q -deformed Whittaker functions and the local Langlands - - PowerPoint PPT Presentation

q-deformed Whittaker functions and the local Langlands correspondence

Sergey OBLEZIN , ITEP March 7, 2013 ICERM (Providence)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 1 / 17

References

1 S. Oblezin, On parabolic Whittaker functions I & II, Lett. Math.

• Phys. 101 & Cent. Eur. J. Math. 10 (2012);

2 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator formalism for

Macdonald polynomials, [math.AG/1204.0926];

3 A. Gerasimov, D. Lebedev, S. Oblezin On a classical limit of

q-deformed Whittaker functions, Lett. Math. Phys., 100 (2012);

4 A. Gerasimov, D. Lebedev, S. Oblezin Parabolic Whittaker functions

and Topological field theories I, Commun. Number Theory Phys. 5 (2011);

5 A. Gerasimov, D. Lebedev, S. Oblezin On q-deformed Whittaker

function I, II & III, Commun. Math. Phys 294 (2010) & Lett. Math. Phys 97 (2011);

6 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator and

Archimedean Hecke algebra, Commun. Math. Phys. 284 (2008) .

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 2 / 17

Whittaker functions

The Gauss (Bruhat) decomposition of G = G(F): G 0 = U− · A · U+ , A =

• ex1, . . . , exN

. Character of B− = U−A with λ = (λ1, . . . , λN) ∈ CN: χλ : B− − → C∗ , χλ(ua) =

N

• i=1

e(λi+ρi) xi . The principal series representation (πλ, Vλ) of G and of U(g): Ind

G

B− χλ =

• f ∈ Fun(G)
• f (bg) = χλ(b) f (g) ,

b ∈ B−

• The Whittaker function Ψλ(g) is a smooth function on X = N−\G

analytic in λ given by Ψλ(g) = eρ(g) ψL , πλ(e−H(g)) ψR

• ,

(1) ψL, ψR ∈ Vλ are defined by character ψ0 : F → C∗ : ψ : U − → C , ψ(u) =

• simple roots

ψ0

• uαi
• .

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 3 / 17

Spherical Whittaker functions

The Iwasawa decomposition of G = G(F): G = K · A · U+ . The spherical Whittaker function Ψλ(z) is a smooth function on H = K\G analytic in λ given by Ψλ(g) = eρ(g) ψK , πλ(e−H(g)) ψR

• ,

(2) with the spherical vector ψK ∈ Vλ.

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 4 / 17

Quantum Toda lattice

In the real case, G = G(R), generators Cr , r = 1, . . . , N of the center ZU(g) define quantum Toda Hamiltonians: Hr · Ψλ(x) := e−ρ(x) ψK , πλ(Cr e−H(x)) ψR

• .

(3) The G(R)-Whittaker function is an eigenfunction: Hr · Ψλ(x) = σr(λ) Ψλ(x) , (4) σr(λ) are r-symmetric functions in λ = (λ1, . . . , λN).

Example

In the case G = GL(2; R) H1 = − ∂ ∂x1 + ∂ ∂x2

• ,

H2 = −2 ∂2 ∂x2

1

+ ∂2 ∂x2

2

• + ex1−x2 ,

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 5 / 17

Example: the GL(2; R)-Whittaker functions

ΨR

λ1, λ2(ex1, ex2) =

• R

dT e

ı λ2(x1+x2−T)+ ı λ1T − 1

• ex1−T +eT−x2

(5) = e

λ1+λ2 2

e

x1+x2 2

K λ1−λ2

• 2

e

x1−x2 2

• .

The Mellin-Barnes integral representation: ΨR

λ1, λ2(ex1, ex2) =

• R−ıǫ

dγ e

ı x2(λ1+λ2−γ)+ ı x1γ

2

• i=1
• λi −γ

Γ

λi − γ

• (6)

Both integral representations can be generalized to GL(N; R) by induction

• ver the rank N, using the Baxter Q-operator formalism, [GLO].

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 6 / 17

Baxter operators for spherical Whittaker functions, [GLO]

One-parameter family of K-biinvariant functions Qs in the Hecke algebra:

• Qs ∗ Ψλ
• (g) =
• G

dh Qs(gh−1) Ψλ(h) = Lp(s; V ) Ψλ(g) .

Theorem

The L-function is the spherical transform of the Baxter operator kernel Lp(s; V ⊗ δ−1/2) =

• A

da Qs(a−1) ϕλ(a) , with spherical function given by ϕλ(g) =

• K

dk eH(kg), λ .

Example

In the case G = GL(1; F): Lp(s; V ) = 1 1 − pλ−s , L∞(s; V ) = h

λ−s Γ

λ − s

• .

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 7 / 17

Explicit formulas: non-Archimedean case

Let ξλ : H(G, K) → C and σλ ⊂ GL(N; C) is the (semisimple) conjugacy class, attached to ξλ via Satake correspondence. The class-one GL(N; F)-Whittaker function associated with ξλ:

1 Ψλ(ug) = ψ(u) Ψλ(ug) ; 2

G

dh Ψλ(gh) φ(h−1) = ξλ(φ) Ψλ(g) for any φ ∈ H(G, K) ;

3 Ψλ(1) = 1 .

The Langlands-Shintani (LS) formula

The class-one GL(N; Qp)-Whittaker function reads ΨQp

λ (pn) =

     p−̺(n) ch Vn pλ1 ...

pλN

• ,

n = (n1 ≥ . . . ≥ nN) 0 , n non-dominant (7)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 8 / 17

Explicit formulas: Archimedean case

The Givental stationary phase integral formula: ΨR

λ(x) =

• C
• k≤n<N

dxnk e−1Fλ(xnk) , C ∼ R

N(N−1) 2

⊂ C

N(N−1) 2

, (8) Fλ(xnk) =

N

• n=1

ıλn

• n
• k=1

xn,k −

n−1

• i=1

xn−1, i

• −
• arrows

etarget(a) −source(a) xN,1

• xN,2

. . .

• xNN

...

• . . .
• .

. .

• x21
• x22
• x11
• (9)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 9 / 17

Baxter operators for Macdonald polynomials, [GLO]

∨

Qq,t

x

· f (Λ) =

• µ∈ZN

∨

Qx(µ, Λ) f (µ) ,

∨

Qx(µ, Λ) = x|µ|−|Λ|ϕµ/Λ , (10) ϕµ/Λ =

N

• i,j=1

i≤j

Γq,tq−1

• tj−iqµi−µj+1

Γq,tq−1

• tj−iqµi−Λj+1 Γq,tq−1
• tj−iqΛi−Λj+1+1

Γq,tq−1

• tj−iqΛi−µj+1+1 ,

Γq,t(z) =

• n≥0

1 − tzqn 1 − zqn , Γq, t(z)×Γq, t−1(qz−1) = t1/2 θ1

• (tz)1/2; q
• θ1
• z1/2; q
• .

Theorem

The Macdonald polynomials are eigenfunctions under the action of (10):

∨

Qq,t

x

· PΛ(z) = L∨

x (z) PΛ(z) ,

L∨

x (z) = N

• i=1

Γq,t(xzi) . (11)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 10 / 17

q-deformed Whittaker functions

ΨR

λ(a)

P q,t

Λ (z) t→0

• t→+∞

Ψq

z(Λ) q→1

• q→0

z=pλ

• ΨQp

λ (a)

Hr · Ψq

z(Λ) = er(x) Ψq z(Λ) .

(12) Hr =

• Ir

r

• k=1
• 1 − qΛik −Λik +1+11−δik+1−ik , 1 TIr ,

TIr =

• i∈Ir

Tq, qΛi ,

Example

In the case GL(2; F): H1 =

• 1 − qΛ1−Λ2+1

T1 + T2 , H2 = T1T2

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 11 / 17

Explicit formulas: q-analog of the LS formula, [GLO]

Ψq

z(pN) =

• GZ

N

• n=1

z|pn|−|pn−1|

n−1

• i=1

(pn,i − pn, i+1)q!

n

• i=1

(pn,i − pn−1, i)q! (pn−1,i − pn, i+1)q! (13) (m)q! := (1 − q) · . . . · (1 − qm) ; when pN = (pN, 1 ≥ . . . ≥ pNN), and Ψq

z(pN) = 0 otherwise.

Summation is over the Gelfand-Zetlin (GZ) patterns: pN,1 pN,2 . . . pNN ... . . . p21 p22 p11 pn+1, k ≥ pnk ≥ pn+1, k+1 , 1 ≤ k ≤ n < N “Uq(glN)-Whittaker function” is a character of Demazure module of glN: Ψq

λ(p) =

∆q(λ)−1 ch Vw(p′) , p = (p1 ≥ . . . ≥ pN) 0 , p non-dominant (14)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 12 / 17

Archimedean limit q → 1, [GLO]

q = e−ε , mε = −

• ε−1 log ε
• Lemma

Let fα(y; ε) :=

• ε−1y + αmε
• q! ,

then as ε → +0 fα(y; ε) ∼

• eA(ε) + e−y + O(ε) ,

α = 1 eA(ε) + O(εα−1) , α > 1 , A(ε) = −π2 6 − 1 2 ln ε 2π .

Theorem

Set pn,k = (n + 1 − 2k)mε + xn,k ε , zn = eı εΛn , 1 ≤ n ≤ k ≤ N , then for the general partition pN: ΨGivental

λ

(xN) = lim

ǫ→+0

• ε

N(N−1) 2

e

(N−1)(N+2) 2

A(ε) Ψq z(pN)

• .

(15)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 13 / 17

Example: pN = (n, . . . , n

m

, 0, . . . , 0)

lim

ǫ→+0

• εm(N−m) e[m(N−m)+1]A(ε) Ψq

z(nm, 0N−m)

• =
• Cm
• k,i

dxnk eFλ(xnk) , Fλ(xk,i) = Fm(λ) −

• arrows

etarget(a) −source(a) x

• xN−m, 1
• . . .
• xN−1, m
• .

. .

• ...
• .

. .

• x11

. . . xm, m

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 14 / 17

Parabolic GL(N; R)-Whittaker functions, [GLO, O]

b+ = hJ + nJ

+ ,

χJ

+ :

nJ

+ −

→ C , ΨJ

λ(x) = e−ρ(x)

ψL , πλ

• e−HJ(x)

ψJ

R

• xi =0

i / ∈J

, (16) x = (x1, . . . , xr) , J =

• J1 ≤ . . . ≤ Jr ≤ N
• .

Theorem

Type J-parabolic GL(N; R)-Whittaker function (14) possesses the Mellin-Barnes integral representation: ΨJ

λ(x) =

• S
• n,k

dγnk

r

• n=1

e

xn

• Jn
• i=1

γJn, i−

Jn+1

• j=1

γJn+1, j

• Jn
• i=1

Jn+1

• j=1

Γ γJn, i−γJn+1, j

• Jn
• i,k=1

i=k

Γ γJn, i−γJn, k

• (17)

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 15 / 17

Parabolic GL(N; R)-Whittaker functions as equivariant volumes of Mhol(D → G/PJ), [GLO]

Let J = (1; N) and consider QMd(P1 → PN−1) ≃ PN(d+1)−1 , which is acted by the group G = S1 × U(N). The G-equivariant symplectic form on QMd(P1 → PN−1) is ωG = ωK¨

ahler + HS1 + λ1HU(1) +. . .+ λNHU(1) ,

U(1)N ⊂ U(N) , The G-equivariant volume of QMd(P1 → PN−1): Zd(x|λ, ) = eωG , [QMd]G =

• C

dγ e

x γ

N

• i=1

d

• n=0

1 γ − λi − n (18) Type A sigma-model provides a regularization of limit d → +∞ Ψ(1; N)

λ1,...,λN(x) =

lim

d→+∞ Zd(x|λ, ) =

• R−ıε

dγ e

x γ

N

• i=1

Γ γ − λi

• (19)

=

• eΩG ,

L+PN−1

G ,

L+PN−1 = lim

d→+∞[QMd] .

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 16 / 17

Archimedean analog of the LS formula, [GLO, O]

Conjecture

The J-parabolic GR-Whittaker function ΨJ

λ(x) can be identified with the

S1 × GK-equivariant volume of Mhol

• D −

→ Fl∨

J

• ,

and possesses the stationary phase integral representation ΨJ

λ(x) =

• CJ
• n,k

dxnk eFJ

λ(xnk) ,

CJ ∼ Rdim FlJ ⊂ Fl∨

J (C) ,

FJ

λ(xnk) is a superpotential in type B sigma-model.

1 Representation theoretic proof: for minimal PJ in type A, [O]; 2 TQFT proof for FlJ = PN−1 via Mirror Symmetry between the two

topological sigma-models in disk D, [GLO].

Sergey OBLEZIN , ITEP q-deformed Whittaker functions and the local Langlands correspondence March 7, 2013 ICERM (Providence) 17 / 17