 0.
Proposition 4.30. Reducibility for G(n) = GSpin(2n) and GSpinE/F (2n).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, F ). Letn ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation of G(n, F )with central character ω = ωπ . Let I(s) = Ind (| det |s σ ⊗ π) denote the parabolically induced
representation of G(m + n, F ).

18

M. ASGARI, J. COGDELL, AND F. SHAHIDI

If n = 0 and m is odd, i.e., non-self-associate parabolic (cf. Remark 2.4), then I(s) isalways irreducible. Otherwise, the following hold:• If σ is not ω −1 -self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is ω −1 -self-dual and L(s, σ, ∧2 ⊗ω −1 ) has a pole at s = 0, then I(s) is irreducible
for 0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (The reducibility
point is the same whether n = 0 or n ≥ 1.)• If σ is ω −1 -self-dual, L(s, σ, Sym2 ⊗ ω −1 ) has a pole at s = 0 and n = 0, then I(s)is reducible for s = 0 and irreducible for s > 0.
• If σ is ω −1 -self-dual, L(s, σ, Sym2 ⊗ω −1 ) has a pole at s = 0, n ≥ 1, and σ appears as
a component of the transfer of π to GL(2n, F ), then I(s) is irreducible for 0 ≤ s < 1,
reducible for s = 1, and irreducible for s > 1.
• If σ is ω −1 -self-dual, L(s, σ, Sym2 ⊗ ω −1 ) has a pole at s = 0, n ≥ 1 and σ does
not appear as a component of the transfer of π, then I(s) is reducible for s = 0 andirreducible for s > 0.
Proposition 4.31. Reducibility for G(n) = UE/F (2n) and UE/F (2n + 1).
Let m ≥ 1 and let σ be an irreducible, unitary, supercuspidal representation of GL(m, E). Letn ≥ 0 and let π be an irreducible, generic, unitary, supercuspidal representation of G(n, F ).
We may consider σ ⊗ π as a representation of M(F ), where M ∼= ResE/F GL(m) × G(n) is
a maximal Levi subgroup in G(m + n). Let I(s) = Ind (| det |s σ ⊗ π) denote the parabolicallyinduced representation of G(m + n, F ). The following hold:• If σ is not conjugate-self-dual, then I(s) is irreducible for s ≥ 0.
• If σ is conjugate-self-dual and L(s, σ, rA ) has a pole at s = 0 when G(n) = UE/F (2n)or L(s, σ, rA ⊗ δE/F ) has a pole at s = 0 when G(n) = UE/F (2n + 1), then I(s) is
irreducible for 0 ≤ s < 1/2, reducible for s = 1/2, and irreducible for s > 1/2. (Thereducibility point is the same whether n = 0 or n ≥ 1.)• If σ is conjugate-self-dual, L(s, σ, rA ⊗ δE/F ) has a pole at s = 0 when G(n) =UE/F (2n) or L(s, σ, rA ) has a pole at s = 0 when G(n) = UE/F (2n + 1), and n = 0,then I(s) is reducible for s = 0 and irreducible for s > 0.
• If σ is conjugate-self-dual, L(s, σ, rA ⊗ δE/F ) has a pole at s = 0 when G(n) =UE/F (2n) or L(s, σ, rA ) has a pole at s = 0 when G(n) = UE/F (2n+1), n ≥ 1, and σappears as a component of the transfer of π to GL(2n, F ) when G(n) = UE/F (2n) or
to GL(2n + 1, F ) when G(n) = UE/F (2n + 1), then I(s) is irreducible for 0 ≤ s < 1,
reducible for s = 1, and irreducible for s > 1.
• If σ is conjugate-self-dual, L(s, σ, rA ⊗ δE/F ) has a pole at s = 0 when G(n) =UE/F (2n) or L(s, σ, rA ) has a pole at s = 0 when G(n) = UE/F (2n + 1), n ≥ 1,and σ does not appear as a component of the transfer of π, then I(s) is reducible
for s = 0 and irreducible for s > 0.
We remark that some of the L-function conditions in the above propositions can also berestated in other ways such as conditions on the parity of the integer m.
References[1] J. Arthur. Unipotent automorphic representations: conjectures. In Orbites unipotentes etreprésentations, II. Astérisque No. 171–172 (1989), 13–71.
[2] J. Arthur. The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups.
Colloquium Publications, 61. American Mathematical Society, Providence, RI, 2013.
[3] M. Asgari. Local L-functions for Split Spinor Groups. Canad. J. Math. 54 (2002), no. 4, 673–693.
[4] M. Asgari, J. Cogdell and F. Shahidi. Rankin-Selberg L-functions for General Spin Groups. In
preparation.

LOCAL TRANSFER AND REDUCIBILITY

19[5] M. Asgari and F. Shahidi. Generic transfer for general spin groups. Duke Math. J., 132 (2006), no1, 137–190.
[6] M. Asgari and F. Shahidi. Generic transfer from GSp(4) to GL(4). Compos. Math. 142 (2006), no.
3, 541–550.
[7] M. Asgari and F. Shahidi. Image of Functoriality for General Spin Groups. Manuscripta Math., 144(2014), no 3, 609–638.
[8] D. Ban and D. Goldberg. R-groups, elliptic representations, and parameters for GSpin groups.
Forum Math. (2014), published online. DOI: 10.1515/forum-2013-0188.
[9] I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive p-adic groups. I. Ann.
Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472.
[10] L. Cai and B. Xu. Residues of Intertwining Operators for U(3, 3) and Base Change. Int. Math. Res.
Not. IMRN (2014), published online. DOI:10.1093/imrn/rnu058.
[11] W. Casselman and F. Shahidi. On irreducibility of standard modules for generic representations.
Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 561–589.
[12] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi. On lifting from classical groups to
GLN . Publ. Math. Inst. Hautes Études Sci. No. 93 (2001), 5–30.
[13] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi. Functoriality for the classicalgroups. Publ. Math. Inst. Hautes Études Sci. No. 99 (2004), 163–233.
[14] J. W. Cogdell, I. I. Piatetski-Shapiro and F. Shahidi. Functoriality for the quasi-split classical groups.
In On Certain L-functions, Clay Math. Proc., 13 (2011), 117–140, AMS.
[15] D. Ginzburg, S. Rallis and D. Soudry. The descent map from automorphic representations of GL(n)to classical groups. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
[16] D. Goldberg. Some results on reducibility for unitary groups and local Asai L-functions. J. Reine
Angew. Math. 448 (1994), 65–95.
[17] D. Goldberg and F. Shahidi. On the tempered spectrum of quasi-split classical groups. Duke Math.
J. 92 (1998), no. 2, 255–294.
[18] D. Goldberg and F. Shahidi. The tempered spectrum of quasi-split classical groups III: The odd
orthogonal groups. Forum Math. 26 (2014), no. 4, 1029–1069.
[19] Harish-Chandra. Collected papers. Vol. IV. 1970–1983. Edited by V. S. Varadarajan. SpringerVerlag, New York, 1984.
[20] V. Heiermann and Y. Kim. On the generic local Langlands correspondence for GSpin groups.
Preprint.
[21] V. Heiermann and G. Muić. On the standard modules conjecture. Math. Z. 255 (2007), no. 4,
847–853.
[22] V. Heiermann and E. Opdam. On the tempered L-functions conjecture. Amer. J. Math. 135 (2013),
no. 3, 777–799.
[23] G. Henniart. Caractérisation de la correspondance de Langlands locale par les facteurs  de paires.
Invent. Math. 113 (1993), no. 2, 339–350.
[24] J. Hundley and E. Sayag. Descent construction for GSpin groups: main results and applications.
Electron. Res. Announc. Math. Sci. 16 (2009), 30–36.
[25] J. Hundley and E. Sayag. Descent construction for GSpin groups. Mem. Amer. Math. Soc. To
appear. Available at arXiv:math.NT/1110.6788 .
[26] H. Jacquet and I. I. Piatetski-Shapiro and J. A. Shalika. Rankin-Selberg convolutions. Amer. J.
Math. 105 (1983), no. 2, 367–464.
[27] H. H. Kim. Automorphic L-functions. In Lectures on automorphic L-functions, 97–201, Fields Inst.
Monogr., 20, Amer. Math. Soc., Providence, RI, 2004.
[28] H. H. Kim. On local L-functions and normalized intertwining operators. Canad. J. Math. 57 (2005),
no. 3, 535–597.
[29] H. H. Kim and W. Kim. On local L-functions and normalized intertwining operators II; quasi-split
groups. In On certain L-functions, volume in honor of F. Shahidi’s 60th birthday, 265–295, Clay
Math. Proc., 13, Amer. Math. Soc., Providence, RI, 2011.
[30] H. H. Kim and M. Krishnamurthy. Stable base change lift from unitary groups to GLn . IMRP Int.
Math. Res. Pap. 2005, no. 1, 1–52.
[31] H. H. Kim and M. Krishnamurthy. Base change lift for odd unitary groups. In Functional analysis
VIII, 116–125, Various Publ. Ser. (Aarhus), 47, Aarhus Univ., Aarhus, 2004.
[32] H. H. Kim and F. Shahidi. Functorial products for GL2 × GL3 and the symmetric cube for GL2 .
With an appendix by Colin J. Bushnell and Guy Henniart. Ann. of Math. (2) 155 (2002), no. 3,
837–893.

20

M. ASGARI, J. COGDELL, AND F. SHAHIDI[33] J. F. Lau. Reducibility of certain induced representations of E6 and E7 . J. Algebra 373 (2013),
183–206.
[34] W. W. Li. On a pairing of Goldberg-Shahidi for even orthogonal groups. Represent. Theory 17(2013), 337–381.
[35] G. Muić. A proof of Casselman-Shahidi’s conjecture for quasi-split classical groups. Canad. Math.
Bull. 44 (2001), no. 3, 298–312.
[36] G. Muić and F. Shahidi. Irreducibility of standard representations for Iwahori-spherical representations. Math. Ann. 312 (1998), no. 1, 151–165.
[37] F. Shahidi. Local coefficients as Artin factors for real groups. Duke Math. J., 52(4):973–1007, 1985.
[38] F. Shahidi. On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. of
Math. (2) 127 (1988), no. 3, 547–584.
[39] F. Shahidi. A proof of Langlands’ conjecture on Plancherel measures; complementary series for
p-adic groups. Ann. of Math. (2) 132 (1990), no. 2, 273–330.
[40] F. Shahidi. On multiplicativity of local factors. In Festschrift in honor of I. I. Piatetski-Shapiro on
the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), 279–289, Israel Math. Conf. Proc.,
3, Weizmann, Jerusalem, 1990.
[41] F. Shahidi. Langlands’ conjecture on Plancherel measures for p-adic groups. In Harmonic analysis on
reductive groups (Brunswick, ME, 1989), 277–295, Progr. Math., 101, Birkhäuser Boston, Boston,MA, 1991.
[42] F. Shahidi. Twisted endoscopy and reducibility of induced representations for p-adic groups. Duke
Math. J. 66 (1992), no. 1, 1–41.
[43] F. Shahidi. The notion of norm and the representation theory of orthogonal groups. Invent. Math.
119 (1995), no. 1, 1–36.
[44] F. Shahidi. Eisenstein series and automorphic L-functions. American Mathematical Society Colloquium Publications, 58. American Mathematical Society, Providence, RI, 2010.
[45] F. Shahidi and S. Spallone. Residues of intertwining operators for SO∗6 as character identities.
Compos. Math. 146 (2010), 772–794.
[46] A. J. Silberger. Introduction to harmonic analysis on reductive p-adic groups. Based on lectures by
Harish-Chandra at the Institute for Advanced Study, 1971–1973. Mathematical Notes, 23. Princeton
University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979.
[47] D. Soudry. On Langlands functoriality from classical groups to GLn . In Automorphic forms. I.
Asterisque No. 298 (2005), 335–390.
[48] S. Spallone. Residues of intertwining operators for classical groups. Int. Math. Res. Not. IMRN
(2008), no. 56.
[49] M. Tadić. Classification of unitary representations in irreducible representations of general linear
group (non-Archimedean case). Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335382.
[50] A. V. Zelevinsky. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210.
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA
E-mail address: asgari@math.okstate.edu
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
E-mail address: cogdell@math.ohio-sate.edu
Mathematics Department, Purdue University, West Lafayette, IN 47907, USA
E-mail address: shahidi@math.purdue.edu