The b-functions of quiver semi-invariants Andr as Cristian L - - PowerPoint PPT Presentation

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

The b-functions of quiver semi-invariants

Andr´ as Cristian L˝

• rincz

Northeastern University

Conference on Geometric Methods in Representation Theory, University of Missouri, Columbia, November 23-25, 2013

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Outline

1

b-functions and prehomogeneous spaces

2

Semi-invariants of quivers

3

b-functions via reflection functors

4

Rational singularities

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

We work over C. Let V be an n-dimensional vector space. Let D the algebra of differential operators on V , i.e. the Weyl algebra D = x1, . . . , xn, ∂1, . . . , ∂n. D[s] := D ⊗C C[s].

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem-Definition (J. Bernstein) Let f ∈ C[x1, . . . , xn] be a non-zero polynomial. Then there is a differential operator P(s) ∈ D[s] and non-zero polynomial b(s) ∈ C[s] such that P(s) · f s+1(x) = b(s) · f s(x)

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem-Definition (J. Bernstein) Let f ∈ C[x1, . . . , xn] be a non-zero polynomial. Then there is a differential operator P(s) ∈ D[s] and non-zero polynomial b(s) ∈ C[s] such that P(s) · f s+1(x) = b(s) · f s(x) The functions b(s) satisfying such a relation form an ideal of C[s], whose monic generator we denote by bf (s). We call bf (s) the b-function (or Bernstein-Sato polynomial) of f .

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Example f = x, then bf = (s + 1) by ∂x · xs+1 = (s + 1) · xs

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Example f = x, then bf = (s + 1) by ∂x · xs+1 = (s + 1) · xs Example f = x2 + y3, then P(s) = 1 12y∂2

x∂y + 1

27∂3

y + s 1

4∂x + 3 8∂2

x

bf (s) = (s + 1)(s + 5 6)(s + 7 6)

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem (M. Kashiwara) All roots of bf (s) are negative rational numbers.

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem (M. Kashiwara) All roots of bf (s) are negative rational numbers. Note that −1 is always a root. One of the various applications of b-functions: Theorem (M. Saito) Assume f is reduced. Then Z(f ) := f −1(0) has rational singularities iff −1 is the largest root of bf (s) and has multiplicity 1. We note that there is a more general result for reduced complete intersections.

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Example Take X = (xij) an n × n generic matrix of variables, and ∂X is the matrix formed by the partial derivatives ∂ ∂xij . Take f = det X, then bf (s) = (s + 1) · · · (s + n) by Cayley’s formula det ∂X · (det X)s+1 = (s + 1) · · · (s + n) · (det X)s

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Example Take X = (xij) an n × n generic matrix of variables, and ∂X is the matrix formed by the partial derivatives ∂ ∂xij . Take f = det X, then bf (s) = (s + 1) · · · (s + n) by Cayley’s formula det ∂X · (det X)s+1 = (s + 1) · · · (s + n) · (det X)s Reason: f ∈ C[V ] is a semi-invariant for a prehomogeneous vector space, i.e. there is an action of a reductive group G on V such that there is a dense orbit, and there is a character σ : G → C∗ s.t. g · f = σ(g) · f

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

In a prehomogeneous vector space there is at most one semi-invariant f (up to constant) for a fixed weight σ.

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

In a prehomogeneous vector space there is at most one semi-invariant f (up to constant) for a fixed weight σ. Let f ∗ ∈ C[V ∗] be the dual semi-invariant of weight σ−1 (which we view as a differential operator).

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

In a prehomogeneous vector space there is at most one semi-invariant f (up to constant) for a fixed weight σ. Let f ∗ ∈ C[V ∗] be the dual semi-invariant of weight σ−1 (which we view as a differential operator). Then the following equation comes for free: f ∗ · f s+1 = b(s) · f s. One can prove b(s) is a polynomial with deg b(s) = deg f , and b(s) is indeed the b-function of f (i.e. its minimal).

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Let Q be a quiver without oriented cycles, β a dimension vector, and consider the group GL(β) :=

• x∈Q0

GL(βx) acting on the representation space Rep(Q, β) :=

• a∈Q1

Hom(Cβta, Cβha). For any two representations V and W , we have Ringel’s exact sequence: 0 → HomQ(V , W )

i

− →

• x∈Q0

Hom(V (x), W (x)) →

dV

W

− →

• a∈Q1

Hom(V (ta), W (ha))

p

− → ExtQ(V , W ) → 0

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Take dimension vectors α, β, such that α, β = 0. For a representation V with dimV = α, we define the semi-invariant cV ∈ SI(Q, β)α,· by cV (W ) := det dV

W

Theorem (H. Derksen - J. Weyman, A. Schofield - M. Van den Bergh) The ring of semi-invariants SI(Q, β) is spanned by the semi-invariants cV , with dim(V ), β = 0.

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

An orbit OW is dense iff ExtQ(W , W ) = 0. Then call β = dimW a prehomogeneous dimension vector, and W generic representation (or partial tilting module). The left perpendicular category ⊥W of a generic rep. is equivalent to the category of representations of a quiver without oriented cycles. Theorem (A. Schofield) Let β be prehomogeneous and W the generic representation, and take V1, . . . , Vk the simple objects of the category ⊥W . Then SI(Q, β) = C[cV1, · · · , cVk].

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Definition (Reflection Functors) For x ∈ Q0 sink (or source), we form a new quiver cxQ by reversing all arrows ending in x. Also define the map cx : Zn → Zn cx(β)y =      βy if x = y, −βx +

• edges x—z

βz if x = y. For an admissible ordering i1, . . . , in of sinks, let c = cin · · · ci1 = −E −1E t be the Coxeter transformation, where E is the Euler matrix of Q.

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem (V. Kac) We have the isomorphisms of rings of semi-invariants: SI(Q, β) ∼ = SI(cxQ, cx(β)), when cx(β)x > 0, SI(Q, β) ∼ = SI(cxQ, cx(β)) ⊗ C[detβx], when cx(β)x = 0, SI(Q, β) ∼ = SI(Q, β − βxǫx), when cx(β)x < 0. These isomorphisms respect weight spaces: SI(Q, β)α,· ∼ = SI(cxQ, cx(β))cx(α),·

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem Let β be a prehomogeneous dimension vector, and f ∈ SI(Q, β)α,· a semi-invariant. Then the b-function satisfies the formula bf (s) = bc(f )(s)

• x∈Q0

[s]c(α)x

βx

[s]c(α)x

c(β)x

where c is the Coxeter transformation. Here we use the notation [s]d

a := a

• i=1

d−1

• j=0

(ds + i + j).

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem Let β be a prehomogeneous dimension vector, and f ∈ SI(Q, β)α,· a semi-invariant. Then the b-function satisfies the formula bf (s) = bc(f )(s)

• x∈Q0

[s]c(α)x

βx

[s]c(α)x

c(β)x

where c is the Coxeter transformation. Here we use the notation [s]d

a := a

• i=1

d−1

• j=0

(ds + i + j).

Further, for a ≤ b we introduce the notation [s]d

a,b := b

• i=b−a+1

d−1

• j=0

(ds + i + j).

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Example D4 : 2

• 1

4

3

• Take cV ∈ SI(Q, β), where V be the indecomposable of dimension

α = (1, 1, 1, 1). Then α, β = 0 gives β = (β1, β2, β3, β4) with β1 + β2 + β3 = 2β4. cV = det   X Iβ4 Y Iβ4 Z Iβ4  

1

β2

• 1

β1

• 1

β4

1

β3

• Andr´

as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities 1

β2

• 1

β1

• 1

β4

1

β3

• 1

β4 − β2

• 1

β4 − β1

• 2

β4

1

β4 − β3

• ∗
• ∗

1

∗

• Andr´

as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities 1

β2

• 1

β1

• 1

β4

1

β3

• 1

β4 − β2

• 1

β4 − β1

• 2

β4

1

β4 − β3

• ∗
• ∗

1

∗

• First, cV = 0 iff βi ≤ β4, for i = 1, 2, 3.

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities 1

β2

• 1

β1

• 1

β4

1

β3

• 1

β4 − β2

• 1

β4 − β1

• 2

β4

1

β4 − β3

• ∗
• ∗

1

∗

• First, cV = 0 iff βi ≤ β4, for i = 1, 2, 3.

Using the inequalities, we can write b(s) as a polynomial b(s) = [s]β4 · [s]β4−β1,β2 · [s]β4−β2,β3 · [s]β4−β3,β1

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

E6 :

2

β5

• 1

β1

• 2

β2

• 3

β6

2

β4

• 1

β3

• with β1 + β2 + β3 + β4 + 2β5 = 3β6 and inequalities

β6 ≤ β4 + β5, β2 + β5, β2 + β4, β1 + β3 + β5 β6 ≥ β1 + β5, β3 + β5.

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

E6 :

2

β5

• 1

β1

• 2

β2

• 3

β6

2

β4

• 1

β3

• with β1 + β2 + β3 + β4 + 2β5 = 3β6 and inequalities

β6 ≤ β4 + β5, β2 + β5, β2 + β4, β1 + β3 + β5 β6 ≥ β1 + β5, β3 + β5. The b-function is b(s) = [s]β1,β1+β4+β5−β6[s]β3,β2+β3+β5−β6[s]2

β6−β5−β3,β2[s]2 β6−β5−β1,β4·

·[s]3

β1+β3+β5−β6,β6[s]β6−β5−β3,β1[s]β1+β3+β5−β6,β3[s]β3,β6−β5·

·[s]2

β6−β3−β1,β2+β4+β5−β6[s]β5

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem Let Q be a Dynkin or Euclidean quiver, and β a prehomogeneous dimension vector. Then all semi-invariants in SI(Q, β) are reducible by reflections to constant functions (hence we can compute their b-functions).

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Theorem Let Q be a Dynkin or Euclidean quiver, and β a prehomogeneous dimension vector. Then all semi-invariants in SI(Q, β) are reducible by reflections to constant functions (hence we can compute their b-functions). Lemma Let Q be a Dynkin quiver, V a sincere generic representation. Then any simple object S in the perpendicular category ⊥V (or V ⊥) satisfies dim S ≤ dim V

Andr´ as Cristian L˝

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The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Corollary Let Q be a Dynkin quiver, then any codimension 1 orbit closure in Rep(Q, β) (i.e. zero set of an irreducible semi-invariant) has rational singularities (in particular, is normal). This is true for Euclidean quivers if the prehomogeneous β is large enough.

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Corollary Let Q be a Dynkin quiver, then any codimension 1 orbit closure in Rep(Q, β) (i.e. zero set of an irreducible semi-invariant) has rational singularities (in particular, is normal). This is true for Euclidean quivers if the prehomogeneous β is large enough. Example

Let Q be the Kronecker quiver 1 2 β = (k · n, k · (n + 1)), α = (n + 1, n + 2) where k, n are arbitrary positive integers. The b-function is b(s) =

n

• i=1

[s]i

2k,(i+1)k

The zero-set has rational singularities iff k > 1.

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Some things to do: Extend arguments of rational singularities to zerosets of more semi-invariants, in particular the null-cone (partially done)

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Some things to do: Extend arguments of rational singularities to zerosets of more semi-invariants, in particular the null-cone (partially done) Investigate the combinatorics behind b(s), in particular the reduction to a polynomial of b(s)

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Some things to do: Extend arguments of rational singularities to zerosets of more semi-invariants, in particular the null-cone (partially done) Investigate the combinatorics behind b(s), in particular the reduction to a polynomial of b(s) Find other techniques that work in wild cases, and other reductive groups (the are already quite a few)

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities

Thank you!

Andr´ as Cristian L˝

• rincz

The b-functions of quiver semi-invariants