The zeta-function of the root system of type G 2 Kohji Matsumoto - PowerPoint PPT Presentation

The zeta-function of the root system of type G 2 Kohji Matsumoto (Nagoya University) (a joint work with Y.Komori and H.Tsumura) 1

V : r -dimensional real vector space � , � : inner product defined on V ∆ : finite reduced root system in V ∆ = ∆ + ∪ ∆ − Ψ = { α 1 , . . . , α r } : fundamental system { λ 1 , . . . , λ r } : fundamental weights (with � α ∨ i , λ j � = δ ij ) P ++ = ⊕ r i =1 N λ i ( N = Z ≥ 1 ) Define the zeta-function of ∆ by 1 ∑ ∏ ζ r ( s , ∆) = � α ∨ , λ � s α λ ∈ P ++ α ∈ ∆ + where s = ( s α ) α ∈ ∆ + ∈ C n , n = | ∆ + | . 2

Examples. Ex 1. ∆ = A 1 ⇒ ∆ + = { α 1 } , P ++ = { mλ 1 | m ∈ N } , 1 , λ � − s = ∑ ∞ m =1 m − s = ζ ( s ) λ ∈ P ++ � α ∨ ⇒ ζ 1 ( s, A 1 ) = ∑ (the Riemann zeta-function) Ex 2. ∆ = A 2 ⇒ ∆ + = { α 1 , α 2 , α 1 + α 2 } , P ++ = { m 1 λ 1 + m 2 λ 2 | m 1 , m 2 ∈ N } , Therefore ∞ ∞ 1 m − s 1 m − s 2 ( m 1 + m 2 ) − s 3 ∑ ∏ ∑ ∑ ζ 2 ( s, A 2 ) = � α ∨ , λ � s α = 1 2 λ ∈ P ++ α ∈ ∆ + m 1 =1 m 2 =1 (the Tornheim double sum) 3

Ex 3. The case s = ( s, s, . . . , s ) g : a semisimple Lie algebra over C ∆ = ∆( g ) ϕ (dim ϕ ) − s Witten zeta-function : ζ W ( s, g ) = ∑ ( ϕ runs over all finite dim irreducible representation of g ) By Weyl’s dimension formula we have ζ W ( s, g ) ∞ ∞ ) − s ( � α ∨ , m 1 λ 1 + · · · + m r λ r � ∑ ∑ ∏ · · · = � α ∨ , λ 1 + · · · + λ r � m 1 =1 m r =1 α ∈ ∆ + s   � α ∨ , λ 1 + · · · + λ r �  ∏ = ζ r (( s, s, . . . , s ) , ∆( g ))  α ∈ ∆ + 4

Ex 4. The Euler-Hoffman-Zagier multiple sum ∞ ∞ n − s 1 ( n 1 + n 2 ) − s 2 · · · ( n 1 + · · · + n r ) − s r ∑ ∑ · · · ζ EHZ,r ( s ) = 1 n 1 =1 n r =1 This can be regarded as a zeta-function of the root system of type C r , which is NOT simply-laced. ∆ + = ∆ l + ∪ ∆ s + ( l : long roots, s : short roots) s l = ( s α ) α ∈ ∆ + , with s α = 0 for any α ∈ ∆ s + Then we find ζ r ( s l , C r ) = ζ EHZ,r ( s ) (It is also possible to understand ζ EHZ,r ( s ) as a zeta-function of the root system of type A r ; cf. Komori-M-Tsumura, Math. Z. 268 (2011)) 5

Special values (at positive integer points) The study of special values is important for both Witten zeta-functions and Euler-Hoffman-Zagier sums ⇒ How are the special values of zeta-functions of root systems? Recall : Witten’s volume formula (coonected with the volumes of certain moduli spaces appearing in quantum gauge theory) ζ W (2 k, g ) = C W (2 k, g ) π 2 kn for k ∈ N , where n = | ∆ + | and C W (2 k, g ) ∈ Q What is the explicit value of C W (2 k, g )? 6

1. A 1 case (the Riemann zeta) ζ (2) = π 2 / 6 , ζ (4) = π 4 / 90 , . . . That is, C W (2 , A 1 ) = 1 / 6 , C W (4 , A 1 ) = 1 / 90 , . . . 2. A 2 case (the Tornheim sum) ζ W ( s, A 2 ) = 2 s ζ 2 (( s, s, s ) , ∆( A 2 )) and Mordell proved ζ 2 ((2 , 2 , 2) , A 2 ) = π 6 / 2835 ⇒ C W (2 , A 2 ) = 4 / 2835 Algorithm of computing C W (2 k, g ) for general case: Szenes (1998), Gunnells-Sczech (2003), Our approach (Around 2005 − ) 7

W = W (∆) : The Weyl group of ∆ Define   ( − 1) − s α  ζ r ( w − 1 s , ∆) , ∑ ∏ S ( s , ∆) =  w ∈ W α ∈ ∆ + ∩ w ∆ − where w − 1 s is defined by ( σ β s ) α = s σ β α for the reflection σ β with respect to (the hyperplane orthogonal to) β (If σ β α ∈ ∆ − , then we understand that s σ β α = s − σ β α ) S ( s , ∆) is the “Weyl group symmetric linear combination” of ζ r ( s , ∆) ⇒ can be expressed as a certain multiple integral involving a product of Lerch-type zeta-functions 8

⇒ When s = k = ( k α ) α ∈ ∆ + ( k α ∈ N ), we have (2 π √− 1) k α   S ( k , ∆) = ( − 1) n  ∏  B k (∆) k α ! α ∈ ∆ + ( B k (∆) : a “root-theoretic” generalization of Bernoulli numbers) In particular, if k α = k β whenever || α || = || β || , then w − 1 k = k , so   ( − 1) − 2 k α  ζ r ( w − 1 2 k , ∆) ∑ ∏ S (2 k , ∆) =  w ∈ W α ∈ ∆ + ∩ w ∆ − ∑ = ζ r (2 k , ∆) = | W | ζ r (2 k , ∆) , and hence w ∈ W (2 π √− 1) 2 k α   ζ r (2 k , ∆) = ( − 1) n  ∏  B 2 k (∆) | W | (2 k α )! α ∈ ∆ + 9

B k (∆) can be calculated via its generating function t k α α ∑ ∏ F ( t , ∆) = B k (∆) ( t = ( t α ) α ∈ ∆ + ) k α ! α ∈ ∆ + k Example: t F ( t, A 1 ) = (classical case) e t − 1 F (( t 1 , t 2 , t 3 ) , A 2 ) t 1 t 2 t 3 e t 1 + t 2 ( e t 3 − t 1 − t 2 − 1) = ( e t 1 − 1)( e t 2 − 1)( e t 3 − 1)( t 3 − t 1 − t 2 ) Now we consider our main target : type G 2 10

Type G 2 : Ψ = { α 1 , α 2 } ∆ + = { α 1 , α 2 , α 1 + α 2 , 2 α 1 + α 2 , 3 α 1 + α 2 , 3 α 1 + 2 α 2 } P ++ = { mλ 1 + nλ 2 | m, n ∈ N } Therefore ζ 2 ( s , G 2 ) = ζ 2 (( s 1 , s 2 , s 3 , s 4 , s 5 , s 6 ) , G 2 ) ∞ ∞ m − s 1 n − s 2 ( m + n ) − s 3 ( m + 2 n ) − s 4 ∑ ∑ = m =1 n =1 × ( m + 3 n ) − s 5 (2 m + 3 n ) − s 6 Now compute the generating function F (( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) , G 2 ) of the Bernoulli numbers B k ( G 2 ) : 11

F (( t 1 , t 2 , t 3 , t 4 , t 5 , t 6 ) , G 2 ) = t 1 t 2 t 3 t 4 t 5 t 6 ( 1 ( e t 1 − 1)( e t 2 − 1)( t 1 + t 2 − t 3 ) − 1 ( t 1 + 2 t 2 − t 4 ) − 1 × × ( t 1 + 3 t 2 − t 5 ) − 1 (2 t 1 + 3 t 2 − t 6 ) − 1 1 ( e t 1 − 1)( e t 3 − 1)( t 1 + t 2 − t 3 ) − 1 ( t 1 − 2 t 3 + t 4 ) − 1 + × (2 t 1 − 3 t 3 + t 5 ) − 1 ( t 1 − 3 t 3 + t 6 ) − 1 e t 1 / 2+ t 4 / 2 + 1 2( e t 1 − 1)( e t 4 − 1)( t 1 / 2 + t 2 − t 4 / 2) − 1 ( t 1 / 2 − t 3 + t 4 / 2) − 1 + × ( t 1 / 2 − 3 t 4 / 2 + t 5 ) − 1 ( t 1 / 2 + 3 t 4 / 2 − t 6 ) − 1 − e 2 t 1 / 3+ t 5 / 3 + e t 1 / 3+2 t 5 / 3 + 1 ( t 1 / 3 − t 4 + 2 t 5 / 3) − 1 3( e t 1 − 1)( e t 5 − 1) × ( t 1 / 3 + t 2 − t 5 / 3) − 1 (2 t 1 / 3 − t 3 + t 5 / 3) − 1 ( t 1 + t 5 − t 6 ) − 1 12

− e t 1 / 3+ t 6 / 3 + e 2 t 1 / 3+2 t 6 / 3 + 1 ( t 1 + t 5 − t 6 ) − 1 3( e t 1 − 1)( e t 6 − 1) × ( t 1 / 3 + t 4 − 2 t 6 / 3) − 1 (2 t 1 / 3 + t 2 − t 6 / 3) − 1 ( t 1 / 3 − t 3 + t 6 / 3) − 1 e t 2 ( e t 2 − 1)( e t 3 − 1)( t 1 + t 2 − t 3 ) − 1 ( t 2 + t 3 − t 4 ) − 1 − × (2 t 2 + t 3 − t 5 ) − 1 ( t 2 + 2 t 3 − t 6 ) − 1 e t 2 ( e t 2 − 1)( e t 4 − 1)( t 1 + 2 t 2 − t 4 ) − 1 ( t 2 + t 3 − t 4 ) − 1 − × ( t 2 + t 4 − t 5 ) − 1 ( t 2 − 2 t 4 + t 6 ) − 1 e t 2 ( e t 2 − 1)( e t 5 − 1)( t 1 + 3 t 2 − t 5 ) − 1 (2 t 2 + t 3 − t 5 ) − 1 + × ( t 2 + t 4 − t 5 ) − 1 (3 t 2 − 2 t 5 + t 6 ) − 1 13

+ e t 2 / 2+ t 6 / 2 + e t 2 2( e t 2 − 1)( e t 6 − 1)( t 1 + 3 t 2 / 2 − t 6 / 2) − 1 × ( t 2 / 2 + t 3 − t 6 / 2) − 1 ( t 2 / 2 − t 4 + t 6 / 2) − 1 (3 t 2 / 2 − t 5 + t 6 / 2) − 1 e t 4 ( e t 3 − 1)( e t 4 − 1)( t 2 + t 3 − t 4 ) − 1 ( t 1 − 2 t 3 + t 4 ) − 1 − × ( t 3 − 2 t 4 + t 5 ) − 1 ( t 3 + t 4 − t 6 ) − 1 e t 5 2( e t 3 − 1)( e t 5 − 1)( t 2 + t 3 / 2 − t 5 / 2) − 1 ( t 3 / 2 − t 4 + t 5 / 2) − 1 + × ( t 1 − 3 t 3 / 2 + t 5 / 2) − 1 (3 t 3 / 2 + t 5 / 2 − t 6 ) − 1 e t 6 ( e t 3 − 1)( e t 6 − 1)(3 t 3 + t 5 − 2 t 6 ) − 1 ( t 2 + 2 t 3 − t 6 ) − 1 + × ( t 3 + t 4 − t 6 ) − 1 ( t 1 − 3 t 3 + t 6 ) − 1 14

e t 5 ( e t 4 − 1)( e t 5 − 1)( t 2 + t 4 − t 5 ) − 1 ( t 3 − 2 t 4 + t 5 ) − 1 − × ( t 1 − 3 t 4 + 2 t 5 ) − 1 (3 t 4 − t 5 − t 6 ) − 1 e t 4 ( e t 4 − 1)( e t 6 − 1)( t 1 + 3 t 4 − 2 t 6 ) − 1 ( t 3 + t 4 − t 6 ) − 1 − × ( t 2 − 2 t 4 + t 6 ) − 1 (3 t 4 − t 5 − t 6 ) − 1 + e t 5 + e 2 t 5 / 3+2 t 6 / 3 + e t 5 / 3+ t 6 / 3 ( t 1 + t 5 − t 6 ) − 1 3( e t 5 − 1)( e t 6 − 1) ) × ( t 3 + t 5 / 3 − 2 t 6 / 3) − 1 ( t 4 − t 5 / 3 − t 6 / 3) − 1 ( t 2 − 2 t 5 / 3 + t 6 / 3) − 1 From this explicit form of the generating function, we can calculate the special values of ζ 2 ( s , G 2 ) 15

23 297904566960 π 12 ζ 2 ((2 , 2 , 2 , 2 , 2 , 2) , G 2 ) = 8165653 1445838676129559305994400000 π 24 ζ 2 ((4 , 4 , 4 , 4 , 4 , 4) , G 2 ) = ζ 2 ((6 , 6 , 6 , 6 , 6 , 6) , G 2 ) 55940539974690617 131888156302530666544150214880458495963616000000 π 36 = Also (since our condition is k α = k β for || α || = || β || ) we have 467 213955059990672000 π 18 ζ 2 ((2 , 4 , 4 , 4 , 2 , 2) , G 2 ) = 16

How to evaluate the values at “odd” integer points? Recall :   ( − 1) − k α  ζ r ( w − 1 k , G 2 ) ∑ ∏ S ( k , G 2 ) =  w ∈ W α ∈ ∆ + ∩ w ∆ − When k α is odd, the signature part appears. For G 2 , it is well-known that | W | = 12, and it is easy to see that | ∆ + ∩ w ∆ − | is odd for 6 elements of W and is even for the other 6 elements of W . Therefore, when k = ( k, k, k, k, k, k ) for some odd positive integer k , all terms on the right-hand side of the above equation are cancelled, and we obtain NO information. 17