Branching laws for non-generic representations Max Gurevich - - PowerPoint PPT Presentation

▶

Nov 07, 2022 157 likes •768 views

Restriction Problem Categorical equivalences On unique irreducible quotients Branching laws for non-generic representations Max Gurevich National University of Singapore June 2018, Catholic University of America, Washington, DC Max Gurevich

SLIDE 1

Restriction Problem Categorical equivalences On unique irreducible quotients

Branching laws for non-generic representations

Max Gurevich

National University of Singapore

June 2018, Catholic University of America, Washington, DC

Max Gurevich Branching laws for non-generic representations

SLIDE 2

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 3

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 4

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 5

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 6

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 7

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Main Problem

F - a p-adic field. π - irreducible smooth representation of GLnpFq. How does π|GLn´1pFq decompose? What are the irreducible quotients of π|GLn´1pFq? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hompπ1|GLn´1pFq, π2q ‰ 0 , for all generic irreducible π1, π2. Goal: What can be said about the non-generic case?One would like to describe the pairs pπ1, π2q for which the Hom space above is non-zero.

Max Gurevich Branching laws for non-generic representations

SLIDE 8

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Tools: Parabolic induction

Representations of tGLnpFqun have an intriguing product structure. Given π1 P Rep GLn1pFq and π2 P Rep GLn2pFq, we can think of π1 b π2 as a representation of the parabolic subgroup P “ ˆ GLn1pFq ˚ GLn2pFq ˙ ă GLn1`n2pFq . π1 ˆ π2 :“ ind

GLn1`n2pFq P

pπ1 b π2q P Rep GLn1`n2pFq

Max Gurevich Branching laws for non-generic representations

SLIDE 9

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Tools: Parabolic induction

Representations of tGLnpFqun have an intriguing product structure. Given π1 P Rep GLn1pFq and π2 P Rep GLn2pFq, we can think of π1 b π2 as a representation of the parabolic subgroup P “ ˆ GLn1pFq ˚ GLn2pFq ˙ ă GLn1`n2pFq . π1 ˆ π2 :“ ind

GLn1`n2pFq P

pπ1 b π2q P Rep GLn1`n2pFq

Max Gurevich Branching laws for non-generic representations

SLIDE 10

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Tools: Parabolic induction

Representations of tGLnpFqun have an intriguing product structure. Given π1 P Rep GLn1pFq and π2 P Rep GLn2pFq, we can think of π1 b π2 as a representation of the parabolic subgroup P “ ˆ GLn1pFq ˚ GLn2pFq ˙ ă GLn1`n2pFq . π1 ˆ π2 :“ ind

GLn1`n2pFq P

pπ1 b π2q P Rep GLn1`n2pFq

Max Gurevich Branching laws for non-generic representations

SLIDE 11

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Tools: Derivatives

For every 0 ď i ď n, there are well-defined functors Rep GLnpFq Ñ Rep GLn´ipFq π ÞÑ πpiq π ÞÑ

piqπ

called left and right Bernstein-Zelevinski derivatives. Derivatives of an irreducible representation are objects of finite length.

Max Gurevich Branching laws for non-generic representations

SLIDE 12

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Tools: Derivatives

For every 0 ď i ď n, there are well-defined functors Rep GLnpFq Ñ Rep GLn´ipFq π ÞÑ πpiq π ÞÑ

piqπ

called left and right Bernstein-Zelevinski derivatives. Derivatives of an irreducible representation are objects of finite length.

Max Gurevich Branching laws for non-generic representations

SLIDE 13

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Filtration

π P Irr GLnpFq Classical results of Bernstein-Zelevinski give a filtration of the GLn´1pFq-representation pπ|GLn´1pFq, Uq t0u “ Un`1 Ă Un Ă ¨ ¨ ¨ Ă U1 “ U , so that each Ui{Ui`1 is well understood.

Max Gurevich Branching laws for non-generic representations

SLIDE 14

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

A question about morphisms

Hompπ1|GLn´1pFq, π2q ‰ 0 ù ñ Hom pUi{Ui`1, π2q ‰ 0 , for some i. From an application of Frobenius reciprocity (a certain adjunction of functors), we can rewrite the above Hom space in another form and obtain: HomGLn´ipFq ´ | det |1{2

F

b πpiq

1 , pi´1qπ2

¯ ‰ 0 . Now, we are asking about Hom spaces of finite-length representations!

Max Gurevich Branching laws for non-generic representations

SLIDE 15

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

A question about morphisms

Hompπ1|GLn´1pFq, π2q ‰ 0 ù ñ Hom pUi{Ui`1, π2q ‰ 0 , for some i. From an application of Frobenius reciprocity (a certain adjunction of functors), we can rewrite the above Hom space in another form and obtain: HomGLn´ipFq ´ | det |1{2

F

b πpiq

1 , pi´1qπ2

¯ ‰ 0 . Now, we are asking about Hom spaces of finite-length representations!

Max Gurevich Branching laws for non-generic representations

SLIDE 16

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

A question about morphisms

Hompπ1|GLn´1pFq, π2q ‰ 0 ù ñ Hom pUi{Ui`1, π2q ‰ 0 , for some i. From an application of Frobenius reciprocity (a certain adjunction of functors), we can rewrite the above Hom space in another form and obtain: HomGLn´ipFq ´ | det |1{2

F

b πpiq

1 , pi´1qπ2

¯ ‰ 0 . Now, we are asking about Hom spaces of finite-length representations!

Max Gurevich Branching laws for non-generic representations

SLIDE 17

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Nice classes of representations

Giving a branching law for all irreducible representations is considered a wild problem. Instead, we will focus on nice subclasses of representations. Motivated by harmonic analysis on automorphic spaces, Arthur attached unitarizable representations to certain (strict) Arthur parameters. We call them (strict) Arthur-type representations.

Max Gurevich Branching laws for non-generic representations

SLIDE 18

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Nice classes of representations

Giving a branching law for all irreducible representations is considered a wild problem. Instead, we will focus on nice subclasses of representations. Motivated by harmonic analysis on automorphic spaces, Arthur attached unitarizable representations to certain (strict) Arthur parameters. We call them (strict) Arthur-type representations.

Max Gurevich Branching laws for non-generic representations

SLIDE 19

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Arthur-type representations

Arthur parameter - combinatorial description: A collection of triples φ “ tpρi, ai, biquiPI , where ρi P Irr GLdipFq are supercuspidal representations, ai, bi P Zą0. Each triple pρi, ai, biq defines a Speh representation πai,bi

ρi

P Irr GLdiaibipFq. Such φ gives rise to πpφq “ ą

iPI

πai,bi

ρi

P Irr GLnpFq , where n “ ř

iPI diaibi.

Max Gurevich Branching laws for non-generic representations

SLIDE 20

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Arthur-type representations

Arthur parameter - combinatorial description: A collection of triples φ “ tpρi, ai, biquiPI , where ρi P Irr GLdipFq are supercuspidal representations, ai, bi P Zą0. Each triple pρi, ai, biq defines a Speh representation πai,bi

ρi

P Irr GLdiaibipFq. Such φ gives rise to πpφq “ ą

iPI

πai,bi

ρi

P Irr GLnpFq , where n “ ř

iPI diaibi.

Max Gurevich Branching laws for non-generic representations

SLIDE 21

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Arthur-type representations

Arthur parameter - combinatorial description: A collection of triples φ “ tpρi, ai, biquiPI , where ρi P Irr GLdipFq are supercuspidal representations, ai, bi P Zą0. Each triple pρi, ai, biq defines a Speh representation πai,bi

ρi

P Irr GLdiaibipFq. Such φ gives rise to πpφq “ ą

iPI

πai,bi

ρi

P Irr GLnpFq , where n “ ř

iPI diaibi.

Max Gurevich Branching laws for non-generic representations

SLIDE 22

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Gan-Gross-Prasad conjecture

Conjecture (Gan-Gross-Prasad) For Arthur parameters φ1, φ2 so that πpφ1q P Irr GLnpFq and πpφ2q P Irr GLn´1pFq, Hom ` πpφ1q|GLn´1pFq, πpφ2q ˘ ‰ 0 holds, if and only if, we can write φ1 “ tpρi, ai, biquiPI , I “ I ` Y I ´ , φ2 “ tpρi, ai, bi ´ 1quiPI ´ Y tpρi, ai, bi ` 1quiPI ` Y tpρj, aj, 1qujPJ

Max Gurevich Branching laws for non-generic representations

SLIDE 23

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Gan-Gross-Prasad conjecture

Conjecture (Gan-Gross-Prasad) For Arthur parameters φ1, φ2 so that πpφ1q P Irr GLnpFq and πpφ2q P Irr GLn´1pFq, Hom ` πpφ1q|GLn´1pFq, πpφ2q ˘ ‰ 0 holds, if and only if, we can write φ1 “ tpρi, ai, biquiPI , I “ I ` Y I ´ , φ2 “ tpρi, ai, bi ´ 1quiPI ´ Y tpρi, ai, bi ` 1quiPI ` Y tpρj, aj, 1qujPJ

Max Gurevich Branching laws for non-generic representations

SLIDE 24

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Results

Theorem (GGP) Conjecture is true when |I| “ 1 (i.e. π1 is Speh representation) or when ai “ 1, @i. Essential reason (for |I| “ 1 case): Highest non-zero derivative of πa,b

ρ

is | det |´1{2 b πa,b´1

ρ

.

Max Gurevich Branching laws for non-generic representations

SLIDE 25

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Results

Theorem (GGP) Conjecture is true when |I| “ 1 (i.e. π1 is Speh representation) or when ai “ 1, @i. Essential reason (for |I| “ 1 case): Highest non-zero derivative of πa,b

ρ

is | det |´1{2 b πa,b´1

ρ

.

Max Gurevich Branching laws for non-generic representations

SLIDE 26

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Results: One direction of GGP fully resolved

Theorem (G.) If π1 “ πpφ1q P Irr GLnpFq and π2 “ πpφ2q P Irr GLn´1pFq are Arthur-type representations with Hom ` π1|GLn´1pFq, π2 ˘ ‰ 0 , then φ1, φ2 must comply with the conditions stated in the GGP conjecture. A similar statement holds when π1, π2 are general unitarizable representations.

Max Gurevich Branching laws for non-generic representations

SLIDE 27

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Results: One direction of GGP fully resolved

Theorem (G.) If π1 “ πpφ1q P Irr GLnpFq and π2 “ πpφ2q P Irr GLn´1pFq are Arthur-type representations with Hom ` π1|GLn´1pFq, π2 ˘ ‰ 0 , then φ1, φ2 must comply with the conditions stated in the GGP conjecture. A similar statement holds when π1, π2 are general unitarizable representations.

Max Gurevich Branching laws for non-generic representations

SLIDE 28

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

Results: Converse direction

Theorem (G.) Conjecture is true when one of π1, π2 is generic.

Max Gurevich Branching laws for non-generic representations

SLIDE 29

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

”Where the dog lies buried”

Derivatives of Speh representations ´ πa,b

ρ

¯piq are all either zero or a given irreducible representation, which we call quasi-Speh. For a given Arthur-type representation π “ πa1,b1

ρ1

ˆ ¨ ¨ ¨ ˆ πak,bk

ρk

, a derivative πpiq is built out of representations of the form ´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq . where i “ j1 ` . . . ` jk. Products of quasi-Speh representations are not irreducible !!

Max Gurevich Branching laws for non-generic representations

SLIDE 30

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

”Where the dog lies buried”

Derivatives of Speh representations ´ πa,b

ρ

¯piq are all either zero or a given irreducible representation, which we call quasi-Speh. For a given Arthur-type representation π “ πa1,b1

ρ1

ˆ ¨ ¨ ¨ ˆ πak,bk

ρk

, a derivative πpiq is built out of representations of the form ´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq . where i “ j1 ` . . . ` jk. Products of quasi-Speh representations are not irreducible !!

Max Gurevich Branching laws for non-generic representations

SLIDE 31

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

”Where the dog lies buried”

Derivatives of Speh representations ´ πa,b

ρ

¯piq are all either zero or a given irreducible representation, which we call quasi-Speh. For a given Arthur-type representation π “ πa1,b1

ρ1

ˆ ¨ ¨ ¨ ˆ πak,bk

ρk

, a derivative πpiq is built out of representations of the form ´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq . where i “ j1 ` . . . ` jk. Products of quasi-Speh representations are not irreducible !!

Max Gurevich Branching laws for non-generic representations

SLIDE 32

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

”Where the dog lies buried”

Derivatives of Speh representations ´ πa,b

ρ

¯piq are all either zero or a given irreducible representation, which we call quasi-Speh. For a given Arthur-type representation π “ πa1,b1

ρ1

ˆ ¨ ¨ ¨ ˆ πak,bk

ρk

, a derivative πpiq is built out of representations of the form ´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq . where i “ j1 ` . . . ` jk. Products of quasi-Speh representations are not irreducible !!

Max Gurevich Branching laws for non-generic representations

SLIDE 33

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

We would like to determine for which Arthur parameters there can be a non-zero morphism in Hom ˆ | det |1{2

F

´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq ,

pj1

1q ´

πa1

1,b1 1

ρ1

¯ ˆ ¨ ¨ ¨ ˆ pj1

kq ´

π

a1

k,b1 k

ρ1

¯¯ . If we knew that non-zero elements of this Hom space factor through a unique quotient and a unique sub-representation, then the GGP rule would follow from a combinatorial argument.

Max Gurevich Branching laws for non-generic representations

SLIDE 34

Restriction Problem Categorical equivalences On unique irreducible quotients Branching Tools (following Bernstein-Zelevinski) GGP conjectures Results Difficulties

We would like to determine for which Arthur parameters there can be a non-zero morphism in Hom ˆ | det |1{2

F

´ πa1,b1

ρ1

¯pj1q ˆ ¨ ¨ ¨ ˆ ´ πak,bk

ρk

¯pjkq ,

pj1

1q ´

πa1

1,b1 1

ρ1

¯ ˆ ¨ ¨ ¨ ˆ pj1

kq ´

π

a1

k,b1 k

ρ1

¯¯ . If we knew that non-zero elements of this Hom space factor through a unique quotient and a unique sub-representation, then the GGP rule would follow from a combinatorial argument.

Max Gurevich Branching laws for non-generic representations

SLIDE 35

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

... and now for something completely different...

Max Gurevich Branching laws for non-generic representations

SLIDE 36

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Affine Hecke algebras

The category of smooth representation of GLnpFq can be decomposed as a product of abelian categories ś

Θ MΘ,

called Bernstein blocks. Each block MΘ is equivalent to the category of modules over a certain intertwiner algbera AΘ. It was shown that AΘ can always be described as tensor products of affine Hecke algebras associated to type A root data. Thus, our problem can be translated (with some care) into a problem on Hom spaces in categories of module over type A affine Hecke algebras. Parabolic induction product can be translated to algebra module induction.

Max Gurevich Branching laws for non-generic representations

SLIDE 37

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Affine Hecke algebras

The category of smooth representation of GLnpFq can be decomposed as a product of abelian categories ś

Θ MΘ,

called Bernstein blocks. Each block MΘ is equivalent to the category of modules over a certain intertwiner algbera AΘ. It was shown that AΘ can always be described as tensor products of affine Hecke algebras associated to type A root data. Thus, our problem can be translated (with some care) into a problem on Hom spaces in categories of module over type A affine Hecke algebras. Parabolic induction product can be translated to algebra module induction.

Max Gurevich Branching laws for non-generic representations

SLIDE 38

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Affine Hecke algebras

The category of smooth representation of GLnpFq can be decomposed as a product of abelian categories ś

Θ MΘ,

called Bernstein blocks. Each block MΘ is equivalent to the category of modules over a certain intertwiner algbera AΘ. It was shown that AΘ can always be described as tensor products of affine Hecke algebras associated to type A root data. Thus, our problem can be translated (with some care) into a problem on Hom spaces in categories of module over type A affine Hecke algebras. Parabolic induction product can be translated to algebra module induction.

Max Gurevich Branching laws for non-generic representations

SLIDE 39

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Affine Hecke algebras

The category of smooth representation of GLnpFq can be decomposed as a product of abelian categories ś

Θ MΘ,

called Bernstein blocks. Each block MΘ is equivalent to the category of modules over a certain intertwiner algbera AΘ. It was shown that AΘ can always be described as tensor products of affine Hecke algebras associated to type A root data. Thus, our problem can be translated (with some care) into a problem on Hom spaces in categories of module over type A affine Hecke algebras. Parabolic induction product can be translated to algebra module induction.

Max Gurevich Branching laws for non-generic representations

SLIDE 40

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Quantum affine Schur-Weyl duality

The classical Schur-Weyl duality can be thought of as an exact functor from module over the algebra CrSns into modules over the algebra UpslNq. This functor can by quantized to move from modules over the Hecke algebra of Sn to the quantum group UqpslNq. The quantum affine Schur-Weyl duality functor Fk,N as defined by Chari-Pressley takes fin.-dim. modules over the affine Hecke algebra GLk to fin.-dim. modules over Uqpˆ slNq. When k ď N, this is a fully faithful exact functor. Our problem now deals with Hom spaces in a category of modules over a quantum affine algebra.

Max Gurevich Branching laws for non-generic representations

SLIDE 41

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Quantum affine Schur-Weyl duality

The classical Schur-Weyl duality can be thought of as an exact functor from module over the algebra CrSns into modules over the algebra UpslNq. This functor can by quantized to move from modules over the Hecke algebra of Sn to the quantum group UqpslNq. The quantum affine Schur-Weyl duality functor Fk,N as defined by Chari-Pressley takes fin.-dim. modules over the affine Hecke algebra GLk to fin.-dim. modules over Uqpˆ slNq. When k ď N, this is a fully faithful exact functor. Our problem now deals with Hom spaces in a category of modules over a quantum affine algebra.

Max Gurevich Branching laws for non-generic representations

SLIDE 42

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Quantum affine Schur-Weyl duality

The classical Schur-Weyl duality can be thought of as an exact functor from module over the algebra CrSns into modules over the algebra UpslNq. This functor can by quantized to move from modules over the Hecke algebra of Sn to the quantum group UqpslNq. The quantum affine Schur-Weyl duality functor Fk,N as defined by Chari-Pressley takes fin.-dim. modules over the affine Hecke algebra GLk to fin.-dim. modules over Uqpˆ slNq. When k ď N, this is a fully faithful exact functor. Our problem now deals with Hom spaces in a category of modules over a quantum affine algebra.

Max Gurevich Branching laws for non-generic representations

SLIDE 43

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Quantum affine Schur-Weyl duality

The classical Schur-Weyl duality can be thought of as an exact functor from module over the algebra CrSns into modules over the algebra UpslNq. This functor can by quantized to move from modules over the Hecke algebra of Sn to the quantum group UqpslNq. The quantum affine Schur-Weyl duality functor Fk,N as defined by Chari-Pressley takes fin.-dim. modules over the affine Hecke algebra GLk to fin.-dim. modules over Uqpˆ slNq. When k ď N, this is a fully faithful exact functor. Our problem now deals with Hom spaces in a category of modules over a quantum affine algebra.

Max Gurevich Branching laws for non-generic representations

SLIDE 44

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Quantum affine Schur-Weyl duality

The classical Schur-Weyl duality can be thought of as an exact functor from module over the algebra CrSns into modules over the algebra UpslNq. This functor can by quantized to move from modules over the Hecke algebra of Sn to the quantum group UqpslNq. The quantum affine Schur-Weyl duality functor Fk,N as defined by Chari-Pressley takes fin.-dim. modules over the affine Hecke algebra GLk to fin.-dim. modules over Uqpˆ slNq. When k ď N, this is a fully faithful exact functor. Our problem now deals with Hom spaces in a category of modules over a quantum affine algebra.

Max Gurevich Branching laws for non-generic representations

SLIDE 45

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Dictionary

A representation π P Irr GLnpFq is generic, if and only if, Fn,2pπq ‰ 0. Images under Fk,N of Speh representations are precisely Kirillov-Reshetekhin modules. Quantum affine Schur-Weyl duality pleasantly transforms the induction product into a tensor product, i.e. Fn1,Npπ1q b Fn2,Npπ2q – Fn1`n2,Npπ1 ˆ π2q,

Max Gurevich Branching laws for non-generic representations

SLIDE 46

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Dictionary

A representation π P Irr GLnpFq is generic, if and only if, Fn,2pπq ‰ 0. Images under Fk,N of Speh representations are precisely Kirillov-Reshetekhin modules. Quantum affine Schur-Weyl duality pleasantly transforms the induction product into a tensor product, i.e. Fn1,Npπ1q b Fn2,Npπ2q – Fn1`n2,Npπ1 ˆ π2q,

Max Gurevich Branching laws for non-generic representations

SLIDE 47

Restriction Problem Categorical equivalences On unique irreducible quotients Bernstein decomposition Quantum affine algebras

Dictionary

A representation π P Irr GLnpFq is generic, if and only if, Fn,2pπq ‰ 0. Images under Fk,N of Speh representations are precisely Kirillov-Reshetekhin modules. Quantum affine Schur-Weyl duality pleasantly transforms the induction product into a tensor product, i.e. Fn1,Npπ1q b Fn2,Npπ2q – Fn1`n2,Npπ1 ˆ π2q,

Max Gurevich Branching laws for non-generic representations

SLIDE 48

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Highest weight modules

There is a triangular decomposition Uqpˆ slNq “ ˆ U´ b ˆ U0 b ˆ U`, similar to the classical setting of universal enveloping algebras. For a module V over Uqpˆ slNq (in the category of our interest), we say that v P V is a highest weight vector, if v is an eigenvector for the algebra ˆ U0 and ˆ U` ¨ v “ 0. A module is said to be a highest weight (or cyclic) module if it is spanned by a highest weight vector. Easy to see that highest weight modules have a unique irreducible quotient!

Max Gurevich Branching laws for non-generic representations

SLIDE 49

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Highest weight modules

There is a triangular decomposition Uqpˆ slNq “ ˆ U´ b ˆ U0 b ˆ U`, similar to the classical setting of universal enveloping algebras. For a module V over Uqpˆ slNq (in the category of our interest), we say that v P V is a highest weight vector, if v is an eigenvector for the algebra ˆ U0 and ˆ U` ¨ v “ 0. A module is said to be a highest weight (or cyclic) module if it is spanned by a highest weight vector. Easy to see that highest weight modules have a unique irreducible quotient!

Max Gurevich Branching laws for non-generic representations

SLIDE 50

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Highest weight modules

There is a triangular decomposition Uqpˆ slNq “ ˆ U´ b ˆ U0 b ˆ U`, similar to the classical setting of universal enveloping algebras. For a module V over Uqpˆ slNq (in the category of our interest), we say that v P V is a highest weight vector, if v is an eigenvector for the algebra ˆ U0 and ˆ U` ¨ v “ 0. A module is said to be a highest weight (or cyclic) module if it is spanned by a highest weight vector. Easy to see that highest weight modules have a unique irreducible quotient!

Max Gurevich Branching laws for non-generic representations

SLIDE 51

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Highest weight modules

There is a triangular decomposition Uqpˆ slNq “ ˆ U´ b ˆ U0 b ˆ U`, similar to the classical setting of universal enveloping algebras. For a module V over Uqpˆ slNq (in the category of our interest), we say that v P V is a highest weight vector, if v is an eigenvector for the algebra ˆ U0 and ˆ U` ¨ v “ 0. A module is said to be a highest weight (or cyclic) module if it is spanned by a highest weight vector. Easy to see that highest weight modules have a unique irreducible quotient!

Max Gurevich Branching laws for non-generic representations

SLIDE 52

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

There and back again

Question: Given k quasi-Speh representations, can they be numbered π1, . . . , πk so that the product of their images Vi “ Fni,Npπiq V1 b ¨ ¨ ¨ b Vk is highest weight ? Can! In particular, this means that π1 ˆ ¨ ¨ ¨ ˆ πk has a unique irreducible quotient. Proof lies in taking the works of Hernandez and Lapid-Minguez to the same setting.

Max Gurevich Branching laws for non-generic representations

SLIDE 53

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

There and back again

Question: Given k quasi-Speh representations, can they be numbered π1, . . . , πk so that the product of their images Vi “ Fni,Npπiq V1 b ¨ ¨ ¨ b Vk is highest weight ? Can! In particular, this means that π1 ˆ ¨ ¨ ¨ ˆ πk has a unique irreducible quotient. Proof lies in taking the works of Hernandez and Lapid-Minguez to the same setting.

Max Gurevich Branching laws for non-generic representations

SLIDE 54

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

There and back again

Question: Given k quasi-Speh representations, can they be numbered π1, . . . , πk so that the product of their images Vi “ Fni,Npπiq V1 b ¨ ¨ ¨ b Vk is highest weight ? Can! In particular, this means that π1 ˆ ¨ ¨ ¨ ˆ πk has a unique irreducible quotient. Proof lies in taking the works of Hernandez and Lapid-Minguez to the same setting.

Max Gurevich Branching laws for non-generic representations

SLIDE 55

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Theorem (Hernandez) For irreducible modules V1, . . . , Vk of Uqpˆ slNq, such that Vi b Vj is highest weight for all i ă j, the product V1 b ¨ ¨ ¨ b Vk is highest weight.

Max Gurevich Branching laws for non-generic representations

SLIDE 56

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Some ideas of proof of GGP conjectures

By Hernandez, enough to show that Vi b Vj is highest weight, for all i ă j. Question reformulated: Is there an order on Speh representations, such that when π1 ă π2, their quasi-Speh derivatives σ1 “ πpt1q

1

, σ2 “ πpt2q

1

would give a highest weight module Fn,Npσ1 ˆ σ2q ? Lapid-Minguez showed that the product of every pair of quasi-Speh (more generally, ladder) representations has a unique irreducible quotient. Moreover, they gave a combinatorial algorithm for computing the isomorphism class

f that quotient.

The image of σ1 ˆ σ2 is highest weight, if and only if, the quotient is given by the Langlands parameter which is the sum

f Langlands parameter of σ1, σ2.

Max Gurevich Branching laws for non-generic representations

SLIDE 57

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Some ideas of proof of GGP conjectures

By Hernandez, enough to show that Vi b Vj is highest weight, for all i ă j. Question reformulated: Is there an order on Speh representations, such that when π1 ă π2, their quasi-Speh derivatives σ1 “ πpt1q

1

, σ2 “ πpt2q

1

would give a highest weight module Fn,Npσ1 ˆ σ2q ? Lapid-Minguez showed that the product of every pair of quasi-Speh (more generally, ladder) representations has a unique irreducible quotient. Moreover, they gave a combinatorial algorithm for computing the isomorphism class

f that quotient.

The image of σ1 ˆ σ2 is highest weight, if and only if, the quotient is given by the Langlands parameter which is the sum

f Langlands parameter of σ1, σ2.

Max Gurevich Branching laws for non-generic representations

SLIDE 58

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Some ideas of proof of GGP conjectures

By Hernandez, enough to show that Vi b Vj is highest weight, for all i ă j. Question reformulated: Is there an order on Speh representations, such that when π1 ă π2, their quasi-Speh derivatives σ1 “ πpt1q

1

, σ2 “ πpt2q

1

would give a highest weight module Fn,Npσ1 ˆ σ2q ? Lapid-Minguez showed that the product of every pair of quasi-Speh (more generally, ladder) representations has a unique irreducible quotient. Moreover, they gave a combinatorial algorithm for computing the isomorphism class

f that quotient.

The image of σ1 ˆ σ2 is highest weight, if and only if, the quotient is given by the Langlands parameter which is the sum

f Langlands parameter of σ1, σ2.

Max Gurevich Branching laws for non-generic representations

SLIDE 59

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Some ideas of proof of GGP conjectures

By Hernandez, enough to show that Vi b Vj is highest weight, for all i ă j. Question reformulated: Is there an order on Speh representations, such that when π1 ă π2, their quasi-Speh derivatives σ1 “ πpt1q

1

, σ2 “ πpt2q

1

would give a highest weight module Fn,Npσ1 ˆ σ2q ? Lapid-Minguez showed that the product of every pair of quasi-Speh (more generally, ladder) representations has a unique irreducible quotient. Moreover, they gave a combinatorial algorithm for computing the isomorphism class

f that quotient.

The image of σ1 ˆ σ2 is highest weight, if and only if, the quotient is given by the Langlands parameter which is the sum

f Langlands parameter of σ1, σ2.

Max Gurevich Branching laws for non-generic representations

SLIDE 60

Restriction Problem Categorical equivalences On unique irreducible quotients Our questions Work of Hernandez Proof

Thank you !

Max Gurevich Branching laws for non-generic representations