Mixed perverse sheaves on flag varieties of Coxeter groups Cristian - PowerPoint PPT Presentation

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Mixed perverse sheaves on flag varieties of Coxeter groups Cristian Vay UNC–CONICET Argentina joint work with P. Achar and S. Riche Almería 2019 On the occasion of Blas 60th birthday Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Motivation 1 Elias-Williamson diagrammatic categories 2 Biequivariant Categories 3 Perverse sheaves 4 Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Coxeter system A finite set S and { m st } s,t ∈ S ∈ N ∪ {∞} such that m ss = 1 and m st = m ts if s � = t . Coxeter group W = � s ∈ S | ( st ) m st = 1 ∀ s, t ∈ S � Examples The symmetric group S n with S = { ( i i + 1) | 1 ≤ i < n } Weyl group of a finite-dimension semisimple Lie algebra Weyl group of an affine Lie algebra Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Hecke algebra H = Z [ v ± 1 ] � H s , s ∈ S | with the following relations � s = ( v − 1 − v ) H s + 1 H 2 and H s H t H s · · · = H t H s H t · · · � �� � � �� � m st m ts ∀ s, t ∈ S with s � = t . Let { H w } w ∈ W be the standard basis of H , H w = H s 1 · · · H s n for any reduced expression of w = s 1 · · · s n ∈ W . Let ( ) : H → H , be the Z -algebra involution induced by v �→ v − 1 H s �→ H − 1 and s Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Kazhdan, Lusztig. Representations of Coxeter groups and Hecke algebras, Invent. Math. (1979). Theorem There exists a unique basis { H w } w ∈ W of H such taht � H w = H w and H w = H w + h x,w H x , x<w with h x,w ∈ v Z [ v ] . Conjectures (actually theorems) The coefficients of h x,w are positives [Kazhdan-Lusztig for Weyl finite and afines groups]. ch L w = � x ≤ w ( − 1) ℓ ( x )+ ℓ ( w ) h x,w (1) ch M x , for a semisimple complex Lie algebra [Beilinson-Bernstein and Brylinsky-Kashiwara]. Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves P : the category of perverse sheaves P is the heart of a t -structura on D b ( G/B, C ) the bounded derived category of B -equivariant complexes (with complex coefficients) on the flag variety G/B of a Kac-Moody The simple objects are IC w , w in the Weyl group of G , the intersection cohomology complexes on the Schubert variety BwB/B . Categorification of the Hecke algebra ∼ � H [ P ] � H w [ IC w ] ✤ Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves R = S ( h ∗ ) with h the Lie algebra of the maximal torus of G . R s = the s -invariant subalgebra of R . S Bim : Soergel Bimodules is the essential image of the hypercohomology H • : P − → R -Bim , which is a fully faithful monoidal functor. Example H • ( IC e ) ≃ R H • ( IC s ) ≃ R ⊗ R s R (1) =: B s . y Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Let S Bim be the idempotent completion of the monoidal subcategory generated by B s , s ∈ S . Algebraic categorification of the Hecke algebra ∼ � H [ S Bim] � H s [ B s ] ✤ The indecomposable objects of S Bim are parametrized by W . Soergel Conjeture (actually theorem [Elias-Williamson]) Let B w be the indecomposable object attached to w ∈ W , then [ B w ] = H w [Soergel for Weyl and dihedral groups; Fiebig-Libedinsky universal Coxeter groups] Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Motivation 1 Elias-Williamson diagrammatic categories 2 Biequivariant Categories 3 Perverse sheaves 4 Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Let h be a realization of ( W, S ) and R = S ( h ∗ ) with gr h ∗ = 2 . Elias-Williamson diagrammatic categories D BS ( h , W ) Objects: B w , for any word w in S . Morphisms: k -graded modules generated by s s s s t s t t t s t s · · · · · · • • f · · · · · · s t s s s t s t s s s s m st even m st odd for any f ∈ R and s, t ∈ S , subject to certain relations Tensor product: B v ⋆ B w = B vw . k is a Noetherian integral domain of finite global dimension s. t. finitely generated proyective modules are free (for instance, k = Z ). Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Example of a relation • • . This implies that B s is self-dual. = = • • Also, it holds that B s ⋆ B s ∼ = B s (1) ⊕ B s ( − 1) . Definition D denotes the autoequivalence in D BS ( h , W ) given by flipping diagrams upside-down. (1) denotes the shift of grading. Elias, Williamson. Soergel Calculus. Represent. Theory (2016). Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Theorem [EW] Assume that k is a field or a complete local ring and let D ( h , W ) be the idempotent completion of D BS ( h , W ) . Then 1 The indescomposable objects are parametrized by W . 2 The assignment [ B s ] �→ H s induces an isomorphism [ D ( h , W )] → H of Z [ v ± 1 ] -algebras. 3 The Soergel conjecture [ B w ] = H w holds for k = R . Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Motivation 1 Elias-Williamson diagrammatic categories 2 Biequivariant Categories 3 Perverse sheaves 4 Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves I ⊂ W closed by the Bruhat order, � D ⊕ BS ,I ( h , W ) = � B w | w ∈ I reduced word � . I = I 0 \ I 1 locally closed, i.e. I 0 and I 1 closed, � D BS ,I ( h , W ) = D ⊕ / D ⊕ BS ,I 0 ( h , W ) / BS ,I 1 ( h , W ) Example D BS , { w } ( h , W ) ∼ = Free fg , Z ( R ) but D BS ,W ( h , W ) �∼ = D BS ( h , W ) Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves I ⊂ W locally closed subset Definition BE I ( h , W ) = K b D ⊕ BS ,I ( h , W ) Example BE { w } ( h , W ) ∼ = D b Mod fg , Z ( R ) and BE W ( h , W ) ∼ = K b D ⊕ BS ( h , W ) Almeria vay

� � � � � Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Recollement or Gluing Theorem J closed finite ⊂ I locally closed. Then there exists a recollement diagram ( i I ( i I I � J ) ! J ) ∗ � BE I ( h , W ) BE J ( h , W ) BE I � J ( h , W ) . ( i I ( i I J ) ∗ I � J ) ∗ J ) ! ( i I ( i I I � J ) ∗ and D interchanges ∗ and ! . Among other things, ∀ F ∈ BE I ( h , W ) there exist distinguished triangles +1 ( i I I � J ) ! ( i I I � J ) ∗ F − → ( i I J ) ∗ ( i I J ) ∗ F → F − − → +1 ( i I J ) ∗ ( i I J ) ! F − → ( i I I � J ) ∗ ( i I I � J ) ∗ F → F − − → Be˘ ılinson, Bernstein, Deligne. Faisceaux pervers, Astérisque (1982). Almeria vay

� Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Example: the singleton case Let w ∈ I minimal, i.e. { w } is closed in I , and x ∈ I � { w } . BE I ( h , W ) BE I � { w } ( h , W ) ( i I I � { w } ) ∗ ( i I I � { w } ) ∗ B x = f · · · → 0 → B w ⊗ R Hom • BS ,I ( h ,W ) ( B w , B x ) → B x → 0 → · · · , D ⊕ This is the cone of f and we have a distinguished triangle [1] I � { w } ) ∗ B x → B w ⊗ R Hom • B x → ( i I BS ,I ( h ,W ) ( B w , B x )[1] − → D ⊕ Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Example: the singleton case Let w ∈ W and s ∈ S such that ws > w . Then s Hom • BS , { w,ws } ( h ,W ) ( B w , B ws ) = R � id B w ⋆ • � D ⊕ and therefore � � [1] i { w,ws } B ws → ∗ B ws → B w � 1 � − → { ws } � � [1] i { w,ws } B w �− 1 � → ! B ws → B ws − → { ws } are distinguished triangles in BE { w,ws } ( h , W ) . Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves t -structure Definition The perverse t -structure in BE I ( h , W ) is defined by p BE I ( h , W ) ≤ 0 = � F | ∀ w ∈ I, ( i I w ) ∗ ( F ) ∈ p BE { w } ( h , W ) ≤ 0 � , p BE I ( h , W ) ≥ 0 = � F | ∀ w ∈ I, ( i I w ) ! ( F ) ∈ p BE { w } ( h , W ) ≥ 0 } . Definition The category of perverse objects is I ( h , W ) = p BE I ( h , W ) ≤ 0 ∩ p BE I ( h , W ) ≥ 0 , P BE the heart of the t -structure. Almeria vay

Motivation Elias-Williamson diagrammatic categories Biequivariant Categories Perverse sheaves Standard and costandard objects BS , { w } ( h , W ) ∼ b w is the canonical object in D ⊕ = Free fg , Z ( R ) . Definition ∆ I w = ( i I ∇ I w = ( i I w ) ! b w and w ) ∗ b w . 1 ∆ I w = ∇ I w = B w if w ∈ I is minimal. 2 ∆ I e = ∇ I e = B ∅ if e ∈ I . • 3 ∆ { e,s } = · · · 0 → B s − − → B ∅ (1) → 0 · · · s • 4 ∇ { e,s } = · · · 0 → B ∅ ( − 1) − − → B s → 0 · · · s 5 D (∆ I w ) = ∇ I w . 6 ∆ J w = ∆ I w if J ⊂ I . Almeria vay