Example Let K = C ( x ) and t C . Consider y ( x ) = t xy ( x ) . - PowerPoint PPT Presentation

Differential representations of SL 2 Alexey Ovchinnikov City University of New York Joint work with: Andrey Minchenko (University of Western Ontario) Goal: Good description for all differential representations of SL 2 . Differential representations of SL2 – page 1/13

Example Let K = C ( x ) and t ∈ C . Consider y ′ ( x ) = t xy ( x ) . The solution looks like x t . A calculation shows that � Z /q Z , t = p/q, ( p, q ) = 1 , x t � � � � Gal K /K = C ∗ , otherwise . This measures algebraic relations. Let now K = C ( x, t ) . What about differential relation with respect to the parameter t ? Differential representations of SL2 – page 2/13

Example continued The differential relations among the solution(s) of ∂ x y ( x, t ) = t xy ( x, t ) are the algebraic relations among y ( x, t ) , ∂ t y ( x, t ) , . . . and, therefore, come from the algebraic relations for the prolonged system. The first prolongation: � � � � � � t 1 ∂ t y ( x, t ) ∂ t y ( x, t ) x x ∂ x = . t y ( x, t ) 0 y ( x, t ) x The group to measure this dependence is � � ∂ t g � � � ∂ t � Gal { ∂ x ,∂ t } = g ∈ C ( t ) \ { 0 } = 0 . g Differential representations of SL2 – page 3/13

Linear differential algebraic groups • k is a differential field: ∂ t : k → k , • G is a subgroup of some GL n ( k ) given as a zero set of a system of differential polynomial equations F = 0 . • Coordinate ring: �� k [ x ij , ∂ t ( x ij ) , . . . , 1 / det( x ij )] ( F, ∂ t F, . . . ) Differential representations of SL2 – page 4/13

Examples • any linear algebraic group, e.g., G m , SL 2 ; � � � � ∂ t g • g ∈ G m | ∂ t = 0 . g • Galois groups of linear ∂ x -differential equations with ∂ t -parameters are linear differential algebraic groups when the field k of ∂ x -constants is differentially closed. (Cassidy, Chatzidakis, Hardouin, Landesman, Singer,. . . ). Differential representations of SL2 – page 5/13

Representations G → GL( V ) a differential polynomial map, V is a finite dimensional vector space. V is called a G -module. • G m ∋ g �→ ( g ) , � � g ∂ t g • G m ∋ g �→ . 0 g Define a new representation on V (1) = ( k ⊕ k ∂ t ) ⊗ k V with ∂ t · a = a · ∂ t + ∂ t ( a ) , a ∈ k , and for v ∈ V g (1 ⊗ v ) := 1 ⊗ gv g ( ∂ t ⊗ v ) := ∂ t ⊗ gv = ∂ t ( g ) · 1 ⊗ v + g · ∂ t ⊗ v. Differential representations of SL2 – page 6/13

Completely reducible representations V is completely reducible if and only if V ∼ = V 1 ⊕ . . . ⊕ V n as G -modules. Let H be a linear algebraic group. Unipotent radical: maximal unipotent normal algebraic subgroup R u ( H ) . H is called reductive if R u ( H ) = { e } . SL 2 is reductive. Theorem 1 Every representation of G is completely reducible if and only if G is conjugate in GL n to a subgroup H ⊂ GL n ( C ) and H is a reductive linear algebraic group. Differential representations of SL2 – page 7/13

Indecomposable representations Is not a direct sum of G -submodules. Every G -module is, therefore, a direct sum of indecomposable G -modules. Every G -module V , dim V = n , can be imbedded into A n , where A is the coordinate ring of G . How to get rid of n and only study submodules of A ? Differential representations of SL2 – page 8/13

Simple socle Only one minimal G -submodule. One can reconstruct any indecomposable V from such G -modules V 1 = span { E 1 , E 2 } and V 2 = span { F 1 , F 2 } :   A 0 B � � � � A B A 1 B 1 , �→  , 0 A 1 B 1   0 C 0 C  0 0 C via the pull-back: V = span { E 1 , F 1 , E 2 + F 2 } . Differential representations of SL2 – page 9/13

Differential algebraic group SL 2 Denote ∂ t by ′ . Coordinate ring: x 11 , . . . , x 22 , x ′ 11 , . . . , x ′ det( x ij ) − 1 , det( x ij ) ′ , . . . � � � � 22 , . . . / . k W. Sit (1975): all differential algebraic subgroups of SL 2 . � � � a ′ � Example: SL 2 ( C ) = ( a ij ) ∈ SL 2 ij = 0 ⊂ SL 2 . Every Zariski dense subgroup of SL 2 is conjugate to SL 2 ( C ) . Differential representations of SL2 – page 10/13

Non-differential SL 2 Let A = k [ x, y ] with the action of SL 2 via � � a b x �→ ax + cy, y �→ bx + dy, ∈ SL 2 . c d Since SL 2 is reductive, every (non-differential) SL 2 -module is a direct sum of irreducible SL 2 -modules. Classical result in the representation theory: Theorem 2 Every irreducible SL 2 -module is isomorphic to a submodule in A . Moreover, every irreducible submodule in A is isomorphic to a submodule of the form: { f ∈ A | f is homogeneous of degree n } . Differential representations of SL2 – page 11/13

Example for differential SL 2 Let R = k [ x, y, x ′ , y ′ , . . . ] with the action of SL 2 via � � a b x ( n ) �→ ( ax + cy ) ( n ) , y ( n ) �→ ( bx + dy ) ( n ) , ∈ SL 2 . c d x 2 , xy, y 2 , x ′ y − xy ′ � is � The subspace V = span SL 2 -invariant, a 2 b 2 a ′ b − ab ′   ab � � 2( a ′ d − b ′ c ) a b 2 ac ad + bc 2 bd   �→   c 2 d 2 c ′ d − cd ′ c d  cd    0 0 0 1 V ∗ has an SL 2 -submodule of dim 1 with trivial action. All such SL 2 -submodules in R are isomorphic to k and always split. Hence, V ∗ does not imbed into R . Differential representations of SL2 – page 12/13

What we have for differential SL 2 Theorem 3 Any idecomposable differential representation of SL 2 with simple socle imbeds into x 11 , . . . , x 22 , x ′ 11 , . . . , x ′ det( x ij ) − 1 , det( x ij ) ′ , . . . � � � � 22 , . . . / . k Any differential representation of SL 2 can be reconstructed from these representation using ⊕ and pull-backs. Differential representations of SL2 – page 13/13