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q -deformed Whittaker functions and the local Langlands correspondence Sergey OBLEZIN , ITEP March 7, 2013 ICERM (Providence) March 7, 2013 ICERM (Providence) 1 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 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) . March 7, 2013 ICERM (Providence) 2 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Whittaker functions The Gauss (Bruhat) decomposition of G = G ( F ): � e x 1 , . . . , e x N � G 0 = U − · A · U + , A = . Character of B − = U − A with λ = ( λ 1 , . . . , λ N ) ∈ C N : � N → C ∗ , e ( λ i + ρ i ) x i . χ λ : B − − χ λ ( ua ) = i =1 The principal series representation ( π λ , V λ ) of G and of U ( g ): � � � � G B − χ λ = f ∈ Fun ( G ) � f ( bg ) = χ λ ( b ) f ( g ) , b ∈ B − Ind The Whittaker function Ψ λ ( g ) is a smooth function on X = N − \ G analytic in λ given by Ψ λ ( g ) = e ρ ( g ) � � ψ L , π λ ( e − H ( g ) ) ψ R , (1) ψ L , ψ R ∈ V λ are defined by character ψ 0 : F → C ∗ : � � � → C , ψ : U − ψ ( u ) = u α i ψ 0 . simple roots March 7, 2013 ICERM (Providence) 3 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Spherical Whittaker functions The Iwasawa decomposition of G = G ( F ): G = K · A · U + . The spherical Whittaker function Ψ λ ( z ) is a smooth function on H = K \ G analytic in λ given by Ψ λ ( g ) = e ρ ( g ) � � ψ K , π λ ( e − H ( g ) ) ψ R , (2) with the spherical vector ψ K ∈ V λ . March 7, 2013 ICERM (Providence) 4 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Quantum Toda lattice In the real case, G = G ( R ), generators C r , r = 1 , . . . , N of the center ZU ( g ) define quantum Toda Hamiltonians: H r · Ψ λ ( x ) := e − ρ ( x ) � � ψ K , π λ ( C r e − H ( x ) ) ψ R . (3) The G ( R )-Whittaker function is an eigenfunction: H r · Ψ λ ( x ) = σ r ( λ ) Ψ λ ( x ) , (4) σ r ( λ ) are r -symmetric functions in λ = ( λ 1 , . . . , λ N ). Example In the case G = GL (2; R ) � ∂ � H 2 = − � 2 � ∂ 2 � + ∂ 2 + ∂ + e x 1 − x 2 , H 1 = − � , ∂ x 2 ∂ x 2 ∂ x 1 ∂ x 2 1 2 March 7, 2013 ICERM (Providence) 5 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Example: the GL (2; R )-Whittaker functions � � e x 1 − T + e T − x 2 � ı � λ 2 ( x 1 + x 2 − T )+ ı � λ 1 T − 1 Ψ R λ 1 , λ 2 ( e x 1 , e x 2 ) = dT e (5) � R � 2 � λ 1+ λ 2 x 1+ x 2 x 1 − x 2 = e e K λ 1 − λ 2 � e . 2 2 2 � The Mellin-Barnes integral representation: � � λ i − γ � 2 � λ i − γ � x 2 ( λ 1 + λ 2 − γ )+ ı ı Ψ R λ 1 , λ 2 ( e x 1 , e x 2 ) = � x 1 γ � � Γ d γ e (6) � i =1 R − ıǫ Both integral representations can be generalized to GL ( N ; R ) by induction over the rank N , using the Baxter Q -operator formalism, [GLO]. March 7, 2013 ICERM (Providence) 6 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Baxter operators for spherical Whittaker functions, [GLO] One-parameter family of K -biinvariant functions Q s in the Hecke algebra: � � � dh Q s ( gh − 1 ) Ψ λ ( h ) = L p ( s ; V ) Ψ λ ( g ) . Q s ∗ Ψ λ ( g ) = G Theorem The L-function is the spherical transform of the Baxter operator kernel � L p ( s ; V ⊗ δ − 1 / 2 ) = da Q s ( a − 1 ) ϕ λ ( a ) , A with spherical function given by � dk e � H ( kg ) , λ � . ϕ λ ( g ) = K Example In the case G = GL (1; F ) : � λ − s � 1 λ − s � Γ L p ( s ; V ) = L ∞ ( s ; V ) = h 1 − p λ − s , . � March 7, 2013 ICERM (Providence) 7 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Explicit formulas: non-Archimedean case Let ξ λ : H ( G , K ) → C and σ λ ⊂ GL ( N ; C ) is the (semisimple) conjugacy class, attached to ξ λ via Satake correspondence. The class-one GL ( N ; F ) -Whittaker function associated with ξ λ : 1 Ψ λ ( ug ) = ψ ( u ) Ψ λ ( ug ) ; 2 � dh Ψ λ ( gh ) φ ( h − 1 ) = ξ λ ( φ ) Ψ λ ( g ) for any φ ∈ H ( G , K ) ; G 3 Ψ λ (1) = 1 . The Langlands-Shintani (LS) formula The class-one GL ( N ; Q p ) -Whittaker function reads  � p λ 1 �   p − ̺ ( n ) ch V n ... , n = ( n 1 ≥ . . . ≥ n N ) Ψ Q p λ ( p n ) = (7)  p λ N  0 , n non-dominant March 7, 2013 ICERM (Providence) 8 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

� � � � � � � � � � Explicit formulas: Archimedean case The Givental stationary phase integral formula: � � dx nk e � − 1 F λ ( x nk ) , N ( N − 1) N ( N − 1) Ψ R C ∼ R ⊂ C λ ( x ) = , (8) 2 2 k ≤ n < N C � n − 1 � � N � n � � e target ( a ) − source ( a ) F λ ( x nk ) = ıλ n x n , k − x n − 1 , i − arrows n =1 k =1 i =1 x N , 1 x N , 2 . . . x NN (9) . ... . . . . . x 21 x 22 x 11 March 7, 2013 ICERM (Providence) 9 / Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence 17

Baxter operators for Macdonald polynomials, [GLO] � ∨ ∨ ∨ Q q , t Q x ( µ, Λ) = x | µ |−| Λ | ϕ µ/ Λ , · f (Λ) = Q x ( µ, Λ) f ( µ ) , (10) x µ ∈ Z N � t j − i q µ i − µ j +1 � � t j − i q Λ i − Λ j +1 +1 � N � Γ q , tq − 1 t j − i q µ i − Λ j +1 � Γ q , tq − 1 ϕ µ/ Λ = � � t j − i q Λ i − µ j +1 +1 � , Γ q , tq − 1 Γ q , tq − 1 i , j =1 i ≤ j � � � ( tz ) 1 / 2 ; q 1 − tzq n Γ q , t ( z ) × Γ q , t − 1 ( qz − 1 ) = t 1 / 2 θ 1 � � Γ q , t ( z ) = 1 − zq n , . z 1 / 2 ; q θ 1 n ≥ 0 Theorem The Macdonald polynomials are eigenfunctions under the action of (10): N � ∨ Q q , t · P Λ ( z ) = L ∨ L ∨ x ( z ) P Λ ( z ) , x ( z ) = Γ q , t ( xz i ) . (11) x i =1 March 7, 2013 ICERM (Providence) 10 Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence / 17

� � � q -deformed Whittaker functions Ψ R λ ( a ) t → 0 q → 1 P q , t � Ψ q Λ ( z ) z (Λ) q → 0 t → + ∞ z = p λ Ψ Q p λ ( a ) H r · Ψ q z (Λ) = e r ( x ) Ψ q z (Λ) . (12) r � � � � 1 − q Λ ik − Λ ik +1 +1 � 1 − δ ik +1 − ik , 1 T I r , H r = T I r = T q , q Λ i , I r k =1 i ∈ I r Example In the case GL (2; F ) : � 1 − q Λ 1 − Λ 2 +1 � H 1 = T 1 + T 2 , H 2 = T 1 T 2 March 7, 2013 ICERM (Providence) 11 Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence / 17

Explicit formulas: q -analog of the LS formula, [GLO] n − 1 � ( p n , i − p n , i +1 ) q ! � � N z | p n |−| p n − 1 | i =1 Ψ q z ( p N ) = (13) � n ( p n , i − p n − 1 , i ) q ! ( p n − 1 , i − p n , i +1 ) q ! GZ n =1 i =1 ( m ) q ! := (1 − q ) · . . . · (1 − q m ) ; when p N = ( p N , 1 ≥ . . . ≥ p NN ), and Ψ q z ( p N ) = 0 otherwise. Summation is over the Gelfand-Zetlin (GZ) patterns: p N , 1 p N , 2 p NN . . . ... p n +1 , k ≥ p nk ≥ p n +1 , k +1 , . . . 1 ≤ k ≤ n < N p 21 p 22 p 11 “ U q ( gl N )-Whittaker function” is a character of Demazure module of � gl N : � ∆ q ( λ ) − 1 ch V w ( p ′ ) , p = ( p 1 ≥ . . . ≥ p N ) Ψ q λ ( p ) = (14) 0 , p non-dominant March 7, 2013 ICERM (Providence) 12 Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence / 17

Archimedean limit q → 1, [GLO] � � q = e − ε , ε − 1 log ε m ε = − Lemma � � ε − 1 y + α m ε Let f α ( y ; ε ) := q ! , then as ε → +0 � e A ( ε ) + e − y + O ( ε ) , A ( ε ) = − π 2 α = 1 6 − 1 2 ln ε f α ( y ; ε ) ∼ , 2 π . e A ( ε ) + O ( ε α − 1 ) , α > 1 Theorem Set p n , k = ( n + 1 − 2 k ) m ε + x n , k z n = e ı ε Λ n , , 1 ≤ n ≤ k ≤ N , ε then for the general partition p N : � � N ( N − 1) ( N − 1)( N +2) A ( ε ) Ψ q Ψ Givental ( x N ) = lim e z ( p N ) (15) ε . 2 2 λ ǫ → +0 March 7, 2013 ICERM (Providence) 13 Sergey OBLEZIN , ITEP q -deformed Whittaker functions and the local Langlands correspondence / 17