The Zelevinsky–Aubert duality for classical groups
Hiraku AtobeHokkaido University(Joint work with Alberto Mı́nguez)

January 28 2021

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

1 / 32

1

The Zelevinsky–Aubert duality

2

The case for GLn (F )3

The case for classical groups

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

2 / 32

1

The Zelevinsky–Aubert duality

2

The case for GLn (F )3

The case for classical groups

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

3 / 32SettingLet
F be a p-adic ﬁeld;
G ∈ {GLn (F ), Sp2n (F ), SO2n+1 (F )} be split;
P = M N be a standard parabolic subgroup of G;
IndG
P (πM ) be the (normalized) parabolically induction ofπM ∈ Rep(G)JacP (π) be the Jacquet module of π ∈ Rep(G);π 7→ [π] be the semisimpliﬁcation.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

4 / 32

Zelevinsky–Aubert duality
For π ∈ Rep(G), deﬁneX

DG (π) :=(−1)dim AM [IndGP (JacP (π))],P =M N

where P runs over all standard parabolic subgroups of G.

Theorem (Aubert 1996)
For any irreducible π, there exists ϵ ∈ {±1} such thatπ̂ := ϵ · DG (π)ˆ∼
is also irreducible. Moreover, π̂= π.
Call π̂ the Zelevinsky–Aubert dual of π.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

5 / 32

History (partial)

Alvis (’79, ’82) and Curtis (’80) deﬁned/studied DG for reductive
group G over a ﬁnite ﬁeld Fq .
Deligne–Lusztig (’83) showed that DG (RTG (θ)) = ±RTG (θ), where
RTG (θ) is the Deligne–Lusztig induction of θ ∈ Irr(T (Fq )). As anapplication, one can determine dim RTG (θ).
Zelevinsky (’80) deﬁned a similar involution for G = GLn (F )generated by Steinberg ↔ trivial. He predicted that it preserves theirreducibility.
After a work of S.-I. Kato (’93), Aubert (’96) deﬁned DG for general
reductive group G over F , and showed that it preserves theirreducibility.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

6 / 32

Motivation and Main ResultConjecture
If π is irreducible and unitary, then so is π̂.
In particular, this duality would produce many unitary representations.
Mœglin–Waldspurger (’86) gave an algorithm to compute π̂ for
G = GLn (F ). Together with Tadić’s result on the unitary dual, it
shows the conjecture in this case.

Main Result (A.-Mı́nguez)
We give an explicit algorithm for π̂ for G = Sp2n (F ), SO2n+1 (F ).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

7 / 32

Known results for G = Sp2n (F ), SO2n+1 (F )It is trivial that if π is supercuspidal, then π̂ = π (Aubert (’96)).
Hanzer (’09) showed that if π is a strongly positive discrete series,
then π̂ is unitary.
Arthur (’13) established an endoscopic classiﬁcation. As aconsequence, he proved that if π is tempered, then π̂ is unitary.
Matić (’17, ’19) gave an explicit formula for π̂ when π is discrete
series which is strongly positive or is in the ﬁrst induction step.
C. Jantzen (’18) gave an algorithm for π̂ of good parity when π is in
the “half-integral case”.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

8 / 32

1

The Zelevinsky–Aubert duality

2

The case for GLn (F )3

The case for classical groups

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

9 / 32

Notation
We explain Jantzen’s algorithm for τ̂ for τ ∈ Irr(GLn (F )).
Let ρ be a supercuspidal representation of GLd (F ).
A segment is a set[x, y]ρ = {ρ| · |x , ρ| · |x−1 , . . . , ρ| · |y },where x, y ∈ R such that x − y ∈ Z≥0 .
Denote by ∆ρ [x, y] (resp. Zρ [y, x]) the unique irreducible
subrepresentation (resp. quotient) of induced representationGLρ| · |x × · · · × ρ| · |y := IndP d(x−y+1)

Hiraku Atobe (Hokkaido)(F )The Zelevinsky–Aubert duality(ρ| · |x ⊠ · · · ⊠ ρ| · |y ).

January 28 2021

10 / 32

Jacquet modules of ∆ρ [x, y] and Zρ [y, x]∆ρ [x, y] is an essentially discrete series representation.
n−1(∆1GL1 (F ) [ n−1
2 , − 2 ] is called a Steinberg representation.)If ρ = 1GL1 (F ) , then Zρ [y, x] is a character of GLx−y+1 (F ).

Proposition (Zelevinsky 1980)
When Pd is a standard parabolic with Levi GLd (F ) × GLd(x−y) (F ),JacPd (∆ρ [x, y]) = ρ| · |x ⊠ ∆ρ [x − 1, y],JacPd (Zρ [y, x]) = ρ| · |y ⊠ Zρ [y + 1, x].
Here, we set ∆ρ [y − 1, y] := 1GL0 (F ) and Zρ [x + 1, x] := 1GL0 (F ) .

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

11 / 32

Derivatives
We introduce the following notion.

Deﬁnition (essentially due to Jantzen and Mı́nguez)
The k-th left ρ-derivative of τ ∈ Rep(GLn (F )) is a semisimple(k)representation Lρ (τ ) satisfying
X[JacPdk (τ )] = ρk ⊠ L(k)τi′ ⊠ τi′′ ,ρ (τ ) +i

where τi′ ⊠ τi′′ ∈ Irr(GLdk (F ) × GLn−dk (F )) such that τi′ 6∼= ρk .
(k)(k+1)(k)

When Lρ (τ ) 6= 0 but Lρ(τ ) = 0, call Lρ (τ ) the highest ρ-derivative.
(k)
Similarly, one can deﬁne the right ρ-derivative Rρ (τ ).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

12 / 32

Properties of derivatives
Let soc(Π) denote the socle of Π, i.e., the maximal semisimple
subrepresentation of Π.

Theorem (Jantzen 2007, Mı́nguez 2009)Let τ be an irreducible representation of GLn (F ).
(k)

The highest ρ-derivative Lρ (τ ) is irreducible.
(k)τ can be recovered from Lρ (τ ) by
τ = soc ρk × L(k)
(τ)
.
ρ(k)

There is an explicit formula for the highest ρ-derivative Lρ (τ ).
There is an explicit formula for soc(ρr × τ ), which is irreducible.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

13 / 32

Algorithm for computing τ̂(k)The analogous results for Rρ are also known.
The derivatives are related with the Zelevinsky dual as follows.

Proposition
For τ ∈ Irr(GLn (F )),(k)bL(k)ρ (τ̂ ) = Rρ (τ ) .

Using this proposition together with explicit formulas for derivatives andsocles, one can compute τ̂ by induction on n.

ExampleWe have Zρ [y, x]b = ∆ρ [x, y] since(1)

Lρ|·|x (∆ρ [x, y]) = ∆ρ [x − 1, y],

Hiraku Atobe (Hokkaido)(1)Rρ|·|x (Zρ [y, x]) = Zρ [y, x − 1].

The Zelevinsky–Aubert duality

January 28 2021

14 / 32

RemarkMœglin–Waldspurger (1986) gave an algorithm to compute theZelevinsky dual τ̂ by a diﬀerent form.
The idea to use derivatives for an algorithm to compute τ̂ is due toJantzen (2007).
There are three explicit formulas for the highest derivatives and soclesgiven by Janzten (2007), Mı́nguez (2009), and Lapid–Mı́nguez (2016).
We will use the explicit formula of Lapid–Mı́nguez, which uses thenotion of best matching functions.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

15 / 32

1

The Zelevinsky–Aubert duality

2

The case for GLn (F )3

The case for classical groups

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

16 / 32

Derivatives

From now, let G = Sp2n (F ) or G = SO2n+1 (F ).
Fix an irreducible supercuspidal representation ρ of GLd (F ).

Deﬁnition(k)

The k-th ρ-derivative of π is a semisimple representation Dρ (π) satisfying
X[JacPdk (π)] = ρk ⊠ Dρ(k) (π) +τi ⊠ πi ,iwhere Pdk is a parabolic of G with Levi GLdk (F ) × G0 , andτi ⊠ πi ∈ Irr(GLdk (F ) × G0 ) such that τi ∼6 ρk .
=(k)

(k+1)

If Dρ (π) 6= 0 but DρHiraku Atobe (Hokkaido)(k)(π) = 0, call Dρ (π) the highest ρ-derivative.

The Zelevinsky–Aubert duality

January 28 2021

17 / 32

Compatibility with the Aubert duality

As in the GLn -case, derivatives are compatible with the Aubert duality.

Proposition
We have(k)Dρ(k) (π̂) = Dρ∨ (π)b.
Here, ρ∨ denotes the contragredient of ρ.
To obtain an algorithm for π̂, we want to solve the following questions:(k)Is the highest derivative Dρ (π) irreducible?
(k)

Can we establish an explicit formula for Dρ (π)?
(k)

Can π be recovered from Dρ (π)?

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

18 / 32

Properties of derivatives

Proposition (Jantzen 2018, Lapid–Tadić 2020, A.–Mı́nguez)
Suppose that ρ is not self-dual. Let π be irreducible representation of G.
(k)The highest derivative Dρ (π) is irreducible.
The socle soc(ρr ⋊ π) is irreducible.
They are related as
π = soc ρk ⋊ Dρ(k) (π) .
(k)Hence, we want explicit formulas for Dρ (π) and for soc(ρr ⋊ π).
From now, ﬁx an irreducible self-dual supercuspidal representation ρ of(k)GLd (F ), and consider ρ| · |x -derivative Dρ|·|x (π) for x ∈ R.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

19 / 32

Langlands classiﬁcation
As a parametrization of Irr(G), we will use the Langlands classiﬁcation.

Theorem (Langlands, Arthur)Any π ∈ Irr(G) can be written as
π = L(∆ρ1 [x1 , y1 ], . . . , ∆ρr [xr , yr ]; πtemp ):= soc (∆ρ1 [x1 , y1 ] × · · · × ∆ρr [xr , yr ] ⋊ π(ϕ, η)) ,whereρi is unitary supercuspidal, and x1 + y1 ≤ · · · ≤ xr + yr < 0;π(ϕ, η) is an irreducible tempered representation.
(ϕ, η) is the tempered L-parameter for π(ϕ, η).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

20 / 32

Jantzen’s algorithm

Jantzen (2018) suggested an algorithm to compute the highest
ρ| · |x -derivative of π for x 6= 0.
The ﬁrst step is to rewrite
π = soc (τ ⋊ πA ) ,whereτ = soc(∆ρ1 [x1 , y1 ] × · · · × ∆ρr [xr , yr ]) with [xi , yi ]ρi 6= [x − 1, −x]ρfor any i;
πA = L(∆ρ [x − 1, −x]t ; π(ϕ, η)) with t ≥ 0.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

21 / 32

Explicit formula for the highest derivatives

π = soc(τ ⋊ πA ) with πA = L(∆ρ [x − 1, −x]t ; π(ϕ, η)).
(a)

(k)

Use the best matching functions for Rρ|·|−x (τ ) and Lρ|·|x (τ ).
(k )Use an A-parameter for Dρ|·|Ax (πA ).
By combining them with Jantzen’s algorithm, we have:

Theorem (A.–Mı́nguez)
Assume that ρ| · |x is not self-dual. Let π ∈ Irr(G). Then we have(k)explicit formulas for the highest ρ| · |x -derivative Dρ|·|x (π) and for thesocle soc((ρ| · |x )r ⋊ π) in terms of the best matching functions andA-parameters.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

22 / 32

The ﬁrst step of the algorithm

We can now state the ﬁrst step of our algorithm.

Algorithm: Step 1
Let π be an irreducible representation of G. If there exist ρ and x ∈ R such(k)that ρ| · |x is not self-dual and π0 := Dρ|·|x (π) 6= 0 for k > 0, then useπ̂ = soc (ρ| · |−x )k ⋊ π̂0
to reduce the computation of π̂ to the one of π̂0 .

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

23 / 32

Problem
Step 1 reduces the problem to the case where π is irreducible such thatDρ(1) (π) 6= 0 =⇒ ρ is self-dual.

(∗)

From now, assume that ρ is self-dual. In this case, the ρ-derivatives(k)(k)Dρ (π) is diﬃcult, and π cannot be recovered from Dρ (π) in general.

ExampleIf σ is supercuspidal such that ρ ⋊ σ = π+ ⊕ π− . ThenDρ(1) (π+ ) = Dρ(1) (π− ) = σ.
But it is easy to see that π̂+ = π− by deﬁnition (Aubert).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

24 / 32

Tempered case
As in this example, we can compute π̂ if π is tempered.

Proposition (A.–Mı́nguez)
Suppose that π = π(ϕ, η) is tempered and satisﬁes (∗). Set{ρ ⊂ ϕ | even mult.} = {ρ1 , . . . , ρr } and yi = max{ di2−1 | ρi ⊠ Sdi ⊂ ϕ}.
Assume that y1 ≥ · · · ≥ yt > 0 = yt+1 = · · · = yr . Thenπ̂ = L(∆ρ1 [0, −y1 ], . . . , ∆ρt [0, −yt ]; π(ϕ′ , η ′ )),where
ϕ′ = ϕ −tMρi ⊠ (S1 ⊕ S2yi +1 )i=1and η ′ (ρ ⊠ Sd ) 6= η(ρ ⊠ Sd )Hiraku Atobe (Hokkaido)⇐⇒ ρ ∈ {ρ1 , . . . , ρr }.

The Zelevinsky–Aubert duality

January 28 2021

25 / 32

New derivatives
The remaining case is that π is non-tempered and satisﬁes (∗).
The key idea to deal with this case is to deﬁne new derivatives.

Deﬁnition (A.-Mı́nguez)(k)Deﬁne the k-th ∆ρ [0, −1]-derivative D∆ρ [0,−1] (π) and the k-th(k)Zρ [0, 1]-derivative DZρ [0,1] (π) as semisimple representations satisfying(k)[JacP2dk (π)] = ∆ρ [0, −1]k ⊠D∆ρ [0,−1] (π)(k)+Zρ [0, 1]k ⊠DZρ [0,1] (π) + (others).
One can deﬁne the notions of the highest ∆ρ [0, −1]-derivatives and thehighest Zρ [0, 1]-derivatives.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert dualityJanuary 28 2021

26 / 32

Properties
These derivatives are substitute for the ρ-derivatives.

Proposition (A.–Mı́nguez)
Suppose that π is irreducible and satisﬁes (∗).
1(k)

The highest ∆ρ [0, −1]-derivative D∆ρ [0,−1] (π) (resp. the highest(k)Zρ [0, 1]-derivative DZρ [0,1] (π)) is irreducible.
The socles soc(∆ρ [0, −1]r ⋊ π) (resp. soc(Zρ [0, 1]r ⋊ π)) isirreducible.
(k)
3 π ∼(π)).
= soc(Zρ [0, 1]k ⋊ D2Zρ [0,1](k)(k)
4 DZ [0,1] (π̂) = D∆ [0,−1] (π)b.
ρρActually, we only assume a weaker condition than (∗).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

27 / 32

The explicit formulas

Moreover:

Theorem (A.–Mı́nguez)
Suppose that π satisﬁes (∗). Then we have explicit formulas for(k)the highest ∆ρ [0, −1]-derivatives D∆ρ [0,−1] (π);(k)

the highest Zρ [0, 1]-derivatives DZρ [0,1] (π);
the socle soc(Zρ [0, 1]r ⋊ π)in terms of matching functions and A-parameters.
Actually, we only assume a weaker condition than (∗).

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

28 / 32

Completion of the algorithm

Algorithm: Step 2 and Step 3
Let π be an irreducible representation satisfying thatDρ(1) (π) 6= 0 =⇒ ρ is self-dual.

(∗)

(k)
If π is non-tempered, can ﬁnd ρ such that π0 := D∆ρ [0,−1] (π) 6= 0 for
k > 0, and usekπ̂ = soc Zρ [0, 1] ⋊ π̂0to reduce the computation of π̂ to the one of π̂0 .
If π is tempered, we have an explicit formula for π̂.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

29 / 32

ExampleNow we set ρ = 1GL1 (F ) , and drop ρ in the notation.
Let π(xη11 , . . . , xηr r ) = π(ϕ, η) with ϕ = ⊕ri=1 S2xi +1 and η(S2xi +1 ) = ηi .

ExampleLet us consider L(∆[0, −2]; π(0− , 0− , 1+ )) ∈ Sp10 (F ), which satisﬁes (∗).
HenceL(∆[0, −2]; π(0− , 0− , 1+ ))(1)

D∆[0,−1]

L(| · |−2 ; π(0− , 0− , 1+ ))D(1)|·|−2π(0− , 0− , 1+ ).
By the explicit formula, π̂(0− , 0− , 1+ ) = L(∆[0, −1]; π(0+ )).
Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

30 / 32

Example

Example (continued)
Now take socles
L(∆[0, −1]; π(0+ ))soc(|·|2 ⋊−)L(∆[0, −2]; π(0+ ))soc(Z[0,1]⋊−)

L(∆[0, −2]; π(0− , 0− , 1+ )).
We conclude that L(∆[0, −2]; π(0− , 0− , 1+ )) is ﬁxed by theZelevinsky–Aubert duality.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

31 / 32

Remarks

RemarkEven if we start at a tempered representation, we need to compute∆ρ [0, −1]-derivatives and the Zρ [0, 1]-socles in general.
As an application of Zρ [0, 1]-derivatives, one can reﬁne Mœglin’s
explicit construction of local A-packets (A.).
In that work, a conjectural “formula” for π̂ for π of Arthur type wasformulated.
Although the explicit formulas for derivatives and for socles arecomplicated, it would be easy to write a computer program.

Hiraku Atobe (Hokkaido)

The Zelevinsky–Aubert duality

January 28 2021

32 / 32