LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIRINNER FORMS
MAHDI ASGARI AND KWANGHO CHOIY

Abstract. We establish the local Langlands conjecture for small rank general spin groups GSpin4 andGSpin6 as well as their inner forms. We construct appropriate L-packets and prove that these L-packets
satisfy the properties expected of them to the extent that the corresponding local factors are available. We
are also able to determine the exact sizes of the L-packets in many cases.

1. Introduction
In this article, we construct L-packets for the split general spin groups GSpin4 , GSpin6 , and their inner
forms over a p-adic field F of characteristic 0, and more importantly, establish their internal structuresin terms of characters of component groups, as predicted by the Local Langlands Conjecture (LLC). Thisestablishes the LLC for the groups in question (cf. Theorems 5.1 and 6.1 and Propositions 5.16 and 6.12).
The construction of the L-packets is essentially an exercise in restriction of representations, thanks to thestructure, as algebraic groups, of the groups we consider; however, proving the properties of the L-packets
requires some deep results of Hiraga-Saito as well as Aubert-Baum-Plymen-Solleveld as we explain below.
Let WF denote the Weil group of F. In a general setting, if G denotes a connected, reductive, linear,algebraic group over F, then the Local Langlands Conjecture asserts that there is a surjective, finite-to-one
map from the set Irr(G) of isomorphism classes of irreducible smooth complex representations of G(F ) to thebset Φ(G) of G-conjugacy
classes of L-parameters, i.e., admissible homomorphisms ϕ : WF × SL2 (C) −→ L G.
L 0b
Here G = G denotes the connected component of the L-group of G, i.e., the complex dual of G [Bor79].
Given ϕ ∈ Φ(G), its fiber Πϕ (G), which is called an L-packet for G, is expected to be controlled by a certainb Furthermore, the map is supposed to preserve certain local
finite group living in the complex dual group G.
factors, such as γ-factors, L-factors, and -factors.
The LLC is already known for several cases: GLn [HT01, Hen00, Sch13], SLn [GK82], U2 and U3 [Rog90],
F -inner forms of GLn and SLn [HS12, ABPS14], GSp4 [GT11], Sp4 [GT10], the F -inner form GSp1,1 of
GSp4 [GT14], the F -inner form Sp1,1 of Sp4 [Cho14c], quasi-split orthogonal and symplectic groups [Art13],
unitary groups [Mok15], and non quasi-split inner forms of unitary groups [KMSW14].
We consider the case of G = GSpin4 , GSpin6 , or one of their non quasi-split F -inner forms. Our approachis based on the study of the restriction of representations from a connected reductive F -group to a closed
subgroup having an identical derived group as G itself. This approach originates in the earlier work on
the LLC for SLn [GK82]. Gelbart and Knapp studied the restriction of representations of GLn to SLn andestablished the LLC for SLn , assuming the LLC for GLn which was later proved [HT01, Hen00, Sch13].
Given an L-parameter ϕ ∈ Φ(SLn ), the L-packet Πϕ (SLn ) is proved to be in bijection with the componentdn ) of the centralizer of the image of ϕ in SL
dn . The multiplicity one property for the restrictiongroup Sϕ (SLfrom GLn to SLn [HS75, Tad92] and the fact that all L-packets of GLn are singletons [HT01, Hen00, Sch13],
are indispensable to establishing the bijection.
Later on, Hiraga and Saito extended the LLC for SLn and the result of Labesse and Langlands [LL79]
for the non-split inner form SL02 of SL2 to the non-split inner form SL0n of SLn [HS12], except for the
representations of SL0n whose liftings to the split GLn are not generic. Those cases were dealt with afterwards
2010 Mathematics Subject Classification. Primary: 22E50; Secondary: 11S37, 20G25, 22E35.
Key words and phrases. local Langlands correspondence of p-adic groups, L-packet, inner form, general spin group, restriction of representations, equality of local factors.
1

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MAHDI ASGARI AND KWANGHO CHOIY

by Aubert, Baum, Plymen, and Solleveld [ABPS14]. The restriction approach is also used in the case of SL0n ,after establishing the LLC for the non-split inner form GL0n of GLn by means of the LLC for GLn and the
local Jacquet-Langlands correspondence [JL70, DKV84, Rog83, Bad08]. However, there is some subtlety inapplying the restriction technique from GL0n to SL0n since, unlike in the case of split SLn , the multiplicity onedn ) of the connected
property fails in this case. Thus, Hiraga and Saito consider a central extension Sϕ,sc (SLbcomponent group PGLn (C) by a certain quotient group Zϕ,sc (SLn ) of the center of SLn (C). They then provedn ) governs the restriction from GL0 tothat the set of irreducible representations of the finite group Sϕ,sc (SLn00SLn and parameterizes the L-packets for SLn . The central extension approach turns out to also include thecase of G = SLn and the previous parameterization of L-packets of SLn .
The LLC for the groups we consider in this paper is related to the LLC for SLn [GK82] and its F -inner
form SL0n [HS12, ABPS14]. Write G for GSpin4 , GSpin6 , or one of their non quasi-split F -inner forms. It
follows from the structure of G as an algebraic group that it is an intermediate group between a product ofSLmi and a product of GLmi or their F -inner forms with suitable integers mi . We give an explicit descriptionof each group structure in Section 2.2. We are then able to utilize the LLC for GLn [HT01, Hen00, Sch13]
and GL0n [HS12]. At the same time, using a theorem of Labesse [Lab85], we are able to define a surjective,
finite-to-one map
L : Irr(G) −→ Φ(G),(1.1)
and construct L-packets Πϕ (G) for each ϕ ∈ Φ(G).
We next study the internal structure of each L-packet. Based on the work of Hiraga and Saito [HS12], wedn ). This
investigate the central extension Sϕ,sc for our case and prove that Sϕ,sc is embedded into Sϕ,sc (SL0is where the internal structures of the L-packets for SLn and SLn are needed. We then prove that there is a
one-to-one correspondence
1−1Πϕ (G) ←→ Irr(Sϕ,sc , ζG ),where Irr(Sϕ,sc , ζG ) denotes the set of irreducible representations of Sϕ,sc with central character ζG corresponding to the group G via the Kottwitz isomorphism [Kot86] (cf. Theorems 5.1 and 6.1). Moreover,using Galois cohomology, we prove that the possible sizes for the L-packet Πϕ (G) are 1, 2, and 4 when
p 6= 2 and 1, 2, 4, and 8 when p = 2 (cf. Propositions 5.5 and 6.4). In the case of G = GSpin4 we arealso able to show that only 1, 2, and 4 occur for any p (see Remarks 5.10 and 5.11). We do this using theclassification of the group of characters stabilizing representations. Further, we describe the group structureof the central extension Sϕ,sc , provide all the sizes of L-packets for GSpin4 and its non quasi-split F -inner
forms, and discuss the multiplicity in restriction. In the case of GSpin6 , unlike that of GSpin4 , we do notclassify the group of characters for GSpin6 nor do we discuss the group structure of Sϕ,sc . This is due to thefact that a full classification of irreducible L-parameters in Φ(SL4 ) is not currently available. Accordingly,
the determination of all the sizes of L-packets for GSpin6 and its non quasi-split F -inner forms as well as
the multiplicity in restriction are not addressed in this paper. These questions will require further study ofthe finite group Sϕ,sc for each L-parameter ϕ for GSpin6 .
Furthermore, in Sections 5.4 and 6.4 we verify that the Plancherel measure is preserved within L-packets
and between inner twisting. Also, the L-map satisfies that an irreducible representation is essentially square
integrable (respectively, tempered) if and only if its L-parameter does not factor through any proper Levi
subgroup (respectively, the image of its L-parameter is bounded).
When G is the split group GSpin4 or GSpin6 , we prove that the local L-, -, and γ-factors are preserved
via the L-map in (1.1). Given τ ∈ Irr(GLr ), r ≥ 1, and σ ∈ Irr(G) which is assumed to be either ψ-generic or
non-supercuspidal if r > 1, we let ϕτ be the L-parameter of τ via the LLC for GLr and let ϕσ = L(σ). Thanks
to the structure theory detailed in Section 2.2, we are able to use results on the generic Langlands functorialtransfer from general spin groups to the general linear groups which are already available [ACS1, AS06, AS14].
As a result, we prove
L(s, τ × σ)

= L(s, ϕτ ⊗ ϕσ ),(s, τ × σ, ψ)= (s, ϕτ ⊗ ϕσ , ψ),γ(s, τ × σ, ψ)= γ(s, ϕτ ⊗ ϕσ , ψ).

(1.2)

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

3

Here, the local factors on the left hand side are those attached by Shahidi [Sha90b, Theorem 3.5] initially togeneric representations and extended to all non-generic non-supercuspidal representations via the Langlands
classification and the multiplicativity of the local factors [Sha90b, §9], and the factors on the right hand sideare Artin local factors associated with the given representations of the Weil-Deligne group of F (cf. Sections
5.5 and 6.5).
Another expected property of the L-packets is that they should satisfy the local character identities of
the theory of (twisted) endoscopy (see [CG14], for example). It is certainly desirable to study this question
for the groups we consider and the L-packets we construct, a task we leave for a future work.
We finally remark that, due to lack of the LLC for the quasi-split special unitary group SUn , our method
is currently limited to split groups GSpin4 , GSpin6 , and their non quasi-split F -inner forms. The case of thequasi-split non-split groups GSpin∗4 and GSpin∗6 will be addressed in a forthcoming work.
The structure of this paper is as follows. In Section 2 we introduce the basic notations and review somebackground material. We also describe the algebraic group structure, F -points, and L-groups of the groups
GSpin4 , GSpin6 , and their non quasi-split F -inner forms, which are the groups under consideration in thispaper. Section 3 states the LLC and the conjectural structure of L-packets in a general setting. In Section 4
we review some well-known results on restriction. We then prove our main results: the LLC for GSpin4 andits non quasi-split F inner forms in Section 5, and the LLC for GSpin6 and its non quasi-split F -inner forms
in Section 6. Furthermore, we describe the possible sizes of L-packets in each case and prove the equality of
local factors via the L-map.
Acknowledgements. This work has benefited from many helpful conversations with Wee Teck Gan andthe authors would like to thank him for all his help. The authors also thank Maarten Solleveld for his
feedback and comments on this work. The first author was partially supported by a Collaborations Grant(#245422) from the Simons Foundation. The second author was partially supported by an AMS-Simons
Travel Grant.
2. The Preliminaries
2.1. Notations and Conventions. Let p be a prime number. We denote by F a p-adic field of characteristic
0, i.e., a finite extension of Qp . Let F̄ be an algebraic closure of F. Denote by OF the ring of integers of F,
and by P the maximal ideal in OF . Let q denote the cardinality of the residue field OF /P.
We denote by WF the Weil group of F and by Γ the absolute Galois group Gal(F̄ /F ). Let G be aconnected, reductive, linear, algebraic group over F. Fixing Γ-invariant splitting data, we define the L-groupb o Γ (see [Bor79, Section 2]).
of G as a semi-direct product L G := G
For an integer i ∈ N and a connected, reductive, algebraic group G over F, we set
H i (F, G) := H i (Gal(F̄ /F ), G(F̄ )),the Galois cohomology of G. For any topological group H, we denote by π0 (H) the group H/H ◦ of connectedcomponents of H, where H ◦ denotes the identity component of H. By Z(H) we will denote the center of
H. We write H D for the group Hom(H, C× ) of all continuous characters. Also, Hder denotes the derived
groups of H. We denote by 1 the trivial character. The cardinality of a finite set S is denoted by |S|. For
two integers x and y, x y means that y is divisible by x. For any positive integer n, we denote by µn thealgebraic group such that µn (R) = {r ∈ R : rn = 1} for any F -algebra R.
Given connected reductive algebraic groups G and G0 over F, we say that G and G0 are F -inner forms
with respect to an F̄ -isomorphism ϕ : G0 → G if ϕ ◦ τ (ϕ)−1 is an inner automorphism (g 7→ xgx−1 ) definedover F̄ for all τ ∈ Gal(F̄ /F ) (see [Bor79, 2.4(3)], [Kot97, p.280]). We often omit the references to F and ϕwhen there is no danger of confusion. We recall that if two F -inner forms G and G0 are quasi-split over F ,then G and G0 are isomorphic over F, [Bor79, Remarks 2.4(3)].
When G and G0 are inner forms of each other, we have L G ∼= L G0 [Bor79, Section 2.4(3)]. In particular,b trivial, we write L G = Gb∼c0 .
if G0 is an inner form of an F -split group G with the action of Γ on G
= L G0 = G
2For positive integers m, n, and d, we let D be a central division algebra of dimension d over F (possibly
D = F, in which case d = 1). Let GLm (D) denote the group of all invertible elements of m × m matrices

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MAHDI ASGARI AND KWANGHO CHOIY

over D. Let SLm (D) be the subgroup of elements in GLm (D) with reduced norm Nrd equal to 1. Note that
there are algebraic groups over F, whose groups of F -points are respectively GLm (D) and SLm (D). By abuseof notation, we shall write GLm (D) and SLm (D) for their algebraic groups over F as well. Note that any
F -inner forms of the split general linear group GLn and the split special linear group SLn are respectivelyof the form GLm (D) and SLm (D), where n = md (see [PR94, Sections 2.2 & 2.3]).
2.2. Group Structures. In this section we describe the structure of the split groups GSpin4 and GSpin6over F as well as their non quasi-split F -inner forms. These are the groups we work with in this paper. The
exact knowledge of the structure of these groups allow us, on the one hand, to use techniques from restrictionof representations to construct L-packets and, on the other hand, make use of generic local transfer from thegeneral spin groups to general linear groups in order to prove preservation of local L-, -, and γ-factors for
our L-packets.
2.2.1. Split Groups. We first give a description of the split groups GSpin4 and GSpin6 in terms of abstractroot data. Let
X2n = Ze0 ⊕ Ze1 ⊕ · · · ⊕ Zen
and let∨
X2n= Ze∗0 ⊕ Ze∗1 ⊕ · · · ⊕ Ze∗2nbe the dual Z-module with the standard Z-pairing between them. We let∆2n = {α1 = e1 − e2 , . . . , αn−1 = en−1 − en , αn = en−1 + en }
and
∨
∗
∗
∨
∗
∗
∨
∗
∗
∗
∆∨
2n = {α1 = e1 − e2 , . . . αn−1 = en−1 − en , αn = en−1 + en − e0 }∨denote the simple roots and coroots, respectively, and let R2n and R2nbe the roots and coroots they generate.
Then∨∨Ψ2n = (X2n , R2n , ∆2n , X2n
, R2n
, ∆∨
2n )
is a based root datum determining the split, connected, reductive group GSpin2n over F. See [ACS2, AS14]
for more details. We are mostly interested in Ψ4 and Ψ6 in this paper.
In fact, the derived group of GSpin2n is isomorphic to Spin2n and we have [AS06, Proposition 2.2] thefollowing isomorphism of algebraic groups over F :GSpin ∼
(2.1)
= (GL1 × Spin )/{(1, 1), (−1, c)},2n2nwhere c denotes the non-trivial element in the center of Spin2n given by∨c = αn−1(−1)αn∨ (−1) = e∗0 (−1)−1 = e∗0 (−1)

in our root data notation.
When n = 2 or 3 we have the accidental isomorphisms
Spin ∼= SL2 × SL2
4

Spin6

∼
= SL4

as algebraic groups over F. The element c will then be identified with (−I2 , −I2 ) ∈ SL2 × SL2 and −I4 ∈ SL4 ,respectively. Therefore, we have the following isomorphisms of algebraic groups over F :GSpin4
GSpin6∼= (GL1 × SL2 × SL2 )/ {(1, I2 , I2 ), (−1, −I2 , −I2 )} ,∼= (GL1 × Spin6 )/ {(1, I4 ), (−1, −I4 )} .

(2.2)
(2.3)

For our purposes here, the following is a more convenient description of these two groups.
Proposition 2.1. As algebraic groups over F we have the following isomorphisms:∼ {(g1 , g2 ) ∈ GL2 × GL2 : det g1 = det g2 }GSpin =
4

GSpin6 ∼= {(g1 , g2 ) ∈ GL1 × GL4 : g12 = det g2 }.

(2.4)
(2.5)

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

5

Proof. We verify these isomorphisms by giving explicit isomorphisms between the respective root data. For(2.4) consider GL2 × GL2 and let T1 be its usual maximal split torus, with T ⊂ T1 denoting the maximal
split torus of
G4 = {(g1 , g2 ) ∈ GL2 × GL2 : det g1 = det g2 } .
∗Using the notation fij and fij, 1 ≤ i, j ≤ 2, for the usual Z-basis of characters and cocharacters of GL2 ×GL2
with the standard Z-pairing between them, the character lattice X ∗ (T ), a quotient of X ∗ (T1 ), and thecocharacter lattice X∗ (T ), a sublattice of X∗ (T1 ), can be given byX ∗ (T ) =Zhf11 , f12 i ⊕ Zhf21 , f22 i
Zhf11 + f12 − f21 − f22 i

andX∗ (T ) = hf11 + f12 − f21 − f22 i⊥ .
Here ⊥ means orthogonal complement with respect to the Z-pairing.
Let R and ∆ denote the roots and simple roots in G4 and let R∨ and ∆∨ be coroots and simple coroots. An
isomorphism of based root data Ψ4 −→ Ψ(G4 ) = (X ∗ (T ), R, ∆, X∗ (T ), R∨ , ∆∨ ) amounts to isomorphismsof Z-modules
ι : X4 −→ X ∗ (T ) and ι∨ : X∗ (T ) −→ X4∨such thathι(x), yi = hx, ι∨ (y)i,x ∈ X4 , y ∈ X∗ (T ),ι(∆4 ) = ∆,andι∨ (∆∨ ) = ∆∨4such that the following diagram commutes:∨

R4 −−−−→ R4∨

x

∨
ιy
ι
∨

R −−−−→ R∨
Setting∆ = {β1 = f11 − f12 , β2 = −f11 − f12 + 2f21 } (mod f11 + f12 − f21 − f22 )
and
∗
∗
∗
∗
∆∨ = {β1∨ = f11− f12, β2∨ = f21
− f22
},

let S denote the 3 × 3 matrix of ι with respect to the Z-bases (e0 , e1 , e2 ) of X4 and (f11 , f12 , f21 )(mod f11 +f12 − f21 − f22 ) of X ∗ (T ), respectively. Similarly, let S ∨ denote the matrix of ι∨ with respect to the Z-bases
∗
∗
∗
∗
∗
∗
(f11
+ f22
, f12
+ f22
, −f21+ f22) of X∗ (T ) and (e∗0 , e∗1 , e∗2 ) of X4∨ , respectively. Assuming that ι(α1 ) = β1 andι(α2 ) = β2 along with the conditions for root data isomorphisms we detailed above, plus the requirement
det S = 1 gives, after some computations, a unique choice for the Z-isomorphisms ι and ι∨ given by


0
0 −1
S =  0 −1 0  and S ∨ = t S.
−1 1
1(Alternatively, we could have assumed that det S = −1 or that ι(α1 ) = β2 , etc., which would have led toslightly different matrices S and S ∨ .) This shows that G4 is indeed a realization, as an algebraic group overF, for GSpin4 .
In fact, using S and S ∨ above, we can arrive at the following explicit realizations. For

 

a
c
t=
,
∈ T, (with ab = cd)bdwe have,e1 (t) = c/b,

e2 (t) = c/a,e0 (t) = 1/c.

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MAHDI ASGARI AND KWANGHO CHOIY

Similarly, for λ ∈ GL1 , we have


 

1
=
,
λ−1
λ−1

  −11λ∗,e2 (λ) =1
λ −1  −1λλ∗e0 (λ) =
,
.
λ−1
λ−1

The proof for the isomorphism between GSpin6 and G6 = (g1 , g2 ) ∈ GL1 × GL4 : g12 = det g2 is similar.
With notations f0 and f1 , f2 , f3 , f4 and their duals for the characters and cocharacters of T1 , the split torus
in GL1 × GL4 , which contains the split torus T of G4 , we can writee∗1 (λ)X ∗ (T ) =1

Zf0 ⊕ Zhf1 , f2 , f3 , f4 i
Zh2f0 − f1 − f2 − f3 − f4 i

andX∗ (T ) = h2f0 − f1 − f2 − f3 − f4 i⊥ .
Again, we set∆ = {β1 = f2 − f3 , β2 = f1 − f2 , β3 = f3 − f4 } (mod 2f0 − f1 − f2 − f3 − f4 )and∆∨ = {β1∨ = f2∗ − f3∗ , β2∨ = f1∗ − f2∗ , β3∨ = f3∗ − f4∗ } .
Now, we let S denote the matrix of ι with respect to (e0 , e1 , e2 , e3 ) and (f0 , f1 , f2 , f3 )(mod 2f0 −f1 −f2 −f3 −f4 )and let S ∨ denote the matrix of ι∨ with respect to (f0∗ + 2f4∗ , f1∗ − f4∗ , f2∗ − f4∗ , f3∗ − f4∗ ) and (e∗0 , e∗1 , e∗2 , e∗3 ).
Assuming ι(αi ) = βi for 1 ≤ i ≤ 3 and the conditions for the root data isomorphim along with the requirement
det S = 1 we get, after similar computations, that
1 −1 −1 −1
−1 1
1
0

and S ∨ = t S = S.
S=
−1 1
0
1
−1 0
1
1
This shows that G6 is indeed a realization, as an algebraic group over F, for GSpin6 .
Again, with the above S we have the following explicit realizations. Fort = (z, diag(a, b, c, d)) ∈ T(with z 2 = abcd)

we havee1 (t)= abz −1e2 (t)= acz −1e3 (t)= bcz −1e0 (t)= dz −1

Similarly, for λ ∈ GL1 , we havee∗1 (λ)=e∗2 (λ)=e∗3 (λ)e∗0 (λ)=
λ−1 , diag(1, 1, λ−1 , λ−1 )
λ−1 , diag(1, λ−1 , 1, λ−1 )
λ−1 , diag(λ−1 , 1, 1, λ−1 )=
λ−1 , diag(λ−1 , λ−1 , λ−1 , λ−1 ) .
LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

7

The proposition implies that we have the following inclusions
SL2 × SL2 ⊂GSpin4 ⊂ GL2 × GL2

(2.6)

SL4 ⊂

GSpin6 ⊂ GL1 × GL4 .

(2.7)

Since
SL2 × SL2 = (GSpin4 )der = (GL2 × GL2 )derSL4 = (GSpin6 )der = (GL1 × GL4 )derthe two groups GSpin4 and GSpin6 fit in the setting (cf. (4.1))
e der ⊆ G ⊆ G,e
Gder = G

(2.8)

with
G = GSpin4 ,e = GL2 × GL2
G

G = GSpin6 ,e = GL1 × GLv ,G

andrespectively.
Moreover, using the surjective map GL2 × GL2 −→ GL1 defined by by (g1 , g2 ) 7→ (det g1 )(det g2 )−1 , theisomorphism (2.4) gives an exact sequence of algebraic groups
1 −→ GSpin4 −→ GL2 × GL2 −→ GL1 −→ 1.

(2.9)

Likewise, using the surjective map GL1 × GL4 −→ GL1 defined by (g1 , g2 ) 7→ g1−2 (det g2 ), the isomorphism(2.5) yields the exact sequence
1 −→ GSpin6 −→ GL1 × GL4 −→ GL1 −→ 1.

(2.10)

2.2.2. Inner Forms. Using the Satake classification [Sat71, p.119] for admissible diagrams of the F -inner
forms, we only have two (up to isomorphism) non quasi-split F -inner forms of GSpin4 , denoted by GSpin2,1
4
and GSpin1,1
4 , whose diagrams are respectively
•
and
•

•

The left diagram gives
SL2 × SL1 (D) ⊂ GSpin42,1 ⊂ GL2 × GL1 (D)(2.11)and the right one gives
SL1 (D) × SL1 (D) ⊂ GSpin1,1(2.12)
4 ⊂ GL1 (D) × GL1 (D).
Here, D denotes the quaternion division algebra over F. (Recall from Section 2.1 that we are writing GLm (D)and SLm (D) for both algebraic groups over F and their F -points, by abuse of notation.)
Similarly, we only have two (up to isomorphism) non quasi-split F -inner forms of GSpin6 , denoted by
1,0
GSpin2,0
6 and GSpin6 , whose diagrams are respectively
•
•
•
and
•
•
The left diagram gives(2.13)
SL2 (D) ⊂ GSpin2,0
6 ⊂ GL1 × GL2 (D)and the right one gives
SL1 (D4 ) ⊂ GSpin1,0(2.14)
6 ⊂ GL1 × GL1 (D4 ).
Here, D4 is a division algebra of dimension 16 over F. We note that the two division algebras D4 and its
opposite D4op of dimension 16 ( with invariants 1/4, −1/4 in Q/Z) have canonically isomorphic multiplicative
groups D4× and (D4op )× .

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MAHDI ASGARI AND KWANGHO CHOIY

Similar to the split forms GSpin4 and GSpin6 , the F -inner forms of GSpin4 and GSpin6 as well as theF -inner forms of SL2 × SL2 , GL2 × GL2 , SL4 , and GL4 all satisfy the setting (2.8).
The arguments giving (2.9) and (2.10) apply again, with det replaced by the reduced norm Nrd, to show
1,1
2,0
1,0
that for F -inner forms GSpin2,1
4 , GSpin4 , GSpin6 , and GSpin6 , we have
1 −→GSpin2,1
4 −→GL2 × GL1 (D) −→ GL1 −→ 1,(2.15)

1 −→GSpin1,1
4 −→GSpin2,0
6 −→GSpin1,0
6 −→

GL1 (D) × GL1 (D) −→ GL1 −→ 1,(2.16)

GL1 × GL2 (D) −→ GL1 −→ 1,(2.17)

GL1 × GL1 (D4 ) −→ GL1 −→ 1.

(2.18)

1 −→1 −→Using (2.15) – (2.18), we have the following isomorphisms of algebraic groups over F, which are analoguesof Proposition 2.1.
∼ {(g1 , g2 ) ∈ GL2 × GL1 (D) : det g1 = Nrd g2 },GSpin2,1 =4∼
GSpin1,1
4 =GSpin2,0 ∼=6∼
GSpin1,0
6 ={(g1 , g2 ) ∈ GL1 (D) × GL1 (D) : Nrd g1 = Nrd g2 },{(g1 , g2 ) ∈ GL1 × GL2 (D) : g12 = Nrd g2 },{(g1 , g2 ) ∈ GL1 × GL1 (D4 ) : g12 = Nrd g2 }.

Finally, recall [AS06, Proposition 2.3] that we have
π0 (Z(GSpin )) ∼= π0 (Z(GSpin )) ∼= Z/2Z.
4

6(2.19)

This also holds for their F -inner forms since F -inner forms have the same center.
2.3. The F -points. We now describe the F -rational points of the split groups GSpin4 and GSpin6 as wellas their non quasi-split inner forms. We start with the following.
Lemma 2.2. Let G be an F -inner form of GSpin2n with n ≥ 1 an integer. Then
H 1 (F, G) = 1.
Proof. We have [AS06, Propositions 2.2 and 2.10]\2n = GSO2n (C)
GSpin
and
\2n )Γ = Z(GSO2n (C)) ∼Z(GSpin= C× .
Here, b denotes the connected component of the L-group. Applying the Kottwitz isomorphism [Kot86,
Theorem 1.2], we can conclude\2n )Γ )D = π0 (C× )D = 1.
H 1 (F, GSpin2n ) ∼= π0 (Z(GSpin
Since the Kottwitz isomorphism is valid for any connected reductive algebraic group over a p-adic field of
characteristic 0, and since F -inner forms of algebraic groups share the same L-groups (cf. Section 2.1), theproof is complete.

Also, recall that H 1 (F, GLn ) = 1 and H 1 (F, Spinn ) = 1 for n ≥ 1 (see [PR94, Lemma 2.8 and Theorem
6.4], for example). It follows from (2.1) that the group GSpin4 (F ) can be described via
1 −→ {±1} −→ F × × Spin4 (F ) −→ GSpin4 (F ) −→ H 1 (F, {±1}) −→ 1or
1 −→ (F × × Spin4 (F ))/{±1} −→ GSpin4 (F ) −→ F × /(F × )2 −→ 1.
Likewise, the group GSpin6 (F ) can be described via
1 −→ {±1} −→ F × × Spin6 (F ) −→ GSpin6 (F ) −→ H 1 (F, {±1}) −→ 1or
1 −→ (F × × Spin6 (F ))/{±1} −→ GSpin6 (F ) −→ F × /(F × )2 −→ 1.

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

9

Using Lemma 2.2, we apply Galois cohomology to (2.9) to obtain
1 −→ GSpin4 (F ) −→ GL2 (F ) × GL2 (F ) −→ F × −→ 1.

(2.20)

GSpin4 (F ) ∼= {(g1 , g2 ) ∈ GL2 (F ) × GL2 (F ) : det g1 = det g2 }.

(2.21)

Thus, we have
Likewise, using Lemma 2.2 and (2.10) we obtain
1 −→ GSpin6 (F ) −→ GL1 (F ) × GL4 (F ) −→ F × −→ 1.

(2.22)

GSpin6 (F ) ∼= {(g1 , g2 ) ∈ GL1 (F ) × GL2 (F ) : (g1 )−2 = det g2 }.

(2.23)

Thus, we have
Similarly, using (2.15) – (2.18) we get
1 −→GSpin2,1
4 (F ) −→GL2 (F ) × GL1 (D) −→ F × −→ 1,

1 −→GSpin1,1
4 (F ) −→GL1 (D) × GL1 (D) −→ F × −→ 1,

1 −→GSpin2,0
6 (F ) −→F × × GL2 (D) −→ F × −→ 1,

1 −→GSpin1,0
6 (F ) −→F × × GL1 (D4 ) −→ F × −→ 1.

We thus have∼
GSpin2,1
4 (F ) = {(g1 , g2 ) ∈ GL2 (F ) × GL1 (D) : det g1 = Nrd g2 },1,1
GSpin4 (F ) ∼= {(g1 , g2 ) ∈ GL1 (D) × GL1 (D) : Nrd g1 = Nrd g2 },2,0GSpin (F ) ∼= {(g1 , g2 ) ∈ F × × GL2 (D) : g 2 = Nrd g2 },6∼
GSpin1,0
6 (F ) =1{(g1 , g2 ) ∈ F

×× GL1 (D4 ) : g12 = Nrd g2 }.

2.4. L-groups. We recall the following descriptions of the L-groups of (the split groups) GSpin4 and GSpin6
from [AS06, Proposition 2.2] and [GT11, Sections 3 and 4]:L\4 = GSO4 (C) ∼GSpin4 = GSpin= (GL2 (C) × GL2 (C))/{(z −1 , z) : z ∈ C× },(2.24)

L\6 = GSO6 (C) ∼GSpin6 = GSpin= (GL1 (C) × GL4 (C))/{(z −2 , z) : z ∈ C× }.

(2.25)

This immediately gives
pr4\4 −→ 1
1 −→ C× −→ GL2 (C) × GL2 (C) −→ GSpin(2.26)

pr6\6 −→ 1.
1 −→ C× −→ GL1 (C) × GL4 (C) −→ GSpin(2.27)

andc 1 in (2.9) and (2.10). Further, the injection C× ,→ GL2 (C) × GL2 (C) is given
Here, C× is considered as GL
by g 7−→ (g −1 I2 , gI2 ) and the injection C× ,→ GL1 (C) × GL4 (C) is given by g 7−→ (g −2 , gI4 ).
The inclusions in (2.6) and (2.7) provide the following surjective maps:
pr4\4  PGL2 (C) × PGL2 (C) = SL\
GL2 (C) × GL2 (C)  GSpin
2 × SL2pr6
d4 .
\6  PGL4 (C) = SL
GL1 (C) × GL4 (C)  GSpin

Since F -inner forms of algebraic groups share the same L-groups (cf. Section 2.1), we haveL

1,1L
GSpin4 = L GSpin2,1
4 = GSpin4

L

1,0
L
GSpin6 = L GSpin2,0
6 = GSpin6 .

10

MAHDI ASGARI AND KWANGHO CHOIY

3. Local Langlands Conjecture in a General Setting
In this section we quickly review some generalities about the Local Langlands Conjecture (LLC). Let G
be a connected, reductive, algebraic group over F. Write G(F ) for the group of F -points of G. Let Irr(G)denote the set of equivalence classes of irreducible, smooth, complex representations of G(F ). By abuse ofnotation, we identify an equivalence class with its representatives. We write Πdisc (G) and Πtemp (G) for thesubsets of Irr(G) consisting, respectively, of discrete series and tempered representations. Moreover, we writeΠscusp (G) for the subset in Irr(G) consisting of supercuspidal ones. Furthermore, we writeΠscusp,unit (G) = Πscusp (G) ∩ Πdisc (G).
Note that we haveΠscusp,unit (G) ⊂ Πdisc (G) ⊂ Πtemp (G) ⊂ Irr(G).
bLet Φ(G) denote the set of G-conjugacy classes of L-parameters, i.e., admissible homomorphisms(3.1)

ϕ : WF × SL2 (C) −→ L G,
b by Cϕ . The center of L G is the(see [Bor79, Section 8.2]). We denote the centralizer of the image of ϕ in GΓΓb . Note that Cϕ ⊃ Z(G)b . We say that ϕ is elliptic if the quotient group Cϕ /Z(G)b ΓΓ-invariant group Z(G)is finite, and ϕ is tempered if ϕ(WF ) is bounded. We denote by Φell (G) and Φtemp (G) the subset of Φ(G)which consist, respectively, of elliptic and tempered L-parameters of G. We set
Φdisc (G) = Φell (G) ∩ Φtemp (G).
Moreover, we write Φsim (G) for the subset in Φ(G) consisting of irreducible ones. Furthermore, letΦsim,disc (G) = Φsim (G) ∩ Φdisc (G).
We then have, in parallel to (3.1),
Φsim,disc (G) ⊂ Φdisc (G) ⊂ Φtemp (G) ⊂ Φ(G).

(3.2)

For any connected reductive group G over F, the Local Langlands Conjecture predicts that there is asurjective finite-to-one map
Irr(G) −→ Φ(G).
This map is supposed to satisfy a number of natural properties. For instance, it preserves certain γ-facotrs,
L-factors, and -factors, which one can attach to both sides. Moreover, considering the fibers of the map, one
can partition Irr(G) into a disjoint union of finite subsets, called the L-packets. Each packet is conjectured
to be characterized by component groups in the L-group, which, for groups we are considering in this paper,
are discussed in Sections 5.2 and 6.2. It is also expected that Φdisc (G) and Φtemp (G) parameterize Πdisc (G)and Πtemp (G), respectively.
b sc the simply connected cover of the derived group Gb der of G,b and by Gb ad the quotient group
Denote by Gbb
G/Z(G). We considerb Γ⊂Gb ad .
Sϕ := Cϕ /Z(G)b sc . We then have an exact sequence
Write Sϕ,sc for the full pre-image of Sϕ in G
b sc ) −→ Sϕ,sc −→ Sϕ −→ 1.
1 −→ Z(G(3.3)

We letSϕ := π0 (Sϕ )Sϕ,sc := π0 (Sϕ,sc )b sc )/(Z(Gb sc ) ∩ S ◦ ).
Zbϕ,sc := Z(Gϕ,scWe then have (see [Art13, (9.2.2)]) a central extension
1 −→ Zbϕ,sc −→ Sϕ,sc −→ Sϕ −→ 1.

(3.4)

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

11b sc ) whose restriction to Z(Gb sc )Γ correNext, let G0 be an F -inner form of G. Fix a character ζG0 of Z(G0sponds to the class of the F -inner form G via the Kottwitz isomorphism [Kot86, Theorem 1.2]. We denoteb sc ). It is
by Irr(Sϕ,sc , ζG0 ) the set of irreducible representations of Sϕ,sc with central character ζG0 on Z(Gexpected that, given an L-parameter ϕ for G0 , there is a bijection between the L-packet Πϕ (G0 ) associatedto ϕ and the set Irr(Sϕ,sc , ζG0 ) [Art06, Section 3]. We note that when G0 = G the character ζG0 is the trivial
character 1 so that
Irr(Sϕ,sc , 1) = Irr(Sϕ ).
b sc ) since Cφ /Z(G)b Γ is finite and Z(G)b ΓIn particular, if ϕ is elliptic, then we have Sϕ = Sϕ and Zbϕ,sc = Z(G◦contains Sφ [Kot84, Lemma 10.3.1]. Thus the exact sequence (3.4) turns out to be the same as (3.3).
4. Review of Results on Restriction
In this section, we review several results about restriction. We refer to [GK82, Lab85, Tad92, HS12] fordetails.
e be con4.1. Results of Gelbart-Knapp, Tadić, and Hiraga-Saito. For this section, we let G and G
nected, reductive, algebraic groups over F satisfying the property thate der ⊆ G ⊆ G,e
Gder = G(4.1)
where the subscript “der” stands for the derived group. Given σ ∈ Irr(G), by [GK82, Lemma 2.3] ande such that
[Tad92, Proposition 2.2], there exists σe ∈ Irr(G)σ ).
σ ,→ ResG
G (e
e

(4.2)

e
e ) to G(F ). It turns
In other words, σ is an irreducible constituent in the restriction ResGσ ) of σe from G(FG (eout, [Tad92, Proposition 2.4 & Corollary 2.5] and [GK82, Lemma 2.1], that Πσ (G) is finite and independente We write Πσ (G) = Πσe (G) for the set of equivalence classes of allof the choice of the lifting σe ∈ Irr(G).
eeGσ ). It is clear that for any irreducible constituents σ1 and σ2 in ResGσ ),irreducible constituents of ResG (eG (ewe have Πσ1 (G) = Πσ2 (G).
Remark 4.1. A member (equivalently all members) of Πσe (G) is supercuspidal, essentially square-integrable,or essentially tempered if and only if σ
e is (see [Tad92, Proposition 2.7]).
e is defined as
We recall that the stabilizer of σ in Gnoe σ := g̃ ∈ G(Fe ) : g̃ σ ∼
G
=σ .
e )/G
e σ acts by conjugation on the set Πσe (G) simply transitively (see [GK82, Lemma
The quotient of G(Fe )/Ge σ and Πσe (G).
2.1(c)]) and there is a bijection between G(FWe also recall the following useful result.

e
Proposition 4.2. ([GK82, Lemma 2.4], [Tad92, Corollary 2.5], and [HS12, Lemma 2.2]) Let σe1 , σe2 ∈ Irr(G).
The following statements are equivalent:e )/G(F ))D such that σ(1) There exists a character χ ∈ (G(Fe1 ∼e2 ⊗ χ;=σ(2) Πσe1 (G) ∩ Πσe2 (G) 6= ∅;(3) Πσe1 (G) = Πσe2 (G).
Since ResGσ ) is completely reducible by [GK82, Lemma 2.1] and [Tad92, Lemma 2.1], we have theG (edecompositionMeResGσ) = m
τ
(4.3)
G (e
e

τ ∈Πσ
e (G)

(see [HS12, Chapter 2]), where the positive integer m denotes the common multiplicity over τ ∈ Πσ (G)e we define[GK82, Lemma 2.1(b)]. Given σe ∈ Irr(G),noe )/G(F ))D : σI(eσ ) := χ ∈ (G(F
e⊗χ∼
e .
(4.4)
=σ

12

MAHDI ASGARI AND KWANGHO CHOIY

Considering the dimension of the C-vector space EndG (ResGσ )), we have (see [Cho14b, Proposition 3.1])
G (ethe equality
m2 · |Πσ (G)| = I(e
σ ).
(4.5)
∼
e ⊗ χ for χ ∈ I(eσ ), we have a non-zero endomorphism Iχ ∈ AutC (Vσe )Following [HS12, Chapter 2], since σe=σsuch thatIχ ◦ (eσ ⊗ χ) = σe ◦ Iχ .
×
For each z ∈ C , we denote by z · idVσe the scalar endomorphism ṽ 7→ z · ṽ for v ∈ Vσe . We identify C×with the subgroup of AutC (Vσe ) consisting of z · idVσe . Define A(eσ ) as the subgroup of AutC (Vσe ) generatedby {Iχ : χ ∈ I(eσ )} and C× . Then the map Iχ 7→ χ induces the exact sequencee1 −→ C× −→ A(eσ ) −→ I(eσ ) −→ 1.

(4.6)

We denote by Irr(A(eσ ), id) the set of isomorphism classes of irreducible smooth representations of the groupA(eσ ) such that z · idVσe ∈ C× acts as the scalar z. By [HS12, Corollary 2.10], we then have a decompositionMVσe ∼ξ  σξ
(4.7)=
ξ∈Irr(A(eσ ),id)as representations of the direct product A(eσ ) × G(F ). It follows that there is a one-to-one correspondence∼ Πσe (G),
Irr(A(eσ ), id) =(4.8)
sending ξ 7→ σξ . We denote by ξσ the preimage of σ via the correspondence (4.8).
4.2. A Theorem of Labesse. We recall a theorem of Labesse in [Lab85] which verifies the existence ofe be connected, reductive, algebraic
a lifting of a given L-parameter in the following setting. Let G and G
groups over F with an exact sequence of connected components of L-groups
be pr b1 −→ Sb −→ G−→ G −→ 1,bebewhere Sb is a central torus in G,
and the surjective homomorphism pr is compatible with Γ-actions on G
andbe
G. Then, Labesse proves in [Lab85, Théorèm 8.1] that for any ϕ ∈ Π(G), there exists ϕe ∈ Π(G) such thatϕ=ϕe ◦ pr.
We note that the analogous result has been proved in [Wei74, Hen80, GT10] for the case G = SLn and
e = GLn .
G
4.3. L-packets for Inner Forms of SLn . In this section we recall some results about the LLC for SLm (D)in [HS12, Chapter 12] and [ABPS14, Section 3]. Throughout this section, let G(F ) = SLm (D) and G∗ (F ) =SLn (F ), where D be a central division algebra of dimension d2 over F with n = md (possibly D = F, inwhich case d = 1).
b trivially, we shall use G
b = L G0 instead of L G = G
b × Γ. Note that G
b=Gc∗ = PGL(C).
Since Γ acts on GLetb
ϕ : WF × SL2 (C) → G
be an L–parameter. Note thatb sc ) = µn (C) and Z(G)b Γ = 1.
Z(GWith notation as in Section 3, we have the exact sequence
1 −→ Zbϕ,sc −→ Sϕ,sc −→ Sϕ −→ 1.

(4.9)

◦
In the case at hand, Zbϕ,sc = µn (C)/(µn (C) ∩ Sϕ,sc). In particular, when ϕ is elliptic, since both Sϕ,sc inSLn (C) and Sϕ in PGLn (C) are finite, the exact sequence (4.9) becomes
1 −→ µn (C) −→ Sϕ,sc −→ Sϕ −→ 1.
b sc ) which corresponds to the inner
Since G is an F –inner form of G∗ , we can fix a character ζG of Z(G∗form G of G via the Kottwitz isomorphism [Kot86, Theorem 1.2]. When D = F (i.e., G = G∗ ) we haveζG = 1.

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

13

Consider the exact sequenceprb −→ 1.
1 −→ C× −→ GLn (C) −→ PGLn (C) = Ge
By the argument in Section 4.2, we have an L-parameter ϕe for Gϕe : WF × SLn (C) → GLn (C)e [HS12], we have a unique irreducible
such that pr ◦ ϕ
e = ϕ (also see [Wei74, Hen80]). By the LLC for Ge associated to the L-parameter ϕrepresentation σe ∈ Irr(G)e and the L-packet Πϕ (G) equals the set Πσe (G)defined in Section 4.1. By [HS12, Lemma 12.6] and [ABPS14, Section 3], there is a homomorphism ΛSLn :Sϕ,sc → A(eσ ) (unique up to one-dimensional characters of Sϕ ) making the following diagram commute:bϕ,sc −−−−→ Sϕ,sc −−−−→ Sϕ −−−−→ 1
1 −−−−→ Z



∼
Λ
ζ
y=
y SLn
yG
1 −−−−→ C× −−−−→ A(eσ ) −−−−→ I(eσ ) −−−−→ 1.
Combining (4.7) and (4.10), we have the following decompositionMMρσ  σ =Vσe ∼ρ  σρ
=σ∈Πϕ (G)(4.10)(4.11)

ρ∈Irr(Sϕ,sc ,ζG )as representations of Sϕ,sc × G(F ). Here, ρσ ∈ Irr(Sϕ,sc , ζG ) is given by ξσ ◦ ΛSLn with ξσ ∈ Irr(A(eσ ), id),and σρ ∈ Πϕ (G) denotes the image of ρ ∈ Irr(Sϕ,sc , ζG ) via the bijection between Πϕ (G) and Irr(Sϕ,sc , ζG ).
5. Local Langlands Correspondence for GSpin4 and its inner forms
In this section we establish the LLC for GSpin4 and all its non quasi-split F -inner forms.
5.1. Construction of L-packets of GSpin4 and Its Inner Forms. It follows from the arguments in
Section 4 on restriction and the group structure (2.6) that given σ ∈ Irr(GSpin4 ), there is a lifting σe ∈Irr(GL2 × GL2 ) such thatGL2 ×GL2
σ ,→ ResGSpin
(e
σ ).
4
By the LLC for GLn [HT01, Hen00, Sch13], we have a unique ϕeσe ∈ Φ(GL2 × GL2 ) corresponding to therepresentation σe. We now define a map
L4 : Irr(GSpin4 ) −→σ7−→

Φ(GSpin4 )(5.1)

pr4 ◦ ϕeσe .

Note that L4 does not depend on the choice of the lifting σ
e. Indeed, if σ
e0 ∈ Irr(GL2 × GL2 ) is another lifting,0 ∼it follows from Proposition 4.2 and (2.20) that σe =σe ⊗ χ for some quasi-character χ on
(GL2 (F ) × GL2 (F ))/GSpin (F ) ∼= F ×.
4

Morover,F× ∼= H 1 (F, C× ),where C× is as in (2.26). The LLC for GL2 × GL2 maps σ
e0 to ϕeσe ⊗ χ (by abuse of notation, employing χfor both the quasi-character and its parameter via Local Class Field Theory). Since pr4 (ϕeσe ⊗ χ) = pr4 (ϕeσe )by (2.26), the map L4 is well-defined.
Moreover, we note that L4 is surjective. Indeed, by Labesse’s Theorem in Section 4.2, ϕ ∈ Φ(GSpin4 )can be lifted to some ϕe ∈ Φ(GL2 × GL2 ). We then obtain σe ∈ Irr(GL2 × GL2 ) via the LLC for GL2 × GL2 .
GL2 ×GL2Thus, any irreducible constituent in the restriction ResGSpin(eσ ) has the image ϕ via the map L4 .
4As expected, for each ϕ ∈ Φ(GSpin4 ), we define the L-packet Πϕ (GSpin4 ) as the set of all inequivalent
irreducible constituents of σenoΠϕ (GSpin ) := Πσe (GSpin ) = σ ,→ ResGL2 ×GL2 (eσ) / ∼
(5.2)
=,
4

4

GSpin4

14

MAHDI ASGARI AND KWANGHO CHOIY

where σe is the unique member in Πϕe(GL2 × GL2 ) and ϕe ∈ Φ(GL2 × GL2 ) is such that pr4 ◦ ϕ
e = ϕ. By the
LLC for GL2 and Proposition 4.2, the fiber does not depends on the choice of ϕ.
e
We define the L-packets for the non quasi-split inner forms similarly. Using the group structure describedin Section 2.2, given σ42,1 ∈ Irr(GSpin2,1
e42,1 ∈ Irr(GL2 × GL1 (D)) such that4 ), there is a lifting σ
GL ×GL (D)21
σ42,1 ,→ ResGSpin
2,1
4

(e
σ42,1 ).

Combining the LLC for GL2 and GL1 (D) [HS12], we have a unique ϕeσe2,1 ∈ Φ(GL2 × GL1 (D)) corresponding4

to the representation σe42,1 . We thus define the map

2,1
L2,1
4 : Irr(GSpin4 ) −→

σ42,1

7−→

Φ(GSpin2,1
4 )

(5.3)

pr4 ◦ ϕeσe2,1 .
4

Again, it follows from the LLC for GL2 and GL1 (D) that this map is well-defined and surjective.
Likewise, for the other F -inner form GSpin1,1
4 of GSpin4 , we have a well-defined and surjective map
1,1
L1,1
4 : Irr(GSpin4 ) −→

σ41,1

7−→

Φ(GSpin1,1
4 )pr4 ◦ ϕ
e

σ
e41,1

(5.4)

.

We similarly define L-packets
2,1Πϕ (GSpin2,1
4 ) = Πσe2,1 (GSpin4 ),ϕ ∈ Φ(GSpin2,1
4 )

(5.5)

1,1
Πϕ (GSpin1,1
4 ) = Πσe1,1 (GSpin4 ),ϕ ∈ Φ(GSpin1,1
4 ).

(5.6)

4

and
4

Again, these L-packet do not depend on the choice of ϕe for similar reasons.
5.2. Internal Structure of L-packets of GSpin4 and Its Inner Forms. In this section we continue toemploy the notation in Section 3. For simplicity of notation, we shall write GSpin] for the split GSpin4 ,1,1
and its non quasi-split F -inner forms GSpin2,1
4 and GSpin4 . Likewise, we shall write SL] and GL] for thecorresponding groups in (2.6), (2.11), and (2.12) so that we haveSL] ⊂ GSpin] ⊂ GL](5.7)

in all cases. From Section 2.4, we recall that\] )ad = PSO4 (C) ∼(GSpin= PGL2 (C) × PGL2 (C),\] )sc = Spin4 (C) ∼(GSpin= SL2 (C) × SL2 (C),\] )sc ) = Z((GSpin\] )sc )Γ ∼Z((GSpin= µ2 (C) × µ2 (C).
d] −→ GSpin\] (cf.
Let ϕ ∈ Φ(GSpin] ) be given. We fix a lifting ϕe ∈ Φ(GL] ) via the surjective map GL
Theorem 4.2). With notation as in Section 3, we haveSϕ ⊂ PSL2 (C) × PSL2 (C),Sϕ,sc ⊂ SL2 (C) × SL2 (C).
We then have (cf. (3.4)) a central extension
1 −→ Zbϕ,sc −→ Sϕ,sc −→ Sϕ −→ 1.

(5.8)\] )sc ) which respectively correspond to GSpin4 , GSpin2,1
Let ζ4 , ζ42,1 , and ζ41,1 be characters on Z((GSpin
4 ,1,1
and GSpin4 via the Kottwitz isomorphism [Kot86, Theorem 1.2]. Note that
ζ4 = 1,

ζ42,1 = 1 × sgn,

where sgn is the non-trivial character of µ2 (C).

and

ζ41,1 = sgn × sgn,

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

15

Theorem 5.1. Given an L-parameter ϕ ∈ Φ(GSpin] ), there is a one-to-one bijection
1−1Πϕ (GSpin] )←→Irr(Sϕ,sc , ζ] ),

σ

7→

ρσ ,

such that we have the following decomposition
Vσe ∼
=

M

ρσ  σσ∈Πϕ (GSpin] )as representations of the direct product Sϕ,sc × GSpin] (F ), where σe ∈ Πϕe(GL] ) is an extension of σ ∈Πϕ (GSpin] ) to GL] (F ) as in Section 4 and ϕe ∈ Φ(GL] ) is a lifting of ϕ ∈ Φ(GSpin] ). Here, ζ] ∈on
ζ4 , ζ42,1 , ζ41,1 according to which inner form GSpin] is.
Proof. We follow the ideas in Section 4.3 and [CL14, Theorem 5.4.1]. Given ϕ ∈ Φ(GSpin] ), we choose alifting ϕe ∈ Φ(GL] ) and obtain the projection ϕ̄ ∈ Φ(SL] ) in the following commutative diagram
3d] = GL2 (C) × GL2 (C)GLϕe

WF × SL2 (C)

ϕ

ϕ̄


/

(5.9)

pr4

\] = GSO4 (C)
GSpin

+

pr
¯

d] = PGL2 (C) × PGL2 (C) .
SL
We then have σe ∈ Πϕe(GL] ) which is an extension of σ ∈ Πϕ (GSpin] ). In addition to (2.26), we also have
pr◦pr
¯
4

1 −→ C× × C× −→ GL2 (C) × GL2 (C) −→ PGL2 (C) × PGL2 (C) −→ 1
Considering the kernels of the projections pr4 and pr¯ ◦ pr4 , we set
X GSpin] (ϕ)
e := {a ∈ H 1 (WF , C× ) : aϕ
e∼
e
= ϕ}
X SL] (ϕ)
e := {a ∈ H 1 (WF , C× × C× ) : aϕ
e∼
e
= ϕ}.
Moreover, by (2.9) and its analogues for the two non quasi-split F -inner forms, we have
∼ F ×.
GL] (F )/GSpin (F ) =]As an easy consequence of Galois cohomology, we also haveGL] (F )/SL] (F ) ∼= F × × F ×.
Set
I GSpin] (eσ ) := {χ ∈ (F × )D ∼eχ ∼e}
= (GL] (F )/GSpin] (F ))D : σ
=σ
SL]
× D
× D ∼
D
∼
I (e
σ ) := {χ ∈ (F ) × (F ) = (GL] (F )/SL] (F )) : σeχ = σe}.
Remark 5.2. Since any character on GLn (F ) (respectively, GLm (D)) is of the form χ ◦ det (respectively,χ ◦ Nrd) for some character χ on F × (see [BH06, Section 53.5]), we often make no distinction between χ andχ ◦ det (respectively, χ ◦ Nrd). Moreover, we note that χ ∈ (GL] (F ))D can be written as χe1  χ
e2 , where χeiwith i = 1, 2, is a character of GL2 (F ) or D× depending on the type of GL] .
It follows from the definitions that
X GSpin] (ϕ)
e ⊂ X SL] (ϕ)
eandI GSpin] (eσ ) ⊂ I SL] (eσ ).

(5.10)We have (F × )D ∼= H 1 (WF , C× ) by the local class field theory. It is also immediate from the LLC for GLn
that
X GSpin] (ϕ)
e ∼σ ) and X SL] (ϕ)
e ∼σ)= I GSpin] (e= I SL] (e16

MAHDI ASGARI AND KWANGHO CHOIY

as groups of characters. Now, with the component group notation S of Section 3, we claim that
I GSpin] (eσ) ∼= Sϕ .

(5.11)

Indeed, by [CL14, Lemma 5.3.4] and the above arguments, this claim follows fromSϕ ∼e
= X GSpin] (ϕ),
since Sϕe is always trivial by the LLC for GL] . Notice that ϕe and ϕ here are respectively φ and φ] in [CL14,
Lemma 5.3.4]. Thus, due to (4.10), (5.10), and(5.11) we haveSϕ ⊂ Sϕ̄ .

(5.12)c ] ) is equal to the image of the disjoint union
Since the centralizer Cϕ̄ (in SLah ∈ GSO4 (C) : hϕ(w)h−1 ϕ(w)−1 = ν(w)ν∈Hom(WF ,C× )from the exact sequencepr¯d] = PGL2 (C) × PGL2 (C) −→ 1,
1 −→ C× −→ GSO4 (C) −→ SL

we haveSϕ ⊂ Cϕ̄ = Sϕ̄ .
So, the pre-images in SL2 (C) × SL2 (C) via the isogeny SL2 (C) × SL2 (C)  PGL2 (C) × PGL2 (C) also satisfy
Sϕ,sc ⊂ Sϕ̄,sc ⊂ SL2 (C) × SL2 (C).
This provides the inclusion of the identity components
◦
◦
Sϕ,sc
⊂ Sϕ̄,sc,(5.13)b sc )/(Z(Gb sc ) ∩ S ◦ ) of Section 3 we have the surjection
and by the definition Zbϕ,sc := Z(Gϕ,scbϕ,sc  Zbϕ̄,sc .
Z(5.14)

Combining (5.12), (5.13), and (5.14), we have the following commutative diagram of component groups:
1 −−−−→ Zbϕ,sc −−−−→ Sϕ,sc −−−−→ Sϕ −−−−→ 1




∩
∩
y
y
y

(5.15)

1 −−−−→ Zbϕ̄,sc −−−−→ Sϕ̄,sc −−−−→ Sϕ̄ −−−−→ 1.
We apply the Hiraga-Saito homomorphism ΛSLn of (4.10) in the case of our SL] , which we denote byΛSL2 ×SL2 . The restrictionΛ] := ΛSL2 ×SL2 |Sϕ,scthen gives the following commutative diagram:
1 −−−−→ Zbϕ,sc −−−−→ Sϕ,sc −−−−→


ζ
Λ
y]
y ]

Sϕ

∩
y

−−−−→ 1
(5.16)

1 −−−−→ C× −−−−→ A(eσ ) −−−−→ I SL] (eσ ) −−−−→ 1.
Note that ζ] is identified with ζG in (4.10) as the character on µ2 (C) × µ2 (C) since both are determinedaccording to SL] .
Similarto A(eσ ) (cf. Section 4), we write AGSpin] (eσ ) for the subgroup of AutC (Vσe ) generated by C×GSpin]
GSpin]and Iχ : χ ∈ I(eσ ) . Hence, A(eσ ) ⊂ A(eσ ). By the definition of ΛSL2 ×SL2 in (4.10) and theLOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

17

commutative diagram (5.15), it is immediate that the image of Λ] is AGSpin] (eσ ). We thus have the following
commutative diagram:
1 −−−−→ Zbϕ,sc −−−−→
Sϕ,sc
−−−−→
Sϕ
−−−−→ 1



∼
Λ
ζ
(5.17)y=y ]y]1 −−−−→ C× −−−−→ AGSpin] (eσ ) −−−−→ I GSpin] (eσ ) −−−−→ 1.
The representation ρσ ∈ Irr(Sϕ,sc , ζ] ) is defined by ξσ ◦ Λ] , where ξσ ∈ Irr(AGSpin] (eσ ), id) is the character asin the decomposition (4.7). Therefore, arguments of Section 4, our construction of L-packets Πϕ (GSpin] ) in
Section 5.1 and diagram (5.17) above give
1−1

1−1

Irr(Sϕ,sc , ζ] ) ←→ Irr(AGSpin] (eσ ), id) ←→ Πϕ (GSpin] ).
We also have the following decomposition
M
Vσe ∼
=

M

ρσ  σ =σ∈Πϕ (GSpin] )(5.18)ρ  σρ ,ρ∈Irr(Sϕ,sc ,ζ] )where σρ denotes the image of ρ via the bijection between Πϕ (GSpin] ) and Irr(Sϕ,sc , ζ] ). Hence, the proofof Theorem 5.1 is complete.

Remark 5.3. Similar to ΛSL2 ×SL2 in Section 4, since Λ] is unique up to Hom(I GSpin] (eσ ), C× ) ∼= Hom(Sϕ , C× ),the same is true for the bijection (5.18).
Remark 5.4. Given ϕ ∈ Φ(GSpin4 ), by Theorem 5.1, we have a one-to-one bijection
1−1

1,2
1,1
Πϕ (GSpin4 ) ∪ Πϕ (GSpin2,1
4 ) ∪ Πϕ (GSpin4 ) ∪ Πϕ (GSpin4 ) ←→ Irr(Sϕ,sc ),2,1∼
where GSpin1,2
4 = {(g1 , g2 ) ∈ (GL1 (D) × GL2 ) : Nrd(g1 ) = det g2 }, which is isomorphic to GSpin4 .

5.3. L-packet Sizes for GSpin4 and Its Inner Forms. In this subsection, we describe the group I GSpin4 (eσ)
for σe ∈ Irr(GL2 × GL2 ) case by case, discuss the group structure of Sϕ,sc , and provide all the sizes of L1,1
packets for GSpin4 , GSpin2,1
using Theorem 5.1. Let us first give the possible cardinalities
4 , and GSpin4for the L-packets of GSpin4 and its inner forms using Galois cohomology.
Proposition 5.5. Let Πϕ (GSpin] ) be an L-packet associated to ϕ ∈ Φ(GSpin] ). Then we haveΠϕ (GSpin] )F × /(F × )2 ,which yields the possible cardinalitiesΠϕ (GSpin] ) =1, 2, 4,
1, 2, 4, 8,

if p 6= 2,
if p = 2.

Proof. We follow the idea of a similar result in the case of Sp4 in [CG15]. The exact sequence of algebraic
groups1 −→ Z(GSpin] ) −→ Z(GL] ) × GSpin] −→ GL] −→ 1,
where the second map is given by multiplication (considering GSpin] ⊂ GL] ), gives a long exact sequence· · · −→ (F × × F × ) × GSpin] (F ) −→ GL] (F ) −→ H 1 (F, Z(GSpin] )) −→ H 1 (F, Z(GL] ) × GSpin] ) −→ 1.
Since H 1 (F, Z(GL] ) × GSpin] ) = 1 by Lemma 2.2 and [PR94, Lemmas 2.8], we haveGL] (F )/ (F × × F × ) × GSpin] (F ) ,→ H 1 (F, Z(GSpin] )).
Also, the exact sequence
1 −→ Z(GSpin] )◦ −→ Z(GSpin] ) −→ π0 (Z(GSpin] )) −→ 1gives
H 1 (F, Z(GSpin] )) ,→ H 1 (F, π0 (Z(GSpin] ))),

18

MAHDI ASGARI AND KWANGHO CHOIY

since, by [AS06, Proposition 2.3], Z(GSpin] )◦ ∼= GL1 and H 1 (F, Z(GSpin] )◦ ) = 1. Combining the above
with (2.19), we haveGL] (F )/ (F × × F × ) × GSpin] (F ) ,→ H 1 (F, π0 (Z(GSpin] ))) ∼(5.19)= H 1 (F, Z/2Z) ∼= F × /(F × )2 .
On the other hand, we know from Section 4 that1−1Πϕ (GSpin] ) ←→ GL] (F )/GL] (F )σe ,→ GL] (F )/ (F × × F × ) × GSpin] (F ) ,where σe ∈ Irr(GL] ) corresponds to a lifting ϕe ∈ Φ(GL] ), via the LLC for GLn and its inner forms, ofϕ ∈ Φ(GSpin] ). Thus the proof is complete.

In what follows, we give an explicit description of the group I GSpin4 (eσ ) for σe ∈ Irr(GL2 × GL2 ) case bycase, and show that among the possible cardinalities in Proposition 5.5, only 1, 2, and 4 do indeed occur (seeRemarks 5.9 - 5.11 and (5.21) - (5.22)). To this end, we verify that I GSpin4 (eσ) ∼= 1, Z/2Z, or (Z/2Z)2 , which
2∼
yields Sϕ = 1, Z/2Z, or (Z/2Z) by (5.11). Thus, due to Theorem 5.1, the size of an L-packet for GSpin4 iseither 1, 2, or 4. Moreover, we show that the central extension Sϕ,sc is isomorphic to (Z/2Z)2 , (Z/2Z)3 , anabelian subgroup of order 16 in Q8 × Q8 , or Q8 × Z/2Z, for irreducible ϕ ∈ Φ(GSpin4 ), and is always abelian
for reducible ϕ ∈ Φ(GSpin4 ). Here Q8 is the quaternion group of order 8. Due to Theorem 5.1, these facts
about Sϕ,sc imply that the size of an L-packet for non-split inner forms GSpin42,1 and GSpin1,1
4 is either 1,
2, or 4.
To describe the group I GSpin4 (eσ ), given σe ∈ Irr(GL2 × GL2 ), we set σe=σe1  σe2 with σ
e1 , σe2 ∈ Irr(GL2 ).
Due to Remark 5.2, we note that χ ∈ (GL2 (F ) × GL2 (F ))D is decomposed into χ
e1  χ
e2 , where χ
ei withi = 1, 2 is a character of GL2 (F ), and we identify χ
ei and χ
ei ◦ det, since any character on GLn (F ) is of theform χe ◦ det for some character χ
e on F × .
Lemma 5.6. Let χ ∈ I GSpin4 (eσ ) be given. Then, χ is of the form
χ
eχ
e−1 ,
where χe ∈ (F × )D .
Proof. Since χ = χ
e1  χ
e2 as above and χ is trivial on GSpin4 (F ), by the structure of GSpin4 (F ) in (2.21),
we haveχ(g1 , g2 ) = χe1 (det g1 )  χe2 (det g2 ) = χe1 (det g1 )eχ2 (det g2 ) = 1
for all (g1 , g2 ) ∈ GL2 (F ) × GL2 (F ) with det g1 = det g2 . Since the determinant map det : GL2 (F ) → F × issurjective, χe1 χe2 (x) must be trivial for all x ∈ F × . Thus, χe1 and χ
e2 are inverse each other.
Proposition 5.7. We have
I GSpin4 (eσ) =I SL2 (e
σ1 ),
if σ
e2 ∼
e1 ηe for some ηe ∈ (F × )D ;
=σ
SL2
SL2
∼
I (e
σ1 ) ∩ I (eσ2 ), if σe2 =6 σe1 ηe for any ηe ∈ (F × )D .

Proof. Since σe=σe1  σe2 with σ
e1 , σe2 ∈ Irr(GL2 ), and by Lemma 5.6, we have∼σeχ = σe ⇐⇒ σ
e1 χ
eσ
e2 χ
e−1 ∼
e1  σe2 ⇐⇒ σei χe∼ei , i = 1, 2.
=σ=σ(5.20)
× D∼
This shows that I(eσ ) = I (eσ1 ) ∩ I (eσ2 ). In particular, if σ
e2 = σe1 ηe for some ηe ∈ (F ) , thenI SL2 (eσ1 ) = I SL2 (eσ2 ). Thus we have, by (5.20), that I GSpin4 (eσ ) = I SL2 (eσ1 ).
GSpin4SL2

SL2Remark 5.8. By the LLC for SL2 ([GK82]), we have I SL2 (eσi ) ∼= π0 (Cϕi ) for ϕi ∈ Φ(SL2 ) correspondingGL2∼
to σi ⊂ ResSL2 (eσi ). Recalling that PGL2 (C) = SO3 (C), it then follows from [GP92, Corollary 6.6] thatI SL2 (eσ1 ) and I SL2 (eσ2 ) consist of quadratic characters. Then, I GSpin4 (eσ ) consists of quadratic characters of×F .
Given ϕ ∈ Φ(GSpin4 ), fix the liftϕe=ϕe1 ⊗ ϕe2 ∈ Φ(GL2 × GL2 )with ϕei ∈ Φ(GL2 ) such that ϕ = pr4 ◦ ϕ,e as in Section 5.1. Letσe=σe1  σe2 ∈ Πϕe(GL2 × GL2 )LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

19

be the unique member such that {eσi } = Πϕei (GL2 ). Recall (see [GT10, Section 5]) that an irreducible LF
parameter φ ∈ Φ(GL2 ) is called primitive if the restriction φ|WF is not of the form IndW
WE θ for some finite
extension E over F. Moreover, φ is called dihedral with respect to (w.r.t.) a quadratic extension E over FF∼∼
if φ|WF ∼= IndW
WE θ or equivalently if (φ|WF ) ⊗ ωE/F = (φ|WF ) (⇔ φ ⊗ ωE/F = φ) for a quadratic characterωE/F corresponding to quadratic E/F via the local class field theory. A primitive representation exists only
when p divides dim φ ([Koc77]). We can now make the following remarks.
Remark 5.9. If σe2 ∼e1 ηe for some ηe ∈ (F × )D , we know that ϕe1 is dihedral w.r.t. one quadratic character=σ(respectively, primitive, dihedral w.r.t. three quadratic characters) if and only if ϕe2 is dihedral (respectively,
primitive, dihedral w.r.t. three quadratic characters).
Remark 5.10. Let ϕ ∈ Φ(GSpin4 ) be irreducible. Then ϕ,e ϕe1 , and ϕe2 are all irreducible. Combining
Proposition 5.7, Remarks 5.8 and 5.9, and [GT10, Porposition 6.3], we conclude the following. Whenσe2 ∼e1 ηe for some ηe ∈ (F × )D , we have
=σ

if ϕ
e1 is primitive or non-trivial on SL2 (C) (hence so is ϕe2 );
 {1},
Z/2Z,
if ϕ
e1 is dihedral w.r.t. one quadratic extension (hence so is ϕe2 );I GSpin4 (eσ) ∼=(Z/2Z)2 , if ϕ
e1 is dihedral w.r.t. three quadratic extensions (hence so is ϕe2 ).
When σe2 ∼6 σe1 ηe for any ηe ∈ (F × )D , an analogous assertion holds, but it would depend on the individual=
parameters and not just whether they are primitive or dihedral. In that case, we have
I GSpin4 (eσ) ∼= {1}, Z/2Z, or (Z/2Z)2 .
Remark 5.11. When ϕei is reducible, σei is either the Steinberg representation twisted by a character oran irreducibly induced representation from the Borel subgroup of GL2 . Since the number of irreducible2σi ) is ≤ 2, we have I SL2 (eσi ) ∼
constituents in ResGL= {1}, or Z/2Z. This implies, by Proposition 5.7, that
SL2 (eI GSpin4 (eσ) ∼= {1}, or Z/2Z.
We now describe the central extension Sϕ,sc for GSpin4 . For irreducible ϕ ∈ Φ(GSpin4 ), from (5.11) andRemark 5.10, we have Sϕ ∼= 1, Z/2Z, or (Z/2Z)2 . Since ϕ is elliptic and Zbϕ,sc = Z(Spin4 (C)) = µ2 (C)×µ2 (C),the exact sequence (5.8) becomes
1 −→ µ2 (C) × µ2 (C) −→ Sϕ,sc ⊂ Spin4 (C) = SL2 (C) × SL2 (C) −→ Sϕ ⊂ PSL2 (C) × PSL2 (C) −→ 1.
When Sϕ ∼= Z/2Z, since (Z/2Z)2 is embedded in= (Z/2Z)2 . When Sϕ ∼= 1, it is obvious that Sϕ,sc ∼Z(Sϕ,sc ), Sϕ,sc is an abelian group of order 8. Since Sϕ,sc ⊂ Sϕ̄,sc from (5.15), using the fact that the groupSϕ̄,sc ∼= (Z/2Z)4 (see the proof of [Cho14b, Proposition 6.5]), we have Sϕ,sc ∼= (Z/2Z)3 . When Sϕ ∼= (Z/2Z)2 ,∼since Sϕ̄,sc = Q8 × Q8 from loc. cit., Sϕ,sc is a subgroup of order 16 in Q8 × Q8 . If Sϕ,sc is non-abelian, sinceZ(Sϕ,sc ) ∼= (Z/2Z)2 , we have Sϕ,sc ∼= Q8 × Z/2Z. It thus follows that Sϕ,sc is an abelian subgroup in Q8 × Q8of order 16, or Q8 × Z/2Z. For reducible ϕ ∈ Φ(GSpin4 ), (5.11) and Remark 5.11 provide Sϕ ∼= 1 or Z/2Z.
Since (Z/2Z)2 is embedded in Z(Sϕ,sc ), Sϕ,sc must be abelian of order 8.
Using Theorem 5.1, we describe the sizes of L-packets for GSpin4 , GSpin42,1 , and GSpin41,1 . Form theprevious paragraph, we note that Sϕ,sc is either abelian or isomorphic to Q8 × Z/2Z. Thus, we have|Πϕ (GSpin4 )| = Πϕ (GSpin42,1 ) = Πϕ (GSpin1,1
4 ) = |Sϕ | = 1, 2, or 4,(5.21)

according to whether Sϕ ∼= {1}, Z/2Z, or (Z/2Z)2 , except when Sϕ,sc ∼= Q8 × Z/2Z, in which case we have1,1|Πϕ (GSpin4 )| = 4, Πϕ (GSpin2,1
4 ) = Πϕ (GSpin4 ) = 1.

(5.22)

Furthermore, using (4.5), the multiplicity in restriction from GL] to GSpin] is 1 except when Sϕ,sc ∼=Q8 × Z/2Z, in which case the multiplicity is 2.
We should mention that, due to Remarks 5.10 and 5.11, the case of Sϕ,sc ∼= Q8 × Z/2Z occurs only if two
conditions hold : σe2 ∼e1 ηe for some ηe ∈ (F × )D , and ϕe1 is dihedral w.r.t. three quadratic extensions (hence=σso is ϕe2 ).

20

MAHDI ASGARI AND KWANGHO CHOIY

5.4. Properties of L-maps for GSpin4 and its inner forms. The L-maps defined in Section
5.1 satisfynoa number of natural and expected properties which we now verify. In what follows, let L] ∈ L4 , L42,1 , L1,1
4
as the case may be.
Proposition 5.12. A representation σ] ∈ Irr(GSpin] ) is essentially square integrable if and only if its
L-parameter ϕσ] := L] (σ] ) does not factor through any proper Levi subgroup of GSO4 (C).
GL]
Proof. By the definition of L] , the representation σ] is an irreducible constituent of the restriction ResGSpin
(e
σ] )
]
for some σe] ∈ Irr(GL] ). As recalled in Remark 4.1, σ] is essentially square integrable representation if andonly if σe] is so. By the LLC for GL] and its inner forms [HT01, Hen00, Sch13, HS12], this is the case if andonly if its parameter ϕeσ] := L(eσ] ) does not factor through any proper Levi subgroup of GL] (C). Finally,
this is the case if and only if ϕσ] does not because the projection pr4 in (2.26) respects Levi subgroups. Remark 5.13. In the same way as in the proof of Proposition 5.12, we have that a given σ] ∈ Irr(GSpin] ) istempered if and only if the image of its L-parameter ϕσ] := L] (σ] ) in GSO4 (C) is bounded.
Due to the fact that restriction of representations preserves the intertwining operator and the Plancherel
measure (see [Cho14a, Section 2.2]), our construction of the L-packets in Section 5.1 gives the following
result.
Proposition 5.14. Let ϕ ∈ Φdisc (GSpin] ) be given. For any σ1 , σ2 ∈ Πϕ (GSpin] ), we have the equality of
the Plancherel measuresµM (ν, τ  σ1 , w) = µM (ν, τ  σ2 , w),(5.23)
where M is an F -Levi subgroup of an F -inner form of GSpin2n of the form of the product of GSpin] andcopies of F -inner forms of GLmi , τ  σ1 , τ  σ2 ∈ Πdisc (M ), ν ∈ a∗M,C , and w ∈ WM with w M = M.
Further, it is a consequence of the equality of the Plancherel measures that the Plancherel measure is alsopreserved between F -inner forms in the following sense. Let GSpin0] be an F -inner form of GSpin] . Givenϕ ∈ Φdisc (GSpin] ), for any σ ∈ Πϕ (GSpin] ) and σ 0 ∈ Πϕ (GSpin0] ), we haveµM (ν, τ  σ, w) = µM 0 (ν, τ 0  σ 0 , w),000(5.24)
00

where M is an F -inner form of M, τ  σ ∈ Πdisc (M ), τ  σ ∈ Πdisc (M ), and τ and τ have the same
L-parameter.
Proof. Since σ1 , σ2 ∈ Πϕ (GSpin] ) are in the same restriction from an irreducible representation from GL] (F )to GSpin] (F ), by [Cho14a, Proposition 2.4], we have (5.23). Similarly, we note that σ and σ 0 have liftings σee and τ 0  σ
e0 have the same Plancherel measures byand σ
e0 in GL] (F ) and GL0] (F ), respectively. Since τ  σ[AC89, Lemma 2.1], again by [Cho14a, Proposition 2.4], we have (5.24).
Finally, we make a remark about the generic representations in L-packets.
Remark 5.15. Consider the case when GSpin] is the split GSpin4 . Since ζ] = 1 now, Theorem 5.1 implies
that we have
1−1Πϕ (GSpin4 ) ←→ Irr(Sϕ ) ∼σ )).
(5.25)= Irr(I GSpin4 (eSuppose that there is a generic representation in Πϕ (GSpin4 ) with respect to a given Whittaker data for
GSpin4 . Then, by [HS12, Chapter 3], we can normalize the bijection (5.25) so that the trivial character1 ∈ Irr(Sϕ ) maps to the generic representation in Πϕ (GSpin6 ).
5.5. Preservation of Local Factors via L4 . We verify that in the case of the split form GSpin4 the map
L4 preserves the local L-, -, and γ-factors when these local factors are defined. On the representation side,these factors are defined via the Langlands-Shahidi method when the representations are generic (or nonsupercuspidal induced from generic via the Langlands classification). On the parameter side, these factors
are Artin factors associated to the representations of the Weil-Deligne group of F.
Since the Langlands-Shahidi method is available for generic data, we do not yet have a counterpart for
the local factors for the non quasi-split F -inner forms of GSpin4 . This is the reason why we are limiting
ourselves to the case of the split form below; however, cf. Remark 5.17 below.

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

21

Proposition 5.16. Let τ be an irreducible admissible representation of GLr (F ), r ≥ 1, and let σ be an irreducible admissible representation of GSpin4 (F ), which we assume to be either ψ-generic or non-supercuspidal
if r > 1. Here, generic is defined with respect to a non-trivial additive character ψ of F in the usual way.
Let ϕτ be the L-parameter of τ via the LLC for GLr (F ) and let ϕσ = L4 (σ). Then,γ(s, τ × σ, ψ)= γ(s, ϕτ ⊗ ϕσ , ψ),L(s, τ × σ)

= L(s, ϕτ ⊗ ϕσ ),(s, τ × σ, ψ)= (s, ϕτ ⊗ ϕσ , ψ).

The local factors on the left hand side are those attached by Shahidi [Sha90b, Theorem 3.5] to the ψ-genericrepresentations of the standard Levi subgroup GLr (F ) × GSpin4 (F ) in GSpin2r+4 (F ) and the standard representation of the dual Levi GLr (C) × GSO(4, C), and extended to all non-generic non-supercuspidal representations via the Langlands classification and the multiplicativity of the local factors [Sha90b, §9]. The factors
on the right hand side are Artin local factors associated to the given representations of the Weil-Delignegroup of F.
Proof. The proof relies on two key properties of the local factors defined by Shahidi, namely that they are
multiplicative (with respect to parabolic induction) and that they are preserved under taking irreducible
constituents upon restriction.
e Mf) be a pair of ambient groups and standard Levi subgroups as in
To be more precise, let (G, M ) and (G,e satisfy (4.1). Moreover, assume that M = G ∩ M
f.
the Langlands-Shahidi machinery. Assume that G and Gf
If σ and σ
e are ψ-generic representations of M (F ) and M (F ), respectively, such thatf(F )M

σ ,→ ResM (F ) (e
σ ),
then
γ(s, σ, r, ψ) = γ(s, σe, r̃, ψ),and similarly for the L- and -factors. Here, by r we denote any of the irreducible constituents of the adjoint
action of the complex dual of M on the Lie algebra of the dual of the unipotent radical of the standard
parabolic of G having M as a Levi. Also, r̃ denotes the corresponding irreducible constituent in the complex
e Below we only need the case where r and r̃ are standard representations.
dual of G.
For a precise description of the multiplicativity property in general we refer to [Sha90a] and to [Asg02,§5] for the specific case of GSpin groups.
Getting back to the proof, if τ σ is non-generic and non-supercuspidal, then it is a quotient of a standard
GL ×GSpin4
module of an induced representation IndP 4(π), where π is an essentially tempered representation of
the standard Levi subgroup of P. However, such a standard Levi only involves GL factors in our case andhence π is generic. As a result, we may define the local factors associated with τ  σ via multiplicativity.
By the above arguments we are reduced to the case where τ and σ are generic supercuspidal. The proof
now follows essentially from the LLC for the general linear groups and (5.9) because
γ(s, τ × σ, ψ)=γ(s, τ × σ
e, ψ)

=

γ(s, ϕτ ⊗ ϕ̃σe , ψ)=γ(s, ϕτ ⊗ (pr4 ◦ ϕ̃σe ), ψ)= γ(s, ϕτ ⊗ ϕσ , ψ),and similarly for the L- and -factors.

Remark 5.17. While we can not make any statement regarding the case of general (non-generic) representations of GSpin4 (F ) or other inner forms due to the current lack of a general satisfactory theory of local
factors in this generality. Our L-packets would satisfy properties analogous to the above if such a satisfactory
local theory becomes available. By satisfactory we refer to local factors that satisfy the natural properties
expected of them such as the so-called “Ten Commandments” for the γ-factors as in [LR05, Theorem 4].

22

MAHDI ASGARI AND KWANGHO CHOIY

6. Local Langlands Correspondence for GSpin6 and its inner forms
In this final section, we establish the LLC for GSpin6 and its non quasi-split F -inner forms. The method
is similar to the case of GSpin4 we presented in Section 5, but with somewhat weaker results on cardinalitiesof L-packets.
6.1. Construction of L-packets of GSpin6 and its inner forms. The discussions on restriction in Section
4 along with the description of the structure of GSpin6 in (2.7) imply that given σ ∈ Irr(GSpin6 ), there is a
lifting σe ∈ Irr(GL1 × GL4 ) such that
GL1 ×GL4
σ ,→ ResGSpin
(e
σ ).
6
By the LLC for GLn [HT01, Hen00, Sch13], we have a unique ϕeσe ∈ Φ(GL1 × GL4 ) corresponding to therepresentation σe.
We now define a map
L6 : Irr(GSpin6 ) −→σ7−→

Φ(GSpin6 )(6.1)

pr6 ◦ ϕeσe .

Note that L4 does not depend on the choice of the lifting σe because if σ
e0 ∈ Irr(GL1 × GL4 ) is another lifting,0 ∼e ⊗ χ for some quasi-character χ on
we have σe =σ(GL1 (F ) × GL4 (F ))/GSpin (F ) ∼= F ×.
6

As before,F× ∼= H 1 (F, C× ),where C× is as in (2.27). The LLC for GL1 × GL4 maps σ
e0 to ϕeσe ⊗ χ, again employing χ for both thequasi-character and its parameter via Local Class Field Theory. Since pr6 (ϕeσe ⊗ χ) = pr6 (ϕeσe ) by (2.27), themap L6 turns out to be well-defined.
As before, L6 is surjective because by Labesse’s Theorem in Section 4.2, ϕ ∈ Φ(GSpin6 ) can be liftedto some ϕe ∈ Φ(GL1 × GL4 ). We then obtain σe ∈ Irr(GL1 × GL4 ) via the LLC for GL1 × GL4 . Thus, any
1 ×GL4
irreducible constituent in the restriction ResGL(eσ ) has the image ϕ via the map L6 .
GSpin6For each ϕ ∈ Φ(GSpin6 ), we define the L-packet Πϕ (GSpin6 ) as the set of all inequivalent irreducible
constituents of σenoGL1 ×GL4
Πϕ (GSpin6 ) := Πσe (GSpin6 ) = σ ,→ ResGSpin
(e
σ
)
/∼
(6.2)
=,
6
where σe is the unique member in Πϕe(GL1 × GL4 ) and ϕe ∈ Φ(GL1 × GL4 ) is such that pr6 ◦ ϕ
e = ϕ. By the
LLC for GL4 and Proposition 4.2, the fiber does not depends on the choice of ϕ.
e
We define the L-packets for the non quasi-split inner forms similarly. Using the group structure describedin Section 2.2, given σ62,0 ∈ Irr(GSpin2,0
e62,0 ∈ Irr(GL1 × GL2 (D)) such that6 ), there is a lifting σ
GL ×GL (D)12
σ62,0 ,→ ResGSpin
2,0
6

(e
σ62,0 ).

Again, combining the LLC for GL4 and GL2 (D) [HS12], we have a unique ϕeσe2,0 ∈ Φ(GL1 × GL2 (D))
corresponding to the representation σ
e62,0 . We thus define the following map
2,0L2,06 : Irr(GSpin6 ) −→

σ62,0

7−→

6

Φ(GSpin2,0
6 )

(6.3)

pr6 ◦ ϕeσe2,0 .
6

Again, it follows from the LLC for GL1 and GL2 (D) that this map is well-defined and surjective.
Likewise, for the other F -inner form GSpin1,0
6 of GSpin6 , we have a well-defined and surjective map
1,0L1,06 : Irr(GSpin6 ) −→

σ61,0

7−→

Φ(GSpin1,0
6 )

(6.4)

pr6 ◦ ϕeσe1,0 .
6

Again, we similarly define L-packets
2,0Πϕ (GSpin2,0
6 ) = Πσe2,0 (GSpin6 ),6

ϕ ∈ Φ(GSpin2,0
6 )

(6.5)

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

23

and
1,0
Πϕ (GSpin1,0
6 ) = Πσe1,0 (GSpin6 ),6

ϕ ∈ Φ(GSpin1,0
6 ).

(6.6)

These L-packet do not depend on the choice of ϕe for similar reasons.
6.2. Internal structure of L-packets of GSpin6 and its inner forms. We continue to employ thenotation of Section 3 in this section. For simplicity of notation, we shall write GSpin[ for the split GSpin6 ,and its non quasi-split F -inner forms GSpin62,0 and GSpin1,0
6 . Likewise, we shall write SL[ and GL[ forcorresponding groups in (2.7), (2.13), and (2.14) so that we haveSL[ ⊂ GSpin[ ⊂ GL[(6.7)

in all cases. Recall from Section 2.4 that\[ )ad = PSO6 (C) ∼(GSpin= PGL4 (C),\[ )sc = Spin6 (C) ∼(GSpin= SL4 (C),\[ )sc ) = Z((GSpin\[ )sc )Γ ∼Z((GSpin= µ4 (C).
d[ −→ GSpin\[ (cf.
Let ϕ ∈ Φ(GSpin[ ) be given. We fix a lifting ϕe ∈ Φ(GL[ ) via the surjective map GL
Theorem 4.2). We haveSϕ ⊂ PSL4 (C),Sϕ,sc ⊂ SL4 (C).
We then have (again by (3.4)) a central extension
1 −→ Zbϕ,sc −→ Sϕ,sc −→ Sϕ −→ 1.

(6.8)\[ )sc ) which respectively correspond to GSpin6 , GSpin2,0Let ζ6 , ζ62,0 , and ζ61,0 be characters on Z((GSpin
6 ,1,0
and GSpin6 via the Kottwitz isomorphism [Kot86, Theorem 1.2]. Note that
ζ6 = 1 ,ζ62,0 = sgn,

and

ζ61,0 = ζ̂4 ,where sgn is the non-trivial character of order 2 on µ4 (C) and ζ̂4 is the non-trivial character of order 4 on
µ4 (C) whose restriction to µ2 equals sgn .
Theorem 6.1. Given an L-parameter ϕ ∈ Φ(GSpin[ ), there is a one-one bijection
1−1Πϕ (GSpin[ )←→Irr(Sϕ,sc , ζ[ ),

σ

7→

ρσ ,

such that we have the following decomposition
Vσe ∼
=

M

ρσ  σσ∈Πϕ (GSpin[ )as representations of the direct product Sϕ,sc × GSpin[ (F ), where σe ∈ Πϕe(GL[ ) is an extension of σ ∈Π(GSpin
)
to
GL
(F
)
as
in
Section
4
and
ϕ
e
∈
Φ(GL
)
is
a
liftingof ϕ ∈ Φ(GSpin[ ). Here, ζ[ ∈
[
[
[ o
nϕ
ζ6 , ζ62,0 , ζ61,0

according to which inner form GSpin[ is.

24

MAHDI ASGARI AND KWANGHO CHOIY

Proof. The idea of the proof is as in the proof of Theorem 5.1. Given ϕ ∈ Φ(GSpin[ ), we choose a liftingϕe ∈ Φ(GL[ ) and obtain the projection ϕ̄ ∈ Φ(SL[ ) in the following commutative diagram4

d[ = GL1 (C) × GL4 (C)GLϕe

WF × SL2 (C)

/

ϕ



pr6

\[ = GSO6 (C)
GSpin
pr
¯


d
SL[ = PGL4 (C) .
*

ϕ̄

(6.9)

We then have σe ∈ Πϕe(GL[ ) which is an extension of σ ∈ Πϕ (GSpin[ ). In addition to (2.27), we also have
pr◦pr
¯
6

1 −→ C× × C× −→ GL1 (C) × GL4 (C) −→ PGL4 (C) −→ 1
Considering the kernels of the projections pr6 and pr¯ ◦ pr6 , we set
X GSpin[ (ϕ)
e := {a ∈ H 1 (WF , C× ) : aϕ
e∼
e
= ϕ}
SL[
1
×
×
X (ϕ)
e := {a ∈ H (WF , C × C ) : aϕ
e∼
e
= ϕ}.
Moreover, by (2.10) and its analogs for the two non quasi-split F -inner forms, we haveGL[ (F )/GSpin (F ) ∼= F ×.
[As an easy consequence of Galois cohomology, we also have
∼ F × × F ×.
GL[ (F )/SL[ (F ) =Set∼σI GSpin[ (eσ ) := {χ ∈ (F × )D ∼eχ =e}
= (GL[ (F )/GSpin[ (F ))D : σ
SL[
× D
× D ∼
D
∼
I (e
σ ) := {χ ∈ (F ) × (F ) = (GL[ (F )/SL[ (F )) : σeχ = σe}.
As in Remark 5.2, we often make no distinction between χ and χ ◦ det (respectively, χ ◦ Nrd). It follows
from the definitions that
X GSpin[ (ϕ)
e ⊂ X SL[ (ϕ)
e× D ∼Recalling (F )1andI GSpin[ (eσ ) ⊂ I SL[ (eσ ).

(6.10)

×= H (WF , C ) by the local class field theory, it is immediate from the LLC for GLn that
X GSpin[ (ϕ)
e ∼σ ) and X SL[ (ϕ)
e ∼σ)= I GSpin[ (e= I SL[ (eas groups of characters. Now, with the component group notation S of Section 3, we claim that
I GSpin[ (eσ) ∼= Sϕ .

(6.11)

Again, by [CL14, Lemma 5.3.4] and above arguments, this claim follows from
∼ X GSpin[ (ϕ),
Sϕ =esince Sϕe is always trivial by the LLC for GL[ . Again note that ϕe and ϕ here are respectively φ and φ] in[CL14, Lemma 5.3.4]. Thus, due to (4.10), (6.10), and(6.11) we haveSϕ ⊂ Sϕ̄ .
d[ ) is equal to the image of the disjoint union
Since the centralizer Cϕ̄ (SLah ∈ GSO6 (C) : hϕ(w)h−1 ϕ(w)−1 = ν(w)ν∈Hom(WF ,C× )from the exact sequencepr¯d[ = PGL4 −→ 1,
1 −→ C× −→ GSO6 (C) −→ SL
we haveSϕ ⊂ Cϕ̄ = Sϕ̄ .

(6.12)

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

25

So, the pre-images in SL4 (C) via the isogeny SL4 (C)  PGL4 (C) also satisfy
Sϕ,sc ⊂ Sϕ̄,sc ⊂ SL4 (C).
This provides the inclusion of the identity components
◦
◦
Sϕ,sc
⊂ Sϕ̄,sc,(6.13)

bϕ,sc (G) of Section 3, we again have the following surjection
and by the definition of Z
bϕ,sc  Z
bϕ,sc (SL[ ).
Z(6.14)

Combining (6.12), (6.13), and (6.14), we again have the commutative diagram of component groups:
1 −−−−→ Zbϕ,sc −−−−→ Sϕ,sc −−−−→ Sϕ −−−−→ 1



∩
∩

y
y
y

(6.15)

1 −−−−→ Zbϕ̄,sc −−−−→ Sϕ̄,sc −−−−→ Sϕ̄ −−−−→ 1.
Now apply Hiraga-Saito’s homomorphism ΛSLn of (4.10) in the case of our SL[ , which we denote by ΛSL4 .
The restriction
Λ[ := ΛSL4 |Sϕ,scthen gives the following commutative diagram:
1 −−−−→ Zbϕ,sc −−−−→ Sϕ,sc −−−−→


ζ
Λ
y[
y [

Sϕ

∩
y

−−−−→ 1
(6.16)

1 −−−−→ C× −−−−→ A(eσ ) −−−−→ I SL[ (eσ ) −−−−→ 1.
Note that ζ[ is identified with ζG in (4.10) as the character on µ4 (C), since both are determined according
to SL[ .
Again, similarto A(eσ ) (cf. Section 4), we write AGSpin[ (eσ ) for the subgroup of AutC (Vσe ) generated
×
GSpin[by C and Iχ : χ ∈ I(eσ ) . Hence, AGSpin[ (eσ ) ⊂ A(eσ ). By the definition of ΛSL4 in (4.10) and thecommutative diagram (6.15), it is immediate that the image of Λ[ is AGSpin[ (eσ ). We thus have the following
commutative diagram:
1 −−−−→ Zbϕ,sc −−−−→

ζ
y[

Sϕ,sc

Λ
y [

−−−−→

Sϕ

∼
y=

−−−−→ 1
(6.17)

1 −−−−→ C× −−−−→ AGSpin[ (eσ ) −−−−→ I GSpin[ (eσ ) −−−−→ 1.
The representation ρσ ∈ Irr(Sϕ,sc , ζ[ ) is defined by ξσ ◦ Λ[ , where ξσ ∈ Irr(AGSpin[ (eσ ), id) is the characteras in the decomposition (4.7). Now, arguments of Section 4, our construction of L-packets Ππ (GSpin[ ) in
Section 6.1 and diagram (6.17) above give
1−1

1−1

Irr(Sϕ,sc , ζ[ ) ←→ Irr(AGSpin[ (eσ ), id) ←→ Πϕ (GSpin[ ).
We also have the following decomposition
M
Vσe ∼
=
σ∈Πϕ (GSpin[ )ρσ  σ =M(6.18)ρ  σρ ,ρ∈Irr(Sϕ,sc ,ζ[ )where σρ denotes the image of ρ via the bijection between Πϕ (GSpin[ ) and Irr(Sϕ,sc , ζ[ ). Hence, the proofof Theorem 6.1 is complete.

Remark 6.2. As before, since Λ[ is unique up to Hom(I GSpin[ (eσ ), C× ) ∼= Hom(Sϕ , C× ), the same is true forthe bijection (6.18).

26

MAHDI ASGARI AND KWANGHO CHOIY

Remark 6.3. Given ϕ ∈ φ(GSpin6 ), by Theorem 6.1, we have a one-to-one bijection between
1−1

0,1
1,0
Πϕ (GSpin6 ) ∪ Πϕ (GSpin2,06 ) ∪ Πϕ (GSpin6 ) ∪ Πϕ (GSpin6 ) ←→ Irr(Sϕ,sc ),where GSpin0,1
is a group isomorphic to GSpin1,0
as can be seen by taking the canonical isomorphism
66op∼
D4 = D4 , as discussed in (2.14).
6.3. L-packet Sizes for GSpin6 and Its Inner Forms. Just as in the case of GSpin4 , we have the following
possible cardinalities for the L-packets of GSpin6 and its inner forms.
Proposition 6.4. Let Πϕ (GSpin[ ) be an L-packet associated to ϕ ∈ Φ(GSpin[ ). Then we have|Πϕ (GSpin[ )|F × /(F × )2 ,which yields the possible cardinalities|Πϕ (GSpin[ )| =1, 2, 4,
1, 2, 4, 8,

if p 6= 2,
if p = 2.

Proof. We proceed similarly as in the proof of Proposition 5.5, again making use of Galois Cohomology. The
exact sequence of algebraic groups
1 −→ Z(GSpin[ ) −→ Z(GL[ ) × GSpin[ −→ GL[ −→ 1gives a long exact sequence· · · −→ (F × × F × ) × GSpin[ (F ) −→ GL[ (F ) −→ H 1 (F, Z(GSpin[ )) −→ H 1 (F, Z(GL[ ) × GSpin[ ) −→ 1.
Since H 1 (F, Z(GL[ ) × GSpin[ ) = 1 by Lemma 2.2 and [PR94, Lemmas 2.8], we haveGL[ (F )/ (F × × F × ) × GSpin[ (F ) ,→ H 1 (F, Z(GSpin[ )).
Also, the exact sequence
1 −→ Z(GSpin[ )◦ −→ Z(GSpin[ ) −→ π0 (Z(GSpin[ )) −→ 1,
we have
H 1 (F, Z(GSpin[ )) ,→ H 1 (F, π0 (Z(GSpin[ ))),
since, by [AS06, Proposition 2.3], Z(GSpin[ )◦ ∼= GL1 and H 1 (F, Z(GSpin[ )◦ ) = 1. Combining the above
with (2.19), we haveGL[ (F )/ (F × × F × ) × GSpin[ (F ) ,→ H 1 (F, π0 (Z(GSpin[ ))) ∼(6.19)= H 1 (F, Z/2Z) ∼= F × /(F × )2 .
On the other hand, we know from Section 4 that1−1Πϕ (GSpin[ ) ←→ GL[ (F )/GL[ (F )σe ,→ GL[ (F )/ (F × × F × ) × GSpin[ (F ) ,where σe ∈ Irr(GL[ ) corresponds to a lifting ϕe ∈ Φ(GL[ ), via the LLC for GLn and its inner forms, ofϕ ∈ Φ(GSpin[ ). This completes the proof.
Next, we give a description of the group I GSpin6 (eσ ) for σe ∈ Irr(GL1 × GL4 ). Unlike the case of GSpin4 weare unable to give a case by case classification (cf. Remark 6.7).
Given σe ∈ Irr(GL1 × GL4 ), we set σ
e = ηe  σ
e0 with ηe ∈ (GL1 (F ))D , σ
e0 ∈ Irr(GL4 ). By Remark 5.2, we noteDthat χ ∈ (GL[ (F )) is decomposed into χ
e1  χ
e2 , where χ
e1 ∈ (GL1 (F ))D and χ
e2 ∈ (GL[ (F ))D . Moreover,D
we identify χ
e2 ∈ (GL4 (F )) and χ
e2 ◦ det, since any character on GLn (F ) is of the form χe ◦ det for somecharacter χ
e on F × .
Lemma 6.5. Any χ ∈ I GSpin6 (eσ ) is of the formχe−2  χe,for some χe ∈ (F × )D .

LOCAL LANGLANDS CONJECTURE FOR p-ADIC GSpin4 , GSpin6 , AND THEIR INNER FORMS

27

Proof. Since χ = χ
e1  χ
e2 as above and χ is trivial on GSpin6 (F ), by the structure of GSpin6 (F ) in (2.23),
we haveχ((g1 , g2 )) = χe1 (g1 )  χe2 (det g2 ) = χe1 (g1 )eχ2 (det g2 ) = 1
2
for all (g1 , g2 ) ∈ GL1 (F ) × GL4 (F ) with (g1 ) = det g2 . Since the determinant map det : GL4 (F ) → F × issurjective, χe1 (eχ2 )2 (x) must be trivial for all x ∈ F × . Thus, we have χ
e1 = (eχ2 )−2 .
Proposition 6.6. We have

I GSpin6 (eσ ) = {χ ∈ I SL4 (eσ0 ) : χ2 = 1}.

Proof. Since σ
e = ηe  σ
e0 with ηe ∈ (GL1 (F ))D and σe0 ∈ Irr(GL4 ), by Lemma 6.5, we have
−2
σ
eχ ∼
e ⇐⇒ ηeχ
e σ
e0 χ
e∼
e0 ⇐⇒ σe0 χe∼e0 and χ
e2 = 1.
=σ= ηe  σ=σThis completes the proof.

GSpin6r

Remark 6.7. Propositions 6.4 and 6.6 imply that I(eσ ) is of the form (Z/2Z) with r = 0, 1, 2 if p 6= 2
and with r = 0, 1, 2, 3 if p = 2. Unlike the case of SL2 (see [GT10, Porposition 6.3]), a full classification of
irreducible L-parameters in Φ(SL4 ) is not currently available. Thus, unlike the case of GSpin4 in Section
5.3 (cf. Remarks 5.10 and 5.11), we do not classify the group I GSpin6 (eσ ) case by case, nor discuss the groupstructure of Sϕ,sc , and accordingly, the description of all the sizes of L-packets for GSpin6 , GSpin2,0
6 , and
GSpin1,0
6 is not available here.
6.4. Proerties of L-maps for GSpin6 and its inner forms. The L-maps defined in Section 6.1 again
satisfy some naturalpropertieso similar to the case of GSpin4 . We now verify those properties. In whatn1,0follows, let L[ ∈ L6 , L2,0
as the case may be.
6 , L4
Proposition 6.8. A representation σ[ ∈ Irr(GSpin[ ) is essentially square integrable if and only if its Lparameter ϕσ[ := L[ (σ[ ) does not factor through any proper Levi subgroup of GSO6 (C).
Proof. The proof is similar to that of Proposition 5.12 so we omit the details.

Remark 6.9. We similarly have that a given σ[ ∈ Irr(GSpin[ ) is tempered if and only if the image of its
L-parameter ϕσ[ := L[ (σ[ ) in GSO6 (C) is bounded.
Again because the restriction of representations preserves the intertwining operator and the Plancherel
measure (see [Cho14a, Section 2.2]), our construction of the L-packets in Section 6.1, gives the following
result.
Proposition 6.10. Let ϕ ∈ Φdisc (GSpin[ ) be given. For any σ1 , σ2 ∈ Πϕ (GSpin[ ), we have the equality of
the Plancherel measuresµM (ν, τ  σ1 , w) = µM (ν, τ  σ2 , w),
where M is an F -Levi subgroup of an F -inner form of GSpin2n of the form of the product of GSpin[ andcopies of F -inner forms of GLmi , τ  σ1 , τ  σ2 ∈ Πdisc (M ), ν ∈ a∗M,C , and w ∈ WM with w M = M.
Further, it is a consequence of the equality of the Plancherel measures that the Plancherel measure is alsopreserved between F -inner forms in the following sense. Let GSpin0[ be an F -inner form of GSpin[ . Givenϕ ∈ Φdisc (GSpin[ ), for any σ ∈ Πϕ (GSpin[ ) and σ 0 ∈ Πϕ (GSpin0[ ) we haveµM (ν, τ  σ, w) = µM 0 (ν, τ 0  σ 0 , w),
where M 0 is an F -inner form of M, τ  σ ∈ Πdisc (M ), τ 0  σ 0 ∈ Πdisc (M 0 ), and τ and τ 0 have the same
L-parameter.
Proof. The proof is similar to that of Proposition 5.14 so we omit the details.

Remark 6.11. Similar to Remark 5.15, in the case where GSpin[ is the split group GSpin6 , we have ζ[ = 1
and Theorem 6.1 implies
1−1\6 )) ∼
Πϕ (GSpin6 ) ←→ Irr(Sϕ (GSpin
σ )).
= Irr(I GSpin6 (e(6.20)

28

MAHDI ASGARI AND KWANGHO CHOIY

Suppose that there is a generic representation in Πϕ (GSpin6 ) with respect to a given Whittaker data for
GSpin6 . Then, by [HS12, Chapter 3], we can normalize the bijection (6.20) above, so that the trivial character1 ∈ Irr(Sϕ ) maps to the generic representation in Πϕ (GSpin6 ).
6.5. Preservation of Local Factors via L6 . As in Section 5.5, we can again verify that in the case of the
split form GSpin6 the map L6 preserves the local L-, -, and γ-factors when these local factors are defined.
We recall that on the representation side, these factors are defined via the Langlands-Shahidi method when
the representations are generic (or non-supercuspidal induced from generic via the Langlands classification)and on the parameter side, they are Artin factors associated to the representations of the Weil-Deligne groupof F.
Since the Langlands-Shahidi method is available for generic data, we do not yet have a counterpart for
the local factors for the non quasi-split F -inner forms of GSpin6 . This is the reason why we are limiting
ourselves to the case of the split form below; however, a remark similar to Remark 5.17 applies again.
Proposition 6.12. Let τ be an irreducible admissible representation of GLr (F ), r ≥ 1, and let σ be an irreducible admissible representation of GSpin6 (F ), which we assume to be either ψ-generic or non-supercuspidal
if r > 1. Here, generic is defined with respect to a non-trivial additive character ψ of F in the usual way.
Let ϕτ be the L-parameter of τ via the LLC for GLr (F ) and let ϕσ = L6 (σ). Then,γ(s, τ × σ, ψ)= γ(s, ϕτ ⊗ ϕσ , ψ),L(s, τ × σ)

= L(s, ϕτ ⊗ ϕσ ),(s, τ × σ, ψ)= (s, ϕτ ⊗ ϕσ , ψ).

The local factors on the left hand side are those attached by Shahidi [Sha90b, Theorem 3.5] to the ψ-genericrepresentations of the standard Levi subgroup GLr (F ) × GSpin6 (F ) in GSpin2r+6 (F ) and the standard representation of the dual Levi GLr (C) × GSO(6, C), and extended to all non-generic non-supercuspidal representations via the Langlands classification and the multiplicativity of the local factors [Sha90b, §9]. The factors
on the right hand side are Artin local factors associated to the given representations of the Weil-Delignegroup of F.
Proof. The proof is similar to that of Proposition 5.16 and we will not repeat it.



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Mahdi Asgari, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A.
E-mail address: asgari@math.okstate.edu
Kwangho Choiy, Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, U.S.A.
E-mail address: kchoiy@siu.edu