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 = Q p ; Consider the following series of groups G n = GL ( n , F ) , in particular G 0 = 1; � G n − 1 v � G n − 1 0 � � Mirabolic P n = = V n ⊂ G n with 0 1 0 1 v = ( v 1 , . . . , v n − 1 ) t ∈ F n , in particular P 1 = 1; Goal of the B-Z 77 Work : Classifying irreducible representations of G n 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 R n Grothendieck group of equivalence classes of admissible representations of G n of finite length R = ⊕ n ≥ 0 R n is a graded algebra Composition series JH 0 ( π ) as the linear combination of irreducible subquotients of π , counting multiplicities For each ordered partition α = ( n 1 , . . . , n r ) , denote by G α = G n 1 × . . . × G n r aligned in the blocks of the diagonal Taking a representation ρ i ∈ R n i , we can define a B-Z product ρ n 1 × . . . × ρ n r = ind G n G α ( ρ n 1 ⊗ . . . ⊗ ρ n r ) Support of an irreducible π ∈ Rep G n : π ∈ JH 0 ( ρ n 1 × . . . × ρ n r ) where ρ i 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 i U ,ψ : Rep M − → Rep G compactly supported induction f compactly supported i U ,ψ ( σ ) = { f : G → V π | f ( umg )= ψ ( u ) ν 1 / 2 ( m ) σ ( m ) f ( g ) } with f ( gk )= f ( g ) for some open compact K ⊂ G r U ,ψ : Rep G − → Rep M Jacquet functors π � v ∈ V π | π ( u ) v − ψ ( u ) v � ⊗ ν − 1 / 2 r U ,ψ ( π ) = If P is a standard parabolic we can denote by r U , 1 the r M , G or r β, ( 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 − : Rep P n − → Rep G n − 1 by Ψ − = r U n , 1 Ψ + : Rep G n − 1 − → Rep P n by Ψ + = i U n , 1 Φ − : Rep P n − → Rep P n − 1 by Φ − ( π ) = r U n ,ψ Φ + : Rep P n − 1 − → Rep P n by Φ + ( π ) = i U n ,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 ∈ Rep P n , 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 on R . Any π ∈ Rep P n can be written as a sum n (Φ + ) ( k − 1 ) ◦ Ψ + ( π ( k ) ) � π = k = 0 These summands appear as the composition factors in the Jordan-Hölder series. Therefore Any irreducible representation π ∈ Rep P n is equivalent to (Φ + ) ( k − 1 ) ◦ Ψ + ( ρ ) where ρ is an irreducible representation of G n − 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 ∈ Rep G n ; 0 ≤ k ≤ n ( π ) ( k ) A (graded) ring homomorphism D ( π ) = � Leibniz rule ( π × σ ) ( k ) = � k i = 0 π ( i ) × σ ( k − i ) Making use of these structures: Building blocks: cuspidal representations if r M , G ( π ) = 0 for any standard Levi subgroup M ⊂ G For irreducible cuspidal representation ρ ∈ Rep G n we have D ( ρ ) = ρ + 1 From above, we have D ( ρ 1 × . . . × ρ r ) = ( ρ 1 + 1 ) . . . ( ρ r + 1 ) Irreducibility criterion : ρ i � = νρ j 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 = r γ, ( n ) ◦ i ( n ) ,β : Rep G β → G γ w ∈ W ( β,γ ) F w where F w = i γ,γ ′ ◦ w ◦ r β ′ ,β F = � Weyl group elements � w ( k ) < w ( l ) if k < l with k , l ∈ same Jordan block of β � W ( β, γ ) = w | 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 F w ( ρ 1 ⊗ . . . ⊗ ρ r ) = � k i ≥ 0 σ ( k 1 , . . . , k r ) , where Matrix B ( w ) = ( | Block β i ∩ w − 1 Block γ j | ) Set β = ( n 1 , . . . , n r ) , γ = ( m 1 , . . . , m s ) , then Each row β i = ( b i , 1 , . . . , b i , s ) a partition of n i Each column β j = ( b 1 , j , . . . , b r , j ) a partition of m j Jordan-Hölder series σ ( l ) σ ( l ) i , 1 ⊗ . . . ⊗ σ ( l ) � � r β i , ( n i ) ( ρ i ) = = i , s i l = 1 l = 1 For any ( k 1 , . . . , k r ) put σ j = σ ( k 1 ) 1 , j ⊗ . . . ⊗ σ ( k r ) irreducible for G γ j r , j Define σ ( k 1 , . . . , k r ) = i ( m 1 ) ,γ 1 σ 1 ⊗ . . . ⊗ i ( m s ) ,γ s σ s 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, . . . , ν k ρ ] , where ρ ∈ Rep G k 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 ρ 2 = ν > 0 ρ 1 Juxtaposed: ∆ 1 ↔ ∆ 2 iff ρ 2 = νρ ′ 1 or ρ 1 = νρ ′ 2 For each segment ∆ , denote by � ∆ � the irreducible sub in the B-Z product ρ × νρ . . . × ν k ρ � ∆ 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 } ↔ Irr G n ↔ � 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 χ 1 , χ 2 { ∆ 1 = [ χ 1 ] , ∆ 2 = [ χ 2 ] } , if χ 1 � = ν ± χ 2 then irreducible { ∆ = [ ν − 1 / 2 χ, ν 1 / 2 χ ] } , get Steinberg χ ⊗ St + χ det , Steinberg is the sub { ∆ 1 = [ ν 1 / 2 χ ] , ∆ 2 = [ ν − 1 / 2 χ ] } , the 1-dimensional representation is the sub Supercuspidal GL ( 3 , F ) Support { χ 1 , χ 2 , χ 3 } , segments ∆ i = [ χ i ] , irreducible in general ∈ { ν − 1 χ 1 , . . . ν 2 , χ 1 } Support { χ 1 , νχ 1 , χ 2 } with χ 2 / { ∆ 1 = [ χ 1 , νχ 1 ] , ∆ 2 = [ χ 2 ] } irreducible { ∆ 1 = [ νχ 1 ] , ∆ 2 = [ χ 1 ] , ∆ 3 = [ χ 2 ] } � [ χ 1 , νχ 1 ] � × � [ χ 2 ] � < � [ χ 1 ] � × � [ νχ 1 ] � × � [ χ 2 ] � Support { χ 1 , νχ 1 , νχ 1 } { ∆ 1 = [ χ 1 , νχ 1 ] , ∆ 2 = [ νχ 1 ] } , { ∆ 1 = [ νχ 1 ] , ∆ 2 = [ χ 1 ] , ∆ 3 = [ νχ 1 ] } � [ χ 1 , νχ 1 ] , [ νχ 1 ] � ⊕ � [ χ 1 ] , [ νχ 1 ] , [ νχ 1 ] � = � [ νχ 1 ] � × � [ χ 1 ] � × � [ νχ 1 ] � Zhuohui Zhang, WIS Israel Bernstein-Zelevinsky Derivative and Their Analogues
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