Whittaker functions and Harmonic analysis

Transcription

1 Whittaker functions and Harmonic analysis Sergey OBLEZIN, Nottingham oody groups & Applications 19 November 2015, KIAS, Seoul

2 1 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator formalism for Macdonald 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 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, Comm. Number Th. Phys. 5 (2011); 5 A. Gerasimov, D. Lebedev, S. Oblezin On q-deformed Whittaker function I, II & III, Comm. Math. Phys 294 (2010) & Lett. Math. Phys 97 (2011); 6 A. Gerasimov, D. Lebedev, S. Oblezin On Baxter Q-operators and their arithmetic implications, Lett. Math. Phys. 88 (2009). 7 A. Gerasimov, D. Lebedev, S. Oblezin Baxter operator and Archimedean Hecke algebra, Comm. Math. Phys. 284 (2008). 8 A. Gerasimov, S Kharchev, D. Lebedev, S. Oblezin On a Gauss-Givental representation of quantum Toda chain wave function, Int. Math. Res. Notices, 2006

3 Topological QFT Representation Theory & HarmonicAnalysis Quantum Integrability Automorphic Forms & Arithmetic Geometry

4 Jacquet s local Whittaker functions The Gauss (Bruhat) decomposition of G = G(F ): For λ = (λ 1,..., λ N ) C N, G 0 = U A U +. χ λ : B = U A C, χ λ (ua) = N a i λ i +ρ i. The principal series representation (π λ, V λ ) : { } Ind G B χ λ = f Fun(G) f (bg) = χ λ (b) f (g), b B The Whittaker function Ψ λ (g) is a smooth function on G(F ) given by Ψ λ (g) = ψ L, π λ (g) ψ R, ψl,r : U ± C, (1) attached to local character ψ : F C and U = w0 1 U +w 0 : ψ R (u) = ψ ( ) ( u αi. ψl (u) = ψ R uw 1 0 simple roots i=1 ) 1

5 Archimedean case: Spherical Whittaker functions The Iwasawa decomposition of G = G(R): G = K A U +, H = K\G. The spherical Whittaker function Ψ sph λ (z) is a smooth function on z H, analytic in λ given by Ψ sph λ (g) = eρ(g) ψ K, π λ (g) ψ R, (2) with the K-invariant (spherical) vector ψ K V λ. 1 Ψ sph λ (k g u) = ψ(u)ψsph λ (g), for all k K and u U + ; 2 D Ψ sph λ (z) = c D(λ)Ψ sph λ (z), for any G-invariant differential operator D on H.

6 Archimedean case: the quantum Toda D-module (Kazhdan, Kostant) For G = G(R), generators C r, r = 1,..., N of the center ZU(g) define quantum Toda Hamiltonians: H r Ψ R λ (ex ) := 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 R λ (ex ) = e r (λ) Ψ R λ (ex ), (4) e r (λ) are r-symmetric functions in λ = (λ 1,..., λ N ). Example: G = GL(2; R) ( H 1 = + ), H 2 = 2( 2 x 1 x 2 x ) x2 2 + e x 1 x 2,

7 Archimedean case: the GL(2; R)-Whittaker functions The Bessel function of the third kind : ( ) Ψ R λ 1, λ 2 (e x 1, e x 2 ) = dt e ı λ 2(x 1 +x 2 T )+ ı λ 1T 1 e x 1 T +e T x 2 R = e λ 1 +λ 2 2 e x 1 +x 2 2 K λ 1 λ 2 The Mellin-Barnes integral representation: Ψ R λ 1, λ 2 (e x 1, e x 2 ) = R ıɛ dγ e ı x 2(λ 1 +λ 2 γ)+ ı x 1γ ( 2 e x1 x2 ) 2. 2 ( λ i γ λi γ ) Γ i=1 (5) (6) Both integral representations can be generalized to GL(N; R) by induction over the rank N, using the Baxter Q-operator formalism, [GLO:08,09,14].

8 Archimedean case: Baxter operator [GLO 08] The Gelfand pair G = GL N (R), K = O N (R) The dual group: G (C) = GL N (C) = the local Hecke algebra H (G, K) Baxter Q-operator and Hecke algebra The one-parameter family of K-bi-invariant functions on G(R), Q s (g) = 2 N N 1 ıs+ det g 2 e πtr(g T g), (7) acting on a spherical Whittaker function produces the L-function: ( Qs Ψ sph ) λ (g) = dh Q s (gh 1 ) Ψ sph λ (h) = L (s; V λ ) Ψ sph λ (g). (8) G The local L-functions for G = GL(1; F ) L p (s; V ) = 1 1 p λ s, L (s; V ) = h λ s Γ ( λ s ).

9 Non-Archimedean case: Spherical Whittaker functions The Gelfand pair G = GL N (Q p ), K = GL N (Z p ) = H p (G, K) The dual group: G (C) = GL N (C) ξ λ : H(G, K) C is a Hecke character; σ λ G (C) is the (semisimple) conjugacy class, Satake-dual to ξ λ ; ψ : U + C is a unipotent character. The class-one GL(N; Q p )-Whittaker function: 1 Ψ Qp λ (kgu) = ψ(u) ΨQp λ (g) ; 2 dh Ψ Qp λ (gh) φ(h 1 ) = ξ λ (φ) Ψ Qp λ (g) for any φ H(G, K) ; G 3 Ψ Qp λ (1) = 1.

10 Non-Archimedean case: the Baxter operator T ωn = characteristic function of K ( ) p Idn 0 0 Id N n K {ω n } N n=1 = fundamental weights of G (C) = GL N (C) Given finite-dimensional ρ V : GL N (C) GL(V ), T V Ψ Qp λ (g) := dh T V (h)ψ Qp λ (gh) = ch V (σ λ ) Ψ Qp λ (g) G p p-adic substitute of the Baxter operator [Piatetski-Shapiro], [GLO 08] Let Q Qp s = n 0 p n s T Sym n C N, then Q Qp s Ψ Qp λ = L p (s, C N ) Ψ Qp λ, (9) where L p (s, C N 1 ) = ( det C N 1 p s ρ C N (σ λ ) ).

11 Non-Archimedean case: The Langlands-Shintani formula L functions Whittaker functions Baxter operators Characters of G Local Langlands Reciprocity Matrix elements Class-one GL(N; Q p )-Whittaker function == GL(N; C)-character Ψ Qp λ (pn ) = ( pλ1 ) p ϱ(n) ch Vn..., n = (n 1... n N ) p λn 0, n non-dominant

12 Archimedean case: Explicit formulas [Givental] [GLO 05] Ψ R ( ) λ e x N = C k n<n dx nk e F λ(x N, x nk ), C R N(N 1) 2 C N(N 1) 2, (10) N ( n n 1 ) F λ (x N, x nk ) = ıλ n x n,k x n 1, i target(a) source(a) e n=1 k=1 i=1 arrows summed over the arrows from Gelfand-Zetlin (GZ) graph: x N,1 x N,2... x NN x N 1, 1... x N 1, N x 11

13 Archimedean case: Explicit formulas [Kharchev-Lebedev] Ψ λn, 1,...,λ NN (e x ) = Q gl N gl N 1 (x N ) Ψ λn 1, 1,...,λ N 1, N 1 (e x 1,..., e x N 1 ) (11) = S 1 k n<n dλ nk N 1 N 1 e ı n=1 ( n+1 x n+1 j=1 ) λ n+1, j n λ n,j j=1 n+1 n ( ıλn,k ıλ Γ n+1, m m=1 k=1 ( ıλn, n=1 Γ s ıλ n, p s p ) ). (12) Contour S : max{im(λ kj )} < min {Im(λ k+1,m)}, k = 1,..., N 1 j m The dual recursion operator ( N Q gl N gl (λ, γ x) = e ı x i=1 N 1 λ i N 1 ) γ j j=1 N N 1 i=1 j=1 ( ıλi ıγ j Γ ).

14 Archimedean case: The dual Baxter operator [GLO 08] One-parameter family of integral operators: ( Q z Ψ(e x ) ) λ = with the integral kernel: Q z (λ, γ) = e ı R N dγ Q z (λ; γ) Ψ γ (e x ) = L (z; x) Ψ λ (e x ) ; (13) N (γ i λ i )z i=1 The eigenvalue, the dual L-function : N ( ıγi ıλ ) j Γ. i,j=1 L (z; x 1,..., x N ) = e 1 ez xn. Recursion == composition of the dual pair of Baxter operators Q gl N gl N = e ıλ Nx N Q gl N 1 λ N Q gl N 1 x N. (14)

15 Unification: Archimedean VS non-archimedean [GLO 09 12] Ψ Qp λ (a) & L p(s) q 0 z=p λ Ψ q z (Λ) & L q y (z) q 1 Ψ R λ (a) & L (s) The q-deformed Whittaker function === q-toda eigenfunction: The q-deformed Baxter operator: H r Ψ q z (Λ) = e r (z) Ψ q z (Λ). (15) Q y Ψ q z = L q y (z) Ψ q z, L q y (z) = Example: G = GL(2) N i=1 n y 1 z i q n (16) H 1 = ( 1 q Λ 1 Λ 2 +1 ) T Λ1 + T Λ2, H 2 = T Λ1 T Λ2

16 Further unification: The Macdonald polynomials Λ q,t = lim N Q(q, t)[x 1,..., x N ] S N, Macdonald s polynomials P q,t Λ (x) = basis in Λ q,t, labeled by partitions Λ (i) P Λ, P µ q,t = 0, iff Λ µ, (ii) P Λ = u Λµ m µ, with u ΛΛ = 1,. µ Λ The main diagramme P q,t Λ (z) t 0 Ψ q z (Λ) q 1 Ψ R λ (a) t + q 0 z=p λ Ψ Qp λ (a)

17 The Ruijsenaars-Macdonald quantum system M r = t r(r 1) 2 Eigenvalue problem: I r i Ir j / Ir tx i x j x i x j T q, xi, T q, x f (x) = f (qx). M r P Λ = χ r (t ϱ q Λ ) P Λ, χ r (y) = I r y i1 y ir. (17) The new scalar product: a, b q,t = 1 d z (z) a(z) b(z 1 ), q C, q < 1, N! T T = { z C N : z i = 1 } N ( ), (z) = Γ q,t z 1 1 i z j, where Γ q,t (y) = 1 tyq n 1 yq n, Γ q, t(z)γ q, t 1(qz 1 ) = t 1/2 θ ( 1 (tz) 1/2 ; q ) ( θ n 0 1 z 1/2 ; q ). i,j=1 i j

18 The dual Ruijsenaars-Macdonald quantum system Normalized Macdonald polynomials Φ Λ (x; q, t) := t ρ(λ) N a,b=1 a<b Γ q,t (t b a q λa λ b ) P q,t Λ (x), Remarkable bispectral symmetry [Koornwinder] The dual Hamiltonians: = t r(n 1) 2 M r Φ Λ ( q µ kρ ; q, q k) = Φ µ ( q Λ kρ ; q, q k). (18) I r i Ir j / Ir j<i the dual eigenvalue problem: 1 t i j+1 q λ j λ i 1 1 t i j q λ j λ i 1 1 t i j 1 q λ j λ i 1 t i j q λ j λ i T q, q λ i, Mr P Λ = χ r (x) P Λ. (19)

19 Baxter operator for Macdonald polynomials [GLO 14] Q γ P(x) = Theorem T d ( ) y Q γ x, y (y) P(y 1 ), γ Z, (20) N ( ) γ N Q γ (x, y) = xi y i Γ q,t (x i y j ). i=1 i,j=1 Macdonald polynomials are eigenfunctions under the action of (14): Q γ P Λ (x) = L γ (λ) P Λ (x), L γ (Λ) = N i=1 Γ q, tq 1(q) Γ q, tq 1(t N i q Λ i γ+1 ), when Λ N γ, and Q γ P Λ (x) = 0 otherwise. Proof uses (q, t)-analog of the Cauchy-Littlewood identity: n m Γ q,t (x i y j ) = P Λ (x) P Λ (y). P Λ, P Λ q,t i=1 j=1 Λ

20 Dual Baxter operator for Macdonald polynomials [GLO 14] Theorem For the Baxter operator with the kernel Q q, t x (µ, Λ) = x µ Λ ϕ µ/λ, Q q,t x P Λ (z) = L x (z) P Λ (z), L x (z) = Proof uses the Pieri formula: where ϕ µ/λ = P Λ P (n) = N i,j=1 i j 1 P (n), P (n) q,t µ i Λ i µ i+1 µ Λ =n N Γ q,t (xz i ), (21) i=1 ϕ µ/λ P µ, ( Γ q,tq 1 t j i q µ i µ j +1 ) ( t j i q ) Γ ( q,tq 1 t j i q Λ i Λ j+1 +1 ) ( µ i Λ j +1 t j i q ) Λ i µ j+1 +1, Γ q,tq 1 Γ q,tq 1

21 The t + limit: q-analog of the LS formula [GLO 10] Ψ q z (p N ) = GZ N n=1 z p n p n 1 n 1 (p n,i p n, i+1 ) q! i=1 (22) n (p n,i p n 1, i ) q! (p n 1,i p n, i+1 ) q! i=1 (m) q! := (1 q)... (1 q m ), summed over the Gelfand-Zetlin (GZ) patterns: p N,1 p N,2... p NN p N 1, 1... p N 1, N p 11 p nk p n 1, k p n, k+1, 1 k n < N U q (gl N )-Whittaker function === character of ĝl N-Demazure module { Ψ q λ (p) = q (λ) 1 ch Vw (p ), p = (p 1... p N ) 0, p non-dominant

22 Archimedean limit q 1 [GLO 12] q = e ε, m ε = [ ε 1 log ε ] Lemma Let f α (y; ε) := ( ε 1 y + αm ε!, then as ε +0 )q f α (y; ε) Theorem Set { e A(ε) + e y + O(ε), α = 1 e A(ε) + O(εα 1), α > 1, A(ε) = π ln ε 2π. p n,k = (n + 1 2k)m ε + x n,k ε, z n = e ı ελn, 1 n k N, then for the general partition p N = ( p N,1 > p N,2 >... > p NN ) : Ψ R ) ] λ( e x N = lim [ε N(N 1) 2 e (N 1)(N+2) 2 A(ε) Ψ q z (p ɛ +0 N ). (23)

23 Example: minimal parabolic, p N [ lim ε m(n m) e [m(n m)+1]a(ε) Ψ q z (n m, 0 N m ) ɛ +0 = (n,..., n, 0,..., 0), [O] }{{} ] m = dx nk e F λ(x nk ), F λ (x k,i ) = F m (λ) arrows target(a) source(a) e is the superpotential in type B sigma-model. x C m k,i x N m, x N 1, m. x x m, m 0

24 Archimedean analog of the LS formula [GLO 11,O] Theorem (in progress) The J-parabolic GL(N; R)-Whittaker function possesses the int rep Ψ J λ (ex ) = e ρ(x) ψ L, π λ ( e H J (x) ) ψ J R xi =0 i / J = S r dγ nk n,k n=1 ( Jn J n+1 xn γ Jn, i γ Jn+1, j i=1 j=1 e ) Jn J n+1 i=1 j=1 J n i,k=1 i k Γ ( γ Jn, i γ Jn+1, j ) Γ ( γ Jn, i γ Jn, k ), (24) and can be identified with the S 1 U(N)-equivariant volume of the space M = Map hol (D, Fl J (C)) of holomorphic maps of the disc D = {z C : z 1} into the complex flag variety Fl J = GL(N, C)/P J.

25 Archimedean Langlands correspondence == Mirror Symmetry [GLO 11] Classical (Type B) integral representation: a s Γ(s) = + dt e Ws(t;a), W s (t; a) = st ae t. M B = Map ( D (C 2, σ, W s ) ). Type A integral representation as an S 1 U(1)-equivariant volume: a (s λ) Γ(s λ) = e Ω + s H S 1 + λh U(1) = vol S 1 U(1)(M A ), Key Observation M A M A = Map hol (D C) is Mirror Symmetric to M B.

26 Further directions: 1 Q q,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 L q,t x (z)-functions.