g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

Transcript

1 Ole Warnaar Department of Mathematics

2 g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i<j k (t i t j ) 2γ dt = k 1 i=0 D = {t R k, 0 t k t 1 1} dt = dt 1 dt k Γ(α + iγ)γ(β + iγ)γ(γ + iγ) Γ ( α + β + (k + i 1)γ ) Γ(γ)

3 The Selberg-like integral C k (t i z) α 1 (t i w) β 1 1 i<j k (t i t j ) 2γ dt (where C is a k-dimensional Pochhammer double loop) arises in the solution of the Knizhnik Zamolodchikov (KZ) equation for the Lie algebra sl 2 = A 1. In Arrangements of hyperplanes and Lie algebra homology, (Invent. Math. 106 (1991), )) Schechtman and Varchenko solved the KZ equation for general simple Lie algebra g in terms of Selberg-like integrals.

4 Simple Lie algebra g of rank n. Root system Φ with simple roots α i, i [n]. Fundamental weights Λ i, i [n]. Bilinear symmetric form (, ) on the dual of the Cartan subalgebra: (α i, Λ j ) = δ ij and ( (α i, α j ) ) n i,j=1 = Cartan matrix of g Two highest weight modules V λ and V µ of highest weight λ and µ where n n λ = λ i Λ i and µ = µ i Λ i

5 α 2 Λ 2 Let ɛ i the ith standard unit vector in R n+1. Then the root system A n is given by {±(ɛ i ɛ j ) 1 i j n + 1} Λ 1 α 1 The root system A 2 with fundamental weights in red.

6 ( (α i, α j ) ) n i,j= = n The Cartan matrix and Dynkin diagram of A n.

7 Attach the set {t j } k1+ +k i j=k 1+ +k i 1 +1 of k i integration variables to the simple root α i (or to the ith node of the Dynkin diagram of g) and write α tj = α i if the variable t j is attached to the root α i. Then the master function is defined as Φ λ,µ (t) = k where k = k k n. t (λ,αt i ) i (1 t i ) (µ,αt i ) 1 i<j k (t i t j ) (αt i,αt j )

8 Let D be an appropriately chosen real integration domain or chain in [0, 1] k. Then the g-selberg integral is S g λ,µ;γ = S g λ,µ;γ (k 1,..., k n ) := Φ λ,µ (t) γ dt D

9 Example 1: g = A 1 λ = (1 α)γ 1 Λ 1 and µ = (1 β)γ 1 Λ 1 S A1 λ,µ;γ = D k t α 1 i (1 t i ) β 1 1 i<j k t i t j 2γ dt Example 2: g = A 2 S A2 λ,µ;γ = D λ = (1 α 1 )γ 1 Λ 1 + (1 α 2 )γ 1 Λ 2 µ = (1 β 1 )γ 1 Λ 1 + (1 β 2 )γ 1 Λ 2 k 1 t α1 1 i k 2 (1 t i ) β1 1 s α2 1 i (1 s i ) β2 1 k 1 k 2 t i t j 2γ s i s j 2γ t i s j γ dt 1 i<j k 1 1 i<j k 2 j=1

10 The Mukhin Varchenko conjecture Mukhin and Varchenko made the following conjecture regarding the g-selberg integral. Let Sing λ,µ [ν] denote the space of singular vectors of weight ν in V λ V µ : Sing λ,µ [ν] := {v V λ V µ : h i v = ν(h i )v, e i v = 0, 1 i n}. Conjecture (Mukhin Varchenko) If Sing λ,µ [λ + µ n k iα i ] is one-dimensional then S g λ,µ;γ evaluates as a product of gamma functions. Remarks on critical points of phase functions and norms of Bethe vectors, Adv. Stud. Pure Math. 27 (2000),

11 Let λ = (1 α)γ 1 Λ n, µ = (1 β 1 )γ 1 Λ (1 β n )γ 1 Λ n Theorem (SOW) If k 1 k 2 k n then n k s λ,µ;γ = S An 1 s r n s=1 k s k s 1 Γ(αδ s,n + (i k s+1 1)γ)Γ(iγ) Γ(γ) Γ(β s + + β r + (i + s r 1)γ) Γ ( αδ r,n + β s + + β r + (i + s r + k r k r+1 2)γ ) A Selberg integral for the Lie algebra A n, Acta Math. to appear.

12 An A 2 Selberg integral Let P λ (X ) = P λ (X ; q, t) be a (normalised) Macdonald polynomial and let (a) λ = (a; q, t) λ = i 1(at 1 i ; q) λi be a q-shifted factorial labelled by a partition. Theorem: A 2 Cauchy-type identity (SOW) Let X = {x 1,..., x n } and Y = {y 1,..., y m }. For abt = q, P λ (X )P µ (Y ) (at m ) λ (bt n ) µ λ,µ = n n (ax i ) (x i ) m j=1 m t µ j (by i ) (y i ) (at j i ) λi µ j (at j i+1 ) λi µ j n m j=1 (x i y j ) (t 1 x i y j ).

13 Special cases: m = 0: q-binomial theorem for Macdonald polynomials n (ax i ) (a) λ P λ (X ) = (x i ) λ n = 0: q-binomial theorem for Macdonald polynomials m (ay i ) (a) λ P λ (Y ) = (y i ) λ a = b = 1, t = q, Y qy : Cauchy identity for Schur functions n m 1 s λ (X )s λ (Y ) = 1 x i y j λ j=1

14 Corollary 1: A 2 Selberg integral Let λ = (1 α 1 )γ 1 Λ 1 + (1 α 2 )γ 1 Λ 2 such that Then k1 S A2 λ,µ;γ = µ = (1 β 1 )γ 1 Λ 1 + (1 β 2 )γ 1 Λ 2 α 1 + α 2 = γ + 1 Γ(α 1 + (i k 2 1)γ)Γ(β 1 + (i 1)γ)Γ(iγ) Γ(α 1 + β 1 + (i + k 1 k 2 2)γ)Γ(γ) k 2 Γ(α 2 + (i 1)γ)Γ(β 2 + (i 1)γ)Γ(iγ) Γ(α 2 + β 2 + (i + k 2 k 1 2)γ)Γ(γ) k 1 Γ(β 1 + β 2 + (i 2)γ) Γ(β 1 + β 2 + (i + k 2 2)γ)

15 Define the n-dimensional q-integral S (n,m) (α 1, α 2, β; k) := For m = 0: n m j=1 S (n,0) (α 1, α 2, β; k) = n [0,1] n (x i q) α2+β+(m n j)k 1 (x i q) α2+β+(m n j+1)k 1 n [0,1] n x α1 1 i (x i q) β (n 1)k 1 1 i<j n x α1 1 i (x i q) β (n 1)k 1 1 i<j n x 2k i (q 1 k x j /x i ) 2k d q x. xi 2k (q 1 k x j /x i ) 2k d q x this is the q-selberg integral of Askey, Habsieger and Kadell.

16 Corollary 2: q-selberg integral transform S (n,m) (α 1, α 2, β; k) = q ζ S (m,n) (α 2, α 1, β; k) n Γ q (β (i 1)k)Γ q (α 1 + (n i)k)γ q (ik + 1) Γ q (α 1 + β + (n m i)k)γ q (k + 1) m Γ q (α 2 + β + (m n i)k)γ q (k + 1) Γ q (β (i 1)k)Γ q (α 2 + (m i)k)γ q (ik + 1) where ( ) ( n m ζ = 2k 2 2k ) + α 1 k ( ) n α 2 k 2 ( ) m 2

17 Summary Long Live The King