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
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Main Problem F - a p -adic field. π - irreducible smooth representation of GL n p F q . How does π | GL n ´ 1 p F q decompose? What are the irreducible quotients of π | GL n ´ 1 p F q ? It is long-known (Jacquet - Piatetski-Shapiro - Shalika) that Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 , for all generic irreducible π 1 , π 2 . Goal: What can be said about the non-generic case?One would like to describe the pairs p π 1 , π 2 q for which the Hom space above is non-zero. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Tools: Parabolic induction Representations of t GL n p F qu n have an intriguing product structure. Given π 1 P Rep GL n 1 p F q and π 2 P Rep GL n 2 p F q , we can think of π 1 b π 2 as a representation of the parabolic subgroup ˆ GL n 1 p F q ˚ ˙ P “ ă GL n 1 ` n 2 p F q . 0 GL n 2 p F q GL n 1 ` n 2 p F q π 1 ˆ π 2 : “ ind p π 1 b π 2 q P Rep GL n 1 ` n 2 p F q P Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Tools: Parabolic induction Representations of t GL n p F qu n have an intriguing product structure. Given π 1 P Rep GL n 1 p F q and π 2 P Rep GL n 2 p F q , we can think of π 1 b π 2 as a representation of the parabolic subgroup ˆ GL n 1 p F q ˚ ˙ P “ ă GL n 1 ` n 2 p F q . 0 GL n 2 p F q GL n 1 ` n 2 p F q π 1 ˆ π 2 : “ ind p π 1 b π 2 q P Rep GL n 1 ` n 2 p F q P Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Tools: Parabolic induction Representations of t GL n p F qu n have an intriguing product structure. Given π 1 P Rep GL n 1 p F q and π 2 P Rep GL n 2 p F q , we can think of π 1 b π 2 as a representation of the parabolic subgroup ˆ GL n 1 p F q ˚ ˙ P “ ă GL n 1 ` n 2 p F q . 0 GL n 2 p F q GL n 1 ` n 2 p F q π 1 ˆ π 2 : “ ind p π 1 b π 2 q P Rep GL n 1 ` n 2 p F q P Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Tools: Derivatives For every 0 ď i ď n , there are well-defined functors Rep GL n p F q Ñ Rep GL n ´ i p F q π p i q ÞÑ π p i q π ÞÑ π 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
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Tools: Derivatives For every 0 ď i ď n , there are well-defined functors Rep GL n p F q Ñ Rep GL n ´ i p F q π p i q ÞÑ π p i q π ÞÑ π 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
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties Filtration π P Irr GL n p F q Classical results of Bernstein-Zelevinski give a filtration of the GL n ´ 1 p F q -representation p π | GL n ´ 1 p F q , U q t 0 u “ U n ` 1 Ă U n Ă ¨ ¨ ¨ Ă U 1 “ U , so that each U i { U i ` 1 is well understood. Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties A question about morphisms Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 ù ñ Hom p U i { U i ` 1 , π 2 q ‰ 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: ´ ¯ | det | 1 { 2 b π p i q 1 , p i ´ 1 q π 2 Hom GL n ´ i p F q ‰ 0 . F Now, we are asking about Hom spaces of finite-length representations! Max Gurevich Branching laws for non-generic representations
Branching Restriction Problem Tools (following Bernstein-Zelevinski) Categorical equivalences GGP conjectures On unique irreducible quotients Results Difficulties A question about morphisms Hom p π 1 | GL n ´ 1 p F q , π 2 q ‰ 0 ù ñ Hom p U i { U i ` 1 , π 2 q ‰ 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: ´ ¯ | det | 1 { 2 b π p i q 1 , p i ´ 1 q π 2 Hom GL n ´ i p F q ‰ 0 . F Now, we are asking about Hom spaces of finite-length representations! Max Gurevich Branching laws for non-generic representations
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