representation of reductive differential algebraic...

Representation of reductivedifferential algebraic groups

Andrey Minchenko∗ and Alexey Ovchinnikov

Department of MathematicsUniversity of Western Ontario∗

Department of MathematicsCUNY Queens College

January 2012

Andrey Minchenko () Representation of reductive differential algebraic groups January, 2012 1 / 18

Contents

1 Basic definitions

2 LAG associated with a reductive LDAG

3 Structure of a reductive LDAG

4 Differential representations of Gmn.

5 Differential representations of SL2.

Contents

1 Basic definitions

k — ordinary differential field with char k = 0U — differentially closed field containing kC — field of constants of U

Definition

A linear differential algebraic group is a Kolchin closed subgroup G ofGLn(U), that is, an intersection of a Kolchin closed subset of Un2

withGLn(U), which is closed under the group operations.

k — ordinary differential field with char k = 0U — differentially closed field containing kC — field of constants of U

Definition

A linear differential algebraic group is a Kolchin closed subgroup G ofGLn(U), that is, an intersection of a Kolchin closed subset of Un2

withGLn(U), which is closed under the group operations.

Definition

Let G be a LDAG. A differential polynomial group homomorphism

r : G → GL(V )

is called a differential representation of G , where V is a finite-dimensionalvector space over k.

Example

ρ : Gm(U)→ GL2(U), g 7→(

g ∂g0 g

), g ∈ U∗.

One can show that the Zariski closure

Hρ := ρ(Gm) =

{(a b0 a

) ∣∣ a ∈ U∗, b ∈ U}∼= Gm ×Ga,

which is not a reductive linear algebraic group.

Definition

r : G → GL(V )

Example

ρ : Gm(U)→ GL2(U), g 7→(

g ∂g0 g

), g ∈ U∗.

Hρ := ρ(Gm) =

{(a b0 a

) ∣∣ a ∈ U∗, b ∈ U}∼= Gm ×Ga,

Definition

r : G → GL(V )

Example

ρ : Gm(U)→ GL2(U), g 7→(

g ∂g0 g

), g ∈ U∗.

Hρ := ρ(Gm) =

{(a b0 a

) ∣∣ a ∈ U∗, b ∈ U}∼= Gm ×Ga,

Definition

A LDAG G ⊂ GLn(U) is called unipotent if it is conjugate to the group ofstrictly upper-triangular matrices.

This definition does not depend on the embedding into GLn.A LDAG G contains a maximal normal unipotent differential algebraicsubgroup Ru(G ).

Definition

The subgroup Ru(G ) is called the unipotent radical of G . G is calledreductive if its unipotent radical is trivial, that is, Ru(G ) = {e}.

Definition

This definition does not depend on the embedding into GLn.

A LDAG G contains a maximal normal unipotent differential algebraicsubgroup Ru(G ).

Definition

Contents

1 Basic definitions

Theorem

Let G ⊂ GLn(U) be a reductive LDAG and

ρ : G → GL(V )

its faithful representation of minimal dimension. Then:

1 The representation ρ is completely reducible.

2 H = ρ(G ) is a reductive LAG.

3 The group H, up to an algebraic isomorphism, does not depend on ρ.

One can show that any completely reducible representation of G extendsto that of H.In particular, IrrepG ↔ IrrepH.

Theorem

ρ : G → GL(V )

One can show that any completely reducible representation of G extendsto that of H.

In particular, IrrepG ↔ IrrepH.

Theorem

ρ : G → GL(V )

One can show that any completely reducible representation of G extendsto that of H.In particular, IrrepG ↔ IrrepH.

Contents

1 Basic definitions

Definition

A connected DAG is called simple if it is non-commutative and has nonontrivial connected normal differential algebraic subgroups.

Definition

A connected DAG is called a torus if it is isomorphic to a Kolchin-densesubgroup of Gm(U)× · · · ×Gm(U).

Theorem (Cassidy)

Let G be a connected Zariski dense differential algebraic subgroup of asimple Chevalley group S. Then either G = S or G is conjugate in S toS(C ).

For example, SLn(U) and SLn(C ) are essentially the only Kolchin-densesubgroups of SLn(U).

Theorem

A connected reductive LDAG is an almost direct product of simple LDAGsand a torus.

Definition

Theorem (Cassidy)

Theorem

Definition

Theorem (Cassidy)

Theorem

Definition

Theorem (Cassidy)

Theorem

Definition

Theorem (Cassidy)

Theorem

Contents

1 Basic definitions

By a subquotient of V , we mean a G -module V1/V2 where V2 ⊂ V1 aresubmodules of V .

The following recalls a way of describing categories of representations inwhich not every representation is a direct sum of irreducibles.

Definition

For any V ∈ Ob(RepG ), denote the set of all simple subquotients of V byJH(V ). For a subset S ⊂ IrrepG , we say that V ∈ Ob(RepG ) isS-isotypic, if JH(V ) ⊂ S .We say that S is splitting if any V is a direct sum U ⊕W , whereJH(U) ⊂ S and JH(W ) ∩ S = ∅.

Theorem

Every element of IrrepGmn is splitting

By a subquotient of V , we mean a G -module V1/V2 where V2 ⊂ V1 aresubmodules of V .The following recalls a way of describing categories of representations inwhich not every representation is a direct sum of irreducibles.

Definition

Theorem

By a subquotient of V , we mean a G -module V1/V2 where V2 ⊂ V1 aresubmodules of V .The following recalls a way of describing categories of representations inwhich not every representation is a direct sum of irreducibles.

Definition

Theorem

Contents

1 Basic definitions

In the classical rational representation theory of the algebraic group SL2 incharacteristic zero, every finite-dimensional SL2-module is a direct sum ofirreducible ones, and each of those is isomorphic to

{xd , xd−1y , . . . , xyd−1, yd

}⊂ k[x , y ],

for some d ≥ 0, where the action of SL2 is:

SL2 3(

a bc d

{x 7→ ax + cy ,

y 7→ bx + dy .

For a term

h = α ·(

x (p1))m1

· . . . ·(

x (pk ))mk

y (q1))n1

· . . . ·(

y (qt))nt

where pi ,mi , qj , nj are non-negative integers, p1 < . . . < pk ,q1 < . . . < qt , and 0 6= α ∈ k, its weight is, by definition,∑

pimi +∑

qjnj . (2)

C = k{cij}1≤i ,j≤2, det = c11c22 − c12c21,

A = k{G} = C/[det− 1], B = C/[det], P = k{x , y}

For a term

h = α ·(

x (p1))m1

· . . . ·(

x (pk ))mk

y (q1))n1

· . . . ·(

y (qt))nt

where pi ,mi , qj , nj are non-negative integers, p1 < . . . < pk ,q1 < . . . < qt , and 0 6= α ∈ k, its weight is, by definition,∑

pimi +∑

qjnj . (2)

C = k{cij}1≤i ,j≤2, det = c11c22 − c12c21,

A = k{G} = C/[det− 1], B = C/[det], P = k{x , y}

For integers d , k ≥ 0, let Pkd ⊂ P be the subspace spanned by the

differential monomials of degree d and weight ≤ k. Note that all Pkd are

SL2-invariant. We have

P0d = Spank

{xd , xd−1y , . . . , yd

}⊂ P.

Ud = Spank

{P0d ,(

xd)′,(

xd−1y)′, . . . ,

(yd)′}⊂ P1

andand Wd = P0

d + (x ′y − xy ′) · P0d−2 ⊂ P1

which are SL2-submodules with Ud being isomorphic to F(P0d

), the

prolongation of P0d .

Theorem

Let V be a differential representation of SL2 that is a non-split extensionof two irreducible representations V1 and V2 of SL2, that is, there is ashort exact sequence

0 −−−−→ V1 −−−−→ V −−−−→ V2 −−−−→ 0,

(hence, V ∈ Rep0SL2). Then there exists d ≥ 1 such that either V or V ∨

is isomorphic to either

1 Ud , in which case dim V = 2d + 2, or

2 Wd , in which case dim V = 2d.

Moreover, U∨d∼= Ud and the G-modules

Ud , Wd , W ∨d , d ≥ 1,

form the complete list of pairwise non-isomorphic G -modules that arenon-trivial extensions of simple modules.

The proof consists of the following steps:

1 embed either V or V ∨ into B using homogeneity,

2 embed the result into P,

3 show that the result is actually inside P1d ,

4 show that P1d has only two submodules with simple socle (Ud and

Wd) that are non-split extensions of two irreducibles.

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