Arthur packets for unipotent representations Andrew Fiori and Qing - - PowerPoint PPT Presentation

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Arthur packets for unipotent representations

• f the p-adic exceptional group G2

Clifton Cunningham

joint work with Andrew Fiori and Qing Zhang University of Calgary

Fields Institute 2020 September 9 Number Theory Seminar

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Abstract

This talk concerns work in progress on a generalization of the notion of local Arthur packets from Arthur-type representations

• f classical groups over p-adic fields to all admissible

representations of all connected reductive algebraic groups over p-adic fields. The main ideas in this direction were described in [CFM+21], building on [Vog93]. In this talk our goal is much more modest: to report on this project for unipotent representations of the exceptional group G2(F) for a p-adic field

• F. We will explain how to use the microlocal geometry of the

moduli space of unramified Langlands parameters to compute what we call Adams-Barbasch-Vogan packets, or ABV-packets for short, for all unipotent representations of G2(F) and how to find the packet coefficients that are required to build stable distributions from ABV-packets. This talk will focus on the case that is the most interesting geometrically and will include a discussion of unipotent representations that are not of Arthur

• type. We will argue that ABV-packets provide the correct

extension of the notion of Arthur packets by explaining that the packet coefficients satisfy the expected endoscopic character identities.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Motivation

We wish to adapt the main local result from Arthur’s book [Art13] to unipotent representations of the the p-adic group G2(F). I reviewed the fundamental properties of A-packets in the introductory talk for this seminar.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Strategy

1 Use [CFM+21] to define candidate packets – what we call

ABV-packets – using a microlocal analysis of a moduli space of unramified Langlands parameters.

2 Show that ABV-packets satisfy endoscopic character identities –

this involves the groups SO4, PGL3, GL2 and, of course, GL1 × GL1.

3 Attach stable distributions to these candidate packets.

Only then should the candidate packets be called Arthur packets! However, even then it should be noted that they correspond to cases not treated in Arthur’s book, both because the group G2(F) is classical and also because not all unipotent representations are of Arthur type.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

The group G2

We view G2 as a Chevalley group scheme with Dynkin diagram with short root γ1 and long root γ2. There are 6 positive roots {γ1, γ2, γ1 + γ1, 2γ1 + γ2, 3γ1 + γ2, 3γ1 + 2γ2} . The dual root system is given by

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Subregular Langlands parameter

• There are five unipotent conjugacy classes in

G2 : O0 < Olong < Oshort < Osub < Oreg.

• Each unipotent conjugacy classes determines a group

homomorphism SL2(C) →

• G2. In particular, we have

ϕsub : SL2(C) → G2.

• Consider the Langlands parameter φ3 : WF × SL2(C) →

G2 defined by φ3(w, x) := ϕsub(x). (Notation will be justified later.)

• The component group of φ3 is non-Abelian:

Aφ3 := π0(Z

G2(φ3)) ∼

= S3.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

The “superunipotent” representation of G2

• The L-packet for φ3 is

Πφ3(G2(F)) = {π3, π̺

3, πε 3},

where

• π3 corresponds to the trivial representation of S3,
• π̺

3 corresponds to the unique 2-dimensional irreducible

representation ̺ of S3, and

• πε

3 corresponds to the sign character ε of S3.

• πε

3 is supercuspidal:

πε

3 := cIndG2(F) G2(OF ) G2[1];

where G2[1] is a cuspidal unipotent representation of G2(Fq).

• All three unipotent representations are tempered. Both π3 and

π̺

3 have nonzero Iwahori-fixed vectors.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

The category of subregular unipotent representations

• This L-packet straddles two blocks in the Bernstein

decomposition of the category of smooth representations of G2(F): Rep(G2(F)) = Rep(G2(F))[T(F),✶] ⊕ · · · ⊕ Rep(G2(F))[G2(F),πε

3 ]

• We can refine this by specifying the relevant cuspidal support,

not just the inertial classes. There is a unique unramified character χsub : T(F) → C× such that Rep(G2(F))sub := Rep(G2(F))(T(F),χsub) ⊕ Rep(G2(F))(G2(F),πε

3 )

contains π3 and π̺

3 in the first block and πε 3 in the second.

• We can identify all six irreducible representations in this

category:

• Rep(G2(F))sub

irred

/equiv = {π0, π1, π2, π3, π̺ 3, πε 3}.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Rep(G2(F))sub partitioned into L-packets

φ2 π2 φ1 π1 φ0 π0 φ3 πε

3

π̺

3

π3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Stabilizing representations and stable distributions

π2 π1 π0 πε

3

π̺

3

π3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Stabilizing representations

ABV-packet L-packet π2 π1 π0 πε

3

π̺

3

π3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Rep(G2(F))sub decomposed into ABV-packets

ψ2 π2 ψ1 π1 ψ0 π0 ψ3 πε

3

π̺

3

π3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

ABV-packets and Aubert duality

• n KRep(G2(F))sub

ψ2 π2 ψ1 π1 ψ0 π0 ψ3 πε

3

π̺

3

π3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Main result

Theorem ([CFZa])

For each Langlands parameter φ : W ′

F × SL2(C) → LG 2 of G2(F)

with subregular infinitesimal parameter there exists a finite set Π

ABV

φ (G2(F)) of irreducible unipotent representations and a function

, φ : Π

ABV

φ (G2(F))

→

• A

ABV

φ

π → , πφ, such that (a) Π

ABV

φ (G2(F))

• A

ABV

φ

Πφ(G2(F))

• Aφ,

π→ ,πφ LLC bijection

(1) where A

ABV

φ (resp.

Aφ) denotes the set of characters of irreducible representations of A

ABV

φ (resp. Aφ).

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Main result

Theorem ([CFZa])

Continuing from above regarding , φ : Π

ABV

φ (G2(F))

→

• A

ABV

φ

π → , πφ, (b) if φ is bounded upon restriction to WF then , φ is bijective and all the representations in Π

ABV

φ (G2(F)) are tempered;

(c) if π ∈ Π

ABV

ψ (G2(F)) is spherical (a.k.a unramified) then

, πφ = ✶, the trivial representation of A

ABV

φ .

Remark

If φ is bounded upon restriction to WF then , φ is not bijective and Π

ABV

φ (G2(F)) contains non-tempered representations.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

L-packets and ABV-packets for all irreducible representations in Rep(G2(F))sub

A − packet L − packet Π

ABV

φ0 (G2(F)) = {π0, π1, πε 3}

Πφ0(G2(F)) = {π0} Π

ABV

φ1 (G2(F)) = {π1, π2, πε 3}

Πφ1(G2(F)) = {π1} Π

ABV

φ2 (G2(F)) = {π2, π̺ 3, πε 3}

Πφ2(G2(F)) = {π2} Π

ABV

φ3 (G2(F)) = {π3, π̺ 3, πε 3}

Πφ3(G2(F)) = {π3, π̺

3, πε 3}

A

ABV

φ0

= S3, Aφ0 = 1, A

ABV

φ1

= S2, Aφ1 = 1 A

ABV

φ2

= S2, Aφ2 = 1 A

ABV

φ3

= S3, Aφ3 = S3.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

The functions , φ : Π

ABV

φ (G2(F)) → Rep(A

ABV

φ )

appearing in the main result

Rep(G2(F))sub Rep(A

ABV

φ0 )

Rep(A

ABV

φ1 )

Rep(A

ABV

φ2 )

Rep(A

ABV

φ3 )

π0 ✶ π1 ̺ ✶ π2 τ ✶ π3 ✶ π̺

3

τ ̺ πε

3

ε ✶ τ ε A

ABV

φ0

= S3, Irrep(A

ABV

φ0 )

= {✶, ̺, ε} A

ABV

φ1

= S2, Irrep(A

ABV

φ1 )

= {✶, τ} A

ABV

φ2

= S2, Irrep(A

ABV

φ2 )

= {✶, τ} A

ABV

φ3

= S3, Irrep(A

ABV

φ3 )

= {✶, ̺, ε}

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Stable distributions

For every Langlands parameter φ with subregular infinitesimal parameter, consider the invariant distribution Θφ :=

• π∈Π

ABV

φ (G2(F))

aψ, πφ Θπ, for aφ ∈ A

ABV

φ .

Theorem ([CFZa], [CFZb])

Suppose Θφ is stable when φ is tempered, unramified .1 Then the distributions Θφ are stable for all unramified Langlands parameters φ. Θφ0 = Θπ0 + 2Θπ1 + Θπε

3

Θφ1 = Θπ1 − Θπ2 + Θπε

3

Θφ2 = Θπ2 − Θπ̺

3 − Θπε 3

Θφ3 = Θπ3 + 2Θπ̺

3 + Θπε 3

1I’m working with Ahmed Moussaoui to prove this hypothesis is satisfied.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Subregular infinitesimal parameter

• The“infinitesimal parameter” λφ : WF →

G2 of a Langlands parameter φ is defined by λφ(w) := φ

• w,
• |w|1/2

|w|−1/2

• .

Set λsub := λφ3; we refer to this as the subregular infinitesimal parameter.

• Up to

G2-conjugacy, there are four Langlands parameters with infinitesimal parameter λsub, each of Arthur type: ψ0(w, x, y) = ϕsub(y), ψ1(w, x, y) = ι

γ1(x) ι γ1+2 γ2(y),

ψ2(w, x, y) = ι

γ1(y) ι γ1+2 γ2(x),

ψ3(w, x, y) = ϕsub(x), where ι

γ1 : SL2 →

G2 embeds into Levi subgroup GL2( γ1) and ι

γ1+2 γ2 : SL2 →

G2 embeds into Levi subgroup GL2( γ1 + 2 γ2).

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Langlands Correspondence

The local Langlands correspondence for representations with subregular infinitesimal parameter: enhanced irreducible L − parameter admissible rep′n (φ0, 1) → π0 (φ1, 1) → π1 (φ2, 1) → π2 (φ3, 1) → π3 (φ3, ̺) → π̺

3

(φ3, ε) → πε

3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Moduli space of Langlands parameters

The moduli space of Langlands parameters with subregular infinitesimal parameter is Xsub =

• G2 × Vsub
• /Hsub

where Hsub := Z

G2(λsub) ∼

= GL2( γ2) and Vsub := {x ∈ g2 | Ad(λsub(Fr))x = qx} = span {X

γ1, X γ1+ γ2, X γ1+2 γ2, X γ1+3 γ2} .

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Langlands parameters as homogeneous cubics in two variables

Let P3[x, y] be the vector space of homogeneous cubics in two variables x, y.

Proposition ([CFZa])

There is a natural isomorphism Vsub → P3[x, y] that defines an equivalence of representations of Hsub on Vsub with the representation det−1 ⊗ Sym3 of GL2 on P3[x, y] given below. Consequently, Per

G2(Xsub) ∼

= PerHsub(Vsub) ∼ = PerGL2(det −1 ⊗ Sym3)

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

A little geometry

• The GL2 action det−1 ⊗ Sym3 on P3[x, y] has the following 4
• rbits:
• C0 = {0} ;
• C1 =
• u3 | u ∈ P1[x, y], p = 0
• ;
• C2 =
• u2u′ | u, u′ ∈ P1[x, y], and u, u′ are linearly indep.
• ;
• C3 = {uu′u′′ | u, u′, u′′ ∈ P1[x, y], and u, u′, u′′ are indep.} .
• The closure relation on the GL2-orbits in P3(x, y) is given by

C0 < C1 < C2 < C3.

• Component groups/fundamental groups and microlocal

fundamental groups: Aφ0 = AC0 = 1 A

ABV

φ0

= Amic

C0 = S3

Aφ1 = AC1 = 1 A

ABV

φ1

= Amic

C1 = S2

Aφ2 = AC2 = 1 A

ABV

φ2

= Amic

C2 = S2

Aφ3 = AC3 = S3 A

ABV

φ3

= Amic

C3 = S3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Langlands correspondence, revisited

P :

• Rep(G2(F))

irred

/equiv

→

• PerGL2(det −1 ⊗ Sym3)

simple

/iso

π0 → IC(✶C0) π1 → IC(✶C1) π2 → IC(✶C2) π3 → IC(✶C3) π̺

3

→ IC(RC3) πε

3

→ IC(EC3) where:

• ✶C0 is the trivial local system on C0, ✶C1 is the trivial local

system on C1, ✶C2 is the trivial local system on C2, ✶C3 is the trivial local system on C3;

• RC3 is the GL2-equivariant local system on C3 corresponding to

representation ̺ of AC3 ∼ = Aφ3 ∼ = S3;

• EC3 is the GL2-equivariant local system on C3 corresponding to

the representation ε of AC3 ∼ = Aφ3 ∼ = S3.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Microlocal vanishing cycles

For any algebraic group H acting on V ∼ = Ad with finitely-many

• rbits, the microlocal vanishing cycles functor

Evs : PerH(V ) → LocH(Λreg) is defined [CFM+21] by EvsCi F :=

• RΦ( | )[−1]F ⊠ ✶!

C ∗

i [dim C ∗

i ]

• |Λreg

i

[− dim Λ], where Λ := T ∗

H(V ) is the conormal variety and Λi := T ∗ CiV . Then

Λreg = ∪

i Λreg i

(disjoint union), so Evs may be decomposed according to Evs = ⊕i EvsCi .

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

The main geometric result

Theorem ([CFZa])

On simple objects, Evs is given by: PerGL2(det−1 ⊗ Sym3)

Evs

− → LocGL2(Λreg) IC(✶C0) → ✶Λreg IC(✶C1) → TΛreg

1

⊕ RΛreg IC(✶C2) → TΛreg

2

⊕ ✶Λreg

1

IC(✶C3) → ✶Λreg

3

IC(RC3) → RΛreg

3

⊕ ✶Λreg

2

IC(EC3) → RΛreg

3

⊕ TΛreg

2

⊕ ✶Λreg

1

⊕ EΛreg

0 ,

where

• ✶Λreg

i

is the constant local system on Λreg

i

for i = 0, 1, 2, 3;

• TΛreg

1

(resp. TΛreg

2 ) is the local system on Λreg

1

(resp. Λreg

2 ) for

the non-trivial character τ of Amic

C1

= S2 (resp. Amic

C2

= S2);

• RΛreg

3

corresponds to the representation ̺ of Amic

C3

= S3;

• EΛreg

3

corresponds to the representation ε of Amic

C3

= S3.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Evs calculated

Evs : PerGL2(det −1 ⊗ Sym3) − → LocGL2(Λreg) PerGL2(P3[x, y]) Loc(Λreg

0 )

Loc(Λreg

1 )

Loc(Λreg

2 )

Loc(Λreg

3 )

IC(✶C0) ✶Λreg IC(✶C1) RΛreg TΛreg

1

IC(✶C2) ✶Λreg

1

TΛreg

2

IC(✶C3) ✶Λreg

3

IC(RC3) ✶Λreg

2

RΛreg

3

IC(EC3) EΛreg TΛreg

1

✶Λreg

2

EΛreg

3

Definition ([CFM+21])

Π

ABV

φi (G2(F)) :=

• π ∈
• Rep(G2(F))

irred

/equiv | EvsCi P(π) = 0

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Normalization/Twisting sheaf

From [CFM+21, Section 7.10], we recall the normalized microlocal vanishing cycles functor NEvsC : PerH(V ) → LocH(T ∗

CΛreg),

defined by NEvsC F := = (EvsC IC(✶C))∨ ⊗ EvsC F. In our case, EvsCi IC(✶Ci) =

• ✶Λreg

i ,

i = 0, 3; TΛreg

i ,

i = 1, 2.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Normalized microlocal vanishing cycles functor

Theorem

On simple objects, the normalized microlocal vanishing cycles functor NEvs is given by: PerGL2(P3[x, y])

NEvs

− → LocGL2(Λreg) IC(✶C0) → ✶Λreg IC(✶C1) → ✶Λreg

1

⊕ RΛreg IC(✶C2) → ✶Λreg

2

⊕ TΛreg

1

IC(✶C3) → ✶Λreg

3

IC(RC3) → RΛreg

3

⊕ TΛreg

2

IC(EC3) → EΛreg

3

⊕ ✶Λreg

2

⊕ TΛreg

1

⊕ EΛreg

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

NEvs calculated

NEvs : PerGL2(det −1 ⊗ Sym3) − → LocGL2(Λreg) PerGL2(P3[x, y]) Loc(Λreg

0 )

Loc(Λreg

1 )

Loc(Λreg

2 )

Loc(Λreg

3 )

IC(✶C0) ✶Λreg IC(✶C1) RΛreg ✶Λreg

1

IC(✶C2) TΛreg

1

✶Λreg

2

IC(✶C3) ✶Λreg

3

IC(RC3) TΛreg

2

RΛreg

3

IC(EC3) EΛreg ✶Λreg

1

TΛreg

2

EΛreg

3

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

, πφ ∈ Rep(A

ABV

φ ) For all Langlands parameters φ with subregular infinitesimal parameter, for all π ∈

• Rep(G2(F))

irred

/equiv, for all a ∈ A

ABV

φ ,

a, πφ := (−1)dim supp P(π)(−1)dim Cφ traceaa−1

φ NEvsCφ P(π),

The proof of the main result now follows by direct inspection of the tables above, using this definition.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Stability

Let Mi denote the standard module for πi for i = 0, 1, 2. Since M0, M1 and M2 are obtained by parabolic induction from representations

• f GL2(F), they are stable standard modules. Write ΘM0, ΘM1 and

ΘM2 for the Harish-Chandra distribution characters attached to these

• representations. These distributions are stable. The distributions Θφi,

for i = 0, 1, 2 are expressed in terms of these four stable distributions, ΘM0, ΘM1, ΘM2 and Θψ3, as follows:     Θφ0 Θφ1 Θφ2 Θφ3     =     1 1 −3 1 1 −2 1 1 −1 1         ΘM0 ΘM1 ΘM2 Θψ3    

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Forthcoming work

(1) In [CFZb] we consider the remaining unipotent representations

• f p-adic G2(F) and calculate the ABV-packets for these

representations.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

Forthcoming work

(2) In [CFZb] we show that ABV-packets for unipotent representations of G2 satisfy the conditions imposed on them by the theory of endoscopy: G2 PGL3 SO4 GL2 GL2 GL1 × GL1 (3) Together with Ahmed Moussaoui, I am working to prove that the tempered distributions, like Θψ3, are stable.

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

A triple of “superunipotent” representations

split unipotent fund′l irred superunipotent Lusztig′s group

• rbit

group rep′n representation class′n G2 G2(a1) S3 ε (sign) cIndG2(F)

G2(OF ) G2[1]

7.33 F4 F4(a3) S4 ε (sign) cIndF4(F)

F4(OF ) F II 4 [1]

7.26 E8 E8(a7) S5 ε (sign) cIndE8(F)

E8(OF ) E II 8 [1]

7.1

Arthur packets for unipotent representations

• f the p-adic

exceptional group G2 Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References

References

[Art13] James Arthur, The endoscopic classification of representations: Orthogonal and symplectic groups, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. [CFM+21] Clifton Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, and Bin Xu, Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples, Memoirs of the American Mathematical Society (in press) (2021). https://arxiv.org/abs/arXiv:1705.01885. [CFZa] Clifton Cunningham, Andrew Fiori, and Qing Zhang, Arthur packets for G(2) and perverse sheaves on cubics. https://arxiv.org/abs/2005.02438. [CFZb] , Arthur packets and endoscopy for unipotent representations

• f the p-adic exceptional group G2. (in preparation).

[Vog93] David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer.

• Math. Soc., Providence, RI, 1993, pp. 305–379.