Twisted endoscopic character relation for
Kaletha’s regular supercuspidal L-packets
Masao Oi
Kyoto University (Hakubi center)

RIMS conference“Automorphic forms, Automorphic representations,Galois representations, and its related topics”(January 29, 2021)

Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

1 / 20Local Langlands correspondence
F : a p-adic ﬁeld.
G: a connected reductive group over F (G := G(F )).
Π(G) := {irreducible admissible representations of G}/∼,
Φ(G) := {L-parameters of G}/∼.

Local Langlands correspondence (LLC)
There exists a natural map with ﬁnite ﬁbers:
LLCG : Π(G) → Φ(G).
In other words, there exists a natural partition of the set Π(G) into subsets(which are ﬁnite, called L-packets) parametrized by L-parameters:⊔−1
Π(G) =ΠG(ΠG
ϕϕ := LLCG (φ)).
ϕ∈Φ(G)

LLC is still conjectural in general, but a number of results have been obtained.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

2 / 20Motivation: comparison of diﬀerent constructions
Approach 1: Specialize the group. For example,– GLN ; Harris–Taylor– quasi-split classical groups (Sp2n , SON , UN ); Arthur, Mok
Approach 2: Specialize the class of representations. For example,– regular supercuspidal representations; Kaletha(works for tamely ramiﬁed groups)(he dropped the regularity recently; arXiv:1912.03274)
Q. Do the two approaches give the same LLC (on their “intersection”)?

Theorem (O.-Tokimoto, 2019)
We assume that p 6= 2. Kaletha’s and Harris–Taylor’s LLC coincide for regularsupercuspidal representations of GLN .
Q. Kaletha’s LLC = Arthur’s (Mok’s) LLC?
Key: Arthur’s LLC is characterized by the twisted endoscopic character relation.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

3 / 20Twisted endoscopic character relation (TECR)θ ↷ G: a “twist” (i.e., rational automorphism preserving a pinning)
H: an endoscopic group for (G, θ)= a quasi-split connected reductive group H over F equipped withˆ Ĥ) = ĜInt(s)◦θ̂,0 for s ∈ Ĝ.
ξˆ: L H ,→ L G such that ξ(The existences of LLCH and LLCG induces a lifting of L-packets:
Lo LLC for G /ΠGϕ9 GOOξ̂◦ϕξ̂endoscopic liftingΠHϕoLLC for H/WF × SL2 (C)

?

ϕ/ LH

Twisted endoscopic character relation
H= an identity between (twisted) characters of representations in ΠG
ϕ and Πϕ ,Gwhich characterizes the map ΠH
ϕ 7→ Πϕ

Goal of my ongoing project
Kaletha’s LLC for regular supercuspidal representations satisﬁes the TECR.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

4 / 20Regular supercuspidal representations
Assume: G is tamely ramiﬁed and p  0.
In 2001, Yu constructed a certain wide class of supercuspidal representations.
Kaletha reparametrized “regular” Yu-data by “tame elliptic regular pairs”;(S, ϑ) consists of– S ⊂ G: a tamely ramiﬁed elliptic maximal torus deﬁned over F ,– ϑ : S(F ) → C× : a “regular” character.

{Yu-data}/∼O?
{reg. Yu-data}/∼
O

1:1

1:1

/ {Yu s.c. rep’ns of G(F )}/∼O?
/ {reg. s.c. rep’ns of G(F )}/∼3

1:1(S,ϑ)7→π(S,ϑ)
{tame ell. reg. pairs}/G(F )-conj.

Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

5 / 20Construction of L-parameters/L-packets(S, ϑ): a tame elliptic regular pair of G.

Construction of an L-parameter ϕ
LLC for S gives φϑ : WF → L S.
Langlands–Shelstad construction gives L jχ : L S ,→ L G.
Then we get an L-parameter φ of G (“regular supercuspidal L-parameter”):ϕϑLjχ

φ : WF −−→ L S −−→ L G.

Construction of an L-packet ΠGϕ
Put J G to be the stable G-conjugacy class of the embedding S ,→ G and
JGG := {j ∈ J G | deﬁned over F }/G-conj.
⇝ for each j ∈ JGG , we get a tame elliptic regular pair (j(S), ϑj ) of G.
PutGΠG
ϕ := {πj | j ∈ JG },Masao Oi (Kyoto Univ.)

πj := π(j(S),ϑj ) .

TECR for Kaletha L-packets

RIMS conference 2021

6 / 20Setting of the problem
G: a quasi-split connected reductive group with θ over F .
H: an endoscopic group for (G, θ).
Suppose: G and H are tamely ramiﬁed and p  0.
φ: a “toral” regular supercuspidal L-parameter of G.
Assume: φ factors through L H ,→ L G.
⇝ φ is regarded as an L-parameter of H (again toral).

ΠGϕO

Kaletha’s
LLC for Go/ϕ/ LGOξ̂want to show ECRΠH
ϕWF × SL2 (C)ϕ

Kaletha’s
LLC for Ho/% ?

LHGG
ΠGϕ is θ-stable, i.e., Πϕ ◦ θ = Πϕ as sets.

Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

7 / 20Precise statement of main result
HMain result at present (ECR for ΠGϕ and Πϕ )For an elliptic strongly regular s-s. element δ ∈ G̃ having a norm in H, we have∑

∆spec
H,G (π)Θ̃π (δ) =π∈ΠGϕ∑ DH (γ)2∑
∆H,G (γ, δ)ΘπH (γ).
2DG̃ (δ)
H

γ∈H/stπH ∈ΠϕΘπH is the Harish-Chandra character of πH ∈ ΠHϕ.
Θ̃π is the θ-twisted character of π.
(We follow the formalism of twisted space (G, G̃); G̃ := G ⋊ θ.)∆spec
H,G (π) ∈ C is an explicit constant (“spectral transfer factor”).
(∆specH,G (π) 6= 0 if and only if π is θ-stable)
γ runs over stable conjugacy classes of norms of δ in H.
DH and DG̃ denote the Weyl discriminants.
∆H,G (γ, δ) is the Langlands–Shelstad–Kottwitz transfer factor.
Strategy: imitate Kaletha’s proof in the standard case (i.e., θ = id).
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

8 / 20Rough outline of Kaletha’s proof
Gdescent (use character formula for G)'standard endoscopy(want to show ECR here)

G0H

reduced to a comparison heredescent (use character formula for H)'H0

Adler–DeBacker–Spice formula describes the characters in terms of adescended group G0 (or H0 ).
Langlands–Shelstad descent gives (G0 , H0 ) a structure of endoscopy.
Waldspurger–Ngô transfer gives a comparison between Lie algebras of G0 and
H0 . For this, we need
Kaletha’s descent lemma, andKaletha’s reﬁnement of ADS formula.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

9 / 20Adler–DeBacker–Spice character formulaLet (S, ϑ) be a toral tame elliptic regular pair of depth r ∈ R>0 .
Let δ ∈ G be an elliptic strongly regular semisimple element.
Take a normal r-approximation δ = δ<r · δ≥r ;- δ<r : “p-adically shallower than r”-part of δ,- δ≥r : “p-adically deeper than r”-part of δ.
- δ≥r ∈ Gδ<r := CentG (δ<r )◦ .

Prop (1st form of ADS formula)Θπ(S,ϑ) (δ) =∑()GδΘσ (gδ<r g −1 ) · µ̂g−1<r
X ∗ g log(δ≥r ) .

g∈S\G/Gδ<rgδ<r g −1 ∈S
G(F )π(S,ϑ) = c-IndKσ∗σ for certain open compact (mod-center) subgroup Kσ .

∗X ∈ (Lie S) is an element representing ϑ|Sr .
Gδµ̂g−1<rX ∗ g is the Fourier transform of the orbital integral on Lie Gδ<r .
(with respect to the Gδ<r (F )-orbit of g −1 X ∗ g)
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

10 / 20

Langlands–Shelstad descent & reduction to Lie algebras
Recall:1:1

GG
ΠG| j : S ,→ G: deﬁned over F }/∼G ,ϕ ←→ JG := {j ∈ J1:1HHΠH| jH : SH ,→ H: deﬁned over F }/∼H .
ϕ ←→ JH := {jH ∈ J

For each j ∈ JGG and jH ∈ JHH , we have∑GδG-side: Θπj (δ) =Θσj (gδ<r g −1 ) · µ̂g−1<rXj∗ g (log(δ≥r ))∑HγH-side: ΘπjH (γ) =ΘσH,jH (hγ<r h−1 ) · µ̂h−1<rX∗h (log(γ≥r ))H,jH

The Langlands–Shelstad descent gives an endoscopic structure on (Hγ<r , Gδ<r ).
Gstandard endoscopy

descent(Gδ<r =: G′Hdescent(standard endoscopy

Hγ<r =: H′Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

11 / 20

Waldspurger–Ngô transfer on Lie algebras
Waldspurger–Ngô transfer (= Lie algebra version of SECR)∑γ(g0 )′

∆H′ ,G′ (Y ∗ , X 0∗ )µ̂G
X ′∗ (X)X ′∗ ∼st X ∗= γ(h0 )∑∆H′ ,G′ (Y, X)∑′µ̂HY ′∗ (Y )Y ′∗ ∼st Y ∗Y /stindex sets: G0 (resp. H 0 )-conj. classes in a G0 (resp. H0 )-conj. class.

Recall: G-side of SECR∑∆specH,G (πj )Θπj (δ) =Gj∈JG∑Gj∈JG∆spec
H,G (πj )∑′Θσj (g δ<r )·µ̂GX ∗,g (log(δ≥r ))jg∈j(S)\G/G′gδ<r g −1 ∈j(S)⇝ We need to consider a descent and transfer of index sets!
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

12 / 20

Transfer of index sets; Kaletha’s descent lemma
The index set of ADS formula is understood as
JGG′ (j) := {k ∈ j | k : S ,→ G0 ⊂ G: deﬁned over F }/∼G′ .
{g ∈ j(S)\G/G0 | gδ<r g −1 ∈ j(S)} ←→ JGG′ (j) : g 7→ Int(g −1 ) ◦ j
1:1Combine JGG with JGG′ and divide it again via stable G0 -conjugacy.
GHGJG′GJG

GJG′

GG0′

GJG′G0H
JH′

H
JHGJG′

H
JH′

H

H0′

H
JH′

H0H
JH′

GH
Kaletha’s descent lemma relates JG′ to JH′ .
′′The Waldspurger–Ngô transfer relates JGG′ to JHH′ .
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

13 / 20Computation of the contributions of shallow parts
1st form of ADS formulaΘπj (δ) =∑()′Θσ (gδ<r g −1 ) · µ̂Gg −1 X ∗ g log(δ≥r ) .
jg∈j(S)\G/G′gδ<r g −1 ∈j(S)Adler–Spice (’09) computed Θσ explicitly.
DeBacker–Spice (’18) sophisticated it based on a root-theoretic language.
Kaletha (’19) rewrote it via endoscopic invariants such as transfer factors.

ADS formula rewritten by KalethaΘπj (δ) =·∑e(G) ε(TG − TG′∗ )·e(G0 )DG (δ)()′∆G,abs(gδ<r g −1 ) · ϑ ◦ j −1 (gδ<r g −1 ) · ι̂Gg −1 X ∗ g log(δ≥r ) .
IIjg∈j(S)\G/G′gδ<r g −1 ∈j(S)⇝ Finally, by putting ∆spec
H,G (πj ) := hinv(jw , j), si, we get SECR.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

14 / 20

Twisted version of the Adler–DeBacker–Spice formulaLet δ ∈ G̃ be an elliptic strongly regular semisimple element.
Suppose: δ has a norm γ ∈ H.
⇝ By transferring γ = γ<r · γ≥r from H to G̃, get δ = δ<r · δ≥r .
(Note: δ<r ∈ G̃ and δ≥r ∈ Gδ<r )1st form of twisted ADS formulaΘ̃π(S,ϑ) (δ) =

∑()Gδ
−1
Θ̃σ (gδ<r g −1 ) · µ̂g−1<r(δ≥r ) .
X ∗ g exp

g∈S\G/Gδ<rg
δ<r ∈S̃This is completely parallel to the standard case!
⇝ can imitate Kaletha’s proof if we have a twisted version of LS descent...?
twisted version of Langlands–Shelstad descent= Waldspurger’s framework “l’endoscopie tordue n’est pas si tordue”.

Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

15 / 20

L’endoscopie tordue n’est pas si tordue
Waldspurger constructed another connected reductive group H̄ relating Gδ<rto Hγ<r via standard and non-standard endoscopy.
Then he proved that
Fourier transforms of orbital integrals are transferred between H′SC and H̄SC ,transfer factor for (H, G̃) is descended to (H̄, G′SC ).

G̃descent(Gδ<r =: G0 o

twisted endoscopy

G0SC
standard endoscopyH̄ oHdescent

Masao Oi (Kyoto Univ.)(Hγ<r =: H0 o
TECR for Kaletha L-packetsH̄SC
non-standard endoscopy

H0SC
RIMS conference 2021

16 / 20

Twisted version of Kaletha’s descent lemma
HG
In the standard case, a map JH′ → JG′ of Kaletha’s descent lemma wasconstructed via admissible isomorphisms (in the sense of LSK):

SH o/S∼=jjH

? _ T♭ o

H0 o
/ T♦  adm. isom.
∼=/ G0

Unfortunately, we cannot simply imitate this construction in the twisted case.
SH o∼=/ Sθ o oSS/ T♦ o o
T♦∃?

jH

H0 o


? _ T♭ oadm. isom.
∼=θ♢⇝ We need to “rigidify” admissible isomorphisms in some way.
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

17 / 20

Waldspurger’s “diagram”
We utilize Waldspurger’s notion of a diagram: (T♭ , T0 , T♦ , T♮ , h, g0 , g1 ).
T♭ oadm. isom.
∼=/ T♦ o oT♦

T0,θ o oT0/ Tθ o oTθ♢Int(g1 )Int(h)

TH oInt(g0 )A diagram encodes information about how an admissible isomorphism isgiven by conjugation (i.e., h, g0 , g1 ).
We can formulate a twisted version of Kaletha’s descent lemma (whichHGrelates JH′ to JG′ ) via diagrams.
⇝ get a comparison of index sets
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

18 / 20

Computation of the contributions of shallow parts
1st form of twisted ADS formulaΘ̃π(S,ϑ) (δ) =

∑

()Gδ
−1
Θ̃σ (gδ<r g −1 ) · µ̂g−1<r(δ≥r ) .
X ∗ g exp

g∈S\G/Gδ<rg
δ<r ∈S̃The contribution of the head is eventually reduced to a computation oftwisted characters of Weil representations of ﬁnite Heisenberg groups.
We upgrade it to “twisted ADSK” by looking at the deﬁnition of ∆ in thetwisted case carefully.

Proposition (twisted version of ADSK formula)Θ̃πj (δ) =·∑e(Gθ ) ε(TGθ − TG′∗ )·e(G0 )
DG̃ (δ)()′∆G̃,abs(gδ<r g −1 ) · ϑ̃ ◦ j̃ −1 (gδ<r g −1 ) · ι̂Gg −1 X ∗ g log(δ≥r ) .
IIjg∈j(S)\G/G′gδ<r g −1 ∈S̃j̃
Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

19 / 20Spectral transfer factors in twisted endoscopy∑∆specH,G (π)Θ̃π (δ) =π∈ΠGϕ∑ DH (γ)2∑
∆H,G (γ, δ)ΘπH (γ),2DG̃ (δ)
H

γ∈H/stπH ∈ΠϕWe put−1() ϑH ◦ j H(γ<r )
∗∗
∆spec
.
H,G (πj ) := ∆I,III γ<r · exp(XH,jH ), δ<r · exp(Xj ) ·−1ϑ̃ ◦ j̃ (δ<r )⇝ In fact, ∆spec
H,G (π) depends only on π.
standard case: ∆spec
H,G (π) = hinv(jw , j), si by Kaletha
Remark. I borrowed this idea from the argument of Mezo in his proof of thetwisted ECR for L-packets of real reductive groups (constructed by Langlands).

Masao Oi (Kyoto Univ.)

TECR for Kaletha L-packets

RIMS conference 2021

20 / 20