Siddhartha Sahi Rutgers University, New Brunswick NJ ...shapiro/Sahi_Berkeley2020.pdfSiddhartha Sahi (Rutgers University) May 15, 2020 2 / 17. Background F root system, fa igsimple

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


Background


Cx [P ] =⟨xλ : λ ∈ P







Background


Cx [P ] =⟨xλ : λ ∈ P







Background


Cx [P ] =⟨xλ : λ ∈ P



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




Background


Cx [P ] =⟨xλ : λ ∈ P



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




Background


Cx [P ] =⟨xλ : λ ∈ P







Background


Cx [P ] =⟨xλ : λ ∈ P







Background


Cx [P ] =⟨xλ : λ ∈ P







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”


Deformation factor





πn (si )(fxλ)

π (si ) (fxλ)= γ

(n)i (λ)



Deformation factor



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)


πn (si )(fxλ)

π (si ) (fxλ)= γ

(n)i (λ)



Deformation factor





πn (si )(fxλ)

π (si ) (fxλ)= γ

(n)i (λ)



Deformation factor





πn (si )(fxλ)

π (si ) (fxλ)= γ

(n)i (λ)



Deformation factor





πn (si )(fxλ)

π (si ) (fxλ)= γ

(n)i (λ)



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)


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


γ(n)i (λ) =

1− v1− vxnαi

x−rn

[−B(λ,αi )

Q(αi )

]αi − 1− xnαi

1− vxnαi


x (1−n)αi




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


γ(n)i (λ) =

1− v1− vxnαi

x−rn

[−B(λ,αi )

Q(αi )

]αi − 1− xnαi

1− vxnαi


x (1−n)αi




Explicit formula


g0 = −1, gjg−j = v−1 for j 6≡ 0


γ(n)i (λ) =

1− v1− vxnαi

x−rn

[−B(λ,αi )

Q(αi )

]αi − 1− xnαi

1− vxnαi


x (1−n)αi




Explicit formula


g0 = −1, gjg−j = v−1 for j 6≡ 0


γ(n)i (λ) =

1− v1− vxnαi

x−rn

[−B(λ,αi )

Q(αi )

]αi − 1− xnαi

1− vxnαi


x (1−n)αi




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”


Baxterization







Baxterization







Baxterization




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



Baxterization




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



Baxterization







Baxterization







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.
























































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





)= 0 with k = v1/2,


}Tiuµ =


(k − k−1)uµ + usiµ if (µ, α∨i ) < 0







)= 0 with k = v1/2,


}Tiuµ =


(k − k−1)uµ + usiµ if (µ, α∨i ) < 0







)= 0 with k = v1/2,


}Tiuµ =


(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





)= 0 with k = v1/2,


}Tiuµ =


(k − k−1)uµ + usiµ if (µ, α∨i ) < 0




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





Pn group algebra: CY := CY [Pn ] = 〈Y µ : µ ∈ Pn〉

Affi ne Hecke algebra H = Hn = 〈H,CY 〉 with relations


Y nαi − 1








Y nαi − 1

)

Induced representation: σ = IndHH (UC ) realized on NC = UC ⊗CY







Y nαi − 1



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 ]





1 if j ∈ Z≥0k−1 if j ∈ nZ<0

−kgj otherwise



ψ (uλ ⊗ Y µ) =1

γ(λ)xλ+µ








1 if j ∈ Z≥0k−1 if j ∈ nZ<0

−kgj otherwise



ψ (uλ ⊗ Y µ) =1

γ(λ)xλ+µ








1 if j ∈ Z≥0k−1 if j ∈ nZ<0

−kgj otherwise



ψ (uλ ⊗ Y µ) =1

γ(λ)xλ+µ








1 if j ∈ Z≥0k−1 if j ∈ nZ<0

−kgj otherwise



ψ (uλ ⊗ Y µ) =1

γ(λ)xλ+µ








1 if j ∈ Z≥0k−1 if j ∈ nZ<0

−kgj otherwise



ψ (uλ ⊗ Y µ) =1

γ(λ)xλ+µ





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

).


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.


Localization

Recall H = 〈H,CY [Pn ]〉, localization Hloc := 〈H,CY (Pn)〉Localization of affi ne group algebra Aloc := 〈C [W ] ,CY (Pn)〉



(k−1 − kY nαi

).




Localization




(k−1 − kY nαi

).




Localization




(k−1 − kY nαi

).




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






























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



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



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



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



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



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


Documents

Siddhartha Sahi Rutgers University, New Brunswick NJ ...shapiro/Sahi_Berkeley2020.pdfSiddhartha Sahi (Rutgers University) May 15, 2020 2 / 17. Background F root system, fa igsimple