Di erential geometry and representations of semi-simple ...Di erential geometry and representations...

Dierential geometry and representations of

semi-simple algebraic groups

Pavel BibikovInstitute of Control Sciences, Moscow, Russia(in collaboration with Valentin Lychagin)

[13.10.2018]

Pavel Bibikov Vrnjacka Banja2018

Introduction

The problem of studying orbit spaces Ω/G for actions G : Ω ofgroups G on spaces Ω is one the most important problems, whichhas a lot of dierent applications in many areas (representationtheory, geometry, dierential equations, etc.).

Most of the cases of this problem fall into the following groups:

Ω is a smooth manifold and G is a Lie group (geometric

situation);

Ω is an algebraic manifold and G is an algebraic Lie group,acting algebraically on Ω (algebraic situation).

Introduction

situation);

Introduction

situation);

Introduction

situation);

Introduction

situation);

In the rst case it was proved by J. L. Koszul and R. Palais, that ifthe action G : Ω is proper and free, then the orbit space Ω/G is asmooth manifold and G-orbits are separated by smooth invariants.

The algebraic case has a very long and interesting history.

G is reductive, Ω is an algebraic manifold ⇒ algebra ofpolynomial invariants C[Ω]G is nite-generated (D. Hilbert,1899).

G is not reductive (14-th Hilbert problem) ⇒ counterexample(Nagata, Steinberg, 1954).

G is semi-simple ⇒ eld of rational invariants C(Ω)G isnite-generated (Rosenlicht, 1956).

Projective action G : Ω ⇒ Geometrical Invariant Theory (D.Mumford, 1960-th).

But it is impossible to use these results to study particularproblems!

Example: classication of binary forms

A binary form is a homogeneous polynomial on C2:

f(x, y) =

n∑k=0

pkxkyn−k,

where pk ∈ C.The space of all binary forms of degree n is denoted by Vn.The action of the group GL2(C) = SL2(C)h C∗:

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

When are two binary forms GL2(C)-equivalent?

f(x, y) =

n∑k=0

pkxkyn−k,

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

f(x, y) =

n∑k=0

pkxkyn−k,

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

f(x, y) =

n∑k=0

pkxkyn−k,

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

f(x, y) =

n∑k=0

pkxkyn−k,

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

f(x, y) =

n∑k=0

pkxkyn−k,

SL2(C) 3 A =

(a11 a12a21 a22

): f(x, y) 7−→ f(a22x−a12y, a11y−a21x),

C∗ 3 λ : f 7−→ λf.

Problem

Classical approach: invariant theory

In algebra we usually consider the action of the group SL2(C).To describe the orbits one can calculate the algebra of polynomial

invariants, i.e. SL2(C)-invariant polynomials I(p0, . . . , pn).Let An := C[Vn]SL2(C) be the invariant algebra.

n = 1: Trivial case:A1 = C.

n = 2: V2 = ax2 + 2bxy + cy2 quadrics;

A2 = C[b2 − ac].

Generator discriminant (= Hessian).

A2 = C[b2 − ac].

History, n = 3

n = 3: Bool, 1841 (debut of the classical invariant theory)

n = 4: Bool, Cayle, Eisinstine, 18401850 (cross-ratio,j-invariant)

n = 5: Hermite (1954): invariant I18 of degree 18, whichcontains more than 800 terms + syzygy

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

6 6 n 6 10, n = 12: Gordan, Shioda, Dixmier, Bedratuke,Brauer, Popovich (18602016).

There is no general approach in classical invariant theory!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

History, n = 3

I4I48+8I38I12−2I24I

28I12−72I4I8I

212−432I312+I

212−16I218 = 0!

New approach

Consider binary forms as solutions of the Euler equationxfx + yfy = nf .

Consider the action of group GL2(C) on this dierential equation!

Let us nd the invariants of this action.

New approach

Necessary denitions

(Jet space.) k-jet of function f in point a ∈ C2:

[f ]ka :=(a, f(a), fx(a), fy(a), fxx(a), fxy(a), fyy(a), . . .

k-jet space Jk := JkC2 with the canonical coordinates

(x, y, u, u10, u01, u20, u11, u02, . . . , uσ),

where uij([f ]ka

)= ∂i+jf

∂ix∂jy(a).

(Euler equation.) Euler dierential equation algebraicmanifold

E := x · u10 + y · u01 = n · u ⊂ J1.

E(k−1) ⊂ Jk prolongations.

Necessary denitions

(x, y, u, u10, u01, u20, u11, u02, . . . , uσ),

where uij([f ]ka

)= ∂i+jf

∂ix∂jy(a).

E := x · u10 + y · u01 = n · u ⊂ J1.

Necessary denitions

(x, y, u, u10, u01, u20, u11, u02, . . . , uσ),

where uij([f ]ka

)= ∂i+jf

∂ix∂jy(a).

E := x · u10 + y · u01 = n · u ⊂ J1.

Necessary denitions

(x, y, u, u10, u01, u20, u11, u02, . . . , uσ),

where uij([f ]ka

)= ∂i+jf

∂ix∂jy(a).

E := x · u10 + y · u01 = n · u ⊂ J1.

Necessary denitions

(x, y, u, u10, u01, u20, u11, u02, . . . , uσ),

where uij([f ]ka

)= ∂i+jf

∂ix∂jy(a).

E := x · u10 + y · u01 = n · u ⊂ J1.

Dierential invariants

Problem

We want to describe the orbits of the action GL2(C) : E(k−1) for

all k.

Dierential invariant of order k is a function

I ∈ C(E(k−1))GL2(C).

Invariant derivation is derivation

∇ = Ad

which commutes with the action of GL2(C).

If I is dierential invariant, then ∇I is also invariant ⇒ one can

get the innite number of invariants from the pair (I,∇).Pavel Bibikov Vrnjacka Banja2018

Problem

all k.

I ∈ C(E(k−1))GL2(C).

∇ = Ad

Problem

all k.

I ∈ C(E(k−1))GL2(C).

∇ = Ad

Algebra of dierential invariants

Function (Hessian)

H =u20u02 − u211

is dierential invariant of order 2.

Derivative

∇ =u01u

dx− u10

is invariant.

Theorem

Algebra of dierential invariants of the action of group GL2(C) is

freely generated by invariant H and derivation ∇.

Function (Hessian)

H =u20u02 − u211

Derivative

∇ =u01u

dx− u10

is invariant.

Theorem

Function (Hessian)

H =u20u02 − u211

Derivative

∇ =u01u

dx− u10

is invariant.

Theorem

Classication

Let f be a binary form. Consider the restrictions

H(f), ∇H(f), ∇2H(f).

They are homogeneous rational functions in variables x, y ⇒ thereis an algebraic dependence between them:

F (H(f),∇H(f),∇2H(f)) = 0.

Theorem

Binary forms f1, f2 of the same degree are GL2(C)-equivalent i

F1 ≡ F2.

Dependence F is a resultant of two polynomials ⇒ this theoremprovides an eective criterion for the equivalence of binary forms.

Classication

H(f), ∇H(f), ∇2H(f).

F (H(f),∇H(f),∇2H(f)) = 0.

Theorem

F1 ≡ F2.

Classication

H(f), ∇H(f), ∇2H(f).

F (H(f),∇H(f),∇2H(f)) = 0.

Theorem

F1 ≡ F2.

Classication

H(f), ∇H(f), ∇2H(f).

F (H(f),∇H(f),∇2H(f)) = 0.

Theorem

F1 ≡ F2.

Classication

H(f), ∇H(f), ∇2H(f).

F (H(f),∇H(f),∇2H(f)) = 0.

Theorem

F1 ≡ F2.

Generalization

Let G be a connected semi-simple complex Lie group, and let

ρλ : G→ GL(V )

be its irreducible representation with highest weight λ.

We want to apply theory of dierential invariants to study theaction G : V .

But where are the functions in this problem?

Main idea: BorelWeilBott theorem.

Generalization

ρλ : G→ GL(V )

Generalization

ρλ : G→ GL(V )

Generalization

ρλ : G→ GL(V )

Generalization

ρλ : G→ GL(V )

1 Let B ⊂ G be Borel group and M := G/B be homogeneouscomplex ag manifold.

2 Consider the action B : G by the right shifts: g 7→ gb−1, whereg ∈ G, b ∈ B.

3 Consider the bundle product E := G×B C = G×C/ ∼, where

(g, c) ∼ (gb−1, χλ(b)c),

and where χλ ∈ X(T ) is the character corresponding to thehighest weight λ of the maximal torus T ⊂ B.

Theorem (BWB)

Consider bundle πλ : E →M , πλ(g, c) = gB and the module

Γ(πλ) of its holomorphic sections. Then representation ρλ is

isomorphic to the action G : Γ(πλ) by left shifts.

(g, c) ∼ (gb−1, χλ(b)c),

Theorem (BWB)

(g, c) ∼ (gb−1, χλ(b)c),

Theorem (BWB)

(g, c) ∼ (gb−1, χλ(b)c),

Theorem (BWB)

(g, c) ∼ (gb−1, χλ(b)c),

Theorem (BWB)

Example: binary forms, G = SL2(C)

Weight λ = nα/2, where α is the positive root of the Liealgebra sl2(C) and n ∈ Z+.

Borel group B = B2(C) consists of upper triangular matrices.

Flag manifold M = SL2/B2 ' CP 1.

The character χλ acts on B in the following way:

(a b0 a−1

)= an.

If we denote the homogeneous coordinates on M by (x : y), thenthe holomorphic sections of bundle πλ are just the homogeneouspolynomials of degree n in variables x and y. Thus, the study ofinvariants of representations of group SL2(C) is reduced to theclassication SL2(C)-orbits of binary forms.

(a b0 a−1

)= an.

(a b0 a−1

)= an.

(a b0 a−1

)= an.

(a b0 a−1

)= an.

(a b0 a−1

)= an.

Using BWB-theorem, one can describe the algebra of dierentialinvariants for the action G : Γ(πλ) and obtain the equivalencecriterion.

Finally, for an arbitrary algebraic action G : Ω on the algebraicmanifold Ω there exists a linearization in the following sense.

According to Sumihiro's linearization theorem, each algebraicG-manifold Ω can be embedded into a G-invariant sub-manifold inan irreducible nite-dimensional G-module V .

Computing the invariants for the linear action G : V and restrictingthem on Ω, we get the invariant on Ω, which separate G-orbits.

This result closes the classical invariant theory.

THANK YOU FOR YOUR ATTENTION!

Di erential geometry and representations of semi-simple ...Di erential geometry and representations...

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