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˝ orincz Northeastern University Conference on Geometric Methods in Representation Theory, University of Missouri, Columbia, November 23-25, 2013 Andr´ as Cristian L˝ orincz The b-functions of quiver semi-invariants

b-functions and prehomogeneous spaces Semi-invariants of quivers b-functions via reflection functors Rational singularities Outline b-functions and prehomogeneous spaces 1 Semi-invariants of quivers 2 b-functions via reflection functors 3 Rational singularities 4 Andr´ as Cristian L˝ orincz 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 = � x 1 , . . . , x n , ∂ 1 , . . . , ∂ n � . D [ s ] := D ⊗ C C [ s ] . Andr´ as Cristian L˝ orincz 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 [ x 1 , . . . , x n ] 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˝ orincz 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 [ x 1 , . . . , x n ] 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 b f ( s ) . We call b f ( s ) the b -function (or Bernstein-Sato polynomial) of f . Andr´ as Cristian L˝ orincz 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 b f = ( s + 1) by ∂ x · x s +1 = ( s + 1) · x s Andr´ as Cristian L˝ orincz 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 b f = ( s + 1) by ∂ x · x s +1 = ( s + 1) · x s Example f = x 2 + y 3 , then P ( s ) = 1 x ∂ y + 1 y + s 1 4 ∂ x + 3 12 y ∂ 2 27 ∂ 3 8 ∂ 2 x b f ( s ) = ( s + 1)( s + 5 6)( s + 7 6) Andr´ as Cristian L˝ orincz 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 b f ( s ) are negative rational numbers. Andr´ as Cristian L˝ orincz 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 b f ( 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 b f ( s ) and has multiplicity 1 . We note that there is a more general result for reduced complete intersections. Andr´ as Cristian L˝ orincz 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 = ( x ij ) an n × n generic matrix of variables, and ∂ X is the ∂ matrix formed by the partial derivatives . ∂ x ij Take f = det X , then b f ( 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˝ orincz 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 = ( x ij ) an n × n generic matrix of variables, and ∂ X is the ∂ matrix formed by the partial derivatives . ∂ x ij Take f = det X , then b f ( 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˝ orincz 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˝ orincz 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˝ orincz 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˝ orincz 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 ) := GL( β x ) x ∈ Q 0 acting on the representation space � Hom( C β ta , C β ha ) . Rep( Q , β ) := a ∈ Q 1 For any two representations V and W , we have Ringel’s exact sequence: i � 0 → Hom Q ( V , W ) − → Hom ( V ( x ) , W ( x )) → x ∈ Q 0 d V p � W − → Hom( V ( ta ) , W ( ha )) − → Ext Q ( V , W ) → 0 a ∈ Q 1 Andr´ as Cristian L˝ orincz 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 dim V = α , we define the semi-invariant c V ∈ SI( Q , β ) � α, ·� by c V ( W ) := det d V 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 c V , with � dim( V ) , β � = 0 . Andr´ as Cristian L˝ orincz 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 O W is dense iff Ext Q ( W , W ) = 0. Then call β = dim W 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 V 1 , . . . , V k the simple objects of the category ⊥ W . Then SI( Q , β ) = C [ c V 1 , · · · , c V k ] . Andr´ as Cristian L˝ orincz 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 ∈ Q 0 sink (or source), we form a new quiver c x Q by reversing all arrows ending in x . Also define the map c x : Z n → Z n  β y if x � = y ,   c x ( β ) y = � − β x + β z if x = y .   edges x — z For an admissible ordering i 1 , . . . , i n of sinks, let c = c i n · · · c i 1 = − E − 1 E t be the Coxeter transformation, where E is the Euler matrix of Q . Andr´ as Cristian L˝ orincz The b-functions of quiver semi-invariants