A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Outline

Prerequisites

Setup and basic questions

Kostant’s first problem

Kostant’s second problem

Source

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Prerequisites: for a broad understanding

I Commutative algebra: Module, ring, algebra, ideal

I Group theory: Group action, fixed point, isotropysubgroup, orbit

I Topology: Open, closed, dense, closure

I Generators and relations: Generating set, relationsamong generators, free objects, linear dependence

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Prerequisites: for a broad understanding

I Commutative algebra: Module, ring, algebra, ideal

I Group theory: Group action, fixed point, isotropysubgroup, orbit

I Topology: Open, closed, dense, closure

I Generators and relations: Generating set, relationsamong generators, free objects, linear dependence

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Prerequisites: for a broad understanding

I Commutative algebra: Module, ring, algebra, ideal

I Group theory: Group action, fixed point, isotropysubgroup, orbit

I Topology: Open, closed, dense, closure

I Generators and relations: Generating set, relationsamong generators, free objects, linear dependence

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Prerequisites: for a broad understanding

I Commutative algebra: Module, ring, algebra, ideal

I Group theory: Group action, fixed point, isotropysubgroup, orbit

I Topology: Open, closed, dense, closure

I Generators and relations: Generating set, relationsamong generators, free objects, linear dependence

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Nice to knows

I More ring theory: Tensor product, prime ideal, radicalideal

I Algebraic geometry: Zariski topology and Hilbert’snullstellensatz

I Lie theory: Lie group, Lie algebra, universal envelopingalgebra, differential operators

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Nice to knows

I More ring theory: Tensor product, prime ideal, radicalideal

I Algebraic geometry: Zariski topology and Hilbert’snullstellensatz

I Lie theory: Lie group, Lie algebra, universal envelopingalgebra, differential operators

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Nice to knows

I More ring theory: Tensor product, prime ideal, radicalideal

I Algebraic geometry: Zariski topology and Hilbert’snullstellensatz

I Lie theory: Lie group, Lie algebra, universal envelopingalgebra, differential operators

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Nice to knows

I More ring theory: Tensor product, prime ideal, radicalideal

I Algebraic geometry: Zariski topology and Hilbert’snullstellensatz

I Lie theory: Lie group, Lie algebra, universal envelopingalgebra, differential operators

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Goals of the talk

I want to:

I Give sufficient conditions for Kostant’s first and (if timepermits) second problem

I Get and give a flavour of invariant theory

I Gain experience in presenting material

I Show how commutative algebra, algebraic geometry,and Lie theory merges

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Goals of the talk

I want to:

I Give sufficient conditions for Kostant’s first and (if timepermits) second problem

I Get and give a flavour of invariant theory

I Gain experience in presenting material

I Show how commutative algebra, algebraic geometry,and Lie theory merges

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Goals of the talk

I want to:

I Give sufficient conditions for Kostant’s first and (if timepermits) second problem

I Get and give a flavour of invariant theory

I Gain experience in presenting material

I Show how commutative algebra, algebraic geometry,and Lie theory merges

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Goals of the talk

I want to:

I Give sufficient conditions for Kostant’s first and (if timepermits) second problem

I Get and give a flavour of invariant theory

I Gain experience in presenting material

I Show how commutative algebra, algebraic geometry,and Lie theory merges

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

A presentation on Kostant’s invariant theorypaper

Vipul Naik

February 21, 2007

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Outline

Prerequisites

Setup and basic questions

Kostant’s first problem

Kostant’s second problem

Source

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 1

Setup: R is a commutative unit ring

I R-algebra: Ring containing R as a subring

I R-module: Abelian group with R action

R algebra =⇒ R moduleQuestion: S is an R-algebra. Hence S is an R module. Is Sfree as an R-module?Remark: A module-theoretic generating set is very differentfrom an algebra-theoretic generating set.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 1

Setup: R is a commutative unit ring

I R-algebra: Ring containing R as a subring

I R-module: Abelian group with R action

R algebra =⇒ R module

Question: S is an R-algebra. Hence S is an R module. Is Sfree as an R-module?Remark: A module-theoretic generating set is very differentfrom an algebra-theoretic generating set.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 1

Setup: R is a commutative unit ring

I R-algebra: Ring containing R as a subring

I R-module: Abelian group with R action

R algebra =⇒ R moduleQuestion: S is an R-algebra. Hence S is an R module. Is Sfree as an R-module?

Remark: A module-theoretic generating set is very differentfrom an algebra-theoretic generating set.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 1

Setup: R is a commutative unit ring

I R-algebra: Ring containing R as a subring

I R-module: Abelian group with R action

R algebra =⇒ R moduleQuestion: S is an R-algebra. Hence S is an R module. Is Sfree as an R-module?Remark: A module-theoretic generating set is very differentfrom an algebra-theoretic generating set.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 2

I k is a field.

I S is a k-algebra.

I G acts as algebra automorphisms on S .

I J = SG is the invariant subring.

Question: S is a J-algebra. Is S free as a J module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 2

I k is a field.

I S is a k-algebra.

I G acts as algebra automorphisms on S .

I J = SG is the invariant subring.

Question: S is a J-algebra. Is S free as a J module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 2

I k is a field.

I S is a k-algebra.

I G acts as algebra automorphisms on S .

I J = SG is the invariant subring.

Question: S is a J-algebra. Is S free as a J module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 2

I k is a field.

I S is a k-algebra.

I G acts as algebra automorphisms on S .

I J = SG is the invariant subring.

Question: S is a J-algebra. Is S free as a J module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 2

I k is a field.

I S is a k-algebra.

I G acts as algebra automorphisms on S .

I J = SG is the invariant subring.

Question: S is a J-algebra. Is S free as a J module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 3

Setup:

I X a set, k a field

I S ≤ kX

I G acts on X , inducing algebra automorphisms on S

I J = SG

Question: Is S free as a J-module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 3

Setup:

I X a set, k a field

I S ≤ kX

I G acts on X , inducing algebra automorphisms on S

I J = SG

Question: Is S free as a J-module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 3

Setup:

I X a set, k a field

I S ≤ kX

I G acts on X , inducing algebra automorphisms on S

I J = SG

Question: Is S free as a J-module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 3

Setup:

I X a set, k a field

I S ≤ kX

I G acts on X , inducing algebra automorphisms on S

I J = SG

Question: Is S free as a J-module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 3

Setup:

I X a set, k a field

I S ≤ kX

I G acts on X , inducing algebra automorphisms on S

I J = SG

Question: Is S free as a J-module?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Exploration

A function is G -invariant iff it is constant on every orbitunder G action.

I IO is the ideal of functions vanishing on orbit O

I RO = IO + k is the ring of functions constant on O

I ⋂O

RO = J

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Exploration

A function is G -invariant iff it is constant on every orbitunder G action.

I IO is the ideal of functions vanishing on orbit O

I RO = IO + k is the ring of functions constant on O

I ⋂O

RO = J

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Exploration

A function is G -invariant iff it is constant on every orbitunder G action.

I IO is the ideal of functions vanishing on orbit O

I RO = IO + k is the ring of functions constant on O

I ⋂O

RO = J

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Exploration

A function is G -invariant iff it is constant on every orbitunder G action.

I IO is the ideal of functions vanishing on orbit O

I RO = IO + k is the ring of functions constant on O

I ⋂O

RO = J

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 4

Setup:

I X is a k-vector space, k a field

I S = k[X ] is the polynomial ring

I G ≤ GL(X ) and acts on S as degree-preservingautomorphisms.

Question: Is S a free module over J?This is Kostant’s First Problem and I will providesufficient conditions for a yes.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 4

Setup:

I X is a k-vector space, k a field

I S = k[X ] is the polynomial ring

I G ≤ GL(X ) and acts on S as degree-preservingautomorphisms.

Question: Is S a free module over J?This is Kostant’s First Problem and I will providesufficient conditions for a yes.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 4

Setup:

I X is a k-vector space, k a field

I S = k[X ] is the polynomial ring

I G ≤ GL(X ) and acts on S as degree-preservingautomorphisms.

Question: Is S a free module over J?This is Kostant’s First Problem and I will providesufficient conditions for a yes.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Setup and question – 4

Setup:

I X is a k-vector space, k a field

I S = k[X ] is the polynomial ring

I G ≤ GL(X ) and acts on S as degree-preservingautomorphisms.

Question: Is S a free module over J?This is Kostant’s First Problem and I will providesufficient conditions for a yes.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Steps towards the solution

Setup: S(Ox) is the ring of functions on Ox as restrictionsof elements of S .

Exploration: Free means absence of nontrivial linearrelations. We relate linear independence in two ways:

I Over J versus over k: Note that all k modules are freeas k is a field.

I On the whole space (that is, in S) versus restricted toan orbit (that is, in S(Ox))

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Steps towards the solution

Setup: S(Ox) is the ring of functions on Ox as restrictionsof elements of S .Exploration: Free means absence of nontrivial linearrelations. We relate linear independence in two ways:

I Over J versus over k: Note that all k modules are freeas k is a field.

I On the whole space (that is, in S) versus restricted toan orbit (that is, in S(Ox))

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Steps towards the solution

Setup: S(Ox) is the ring of functions on Ox as restrictionsof elements of S .Exploration: Free means absence of nontrivial linearrelations. We relate linear independence in two ways:

I Over J versus over k: Note that all k modules are freeas k is a field.

I On the whole space (that is, in S) versus restricted toan orbit (that is, in S(Ox))

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Steps towards the solution

Setup: S(Ox) is the ring of functions on Ox as restrictionsof elements of S .Exploration: Free means absence of nontrivial linearrelations. We relate linear independence in two ways:

I Over J versus over k: Note that all k modules are freeas k is a field.

I On the whole space (that is, in S) versus restricted toan orbit (that is, in S(Ox))

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Steps towards the solution

Setup: S(Ox) is the ring of functions on Ox as restrictionsof elements of S .Exploration: Free means absence of nontrivial linearrelations. We relate linear independence in two ways:

I Over J versus over k: Note that all k modules are freeas k is a field.

I On the whole space (that is, in S) versus restricted toan orbit (that is, in S(Ox))

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Statement of the major result

Setup:

I X = kn, S = k[X ], G ≤ GL(X ), J = SG

I J+ ≡ positive degree part of J

I H is a graded subspace complement to J+S

Basic result:JH = S

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Statement of the major result

Setup:

I X = kn, S = k[X ], G ≤ GL(X ), J = SG

I J+ ≡ positive degree part of J

I H is a graded subspace complement to J+S

Basic result:JH = S

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Statement of the major result

Setup:

I X = kn, S = k[X ], G ≤ GL(X ), J = SG

I J+ ≡ positive degree part of J

I H is a graded subspace complement to J+S

Basic result:JH = S

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Outline

Prerequisites

Setup and basic questions

Kostant’s first problem

Kostant’s second problem

Source

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Restatement of result – and steps to proof

Result (stated without proof): The following areequivalent.

1. J ⊗ H → S (given as f ⊗ g 7→ fg) is an isomorphism.

2. S is free over J.

3. Let M be a k submodule intersecting J+S trivially.Then for elements of M, k-linear independence equalsJ-linear independence.

Goal: To determine when equivalent condition (2) holds.Approach: Try to determine when equivalent condition (3)holds.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Restatement of result – and steps to proof

Result (stated without proof): The following areequivalent.

1. J ⊗ H → S (given as f ⊗ g 7→ fg) is an isomorphism.

2. S is free over J.

3. Let M be a k submodule intersecting J+S trivially.Then for elements of M, k-linear independence equalsJ-linear independence.

Goal: To determine when equivalent condition (2) holds.Approach: Try to determine when equivalent condition (3)holds.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Restatement of result – and steps to proof

Result (stated without proof): The following areequivalent.

1. J ⊗ H → S (given as f ⊗ g 7→ fg) is an isomorphism.

2. S is free over J.

3. Let M be a k submodule intersecting J+S trivially.Then for elements of M, k-linear independence equalsJ-linear independence.

Goal: To determine when equivalent condition (2) holds.Approach: Try to determine when equivalent condition (3)holds.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Restatement of result – and steps to proof

Result (stated without proof): The following areequivalent.

1. J ⊗ H → S (given as f ⊗ g 7→ fg) is an isomorphism.

2. S is free over J.

3. Let M be a k submodule intersecting J+S trivially.Then for elements of M, k-linear independence equalsJ-linear independence.

Goal: To determine when equivalent condition (2) holds.

Approach: Try to determine when equivalent condition (3)holds.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Restatement of result – and steps to proof

Result (stated without proof): The following areequivalent.

1. J ⊗ H → S (given as f ⊗ g 7→ fg) is an isomorphism.

2. S is free over J.

3. Let M be a k submodule intersecting J+S trivially.Then for elements of M, k-linear independence equalsJ-linear independence.

Goal: To determine when equivalent condition (2) holds.Approach: Try to determine when equivalent condition (3)holds.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Proof programme begins

Programme: We prove equivalent condition (3): If Mintersects J+S trivially, nontrivial J-linear relation givesnontrivial k-linear relation. We take the orbit route

1. nontrivial J-linear relation =⇒ nontrivial k-linearrelation restricted to orbit

2. nontrivial k-linear relation restricted to orbit =⇒nontrivial k-linear relation

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Naive proof idea for part (1)

Claim: nontrivial J-linear relation over S =⇒ nontrivial krelation on restriction to orbit

Naive Proof Idea: Map original relation via quotient mapto a k-linear relation.Bug: Nontrivial relation may restrict to trivial relation.Proper proof: Observe that the set of points at which therelation restricts to a trivial one is open. How? Look at thenext slide.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Naive proof idea for part (1)

Claim: nontrivial J-linear relation over S =⇒ nontrivial krelation on restriction to orbitNaive Proof Idea: Map original relation via quotient mapto a k-linear relation.

Bug: Nontrivial relation may restrict to trivial relation.Proper proof: Observe that the set of points at which therelation restricts to a trivial one is open. How? Look at thenext slide.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Naive proof idea for part (1)

Claim: nontrivial J-linear relation over S =⇒ nontrivial krelation on restriction to orbitNaive Proof Idea: Map original relation via quotient mapto a k-linear relation.Bug: Nontrivial relation may restrict to trivial relation.

Proper proof: Observe that the set of points at which therelation restricts to a trivial one is open. How? Look at thenext slide.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Naive proof idea for part (1)

Claim: nontrivial J-linear relation over S =⇒ nontrivial krelation on restriction to orbitNaive Proof Idea: Map original relation via quotient mapto a k-linear relation.Bug: Nontrivial relation may restrict to trivial relation.Proper proof: Observe that the set of points at which therelation restricts to a trivial one is open. How? Look at thenext slide.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (1)

Claim: Take a collection fi of functions. Locate the set ofpoints x such that fi |Ox are linearly independent. This set isopen.Proof Idea: Convert the linear independence of fi s on theorbit to linear independence of columns constructed at thepoint. This is the matrix D = dij with dij = pi .fj .Ingredient: The fact that a function being zero on thewhole orbit is captured by all derivatives being zero at onepoint. That is, complex analytic functions have Taylorexpansions.Limitation: In the form presented, this proof works onlyover C.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2)

Claim: Under certain conditions, a nontrivial k-linearrelation over an orbit =⇒ nontrivial k-linear relation overwhole space (X )

Ingredients:

I Nontrivial linear relation over subset =⇒ nontriviallinear relation over Zariski closure

I Function vanishes on the zero set of a “Galois closed”ideal =⇒ Function is in ideal

Special fact: When k is algebraically closed, a “radicalideal” is a “Galois-closed” ideal, courtesy Hilbert’snullstellensatz

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2)

Claim: Under certain conditions, a nontrivial k-linearrelation over an orbit =⇒ nontrivial k-linear relation overwhole space (X )Ingredients:

I Nontrivial linear relation over subset =⇒ nontriviallinear relation over Zariski closure

I Function vanishes on the zero set of a “Galois closed”ideal =⇒ Function is in ideal

Special fact: When k is algebraically closed, a “radicalideal” is a “Galois-closed” ideal, courtesy Hilbert’snullstellensatz

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2)

Claim: Under certain conditions, a nontrivial k-linearrelation over an orbit =⇒ nontrivial k-linear relation overwhole space (X )Ingredients:

I Nontrivial linear relation over subset =⇒ nontriviallinear relation over Zariski closure

I Function vanishes on the zero set of a “Galois closed”ideal =⇒ Function is in ideal

Special fact: When k is algebraically closed, a “radicalideal” is a “Galois-closed” ideal, courtesy Hilbert’snullstellensatz

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2)

Claim: Under certain conditions, a nontrivial k-linearrelation over an orbit =⇒ nontrivial k-linear relation overwhole space (X )Ingredients:

I Nontrivial linear relation over subset =⇒ nontriviallinear relation over Zariski closure

I Function vanishes on the zero set of a “Galois closed”ideal =⇒ Function is in ideal

Special fact: When k is algebraically closed, a “radicalideal” is a “Galois-closed” ideal, courtesy Hilbert’snullstellensatz

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2) (contd)

Setup: Let P be the Zariski cone of J+S .

Claim: If J+S is a radical ideal, k = C and O is a denseorbit in P, nontrivial k-linear relation in M over O =⇒nontrivial k-linear relation over X .Proof Idea: Apply the ingredients on the previous slide.Will become clearer when done on the blackboard.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2) (contd)

Setup: Let P be the Zariski cone of J+S .Claim: If J+S is a radical ideal, k = C and O is a denseorbit in P, nontrivial k-linear relation in M over O =⇒nontrivial k-linear relation over X .

Proof Idea: Apply the ingredients on the previous slide.Will become clearer when done on the blackboard.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof for part (2) (contd)

Setup: Let P be the Zariski cone of J+S .Claim: If J+S is a radical ideal, k = C and O is a denseorbit in P, nontrivial k-linear relation in M over O =⇒nontrivial k-linear relation over X .Proof Idea: Apply the ingredients on the previous slide.Will become clearer when done on the blackboard.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The exact claim

Summary: We have obtained sufficient conditions for S tobe free over J.

Overall claim: Let k = C, J+S be a radical ideal, and O bea dense orbit in P. Then, if M ∩ J+S = 0, J-linearindependence equals k-linear independence for elements inM.Ingredients:

I C is algebraically closed and J+S is a radical ideal =⇒J+S is Galois closed.

I The Lie algebra structure of C was used in transferringan orbit-related question to a point-related question.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The exact claim

Summary: We have obtained sufficient conditions for S tobe free over J.Overall claim: Let k = C, J+S be a radical ideal, and O bea dense orbit in P. Then, if M ∩ J+S = 0, J-linearindependence equals k-linear independence for elements inM.

Ingredients:

I C is algebraically closed and J+S is a radical ideal =⇒J+S is Galois closed.

I The Lie algebra structure of C was used in transferringan orbit-related question to a point-related question.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The exact claim

Summary: We have obtained sufficient conditions for S tobe free over J.Overall claim: Let k = C, J+S be a radical ideal, and O bea dense orbit in P. Then, if M ∩ J+S = 0, J-linearindependence equals k-linear independence for elements inM.Ingredients:

I C is algebraically closed and J+S is a radical ideal =⇒J+S is Galois closed.

I The Lie algebra structure of C was used in transferringan orbit-related question to a point-related question.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Outline

Prerequisites

Setup and basic questions

Kostant’s first problem

Kostant’s second problem

Source

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Statement of the problem

Setup: k, X , S , J, J+SQuestion: Let H be a module-theoretic complement to J+Sin S . How does the map γx : H → S(Ox) look?

Partial answer: It is surjective. It is an “isomorphism” incase any function in H vanishing on P is zero.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Statement of the problem

Setup: k, X , S , J, J+SQuestion: Let H be a module-theoretic complement to J+Sin S . How does the map γx : H → S(Ox) look?Partial answer: It is surjective. It is an “isomorphism” incase any function in H vanishing on P is zero.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Conditions for quasi-regularity

Recall: We already proved that if Ox is dense in P, the mapγx is injective.

Definition: If Ox is dense in P, x is termed regular(defined). Ifthe saturation of Ox under the C∗ action is dense in P, x istermed quasi-regular(defined).Question: Is γx an isomorphism for quasi-regular x (the wayit is for regular x)?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Conditions for quasi-regularity

Recall: We already proved that if Ox is dense in P, the mapγx is injective.Definition: If Ox is dense in P, x is termed regular(defined). Ifthe saturation of Ox under the C∗ action is dense in P, x istermed quasi-regular(defined).

Question: Is γx an isomorphism for quasi-regular x (the wayit is for regular x)?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Conditions for quasi-regularity

Recall: We already proved that if Ox is dense in P, the mapγx is injective.Definition: If Ox is dense in P, x is termed regular(defined). Ifthe saturation of Ox under the C∗ action is dense in P, x istermed quasi-regular(defined).Question: Is γx an isomorphism for quasi-regular x (the wayit is for regular x)?

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

The proof

Proof ingredients:

I The set of regular points is Euclidean nonempty open(nonvanishing minors) and the saturation of any Ox fora regular point x is dense. So they intersect.

I The set of x for which γx is isomorphism is closed underC∗ action.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Miscellaneous comments

I A general notion of Galois correspondence

I Other questions about finitely generated and aboutfree.

I A natural complement to J+S , namely the space ofharmonic polynomials.

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

Outline

Prerequisites

Setup and basic questions

Kostant’s first problem

Kostant’s second problem

Source

A presentation onKostant’s invariant

theory paper

Vipul Naik

Prerequisites

Setup and basicquestions

Kostant’s firstproblem

Kostant’s secondproblem

Source

This is based on the paper “Lie Group Representations ofPolynomial Rings” by Bertram Kostant. The paper isavailable via JSTOR to paid subscribers at:http://links.jstor.org/sici?sici=0002-9327(196307)85:3%3C327:LGROPR%3E2.0.CO%3B2-1This is a modified version of a presentation I have as part ofmy Visiting Students’ Research Programme at TIFR, underthe guidance of Professor Dipendra Prasad.