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

Differential representations of SL2

Alexey Ovchinnikov

City University of New York

Joint work with: Andrey Minchenko (University of Western Ontario) Goal: Good description for all differential representations of SL2.

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 xt. A calculation shows that

Gal

• K
• xt

/K

• =
• Z/qZ,

t = p/q, (p, q) = 1, C∗,

• therwise.

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

∂xy(x, t) = t xy(x, t)

are the algebraic relations among

y(x, t), ∂ty(x, t), . . .

and, therefore, come from the algebraic relations for the prolonged system. The first prolongation:

∂x

• ∂ty(x, t)

y(x, t)

• =
• t

x 1 x t x

∂ty(x, t) y(x, t)

• .

The group to measure this dependence is

Gal{∂x,∂t} =

• g ∈ C(t) \ {0}
• ∂t

∂tg g

• = 0
• .

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 GLn(k) given as a zero set of a

system of differential polynomial equations F = 0.

• Coordinate ring:

k[xij, ∂t(xij), . . . , 1/ det(xij)]

• (F, ∂tF, . . .)

Differential representations of SL2 – page 4/13

Examples

• any linear algebraic group, e.g., Gm, SL2;
• g ∈ Gm | ∂t
• ∂tg

g

• = 0
• .
• 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.

• Gm ∋ g → (g),
• Gm ∋ g →
• g

∂tg 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 ∼ = V1 ⊕ . . . ⊕ Vn as G-modules.

Let H be a linear algebraic group. Unipotent radical: maximal unipotent normal algebraic subgroup Ru(H).

H is called reductive if Ru(H) = {e}. SL2 is reductive.

Theorem 1 Every representation of G is completely reducible if and only if G is conjugate in GLn to a subgroup

H ⊂ GLn(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 An, 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 V1 = span{E1, E2} and V2 = span{F1, F2}:

• A

B C

• ,
• A1

B1 C

• →

   A B A1 B1 C    ,

via the pull-back: V = span{E1, F1, E2 + F2}.

Differential representations of SL2 – page 9/13

Differential algebraic group SL2

Denote ∂t by ′. Coordinate ring:

k

• x11, . . . , x22, x′

11, . . . , x′ 22, . . .

• /
• det(xij) − 1, det(xij)′, . . .
• .
• W. Sit (1975): all differential algebraic subgroups of SL2.

Example: SL2(C) =

• (aij) ∈ SL2
• a′

ij = 0

• ⊂ SL2.

Every Zariski dense subgroup of SL2 is conjugate to SL2(C).

Differential representations of SL2 – page 10/13

Non-differential SL2

Let A = k[x, y] with the action of SL2 via

x → ax + cy, y → bx + dy,

• a

b c d

• ∈ SL2.

Since SL2 is reductive, every (non-differential) SL2-module is a direct sum of irreducible SL2-modules. Classical result in the representation theory: Theorem 2 Every irreducible SL2-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 SL2

Let R = k[x, y, x′, y′, . . .] with the action of SL2 via

x(n) → (ax + cy)(n), y(n) → (bx + dy)(n),

• a

b c d

• ∈ SL2.

The subspace V = span

• x2, xy, y2, x′y − xy′ is

SL2-invariant,

• a

b c d

• →

     a2 ab b2 a′b − ab′ 2ac ad + bc 2bd 2(a′d − b′c) c2 cd d2 c′d − cd′ 1      V ∗ has an SL2-submodule of dim 1 with trivial action. All

such SL2-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 SL2

Theorem 3 Any idecomposable differential representation of

SL2 with simple socle imbeds into k

• x11, . . . , x22, x′

11, . . . , x′ 22, . . .

• /
• det(xij) − 1, det(xij)′, . . .
• .

Any differential representation of SL2 can be reconstructed from these representation using ⊕ and pull-backs.

Differential representations of SL2 – page 13/13