Bernstein-Zelevinsky Derivative and Their Analogues AFW Workshop, - - PowerPoint PPT Presentation

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Bernstein-Zelevinsky Derivative and Their Analogues

AFW Workshop, Duquesne U Pittsburgh Zhuohui Zhang, WIS Israel March 10, 2019

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Setup (Bernstein-Zelevinsky 77)

Fix a non-archimedian local field F, for example F = Qp; Consider the following series of groups

Gn = GL(n, F), in particular G0 = 1; Mirabolic Pn = Gn−1 v

1

• =

Gn−1 0

1

• Vn ⊂ Gn with

v = (v1, . . . , vn−1)t ∈ F n, in particular P1 = 1;

Goal of the B-Z 77 Work:

Classifying irreducible representations of Gn Calculating the composition series of the restrictions of parabolically induced representations

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Notations

We denote by Rn Grothendieck group of equivalence classes of admissible representations of Gn of finite length R = ⊕n≥0Rn is a graded algebra Composition series JH0(π) as the linear combination of irreducible subquotients of π, counting multiplicities For each ordered partition α = (n1, . . . , nr), denote by Gα = Gn1 × . . . × Gnr aligned in the blocks of the diagonal Taking a representation ρi ∈ Rni, we can define a B-Z product ρn1 × . . . × ρnr = indGn

Gα(ρn1 ⊗ . . . ⊗ ρnr )

Support of an irreducible π ∈ RepGn: π ∈ JH0(ρn1 × . . . × ρnr ) where ρi cuspidal, will appear as a sub if the factors are permuted correctly This product on R is commutative under ×

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Specification of Functors

P = MU a parabolic with ψ a character on U normalized by M ν is the determinant character on M iU,ψ : RepM − → RepG compactly supported induction iU,ψ(σ) = {f : G → Vπ |

f compactly supported f (umg)=ψ(u)ν1/2(m)σ(m)f (g) with f (gk)=f (g) for some open compact K⊂G

} rU,ψ : RepG − → RepM Jacquet functors rU,ψ(π) = π v ∈ Vπ | π(u)v − ψ(u)v ⊗ ν−1/2 If P is a standard parabolic we can denote by rU,1 the rM,G or rβ,(n) if the parabolic is given by a partition β of n.

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Specification of Functors

We fix an additive character ψ on F, then we can define Ψ− : RepPn − → RepGn−1 by Ψ− = rUn,1 Ψ+ : RepGn−1 − → RepPn by Ψ+ = iUn,1 Φ− : RepPn − → RepPn−1 by Φ−(π) = rUn,ψ Φ+ : RepPn−1 − → RepPn by Φ+(π) = iUn,ψ What really makes a difference between these functors are the normalizers of the character ψ.

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Multiplication Table of Functors

All these functors are exact Adjunctions: Ψ− ⊣ Ψ+ and Φ+ ⊣ Φ− Φ− ◦ Ψ+ = 0 and Ψ− ◦ Φ+ = 0 Φ− ◦ Φ+ = ✶ and Ψ− ◦ Ψ+ = ✶ ✶ = Φ+ ◦ Φ− + Ψ+ ◦ Ψ−

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Derivatives

Derivatives: For π ∈ RepPn, define π(k) = Ψ− ◦ (Φ−)k−1(τ) These functors maps finite length admissible representations to finite length admissible representations, and can be considered as matrices

• n R.

Any π ∈ RepPn can be written as a sum π =

n

• k=0

(Φ+)(k−1) ◦ Ψ+(π(k)) These summands appear as the composition factors in the Jordan-Hölder series. Therefore Any irreducible representation π ∈ RepPn is equivalent to (Φ+)(k−1) ◦ Ψ+(ρ) where ρ is an irreducible representation of Gn−k.

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Additional Structures on R

Coproduct c(π) =

0≤k≤n r(k,n−k),(n)(π) for π ∈ RepGn;

A (graded) ring homomorphism D(π) =

0≤k≤n(π)(k)

Leibniz rule (π × σ)(k) = k

i=0 π(i) × σ(k−i)

Making use of these structures: Building blocks: cuspidal representations if rM,G(π) = 0 for any standard Levi subgroup M ⊂ G For irreducible cuspidal representation ρ ∈ RepGn we have D(ρ) = ρ + 1

From above, we have D(ρ1 × . . . × ρr) = (ρ1 + 1) . . . (ρr + 1)

Irreducibility criterion: ρi = νρj for any pairs of i, j

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Geometric Lemma

Calculation of the functor F = rγ,(n) ◦ i(n),β : RepGβ → Gγ F =

w∈W (β,γ) Fw where Fw = iγ,γ′ ◦ w ◦ rβ′,β

Weyl group elements W (β, γ) =

• w|

w(k)<w(l) if k<l with k,l∈same Jordan block of β w−1(k)<w−1(l) if k<l with k,l∈same Jordan block of γ

• Refined blocks γ′ = γ ∩ w(β) < γ, β′ = β ∩ w −1(γ) < β

Fw(ρ1 ⊗ . . . ⊗ ρr) =

ki≥0 σ(k1, . . . , kr), where

Matrix B(w) = (|Blockβ

i ∩ w −1Blockγ j |)

Set β = (n1, . . . , nr), γ = (m1, . . . , ms), then

Each row βi = (bi,1, . . . , bi,s) a partition of ni Each column βj = (b1,j, . . . , br,j) a partition of mj Jordan-Hölder series rβi ,(ni )(ρi) =

• l=1

σ(l)

i

=

• l=1

σ(l)

i,1 ⊗ . . . ⊗ σ(l) i,s

For any (k1, . . . , kr) put σj = σ(k1)

1,j ⊗ . . . ⊗ σ(kr ) r,j

irreducible for Gγj Define σ(k1, . . . , kr) = i(m1),γ1σ1 ⊗ . . . ⊗ i(ms),γs σs

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Bernstein-Zelevinsky Classification

Segments of cuspidal representations ∆ = [ρ, νρ, . . . , νkρ], where ρ ∈ RepGk is a cuspidal representation, character ν = | det |. A Multiset a of segments is a set Ω of segments with a multiplicity function ϕ : Ω − → Z+ Linked: ∆1 ∼ ∆2 iff they don’t contain one another, and ∆1 ∪ ∆2 is a segment

Precedes ∆1 < ∆2 if ρ2 = ν>0ρ1 Juxtaposed: ∆1 ↔ ∆2 iff ρ2 = νρ′

1 or ρ1 = νρ′ 2

For each segment ∆, denote by ∆ the irreducible sub in the B-Z product ρ × νρ . . . × νkρ ∆1 × . . . × ∆r,where ∆i are cuspidal segments, is irreducible iff. ∆i ≁ ∆j

If the segments are linked, then its irreducible constituents consist of the multiset of segments obtained from elementary operations: replace {∆1, ∆2} by {∆∪, ∆∩}

There is a bijection between O = {finite multiset of segments in C } ↔ IrrGn ϕ ↔ ϕ

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Examples

GL(2, F), induce from characters χ1, χ2

{∆1 = [χ1], ∆2 = [χ2]}, if χ1 = ν±χ2 then irreducible {∆ = [ν−1/2χ, ν1/2χ]}, get Steinberg χ ⊗ St + χ det, Steinberg is the sub {∆1 = [ν1/2χ], ∆2 = [ν−1/2χ]}, the 1-dimensional representation is the sub Supercuspidal

GL(3, F)

Support {χ1, χ2, χ3}, segments ∆i = [χi], irreducible in general Support {χ1, νχ1, χ2} with χ2 / ∈ {ν−1χ1, . . . ν2, χ1}

{∆1 = [χ1, νχ1], ∆2 = [χ2]} irreducible {∆1 = [νχ1], ∆2 = [χ1], ∆3 = [χ2]} [χ1, νχ1] × [χ2] < [χ1] × [νχ1] × [χ2]

Support {χ1, νχ1, νχ1}

{∆1 = [χ1, νχ1], ∆2 = [νχ1]}, {∆1 = [νχ1], ∆2 = [χ1], ∆3 = [νχ1]} [χ1, νχ1], [νχ1] ⊕ [χ1], [νχ1], [νχ1] = [νχ1] × [χ1] × [νχ1]

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Archimedian Analogues [Aizenbud-Gourevitch-Sahi]?

Definition of Φ− and Ψ− Functors What category? M (G) smooth, admissible, Fréchet representations

• f moderate growth [Casselman 89, Wallach 92].

This category is equivalent to the category MHC(G) of Harish-Chandra modules (Casselman-Wallach, written in Bernstein-Krötz) For π ∈ M (G),

Φ−(π) =

π v∈Vπ|π(u)v−ψ(u)v ⊗ | det |−1/2 from M (Pn) to M (Pn−1)

Ψ−(π) = liml

π π(β)v|v∈Vπ,β∈U(vn)≤l from M (Pn) to M (Gn−1)

Derivative: D(k)(π) = Ψ− ◦ (Φk−1)(π) : M (Gn) → M (Gn−k). Pre-derivative: E (k)(π) = Φk−1(π) : M (Gn) → M (Pn−k+1)

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Archimedian Analogues [Aizenbud-Gourevitch-Sahi]?

Annihilator varieties: Vπ = SpecGrAnn(π) is a union of closures

• f nilpotent orbits ⊂ g∗ ∼

= g Wavefront set: WF(π) = ∪u,v∈VπWFeπ(g)u, v the union of wavefront sets (as distributions) of matrix coefficients, WF(π) ⊂ N the nilpotent cone of g∗ Depth: minimal degree d such that X d = 0 for all X ∈ Vπ Depth(π) = highest order of derivative. Denote by M ≤d(G) the subcategory with representations with depth ≤ d

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Archimedian Derivatives

One of the main problems is admissibility. Also, it is unknown if the derivatives are exact functors in general. But:

AGS15: (·)(d) is an exact functor M ≤d(G) − → M (Gn−d), for arbitrary k exactness is unknown Another AGS15: E k is an exact functor M (Gn) → M (pn−k+1)

A calculation of pre-derivatives [AGS 15]: for a partition (n1, . . . , nk)

• f n we have E k(χ1 × . . . × χk) = (χ1)|Gn1−1 × . . . × (χk)|Gnk −1

Depth and wave-front set [GS 15]: if π irreducible, with WF(π) = O(n1,...,nk), then Depth(π) = n1 and WF(E n1(π)) = O(n2,...,nk)

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Application: Whittaker models [Gomez, Gourevitch, Sahi]

M≤O(Gn) the subcategory of representations with wave-front set in the closure of O, and M<O(Gn) that with wave-front set NOT containing O Whittaker models W(λ1,...,λk)(π) = (E λk . . . E λ1(π))∗ as a functor from the quotient category

W(λ1,...,λk ) is an exact faithful functor from M ≤(λ1,...,λk )(Gn)/M <(λ1,...,λk )(Gn) to finite dim. vector spaces If π is irreducible unitary representation, or monomial representation, and π / ∈ M <(λ1,...,λk )(Gn), then W(λ1,...,λk )(π) one-dimensional

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Automorphic Analogues? [ZZ Ongoing]

Candidates for the Ψ− and Φ− operations: Fourier-Whittaker coefficients Ω−φ(g) =

• U(F)\U(A) φ(ng)ψ(n)dn

Gives Φ− and Ψ− respectively depending on whether ψ = 0 Candidate for Ψ+ and Φ+? Eisenstein series: E (φ, g) =

• γ∈Pk−1(Q)\Gk−1(Q)

φ(γg) Consider the set V = {1, . . . , n − 1} and its subset S ⊂ V , we can translate the subset S into a sequence of derivative and pre-derivative operations as follows S = (1, 1, 0, 0, 1, 1) → DS = E 2 ◦ D ◦ D3 and S = (1, 1, 0, 0, 1, 1) → ES = E1E2E5E6 Then there is an expansion ✶ =

• S⊂V

ESDS

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

How to make it work?

Refinement of Fourier-Whittaker Coefficients: in fact a lot of the Fourier-Whittaker coefficients determine each other. For example on GL(4) W0,1,1(g) +

• q∈Q∗

W1,1,1( 1

1 1 q 1

• g) =
• A2 W1,1,0(

1

1 1 s t 1

• g)dsdt+
• q∈Q∗
• A2 W1,1,1(

1

1 1 q 1

1

1 1 s t 1

• g)dsdt

(Gourevitch, Gustafsson, Kleinschmidt, Person, Sahi 19) A refinement of Fourier expansion formula of simply-laced split group for minimal/next-to-minimal automorphic functions. (Wang 16) Analogue of the "geometric lemma": CTP′ ◦ EisP =

• w∈WM,M′

(EisM′

wM1w −1 ◦ RG M1,w ◦ CT M M1)

where RG

M1,w is the intertwining operator for induced representations

from the parabolic M′

Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues

B-Z Classification for p-adic Groups Archimedian Analogue Automorphic Analogues

Thank You

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Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues