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
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