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Whittaker functions and Harmonic analysis
Sergey OBLEZIN , Nottingham
Eisenstein series on Kac-Moody groups & Applications
19 November 2015 , KIAS, Seoul
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 1
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1 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator formalism forMacdonald polynomials, Lett. Math.Phys. 104 (2014);
2 S. Oblezin, On parabolic Whittaker functions I & II, Lett. Math.Phys. 101 & Cent. Eur. J. Math. 10 (2012);
3 A. Gerasimov, D. Lebedev, S. Oblezin On a classical limit ofq-deformed Whittaker functions, Lett. Math. Phys., 100 (2012);
4 A. Gerasimov, D. Lebedev, S. Oblezin Parabolic Whittaker functionsand Topological field theories I, Comm. Number Th. Phys. 5 (2011);
5 A. Gerasimov, D. Lebedev, S. Oblezin On q-deformed Whittakerfunction I, II & III, Comm. Math. Phys 294 (2010) & Lett. Math.Phys 97 (2011);
6 A. Gerasimov, D. Lebedev, S. Oblezin On Baxter Q-operators andtheir arithmetic implications, Lett. Math. Phys. 88 (2009) .
7 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator andArchimedean Hecke algebra, Comm. Math. Phys. 284 (2008) .
8 A. Gerasimov, S Kharchev, D. Lebedev, S. Oblezin On aGauss-Givental representation of quantum Toda chain wave function,Int. Math. Res. Notices, 2006
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 2
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TopologicalQFT
"*
ss ++
SS
""
QuantumIntegrability
t|
KK
||
Representation Theory&
HarmonicAnalysis
Automorphic Forms&
Arithmetic Geometry
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 3
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Jacquet’s local Whittaker functionsThe Gauss (Bruhat) decomposition of G = G (F ):
G 0 = U− · A · U+ .
For λ = (λ1, . . . , λN) ∈ CN ,
χλ : B− = U−A −→ C∗ , χλ(ua) =N∏i=1
|ai |λi+ρi .
The principal series representation (πλ, Vλ) :
IndGB− χλ =
f ∈ Fun(G )∣∣∣ f (bg) = χλ(b) f (g) , b ∈ B−
The Whittaker function Ψλ(g) is a smooth function on G (F ) given by
Ψλ(g) =⟨ψL , πλ(g)ψR
⟩, ψL,R : U± −→ C∗ , (1)
attached to local character ψ : F → C∗ and U− = w−10 U+w0 :
ψR(u) =∏
simple roots
ψ(uαi
). ψL(u) = ψR
(uw−1
0
)−1
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 4
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Archimedean case: Spherical Whittaker functions
The Iwasawa decomposition of G = G (R):
G = K · A · U+ , H = K\G .
The spherical Whittaker function Ψsphλ (z) is a smooth function on
z ∈ H, analytic in λ given by
Ψsphλ (g) = eρ(g)
⟨ψK , πλ(g)ψR
⟩, (2)
with the K -invariant (spherical) vector ψK ∈ Vλ.
1 Ψsphλ (k · g · u) = ψ(u)Ψsph
λ (g) , for all k ∈ K and u ∈ U+ ;
2 D ·Ψsphλ (z) = cD(λ)Ψsph
λ (z) ,for any G -invariant differential operator D on H.
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 5
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Archimedean case: the quantum Toda D-module(Kazhdan, Kostant)
For G = G (R), generators Cr , r = 1, . . . ,N of the center ZU(g) definequantum Toda Hamiltonians:
Hr ·ΨRλ (ex) := e−ρ(x)
⟨ψK , πλ(Cr e−H(x))ψR
⟩. (3)
The G (R)-Whittaker function is an eigenfunction:
Hr ·ΨRλ (ex) = er (λ) ΨR
λ (ex) , (4)
er (λ) are r -symmetric functions in λ = (λ1, . . . , λN).
Example: G = GL(2; R)
H1 = −~( ∂
∂x1+
∂
∂x2
), H2 = −~2
( ∂2
∂x21
+∂2
∂x22
)+ ex1−x2 ,
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 6
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Archimedean case: the GL(2; R)-Whittaker functions
The Bessel function “of the third kind”:
ΨRλ1, λ2
(ex1 , ex2) =
∫RdT e
ı~λ2(x1+x2−T )+ ı
~λ1T − 1~
(ex1−T +eT−x2
)(5)
= eλ1+λ2
2 ex1+x2
2 Kλ1−λ2~
(2
~e
x1−x22).
The Mellin-Barnes integral representation:
ΨRλ1, λ2
(ex1 , ex2) =
∫R−ıε
dγ eı~ x2(λ1+λ2−γ)+ ı
~ x1γ2∏
i=1
~λi−γ
~ Γ(λi − γ
~
)(6)
Both integral representations can be generalized to GL(N; R) by inductionover the rank N, using the Baxter Q-operator formalism, [GLO:08,09,14].
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 7
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Archimedean case: Baxter operator [GLO’08]The Gelfand pair G = GLN(R), K = ON(R)
=⇒ the local Hecke algebra H∞(G , K )
The dual group: G∨(C) = GLN(C)
Baxter Q-operator and Hecke algebra
The one-parameter family of K -bi-invariant functions on G (R),
Qs(g) = 2N | det g |ıs+N−12 e−πTr(g
T g) , (7)
acting on a spherical Whittaker function produces the L-function:(Qs ∗Ψsph
λ
)(g) =
∫G
dh Qs(gh−1) Ψsphλ (h) = L∞(s; Vλ) Ψsph
λ (g) . (8)
The local L-functions for G = GL(1; F )
Lp(s; V ) =1
1− pλ−s, L∞(s; V ) = h
λ−s~ Γ
(λ− s
~
).
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 8
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Non-Archimedean case: Spherical Whittaker functions
The Gelfand pair G = GLN(Qp), K = GLN(Zp) =⇒ Hp(G , K )
The dual group: G∨(C) = GLN(C)
ξλ : H(G , K )→ C is a Hecke character;
σλ ⊂ G∨(C) is the (semisimple) conjugacy class, Satake-dual to ξλ;
ψ : U+ −→ C× is a unipotent character.
The class-one GL(N; Qp)-Whittaker function:
1 ΨQp
λ (kgu) = ψ(u) ΨQp
λ (g) ;
2∫G
dh ΨQp
λ (gh)φ(h−1) = ξλ(φ) ΨQp
λ (g) for any φ ∈ H(G , K ) ;
3 ΨQp
λ (1) = 1 .
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 9
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Non-Archimedean case: the Baxter operator
Tωn = characteristic function of K ·(
p·Idn 00 IdN−n
)· K
ωnNn=1 = fundamental weights of G∨(C) = GLN(C)
Given finite-dimensional ρV : GLN(C) −→ GL(V ),
TV ·ΨQp
λ (g) :=
∫Gp
dh TV (h)ΨQp
λ (gh) = chV (σλ) ΨQp
λ (g)
p-adic substitute of the Baxter operator [Piatetski-Shapiro], [GLO’08]
Let QQps =
∑n≥0
p−n s TSymnCN , then
QQps ·ΨQp
λ = Lp(s, CN) ΨQp
λ , (9)
where Lp(s, CN) = 1
detCN(
1− p−sρCN (σλ)) .
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 10
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Non-Archimedean case: The Langlands-Shintani formulaL− functions
\\
Baxteroperators
Whittaker functions
19
Characters of G∨
19
oo Local Langlands
Reciprocity// Matrix elements
%-
Class-one GL(N ; Qp)-Whittaker function == GL(N ; C)-character
ΨQp
λ (pn) =
p−%(n) chVn
(pλ1
. . .pλN
), n = (n1 ≥ . . . ≥ nN)
0 , n non-dominant
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 11
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Archimedean case: Explicit formulas [Givental] [GLO’05]
ΨRλ
(exN)
=
∫C
∏k≤n<N
dxnk eFλ(xN , xnk ) , C ∼ RN(N−1)
2 ⊂ CN(N−1)
2 , (10)
Fλ(xN , xnk) =N∑
n=1
ıλn
( n∑k=1
xn,k−n−1∑i=1
xn−1, i
)−∑
arrowsetarget(a)− source(a)
summed over the arrows from Gelfand-Zetlin (GZ) graph:
xN,1
$$
xN,2 . . .
$$
xNN
xN−1, 1
::
##
. . .
xN−1,N−1
::
. . .
==
!!
...
;;
x11
>>
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 12
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Archimedean case: Explicit formulas [Kharchev-Lebedev]
ΨλN, 1,...,λNN (ex) =∨QglNglN−1
(xN) ∗ΨλN−1, 1,...,λN−1,N−1(ex1 , . . . , exN−1) (11)
=
∫S
∏1≤k≤n<N
dλnk e− ı
~
N−1∑n=1
xn+1
(n+1∑j=1
λn+1, j −n∑
j=1λn,j
)×
×N−1∏n=1
n+1∏m=1
n∏k=1
Γ(ıλn,k−ıλn+1,m
~
)∏s 6=p
Γ(ıλn, s−ıλn, p
~
) . (12)
Contour S : maxjIm(λkj) < min
mIm(λk+1,m) , k = 1, . . . ,N − 1
The dual recursion operator
∨Q
glNglN−1
(λ, γ|x) = e− ı
~ x( N∑i=1
λi −N−1∑j=1
γj
)×
N∏i=1
N−1∏j=1
Γ( ıλi − ıγj
~
).
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 13
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Archimedean case: The dual Baxter operator [GLO’08]One-parameter family of integral operators:( ∨Qz ∗Ψ(ex)
)λ
=
∫RN
dγ∨
Qz(λ; γ) Ψγ(ex) = L∨(z ; x) Ψλ(ex) ; (13)
with the integral kernel:
∨Qz(λ, γ) = e
− ı~
N∑i=1
(γi −λi )z N∏i ,j=1
Γ( ıγi − ıλj
~
).
The eigenvalue, the “dual L-function”:
L∨(z ; x1, . . . , xN) = e−1~ ez−xN .
Recursion == composition of the dual pair of Baxter operators
QglNglN
= eıλNxN ×QglN−1
λN ∨QglN−1
xN . (14)
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 14
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Unification: Archimedean VS non-Archimedean [GLO’09’12]
Ψqz (Λ) & Lq
y (z)
q→1
))
q→0
z=pλ
uu
ΨQp
λ (a) & Lp(s) ΨRλ (a) & L∞(s)
The q-deformed Whittaker function === q-Toda eigenfunction:
Hr ·Ψqz (Λ) = er (z) Ψq
z (Λ) . (15)
The q-deformed Baxter operator:
Qy ∗Ψqz = Lq
y (z) Ψqz , Lq
y (z) =N∏i=1
∏n≥0
1
1− y−1ziqn(16)
Example: G = GL(2)
H1 =(1− qΛ1−Λ2+1
)TΛ1 + TΛ2 , H2 = TΛ1TΛ2
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 15
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Further unification: The Macdonald polynomials
Λq,t =←−limN
Q(q, t)[x1, . . . , xN ]SN ,
Macdonald’s polynomials Pq,tΛ (x) = basis in Λq,t , labeled by partitions Λ
(i) 〈PΛ, Pµ〉q,t = 0 , iff Λ 6= µ ,
(ii) PΛ =∑µ≥Λ
uΛµmµ , with uΛΛ = 1 , .
The main diagramme
ΨRλ (a)
P q,tΛ (z)
t→0
&&
t→+∞
88Ψq
z (Λ)
q→166
q→0
z=pλ ((Ψ
Qp
λ (a)
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 16
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The Ruijsenaars-Macdonald quantum system
Mr = tr(r−1)
2
∑Ir
∏i∈Irj /∈Ir
txi − xjxi − xj
Tq, xi , Tq, x · f (x) = f (qx) .
Eigenvalue problem:
Mr · PΛ = χr (t%qΛ) PΛ , χr (y) =∑Ir
yi1 · · · yir . (17)
The new scalar product:
〈a, b〉′q,t =1
N!
∮T
d×z ∆(z) a(z) b(z−1) , q ∈ C , |q| < 1 ,
T =
z ∈ CN : |zi | = 1, ∆(z) =
N∏i,j=1i 6=j
Γq,t
(z−1i zj
)−1,
where
Γq,t(y) =∏n≥0
1− tyqn
1− yqn, Γq, t(z)Γq, t−1(qz−1) = t1/2 θ1
((tz)1/2; q
)θ1
(z1/2; q
) .
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 17
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The dual Ruijsenaars-Macdonald quantum systemNormalized Macdonald polynomials
ΦΛ(x ; q, t) := tρ(Λ)N∏
a,b=1a<b
Γq,t(tb−aqλa−λb)× Pq,tΛ (x) ,
Remarkable bispectral symmetry [Koornwinder]
ΦΛ
(qµ−kρ; q, q−k
)= Φµ
(qΛ−kρ; q, q−k
). (18)
The dual Hamiltonians:
M∨r = tr(N−1)
2
∑Ir
∏i∈Irj /∈Irj<i
1− t i−j+1qλj−λi−1
1− t i−jqλj−λi−1
1− t i−j−1qλj−λi
1− t i−jqλj−λiTq, qλi ,
the dual eigenvalue problem:
M∨r · PΛ = χr (x) PΛ . (19)
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 18
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Baxter operator for Macdonald polynomials [GLO’14]
Qγ · P(x) =
∫T
d×y Qγ
(x , y)
∆(y) P(y−1) , γ ∈ Z , (20)
Qγ(x , y) =N∏i=1
(xiyi)γ N∏
i ,j=1
Γq,t(xiyj) .
Theorem
Macdonald polynomials are eigenfunctions under the action of (14):
Qγ · PΛ(x) = Lγ(λ) PΛ(x) , Lγ(Λ) =N∏i=1
Γq, tq−1(q)
Γq, tq−1(tN−iqΛi−γ+1),
when ΛN ≥ γ, and Qγ · PΛ(x) = 0 otherwise.
Proof uses (q, t)-analog of the Cauchy-Littlewood identity:n∏
i=1
m∏j=1
Γq,t(xiyj) =∑
Λ
PΛ(x) PΛ(y)
〈PΛ, PΛ〉q,t.
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 19
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Dual Baxter operator for Macdonald polynomials [GLO’14]
Theorem
For the Baxter operator with the kernel Qq, tx (µ,Λ) = x |µ−Λ|ϕµ/Λ ,
Qq,tx · PΛ(z) = L∨x (z) PΛ(z) , L∨x (z) =
N∏i=1
Γq,t(xzi ) , (21)
Proof uses the Pieri formula:
PΛ × P(n) =1
〈P(n), P(n)〉q,t
∑µi≥Λi≥µi+1|µ−Λ|=n
ϕµ/Λ Pµ ,
where
ϕµ/Λ =N∏
i,j=1i≤j
Γq,tq−1
(t j−iqµi−µj+1
)Γq,tq−1
(t j−iqµi−Λj+1
) Γq,tq−1
(t j−iqΛi−Λj+1+1
)Γq,tq−1
(t j−iqΛi−µj+1+1
) ,Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysis
Eisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 20/ 26
The t → +∞ limit: q-analog of the LS formula [GLO’10]
Ψqz (p
N) =
∑GZ
N∏n=1
z|p
n|−|p
n−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!
(22)
(m)q! := (1− q) · . . . · (1− qm) ,summed over the Gelfand-Zetlin (GZ) patterns:
pN,1 pN,2 . . . pNN
pN−1, 1 . . . pN−1,N−1
. . .. . .
p11
pnk ≥ pn−1, k ≥ pn, k+1 ,1 ≤ k ≤ n < N
Uq(glN)-Whittaker function === character of glN-Demazure module
Ψqλ(p) =
∆q(λ)−1 chVw (p′) , p = (p1 ≥ . . . ≥ pN)
0 , p non-dominant
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 21
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Archimedean limit q → 1 [GLO’12]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) = −π
2
6− 1
2ln
ε
2π.
Theorem
Set
pn,k = (n + 1− 2k)mε +xn,kε, zn = eı εΛn , 1 ≤ n ≤ k ≤ N ,
then for the general partition pN
=(pN,1 > pN,2 > . . . > pNN
):
ΨRλ
(exN)
= limε→+0
[ε
N(N−1)2 e
(N−1)(N+2)2
A(ε) Ψqz (p
N)]. (23)
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 22
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Example: minimal parabolic, pN
= (n, . . . , n︸ ︷︷ ︸m
, 0, . . . , 0), [O]
limε→+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) −∑
arrowsetarget(a)− source(a)
is the superpotential in type B sigma-model.
x
xN−m, 1
// . . .
// xN−1,m
...
// . . .
// ...
x11
// . . . // xm,m // 0
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 23
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Archimedean analog of the LS formula [GLO’11,O]
Theorem (in progress)
The J-parabolic GL(N; R)-Whittaker function possesses the int rep
ΨJλ(ex) = e−ρ(x)
⟨ψL , πλ
(e−H
J(x))ψJR
⟩∣∣∣ xi=0
i /∈J
=
∫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=1i 6=k
Γ(γJn, i−γJn, k
~) , (24)
and can be identified with the S1 × U(N)-equivariant volume of the space
M = Maphol(D, Fl∨J (C))
of holomorphic maps of the disc D = z ∈ C : |z | ≤ 1 into the complexflag variety FlJ = GL(N,C)/PJ .
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 24
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Archimedean Langlandscorrespondence == Mirror Symmetry [GLO’11]
• Classical (Type B) integral representation:
a−s Γ(s) =
+∞∫−∞
dt eWs(t;a) , Ws(t; a) = st − aet .
MB = Map(
D → (C2, σ, Ws)).
• Type A integral representation as an S1 × U(1)-equivariant volume:
a−(s−λ) Γ(s − λ) =
∫MA
eΩ + s HS1 + λHU(1) = volS1×U(1)(MA) ,
Key Observation
MA = Maphol(D → C) is Mirror Symmetric to MB .
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 25
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Further directions:
1 Qq,t-operators VS Hecke type algebras;
2 Extension to other types of G ;
3 Applications to the Langlands-Shahidi method;
4 TQFT VS Macdonald polynomials;
5 Automorphic and arithmetic interpretations of the Lq,tx (z)-functions.
Sergey OBLEZIN , Nottingham Whittaker functions and Harmonic analysisEisenstein series on Kac-Moody groups & Applications 19 November 2015 , KIAS, Seoul 26
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