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A note on the simplicity and the universal covering of some Kac-Moody group A note on the simplicity and the universal covering of some Kac-Moody group Jun Morita Institute of Mathematics, University of Tsukuba, Japan Fields Institute, March 25-29, 2013

A note on the simplicity and the universal covering of some Kac-Moody group Contents Contents Recent Topic - Simplicity - Notation Presentation Universal Covering Remark Schur Multiplier Conclusion

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic § Recent Topic - Simplicity -

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic 2009, P.E. Caprace - B. R´ emy Theorem Let A be an n × n indecomposable GCM, and F q a finite field with q = p ℓ elements. Let G u ( A , F q ) be the universal Kac-Moody group over F q of type A, and G ′ u ( A , F q ) = [ G u ( A , F q ) , G u ( A , F q )] its derived subgroup. We suppose that A is not of affine type, and q ≥ n > 2 . Then G ′ u ( A , F q ) is simple modulo its center.

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic 2012, P.E. Caprace - B. R´ emy Theorem ( 2 ) − a Let A = be a 2 × 2 hyperbolic GCM, − 1 2 that is, a > 4 , and F q a finite field with q > 3 . Let G u ( A , F q ) be the universal Kac-Moody group over F q of type A. Then G u ( A , F q ) is simple modulo its center.

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic Remaining Case (with B. R´ emy) Theorem ( 2 ) − a Let A = be a 2 × 2 hyperbolic GCM − b 2 satisfying ab > 4 with a > 1 and b > 1 , and F the algebraic closure of a finite field F p . Let G u ( A , F ) be the universal Kac-Moody group over F of type A. Then G u ( A , F ) is simple modulo its center.

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic Uniformization Theorem Let A be an indecomposable GCM, and F the algebraic closure of a finite field F p . Let G u ( A , F ) be the universal Kac-Moody group over F of type A, and G ′ u ( A , F ) its derived subgroup. We suppose that A is not of affine type. Then G ′ u ( A , F ) is simple modulo its center.

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic Simple Group with Trivial Schur Multiplier Theorem Let A be an indecomposable GCM, and F the algebraic closure of a finite field F p . Then the following two conditions are equivalent. (1) det ( A ) = ± p c for some c ≥ 0 . (2) G u ( A , F ) is a simple group with trivial Schur multiplier.

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic Rank 2 Case Example Let F be the algebraic closure of a finite field F p . Then the following groups are simple groups with trivial Schur multipliers. ( 2 ) − 2 (1) G u ( , F ) , p = 2 − 3 2 ( 2 ) − 3 (2) G u ( , F ) , p = 5 − 3 2 ( ) 2 − 5 (3) G u ( , F ) , p = 3 − 17 2

A note on the simplicity and the universal covering of some Kac-Moody group Recent Topic Anther Infinite Field Question Let A be an indecomposable non-finite & non-affine GCM, and F another infinite field. (1) Is G ′ u ( A , F ) simple modulo its center ? (2) Especially how about G ′ u ( A , C ) ?

A note on the simplicity and the universal covering of some Kac-Moody group Notation § Notation

A note on the simplicity and the universal covering of some Kac-Moody group Notation Set Up I Let A be an n × n GCM, and put n ′ = corank ( A ). We let G u ( A , − ) denote the so-called Tits group functor associated with A . Let G u ( A , F ) be the universal Kac-Moody group over F of type A , and G ′ u ( A , F ) the derived subgroup of G u ( A , F ). There is ∃ an embedding : T = Hom ( Z n + n ′ , F × ) ֒ → G u ( A , F ).

A note on the simplicity and the universal covering of some Kac-Moody group Notation Set Up II Let g be the Kac-Moody algebra over C of type A , and ∆ re the set of real roots. For each α ∈ ∆ re , there is a group homomorphism x α : F ֒ → G u ( A , F ). Put U α = Im( x α ) = { x α ( t ) | t ∈ F } . Then, G u ( A , F ) = � T , U α | α ∈ ∆ re � , G ′ u ( A , F ) = � U α | α ∈ ∆ re � , G ′ ad ( A , F ) = G ′ u ( A , F ) / Z ( G ′ u ( A , F )), G u ( A , F ) = G ′ u ( A , F ) if det ( A ) � = 0, G ad ( A , F ) = G ′ ad ( A , F ) if det ( A ) � = 0.

A note on the simplicity and the universal covering of some Kac-Moody group Presentation § Presentation

A note on the simplicity and the universal covering of some Kac-Moody group Presentation 1986, J. Tits Theorem ( G ′ u -version) The group G ′ u ( A , F ) is presented by the generators x α ( t ) with α ∈ ∆ re and t ∈ F, and the following defining relations: (A) x α ( s ) x α ( t ) = x α ( s + t ) , (B) [ x α ( s ) , x β ( t )] = ∏ x i α + j β ( N α,β, i , j s i t j ) , (B ′ ) w α ( u ) x β ( t ) w α ( − u ) = x β ′ ( t ′ ) , (C) h α ( u ) h α ( v ) = h α ( uv ) .

A note on the simplicity and the universal covering of some Kac-Moody group Presentation Condition for (B) Let g = ⊕ α ∈ ∆ g α be the root space decomposition, where g α = { x ∈ g | [ h , x ] = α ( h ) x ( ∀ h ∈ h ) } , ∆ = { α ∈ h ∗ | g α � = 0 } , g 0 = h . Put Q α,β = { i α + j β | i , j ∈ Z > 0 } ∩ ∆. Then, we have Q α,β x i α + j β ( N α,β, i , j s i t j ) (B) [ x α ( s ) , x β ( t )] = ∏ whenever Q α,β ⊂ ∆ re .

A note on the simplicity and the universal covering of some Kac-Moody group Presentation Relation (B), 1987, J. M. Theorem There are essentially five type relations in (B). [ x α ( s ) , x β ( t )] = 1 [ x α ( s ) , x β ( t )] = x α + β ( ± ( r + 1) st ) r = max { i ∈ Z | β − i α ∈ ∆ re } [ x α ( s ) , x β ( t )] = x α + β ( ± st ) x 2 α + β ( ± s 2 t ) [ x α ( s ) , x β ( t )] = x α + β ( ± 2 st ) x 2 α + β ( ± 3 s 2 t ) · x α +2 β ( ± 3 st 2 ) [ x α ( s ) , x β ( t )] = x α + β ( ± st ) x 2 α + β ( ± s 2 t ) · x 3 α + β ( ± s 3 t ) x 3 α +2 β ( ± 2 s 3 t 2 )

A note on the simplicity and the universal covering of some Kac-Moody group Presentation About (B ′ ), (C) For u , v ∈ F × , we put w α ( u ) = x α ( u ) x − α ( − u − 1 ) x α ( u ), h α ( u ) = w α ( u ) w α ( − 1). Then, (B ′ ) w α ( u ) x β ( t ) w α ( − u ) = x β ′ ( t ′ ), (C) h α ( u ) h α ( v ) = h α ( uv ), where h α is the coroot of α and β ′ = β − β ( h α ) α, t ′ = ± u − β ( h α ) t .

A note on the simplicity and the universal covering of some Kac-Moody group Presentation SL 2 ( F ) For each α ∈ ∆ re , there is a group isomorphism ≃ ϕ α : � U α , U − α � − → SL 2 ( F ) satisfying ( 1 t ( 1 0 ) ) x α ( t ) �→ , x − α ( t ) �→ , 0 1 t 1 ( ) 0 u w α ( u ) �→ , − u − 1 0 ( u ) 0 h α ( u ) �→ . 0 u − 1

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering § Universal Covering

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering Central Extension A group epimorphism E − → G is called an extension, and an extension E − → G is called a central extension if Ker [ E − → G ] ⊂ Z ( E ).

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering Universal Covering A central extension E − → G is called a universal covering (or a universal central extension) if for any central extension E ′ − → G , there uniquely exists a → E ′ such that the group homomorphism E − following diagram is commutative. E → G ↓ ր E ′

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering Steinberg Group The Steinberg group St ( A , F ) over a field F of type A is defined to be the group generated by ˆ x α ( t ) for all α ∈ ∆ re and t ∈ F with the defining relations corresponding to (A), (B), (B ′ ).

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering 1990, J. M. - U. Rehmann Theorem Let A be a GCM, and F an infinite field. Then, St ( A , F ) is a universal covering of G ′ u ( A , F ) , which is induced by ˆ x α ( t ) �→ x α ( t ) .

A note on the simplicity and the universal covering of some Kac-Moody group Universal Covering Application Theorem Let A be an indecomposable GCM, and F the algebraic closure of a finite field F p . We suppose that A is not of affine type. Then, G ′ u ( A , F ) is a universal covering of a simple group G ′ ad ( A , F ) .

A note on the simplicity and the universal covering of some Kac-Moody group Remark § Remark

A note on the simplicity and the universal covering of some Kac-Moody group Remark Remark I Let A be an n × n GCM, and Π = { α 1 , . . . , α n } the set of simple roots. The principal divisors of A is denoted by π ( A ) = ( d 1 , · · · , d n ), and we put Γ = ⊕ n i =1 Z / d i Z . Then, for a field F , we have Z ( G ′ u ( A , F )) = { h α 1 ( u 1 ) · · · h α n ( u n ) | u a 1 j 1 · · · u a nj = 1 , ∀ j } n ≃ Hom (Γ , F × ).