Presenting and Extending Hecke Endomorphism Algebras

Presenting and Extending Hecke Endomorphism Algebras Jie Du University of New South Wales Shanghai Conference on Representation Theory 7-11 December 2015 1 / 27

The Hecke Endomorphism Algebra The (equal parameter) Hecke algebra H q associated with a Coxeter system (W, S) is presented as the Z[q]-algebra generated by T s, s S, subject to the relations the Hecke relations: (H1) Ts 2 = (q 1)T s + q; (H2) T s T t T s... = T }{{} t T s T t..., where m }{{} s,t is the order of st. m s,t m s,t Replacing q by q cs, the unequal parameter Hecke algebra can be defined similarly. These algebras naturally appear in the theory of finite groups of Lie type. Definition Let Λ = Λ(W, S) = {I S W I is finite} x I = w W I T w, (I Λ). The Z[q]-algebra S q = End Hq ( I Λ x I H q ) is called the Hecke endomorphism algebra associated to the Coxeter group W. 2 / 27

Motivation In the categorification of H q by Soergel bimodules, KL generators are used and the category is presented by generators and relations. [Elias Williamson] Q: Presenting S q directly by Hecke type relations. If (W, S) is a Weyl group, then S q has a stratification relative to a quasi-poset structure on Λ. Conjecture: There exists a preorder on the set Ω of left cells in W, strictly compatible with its partition Ω into two-sided cells, and a right H-module X such that the following statements hold: (1) X has an finite filtration with sections of the form S ω, ω Ω. (2) Let T + := ( I Λ x I H q ) X and put { S + q := End Hq (T + ), + (ω) := Hom Hq (S ω, T + ), for any ω Ω. Then, for any commutative, Noetherian Z-algebra R, the set { + (ω) R } ω Ω is a strict stratifying system for S + q,r-mod relative to the quasi-poset (Ω, ). 3 / 27

References G. Williamson, Singular Soergel bimodules, IMRN, 2011, 4555 4632. J. Du, B. Jensen and X. Su, Presenting Hecke endomorphism algebras by Hasse quivers with relations, arxiv:1511.04135. J. Du, B. Parshall and L. Scott, Stratifying endomorphism algebras associated to Hecke algebras, J. Algebra, 203 (1998), 169 210. V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, On the category O for rational Cherednik algebras, Invent. math. 156 (2003), 617 651. R. Rouquier, q-schur algebras and complex reflection groups, Moscow Math. J. 8 (2008), 119 158. J. Du, B. Parshall and L. Scott, Extending Hecke endomorphism algebras, Pacific J. Math. (Steinberg memorial issue), to appear (arxiv:1501.06481). 4 / 27

KL generators For w W, let C + w = y w P y,w T y. Then {C w + } w W is a new basis for H q, x I = C w + I, and { C s + C w + (q + 1)C w +, if sw < w, = C sw + + q y<w,sy<y µ(y, w)q 1 2 (l(w) l(y) 1) C y +, otherwise, where µ(y, w)q 1 2 (l(w) l(y) 1) is the leading term of P y,w. For a reduced expression w = (s 1, s 2..., s l ) for w, let C + w = C + s 1 C + s 2 C + s l. Then there exist a y,w Z[q] such that C + w = C + w + q y<w a y,w C + y, ( relation) where y < w denotes a subsequence of w obtained by removing some terms in the sequence. 5 / 27

Relations in KL generators Let x s = C + s = T s + 1. The elements x s, s S generate H q. For {s, t} Λ and m m s,t, let T [m]t = T t T s T }{{} t and T t[m] = T t T s T t. }{{} m m Define x [m]t, x t[m] similarly. Let x (1) (s,t) = x t and x (m) m 1 (s,t) = 1 + (T [i]s + T [i]t ) + T [m]t for m 2. i=1 In terms of KL generators, we have m 1 2 x (m) (s,t) = i=0 ( m i 1 i ) ( q) i x [m 2i]t. The Hecke algebra H q can be presented by x s, s S, and (a) Quasi-idempotent relations: (x s ) 2 = (q + 1)x s and (b) Braid relations: x (ms,t) (s,t) = x (ms,t) (t,s) for all s, t S. 6 / 27

Notation For each I Λ and every total ordering on I : I : = I 0 I 1 I 2 I m = I, where I J I J, I = J 1, fix a reduced expression w I = s l s l 1 s 1 such that w Ii = s ji s ji 1 s 1 for all i = 1,..., m. Let w I = (s l, s l 1,, s 1 ) and let x w I = x sl x sl 1 x s1. Then the relation above gives the following. Lemma For any I Λ, any ordering I as above, and any reduced expression w I, there exist a y,w I Z[q] such that x I = x w I + q a y,w I x y. y<w I 7 / 27

The Hasse quiver Consider the poset Λ with the inclusion relation which can be describe by its Hasse graph (or diagram). Replacing every edge in the Hasse graph of the poset Λ by a pair of arrows in opposite direction yields a quiver Q, the Hasse quiver of W. Example If W = S 4, then the Hasse quiver Q has the form Q : {1} {1, 2} {2} {1, 3} {3} {2, 3} Here, for I J, K in Λ, down arrows: δ J,I ; up arrows: υ I,J ; length 0 path: e K. {1, 2, 3} 8 / 27

Let Q q := Z[q]Q be the path algebra of Q over Z[q]. For each singleton {s} Λ, let χ s = υ,{s} δ {s}, Q q. For each I Λ, a total ordering I on I, and a reduced expression w I, let where χ I = χ w I + q a y,w I χ y and ρ I = χ I π(i ), y<w I χw = χ sl χ sl 1 χ s1 if w = (s j, s l 1,, s 1 ), a y,w I Z[q] are given in the lemma above, and π(i ) is the Poincaré polynomial of WI. 9 / 27

Hasse Presentation of RS q Let R = { f 1 + qg } f, g Z[q] = Theorem (DJS := D.-Jensen-Su) { f h }. f, h Z[q], h(0) = 1 Let J be the subset of RQ = RQ q consisting of the following relations (J1) Quasi-idempotent relations: δ J,I υ I,J π(j) π(i ) e J, I J in Λ. (J2) Sandwich relations: υ I,J υ J,K υ I,J υ J,K and δ K,J δ J,I δ K,J δ J,I for all I, J, J, K Λ with I J K and I J K. (J3) Extended braid relations: υ,i1... υ Im 1,I δ I,Im 1... δ I2,I 1 δ I1, χ I for all I Λ and given ordering I. Then RS q = RQ/ J. 10 / 27

Idea of the proof If we put δ J,I = π(i ) π(j) δ J,I, then relations (J1) and (J3) give ( J1) and ( J3) below. Hence, J generates in RQ q the ideal J, where J consists of the following relations: ( J1) Idempotent relations: for all I, J Λ with I J; ( J2) Sandwich relations: δ J,I υ I,J = e J υ I,J υ J,K = υ I,J υ J,K and δ K,J δj,i = δ K,J δj,i for all I, J, J, K Λ with I J K and I J K; ( J3) Extended braid relations: υ,i1... υ Im 1,I δ I,Im 1... δ I2,I 1 δ I1, = ρ I for all I Λ and ordering I : = I 0 I 1 I m = I. 11 / 27

Torsion relations Over Z[q], we have a surjective algebra homomorphism ψ : Q q / J S q, and ker(ψ) = K/ J consists of R-torsion elements. Proposition Let p = i f ip i be a linear combination of paths from I to J, where each f i Z[q]. If υ,j pδ I, = 0 in Q q / J, then p is R-torsion. Further, the ideal K is generated by all such R-torsion elements (and J ). Example (the dihedral case) Q : υ {1} υ {2} δ {1} δ {2} {1} {2} δ {1,2},{1} δ {1,2},{2} υ {1},{1,2} {1, 2} υ {2},{1,2} 12 / 27

Presenting S q (I n ) over Z[q] Theorem (Elias, DJS) The Z[q]-algebra S q (I n ) is isomorphic to the quotient algebra of the path algebra Q q modulo the following relations. (1) Quasi-idempotent relations: for i = 1, 2, (i) δ {i} υ {i} = (1 + q)e {i} ; (ii) δ {1,2},{i} υ {i},{1,2} = (1 + q + + q n 1 )e {1,2}. (2) Sandwich relations: (i) υ {1} υ {1},{1,2} = υ {2} υ {2},{1,2} (ii) δ {1,2},{1} δ {1} = δ {1,2},{2} δ {2} ; (3) Refined braid relations: for (s, t) = (1, 2) or (2, 1). ( n (i) υ {t},{1,2} δ {1,2},{s} = δ {t} j=2 bn j χ [j 2]t) υ{s}, where χ [0]t = e, if n is even; ( n (ii) υ {s},{1,2} δ {1,2},{s} = δ {s} j=3 bn j χ ) [j 2]t υ{s} + ( q) n 1 2 e {s}, if n is odd. ( ) j + s 1 Here bj m is equal to ( q) j 1 s if s = m j 2 is an integer, and 0 otherwise, given by x (m) (s,t) = m j=1 bm j x [j]t. 13 / 27

Presenting S q (S 4 ) over Z[q] Theorem (DJS) The Z[q]-algebra S q (S 4 ) is isomorphic to the quotient algebra of the path algebra Q q modulo the quasi-idempotent and sandwich relations together with the following four sets of relations (T1) δ {1}, χ 2 υ,{1} = υ {1}{1,2} δ {1,2},{1} + qe {1} ; (T2) δ {1}, υ,{3} = υ {1},{1,3} δ {1,3},{3} (T3) δ {1,2},{1} υ {1},{1,3} δ {1,3},{1} υ {1},{1,2} = υ {1,2},S δ S,{1,2} + q(q + 1)e {1,2} ; (T4) δ {1,2},{2} υ {2},{2,3} δ {2,3},{3} υ {3},{1,3} = υ {1,2},S δ S,{1,3} + qδ {1,2},{1} υ {1},{1,3}. NB: In the Hasse quiver, every path can be reversed. This defines an anti-involution τ on Q q. Also, every graph automorphism induces an algebra automorphism σ on Q q. Moreover, we may speak of subgraph automorphisms. Each set above is obtained by applying τ and graph or subgraph automorphisms to the relation. 14 / 27

Extending Hecke endomorphism algebras An ideal J in an R-algebra A as above is called a stratifying ideal provided that J is an R-direct summand of A and, for A/J-modules M, N, inflation defines an isomorphism of Ext-groups. Ext n A/J (M, N) Ext n A (M, N), n 0. A stratification of length n of A is a sequence 0 = J 0 J 1 J n = A of stratifying ideals of A. If, in addition, each J i /J i 1 is a projective A/J i 1 -module, then the stratification is standard. Theorem (D.-Parshall-Scott =: DPS, 98) If S > 1, then S q has a stratification of length 3. We conjectured the existence of a standard stratification for an enlarged S q in terms of two-sided calls. We have established a local version. 15 / 27

KL basis and Lusztig s function a Assume now (W, S) is a Weyl group and let Z = Z[t, t 1 ] and let H = ZH q and S = ZS q (q = t 2 ). For w W, let C w = t l(w) C + w and C xc y = z W h x,y,z C z. Using the preorders L and R on W, the positivity of the coefficients of the h x,y,z implies h x,y,z 0 = z L y, z R x Define the Lusztig function a : W N as follows. For z W, let a(z) be the smallest nonnegative integer such that t a(z) h x,y,z N[t] for all x, y W. For x, y, z W, let γ x,y,z be the coefficient of t a(z) in h x,y,z 1. Then γ x,y,z 0 = x L y 1, y L z 1, z L x 1. 16 / 27

The asymptotic algebra J The asymptotic form J of H is a ring with Z-basis {j x x W } and multiplication j x j y = z γ x,y,z 1j z. Following Lusztig, define a Z-algebra homomorphism ϖ : H J Z = J Z, C w h w,d,z j z, z W d D a(d)=a(z) where D is the set of distinguished involutions in W. In particular, ϖ Q(t) becomes an isomorphism ϖ = ϖ Q(t) : H Q(t) Also, ϖ induces a monomorphism JQ(t). ϖ = ϖ Q[t,t 1 ] : H Q[t,t 1 ] J Q[t,t 1 ] = J Q Q[t, t 1 ]. Moreover, base change to Q[t, t 1 ]/(t 1) induces an isomorphism ϖ = ϖ Q : QW J Q 17 / 27

Specht and Dual Specht modules For the irreducible (left) J Q -module identified with E Irr(QW ), the (left) H Q[t,t 1 ]-module S(E) := ϖ (E Q[t, t 1 ]) = ϖ (E Q[t,t 1 ]) is called here a dual Specht module for H Q[t,t 1 ]. Note that S(E) = E Q[t,t 1 ] as a Q[t, t 1 ]-module. Therefore, S(E) is a free Q[t, t 1 ]-module. The dual S E = Hom Q[t,t 1 ](S(E), Q[t, t 1 ]) is called a Specht module. For base change to a commutative, Noetherian Q[t, t 1 ]-algebra R, set { S R (E) := S(E) Q[t,t 1 ] R, S E,R := S E Q]t,t 1 ] R = Hom R (S R (E), R). 18 / 27

A result of Ginzburg Guay Opdam Rouquier Let Φ 2e (t) denote the (cyclotomic) minimum polynomial for a primitive 2eth root of unity ζ = exp(2πi/2e) C. Fix a modular system (K, Q, k) by letting Q := Q[t, t 1] ] p, where p = (Φ 2e (t)); K := Q(t), the fraction field of Q; k := Q/m = Q( ζ), the residue field of Q. For R = Q, let S(E) := S Q (E) and S E := S E,Q. Proposition (GGOR) Assume that e 2. Suppose E = E are irreducible QW -modules. Then, for a sorting function f, Hom Hk (S k (E), S k (E )) 0 = f (E) < f (E ), Hom Hk (S k (E), S k (E)) = k, Ext 1 H( S(E), S(E )) 0 = f (E) < f (E ), and Ext 1 H( S(E), S(E)) = 0. 19 / 27

The sorting function f The sorting function f : irr(qw ) N is defined by f (E) = a E + A E = a(e) + N a(e det), where the generic degree d E = bt a E + + ct A E, with bc 0, a E A E. We can now define the (somewhat subtle) preorder f on Ω by setting ω f ω (for ω, ω Ω) if and only if either f (ω) < f (ω ), or ω and ω lie in the same two-sided cell. Note that the equivalence classes of the preorder f identify with the set of two-sided cells thus, f is strictly compatible with the set of two-sided cells. 20 / 27

Left cell and Dual left cell modules Parallel to S(E) and S E, we define, for each ω Ω, the left cell module S(ω) := H Lω /H < Lω H mod and the dual left cell module S ω := Hom Z (S(ω), Z) mod H. Define similarly the module over Q: S(ω) := S(ω) Q, Sω := S ω Q, ω Ω. Lusztig and GGOR tell us: Lemma There is an H-module isomorphism σ : ϖ (J ω Z) S(ω) induced by the map σ : J Z H, j y C y. In particular, S(ω) is a direct sum of modules S(E) for some E Irr(QW ). Moreover, for left cells ω, ω (and e 2), we have Ext 1 H( S ω, S ω ) 0 = f (ω) > f (ω ). 21 / 27

Preliminary lemma 1 Let R be a commutative ring and let C be an abelian R-category. Let M, Y C, and suppose that Ext 1 C (M, Y ) is generated as an R-module by elements ɛ 1,, ɛ m. Let χ := i ɛ i Ext 1 C (M m, Y ) correspond to the short exact sequence 0 Y X M m 0. Lemma (Homological vanishing) The map Ext 1 C (M, Y ) Ext1 C (M, X ), induced by the inclusion Y X, is the zero map. If Ext 1 C (M, M) = 0, then Ext1 C (M, X ) = 0. 22 / 27

Preliminary lemma 2 Lemma (Property of KL basis) Let N M be left ideals in H R, with each spanned by the Kazhdan Lusztig basis elements C y that they contain. Let s S be a simple reflection and assume either N = 0 or that q + 1 is not a zero divisor in R. Suppose 0 x M/N satisfies T s x = qx.then x is represented in M by an R-linear combination of Kazhdan Lusztig basis elements C y with sy < y. 23 / 27

The H-module X ω For ω Ω, we iteratively construct an H-module X ω, filtered by dual left cell modules, such that S ω X ω is the lowest nonzero filtration term, and Ext 1 H( S ω, X ω ) = 0 for all left cells ω. For j N, let Ω j = {ν Ω f (ν) = j}. Fix i = f (ω). Suppose Ext 1 H( S τ, S ω ) 0 for some τ Ω. Then, by the lemma, f (τ) > f (ω) = i. Assume f (τ) = j is minimal with this property. Since Q is a DVR and Ext 1 H( S τ, S ω ) is finitely generated, it follows that Ext 1 H( S τ, S ω ) is a direct sum of m τ ( 0) nonzero cyclic Q-modules. Let Ỹτ mτ be the extension of S τ by S ω, constructed by using generators for the cyclic modules. Then the homological vanishing lemma implies Ext 1 H( S τ, Ỹτ ) = 0. Continue to construct for certain j 1 < j 2 < < j r m τ1 m τ 1 Ω j1 S τ2 τ 1 τ Y j1 =, Y j1,j 2 = 2 Ω j2 S τ 2,, Y j1,...,j r = S X ω. ω Y j1 24 / 27

Let Ω be the set of all left cells that do not contain the longest element of a parabolic subgroup. Put T = λ S x λ H and X = ω Ω Xω. Theorem (DPS, 15) Assume that e 2. Let T + = T X, S + = End H( T + ) and (ω) = Hom H( S ω, T + ) for ω Ω. Then:- { (ω)} ω Ω is a strict stratifying system for the category S + -mod with respect to the quasi-poset (Ω, f ). Hence, S + has a standard stratification of length # Ω, the number of two-sided cells. 25 / 27

Identification of S + Theorem (DPS, 15) Assume that e 2. The Q-algebra S + is quasi-hereditary, with standard modules (E) = Hom H( S E, T + ), E Irr(QW ), and partial order < f. In order to use the covering theory of Rouquier, we consider the ring R := (C[t, t 1 ] (t ζ) ) We have the following category equivalence. Theorem (DPS, 15) Assume that e 2. The category of left modules over the base-changed algebra S + R := S + Q R is equivalent to the R-category O of modules for the rational Cherednik R-algebra associated to W. 26 / 27

THANK YOU 27 / 27