I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Positivity of canonical bases of quantum
coideal algebras and geometry of flag varieties
Weiqiang Wang
University of Virginia
Kyoto, 1/23/2015

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

References[Bao-W] A new approach to Kazhdan-Lusztig theory of type B
via quantum symmetric pairs, arXiv:1310.0103.
[Bao-Kujawa-Yiqiang Li-W] Geometric Schur duality of classicaltype, arXiv:1404.4000v2.
[Li-W] Positivity vs negativity of canonical bases,
arXiv:1501.00688.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Plan B(1) iSchur duality with Hecke algebra of type B(2) iCanonical basis (iCB) of quantum coideal algebras(*) (Almost Skipped) iCB character formula (=Kazhdan-Lusztig
theory) for category O of (super) type B(3) Geometric realization via partial flag varieties of type B(4) Positivity of iCB and transfer maps
Plan A (what I will not talk much):(1) Jimbo(2) Lusztig-Kashiwara(*) Brundan Conjecture 2002, Cheng-Lam-W Theorem 2012
(3) Beilinson-Lusztig-MacPherson 1990 (4) McGerty 2012

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality• Let U = hEi , Fi , Ki±1 i be the quantum group of type glN .
• ∃ a bar-involution on U such that q = q −1 , E i = Ei , F i = Fi
and K i = Ki−1 .
• Let V be the natural representation of U.
Then V⊗d is a U-module, via the coproduct ∆ : U → U ⊗ U.
• Let HSd = hHi , 1 ≤ i ≤ d − 1i be Hecke algebra of type A.
There is a bar-involution on HSd such that H i = Hi−1 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality• Let U = hEi , Fi , Ki±1 i be the quantum group of type glN .
• ∃ a bar-involution on U such that q = q −1 , E i = Ei , F i = Fi
and K i = Ki−1 .
• Let V be the natural representation of U.
Then V⊗d is a U-module, via the coproduct ∆ : U → U ⊗ U.
• Let HSd = hHi , 1 ≤ i ≤ d − 1i be Hecke algebra of type A.
There is a bar-involution on HSd such that H i = Hi−1 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality• Let U = hEi , Fi , Ki±1 i be the quantum group of type glN .
• ∃ a bar-involution on U such that q = q −1 , E i = Ei , F i = Fi
and K i = Ki−1 .
• Let V be the natural representation of U.
Then V⊗d is a U-module, via the coproduct ∆ : U → U ⊗ U.
• Let HSd = hHi , 1 ≤ i ≤ d − 1i be Hecke algebra of type A.
There is a bar-involution on HSd such that H i = Hi−1 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality• Let U = hEi , Fi , Ki±1 i be the quantum group of type glN .
• ∃ a bar-involution on U such that q = q −1 , E i = Ei , F i = Fi
and K i = Ki−1 .
• Let V be the natural representation of U.
Then V⊗d is a U-module, via the coproduct ∆ : U → U ⊗ U.
• Let HSd = hHi , 1 ≤ i ≤ d − 1i be Hecke algebra of type A.
There is a bar-involution on HSd such that H i = Hi−1 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality

HSd

U
V⊗d

• [Jimbo] Action of U via coproduct and action of HSd via
R-matrix on V⊗d form double centralizers.
• Schur-Jimbo duality is a decaf of category O of gl(d).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality

HSd

U
V⊗d

• [Jimbo] Action of U via coproduct and action of HSd via
R-matrix on V⊗d form double centralizers.
• Schur-Jimbo duality is a decaf of category O of gl(d).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Schur duality

HSd

U
V⊗d

• [Jimbo] Action of U via coproduct and action of HSd via
R-matrix on V⊗d form double centralizers.
• Schur-Jimbo duality is a decaf of category O of gl(d).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Motivation

Motivating problem: Develop a Kazhdan-Lusztig theory inCategory O of Lie superalgebras (osp), say of type B.
• The Hecke algebra of type Bd , HBd = h, HSd , H0 i, acts
naturally on V⊗d :if we choose the standard basis {vi } of V to run over indices ofthe form [−a, a], then H0 acts on the first tensor factor by(v−i ,if i > 0,
H0 : vi 7→−1
v−i + (q − q )vi , if i < 0.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Motivation

Motivating problem: Develop a Kazhdan-Lusztig theory inCategory O of Lie superalgebras (osp), say of type B.
• The Hecke algebra of type Bd , HBd = h, HSd , H0 i, acts
naturally on V⊗d :if we choose the standard basis {vi } of V to run over indices ofthe form [−a, a], then H0 acts on the first tensor factor by(v−i ,if i > 0,
H0 : vi 7→−1
v−i + (q − q )vi , if i < 0.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Motivation

Motivating problem: Develop a Kazhdan-Lusztig theory inCategory O of Lie superalgebras (osp), say of type B.
• The Hecke algebra of type Bd , HBd = h, HSd , H0 i, acts
naturally on V⊗d :if we choose the standard basis {vi } of V to run over indices ofthe form [−a, a], then H0 acts on the first tensor factor by(v−i ,if i > 0,
H0 : vi 7→−1
v−i + (q − q )vi , if i < 0.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

A double centralizer question• Question: What quantum algebra centralizes HBd ?

HSd

U
V⊗d
Uı

HBd

• Idea behind: this (Uı , HBd )-duality (with this very HBd -action)
serves as a decaf of category O of so(2d + 1).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

A double centralizer question• Question: What quantum algebra centralizes HBd ?

HSd

U
V⊗d
Uı

HBd

• Idea behind: this (Uı , HBd )-duality (with this very HBd -action)
serves as a decaf of category O of so(2d + 1).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

Quantum coideal subalgebras• Uı comes in 2 forms, depending on the parity of N (ignore!)• ∃ a presentation Uı = {ei , fi , ...}i>0 , but “bad" Serre relations;∃ an imbedding ı : Uı ,→ U, e.g., ei → Ei + Ki−1 F−i .
• Uı is a coideal subalgebra of U, i.e., ∆ : Uı → Uı ⊗ U.
• The algebra Uı admits a bar-involution ψı , ψı (ei ) = ei , ...
(This bar map was independently noted by Ehrig-Stroppel)• (U, Uı ) is an example of quantum symmetric pairs [Noumi,
Letzter, Kolb] in different presentations(the involution ı is rotation by 180 degrees).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiSchur duality

iSchur duality

Theorem 1 (Bao-W2012)
The actions of Uı and HBm on V⊗m form double centralizers.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Canonical basis• Let L(λ) be the simple U-module with h.wt. λ ∈ X + .
• L(λ) admits a bar-involution, which is compatible with thebar-involution on U.
• [Lusztig, Kashiwara] L(λ) admit a canonical/crystal basis(CB).
• Via ı : Uı → U, L(λ) becomes a Uı -module.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Canonical basis• Let L(λ) be the simple U-module with h.wt. λ ∈ X + .
• L(λ) admits a bar-involution, which is compatible with thebar-involution on U.
• [Lusztig, Kashiwara] L(λ) admit a canonical/crystal basis(CB).
• Via ı : Uı → U, L(λ) becomes a Uı -module.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Canonical basis• Let L(λ) be the simple U-module with h.wt. λ ∈ X + .
• L(λ) admits a bar-involution, which is compatible with thebar-involution on U.
• [Lusztig, Kashiwara] L(λ) admit a canonical/crystal basis(CB).
• Via ı : Uı → U, L(λ) becomes a Uı -module.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Canonical basis• Let L(λ) be the simple U-module with h.wt. λ ∈ X + .
• L(λ) admits a bar-involution, which is compatible with thebar-involution on U.
• [Lusztig, Kashiwara] L(λ) admit a canonical/crystal basis(CB).
• Via ı : Uı → U, L(λ) becomes a Uı -module.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Quasi-R-matrix• Lusztig’s quasi-R-matrix Θ is a variant of Drinfeld’s R-matrix.
• The coproduct ∆ : U → U ⊗ U does not commute with the barmaps, i.e., ∆ 6= ∆.
• Θ intertwines the coproduct ∆ and ∆, i.e., Θ∆ = ∆Θ. iIt leads
to a bar-involution and CBs in tensor product U-moduleL(λ1 ) ⊗ · · · ⊗ L(λ` ) [Lusztig].
• The bar map on Uı is not compatible with the bar map on U,
i.e., ı(ψı (u)) 6= ı(u), ∀u (recall ei → Ei + Ki−1 F−i ).
• One key theorem of [Bao-W] is the existence of Υ whichintertwines the imbedding ı : Uı → U and its bar-conjugate ı.
• This leads to a quasi-R-matrix Θı which intertwines∆ : Uı → Uı ⊗ U and its bar-conjugate; Θı 6= Θ|Uı .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

ı-canonical bases in tensor products
Theorem 2 (Bao-W)
1

2

The Θı induces a bar involution ψı on the tensor product
Uı -module L(λ1 ) ⊗ · · · ⊗ L(λ` ), and in particular on L(λ).
This tensor module admits an ı-canonical basis (iCB),
which is ψı -invariant and whose transition matrix to
Lusztig’s CB is uni-upper-triangular with coeff. in qZ[q].
In particular, L(λ) admits an iCB.

• The three bar-involutions Uı  V⊗dψı (uMh) = ψı (u)ψı (M)h̄,

HBd are all compatible:

for u ∈ Uı , M ∈ V⊗d , h ∈ HBd .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

ı-canonical bases in tensor products
Theorem 2 (Bao-W)
1

2

The Θı induces a bar involution ψı on the tensor product
Uı -module L(λ1 ) ⊗ · · · ⊗ L(λ` ), and in particular on L(λ).
This tensor module admits an ı-canonical basis (iCB),
which is ψı -invariant and whose transition matrix to
Lusztig’s CB is uni-upper-triangular with coeff. in qZ[q].
In particular, L(λ) admits an iCB.

• The three bar-involutions Uı  V⊗dψı (uMh) = ψı (u)ψı (M)h̄,

HBd are all compatible:

for u ∈ Uı , M ∈ V⊗d , h ∈ HBd .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

ı-canonical bases in tensor products
Theorem 2 (Bao-W)
1

2

The Θı induces a bar involution ψı on the tensor product
Uı -module L(λ1 ) ⊗ · · · ⊗ L(λ` ), and in particular on L(λ).
This tensor module admits an ı-canonical basis (iCB),
which is ψı -invariant and whose transition matrix to
Lusztig’s CB is uni-upper-triangular with coeff. in qZ[q].
In particular, L(λ) admits an iCB.

• The three bar-involutions Uı  V⊗dψı (uMh) = ψı (u)ψı (M)h̄,

HBd are all compatible:

for u ∈ Uı , M ∈ V⊗d , h ∈ HBd .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

ı-canonical bases in tensor products
Theorem 2 (Bao-W)
1

2

The Θı induces a bar involution ψı on the tensor product
Uı -module L(λ1 ) ⊗ · · · ⊗ L(λ` ), and in particular on L(λ).
This tensor module admits an ı-canonical basis (iCB),
which is ψı -invariant and whose transition matrix to
Lusztig’s CB is uni-upper-triangular with coeff. in qZ[q].
In particular, L(λ) admits an iCB.

• The three bar-involutions Uı  V⊗dψı (uMh) = ψı (u)ψı (M)h̄,

HBd are all compatible:

for u ∈ Uı , M ∈ V⊗d , h ∈ HBd .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

ı-canonical bases in tensor products
Theorem 2 (Bao-W)
1

2

The Θı induces a bar involution ψı on the tensor product
Uı -module L(λ1 ) ⊗ · · · ⊗ L(λ` ), and in particular on L(λ).
This tensor module admits an ı-canonical basis (iCB),
which is ψı -invariant and whose transition matrix to
Lusztig’s CB is uni-upper-triangular with coeff. in qZ[q].
In particular, L(λ) admits an iCB.

• The three bar-involutions Uı  V⊗dψı (uMh) = ψı (u)ψı (M)h̄,

HBd are all compatible:

for u ∈ Uı , M ∈ V⊗d , h ∈ HBd .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB: examples• Recall V has a standard basis {vi }i∈[−a,a] , same as CB.
• The iCB of V is given by(vi ,vi + qv−i ,if i ≥ 0,
if i < 0.

• Another example of iCB (blue parts are CB) in V ⊗ V∗ :∗∗∗∗(v3/2 ⊗ v3/2
+ qv1/2 ⊗ v1/2
) + q(v1/2 ⊗ v−1/2+ qv3/2 ⊗ v−3/2)• Note (2 levels of) positivity in above examples!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB, againne• The standard divided power is e(n) = [n]!
.

• Here are some new “divided powers" in a coideal algebra
setting when N is even: for a ≥ 1,t(t − [2 − 2a])(t − [4 − 2a]) · · · (t − [2a − 4])(t − [2a − 2]),[2a]!
(t − [−2a])(t − [2 − 2a]) · · · (t − [2a − 2])(t − [2a])t h2a+1i =.
[2a + 1]!
t h2ai =• These formulas encode the KL polynomials of Hermitiansymmetric pair (Bn , An−1 ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB, againne• The standard divided power is e(n) = [n]!
.

• Here are some new “divided powers" in a coideal algebra
setting when N is even: for a ≥ 1,t(t − [2 − 2a])(t − [4 − 2a]) · · · (t − [2a − 4])(t − [2a − 2]),[2a]!
(t − [−2a])(t − [2 − 2a]) · · · (t − [2a − 2])(t − [2a])t h2a+1i =.
[2a + 1]!
t h2ai =• These formulas encode the KL polynomials of Hermitiansymmetric pair (Bn , An−1 ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

iCB 6= CB, againne• The standard divided power is e(n) = [n]!
.

• Here are some new “divided powers" in a coideal algebra
setting when N is even: for a ≥ 1,t(t − [2 − 2a])(t − [4 − 2a]) · · · (t − [2a − 4])(t − [2a − 2]),[2a]!
(t − [−2a])(t − [2 − 2a]) · · · (t − [2a − 2])(t − [2a])t h2a+1i =.
[2a + 1]!
t h2ai =• These formulas encode the KL polynomials of Hermitiansymmetric pair (Bn , An−1 ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Application: Kazhdan-Lusztig theory of (super) type B

The iCB on V⊗m ⊗ (V∗ )⊗n can be expanded with respect to theı (q) ∈ Z[q].
simple tensor basis, with coefficients tλµ
Theorem 312

ı (q) ∈ Z [q];(Positivity) tλµ≥0ı (q) are KL polynomials for BGG
The polynomials tλµcategory O of the ortho-symplectic Lie superalgebraosp(2m + 1|2n). (Take N = ∞.)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Application: Kazhdan-Lusztig theory of (super) type B

The iCB on V⊗m ⊗ (V∗ )⊗n can be expanded with respect to theı (q) ∈ Z[q].
simple tensor basis, with coefficients tλµ
Theorem 312

ı (q) ∈ Z [q];(Positivity) tλµ≥0ı (q) are KL polynomials for BGG
The polynomials tλµcategory O of the ortho-symplectic Lie superalgebraosp(2m + 1|2n). (Take N = ∞.)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Application: Kazhdan-Lusztig theory of (super) type B

The iCB on V⊗m ⊗ (V∗ )⊗n can be expanded with respect to theı (q) ∈ Z[q].
simple tensor basis, with coefficients tλµ
Theorem 312

ı (q) ∈ Z [q];(Positivity) tλµ≥0ı (q) are KL polynomials for BGG
The polynomials tλµcategory O of the ortho-symplectic Lie superalgebraosp(2m + 1|2n). (Take N = ∞.)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

Remarks on CB/KL character formula

Why CB (re)formulation for Kazhdan-Lusztig theories?
• Weyl group/Hecke algebra do not control linkage in super O• Motivation from a formulation of CB character formula forgl(m|n) (Brundan conjecture 2002, Cheng-Lam-W Theorem2012) – no new CB is needed• n = 0. an iCB reformulation of KL conjecture for so2m+1(Theorem of Beilinson-Bernstein, Brylinski-Kashiwara).
• iCB for V (e.g. v−i + qvi for i > 0) corresponds to tilting
modules for so3 ∼= sl2 .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

An intermediate summary• ∃ iSchur duality between Uı and Hecke algebra of type B.
• ∃ new Canonical bases (iCB) for (tensor) modules of Uı(Uı is not a Drinfeld-Jimbo quantum group).
• iCB allows to formulate a KL theory for the ortho-symplectic
Lie superalgebras, an open problem since 1970’s.
• Recall KL theory [of type B] has much to do flag varieties [oftype B] ...

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

An intermediate summary• ∃ iSchur duality between Uı and Hecke algebra of type B.
• ∃ new Canonical bases (iCB) for (tensor) modules of Uı(Uı is not a Drinfeld-Jimbo quantum group).
• iCB allows to formulate a KL theory for the ortho-symplectic
Lie superalgebras, an open problem since 1970’s.
• Recall KL theory [of type B] has much to do flag varieties [oftype B] ...

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

An intermediate summary• ∃ iSchur duality between Uı and Hecke algebra of type B.
• ∃ new Canonical bases (iCB) for (tensor) modules of Uı(Uı is not a Drinfeld-Jimbo quantum group).
• iCB allows to formulate a KL theory for the ortho-symplectic
Lie superalgebras, an open problem since 1970’s.
• Recall KL theory [of type B] has much to do flag varieties [oftype B] ...

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

An intermediate summary• ∃ iSchur duality between Uı and Hecke algebra of type B.
• ∃ new Canonical bases (iCB) for (tensor) modules of Uı(Uı is not a Drinfeld-Jimbo quantum group).
• iCB allows to formulate a KL theory for the ortho-symplectic
Lie superalgebras, an open problem since 1970’s.
• Recall KL theory [of type B] has much to do flag varieties [oftype B] ...

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Canonical basis

An intermediate summary• ∃ iSchur duality between Uı and Hecke algebra of type B.
• ∃ new Canonical bases (iCB) for (tensor) modules of Uı(Uı is not a Drinfeld-Jimbo quantum group).
• iCB allows to formulate a KL theory for the ortho-symplectic
Lie superalgebras, an open problem since 1970’s.
• Recall KL theory [of type B] has much to do flag varieties [oftype B] ...

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric setting• There is a geometric construction of U̇q (glN ) and CB onU̇q (glN ) using partial flag varieties of type A.
[Beilinson-Lusztig-McPherson 1990]• There is a geometric realization of Schur-Jimbo duality
between U̇q (glN ) and HSd [Grojnowski-Lusztig1992]• (Old) Question: What is the quantum algebras (and duality)
behind partial flag varieties of classical type?
Answers (2014):[Bao-Kujawa-Yiqiang Li-W] type B and C.
[Zhaobing Fan-Yiqiang Li] type D.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric setting• There is a geometric construction of U̇q (glN ) and CB onU̇q (glN ) using partial flag varieties of type A.
[Beilinson-Lusztig-McPherson 1990]• There is a geometric realization of Schur-Jimbo duality
between U̇q (glN ) and HSd [Grojnowski-Lusztig1992]• (Old) Question: What is the quantum algebras (and duality)
behind partial flag varieties of classical type?
Answers (2014):[Bao-Kujawa-Yiqiang Li-W] type B and C.
[Zhaobing Fan-Yiqiang Li] type D.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric setting• There is a geometric construction of U̇q (glN ) and CB onU̇q (glN ) using partial flag varieties of type A.
[Beilinson-Lusztig-McPherson 1990]• There is a geometric realization of Schur-Jimbo duality
between U̇q (glN ) and HSd [Grojnowski-Lusztig1992]• (Old) Question: What is the quantum algebras (and duality)
behind partial flag varieties of classical type?
Answers (2014):[Bao-Kujawa-Yiqiang Li-W] type B and C.
[Zhaobing Fan-Yiqiang Li] type D.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric setting• There is a geometric construction of U̇q (glN ) and CB onU̇q (glN ) using partial flag varieties of type A.
[Beilinson-Lusztig-McPherson 1990]• There is a geometric realization of Schur-Jimbo duality
between U̇q (glN ) and HSd [Grojnowski-Lusztig1992]• (Old) Question: What is the quantum algebras (and duality)
behind partial flag varieties of classical type?
Answers (2014):[Bao-Kujawa-Yiqiang Li-W] type B and C.
[Zhaobing Fan-Yiqiang Li] type D.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric setting• There is a geometric construction of U̇q (glN ) and CB onU̇q (glN ) using partial flag varieties of type A.
[Beilinson-Lusztig-McPherson 1990]• There is a geometric realization of Schur-Jimbo duality
between U̇q (glN ) and HSd [Grojnowski-Lusztig1992]• (Old) Question: What is the quantum algebras (and duality)
behind partial flag varieties of classical type?
Answers (2014):[Bao-Kujawa-Yiqiang Li-W] type B and C.
[Zhaobing Fan-Yiqiang Li] type D.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATION

Geometric dualityıSchur duality recalled

Recall iSchur duality
Uı  Sdı

 V⊗d

HBd .

Here Sdı := EndHB (V⊗d ) is a q-Schur-type algebra.
d

C ANONICAL BASIS , II

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Type B flag varieties

Set N = 2n + 1, D = 2d + 1. Let q be an odd prime power.
• Fix a non-degenerate symmetric bilinear form Q on FDq.
• X : the variety of N-step isotropic flags (i.e. V−i = Vi⊥ ):V = (0 = V−n−1 ⊆ · · · ⊆ V−1 ⊆ V1 ⊆ · · · ⊆ Vn+1 = FDq)
• B: the complete flag variety of type Bd .
• O(D)-orbits on X × X , X × B and B × B are parameterized bycertain matrices (independent of q = |Fq |).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Type B flag varieties

Set N = 2n + 1, D = 2d + 1. Let q be an odd prime power.
• Fix a non-degenerate symmetric bilinear form Q on FDq.
• X : the variety of N-step isotropic flags (i.e. V−i = Vi⊥ ):V = (0 = V−n−1 ⊆ · · · ⊆ V−1 ⊆ V1 ⊆ · · · ⊆ Vn+1 = FDq)
• B: the complete flag variety of type Bd .
• O(D)-orbits on X × X , X × B and B × B are parameterized bycertain matrices (independent of q = |Fq |).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Type B flag varieties

Set N = 2n + 1, D = 2d + 1. Let q be an odd prime power.
• Fix a non-degenerate symmetric bilinear form Q on FDq.
• X : the variety of N-step isotropic flags (i.e. V−i = Vi⊥ ):V = (0 = V−n−1 ⊆ · · · ⊆ V−1 ⊆ V1 ⊆ · · · ⊆ Vn+1 = FDq)
• B: the complete flag variety of type Bd .
• O(D)-orbits on X × X , X × B and B × B are parameterized bycertain matrices (independent of q = |Fq |).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Type B flag varieties

Set N = 2n + 1, D = 2d + 1. Let q be an odd prime power.
• Fix a non-degenerate symmetric bilinear form Q on FDq.
• X : the variety of N-step isotropic flags (i.e. V−i = Vi⊥ ):V = (0 = V−n−1 ⊆ · · · ⊆ V−1 ⊆ V1 ⊆ · · · ⊆ Vn+1 = FDq)
• B: the complete flag variety of type Bd .
• O(D)-orbits on X × X , X × B and B × B are parameterized bycertain matrices (independent of q = |Fq |).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Convolution algebras• AO(D) (X × X ): the space of O(D)-invariant A -valued
functions on X × X . (A = Z[q, q −1 ])• Similarly define AO(D) (X × B) and AO(D) (B × B).
• AO(D) (X × X ) and AO(D) (B × B) are A -algebras by
convolution products.
• The convolution products also produce the following duality(i.e. double centralizing action):
AO(D) (X × X )AO(D) (X × B)  AO(D) (B × B)

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Convolution algebras• AO(D) (X × X ): the space of O(D)-invariant A -valued
functions on X × X . (A = Z[q, q −1 ])• Similarly define AO(D) (X × B) and AO(D) (B × B).
• AO(D) (X × X ) and AO(D) (B × B) are A -algebras by
convolution products.
• The convolution products also produce the following duality(i.e. double centralizing action):
AO(D) (X × X )AO(D) (X × B)  AO(D) (B × B)

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Convolution algebras• AO(D) (X × X ): the space of O(D)-invariant A -valued
functions on X × X . (A = Z[q, q −1 ])• Similarly define AO(D) (X × B) and AO(D) (B × B).
• AO(D) (X × X ) and AO(D) (B × B) are A -algebras by
convolution products.
• The convolution products also produce the following duality(i.e. double centralizing action):
AO(D) (X × X )AO(D) (X × B)  AO(D) (B × B)

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Convolution algebras• AO(D) (X × X ): the space of O(D)-invariant A -valued
functions on X × X . (A = Z[q, q −1 ])• Similarly define AO(D) (X × B) and AO(D) (B × B).
• AO(D) (X × X ) and AO(D) (B × B) are A -algebras by
convolution products.
• The convolution products also produce the following duality(i.e. double centralizing action):
AO(D) (X × X )AO(D) (X × B)  AO(D) (B × B)

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

The algebra Kı

Similar to [BLM90], we show
• Structure constants in AO(D) (X × X ) stabilize as D → ∞,which gives rise to a “limit" algebra Kı .
• ∃ a natural surjective algebra homomorphism φd :φdKı  AO(D) (X × X )AO(D) (X × B).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

The algebra Kı

Similar to [BLM90], we show
• Structure constants in AO(D) (X × X ) stabilize as D → ∞,which gives rise to a “limit" algebra Kı .
• ∃ a natural surjective algebra homomorphism φd :φdKı  AO(D) (X × X )AO(D) (X × B).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

The algebra Kı

Similar to [BLM90], we show
• Structure constants in AO(D) (X × X ) stabilize as D → ∞,which gives rise to a “limit" algebra Kı .
• ∃ a natural surjective algebra homomorphism φd :φdKı  AO(D) (X × X )AO(D) (X × B).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

The algebra Kı

Similar to [BLM90], we show
• Structure constants in AO(D) (X × X ) stabilize as D → ∞,which gives rise to a “limit" algebra Kı .
• ∃ a natural surjective algebra homomorphism φd :φdKı  AO(D) (X × X )AO(D) (X × B).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

The algebra Kı

Similar to [BLM90], we show
• Structure constants in AO(D) (X × X ) stabilize as D → ∞,which gives rise to a “limit" algebra Kı .
• ∃ a natural surjective algebra homomorphism φd :φdKı  AO(D) (X × X )AO(D) (X × B).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , II

Geometric duality

Geometric iSchur duality
Theorem 4 (Bao-Kujawa-Li-W2014)
U̇ı 

Sdı

V⊗d

∼
=

∼
=

∼
=

Kı  AO(D) (X × X )



HBd
∼
= [Iwahori]

AO(D) (X × B)  AO(D) (B × B)

Here U̇ı is the modified (or idempotented) version of the coideal
subalgebra Uı . (Sloppy with notation on base change above.)• Recall ı : Uı → U, ei 7→ Ei + Ki−1 F−i . This embedding agrees
with geometry (another example of positivity!)I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

iCanonical Basis of U̇ı
Facts:• AO(D) (X × X ) admits a [geometric] bar involution and acanonical basis with positivity {A}d , where A runs over
Ξd := {A ∈ MatN×N (Z≥0 ) | |A| = D = 2d + 1, Aı = A}
(Here ı denotes the involution of rotation by 180 degree.)• The bar involutions stabilize as D → ∞, and lifts to thealgebra Kı .
Recall Q(q) ⊗ Kı ∼= U̇ı .
Theorem 5 (BKLW2014)
The algebra Kı , and hence U̇ı , admits a stably CB {A}, wheree := {(aij ) ∈ MatN×N (Z ) | aij ≥ 0 (i 6= j), |A| is odd}.
A∈ΞI S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

iCanonical Basis of U̇ı
Facts:• AO(D) (X × X ) admits a [geometric] bar involution and acanonical basis with positivity {A}d , where A runs over
Ξd := {A ∈ MatN×N (Z≥0 ) | |A| = D = 2d + 1, Aı = A}
(Here ı denotes the involution of rotation by 180 degree.)• The bar involutions stabilize as D → ∞, and lifts to thealgebra Kı .
Recall Q(q) ⊗ Kı ∼= U̇ı .
Theorem 5 (BKLW2014)
The algebra Kı , and hence U̇ı , admits a stably CB {A}, wheree := {(aij ) ∈ MatN×N (Z ) | aij ≥ 0 (i 6= j), |A| is odd}.
A∈ΞI S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

iCanonical Basis of U̇ı
Facts:• AO(D) (X × X ) admits a [geometric] bar involution and acanonical basis with positivity {A}d , where A runs over
Ξd := {A ∈ MatN×N (Z≥0 ) | |A| = D = 2d + 1, Aı = A}
(Here ı denotes the involution of rotation by 180 degree.)• The bar involutions stabilize as D → ∞, and lifts to thealgebra Kı .
Recall Q(q) ⊗ Kı ∼= U̇ı .
Theorem 5 (BKLW2014)
The algebra Kı , and hence U̇ı , admits a stably CB {A}, wheree := {(aij ) ∈ MatN×N (Z ) | aij ≥ 0 (i 6= j), |A| is odd}.
A∈ΞI S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

iCanonical Basis of U̇ı
Facts:• AO(D) (X × X ) admits a [geometric] bar involution and acanonical basis with positivity {A}d , where A runs over
Ξd := {A ∈ MatN×N (Z≥0 ) | |A| = D = 2d + 1, Aı = A}
(Here ı denotes the involution of rotation by 180 degree.)• The bar involutions stabilize as D → ∞, and lifts to thealgebra Kı .
Recall Q(q) ⊗ Kı ∼= U̇ı .
Theorem 5 (BKLW2014)
The algebra Kı , and hence U̇ı , admits a stably CB {A}, wheree := {(aij ) ∈ MatN×N (Z ) | aij ≥ 0 (i 6= j), |A| is odd}.
A∈ΞI S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Negativity of stably CB• Recall for type A one hasU̇(slN )vs U̇(glN )• (Recall N = 2n + 1.) ∃ 2 versions of modified coideal
algebras:U̇ı (slN ) vs U̇ı (glN ) = U̇ı• [Li-W15] The stably canonical basis of U̇(glN ) does not havepositive structure constants;
Similar negativity for U̇ı (glN ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Negativity of stably CB• Recall for type A one hasU̇(slN )vs U̇(glN )• (Recall N = 2n + 1.) ∃ 2 versions of modified coideal
algebras:U̇ı (slN ) vs U̇ı (glN ) = U̇ı• [Li-W15] The stably canonical basis of U̇(glN ) does not havepositive structure constants;
Similar negativity for U̇ı (glN ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Negativity of stably CB• Recall for type A one hasU̇(slN )vs U̇(glN )• (Recall N = 2n + 1.) ∃ 2 versions of modified coideal
algebras:U̇ı (slN ) vs U̇ı (glN ) = U̇ı• [Li-W15] The stably canonical basis of U̇(glN ) does not havepositive structure constants;
Similar negativity for U̇ı (glN ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Negativity of stably CB• Recall for type A one hasU̇(slN )vs U̇(glN )• (Recall N = 2n + 1.) ∃ 2 versions of modified coideal
algebras:U̇ı (slN ) vs U̇ı (glN ) = U̇ı• [Li-W15] The stably canonical basis of U̇(glN ) does not havepositive structure constants;
Similar negativity for U̇ı (glN ).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Transfer map

ı
The transfer map φıd+N,d : Sd+N
→ Sdı is the composition
∼
=

ı
Sd+N
−→ EndHB

d+N(V⊗(d+N) )−→EndHB ×HS (V⊗(d+N) )
d

N

∼
=

1⊗χ

−→ EndHB (V⊗d ) −→ Sdı ,dwhere χ is a “sign" homomorphism.
The algebra U̇ı can be thought as an inverse limit of the family{Sdı }d≥1 , i.e., ∃ homomorphism φd : U̇ı (slN ) → Sdı , compatible
with the transfer map φıd+N,d .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Transfer map

ı
The transfer map φıd+N,d : Sd+N
→ Sdı is the composition
∼
=

ı
Sd+N
−→ EndHB

d+N(V⊗(d+N) )−→EndHB ×HS (V⊗(d+N) )
d

N

∼
=

1⊗χ

−→ EndHB (V⊗d ) −→ Sdı ,dwhere χ is a “sign" homomorphism.
The algebra U̇ı can be thought as an inverse limit of the family{Sdı }d≥1 , i.e., ∃ homomorphism φd : U̇ı (slN ) → Sdı , compatible
with the transfer map φıd+N,d .

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity of iCB under a transfer map

Theorem 6 (Li-W15)
The transfer map sends each CB element to a sum of CB
elements with coefficients in Z≥0 [q, q −1 ].
The transfer map does preserve CB of ıSchur algebras
asymptotically; cf. [McGerty12] in type A.
For A ∈ Ξd0 (recall |A| = 2d + 1), set 2p A := A + 2pI ∈ Ξd0 +pN .
Theorem 7 (Li-W15)
For each A ∈ Ξd0 , we have φıd,d−N ({2p A}d ) = {2p−2 A}d−N , forp  0 (where d = d0 + pN).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity of iCB under a transfer map

Theorem 6 (Li-W15)
The transfer map sends each CB element to a sum of CB
elements with coefficients in Z≥0 [q, q −1 ].
The transfer map does preserve CB of ıSchur algebras
asymptotically; cf. [McGerty12] in type A.
For A ∈ Ξd0 (recall |A| = 2d + 1), set 2p A := A + 2pI ∈ Ξd0 +pN .
Theorem 7 (Li-W15)
For each A ∈ Ξd0 , we have φıd,d−N ({2p A}d ) = {2p−2 A}d−N , forp  0 (where d = d0 + pN).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity of iCB under a transfer map

Theorem 6 (Li-W15)
The transfer map sends each CB element to a sum of CB
elements with coefficients in Z≥0 [q, q −1 ].
The transfer map does preserve CB of ıSchur algebras
asymptotically; cf. [McGerty12] in type A.
For A ∈ Ξd0 (recall |A| = 2d + 1), set 2p A := A + 2pI ∈ Ξd0 +pN .
Theorem 7 (Li-W15)
For each A ∈ Ξd0 , we have φıd,d−N ({2p A}d ) = {2p−2 A}d−N , forp  0 (where d = d0 + pN).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity of iCB under a transfer map

Theorem 6 (Li-W15)
The transfer map sends each CB element to a sum of CB
elements with coefficients in Z≥0 [q, q −1 ].
The transfer map does preserve CB of ıSchur algebras
asymptotically; cf. [McGerty12] in type A.
For A ∈ Ξd0 (recall |A| = 2d + 1), set 2p A := A + 2pI ∈ Ξd0 +pN .
Theorem 7 (Li-W15)
For each A ∈ Ξd0 , we have φıd,d−N ({2p A}d ) = {2p−2 A}d−N , forp  0 (where d = d0 + pN).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity again

The asymptotic property of transfer maps allows us to define
the canonical basis Bı (slN ) for U̇ı (slN ), parametrized bye ∼, where the relation ∼ on Ξ
e is defined by A ∼ A + 2I.
A ∈ Ξ/
Theorem 8 (Li-W15)
The structure constants for the algebra U̇ı (slN ) with respect tothe iCB Bı (slN ) are in Z≥0 [q, q −1 ].

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positivity again

The asymptotic property of transfer maps allows us to define
the canonical basis Bı (slN ) for U̇ı (slN ), parametrized bye ∼, where the relation ∼ on Ξ
e is defined by A ∼ A + 2I.
A ∈ Ξ/
Theorem 8 (Li-W15)
The structure constants for the algebra U̇ı (slN ) with respect tothe iCB Bı (slN ) are in Z≥0 [q, q −1 ].

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positiivity again, and again• The iCB Bı (slN ) satisfies (and in turn is characterized by) thealmost orthonormality with respect to a geometric bilinear form.
• The geometric bilinear form is positive with respect to the iCB.
• Recall N = 2n + 1. We have U̇(sln ) ⊂ U̇ı (slN ), and that
Bı (slN ) ∩ U̇(sln ) is the CB of U̇(sln ).
• The action of U̇(sln ) (resp., the action of Sdı ) on V⊗d withrespect to the corresponding iCB is positive.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positiivity again, and again• The iCB Bı (slN ) satisfies (and in turn is characterized by) thealmost orthonormality with respect to a geometric bilinear form.
• The geometric bilinear form is positive with respect to the iCB.
• Recall N = 2n + 1. We have U̇(sln ) ⊂ U̇ı (slN ), and that
Bı (slN ) ∩ U̇(sln ) is the CB of U̇(sln ).
• The action of U̇(sln ) (resp., the action of Sdı ) on V⊗d withrespect to the corresponding iCB is positive.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positiivity again, and again• The iCB Bı (slN ) satisfies (and in turn is characterized by) thealmost orthonormality with respect to a geometric bilinear form.
• The geometric bilinear form is positive with respect to the iCB.
• Recall N = 2n + 1. We have U̇(sln ) ⊂ U̇ı (slN ), and that
Bı (slN ) ∩ U̇(sln ) is the CB of U̇(sln ).
• The action of U̇(sln ) (resp., the action of Sdı ) on V⊗d withrespect to the corresponding iCB is positive.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Positiivity again, and again• The iCB Bı (slN ) satisfies (and in turn is characterized by) thealmost orthonormality with respect to a geometric bilinear form.
• The geometric bilinear form is positive with respect to the iCB.
• Recall N = 2n + 1. We have U̇(sln ) ⊂ U̇ı (slN ), and that
Bı (slN ) ∩ U̇(sln ) is the CB of U̇(sln ).
• The action of U̇(sln ) (resp., the action of Sdı ) on V⊗d withrespect to the corresponding iCB is positive.

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Summary• ∃ new Canonical bases (iCB) for coideal algebra U̇ı and its
modules (Uı is not a Drinfeld-Jimbo quantum algebra.)• Special cases of these iCB allow to formulate and solve KL
conjecture for osp(2m + 1|2n), an open problem since 1970’s.
• The U̇ı and iCB admit geometric realization, generalizing the1990 BLM construction for type A.
• There are many different types of positivity for the iCB(more conjectural than proven).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Summary• ∃ new Canonical bases (iCB) for coideal algebra U̇ı and its
modules (Uı is not a Drinfeld-Jimbo quantum algebra.)• Special cases of these iCB allow to formulate and solve KL
conjecture for osp(2m + 1|2n), an open problem since 1970’s.
• The U̇ı and iCB admit geometric realization, generalizing the1990 BLM construction for type A.
• There are many different types of positivity for the iCB(more conjectural than proven).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Summary• ∃ new Canonical bases (iCB) for coideal algebra U̇ı and its
modules (Uı is not a Drinfeld-Jimbo quantum algebra.)• Special cases of these iCB allow to formulate and solve KL
conjecture for osp(2m + 1|2n), an open problem since 1970’s.
• The U̇ı and iCB admit geometric realization, generalizing the1990 BLM construction for type A.
• There are many different types of positivity for the iCB(more conjectural than proven).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

Summary• ∃ new Canonical bases (iCB) for coideal algebra U̇ı and its
modules (Uı is not a Drinfeld-Jimbo quantum algebra.)• Special cases of these iCB allow to formulate and solve KL
conjecture for osp(2m + 1|2n), an open problem since 1970’s.
• The U̇ı and iCB admit geometric realization, generalizing the1990 BLM construction for type A.
• There are many different types of positivity for the iCB(more conjectural than proven).

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!

I S CHUR DUALITYI C ANONICAL BASISG EOMETRIC REALIZATIONC ANONICAL BASIS , IIiCanonical basis

One more thing
What is next
• iCanonical Bases for [a class of] quantum symmetric pairs• Affinization of q-coideal algebras, iSchur duality, and iCB(via classical type affine flag variety, Steinberg variety, ...)
• iCategorification• Enhance the “locally type A" philosophy of Nakajima andKhovanov-Lauda-Rouquier to “locally type A with involution" ?!

Thank You!