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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).
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).
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).
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).
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).
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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!
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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?
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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).
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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
34I
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!
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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
dx+B
d
dy,
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
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
dx+B
d
dy,
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
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
dx+B
d
dy,
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
Algebra of dierential invariants
Function (Hessian)
H =u20u02 − u211
u2
is dierential invariant of order 2.
Derivative
∇ =u01u
d
dx− u10
u
d
dy
is invariant.
Theorem
Algebra of dierential invariants of the action of group GL2(C) is
freely generated by invariant H and derivation ∇.
Pavel Bibikov Vrnjacka Banja2018
Algebra of dierential invariants
Function (Hessian)
H =u20u02 − u211
u2
is dierential invariant of order 2.
Derivative
∇ =u01u
d
dx− u10
u
d
dy
is invariant.
Theorem
Algebra of dierential invariants of the action of group GL2(C) is
freely generated by invariant H and derivation ∇.
Pavel Bibikov Vrnjacka Banja2018
Algebra of dierential invariants
Function (Hessian)
H =u20u02 − u211
u2
is dierential invariant of order 2.
Derivative
∇ =u01u
d
dx− u10
u
d
dy
is invariant.
Theorem
Algebra of dierential invariants of the action of group GL2(C) is
freely generated by invariant H and derivation ∇.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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:
χλ
(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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
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.
Pavel Bibikov Vrnjacka Banja2018
THANK YOU FOR YOUR ATTENTION!
Pavel Bibikov Vrnjacka Banja2018