On the Langlands parameter of a simple supercuspidal representation: odd orthogonal groups

0.21 MB·English

by  Moshe Adrian

ON THE LANGLANDS PARAMETER OF A SIMPLE
SUPERCUSPIDAL REPRESENTATION: ODD
ORTHOGONAL GROUPS
5
1
0
MOSHE ADRIAN
2

n
a Abstract. In this work, we explicitly compute a certain family
J of twisted gamma factors of a simple supercuspidalrepresentation

9 π ofa p-adicoddorthogonalgroup. These computations,together
2
with analogouscomputations for generallinear groups carried out

in previous workwithLiu [AL14], allow us to give a predictionfor
]
T theLanglandsparameterofπ. Ifweassumethe“depth-preserving
R conjecture”, we prove that our prediction is correct if p is suffi-
. ciently large.
h
t
a
m
[
Contents

1
v 1. Introduction 1
0
Acknowledgements 5
0
5 2. Notation 5
7
3. The local functional equation for odd orthogonal groups 6
0
. 4. The simple supercuspidal representations of GL 9
1 n
0 5. The simple supercuspidal representations of SO2ℓ+1 10
5 6. The computation of Φ(W,f ) 12
s
1
7. The computation of Φ∗(W,f ) 13
: s
v
8. Depth-preservation 15
i
X References 16
r
a
1. Introduction
Let G be a connected reductive group defined over a p-adic field F.
Recently, Gross, Reeder, andYu[GR10,RY13]haveconstructed aclass
Date: January 30, 2015.
2010 Mathematics Subject Classification. Primary 11S37, 22E50; Secondary
11F85, 22E55.
Key words and phrases. Simple supercuspidal, Local Langlands Conjecture.
MA is is supportedin partby postdoc researchfunding fromthe Department of
Mathematics at University of Toronto.
1
2 MOSHEADRIAN
of supercuspidal representations of G = G(F), called simple supercus-
pidal representations. These are the supercuspidal representations of
G of minimal positive depth.
Of interest is the Langlands correspondence for simple supercuspidal
representations of G. Recall that the Langlands correspondence (or
LLC) is a certain conjectural finite-to-one map
Π(G) → Φ(G)
from equivalence classes of irreducible admissible representations of G
toequivalence classes ofLanglandsparametersofG. Inthepast several
years there has been much activity on the Langlands correspondence.
DeBacker and Reeder [DR09] have constructed a correspondence for
depth zero supercuspidal representations of unramified p-adic groups,
and Reeder [R08] extended this construction to certain positive depth
supercuspidal representations of unramified groups. Recently, under a
mild assumption on the residual characteristic, Kaletha [K13] has con-
structed a correspondence for simple supercuspidal representations of
p-adicgroups, andhasverifiedthathiscorrespondencesatisfiesmanyof
the expected properties of the Langlands correspondence. Kaletha has
subsequently [K15] extended these results to epipelagic supercuspidal
representations of p-adic groups.
Our present work is the first in a series of papers dedicated to ex-
plicitly determining the Langlands parameter ϕ of a simple super-
π
cuspidal representation π of a classical group G, using the theory
of gamma factors. Let π and τ be a pair of irreducible representa-
tions of G and GL , where G is either SO , SO or Sp . Here,
n 2ℓ 2ℓ+1 2ℓ
we assume that SO is split and SO is quasi-split. Fix a non-
2ℓ+1 2ℓ
trivial character ψ of F. The Rankin-Selberg integral for G × GL
n
was constructed, in different settings, in a series of works including
[GPSR87, G90, GPSR97, GRS98, Sou93]. If G is orthogonal, we de-
note this Rankin-Selberg integral by Φ(W,f ), and Φ(W,φ,f ) other-
s s
wise. Here, W is a Whittaker function for π, f is a certain holomor-
s
phic section of a principal series (induced from τ) depending on the
complex parameter s, and φ is a Schwartz function. Applying a stan-
dard normalized intertwining operator M∗(τ,s) to f , one obtains a
s
similar integral Φ∗(W,f ) = Φ(W,M∗(τ,s)f ) if G is orthogonal (and
s s
Φ∗(W,φ,f ) = Φ(W,φ,M∗(τ,s)f ) otherwise) related to Φ(W,f ) (or
s s s
Φ(W,φ,f )) by a functional equation
s
γ(s,π ×τ,ψ)Φ(W,f ) = Φ∗(W,f ) if G = SO
s s 2ℓ+1
γ(s,π ×τ,ψ)Φ(W,f ) = c(s,ℓ,τ,γ)Φ∗(W,f ) if G = SO
s s 2ℓ

γ(s,π ×τ,ψ)Φ(W,φ,f ) = Φ∗(W,φ,f ) if G = Sp
 s s 2ℓ

LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 3
For an accessible paper discussing all of these functional equations,
we refer the reader to the recent work of Kaplan [K14ii].
The term γ(s,π×τ,ψ) is known as the gamma factor of π×τ, and
is the key ingredient in determining ϕ in this paper. To determine ϕ ,
π π
one must locate the poles of the gamma factors γ(s,π ×τ,ψ) where τ
ranges over the supercuspidal representations of various general linear
groups. The location of poles will determine the isobaric constituents
of a specific functorial lift Π to a general linear group GL, the lift cor-
responding to the standard L-homomorphism LG → LGL (see [ACS14,
§2]). In the case of simple supercuspidal representations π of a classical
group G, it is expected that the isobaric constituents of Π are either
tamely ramified characters of GL , or simple supercuspidal representa-
1
tions (for p large, this can be seen from the so-called “depth-preserving
conjecture”; see §8). We can then use the explicit local Langlands cor-
respondence for simple supercuspidal representations of general linear
groups (see [AL14, BH10, BH13]) to determine ϕ .
π
While describing the Langlands correspondence explicitly is in gen-
eral a difficult task, in the simple supercuspidal setting it turns out to
be tractable. To explain our approach, we first make some observa-
tions. If π is a simple supercuspidal representation of a classical group,
it is expected (as remarked earlier) that its Langlands parameter ϕ
π
decomposes as an irreducible subrepresentation ϕ (that corresponds
π,1
toasimple supercuspidal representation ofageneral lineargroupunder
LLC) and a (possibly empty) direct sum ϕ of tamely ramified one-
π,2
dimensional subrepresentations. Using this prediction as our guide, we
can describe our method to fully determine ϕ in two steps. The first
π
is to compute the gamma factor γ(s,π ×τ,ψ), where τ is any tamely
ramified character of GL . This computation will determine ϕ . By
1 π,2
multiplicativity of gamma factors, it will also yield a prediction for a
specific simple supercuspidal representation σ (whose associated Lang-
lands parameter is ϕ ) that is an isobaric constituent of Π, leading
π,1
to step two. In step two, one computes γ(s,π×σ,ψ) in order to prove
that σ is indeed such a constituent. We can then use LLC for sim-
ple supercuspidal representations of general linear groups to determine
ϕ .
π,1
We now give more details on these steps in the case of G = SO ,
2ℓ+1
which is the central concern of this paper. The simple supercuspidal
representations of G are parameterized by two pieces of data: a choice
of a uniformizer ̟ in F, and a sign. More explicitly (see §5), let χ
be an affine generic character of the pro-unipotent radical I+ of an
Iwahori I. The choice of a uniformizer ̟ in F determines an element
g in G which normalizes I and stabilizes χ. We can extend χ to
χ
4 MOSHEADRIAN
K = hg iI+ in two different ways, since g2 = 1, and π = IndGχ is
χ χ K
simple supercuspidal.
Let τ be a tamely ramified character of GL . Our main result (The-
1
orem 7.2) is
Theorem 1.1.
γ(s,π ×τ,ψ) = χ(g )τ(−̟)q1/2−s.
χ
In particular, since there are no poles in γ(s,π × τ,ψ), there can
be no tamely ramified character of the Weil group W appearing as
F
a summand of ϕ . In particular, it is therefore expected that ϕ is
π π
irreducible. Therefore, on the one hand, the computation of γ(s,π ×
τ,ψ) determines no part of ϕ . On the other hand, it does tell us
π
precisely what the Langlands parameter should be, as follows.
By the results of [AL14, BH13] (see in particular [AL14, Remark
3.18]), there exists a unique irreducible 2ℓ-dimensional representation
ϕ : W → GL(2ℓ,C) with trivial determinant, whose gamma factor
F
γ(s,ϕ×τ,ψ) equals χ(g )τ(−̟)q1/2−s for every tamely ramified char-
χ
acter τ (we give more details on this in §4). The Langlands parameter
ϕ has been explicitly described in [AL14, BH10, BH13]. In the case
that p ∤ 2ℓ, let us briefly recall this description (here we follow [AL14]).
Let ̟ be a uniformizer of F, let ̟ be a 2ℓth root of ̟, and set
E
E = F(̟ ). Relative to the basis
E
̟2ℓ−1,̟2ℓ−2,··· ,̟ ,1
E E E
of E/F, we have an embedding
ι : E× ֒→ GL .
2ℓ
Define a character ξ of E× by setting ξ| = λ ◦ ι, where λ(A) :=
1+pE
ψ(A +A +...+A +1A ),forA = (A) ∈ GL . Moreover, we
12 23 2ℓ−1,2ℓ ̟ 2ℓ,1 ij 2ℓ
define ξ|kF× ≡ (κE/F|kF×)−1, where κE/F = det(IndWWFE(1E)), 1E denotes
the trivial character of W , and k is the residue field of F. Finally,
E F
we set
ξ(̟ ) = χ(g )λ (ψ)−1,
E χ E/F
where λ (ψ) is the Langlands constant (see [BH06, §34.3]). Then
E/F
ϕ = IndWFξ. The proof that ϕ = ϕ boils down to the computation of
WE π
γ(s,π × Π,ψ), where Π is the simple supercuspidal representation of
GL corresponding to ϕ under LLC, and will be carried out in future
2ℓ
work. This is our “step two” described above. On the other hand, if
we assume the “depth-preserving conjecture”, we can easily prove (see
Corollary 8.1) that ϕ = ϕ for p sufficiently large.
π
LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 5
The cases of SO and Sp are in some sense similar. For exam-
2ℓ 2ℓ
ple, suppose that G = SO . Let π be a simple supercuspidal rep-
2ℓ
resentation of G, and let ϕ be its Langlands parameter. Again, it
π
is expected that ϕ decomposes as an irreducible subrepresentation
π
ϕ (that corresponds to a simple supercuspidal representation under
π,1
LLC) and a (possibly empty) direct sum ϕ of tamely ramified one-
π,2
dimensional subrepresentations (again, for p large, this can be seen
from the “depth-preserving conjecture”). It is known that all sum-
mands must be self-dual. Therefore the computation of γ(s,π ×τ,ψ),
where τ is a quadratic and tamely ramified character of GL will de-
1
termine ϕ . The computation of γ(s,π × τ,ψ), where τ is tamely
π,2
ramified and not quadratic, will give a specific prediction (as in the
case of SO ) for ϕ . Then one must perform “step two” by cal-
2ℓ+1 π,1
culating γ(s,π × Π,ψ), where Π is the unique simple supercuspidal
representation (with trivial central character) of a certain general lin-
eargroupsuch thatγ(s,Π×τ,ψ) = γ(s,π×τ,ψ)foralltamely ramified
characters τ of GL .
1
We now summarize the contents of the sections of this paper. In
§3, we recall the functional equation for odd orthogonal groups as in
[Sou93]. In §4, we recall a construction of simple supercuspidal repre-
sentations of GL as well as their tamely ramified GL -twisted gamma
n 1
factors. In §5, we give a construction of simple supercuspidal represen-
tations of odd orthogonal groups that may be used in computing the
Rankin-Selberg integrals from §3. In §6 and §7, we compute a family of
Rankin-Selberg integralsthatallowustoexplicitly compute thetamely
ramified GL -twisted gamma factors of a simple supercuspidal repre-
1
sentation of SO . The main result on the values of these gamma
2ℓ+1
factors is Theorem 7.2, which allows us to predict the Langlands pa-
rameter of a simple supercuspidal representation of an odd orthogonal
group. Finally, in §8, we show that if we assume a conjecture on depth
preservation, thenourprediction fortheLanglandsparameter isindeed
the correct Langlands parameter.
Acknowledgements. This paper has benefited from conversations
with Benedict Gross, Tasho Kaletha, and Eyal Kaplan. We thank
them all.
2. Notation
Let F be a p-adic field, with ring of integers o, maximal ideal p,
k = o/p, let q denote the cardinality of k , and fix a uniformizer ̟.
F F
We normalize measures dv and d×v on F and F× by taking vol(o) = q1
2
and vol(o×) = 1. Throughout, we fix a nontrivial additive character
6 MOSHEADRIAN
ψ : F → C× that is trivial on p but nontrivial on o. Given a connected
reductive group G defined over F, let G = G(F). We also let W
F
denote the Weil group of F.
3. The local functional equation for odd orthogonal
groups
In this section we recall the functional equation for odd orthogonal
groups, as in [Sou93, §1.3, §9.6, §10.1].1
1
Let J denote the ℓ×ℓ matrix ... . We define
ℓ
 
1
 
(3.1) SO = g ∈ GL : det(g) = 1,tgJ g = J ,
2ℓ+1 2ℓ+1 2ℓ+1 2ℓ+1
(cid:8) (cid:9)
(3.2) SO = g ∈ GL : det(g) = 1,tgJ g = J .
2ℓ 2ℓ 2ℓ 2ℓ
LetT bethesplit(cid:8)maximaltorusofSO . Let∆ b(cid:9)ethestandard
SO2ℓ 2ℓ SO2ℓ
set of simple roots of SO , so that ∆ = {ǫ −ǫ ,...,ǫ −ǫ ,ǫ +
2ℓ SO2ℓ 1 2 ℓ−1 ℓ ℓ−1
ǫ }, where ǫ (t) = t is the i-th coordinate function of t ∈ T . Let
ℓ i i SO2ℓ
Q be the standard maximal parabolic subgroup of SO corresponding
ℓ 2ℓ
to ∆\{ǫ +ǫ }. The Levi part of Q is isomorphic to GL , and we
ℓ−1 ℓ ℓ ℓ
denote its unipotent radical by U .
ℓ
Let T be the split maximal torus of SO . let ∆ be
SO2ℓ+1 2ℓ+1 SO2ℓ+1
the standard set of simple roots of SO , so that ∆ = {ǫ −
2ℓ+1 SO2ℓ+1 1
ǫ ,...,ǫ −ǫ ,ǫ }. The highest root is ǫ +ǫ . Let U denote the
2 ℓ−1 ℓ ℓ 1 2 SO2ℓ+1
subgroup of upper triangular unipotent matrices.
Let ψ be a nontrivial additive character of F. We let U denote
GLℓ
the maximal unipotent subgroup of GL consisting of upper triangular
ℓ
matrices. We also denote by ψ the standard nondegenerate Whittaker
character of
ℓ−1
ψ(u) = ψ u ,
i,i+1
!
i=1
X
for u = (u ) ∈ U .
ij GLℓ
Let τ be a generic representation of GL . We denote by W(τ,ψ) the
ℓ
Whittaker model of τ with respect to ψ.
For s ∈ C, define
VSO2ℓ(W(τ,ψ),s) = IndSO2ℓ(W(τ,ψ)|det|s−1/2).
Qℓ Qℓ
1Thereisactuallyatypoin[Sou93]asobservedin[K14ii,Remark3.1]. However,
this does not affect our present work (see Remark 3.2).
LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 7
Thus, afunctionf inVSO2ℓ(W(τ,ψ),s)isasmoothfunctiononSO ×
s Qℓ 2ℓ
GL , where for any g ∈ SO , m ∈ Q , and u in the unipotent radical
ℓ 2ℓ ℓ
U , we have
ℓ
(3.3) fs((mug,a) = |det(m)|s+n−22fs(g,am),
and the mapping a 7→ f (g,a) belongs to W(τ,ψ).
s
We now let ℓ,n ∈ N such that ℓ ≥ n. Let U denote the sub-
SO2ℓ+1
group of upper triangular unipotent matrices in SO . We use the
2ℓ+1
notation ψ again to define a non-degenerate character on U by
SO2ℓ+1
ℓ
ψ(u) = ψ u .
i,i+1
!
i=1
X
Let π be a generic representation of SO with respect to ψ. We
2ℓ+1
shall denote by W(π,ψ) the Whittaker model of π with respect to ψ.
Let τ be a generic representation of GL with respect to ψ.
n
In computing γ-factors, we will be interested in the following local
integrals (see [Sou93]):
Φ(W,f )
s
:= W(xj (h))f (h,I )dxdh,
n,ℓ s n
ZUSO2n\SO2nZX(n,ℓ)
where W ∈ W(π,ψ), f ∈ VSO2n(W(τ,ψ−1),s), and where dh is a fixed
s Qn
right-invariant Haar measure on the quotient space U \SO , and
SO2n 2n
dx is the product measure inherited from dv.
Here, j : SO → SO is the map
n,ℓ 2n 2ℓ+1
A B
A B
7→ I ∈ SO ,
C D  2(ℓ−n)+1  2ℓ+1
(cid:18) (cid:19) C D
 
I
n
y I
ℓ−n
 
X = 1 ∈ SO .
(n,ℓ) 2ℓ+1
 I 
ℓ−n
 
 y′ I 
n
 
 
Note that y ∈ Fℓ−1 is a column vector, and y′ ∈ Fℓ−1 is a row vector.
Moreover, the notation y′ means that these coordinates are determined
by the coordinates of y, according to the matrix defining SO .
2ℓ+1
Now let n be odd. We define an intertwining operator by
8 MOSHEADRIAN
M(τ,s)f (h,a) = f (uw−1h,a)du,
s s n
ZUn
where
1
I
w = n I ∈ SO ,
n I  2(n−1)  2n
(cid:18) n (cid:19) 1
 
and where U is the opposite unipotent radical to U .
n n
Then we will also be interested in the integrals (see [Sou93, §9.6])
Φ∗(W,f ), where
s
Φ∗(W,f )
s
:= γ(2s−1,τ,∧2,ψ) W(cˆ xj (h)δ ω′)M(τ,s)f (ωh,b∗)dxdh.
n,ℓ n,ℓ o s n
ZUSO2n\SO2nZX(n,ℓ)
Here, if g ∈ GL , then g∗ = J tg−1J . Moreover,
n n n
cˆ = diag(I ,−I ,1,−I ,I ) ∈ SO ,
n,ℓ n ℓ−n ℓ−n n 2ℓ+1
δ = diag(I ,−1,I ),
o ℓ ℓ
b = diag(1,−1,...,−1,1) ∈ GL ,
n n
I
n−1
1
 
ω′ = I ,
2(ℓ−n)+1
 1 
 
 I 
n−1
 
 
I
n−1
0 1
ω =  .
1 0
 I 
n−1
 
 
We then have the functional equation for odd orthogonal groups.
Theorem 3.1. [Sou93, §10.1]
(3.4) γ(s,π ×τ,ψ)Φ(W,f ) = Φ∗(W,f ).
s s
Remark 3.2. In [K14ii, Remark 3.1] it was observed that there is a
typo in [Sou93] having to do with the Whittaker space for π. However,
as noted in the same remark, this typo only needs to be fixed in the case
that l < n. Since we are at present only considering n = 1, we may
still follow the notation and integrals from [Sou93].
LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 9
4. The simple supercuspidal representations of GL
n
In this section we review the definition of a simple supercuspidal
representation of GL , as in [AL14] (see also [KL15]). We then recall
n
the values of a family of its twisted gamma factors, in order to predict
the Langlands parameter of a simple supercuspidal of SO .
2ℓ+1
Let GLn = GLn(F), Z the center of G, K = GLn(o) the standard
maximal compact subgroup, I the standard Iwahori subgroup, and I+
its pro-unipotent radical. Fix a nontrivial additive character ψ of F
that is trivial on p and nontrivial on o. Fix a uniformizer ̟ in F.
Let U denote the unipotent radical of the standard upper triangular
Borel subgroup of GL . For any u ∈ U, we recall from §3 the standard
n
nondegenerate Whittaker character ψ of U:
n−1
ψ(u) = ψ u .
i,i+1
!
i=1
X
Set H = ZI+, and fix a character ω of Z ∼= F×, trivial on 1 + p.
For t ∈ o×/(1 + p), we define a character χ : H → C× by χ(zk) =
ω(z)ψ(r +r ...+r +r ) for
1 2 n−1 n
x r ∗ ···
1 1
∗ x r ···
2 2
k =  ... ... ... 
 ∗ r 
 n−1 
 ̟r ··· x 
 n n 
 
The compactly induced representation π := cIndGχ is then a direct
χ H
sum of n distinct irreducible supercuspidal representations of GL .
n
They are parameterized by ζ, where ζ is a complex nth root of ω(̟),
as follows. Set
0 1
1
g =  ... 
χ
 1 
 
 ̟ 0 
 
 
and set H′ = hg iH. Then the summands of π are
χ χ
σζ := cIndG χ
χ H′ ζ
where χ (gjh) = ζjχ(h), as ζ runs over the complex nth roots of ω(̟).
ζ χ
The σζ are called the simple supercuspidal representations of GL .
χ n
10 MOSHEADRIAN
Finally, for the simple supercuspidal representation σζ, we may de-
χ
fine a Whittaker function on GL by setting
n
ψ(u)χ (h′) if g = uh′ ∈ UH′
W(g) = ζ
0 else
(cid:26)
This function is well-defined, by definition of ψ and χ . Using this
ζ
Whittaker function, it turns out to be not so difficult to compute
the Rankin-Selberg integrals that arise in the definition of the twisted
gamma factors γ(s,σζ ×τ,ψ), where τ is a tamely ramified character
χ
of F×. In §5, we will define an analogous Whittaker function for a
simple supercuspidal representation π of SO , which will allow us
2ℓ+1
to compute the relevant Rankin-Selberg integrals for π.
In [AL14], we explicitly computed γ(s,σζ ×τ,ψ):
χ
Lemma 4.1. [AL14, Lemma 3.14] Let τ be a tamely ramified character
of GL . Then
1
γ(s,σζ ×τ,ψ) = τ(−1)n−1τ(̟)χ(g )q1/2−s
χ χ
This result allows us to conclude (see [AL14, Remark 3.18]) that
a simple supercuspidal of GL , up to central character, can be dis-
n
tinguished from any other supercuspidal of GL by the above twisted
n
γ-factors.
5. The simple supercuspidal representations of SO
2ℓ+1
In this section we construct the simple supercuspidal representations
of SO , and define an explicit Whittaker function for each simple
2ℓ+1
supercuspidal.
Let T = T denote the split maximal torus of SO . Asso-
SO2ℓ+1 2ℓ+1
ciated to T we have the set of affine roots Ψ. Let X∗(T) denote the
character lattice of T, let T be the maximal compact subgroup of T ,
0 0
and set
T = ht ∈ T : λ(t) ∈ 1+p ∀λ ∈ X∗(T)i.
1 0
To each α ∈ Ψ we have an associated affine root group U in G.
α
Associated to the standard set of simple roots ∆ we have a set of
SO2ℓ+1
simple affine Π and positive affine roots Ψ+. Set
I = hT ,U : α ∈ Ψ+i,
0 α
I+ = hT ,U : α ∈ Ψ+i,
1 α
I++ = hT ,U : α ∈ Ψ+ \Πi.
1 α
In terms of the Moy-Prasad filtration, if x denotes the barycenter of
the fundamental alcove, then I = G = G ,I+ = G = G , and
x x,0 x+ x,1
2ℓ
I++ = G .
x,1+
2ℓ