The modular Weyl-Kac character formula
TTHE MODULAR WEYL–KAC CHARACTER FORMULA
CHRIS BOWMAN, AMIT HAZI, AND EMILY NORTON
Abstract.
We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated in one degree. Each of these homogeneousrepresentations is one-dimensional and can be cohomologically constructed via a BGG resolu-tion involving every (infinite dimensional) standard representation of the category. We hencedetermine the complete first row of the inverse parabolic p -Kazhdan–Lusztig matrix for anarbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl–Kaccharacter formula to all Coxeter systems (and their parabolics) and proves that this generalisedformula is rigid with respect to base change to an arbitrary field. Introduction
The discovery of counterexamples to the expected bounds of Lusztig’s conjecture was anearthquake in representation theory. It marked the beginning of a new era of Lie theory,in which diagrammatic Hecke categories play centre stage in our attempts to understand thestructure of algebraic groups in terms of parabolic “ p -Kazhdan–Lusztig polynomials”. There areprecious few general results concerning either the simple representations of these diagrammaticHecke categories, or the underlying combinatorics of parabolic p -Kazhdan–Lusztig polynomials.Given W an arbitrary Coxeter group and P an arbitrary parabolic subgroup, we classify andconstruct the homogeneous (simple) representations of the anti-spherical Hecke category H P \ W (note that H P \ W is a k -linear graded category and so this notion makes sense). We prove thata H P \ W -module is homogeneous if and only if it is one-dimensional if and only if it is the simple L (1 P \ W ) labelled by the identity coset 1 P \ W ∈ P W (and we provide a basis of L (1 P \ W ) by wayof Libedinsky’s light-leaves construction). Theorem A.
The anti-spherical Hecke category H P \ W admits a unique homogeneous simplemodule, L (1 P \ W ) , labelled by the identity element of the Hecke algebra. This simple module isa one-dimensional quotient of the infinite-dimensional standard H P \ W -module ∆(1 P \ W ) . Concurrently, we provide a cohomological construction of the unique homogeneous H P \ W -module by way of a BGG resolution. Within this BGG resolution, every one of the (infinite-dimensional) standard H P \ W -representations ∆( w ) for w ∈ P W appears in degree as dictatedby the length function on the underlying Hecke algebra. Our BGG resolutions allow us tocalculate the complete first row of the (inverse) parabolic p -Kazhdan–Lusztig matrix for W an arbitrary Coxeter group and P an arbitrary parabolic subgroup. This provides the firstfamily of explicit (inverse) p -Kazhdan–Lusztig polynomials to admit a uniform description acrossarbitrary Coxeter groups and their parabolic subgroups. In the case that W is an affine Weylgroup and P is the maximal finite parabolic subgroup this gives new character formulas forrepresentations of the corresponding algebraic groups through [AJS94, AMRW19]. Theorem B.
Associated to the unique homogeneous simple H P \ W -module, L (1 P \ W ) , we havea complex C • (1 P \ W ) = (cid:76) w ∈ P W ∆( w ) (cid:104) (cid:96) ( w ) (cid:105) with differential given by an alternating sum overall “simple reflection homomorphisms”. This complex is exact except in degree zero, where H ( C • (1 P \ W )) = L (1 P \ W ) . We hence vastly generalise the Weyl–Kac character formula to allCoxeter systems (and their parabolics) and to arbitrary fields via the formula [ L (1 P \ W )] = (cid:88) w ∈ P W ( − v ) (cid:96) ( w ) [∆( w )] and thus conclude that the first row of the inverse parabolic p -Kazhdan–Lusztig matrix hasentries ( − v ) (cid:96) ( w ) regardless of the characteristic p (cid:62) . a r X i v : . [ m a t h . R T ] J un CHRIS BOWMAN, AMIT HAZI, AND EMILY NORTON
Specialising to the case of (affine) Weyl groups, our character formulas and resolutions have along history. For finite Weyl groups, Bernstein–Gelfand–Gelfand constructed their eponymousresolutions in the context of finite dimensional Lie algebras [BGG75]. For Kac–Moody Liealgebras these were the subject of Kac–Kazhdan’s conjecture [KK79] (over C ) which was verifiedby Wakimoto (for W = (cid:98) S [Wak86]), Hayashi (for classical type [Hay88]) and Feigin, Frenkel,and Ku (in full generality [Ku89, FF92]) and was extended to arbitrary fields by Mathieu[Mat96] and subsequently reproven by Arakawa using W -algebras [Ara06, Ara07]. For parabolicsubgroups of finite Weyl groups, our resolutions were first constructed in [Lep77] and went onto have spectacular applications in the study of the Laplacian on Euclidean space [Eas05]. Forthe infinite dihedral Weyl group with two generators, these resolutions were generalised to theVirasoro and blob algebras of algebraic statistical mechanics [Fel89, MS94, GJSV13]. For W thefinite symmetric group and P a maximal parabolic, these resolutions were one of the highlightsof Brundan–Stroppel’s founding work on categorical representation theory [BS10, BS11]. For W the affine symmetric group and P the maximal finite parabolic and k = C , Theorem B provesBerkesch–Griffeth–Sam’s conjecture [BGS14] (recently verified by the first and third authors andJos´e Simental, [BNS]) concerning BGG resolutions of unitary modules for Cherednik algebras.Kazhdan–Lusztig conjectured that much of combinatorial Lie theory should generalise be-yond the realm of Weyl groups (where our resolutions admit the aforementioned geometricrealisations) to arbitrary Coxeter groups. Hecke categories provide the structural perspective inwhich the Kazhdan–Lusztig conjecture was finally proven [EW14] and serve as the archetypalsetting for studying all Lie theoretic objects. In this light, Theorem B provides the prototype forall these BGG resolutions — the instances for Lie groups, Kac–Moody Lie algebras, and theirparabolic analogues are simply the examples for which a classical geometric structure exists. Acknowledgements.
We would like to thank George Lusztig for his comments on a previousversion of this manuscript. The diagrammatic Hecke categories
We begin by recalling the basics of diagrammatic Hecke categories. Almost everything fromthis section is lifted from Elias–Williamson’s original paper [EW16] or is an extension of theirresults to the parabolic setting [LW]. In this paper, we suppress mention of category theoreticterminology because it jars slightly with our representation theoretic aims. Instead of thedefinition of the Hecke category via objects and morphisms, we favour the language of Brundan–Stroppel’s “locally unital associative algebras”, see [BS18, Remark 2.3] for more details.1.1.
Coxeter systems.
Let (
W, S ) be a Coxeter system: W is the group generated by thefinite set S subject to the relations ( στ ) m στ = 1 for m στ ∈ N ∪ {∞} satisfying m στ = m τσ andthat m στ = 1 if and only if σ = τ ∈ S . Let (cid:96) : W → N be the corresponding length function.Let L = Z [ v, v − ] be the ring of Laurent polynomials with integer coefficients in one variable v .Consider S P ⊆ S an arbitrary subset and ( P, S P ) its corresponding Coxeter group. We saythat ( P, S P ) is the parabolic subgroup corresponding to the set S P ⊆ S . We denote by P W ⊆ W the set of minimal coset representatives in P \ W .We define a subword of w = σ σ · · · σ (cid:96) to be a sequence t = ( t , t , . . . , t (cid:96) ) ∈ { , } (cid:96) and we set w t := σ t σ t · · · σ t (cid:96) (cid:96) and we emphasise that s i = 1 ∅ ∈ W . We let (cid:54) denote the (strong) Bruhatorder on W . Namely y (cid:54) w if for some (or equivalently, every) reduced expression w thereexists a subword t and a reduced expression y such that w t = y . We let exp (cid:96)P ( w ) denote the setof all expressions w = σ σ · · · σ (cid:96) of w of length (cid:96) such that σ . . . σ k ∈ P W for each 1 (cid:54) k (cid:54) (cid:96) ,we let exp P ( w ) = ∪ (cid:96) (cid:62) exp (cid:96)P ( w ), and exp (cid:96)P = ∪ w ∈ W exp (cid:96)P ( w ). We set rexp P ( w ) := exp (cid:96) ( w ) P ( w ).1.2. Bi-coloured quantum numbers and Cartan matrices.
We define the the x - and y -bicoloured quantum numbers as follows. First set[0] x = [0] y = 0 [1] x = [1] y = 1 [2] x = x [2] y = y HE MODULAR WEYL–KAC CHARACTER FORMULA 3 and then inductively define[2] x [ k ] y = [ k + 1] x + [ k − x [2] y [ k ] x = [ k + 1] y + [ k − y . When k is odd, [ k ] x = [ k ] y . The following definition allows one to speak of Cartan matrices ofCoxeter groups. Definition 1.1.
Let k be a commutative ring. A balanced realisation of ( W, S ) over k is a free,finite rank k -module h , together with subsets { α ∨ σ | σ ∈ S } ⊂ h and { α σ | σ ∈ S } ⊂ h ∗ =Hom k ( h , k ) , satisfying:(1) (cid:104) α ∨ σ , α σ (cid:105) = 2 for all σ ∈ S ;(2) the assignment σ ( v ) = v − (cid:104) v, α σ (cid:105) α ∨ σ for all v ∈ h yields a representation of W ;(3) Set x = −(cid:104) α ∨ σ , α τ (cid:105) and y = −(cid:104) α ∨ τ , α σ (cid:105) . For every σ, τ ∈ S , we require that [ m στ ] x = [ m στ ] y = 0 [ m στ − x = [ m στ − y = 1 We define the associated
Cartan matrix to be the | S | × | S | -matrix ( (cid:104) α ∨ σ , α τ (cid:105) ) σ,τ ∈ S . Soergel graphs.
Let (
W, S ) denote an arbitrary Coxeter system with S finite. Given σ ∈ S we define the monochrome Soergel generators to be the framed graphs ∅∅ = σσ = spot ∅ σ = fork σσσ =and given any σ, τ ∈ S with m στ = m we have the bi-chrome generatorbraid τσστ ( m ) = . . .. . . braid τσστ ( m ) = . . .. . . for m odd, or even respectively. Here the northern edges are coloured with the sequence τ στ σ . . . στ (cid:124) (cid:123)(cid:122) (cid:125) m times τ στ σ . . . τ σ (cid:124) (cid:123)(cid:122) (cid:125) m times for m odd or even respectively and the southern edges are coloured στ στ . . . τ σ (cid:124) (cid:123)(cid:122) (cid:125) m times στ στ . . . στ (cid:124) (cid:123)(cid:122) (cid:125) m times for m odd or even respectively. We define the northern/southern reading word of a Soergelgenerator (or its dual) to be word in the alphabet S obtained by reading the colours of thenorthern/southern edge of the frame respectively (ignoring any ∅ symbols). Pictorially, we definethe duals of these generators to be the graphs obtained by reflection through their horizontalaxes. Non-pictorially, we simply swap the sub- and superscripts. We sometimes denote dualityby ∗ . For example, the dual of the fork generator is pictured as follows fork σσσ = . Given any two (dual) Soergel generators D and D (cid:48) we define D ⊗ D (cid:48) to be the diagram obtainedby horizontal concatenation (and we extend this linearly). The northern/southern colour se-quence of D ⊗ D (cid:48) is the concatenation of those of D and D (cid:48) ordered from left to right. Given anytwo (dual) Soergel generators, we define their product D ◦ D (cid:48) to be the vertical concatenationof D on top of D (cid:48) if the southern reading word of D is equal to the northern reading word of D (cid:48) and to be zero otherwise. Finally, we define a Soergel graph to be any graph obtained byrepeated horizontal and vertical concatenation of the Soergel generators and their duals.
CHRIS BOWMAN, AMIT HAZI, AND EMILY NORTON
Some specific graphs.
For w = σ . . . σ (cid:96) an expression, we define w = σ ⊗ σ ⊗ · · · ⊗ σ (cid:96) and given k > σ, τ ∈ S we set kστ = σ ⊗ τ ⊗ σ ⊗ τ . . . to be the alternately coloured idempotent on k strands (so that the final strand is σ - or τ -coloured if k is odd or even respectively). Given σ, τ ∈ S with m στ = m even, let w = ρ · · · ρ k ( στ · · · στ ) ρ m + k +1 · · · ρ (cid:96) w = ρ · · · ρ k ( τ σ · · · τ σ ) ρ m + k +1 · · · ρ (cid:96) be two reduced expressions for w ∈ W . We say that w and w are adjacent and we set braid ww = ρ ⊗ · · · ⊗ ρ k ⊗ braid σττσ ( m ) ⊗ ρ m + k +1 ⊗ · · · ⊗ ρ (cid:96) (similarly for m odd). Now, given a sequence of adjacent reduced expressions, w = w (1) , w (2) , . . . , w ( q ) = w and the value q is minimal such that this sequence exists, then we set braid ww = (cid:89) (cid:54) p The diagrammatic Hecke categories. Let ( W, S ) be a Coxeter system with Cartanmatrix ( (cid:104) α ∨ σ , α τ (cid:105) ) σ,τ ∈ S . Set x = −(cid:104) α ∨ σ , α τ (cid:105) and y = −(cid:104) α ∨ τ , α σ (cid:105) . In order to save space, we set jw στσσσσ = ( σ ⊗ spot τ ∅ ⊗ σ )( fork σσσ ⊗ σ ) wj σσσστσ = ( σ ⊗ fork σσσ )( σ ⊗ spot ∅ τ ⊗ σ )We are now ready to inductively define the Jones–Wenzl projectors JW kστ and JW k +1 στ as follows JW k +1 στ = JW kστ ⊗ σ + [2 k − x [2 k ] x ( JW kστ ⊗ σ )( k − στ ⊗ jw στσσσσ )( JW k − στ ⊗ σσ )( k − στ ⊗ wj σσσστσ )( JW kστ ⊗ σ )and JW kστ = JW k − στ ⊗ τ + [2 k − x [2 k − x ( JW k − στ ⊗ τ )( k − στ ⊗ jw τσττττ )( JW k − στ ⊗ ττ )( k − στ ⊗ wj ττττστ )( JW k − στ ⊗ τ ) . We remark that in each case the leftmost strand is coloured with σ and the second term hascoefficient equal to an x -bicoloured quantum integers. The pictorial version of the first recursion(for 2 k + 1 odd) is given in Figure 1................. JW k +1 στ = ................ JW kστ + [2 k − x [2 k ] x ................ JW kστ JW k − στ JW kστ Figure 1. The Jones–Wenzl projector JW k +1 στ The elements JW kτσ and JW k +1 τσ are the same as the above except with the inverted colourpattern and coefficients equal to y -bicoloured quantum integers. Specifically, we have that JW k +1 τσ = JW kτσ ⊗ τ + [2 k − y [2 k ] y ( JW kτσ ⊗ τ )( k − τσ ⊗ jw τσττττ )( JW k − τσ ⊗ ττ )( k − τσ ⊗ wj ττττστ )( JW kτσ ⊗ τ )and JW kτσ = JW k − τσ ⊗ σ + [2 k − y [2 k − y ( JW k − τσ ⊗ σ )( k − τσ ⊗ jw στσσσσ )( JW k − τσ ⊗ σσ )( k − τσ ⊗ wj σσσστσ )( JW k − τσ ⊗ σ ) . HE MODULAR WEYL–KAC CHARACTER FORMULA 5 Definition 1.2. Let ( W, S ) be a Coxeter system. We define H W to be the locally-unital associa-tive Z -algebra spanned by all Soergel-graphs with multiplication given by vertical concatenationof diagrams modulo the following local relations and their duals. σ τ = δ σ,τ σ ∅ σ = 0 ∅ = ∅ ∅ spot ∅ σ σ = spot ∅ σ σ fork σσσ σσ = fork σσσ mτσ braid τσστ ( m ) mστ = braid τσστ ( m ) For each σ ∈ S we have monochrome relations ( spot ∅ σ ⊗ σ ) fork σσσ = σ ( σ ⊗ fork σσσ )( fork σσσ ⊗ σ ) = fork σσσ fork σσσ fork σσσ fork σσσ = 0 ( spot ∅ σ spot σ ∅ ) ⊗ σ + σ ⊗ ( spot ∅ σ spot σ ∅ ) = 2( spot σ ∅ spot ∅ σ ) For every ordered pair ( σ, τ ) ∈ S with σ (cid:54) = τ , the bi-chrome relations: The two-colour barbell, ( spot ∅ τ spot τ ∅ ) ⊗ σ − σ ⊗ ( spot ∅ τ spot τ ∅ ) = (cid:104) α ∨ σ , α τ (cid:105) (( spot σ ∅ spot ∅ σ ) − σ ⊗ ( spot ∅ σ spot σ ∅ )) . If m = m στ < ∞ we also have the fork-braid relations braid τσ ··· στστ ··· τσ ( fork σσσ ⊗ m − τσ )( σ ⊗ braid στ ··· τστσ ··· στ ) = ( m − τσ ⊗ fork τττ )( braid τσ ··· στστ ··· τσ ⊗ τ ) braid τσ ··· τσστ ··· στ ( fork σσσ ⊗ m − τσ )( σ ⊗ braid στ ··· σττσ ··· τσ ) = ( m − τσ ⊗ fork σσσ )( braid τσ ··· τσστ ··· στ ⊗ σ ) for m odd and even, respectively. We require the cyclicity relation, ( mτσ ⊗ ( spot ∅ σ fork σσσ ))( τ ⊗ braid σττσ ( m ) ⊗ σ )(( fork τττ spot τ ∅ ) ⊗ mστ ) = braid τσ...στστ ··· τσ ( mτσ ⊗ ( spot ∅ τ fork τττ ))( τ ⊗ braid σττσ ( m ) ⊗ τ )(( fork τττ spot τ ∅ ) ⊗ mστ ) = braid τσ...τσστ ··· στ . for m odd or even, respectively. We have the Jones–Wenzl relations ( m − τσ ⊗ spot ∅ τ ) braid τσστ ( m ) = ( spot ∅ σ ⊗ m − τσ ) JW mστ ( m − τσ ⊗ spot ∅ σ ) braid τσστ ( m ) = ( spot ∅ σ ⊗ m − τσ ) JW mστ for m odd or even, respectively. For ( σ, τ , ρ ) ∈ S with m σρ = m ρτ = 2 and m στ = m , we have ( braid τσστ ( m ) ⊗ ρ ) braid στ ··· σρρστ ··· σ = braid τσ ··· τρρτσ ··· τ ( ρ ⊗ braid τσστ ( m )) . We have the three Zamolodchikov relations: for a type A triple σ, τ , ρ ∈ S with m στ = 3 = m σρ and m σρ = 2 we have that braid ρσρτσρσρστσρ braid σρστσρσρτστρ braid σρτστρστρσρτ braid στρσρτστσρστ braid στσρσττστρστ braid τστρσττσρτστ = braid ρσρτσρρστρσρ braid ρστρσρρστσρσ braid ρστσρσρτστρσ braid ρτστρστρσρτσ braid τρσρτστσρστσ braid τσρστστσρτστ . For a type B triple σ, τ , ρ ∈ S such that m σρ = 4 , m τρ = 2 , m στ = 3 , we have that braid ρσρτστρστρσρστσρστ braid ρσρστσρστσρσρτσρστ braid σρσρτσρστσρστρσρστ braid σρστρσρστσρστσρσρτ braid σρστσρσρτσρτστρσρτ braid σρτστρσρτστρσρτστρ × braid στρσρτστρστρσρστσρ braid στρσρστσρστσρσρτσρ braid στσρσρτσρτστρσρτσρ braid τστρσρτσρτσρτστρσρ = braid ρσρτστρστρσρτσρτστ braid ρσρτσρτστρστρσρστσ braid ρστρσρστσρστσρσρτσ braid ρστσρσρτσρτστρσρτσ braid ρτστρσρτστρσρτστρσ braid τρσρτστρστρσρστσρσ × braid τρσρστσρστσρσρτσρσ braid τσρσρτσρστσρστρσρσ braid τσρστρσρστσρστσρσρ braid τσρστσρσρτσρτστρσρ and for a type H triple σ, τ , ρ ∈ S such that m σρ = 2 , m τρ = 5 , m στ = 3 , we have a final H relation , for we refer to [EW16, Definition 5.2] . Further, we require the bifunctoriality relation (cid:0) ( D ◦ x ) ⊗ ( D ◦ y ) (cid:1)(cid:0) ( x ◦ D ) ⊗ ( y ◦ D ) (cid:1) = ( D ◦ x ◦ D ) ⊗ ( D ◦ y ◦ D ) and the monoidal unit relation ∅ ⊗ D = D = D ⊗ ∅ for all diagrams D , D , D , D and all words x, y . Finally, we require the (non-local) cyclotomicrelation spot ∅ σ spot σ ∅ ⊗ w = 0 for all w ∈ exp ( w ) , w ∈ W, and all σ ∈ S . To the authors’ knowledge, this relation has not been explicitly determined (but can be given more computingpower). We invite the reader to either believe that this can be written down (as is now standard in this area)or to read all results in this paper “modulo” any Coxeter group W with a parabolic subgroup of type H . CHRIS BOWMAN, AMIT HAZI, AND EMILY NORTON Remark 1.3. The cyclotomic relation amounts to considering Soergel modules instead of Soergelbimodules, or equivalently, to considering finite dimensional k -modules rather than modules offinite rank over the polynomial ring generated by the “barbells”, spot ∅ σ spot σ ∅ , for σ ∈ S . = = + Figure 2. The fork-braid and Jones–Wenzl relations for m στ = 3.= Figure 3. The Zamolodchikov relation for A . Definition 1.4. Given S P ⊆ S we define H P \ W to be the quotient of H W by the (non-local)relation σ ⊗ w = 0 for all σ ∈ S P ⊆ S and w ∈ exp ( w ) for w ∈ W . Parabolic light leaves tableaux. We now recall the combinatorics of tableaux for dia-grammatic Hecke categories. The cellular poset will be given by the set P (cid:54) w = { x | x ∈ P W and x (cid:54) w } partially ordered by the Bruhat order. We will then fix a choice of reduced expression x (cid:54) w for every x ∈ P (cid:54) w . Given t a subword of w ∈ exp (cid:96)P ( w ), we define Shape k ( t ) = σ t σ t . . . σ t k k for 1 (cid:54) k (cid:54) (cid:96) . Fix w = σ . . . σ (cid:96) ∈ exp (cid:96)P ( w ). In the non-parabolic case, the indexing set of“tableaux of shape x and weight w ” will then be given byStd (cid:54) w ( x ) = { t | Shape (cid:96) ( t ) = x } and we define the set of “parabolic tableaux of shape x and weight w ” to beStd P (cid:54) w ( x ) = { t | Shape k ( t ) σ k +1 ∈ P W for 0 (cid:54) k < (cid:96) and Shape (cid:96) ( t ) = x } ⊆ Std (cid:54) w ( x ) . Finally, we take the union over all possible “weights” to obtain the indexing set of tableauxStd P (cid:54) (cid:96) ( x ) = (cid:91) w ∈ P - exp (cid:96)P ( w ) w ∈ P W Std P (cid:54) w ( x ) . Given x < xτ and t ∈ Std P (cid:54) w ( x ), we define t + = ( t , . . . , t q , ∈ Std (cid:54) wτ ( xτ ) t − = ( t , . . . , t q , ∈ Std (cid:54) wτ ( x )and this will be the backbone of how we grow the cellular bases as we let (cid:96) increase. HE MODULAR WEYL–KAC CHARACTER FORMULA 7 Idempotent truncations and cellular bases. We can decompose the the diagrammaticanti-spherical Hecke category in the following manner, H P \ W = (cid:77) x ∈ exp P ( x ) y ∈ exp P ( y ) x,y ∈ W x H P \ W y and hence regard this algebra as a locally unital associative algebra in the sense of [BS18,Remark 2.3]. This formalism allows one to truncate all important questions about the infi-nite dimensional algebra and instead consider them as being glued together, using standardhighest-weight theory, from the (infinitely many) finite-dimensional idempotent pieces. For ourpurposes, the x H P \ W y are a little “too small”. Instead we prefer to use co-saturated sums ofthese idempotents (in the Bruhat ordering) as follows (cid:54) w = (cid:88) x (cid:54) w x for any fixed expression w ∈ exp (cid:96)P ( w ) for w ∈ W for (cid:96) (cid:62) 0. In particular, we will wish toconsider the subalgebras (cid:54) w H P \ W (cid:54) w and we will glue these together in order to understand H P \ W . We begin by constructing acellular basis of (cid:54) w (cid:48) H P \ W (cid:54) w (cid:48) for each w (cid:48) ∈ exp (cid:96) +1 P ( w (cid:48) ) and w (cid:48) ∈ W given cellular bases of thealgebras (cid:54) w H P \ W (cid:54) w for all w ∈ exp (cid:96)P ( w ) and w ∈ W . For any fixed expression w ∈ exp (cid:96)P ( w ),we have an embedding (cid:54) w H P \ W (cid:54) w (cid:44) → (cid:54) wτ H P \ W (cid:54) wτ (1.1)given by : D (cid:55)→ D ⊗ τ . We consider each of these embeddings in turn (for all τ ∈ S ). If yτ > y ,then for any y ∈ rexp P ( y ), y + ∈ rexp P ( yτ ), y − ∈ rexp P ( y ) and t ∈ Std (cid:54) w ( y ) we define c t + = braid y + yτ ( c t ⊗ τ ) c t − = braid y − y ( c t ⊗ spot ∅ τ ) . If yτ = y (cid:48) < y , then y (cid:48) τ ∈ rexp P ( y ). For any y + ∈ rexp P ( y ), y − ∈ rexp P ( yτ ) and t ∈ Std (cid:54) w ( y (cid:48) τ )we define c t + = braid y + y (cid:48) τ ( y (cid:48) ⊗ fork τττ )( c t ⊗ ττ ) c t − = braid y − y (cid:48) ( y (cid:48) ⊗ spot ∅ τ fork τττ )( c t ⊗ ττ ) Theorem 1.5 ([EW16, Section 6.4] and [LW, Theorem 7.3]) . Fix some reduced expression x for each x ∈ P (cid:54) w . The algebra (cid:54) w H P \ W (cid:54) w is a finite-dimensional, quasi-hereditary algebrawith graded cellular basis { c st | s , t ∈ Std P (cid:54) w ( x ) , x ∈ P (cid:54) w } (1.2) with respect to the Bruhat ordering on P (cid:54) w and anti-involution ∗ .Proof. That (1.2) provides a graded cellular basis is proven in [EW16, Section 6.4] and [LW,Theorem 7.3]. (In the latter paper, cellularity is not mentioned directly and instead it is merelystated that (1.2) provides a basis; however, the proof from the former paper that this basisis cellular goes through unchanged.) The only point of the theorem which is not explicitlystated in [EW16, Section 6.4] and [LW, Theorem 7.3] is that the algebra is quasi-hereditary.However, this is immediate from the fact that each layer of the cellular basis contains (at leastone) idempotent c ss = x for s the unique tableau in Std (cid:54) x ( x ) ⊆ Std (cid:54) w ( x ). (cid:3) We reiterate that we have chosen to fix expressions x, y for each x, y ∈ P (cid:54) w . We defineone-sided ideals H (cid:54) xP \ W = H P \ W (cid:54) x H Through the embedding of equation (1.1) weobtain a corresponding restriction functorRes (cid:54) wτ (cid:54) w : (cid:54) wτ H P \ W (cid:54) wτ − mod −−→ (cid:54) w H P \ W (cid:54) w − mod . Theorem 1.5 has the following immediate corollary. Corollary 1.6. Assume that y ∈ P (cid:54) w . If y > yτ (cid:54)∈ P W , then Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( y )) = 0 . Thus we may now assume that x = yτ > y , with x, y ∈ P (cid:54) w . We let x and y be reducedexpressions for x and y respectively. We have that → { c t + | t ∈ Std (cid:54) w ( y ) } → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( x )) → { c t + | t ∈ Std (cid:54) w ( x ) } → and → { c t − | t ∈ Std (cid:54) w ( y ) } → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( y )) → { c t − | t ∈ Std (cid:54) w ( x ) } → where in both cases the submodule is isomorphic to ∆ (cid:54) w ( y ) and the quotient module is isomor-phic to ∆ (cid:54) w ( x ) .Proof. The yτ (cid:54)∈ P W case follows immediately from the fact ( (cid:54) w ⊗ τ )∆ (cid:54) wτ ( y ) = 0. The maps∆ (cid:54) w ( y ) (cid:44) → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( x )) φ : c t (cid:55)→ c t ⊗ τ ∆ (cid:54) w ( y ) (cid:44) → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( y )) φ : c t (cid:55)→ c t ⊗ spot ∅ τ for t ∈ Std (cid:54) w ( y ) are injective ( (cid:54) w ⊗ τ ) H P \ W ( (cid:54) w ⊗ τ )-homomorphisms by construction.Similarly, the maps∆ (cid:54) w ( x ) → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( x )) /φ (∆ (cid:54) w ( y )) : c t (cid:55)→ ( c t ⊗ τ )( y ⊗ fork τττ )∆ (cid:54) w ( x ) → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( y )) /φ (∆ (cid:54) w ( y )) : c t (cid:55)→ ( c t ⊗ τ )( y ⊗ ( fork τττ spot τ ∅ )for t ∈ Std (cid:54) w ( x ) are ( (cid:54) w ⊗ τ ) H P \ W ( (cid:54) w ⊗ τ )-isomorphisms by construction (as we havesimply multiplied on the right by an element of the algebra). (cid:3) HE MODULAR WEYL–KAC CHARACTER FORMULA 9 p -Kazhdan–Lusztig polynomials. The categorical (rather than geometric) definitionof the (parabolic) p -Kazhdan–Lusztig polynomials is given via the diagrammatic character of[EW16, Definition 6.23] and [LW, Section 8] (for the non-parabolic and parabolic cases, respec-tively). In the language of this paper, the definition of the parabolic or more specifically the anti-spherical p -Kazhdan–Lusztig polynomial , p n x,y ( v ) for x, y ∈ P W , is as follows, p n x,y ( v ) := (cid:88) k ∈ Z dim t (Hom (cid:54) w H P \ W (cid:54) w ( P (cid:54) w ( x ) , ∆ (cid:54) w ( y )) = (cid:88) k ∈ Z [∆ (cid:54) w ( y ) : L (cid:54) w ( x ) (cid:104) k (cid:105) ] v k for any x, y (cid:54) w and x ∈ rex P ( x ) , y ∈ rex P ( y ), w ∈ rex P ( w ) are arbitrary (here the role playedby w ∈ W is merely to allows us to work with finite-dimensional projective modules). Weclaim no originality in this observation and refer to [Pla17, Theorem 4.8] for more details. Theparabolic p -Kazhdan–Lusztig polynomials are recorded in the | P W | × | P W | -matrix D P \ W = (cid:0) p n x,y ( v ) (cid:1) x,y ∈ P W and we set D − P \ W = (cid:0) p n − x,y ( v ) (cid:1) x,y ∈ P W to be the inverse of this matrix (which exists, as D P \ W is lower uni-triangular). The non-parabolic ( p -)Kazhdan–Lusztig polynomials are obtained by setting P = { W } (cid:54) W .2. The classification and construction ofhomogeneous H P \ W -modules It is, in general, a hopeless task to attempt to understand all p -Kazhdan–Lusztig polynomialsor to understand all simple H P \ W -modules. In particular, it was shown in [Wil17] that one canembed certain number-theoretic questions (for which no combinatorial solution could possiblybe hoped to exist) into the p -Kazhdan–Lusztig matrices of affine symmetric groups.Thus we restrict our attention to classes of modules which we can hope to understand. Overthe complex numbers, the first port of call would be to attempt to understand the unitary mod-ules; for Lie groups this ongoing project is Vogan’s famous Atlas of Lie groups. Over arbitraryfields, the notion of unitary no longer makes sense; however, for graded algebras the homoge-neous representations seem to provide a suitable replacement. For quiver Hecke algebras, thehomogeneous representations were classified and constructed by Kleshchev–Ram [KR12]. For(quiver) Hecke algebras of symmetric groups, the notions of unitary and homogeneous repre-sentations coincide over the complex field [BNS, Theorem 8.1] and the beautiful cohomologicaland structural properties of these (homogeneous) representations are entirely independent ofthe field [BNS, KR12].In this section, we fix W an arbitrary Coxeter group and fix P an arbitrary parabolic subgroupand we classify and construct the homogeneous representations of the diagrammatic Heckecategory H P \ W . We first provide a cohomological construction of the module L (1 P \ W ) via aBGG resolution. This cohomological construction allows us to immediately deduce a basis-theoretic construction of L (1 P \ W ), from which we easily read-off the fact that L (1 P \ W ) ishomogeneous. We then prove that L ( w ) is inhomogeneous for any 1 (cid:54) = w ∈ P W . Definition 2.1. Given w, y ∈ P W , we say that ( w, y ) is a Carter–Payne pair if y (cid:54) w and (cid:96) ( y ) = (cid:96) ( w ) − . We let CP (cid:96) denote the set of Carter–Payne pairs ( w, y ) with (cid:96) ( w ) = (cid:96) ∈ N . For P ⊆ W an affine Weyl group an its maximal finite parabolic subgroup, the followingfamily of homomorphisms were first considered (in the context of algebraic groups) by Carter–Payne in [CP80]. Theorem 2.2. Let ( w, y ) be a Carter–Payne pair. Pick an arbitrary w = σ . . . σ (cid:96) and supposethat y = σ . . . σ p − (cid:98) σ p σ p +1 . . . σ (cid:96) is the subexpression of y obtained by deleting precisely oneelement σ p ∈ S . We have that Hom H P \ W (∆( w ) , ∆( y )) is v -dimensional and spanned by the map ϕ wy ( c t ) = c t ( σ ··· σ p − ⊗ spot σ p ∅ ⊗ σ p +1 ··· σ (cid:96) ) for t ∈ Std( w ) .Proof. For ease of exposition, we first truncate to the algebra (cid:54) w H P \ W (cid:54) w . We have that∆ (cid:54) w ( y ) = Span Z { σ ··· σ p − ⊗ spot σ p ∅ ⊗ σ p +1 ··· σ (cid:96) } by Theorem 1.5. Moreover this space is of strictly positive degree, namely v . Whereas, thecharacter of the simple head, L ( y ) of ∆( y ), is invariant under swapping v and v − by [HM10,2.18 Proposition]. Therefore∆ (cid:54) w ( y ) = rad(∆ (cid:54) w ( y )) and L (cid:54) w ( y ) = 0 . (2.1)By our assumption that ( w, y ) is a Carter–Payne pair, there does not exist an x ∈ W such that y < x < w . We now apply this assumption twice. Firstly, we note that [∆ (cid:54) w ( y ) : L (cid:54) w ( x )] (cid:54) = 0implies that y (cid:54) x (cid:54) w . Putting this together with 2.1 we have that rad(∆ (cid:54) w ( y )) = L (cid:54) w ( w ) (cid:104) (cid:105) and the graded decomposition number is equal todim v (Hom (cid:54) w H P \ W (cid:54) w ( P (cid:54) w ( w ) , ∆ (cid:54) w ( y ))) = (cid:88) k ∈ Z [∆ (cid:54) w ( y ) : L (cid:54) w ( w ) (cid:104) k (cid:105) ] = v . Now applying our assumption again, we conclude that this homomorphism factors through theprojection P (cid:54) w ( w ) → ∆ (cid:54) w ( w ) by highest weight theory and so we havedim v (Hom (cid:54) w H P \ W (cid:54) w ( P (cid:54) w ( w ) , ∆ (cid:54) w ( y ))) = dim v (Hom (cid:54) w H P \ W (cid:54) w (∆ (cid:54) w ( w ) , ∆ (cid:54) w ( y )))and thus the result follows. (cid:3) We set P (cid:96) = { w ∈ P W | (cid:96) ( w ) = (cid:96) } for each (cid:96) ∈ N . Following a construction going back towork of Bernstein–Gelfand–Gelfand and Lepowsky [BGG75, GL76], we are going to define acomplex of graded H P \ W -modules · · · −→ ∆ δ −→ ∆ δ −→ ∆ δ −→ , (2.2)where ∆ (cid:96) := (cid:77) w ∈P (cid:96) ∆( w ) (cid:104) (cid:96) ( w ) (cid:105) . (2.3)We will refer to this as the BGG complex . We momentarily assume that P = 1 (cid:54) W is thetrivial parabolic (so that P W = W ). We call any quadruple w (cid:62) x, y (cid:62) z for w, x, y, z ∈ W with (cid:96) ( w ) = (cid:96) ( z ) + 2 a diamond . For such a diamond we have homomorphisms of H W -modules∆( x )∆( y )∆( w ) ∆( z ) ϕ wx ϕ wy ϕ xz ϕ yz given by our Carter–Payne homomorphisms. By an easy variation on [BGG75, Lemma 10.4],it is possible to pick a sign ε ( s, t ) for each of the four Carter–Payne pairs such that for everydiamond the product of the signs associated to its four arrows is equal to − H P \ W . If x, y ∈ P W then thediamond above is preserved in the sense that ∆( w ), ∆( x ), ∆( y ), ∆( z ) (cid:54) = 0 and the Carter–Payne homomorphisms (and their signs) go through unchanged. We now assume that one of HE MODULAR WEYL–KAC CHARACTER FORMULA 11 x, y (cid:54)∈ P W and without loss of generality we suppose that y (cid:54)∈ P W . In this case our diamond“collapses” in the sense that ∆( x )0∆( w ) ∆( z ) ϕ wx ϕ xz and we instead refer to this as a strand in our complex and we denote this by∆( x )∆( w ) ∆( z ). ϕ wx ϕ xz We can now define the H P \ W -differential δ (cid:96) : ∆ (cid:96) → ∆ (cid:96) − for (cid:96) (cid:62) ε ( s, t ) ϕ st : ∆( s ) (cid:104) (cid:96) (cid:105) → ∆( t ) (cid:104) (cid:96) − (cid:105) over all Carter–Payne pairs ( s, t ) ∈ CP (cid:96) . We set C • (1 P \ W ) = (cid:76) (cid:96) (cid:62) ∆ (cid:96) (cid:104) (cid:96) (cid:105) together with thedifferential ( δ (cid:96) ) (cid:96) (cid:62) . Proposition 2.3. We have that Im( δ (cid:96) +1 ) ⊆ ker( δ (cid:96) ) , in other words C • (1 P \ W ) is a complex.Proof. The (cid:96) = 0 case follows trivially from the fact that the highest weight ordering on P W is given by the Bruhat order. We now assume that (cid:96) > (cid:54) w H P \ W (cid:54) w for w = σ σ · · · σ (cid:96) a fixed reduced expression of w ∈ W . Let z beany subexpression of w of length (cid:96) − 2. Such a subexpression is obtained by deleting a pair σ p and σ q for some 1 (cid:54) p < q (cid:54) (cid:96) . There are precisely two ways of doing this: namely, we are freeto either first delete the p th term (to first obtain some y (cid:54) w of length (cid:96) − 1) followed by the q th term (to hence obtain z of length (cid:96) − 2) or vice vera (in which case we first delete the q thterm to obtain some x (cid:54) w of length (cid:96) − p th term to obtain z of length (cid:96) − x, y ∈ P W then this provides the diamonds of our complex. If either x or y doesnot belong to P W then this provides a strand in our complex. Thus it remains to show that ϕ wx ϕ xz = ϕ wy ϕ yz for w > x, y > z (note that if one of x, y (cid:54)∈ P W then this implies the composition along thestrand is zero) at which point the result follows by a standard argument for BGG complexes.Thus it remains to show that( σ ··· σ p − ⊗ spot σ p ∅ ⊗ σ p +1 ··· σ (cid:96) )( σ ··· σ q − ⊗ spot σ q ∅ ⊗ σ q +1 ··· σ (cid:96) )is equal to ( σ ··· σ q − ⊗ spot σ q ∅ ⊗ σ q +1 ··· σ (cid:96) )( σ ··· σ p − ⊗ spot σ p ∅ ⊗ σ p +1 ··· σ (cid:96) ) . This is immediate from the graph isotopy relation and so the result follows. (cid:3) We have already encountered one drawback of the restriction functors from the previoussection: they kill any standard module ∆ (cid:54) w ( x ) such that xτ (cid:54)∈ P W (and therefore the simplehead is also killed). To remedy this, we define slightly larger algebras (cid:54) (cid:96) H P \ W (cid:54) (cid:96) for (cid:54) (cid:96) = (cid:88) w ∈ exp (cid:96)P ( w ) w ∈ W (cid:54) w and we define Res (cid:96) +1 (cid:96) : (cid:54) (cid:96) +1 H P \ W (cid:54) (cid:96) +1 → (cid:54) (cid:96) H P \ W (cid:54) (cid:96) to be the functorRes (cid:96) +1 (cid:96) = (cid:77) w ∈ exp (cid:96)P wτ ∈ exp (cid:96) +1 P Res (cid:54) wτ (cid:54) w . Lemma 2.4. Let P \ W (cid:54) = x, w ∈ P W and suppose that x (cid:54) w . We have that Res (cid:54) wτ (cid:54) w ( L (cid:54) wτ ( x )) (cid:54) = 0 for some τ ∈ S and w ∈ exp P ( w ) . Therefore Res (cid:96) +1 (cid:96) ( (cid:54) (cid:96) +1 L ( x )) = 0 implies x = 1 P \ W .Proof. For 1 P \ W (cid:54) = x ∈ P W , there exists some τ ∈ S such that xτ = x (cid:48) < x and x (cid:48) ∈ P W .In which case, x = x (cid:48) τ is a reduced expression for x and x (cid:48) τ ∈ L ( x ). Our assumption that xτ = x (cid:48) (cid:54) x (cid:54) w (cid:54) wτ implies that the preimage of x (cid:48) τ ∈ H (cid:54) wτ under the map of (1.1) isequal to 0 (cid:54) = x (cid:48) ∈ H (cid:54) w and so the result follows. (cid:3) We are now ready to define the BGG resolution of the H P \ W -module L (1 P \ W ). For W theaffine symmetric group, P the maximal finite parabolic and k = C , the existence of these BGGresolutions was conjectured by Berkesch–Griffeth–Sam in [BGS14]. This conjecture was provenby way of the KZ -functor in the context of the quiver Hecke algebras of type A (by the first andthird authors with Jos´e Simental, [BNS]). In type A , the diagrammatic Hecke categories and(truncations of) quiver Hecke algebras were recently shown to be isomorphic in [BCH]. Thusthe following theorem generalises the BGG resolutions [BGS14, BNS] to all Coxeter groups, W ,and all parabolic subgroups, P , and arbitrary fields, k . Theorem 2.5. Fix W an arbitrary Coxeter group and fix P an arbitrary parabolic subgroup.The H P \ W -complex C • (1 P \ W ) is exact except in degree zero, where H ( C • (1 P \ W )) = L (1 P \ W ) . Moreover, we have that L (1 P \ W ) = k { c s | Shape k ( s ) = P \ W for all k (cid:62) } . Proof. By applying the restriction functor to Proposition 2.3, we have thatRes (cid:96) +1 (cid:96) ( (cid:54) (cid:96) +1 C • (1 P \ W ))forms a complex of (cid:54) (cid:96) H P \ W (cid:54) (cid:96) -modules. Moreover, we can idempotent-truncate D τ,w • (1 P \ W ) = ( (cid:54) w ⊗ τ )(Res (cid:96) +1 (cid:96) ( (cid:54) (cid:96) +1 C • (1 P \ W ))) (2.4)and hence obtain a complex of (cid:54) w H P \ W (cid:54) w -modules (through the identification of H (cid:54) w (cid:44) →H (cid:54) wτ ). Let x, y ∈ P W with x = yτ > y . For y ∈ P W , we have that already seen that0 → { c t + | t ∈ Std (cid:54) w ( y ) } → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( x )) → { c t + | t ∈ Std (cid:54) w ( x ) } → → { c t − | t ∈ Std (cid:54) w ( y ) } → Res (cid:54) wτ (cid:54) w (∆ (cid:54) wτ ( y )) → { c t − | t ∈ Std (cid:54) w ( x ) } → (cid:54) w ( y ) and the quotient module is isomor-phic to ∆ (cid:54) w ( x ). We now simply note that ϕ xy ( c t + ) = ( y ⊗ spot τ ∅ ) c t + = c t − for any t ∈ Std (cid:54) w ( x ) or t ∈ Std (cid:54) w ( y ) by definition. Therefore, we have that (cid:0) Res (cid:54) wτ (cid:54) w ◦ ϕ xy (cid:1) = id x (cid:104) (cid:105) + id y (cid:104) (cid:105) for x, y ∈ P W where id z (cid:104) (cid:105) ∈ Hom (cid:54) w H P \ W (cid:54) w (∆ (cid:54) w ( z ) (cid:104) (cid:96) ( z ) (cid:105) , ∆ (cid:54) w ( z ) (cid:104) (cid:96) ( z ) + 1 (cid:105) )is simply the graded shift of the identity map for z = x, y for x, y ∈ W . This implies that D τ,w • (1 P \ W ) = (cid:77) yτ>yy (cid:54) w (cid:0) ∆( y ) (cid:104) (cid:96) ( y ) (cid:105) ⊕ ∆( y ) (cid:104) (cid:96) ( y ) + 1 (cid:105) (cid:1) with differentialRes (cid:54) wτ (cid:54) w ◦ δ (cid:96) = (cid:88) ( x,y ) ∈ CP (cid:96) x = yτ ( id x (cid:104) (cid:105) + id y (cid:104) (cid:105) ) + (cid:88) ( s,t ) ∈ CP (cid:96) s (cid:54) = tτ (Res (cid:54) wτ (cid:54) w ◦ ϕ st ) . HE MODULAR WEYL–KAC CHARACTER FORMULA 13 Thus we have that H j (( (cid:54) w ⊗ τ )Res (cid:54) wτ (cid:54) w ( (cid:54) (cid:96) +1 C • (1 P \ W )) = 0for all j (cid:62) 0. Now, summing over all τ ∈ S , w ∈ W , and w ∈ exp (cid:96)P ( w ) we deduce thatRes (cid:96) +1 (cid:96) ( (cid:54) (cid:96) +1 C • (1 P \ W ))forms a complex with zero homology in every degree. By Lemma 2.4, we have that restrictionkills no simple H P \ W -module L ( w ) for 1 (cid:54) = w ∈ P W . Moreover,Head( (cid:54) (cid:96) +1 ∆(1 P \ W )) = (cid:54) (cid:96) +1 L (1 P \ W ) (cid:54)⊂ Im( δ )and [ (cid:96) +1 ∆( w ) : (cid:96) +1 L (1 P \ W )] = 0 for 1 P \ W (cid:54) = w ∈ P W simply because the highest weightstructure on H P \ W is given by the Bruhat order. Therefore H j ( (cid:54) (cid:96) +1 C • (1 P \ W )) = (cid:40) (cid:54) (cid:96) +1 L (1 P \ W ) if j = 00 otherwise . Finally, we have proven that L (1 P \ W ) is killed by multiplication by the idempotent τ at the (cid:96) th point for any (cid:96) (cid:62) τ ∈ S. Thus L (1 P \ W ) is spanned by c s for s the emptytableau, as required. (cid:3) We immediately deduce the following corollary, which is new even for the classical (inverse)parabolic and non-parabolic Kazhdan–Lusztig polynomials (in other words, for k the complexfield). Indeed, this seems to be the first non-trivial family of parabolic ( p − )Kazhdan–Lusztigpolynomials which admits a uniform construction across all Coxeter groups and all parabolicsubgroups. Corollary 2.6 (The Weyl–Kac character formula for Coxeter groups) . In the graded Grothendieckgroup of H P \ W , we have that [ L (1 P \ W )] = (cid:88) w ∈ P W ( − v ) (cid:96) ( w ) [∆( w )] Thus the complete first row of the inverse p -Kazhdan–Lusztig matrix is given by p n − ,w = ( − v ) (cid:96) ( w ) for all w ∈ P W . Theorem 2.7. The module L (1 P \ W ) is both the unique homogeneous H P \ W -module and theunique 1-dimensional H P \ W -module.Proof. That the module L (1 P \ W ) is homogeneous is clear (as it is 1-dimensional). We nowprove the converse, namely for any 1 (cid:54) = w ∈ P W we show that L ( w ) is inhomogeneous and ofdimension strictly greater than 1. Let 1 (cid:54) = w ∈ P W and choose τ such that wτ = w (cid:48) < w . ByTheorem 1.5, the elements w (cid:48) ⊗ spot τ ∅ ⊗ τ w (cid:48) ⊗ fork τττ (2.5)span ∆ (cid:54) wττ ( w ). The former is homogeneous of degree − w (cid:48) ⊗ spot ∅ τ ⊗ τ )( w (cid:48) ⊗ fork τττ ) = w (cid:48) and ( w (cid:48) ⊗ spot ∅ τ ⊗ τ )( w (cid:48) ⊗ spot ∅ τ ⊗ τ ) = w (cid:48) ⊗ spot ∅ τ spot τ ∅ ⊗ τ = 0 (mod (cid:54) wττ H We recall from the introduction that the conjecture of Berkesch–Griffeth–Sam(or rather, its equivalent formulation for homogenous representations of quiver Hecke algebras)can be deduced from Theorem B. This might be surprising to the reader familiar with the ho-mogeneous representations of quiver Hecke algebras. In [KR12] it is shown that there are upto e − distinct homogeneous representations of any block of the quiver Hecke algebras (andfor sufficiently large rank, there are precisely e − such representations for a “regular block”).Whereas, in this paper we have seen that there is precisely one homogenous representation of H P \ W for S h = P ⊂ W = (cid:98) S h for h ∈ N . Therefore, one might think that there are “more”homogenous representations of the quiver Hecke algebra. However, for each (cid:54) h < e there isan isomorphism between a finite truncation of H P \ W and the Serre quotient of the quiver Heckealgebra corresponding to the set of partitions with at most h columns [BCH] . Through these iso-morphisms, one can obtain the e − distinct BGG resolutions of the e − distinct homogenoussimple modules of the quiver Hecke algebra predicted by Berkesch–Griffeth–Sam [BGS14] . We now provide an elementary infinite family of simple modules which do not admit BGGresolutions, in order to bolster our claim in the introduction that such resolutions are “rare”.In [BGG75] an example of such a simple for W = S is given. Theorem 2.9. Let k be a field of positive characteristic. Let W denote an affine Weyl group andlet P ⊂ W be the maximal finite parabolic. There exists infinitely many simple H P \ W -moduleswhich do not admit BGG resolutions.Proof. It is enough to consider the case S = P ⊂ W = (cid:98) S . The Coxeter presentation of W is (cid:104) σ, τ | τ = σ = 1 (cid:105) and we let P denote the finite parabolic generated by the reflection τ . Wewill provide an infinite family of examples of x ∈ W such that rad(∆( x )) is not generated by thehomomorphic images of standard modules, thus showing that each such L ( x ) does not admit aBGG resolution. We set x = ( στ ) np − , y = ( στ ) np +1 , and z = ( στ ) np . Suppose that L ( y ) is asubquotient of ∆( x ) and that L ( y ) belongs to the submodule generated by the homomorphicimages of standard modules. Then L ( y ) must be in the image of a homomorphism from ∆( y )or ∆( z ) by highest weight theory. We have that x (cid:54) z (cid:54) y are a strand of Carter–Payne pairsand so L ( y ) does not belong to the image of a homomorphism from ∆( z ). We claim thatdim t (Hom H P \ W (∆( y ) , ∆( x )) = 0 (cid:88) k ∈ Z [∆( x ) : L ( y ) (cid:104) k (cid:105) ] = t . (2.6)and thus the result follows, once we have established the claim. We first truncate to consider (cid:54) στστ H P \ W (cid:54) στστ . The module ∆ (cid:54) y ( x ) is ( np − k − στ ⊗ fork σσσ ⊗ p − − kτσ )( kστ ⊗ spot τ ∅ ⊗ p − kστ ) ( k − στ ⊗ fork τ ∅ ττ ⊗ p − − kστ )( kστ ⊗ spot σ ∅ ⊗ p − kτσ )for 1 (cid:54) k < np for k odd and k even, respectively. The Gram-matrix of the cell-form of thismodule has − − A np − ). The determinant of thismatrix is np and so there is a subquotient isomorphic to L ( y ) in ∆( x ). The claim concerningthe degree (which is not required for the proof of the proposition) follows from the fact that allbasis elements of this weight are concentrated in degree zero. (cid:3) Remark 2.10. Through the isomorphism of [BCH] , the claim of equation (2.6) can be rephrasedas a question concerning decomposition numbers and homomorphisms for the symmetric group S np + p in characteristic p > . In which case equation (2.6) is equivalent to dim(Hom k S np p ( S ( np + p ) , S ( np , p )) = 0 [ S ( np , p ) : D ( np + p )] = 1 , where S ( λ ) is the Specht module and D ( µ ) is the simple head for λ, µ partitions (the latter p -regular) of np + p . This example was already known to Gordon James in [Jam78, 24.5Examples] for p = 2 and the general case is similar. In particular, the rephrased claim followsimmediately from [Jam78, 24.4 Theorem] and [Jam78, 24.15 Theorem] . HE MODULAR WEYL–KAC CHARACTER FORMULA 15 References [AJS94] H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a p th root of unityand of semisimple groups in characteristic p : independence of p , Ast´erisque (1994).[AMRW19] P. Achar, S. Makisumi, S. Riche, and G. Williamson, Koszul duality for Kac-Moody groups and charactersof tilting modules , J. Amer. Math. Soc. (2019), no. 1, 261–310.[Ara06] T. Arakawa, A new proof of the Kac-Kazhdan conjecture , Int. Math. Res. Not. (2006).[Ara07] , Representation theory of W -algebras , Invent. Math. (2007), no. 2, 219–320.[BGS14] C. Berkesch Zamaere, S. Griffeth, and S. V. Sam, Jack polynomials as fractional quantum Hall states andthe Betti numbers of the ( k + 1) -equals ideal , Comm. Math. Phys. (2014), no. 1, 415–434.[BCH] C. Bowman, A. Cox, and A. Hazi, Path isomorphisms between quiver Hecke and diagrammatic Bott–Samelsonendomorphism algebras , arXiv:2005.02825.[BGG75] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Differential operators on the base affine space and a studyof g -modules , 21–64.[BNS] C. Bowman, E. Norton, and J. Simental, Characteristic-free bases and BGG resolutions of unitary simplemodules for quiver Hecke and Cherednik algebras , arXiv:1803.08736 , preprint.[BS10] J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra. II. Koszulity ,Transform. Groups (2010), no. 1, 1–45.[BS11] , Highest weight categories arising from Khovanov’s diagram algebra III: category O , Represent. The-ory (2011), 170–243.[BS18] , Semi-infinite highest weight categories , arXiv1808.08022 (2018).[CP80] R. W. Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules , Math.Proc. Cambridge Philos. Soc. (1980), no. 3, 419–425.[Eas05] M. Eastwood, Higher symmetries of the Laplacian , Ann. of Math. (2) (2005), no. 3, 1645–1665.[EH04] T. J. Enright and M. Hunziker, Resolutions and Hilbert series of determinantal varieties and unitary highestweight modules , J. Algebra (2004), 608–639.[EW14] B. Elias and G. Williamson, The Hodge theory of Soergel bimodules , Ann. of Math. (2) (2014), 1089–1136.[EW16] , Soergel calculus , Represent. Theory (2016), 295–374. MR 3555156[Fel89] G. Felder, BRST approach to minimal models , Nuclear Phys. B (1989), no. 1, 215–236.[FF92] B. Feigin and E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikiui algebras , Infiniteanalysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992.[GJSV13] A. Gainutdinov, J. Jacobsen, H. Saleur, and R. Vasseur, A physical approach to the classification of inde-composable Virasoro representations from the blob algebra , Nuclear Phys. B (2013), 614–681.[GL76] H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas , Invent. Math. (1976), no. 1, 37–76.[Hay88] T. Hayashi, Sugawara operators and Kac-Kazhdan conjecture , Invent. Math. (1988), no. 1, 13–52.[HM10] J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A , Adv. Math. (2010), no. 2, 598–642.[Jam78] G. D. James, The Representation Theory of the Symmetric Groups , Lecture Notes in Mathematics, vol. 682,Springer, Berlin, 1978.[KK79] V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Liealgebras , Adv. in Math. (1979), no. 1, 97–108.[KR12] A. Kleshchev and A. Ram, Homogeneous representations of Khovanov–Lauda algebras , J. Eur. Math. Soc. (2012), 1293–1306.[Ku89] J. Ku, Structure of the Verma module M ( − ρ ) over Euclidean Lie algebras , J. Algebra (1989), 367–387.[Las78] A. Lascoux, Syzygies des vari´et´es d´eterminantales , Adv. in Math. (1978), no. 3, 202–237.[Lep77] J. Lepowsky, A generalization of Bernstein-Gelfand-Gelfand’s resolution , J. Algebra (1977) 496–511.[LW] N. Libedinsky and G. Williamson, The anti-spherical category , arXiv:1702.00459.[Mat96] O. Mathieu, On some modular representations of affine Kac-Moody algebras at the critical level , CompositioMath. (1996), no. 3, 305–312.[MS94] P. Martin and H. Saleur, The blob algebra and the periodic Temperley–Lieb algebra , Lett. Math. Phys. (1994), no. 3, 189–206.[Pla17] D. Plaza, Graded cellularity and the monotonicity conjecture , J. Algebra (2017), 324–351.[Wak86] M. Wakimoto, Fock representations of the affine Lie algebra A (1)1 , Comm. Math. Phys. (1986).[Wil17] G. Williamson, Schubert calculus and torsion explosion , J. Amer. Math. Soc. (2017), 1023–1046. School of Mathematics, Statistics and Actuarial Science University of Kent, Canterbury, UK E-mail address : [email protected] Department of Mathematics, City, University of London, London, UK E-mail address : [email protected] FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany. E-mail address ::