Post on 18-Jul-2020
Metaplectic representations of Hecke algebrasJoint work with Jasper Stokman and Vidya Venkateswaran
Siddhartha SahiRutgers University, New Brunswick NJ
Representation Theory Seminar, UC Berkeley
Siddhartha Sahi (Rutgers University) May 15, 2020 1 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .
πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2c
Defined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Background
Φ root system, {αi} simple roots, W Weyl group, P weight lattice
Cx [P ] =⟨xλ : λ ∈ P
⟩= group algebra, Cx (P) fraction field
W acts on Cx [P ] and on Cx (P) by the reflection representation π
In fact π is the first in a family π1 = π,π2,π3, . . .πn depends on a bilinear form B and parameters v , g1, . . . , gbn/2cDefined by Chinta-Gunnels, motivated by work of Kazhdan-Patterson
Arises in number theory (Weyl group multiple Dirichlet series)
In the application to WMDS, the gi are certain Gauss sums
Siddhartha Sahi (Rutgers University) May 15, 2020 2 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}
Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Deformation factor
Let B be a W -invariant symmetric form on the weight lattice P
Assume Q (λ) = 12B (λ,λ) ∈ Z, and gcd (n,Q (α)) = 1 for all α ∈ Φ
Let Pn := {λ ∈ P : B (λ, α) ≡ 0 mod n for all α ∈ Φ}Cx (P) is spanned by elements fxλ with λ ∈ P, f ∈ Cx (Pn)
The πn action of the simple relection si = sαi ∈ W satisfies
πn (si )(fxλ)
π (si ) (fxλ)= γ
(n)i (λ)
Here γ(n)i : P/Pn → Cx (P) is a certain “deformation factor”
Siddhartha Sahi (Rutgers University) May 15, 2020 3 / 17
Explicit formula
Let rn [a] ∈ {0, 1, . . . , n− 1} denote the remainder of a mod n
rn extends naturally to rational numbers a/b with gcd (n, b) = 1Fix parameters v and {gj : j ∈ Z/nZ} satisfying
g0 = −1, gjg−j = v−1 for j 6≡ 0
The deformation factor is
γ(n)i (λ) =
1− v1− vxnαi
x−rn
[−B(λ,αi )
Q(αi )
]αi − 1− xnαi
1− vxnαi
g−B(λ,αi )gQ(αi )
x (1−n)αi
Theorem (Chinta-Gunnels)
The above formula defines a representation πn of W on Cx (P)
Siddhartha Sahi (Rutgers University) May 15, 2020 4 / 17
Explicit formula
Let rn [a] ∈ {0, 1, . . . , n− 1} denote the remainder of a mod nrn extends naturally to rational numbers a/b with gcd (n, b) = 1
Fix parameters v and {gj : j ∈ Z/nZ} satisfying
g0 = −1, gjg−j = v−1 for j 6≡ 0
The deformation factor is
γ(n)i (λ) =
1− v1− vxnαi
x−rn
[−B(λ,αi )
Q(αi )
]αi − 1− xnαi
1− vxnαi
g−B(λ,αi )gQ(αi )
x (1−n)αi
Theorem (Chinta-Gunnels)
The above formula defines a representation πn of W on Cx (P)
Siddhartha Sahi (Rutgers University) May 15, 2020 4 / 17
Explicit formula
Let rn [a] ∈ {0, 1, . . . , n− 1} denote the remainder of a mod nrn extends naturally to rational numbers a/b with gcd (n, b) = 1Fix parameters v and {gj : j ∈ Z/nZ} satisfying
g0 = −1, gjg−j = v−1 for j 6≡ 0
The deformation factor is
γ(n)i (λ) =
1− v1− vxnαi
x−rn
[−B(λ,αi )
Q(αi )
]αi − 1− xnαi
1− vxnαi
g−B(λ,αi )gQ(αi )
x (1−n)αi
Theorem (Chinta-Gunnels)
The above formula defines a representation πn of W on Cx (P)
Siddhartha Sahi (Rutgers University) May 15, 2020 4 / 17
Explicit formula
Let rn [a] ∈ {0, 1, . . . , n− 1} denote the remainder of a mod nrn extends naturally to rational numbers a/b with gcd (n, b) = 1Fix parameters v and {gj : j ∈ Z/nZ} satisfying
g0 = −1, gjg−j = v−1 for j 6≡ 0
The deformation factor is
γ(n)i (λ) =
1− v1− vxnαi
x−rn
[−B(λ,αi )
Q(αi )
]αi − 1− xnαi
1− vxnαi
g−B(λ,αi )gQ(αi )
x (1−n)αi
Theorem (Chinta-Gunnels)
The above formula defines a representation πn of W on Cx (P)
Siddhartha Sahi (Rutgers University) May 15, 2020 4 / 17
Explicit formula
Let rn [a] ∈ {0, 1, . . . , n− 1} denote the remainder of a mod nrn extends naturally to rational numbers a/b with gcd (n, b) = 1Fix parameters v and {gj : j ∈ Z/nZ} satisfying
g0 = −1, gjg−j = v−1 for j 6≡ 0
The deformation factor is
γ(n)i (λ) =
1− v1− vxnαi
x−rn
[−B(λ,αi )
Q(αi )
]αi − 1− xnαi
1− vxnαi
g−B(λ,αi )gQ(αi )
x (1−n)αi
Theorem (Chinta-Gunnels)
The above formula defines a representation πn of W on Cx (P)
Siddhartha Sahi (Rutgers University) May 15, 2020 4 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra H
H is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localization
We construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Baxterization
The original C-G proof involves a computer check of braid relations
It is desirable to have a more conceptual proof (as noted by C-G)
In https://arxiv.org/abs/1808.01069 we provide such an argument
Involves the affi ne Weyl group W ≈W n P and its Hecke algebra HH is isomorphic to group algebra C [W ] after suitable localizationWe construct a representation vn of H and obtain πn by localization
This localization procedure is sometimes called “Baxterization”
Siddhartha Sahi (Rutgers University) May 15, 2020 5 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
Metaplectic polynomials
In work in progress, we extend vn to the double affi ne Hecke algebra
The extension of v1 is the usual "polynomial" representation
The well-known Macdonald polynomials Eλ arise naturally via v1
vn leads to a natural family of "metaplectic" polynomials E(n)λ
The polynomials E (n)λ appear to share many features with Eλ
It would be interesting to understand which properties generalize!
We now explain our construction.
Siddhartha Sahi (Rutgers University) May 15, 2020 6 / 17
The reflection representation
Finite Hecke algebra H: generators Ti satisying braid relations
Quadratic relations: (Ti − k)(Ti + k−1
)= 0 with k = v1/2,
Reflection representation: Vector space U, basis{uµ : µ ∈ P
}Tiuµ =
usiµ if (µ, α∨i ) > 0kuµ if (µ, α∨i ) = 0
(k − k−1)uµ + usiµ if (µ, α∨i ) < 0
Let C := {λ ∈ P | (λ, α∨) ≤ n ∀ α ∈ Φ}; C is W -stableUC :=
⊕λ∈C Cuλ is an H-submodule of U
Siddhartha Sahi (Rutgers University) May 15, 2020 7 / 17
The reflection representation
Finite Hecke algebra H: generators Ti satisying braid relations
Quadratic relations: (Ti − k)(Ti + k−1
)= 0 with k = v1/2,
Reflection representation: Vector space U, basis{uµ : µ ∈ P
}Tiuµ =
usiµ if (µ, α∨i ) > 0kuµ if (µ, α∨i ) = 0
(k − k−1)uµ + usiµ if (µ, α∨i ) < 0
Let C := {λ ∈ P | (λ, α∨) ≤ n ∀ α ∈ Φ}; C is W -stableUC :=
⊕λ∈C Cuλ is an H-submodule of U
Siddhartha Sahi (Rutgers University) May 15, 2020 7 / 17
The reflection representation
Finite Hecke algebra H: generators Ti satisying braid relations
Quadratic relations: (Ti − k)(Ti + k−1
)= 0 with k = v1/2,
Reflection representation: Vector space U, basis{uµ : µ ∈ P
}Tiuµ =
usiµ if (µ, α∨i ) > 0kuµ if (µ, α∨i ) = 0
(k − k−1)uµ + usiµ if (µ, α∨i ) < 0
Let C := {λ ∈ P | (λ, α∨) ≤ n ∀ α ∈ Φ}; C is W -stableUC :=
⊕λ∈C Cuλ is an H-submodule of U
Siddhartha Sahi (Rutgers University) May 15, 2020 7 / 17
The reflection representation
Finite Hecke algebra H: generators Ti satisying braid relations
Quadratic relations: (Ti − k)(Ti + k−1
)= 0 with k = v1/2,
Reflection representation: Vector space U, basis{uµ : µ ∈ P
}Tiuµ =
usiµ if (µ, α∨i ) > 0kuµ if (µ, α∨i ) = 0
(k − k−1)uµ + usiµ if (µ, α∨i ) < 0
Let C := {λ ∈ P | (λ, α∨) ≤ n ∀ α ∈ Φ}; C is W -stable
UC :=⊕
λ∈C Cuλ is an H-submodule of U
Siddhartha Sahi (Rutgers University) May 15, 2020 7 / 17
The reflection representation
Finite Hecke algebra H: generators Ti satisying braid relations
Quadratic relations: (Ti − k)(Ti + k−1
)= 0 with k = v1/2,
Reflection representation: Vector space U, basis{uµ : µ ∈ P
}Tiuµ =
usiµ if (µ, α∨i ) > 0kuµ if (µ, α∨i ) = 0
(k − k−1)uµ + usiµ if (µ, α∨i ) < 0
Let C := {λ ∈ P | (λ, α∨) ≤ n ∀ α ∈ Φ}; C is W -stableUC :=
⊕λ∈C Cuλ is an H-submodule of U
Siddhartha Sahi (Rutgers University) May 15, 2020 7 / 17
Affi ne Hecke algebra
Φn := {nα : α ∈ Φ} is a root system, with weight lattice
Pn := {µ ∈ P : B (µ, α) ≡ 0 mod n for all α ∈ Φ}
Pn group algebra: CY := CY [Pn ] = 〈Y µ : µ ∈ Pn〉Affi ne Hecke algebra H = Hn = 〈H,CY 〉 with relations
TiY µ − Y siµTi = (k − k−1)(Y siµ − Y µ
Y nαi − 1
)Induced representation: σ = IndHH (UC ) realized on NC = UC ⊗CY
Siddhartha Sahi (Rutgers University) May 15, 2020 8 / 17
Affi ne Hecke algebra
Φn := {nα : α ∈ Φ} is a root system, with weight lattice
Pn := {µ ∈ P : B (µ, α) ≡ 0 mod n for all α ∈ Φ}
Pn group algebra: CY := CY [Pn ] = 〈Y µ : µ ∈ Pn〉
Affi ne Hecke algebra H = Hn = 〈H,CY 〉 with relations
TiY µ − Y siµTi = (k − k−1)(Y siµ − Y µ
Y nαi − 1
)Induced representation: σ = IndHH (UC ) realized on NC = UC ⊗CY
Siddhartha Sahi (Rutgers University) May 15, 2020 8 / 17
Affi ne Hecke algebra
Φn := {nα : α ∈ Φ} is a root system, with weight lattice
Pn := {µ ∈ P : B (µ, α) ≡ 0 mod n for all α ∈ Φ}
Pn group algebra: CY := CY [Pn ] = 〈Y µ : µ ∈ Pn〉Affi ne Hecke algebra H = Hn = 〈H,CY 〉 with relations
TiY µ − Y siµTi = (k − k−1)(Y siµ − Y µ
Y nαi − 1
)
Induced representation: σ = IndHH (UC ) realized on NC = UC ⊗CY
Siddhartha Sahi (Rutgers University) May 15, 2020 8 / 17
Affi ne Hecke algebra
Φn := {nα : α ∈ Φ} is a root system, with weight lattice
Pn := {µ ∈ P : B (µ, α) ≡ 0 mod n for all α ∈ Φ}
Pn group algebra: CY := CY [Pn ] = 〈Y µ : µ ∈ Pn〉Affi ne Hecke algebra H = Hn = 〈H,CY 〉 with relations
TiY µ − Y siµTi = (k − k−1)(Y siµ − Y µ
Y nαi − 1
)Induced representation: σ = IndHH (UC ) realized on NC = UC ⊗CY
Siddhartha Sahi (Rutgers University) May 15, 2020 8 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Metaplectic representation
Recall v = k2 and {gj : j ∈ Z/nZ}: g0 = −1, gjg−j = v−1 for j 6≡ 0
For j ∈ Z define γj =
1 if j ∈ Z≥0k−1 if j ∈ nZ<0
−kgj otherwise
For λ ∈ C define γ(λ) := ∏α∈Φ+ γQ(α)(λ,α∨)
Define a surjective map NC = UC ⊗CY [Pn ]ψ−→ Cx [P ] by
ψ (uλ ⊗ Y µ) =1
γ(λ)xλ+µ
Theorem (S.-Stokman-Venkateswaran)
The kernel of ψ is stable under (σ,H)
Metaplectic rep. vn: Quotient action of H on Cx [P ]
Siddhartha Sahi (Rutgers University) May 15, 2020 9 / 17
Localization
Recall H = 〈H,CY [Pn ]〉, localization Hloc := 〈H,CY (Pn)〉
Localization of affi ne group algebra Aloc := 〈C [W ] ,CY (Pn)〉
Theorem (Localization)
The identity on CY (Pn) extends to an algebra isomorphism Aloc ≈ Hloc ,with si 7→ ci (Y ) (Ti − k) + 1 with ci (Y ) = (1− Y nαi ) /
(k−1 − kY nαi
).
Theorem (S.-Stokman-Venkateswaran)
The metaplectic representation v of H on Cx [P ] extends (uniquely) to arepresentation vloc of Hloc on Cx (P). The action of W induced by theisomorphism Aloc ≈ Hloc coincides with the CG action.
Siddhartha Sahi (Rutgers University) May 15, 2020 10 / 17
Localization
Recall H = 〈H,CY [Pn ]〉, localization Hloc := 〈H,CY (Pn)〉Localization of affi ne group algebra Aloc := 〈C [W ] ,CY (Pn)〉
Theorem (Localization)
The identity on CY (Pn) extends to an algebra isomorphism Aloc ≈ Hloc ,with si 7→ ci (Y ) (Ti − k) + 1 with ci (Y ) = (1− Y nαi ) /
(k−1 − kY nαi
).
Theorem (S.-Stokman-Venkateswaran)
The metaplectic representation v of H on Cx [P ] extends (uniquely) to arepresentation vloc of Hloc on Cx (P). The action of W induced by theisomorphism Aloc ≈ Hloc coincides with the CG action.
Siddhartha Sahi (Rutgers University) May 15, 2020 10 / 17
Localization
Recall H = 〈H,CY [Pn ]〉, localization Hloc := 〈H,CY (Pn)〉Localization of affi ne group algebra Aloc := 〈C [W ] ,CY (Pn)〉
Theorem (Localization)
The identity on CY (Pn) extends to an algebra isomorphism Aloc ≈ Hloc ,with si 7→ ci (Y ) (Ti − k) + 1 with ci (Y ) = (1− Y nαi ) /
(k−1 − kY nαi
).
Theorem (S.-Stokman-Venkateswaran)
The metaplectic representation v of H on Cx [P ] extends (uniquely) to arepresentation vloc of Hloc on Cx (P). The action of W induced by theisomorphism Aloc ≈ Hloc coincides with the CG action.
Siddhartha Sahi (Rutgers University) May 15, 2020 10 / 17
Localization
Recall H = 〈H,CY [Pn ]〉, localization Hloc := 〈H,CY (Pn)〉Localization of affi ne group algebra Aloc := 〈C [W ] ,CY (Pn)〉
Theorem (Localization)
The identity on CY (Pn) extends to an algebra isomorphism Aloc ≈ Hloc ,with si 7→ ci (Y ) (Ti − k) + 1 with ci (Y ) = (1− Y nαi ) /
(k−1 − kY nαi
).
Theorem (S.-Stokman-Venkateswaran)
The metaplectic representation v of H on Cx [P ] extends (uniquely) to arepresentation vloc of Hloc on Cx (P). The action of W induced by theisomorphism Aloc ≈ Hloc coincides with the CG action.
Siddhartha Sahi (Rutgers University) May 15, 2020 10 / 17
Double affi ne Hecke algebras
vn can be extended to the double affi ne Hecke algebra (preprint)
We get an action of “metaplectic”Cherednik operators andpolynomials E (n)λ generalizing Macdonald polynomials Eλ.
The E (n)λ specialize to Whittaker functions on metaplectic coveringgroups, which are related to p-parts of WMDS, studied byChinta-Gunnels and McNamara
This generalizes the relation between Whittaker functions andMacdonald polynomials via the Casellman-Shalika formula
We give some tables of metaplectic polynomials, here ε = ±1
Siddhartha Sahi (Rutgers University) May 15, 2020 11 / 17
Double affi ne Hecke algebras
vn can be extended to the double affi ne Hecke algebra (preprint)
We get an action of “metaplectic”Cherednik operators andpolynomials E (n)λ generalizing Macdonald polynomials Eλ.
The E (n)λ specialize to Whittaker functions on metaplectic coveringgroups, which are related to p-parts of WMDS, studied byChinta-Gunnels and McNamara
This generalizes the relation between Whittaker functions andMacdonald polynomials via the Casellman-Shalika formula
We give some tables of metaplectic polynomials, here ε = ±1
Siddhartha Sahi (Rutgers University) May 15, 2020 11 / 17
Double affi ne Hecke algebras
vn can be extended to the double affi ne Hecke algebra (preprint)
We get an action of “metaplectic”Cherednik operators andpolynomials E (n)λ generalizing Macdonald polynomials Eλ.
The E (n)λ specialize to Whittaker functions on metaplectic coveringgroups, which are related to p-parts of WMDS, studied byChinta-Gunnels and McNamara
This generalizes the relation between Whittaker functions andMacdonald polynomials via the Casellman-Shalika formula
We give some tables of metaplectic polynomials, here ε = ±1
Siddhartha Sahi (Rutgers University) May 15, 2020 11 / 17
Double affi ne Hecke algebras
vn can be extended to the double affi ne Hecke algebra (preprint)
We get an action of “metaplectic”Cherednik operators andpolynomials E (n)λ generalizing Macdonald polynomials Eλ.
The E (n)λ specialize to Whittaker functions on metaplectic coveringgroups, which are related to p-parts of WMDS, studied byChinta-Gunnels and McNamara
This generalizes the relation between Whittaker functions andMacdonald polynomials via the Casellman-Shalika formula
We give some tables of metaplectic polynomials, here ε = ±1
Siddhartha Sahi (Rutgers University) May 15, 2020 11 / 17
Double affi ne Hecke algebras
vn can be extended to the double affi ne Hecke algebra (preprint)
We get an action of “metaplectic”Cherednik operators andpolynomials E (n)λ generalizing Macdonald polynomials Eλ.
The E (n)λ specialize to Whittaker functions on metaplectic coveringgroups, which are related to p-parts of WMDS, studied byChinta-Gunnels and McNamara
This generalizes the relation between Whittaker functions andMacdonald polynomials via the Casellman-Shalika formula
We give some tables of metaplectic polynomials, here ε = ±1
Siddhartha Sahi (Rutgers University) May 15, 2020 11 / 17
Some Metaplectic polynomials for GL(3)
Formulas for E (n)λ (x), 1 ≤ n ≤ 5 and λ ∈ Z3 of weight at most 2.
E (1)(0,0,0)(x) = 1
E (2)(0,0,0)(x) = 1
E (3)(0,0,0)(x) = 1
E (4)(0,0,0)(x) = 1
E (5)(0,0,0)(x) = 1
E (1)(1,0,0)(x) = x1
E (2)(1,0,0)(x) = x1
E (3)(1,0,0)(x) = x1
E (4)(1,0,0)(x) = x1
E (5)(1,0,0)(x) = x1
Siddhartha Sahi (Rutgers University) May 15, 2020 12 / 17
Some Metaplectic polynomials for GL(3)
E (1)(0,1,0)(x) =
(k−1)(k+1)k 4q−1 x1 + x2
E (2)(0,1,0)(x) =
(k−1)(k+1)k (kq2+ε)
x1 + x2
E (3)(0,1,0)(x) =
(k−1)(k+1)g1k 4g 31 q
3+1 x1 + x2
E (4)(0,1,0)(x) =
(k−1)(k+1)g1k 4g 31 q
4+1 x1 + x2
E (5)(0,1,0)(x) =
(k−1)(k+1)g1k 4g 31 q
5+1 x1 + x2
E (1)(0,0,1)(x) =
(k−1)(k+1)qk 2−1 x1 +
(k−1)(k+1)qk 2−1 x2 + x3
E (2)(0,0,1)(x) = −
(k−1)(k+1)k (k+εq2) x1 +
(k−1)(k+1)q2+εk x2 + x3
E (3)(0,0,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
3+1 x1 +(k−1)(k+1)g1k 2g 31 q
3+1 x2 + x3
E (4)(0,0,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
4+1 x1 +(k−1)(k+1)g1k 2g 31 q
4+1 x2 + x3
E (5)(0,0,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
5+1 x1 +(k−1)(k+1)g1k 2g 31 q
5+1 x2 + x3
Siddhartha Sahi (Rutgers University) May 15, 2020 13 / 17
Some Metaplectic polynomials for GL(3)
E (1)(0,1,1)(x) =
(k−1)(k+1)qk 2−1 x1x2 +
(k−1)(k+1)qk 2−1 x3x1 + x3x2
E (2)(0,1,1)(x) = −
(k−1)(k+1)k (k+εq2) x1x2 +
(k−1)(k+1)q2+εk x3x1 + x3x2
E (3)(0,1,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
3+1 x1x2 +(k−1)(k+1)g1k 2g 31 q
3+1 x3x1 + x3x2
E (4)(0,1,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
4+1 x1x2 +(k−1)(k+1)g1k 2g 31 q
4+1 x3x1 + x3x2
E (5)(0,1,1)(x) = −
(k−1)(k+1)g 21k 2g 31 q
5+1 x1x2 +(k−1)(k+1)g1k 2g 31 q
5+1 x3x1 + x3x2
E (1)(1,0,1)(x) =
(k−1)(k+1)k 4q−1 x1x2 + x3x1
E (2)(1,0,1)(x) =
(k−1)(k+1)k (kq2+ε)
x1x2 + x3x1
E (3)(1,0,1)(x) =
(k−1)(k+1)g1k 4g 31 q
3+1 x1x2 + x3x1
E (4)(1,0,1)(x) =
(k−1)(k+1)g1k 4g 31 q
4+1 x1x2 + x3x1
E (5)(1,0,1)(x) =
(k−1)(k+1)g1k 4g 31 q
5+1 x1x2 + x3x1
Siddhartha Sahi (Rutgers University) May 15, 2020 14 / 17
Some Metaplectic polynomials for GL(3)
E (1)(1,1,0)(x) = x1x2
E (2)(1,1,0)(x) = x1x2
E (3)(1,1,0)(x) = x1x2
E (4)(1,1,0)(x) = x1x2
E (5)(1,1,0)(x) = x1x2
E (1)(2,0,0)(x) = x
21 +
q(k−1)(k+1)qk 2−1 x1x2 +
q(k−1)(k+1)qk 2−1 x3x1
E (2)(2,0,0)(x) = x
21
E (3)(2,0,0)(x) = x
21
E (4)(2,0,0)(x) = x
21
E (5)(2,0,0)(x) = x
21
Siddhartha Sahi (Rutgers University) May 15, 2020 15 / 17
Some Metaplectic polynomials for GL(3)
E (1)(0,2,0)(x) =
(k−1)(k+1)(qk 2−1)(qk 2+1)x
21 + x
22 +
(k−1)(k+1)(k 4q2+qk 2−q−1)(qk 2+1)(qk 2−1)2 x1x2 +
(k−1)2(k+1)2q(qk 2+1)(qk 2−1)2 x3x1 +
q(k−1)(k+1)qk 2−1 x3x2
E (2)(0,2,0)(x) =
(k−1)(k+1)(q2k 2−1)(q2k 2+1)x
21 + x
22
E (3)(0,2,0)(x) =
(k−1)(k+1)g 21k 2g 31+q
6 x21 + x22
E (4)(0,2,0)(x) =
(k−1)(k+1)k (q8k+ε)
x21 + x22
E (5)(0,2,0)(x) =
(k−1)(k+1)g2k 4g 32 q
10+1 x21 + x22
Siddhartha Sahi (Rutgers University) May 15, 2020 16 / 17
Some Metaplectic polynomials for GL(3)
E (1)(0,0,2)(x) =
(k−1)(k+1)(kq−1)(kq+1)x
21 +
(k−1)(k+1)(kq−1)(kq+1)x
22 + x
23 +
(q+1)(k−1)2(k+1)2(kq−1)(kq+1)(qk 2−1)x1x2 +
(q+1)(k−1)(k+1)(kq−1)(kq+1) x3x1 +
(q+1)(k−1)(k+1)(kq−1)(kq+1) x3x2
E (2)(0,0,2)(x) =
(k−1)(k+1)(kq2−1)(kq2+1)x
21 +
(k−1)(k+1)(kq2−1)(kq2+1)x
22 + x
23
E (3)(0,0,2)(x) = −
(k−1)(k+1)g1k 4g 31+q
6 x21 +(k−1)(k+1)k 2g 21
k 4g 31+q6 x22 + x
23
E (4)(0,0,2)(x) = −
(k−1)(k+1)k (εq8+k ) x
21 +
(k−1)(k+1)q8+εk x22 + x
23
E (5)(0,0,2)(x) = −
(k−1)(k+1)g 22k 2g 32 q
10+1 x21 +(k−1)(k+1)g2k 2g 32 q
10+1 x22 + x23
Siddhartha Sahi (Rutgers University) May 15, 2020 17 / 17