On cyclic quiver parabolic Kostka-Shoji polynomials Daniel Orr* - PowerPoint PPT Presentation

On cyclic quiver parabolic Kostka-Shoji polynomials Daniel Orr* Mark Shimozono AMS Southeastern Sectional Meeting University of Florida November 3, 2019 1 / 11

Lusztig’s t -analog of weight multiplicity G complex reductive group X + ( G ) ⊂ X ( G ) (dominant) integral G -weights with respect to B ⊂ G Two bases of R ( G )[ t ] = ( Z [ t ] X ( G )) W ∼ = spherical AHA: (1) s λ = ch V ( λ ) (2) P µ = Macdonald spherical function (Hall-Littlewood polynomial) s λ = � µ ≤ λ K λµ ( t ) P µ Theorem (Lusztig) K λµ ( t ) ∈ Z ≥ 0 [ t ] for any λ, µ ∈ X + ( G ) . K λµ (1) = dim V ( λ ) µ 2 / 11

“Dual” approach [Broer, R. Brylinski, Hesselink] L µ = G × B − C µ line bundle on X = G/B − π : T ∗ X → X Definition (Hall-Littlewood series) χ µ = χ G × C × ( π ∗ L µ ) ∈ R ( G )[[ t ]] Theorem λ ≥ µ K λµ ( t ) s λ for any λ, µ ∈ X + ( G ) . χ µ = � Theorem [Broer] For µ ∈ X + ( G ) , one has H p ( T ∗ X, π ∗ L µ ) = 0 for all p > 0 . 3 / 11

Lascoux-Sch¨ utzenberger formula G = GL n For λ, µ ∈ X + pol ( G ) polynomial weights (partitions with at most n parts), the K λµ ( t ) are Kostka-Foulkes polynomials . Theorem [Lascoux-Sch¨ utzenberger] T t charge ( T ) where T runs over all semistandard Young K λµ ( t ) = � tableaux of shape λ and weight µ . 4 / 11

Quiver generalization [OS] Q = ( I, Ω) quiver i = ( i 1 , . . . , i ℓ ) ∈ I ℓ a = ( a 1 , . . . , a ℓ ) ∈ Z ℓ ≥ 0 d = d ( i , a ) = ( d i ) i ∈ I ∈ Z I ≥ 0 given by d i = � i k = i a k Definition [Lusztig] Z i , a = set of all pairs ( F • , x ) where x ∈ Rep d ( Q ) and ⊕ i ∈ I C d i = F 1 ⊃ F 2 ⊃ · · · ⊃ F ℓ ⊃ F ℓ +1 = 0 is a flag of I -graded subspaces such that x ( F k ) ⊂ F k +1 and F k /F k +1 has dimension a k at i k and zero elsewhere. π → X i , a is a G = � Z i , a i ∈ I GL d i -equivariant vector bundle over a product of partial flag varieties X i , a . 5 / 11

Quiver generalization [OS] (cont’d) µ = ( µ 1 , . . . , µ ℓ ) where µ k ∈ X ( GL a k ) for all k � G -equivariant vector bundle W µ on X i , a Definition [OS] Quiver Hall-Littlewood series χ i , a µ = χ G × C × ( Z i , a , π ∗ W µ ) ∈ R ( G )[[ t ]] Quiver Kostka-Shoji polynomials � � χ i , a � K i , a λ = ( λ i ) i ∈ I ∈ X + µ = λ, µ ( t ) s λ pol ( G ) λ 6 / 11

Special cases of quiver Kostka-Shoji polynomials Jordan quiver : (parabolic) Kostka-Foulkes polynomials Cyclic quiver (affine type A) : Kostka polynomials for complex reflection groups defined by Shoji (limit symbol case); intersection cohomology of enhanced nilpotent cone [Achar-Henderson] Directed path (type A) : truncated Littlewood-Richardson coefficients [W. Craig] 7 / 11

Higher vanishing conjecture Conjecture [OS] If µ concatenates to a dominant G -weight, then H p ( Z i , a , π ∗ W µ ) = 0 for all p > 0 . Known cases Jordan quiver, a k ≡ 1 [Broer] Cyclic quiver, a k ≡ 1 [Panyushev, Finkelberg-Ionov] Any quiver, sufficiently dominant µ [Panyushev] 8 / 11

Combinatorial positivity We say that combinatorial positivity holds for ( i , a , µ ) if K i , a λ, µ ( t ) ∈ Z ≥ 0 [ t ] for all λ ∈ X + ( G ) . For the Jordan quiver , combinatorial positivity is known when µ is a sequence of rectangles [Shimozono]. The parabolic Kostka polynomials count graded multiplicites in: Kirillov-Reshetikhin modules (twisted) functions on nilpotent orbit closures 9 / 11

Cyclic quiver, rectangles at a single vertex I = Z /r Z Ω = { ( i, i + 1) : i ∈ I }  0 · · · r − 2 r − 1 0 · · · r − 2 r − 1 i  a η 1 · · · η 1 η 1 η 2 · · · η 2 η 2 · · · ν η 1 ν η 2 0 · · · 0 0 · · · 0  µ 1 2 η = ( η 1 , . . . , η s ) arbitrary heights ν = ( ν 1 ≥ · · · ≥ ν s ) decreasing widths Theorem [OS] Combinatorial positivity holds for ( i , a , µ ) above. For any λ ∈ X + ( G ) , K i , a � t charge ( T ) λ, µ ( t ) = T where T = ( T i ) i ∈ I runs over “Littlewood-Richardson multitableaux” of shape λ = ( λ i ) i ∈ I . 10 / 11

Related results, observations, and conjectures 1 If the rectangles are columns , i.e., ν = (1 , . . . , 1) and η dominant: quiver Kostka-Shoji polynomials give irreducible multiplicites in graded induction from S n to Γ n = ( Z /r Z ) n ⋊ S n of Garsia-Procesi S n -module R η ( n = | η | ). 2 In the setting of (1), quiver Hall-Littlewood functions arise from q = 0 specialization of Haiman’s wreath Macdonald function H η ( q, t ) . 3 For cyclic quivers and any ( i , a , µ ) satisfying dominance, we conjecture a positive combinatorial “charge” formula for K i , a λ, µ ( t ) over “catabolizable tableaux.” 11 / 11