Representations of Rational Cherednik Algebras in Positive Characteristic
aa r X i v : . [ m a t h . R T ] N ov REPRESENTATIONS OF RATIONAL CHEREDNIK ALGEBRAS IN POSITIVECHARACTERISTIC
MARTINA BALAGOVI´C, HARRISON CHEN
Abstract.
We study rational Cherednik algebras over an algebraically closed field of positive characteristic.We first prove several general results about category O , and then focus on rational Cherednik algebrasassociated to the general and special linear group over a finite field of the same characteristic as the underlyingalgebraically closed field. For such algebras we calculate the characters of irreducible representations withtrivial lowest weight. Introduction
Given an algebraically closed field k , a finite-dimensional k -vector space h , a finite group G ⊆ GL ( h )generated by reflections, a constant parameter t ∈ k and a collection of constants c s ∈ k labeled by conjugacyclasses of reflections s in G , the rational Cherednik algebra H t,c ( G, h ) is a certain non-commutative infinite-dimensional associative algebra over k , which deforms the semidirect product of the group algebra k G andthe symmetric algebra on h and h ∗ , S ( h ∗ ⊕ h ). Rational Cherednik algebras have been extensively studiedsince the early 1990s, and most efforts have focused on the case when the underlying field k is the field ofcomplex numbers. This paper is one of the first attempts to study their representation theory in the casewhere k is an algebraically closed field of finite characteristic p .The parameter t can be rescaled by a nonzero constant, producing two families of algebras with differenttypes of behavior: one for t = 0 and one for t = 0, the latter being equivalent to t = 1.In characteristic zero, one commonly defines a category of H t,c ( G, h )-representations called category O ,analogous to category O in Lie theory. It is generated (under taking subquotients and extensions) by standardor Verma modules M t,c ( τ ), which are parametrized by irreducible representations τ of the finite group G .Verma modules admit a contravariant form B , such that the kernel of the form is the unique maximal propersubmodule J t,c ( τ ) of M t,c ( τ ). The quotients L t,c ( τ ) = M t,c ( τ ) / Ker B comprise all irreducible modules in O . In positive characteristic we define category O = O t,c in a way that allows us to formulate and proveanalogues of the properties and results in characteristic zero. One significant difference is that, while incharacteristic zero and for generic choice of parameters Verma modules M t,c ( τ ) are irreducible, this neverhappens in positive characteristic. The reason is that the algebra H t,c ( G, h ) has a large center, so the module M t,c ( τ ) always has a large submodule. To account for this, we define baby Verma modules N t,c ( τ ), whichare quotients of Verma modules by the action of a certain large central subspace of H t,c ( G, h ). Baby Vermamodules are finite dimensional; consequently, all the irreducible quotients L t,c ( τ ) are finite-dimensional, andwe define category O to be the category of finite dimensional graded modules. It contains all the babyVerma modules, but not the Verma modules. This is analogous to the situation in the representation theoryof Lie algebras in positive characteristic and to the study of the rational Cherednik algebras H ,c ( G, h ) incharacteristic zero.We prove the standard theorems about category O . Namely, we show that any irreducible object, up tograding shifts, is isomorphic to some L t,c ( τ ), and that the baby Verma modules admit a contravariant formwith the usual properties. We define characters and show that for t = 0 and generic c , the characters of L t,c ( τ ) are of a specific form, depending on the structure of a certain reduced module R t,c ( τ ). We calculatean upper bound for the dimension of irreducible modules.From here, we turn to investigate the characters of L t,c ( τ ) for t = 0 and t = 1 and all values of c forspecific classes of groups G . Over a field k of characteristic p , we study the rational Cherednik algebraassociated to the general and special linear group over a finite field, G = GL n ( F q ) and G = SL n ( F q ), for q = p r . These groups are reflection groups which have no counterpart in characteristic zero. e describe the characters of the irreducible modules L t,c (triv) associated to the trivial representation of GL n ( F q ), for all c (Theorems 4.3 and 4.10). One remarkable property is that for sufficiently large n and p ,the character of L ,c (triv) does not depend on the specific choice of c . This phenomenon does not arise incharacteristic zero.For n, p, r large enough, the structure (specifically, the conjugacy classes) of GL n ( F q ) and SL n ( F q ) issimilar enough to enable us to obtain the characters of the irreducible modules L t,c (triv) for SL n ( F q ) as acorollary of the corresponding results for GL n ( F q ). We specify the exact conditions on n, p, r being “largeenough” and calculate the remaining characters, thus obtaining the complete classification of L t,c (triv) forthe rational Cherednik algebra associated to SL n ( F q ). The main results about SL n ( F q ) are Theorems 5.6and 5.11.In the first phase of this work we have gathered information about the structure of irreducible represen-tations through calculations of the contravariant form B in the algebra software MAGMA [2]. We used thisdata to formulate conjectures and to settle the cases of small groups, which are often different from thegeneral situation (for GL n ( F p r ), “small” refers to small n, p, r ).This is the first paper in a series of two that we are planning on this topic. The second paper will bededicated to studying, in more detail, category O for the rational Cherednik algebras associated to the group GL ( F p ), and calculating the characters of irreducible representations L t,c ( τ ) for all τ .The roadmap of this paper is as follows. Section 2 contains definitions of rational Cherednik algebras,Dunkl operators, baby Verma modules, category O , and general results analogous to the ones in characteristiczero. In Section 3 we define characters of objects in category O and discuss characters of irreducible modulesfor generic c . In Section 4 we study characters of irreducible modules associated to the trivial lowest weightfor G = GL n ( F q ), and offer the complete classification for all t and c . In Section 5 we solve the same problemfor G = SL n ( F q ). The main results are Theorems 4.3, 4.10, 5.6 and 5.11. Appendix A contains data andconjectures about characters of rational Cherednik algebras associated to orthogonal groups over a finitefield, which might be used for further work.2. Definitions and basic properties
Notation.
Let k be an algebraically closed field of characteristic p , F q a finite field of q = p r elements, G a finite group, k G its group algebra, h a faithful n -dimensional representation of G over k , and h ∗ its dualrepresentation. We may regard G as a subgroup of GL ( h ). We will often let { x , . . . , x n } denote a basis of h ∗ and { y , . . . , y n } the dual basis of h . Let ( · , · ) be the canonical pairing h ⊗ h ∗ → k or h ∗ ⊗ h → k .For any vector space V let T V and SV denote the tensor and symmetric algebra of V over k , and S i V the homogeneous subspace of SV of degree i . For a graded vector space M , let M i denote the i -th gradedpiece, and M [ j ] the same vector space with the grading shifted by j , meaning M [ j ] i = M i + j . For M = ⊕ M i a graded vector space, define its Hilbert series asHilb M ( z ) = X i dim M i z i . For a filtered module F , let gr( F ) denote the associated graded module. For an associative algebra A , a, b ∈ A , S ⊆ A , let [ a, b ] = ab − ba be the usual commutator, and let h S i be the ideal generated by thesubset S .For V some space of polynomials and m = p a a power of the characteristic p , we define V m to be the set { f m | f ∈ V } . If V ⊂ k [ x , . . . x n ] is graded and the Hilbert series of k [ x , . . . x n ] /V is h ( z ), then the Hilbertseries of k [ x , . . . x n ] / h V m i ∼ = k [ x , . . . x n ] ⊗ k [ x m ,...x mn ] ( k [ x m , . . . x mn ] / h V m i ) is h ( z m ) (cid:16) − z m − z (cid:17) n . For λ ∈ F q , λ = 0, let d λ be the following elements of GL n ( F q ): for λ = 1 , d λ = λ − · · ·
00 1 0 · · ·
00 0 1 · · · · · · , d = · · ·
00 1 0 · · ·
00 0 1 · · · · · · . .2. Reflection groups.Definition 2.1.
An element s ∈ GL ( h ) is a reflection if the rank on h of 1 − s is 1. A finite subgroup G ⊂ GL ( h ) is a reflection group if it is generated by reflections. Example 2.2.
Let h = k n and F q ⊆ k a finite subfield. The group GL n ( F q ) is a finite subgroup of GL ( h ),generated by conjugates of d λ . All d λ and their conjugates are reflections, so GL n ( F q ) is a reflection group.This is the main example, in the sense that for k the algebraic closure of F p , and for any reflection group G ⊆ GL ( h ) one can construct the finite field of coefficients of G , which is some finite field of characteristic p , and view G as a subgroup of GL n ( F q ). Let h F = F nq be the corresponding F q -form of h = k n preserved by G . Reflections in G are elements that are conjugate in GL ( h ) to some d λ . If G = GL n ( F q ), there are q conjugacy classes of reflections, with representatives d λ . If G is a proper subgroup of GL n ( F q ), it is possiblefor elements of G to be conjugate in GL ( h ) but not in G , and it is thus possible to have reflections whichare not G -conjugate to any d λ , and to have reflections with the same eigenvalues which are not in the sameconjugacy class.A reflection s ∈ G is called a semisimple reflection if it is semisimple as an element of GL ( h ); such elementsare conjugate in GL ( h ) to some d λ with λ = 1. A reflection s ∈ G is called a unipotent reflection if it isunipotent as an element of GL ( h ); such an element is conjugate in GL ( h ) to d , and has the property that s p = 1. Note that in characteristic zero, a unipotent reflection generates an infinite subgroup, so consideringunipotent reflections is unique to working in positive characteristic.The group G acts on h , and we define the structure of the dual representation on h ∗ in the standardway: for x ∈ h ∗ , y ∈ h , g ∈ G , let ( g . y, x ) = ( y, g − . x ). After choosing bases of h and h ∗ , we see that g ∈ GL n ( F q ) ⊆ GL ( h ) acts in the dual representation as a matrix ( g − ) t ∈ GL n ( F q ) ⊆ GL ( h ∗ ). Consequently, g is a reflection on h with eigenvalue λ − if and only if it is a reflection on h ∗ with eigenvalue λ . Proposition 2.3.
There exists a bijection between the set of reflections in GL ( h F ) and the set of all vectors α ⊗ α ∨ = 0 in h ∗ F ⊗ h F such that ( α, α ∨ ) = 1 . The reflection s corresponding to α ⊗ α ∨ acts:on h ∗ by s . x = x − ( α ∨ , x ) α on h by s . y = y + ( y, α )1 − ( α, α ∨ ) α ∨ . If ( α, α ∨ ) = 0 , such a reflection s is semisimple, acting on h ∗ with eigenvalue of multiplicity n − andeigenvalue λ = 1 − ( α ∨ , α ) = 1 of multiplicity . If ( α, α ∨ ) = 0 , the reflection is unipotent, acting on h ∗ witheigenvalue of multiplicity n − and one Jordan block of size .Proof. For any reflection s , the operator 1 − s on h ∗ F has rank one, and can thus be represented as α ⊗ α ∨ ∈ h ∗ F ⊗ h F , so that for all x ∈ h ∗ , (1 − s ) . x = ( α ∨ , x ) α. Since s is invertible, 0 = s . α = (1 − ( α ∨ , α )) α , so ( α ∨ , α ) = 1 and 1 − ( α ∨ , α ) is the eigenvalue of s on h ∗ that is not 1. The formula for the action of s on h in terms of α ⊗ α ∨ follows from the one for the action of s on h ∗ and the definition of dual representation.Conversely, given any α ⊗ α ∨ ∈ h ∗ F ⊗ h F , the above formulas define dual reflections on h F and h ∗ F .Any vector x in the kernel of α ∨ is an eigenvector of s with eigenvalue 1. If ( α, α ∨ ) = 0, then α is aneigenvector with eigenvalue 1 − ( α ∨ , α ) = 1, and s is semisimple. If ( α, α ∨ ) = 0, then s i . x = x − i ( α ∨ , x ) α ,so s p = 1 and s is unipotent. (cid:3) Let S ⊆ G be the set of all reflections in G . Let α s and α ∨ s denote a choice of vectors from the propositionwhich correspond to a reflection s . Let λ s = 1 − ( α ∨ s , α s ); then s has an eigenvalue λ s on h ∗ and λ − s on h .In GL ( h F ), s is conjugate to the d λ − s from Subsection 2.1.2.3. Rational Cherednik algebras.
Let t ∈ k . Let c : S → k be a conjugation invariant function on theset of reflections, which we write as s c s . efinition 2.4. The rational Cherednik algebra H t,c ( G, h ) is the quotient of the semidirect product k G ⋉ T ( h ⊕ h ∗ ) by the ideal generated by relations:[ x, x ′ ] = 0 , [ y, y ′ ] = 0 , [ y, x ] = ( y, x ) t − X s ∈ S c s ((1 − s ) . x, y ) s, for all x, x ′ ∈ h ∗ , y, y ′ ∈ h . Note that for g ∈ G and y ∈ h , we use notation gy for multiplication in the algebra, and g . y for the actionfrom the representation; they are related by gyg − = g . y .The parameters t and c can be simultaneously rescaled, in the sense that H t,c ( G, h ) ∼ = H at,ac ( G, h ) for any a ∈ k × (the isomorphism sends generators x ∈ h ∗ to tx and fixes h and G ). This implies that it is enoughto study two cases with respect to t , t = 0 and t = 1. We are mostly interested in the t = 1 case.It is clear from the definition that any element of the algebra H t,c ( G, h ) can be written as a linearcombination of gx a · · · x a n n y b · · · y b n n for some g ∈ G and some integers a i , b i ≥
0. The common questionasked for algebras defined in such a way by generators and relations is whether this set is also linearlyindependent. The answer in the case of Cherednik algebras is positive, as proven in [6], Theorem 2.1.
Theorem 2.5 (PBW Theorem for Cherednik algebras) . The set { gx a · · · x a n n y b · · · y b n n | g ∈ G, a i , b i ≥ } is a basis for H t,c . We will need a localization lemma, which is proved as in characteristic zero (see [4]). Let h reg be thesubspace of h consisting of elements that are not contained in any reflection hyperplane, meaning they arenot fixed by any s ∈ S . Let D ( h reg ) be the algebra of differential operators on h reg . Let H loc t,c ( G, h ) bethe localization of H t,c ( G, h ) as k [ h ] = S h ∗ -module away from the reflection hyperplanes. Define the mapΘ t,c : H t,c ( G, h ) → k G ⋉ D ( h reg ) by Θ t,c ( x ) = x , Θ t,c ( y ) = t∂ y − P s ∈ S c s ( α s ,y ) α s (1 − s ), Θ t,c ( g ) = g for x ∈ h ∗ , y ∈ h , g ∈ G . Lemma 2.6 (Localization lemma) . The induced map of localizations Θ t,c : H loc t,c → k G ⋉ D ( h reg ) is an isomorphism of algebras. Verma Modules M t,c ( τ ) and Dunkl operators.Definition 2.7. Let τ be an irreducible finite-dimensional representation of G . Define a k G ⋉ S h -modulestructure on it by requiring the h -action on τ to be zero. The Verma module is the induced H t,c ( G, h )-module M t,c ( G, h , τ ) = H t,c ( G, h ) ⊗ k G ⋉ S h τ . We will write M t,c ( τ ) instead of M t,c ( G, h , τ ) when it is clear what are G and h .As it is an induced module, it satisfies the following universal mapping property: Lemma 2.8 (Frobenius reciprocity) . Let M be an H t,c ( G, h ) -module. Let τ ⊂ M be a G -submodule on which h ⊆ H t,c ( G, h ) acts as zero. Then there is a unique homomorphism φ : M t,c ( τ ) → M of H t,c ( G, h ) -modulessuch that φ | τ is the identity. Define a Z − grading on H t,c ( G, h ) by letting x ∈ h ∗ have degree 1, y ∈ h have degree -1, and g ∈ G havedegree 0. We will denote by the subscript + the positive degree elements of a graded module.By the PBW Theorem 2.5, M t,c ( τ ) ∼ = S h ∗ ⊗ τ as k -vector spaces. The grading on M t,c ( τ ) by degree of S h ∗ is compatible with the grading on H t,c ( G, h ) defined above, and we consider it as a graded module.Through the identification M t,c ( τ ) ∼ = S h ∗ ⊗ τ , the action of the generators of H t,c ( G, h ) can be explicitlywritten as follows. Let f ⊗ v ∈ S h ∗ ⊗ τ ∼ = M t,c ( τ ), x ∈ h ∗ , y ∈ h and g ∈ G . Then x . ( f ⊗ v ) = ( xf ) ⊗ v,g . ( f ⊗ v ) = g . f ⊗ g . vy . ( f ⊗ v ) = t∂ y ( f ) ⊗ v − X s ∈ S c s ( y, α s ) α s (1 − s ) . f ⊗ s . v. he operators on M t,c ( τ ) corresponding to the action of y ∈ h , given by D y = t∂ y ⊗ − X s ∈ S c s ( y, α s ) α s (1 − s ) ⊗ s, are called Dunkl operators .We say a homogeneous element v ∈ M t,c ( τ ) is singular if D y v = 0 for all y ∈ h . Any such element ofpositive degree generates a proper H t,c ( G, h ) submodule. By Lemma 2.8, this submodule is isomorphic to aquotient of M t,c (( k G ) v ). Remark 2.9.
The more common definition of the Dunkl operator in characteristic zero is D y = t∂ y ⊗ − X s ∈ S c s y, α s )(1 − λ s ) α s (1 − s ) ⊗ s. The difference is due to a different convention in normalization of α s and α ∨ s . As explained in Proposition2.3, for a given reflection s , the vectors α s and α ∨ s are determined uniquely up to multiplication by nonzeroscalars. We choose their mutual normalization so that ( α, α ∨ ) = 1 − λ s , while the common normalizationin characteristic zero is ( α, α ∨ ) = 2. That brings about an additional factor of − λ s in some formulas. Thereason for choosing a non-standard convention is that in characteristic p there are unipotent reflections, forwhich λ s = 1 and ( α, α ∨ ) = 0.2.5. Contravariant Form.
The results from this section can be found in Section 3.11 of [4].There is an analogue of Shapovalov form on Verma modules. First, for any graded H t,c ( G, h )-module M with finite dimensional graded pieces, define its restricted dual M † to be the graded module whose i − th graded piece it the dual of the i − th graded piece of M . It is a left module for the opposite algebra H t,c ( G, h ) opp of H t,c ( G, h ). Let ¯ c : S → k be the function ¯ c ( s ) = c ( s − ). There is a natural isomorphism H t,c ( G, h ) opp → H t, ¯ c ( G, h ∗ ) that is the identity on h and h ∗ , and sends g g − for g ∈ G , making M † a H t, ¯ c ( G, h ∗ )-module. Definition 2.10.
Let τ be an an irreducible finite-dimensional representation of G . By Lemma 2.8, there isa unique homomorphism φ : M t,c ( G, h , τ ) → M t, ¯ c ( G, h ∗ , τ ∗ ) † which is the identity in the lowest graded piece τ . By adjointness, it is equivalent to the contravariant form pairing B : M t,c ( G, h , τ ) × M t, ¯ c ( G, h ∗ , τ ∗ ) → k . Proposition 2.11.
The contravariant form B satisfies the following properties. a) It is G -invariant: for f ∈ M t,c ( τ ) , h ∈ M t, ¯ c ( τ ∗ ) , B ( g . f, g . h ) = B ( f, h ) . b) For x ∈ h ∗ , f ∈ M t,c ( τ ) , and h ∈ M t, ¯ c ( τ ∗ ) , B ( xf, h ) = B ( f, D x ( h )) . c) For y ∈ h , f ∈ M t,c ( τ ) , and h ∈ M t, ¯ c ( τ ∗ ) , B ( f, yh ) = B ( D y ( f ) , h ) . d) The form is zero on elements in different degrees: if f ∈ M t,c ( τ ) i and h ∈ M t, ¯ c ( τ ∗ ) j , i = j , then B ( f, h ) = 0 . e) The form is the canonical pairing of τ and τ ∗ in the zeroth degree: for v ∈ τ = M t,c ( τ ) , f ∈ τ ∗ = M t, ¯ c ( τ ∗ ) , B ( v, f ) = ( v, f ) . As B respects the grading of M t,c ( τ ) and M t, ¯ c ( τ ∗ ), we can think of it as a collection of bilinear forms ontheir finite-dimensional graded pieces. Let B i be the restriction of B to the M t,c ( τ ) i ⊗ M t, ¯ c ( τ ∗ ) i . We defineKer B to be Ker φ ⊆ M t,c ( G, h , τ ). Singular vectors of positive degree in M t,c ( τ ) are in Ker B , and so arethe submodules generated by them.2.6. Baby Verma modules N t,c ( τ ) and irreducible modules L t,c ( τ ) .Definition 2.12. For τ an irreducible G -representation, define the H t,c ( G, h )-representation L t,c ( τ ) = L t,c ( G, h , τ ) as the quotient M t,c ( G, h , τ ) / Ker B .The modules L t,c ( τ ) are graded. We are going to show that they are irreducible, as in characteristiczero. A notable difference is that, while in characteristic zero, for generic t and c , the module M ,c ( τ ) isirreducible and hence equal to L ,c ( τ ), in characteristic p or for t = 0 this never happens. On the contrary,all L t,c ( τ ) are finite dimensional, and M t,c ( τ ) always have a large submodule. Because of that, we sometimesprefer using baby Verma modules defined below instead of Verma modules. The definition and the name re analogous to the ones used in Lie theory in characteristic p and to the notation used in [5] for rationalCherednik algebras in characteristic zero at t = 0.Let (( S h ∗ ) G ) + denote the subspace of G -invariants in S h ∗ of positive degree. At t = 1, the subspace(( S h ∗ ) G ) p + is central in H ,c ( G, h ), and (( S h ∗ ) G ) p + M ,c ( τ ) is a proper submodule of M ,c ( τ ). Definition 2.13.
The baby Verma module for the algebra H ,c ( G, h ) is the quotient N ,c ( τ ) = N ,c ( G, h , τ ) = M ,c ( τ ) / (( S h ∗ ) G ) p + M ,c ( τ ) . Since ( S h ∗ ) G is graded, N ,c ( τ ) is a graded module. The subspace (( S h ∗ ) G ) p + M ,c ( τ ) is contained inKer B . To see this, let Z ∈ (( S h ∗ ) G ) p + be an arbitrary homogeneous element of positive degree m , v ∈ τ and y ∈ h arbitrary. Then D y ( Z ⊗ v ) = ( yZ ) . v = ( Zy ) . v = Z . ( y . v ) = 0, so Z ⊗ v is singular and therefore inKer B .Because of this, the form B descends to the N ,c ( τ ), and L ,c ( τ ) can be alternatively realized as N ,c ( τ ) / Ker B .To define baby Verma modules at t = 0, we use that (( S h ∗ ) G ) + is central in H ,c ( G, h ), so (( S h ∗ ) G ) + M ,c ( τ )is a proper submodule of M ,c ( τ ). Definition 2.14.
The baby Verma module for the algebra H ,c ( G, h ) is the quotient N ,c ( τ ) = N ,c ( G, h , τ ) = M ,c ( τ ) / (( S h ∗ ) G ) + M ,c ( τ ) . By the same arguments as above, it is graded, the form B descends to it, and L ,c ( τ ) can be alternativelyrealized as a quotient of L ,c ( τ ) by the kernel of B .Next, we turn to the basic properties of modules L t,c ( τ ) and N t,c ( τ ). We will need the following lemma,which is a consequence of the Hilbert-Noether Theorem and can be found in [9] as Corollary 2.3.2. Lemma 2.15.
For any finite group G , field F , and a finite dimensional F [ G ] -module h , the algebra ofinvariants ( S h ) G is finitely generated over F , and S h is a finite integral extension of ( S h ) G . The following proposition is unique to fields of positive characteristic.
Proposition 2.16.
All N t,c ( τ ) , and thus L t,c ( τ ) , are finite dimensional.Proof. As explained in section 2.1, the Hilbert series of a baby Verma module is defined asHilb N ,c ( τ ) ( z ) = X i dim N ,c ( τ ) i z i , Hilb N ,c ( τ ) ( z ) = X i dim N ,c ( τ ) i z i . The series at t = 0 and t = 1 are related byHilb N ,c ( τ ) ( z ) = (cid:18) − z p − z (cid:19) n Hilb N ,c ( τ ) ( z p ) . Because of this formula, and because L t,c ( τ ) is a quotient of N t,c ( τ ), it is enough to prove the propositionfor N ,c ( τ ).The representation τ is finite-dimensional, so M ,c ( τ ) ∼ = S h ∗ ⊗ τ is a finite module over S h ∗ . By Lemma2.15, S h ∗ is a finite module over ( S h ∗ ) G .For any commutative ring R , maximal ideal m , and finite R -module M , the quotient M/ m M is a finite-dimensional vector space over R/ m . Applying this to m = (( S h ∗ ) G ) + , R = (( S h ∗ ) G ), and M = M ,c ( τ ), itfollows that N ,c ( τ ) is finite dimensional over k . (cid:3) Remark 2.17.
As in characteristic zero, the module L t,c ( τ ) is irreducible. In characteristic zero one showsthis by showing that Ker B is the sum of all graded proper submodules of M t,c ( τ ), and that there is a naturalinner Z -grading on M t,c ( τ ), and that all submodules are graded. In characteristic p this fails, as there isonly a natural inner Z /p Z -grading. There exist submodules of M t,c ( τ ) which are not Z -graded: for example,for any f ∈ ( S h ∗ ) G , the subspace S h ∗ (1 + f p ) ⊗ τ is a proper submodule. In fact, the sum of all propersubmodules of M t,c ( τ ) is the whole M t,c ( τ ) (the sum of all submodules of the form S h ∗ (1 + f p ) ⊗ τ equals M t,c ( τ )). However, the situation is better if we consider only graded submodules, or if we let baby Vermamodules take over the role of Verma modules. This is explained more precisely by the following results.To show irreducibility of L t,c ( τ ), we will need the following form of Nakayama’s lemma. Recall that the Jacobson radical of a commutative ring R , denoted rad( R ), is the maximal ideal that annihilates all simplemodules, or equivalently, the intersection of all maximal ideals. Also recall that k [[ x , . . . , x n ]] is local, sorad( k [[ x , . . . , x n ]]) = h x , . . . , x n i . emma 2.18 (Nakayama) . Let R be a commutative ring, I ⊂ rad ( R ) an ideal, and M a finitely generated R -module. Let m , . . . , m n ∈ M be such that their projections generate M/IM over
R/I . Then, m , . . . , m n generate M over R . Lemma 2.19.
Let L t,c ( τ ) + be the positively graded part of L t,c ( τ ) . If v , . . . , v m ∈ L t,c ( τ ) are such thattheir projections ¯ v , . . . , ¯ v m ∈ L t,c ( τ ) /L t,c ( τ ) + ∼ = τ span τ over k , then v , . . . , v m generate L t,c ( τ ) as an S h ∗ module.Proof. This is a direct application of Nakayama’s lemma, with R = k [[ x , . . . , x n ]], M = L t,c ( τ ), and I = h x , . . . , x n i = rad( R ). By Lemma 2.18, v , . . . , v m generate L t,c ( τ ) as a k [[ x , . . . , x n ]]-module. Since L t,c ( τ ) is finite-dimensional, an infinite power series really acts on M as a finite polynomial. (cid:3) Proposition 2.20.
The L t,c ( τ ) are irreducible for every t, c and τ .Proof. Let f be any nonzero element of L t,c ( τ ). We claim that it generates L t,c ( τ ) as an H t,c ( G, h ) module.If the projection ¯ f of f to L t,c ( τ ) ∼ = τ is nonzero, then the set of G -translates of ¯ f spans the irreduciblerepresentation τ , so by Lemma 2.19 the set of G -translates of f generates L t,c ( τ ) as an S h ∗ module, and f generates L t,c ( τ ) as an H t,c ( G, h )-module.If the projection of f to L t,c ( τ ) ∼ = τ is zero, write f = f + · · · + f d , with f i ∈ L t,c ( τ ) i . The form B is non-degenerate on L t,c ( τ ), so f / ∈ Ker B ; it respects the grading so there is some r > f r Ker B . Theform is bilinear, so there exist a monomial y a · · · y a n n ∈ S r h and v ∈ τ ∗ such that B ( f r , y a · · · y a n n v ) = 0. Bycontravariance of B , and writing D i for D y i , this is equal to 0 = B ( D a · · · D a n n f r , v ) = B ( D a · · · D a n n f, v ).So, D a · · · D a n n f is a nonzero element of L t,c ( τ ), with a nonzero projection to L t,c ( τ ) ∼ = τ . By the previ-ous reasoning, D a · · · D a n n f generates L t,c ( τ ) as an H t,c ( G, h )-module, and thus f generates L t,c ( τ ) as an H t,c ( G, h )-module. (cid:3) Corollary 2.21.
A Verma module M t,c ( τ ) has a unique maximal graded submodule.Proof. Consider the sum of all graded submodules. None of these submodules have elements in L t,c ( τ ) ∼ = τ ,since such elements generate the entire module M t,c ( τ ), so their sum is a proper submodule of M t,c ( τ ). (cid:3) Corollary 2.22.
A baby Verma module N t,c ( τ ) has a unique maximal submodule.Proof. Let N be any proper submodule, and f ∈ N arbitrary nonzero element. Write f = f + · · · + f d ,with f i in the i -th graded piece. Baby Verma modules are finite-dimensional, N is a proper submodule,so a similar argument as in Proposition 2.20 implies that f = 0. Thus, any proper submodule has zeroprojection to the zeroth graded piece, and so the sum of all proper submodules is still proper. (cid:3) The unique maximal graded submodule of M t,c ( τ ) descends to the unique maximal submodule of N t,c ( τ ),which we will call ¯ J t,c ( τ ). The following corollary follows by the irreducibility of M t,c ( τ ) / Ker B . Corollary 2.23.
The kernel of B is J t,c ( τ ) . Thus, L t,c ( τ ) = M t,c ( τ ) /J t,c ( τ ) = M t,c ( τ ) / Ker B ∼ = N t,c ( τ ) / Ker B = N t,c ( τ ) / ¯ J t,c ( τ ) . Category O . We now define the category O of H t,c ( G, h ) modules. The definition, which is somewhatdifferent than in characteristic zero, is justified by Remark 2.17 and Proposition 2.25. Definition 2.24.
The category O t,c ( G, h ) is the category of Z -graded H t,c ( G, h )-modules which are finite-dimensional over k .We usually write O t,c or O instead of O t,c ( G, h ) when it is clear what the arguments are. Proposition 2.25.
For every irreducible L ∈ O t,c ( G, h ) , there is a unique irreducible G -representation τ and i ∈ Z such that L ∼ = L t,c ( G, h , τ )[ i ] .Proof. Let L ∈ O t,c be any irreducible module in category O . It is graded and finite-dimensional, so theremust be a lowest graded piece L i . Without loss of generality, we can shift indices so that the lowest gradedpiece is in degree zero. Further, if the degree zero part L , which is a G -representation, is reducible, thenany proper G -subrepresentation of L generates a proper H t,c ( G, h )-subrepresentation of L . So, L ∼ = τ forsome irreducible G -representation τ . By Proposition 2.8, there exists a nonzero graded homomorphism φ : t,c ( τ ) → L . Since L is irreducible, this homomorphism is surjective, and L is isomorphic to M t,c ( τ ) / Ker( φ ).Since J t,c ( τ ) is the unique maximal graded submodule, Ker( φ ) = J t,c ( τ ) and the result follows. Uniquenessfollows from the fact that L t,c ( τ ) i ∼ = τ . (cid:3) A lemma about finite fields.Lemma 2.26.
Let q = p r be a prime power. Let f ∈ k [ x , x , . . . , x n ] be a polynomial in n variables, forwhich there exists a variable x i such that deg x i ( f ) < q − . Then X x ,...x n ∈ F q f ( x , . . . x n ) = 0 . Proof.
It is enough to prove the claim for monomials of degree m < q − f = x m and S m := P i ∈ F q i m . For every j ∈ F q , j m S m = P i ∈ F q ( ij ) m = S m , which is equivalent to (1 − j m ) S m = 0 forall j = 0. As m < q −
1, there exists some j such that 1 − j m = 0, and so S m = 0. (cid:3) Remark 2.27.
In particular, the assumptions of the lemma are satisfied by all f such that deg( f ) < n ( q − . Characters
Definition and basic properties.Definition 3.1.
Let K ( G ) be the Grothendieck group of the category of finite dimensional representationsof G over k . For M = ⊕ i M i any graded H t,c ( G, h ) module with finite dimensional graded pieces, define itscharacter to be the power series in formal variables z, z − with coefficients in K ( G ) χ M ( z ) = X i [ M i ] z i , and recall we defined its Hilbert series asHilb M ( z ) = X i dim( M i ) z i . If M is in category O , it is finite dimensional and its character is in K ( G )[ z, z − ].The character of M t,c ( τ ) is χ M t,c ( τ ) ( z ) = X i ≥ [ S i h ∗ ⊗ τ ] z i , and its Hilbert series is Hilb M t,c ( τ ) ( z ) = dim( τ )(1 − z ) n . The character of N t,c ( τ ) depends on whether t = 0 or t = 0; these cases are related by χ N ,c ( τ ) ( z ) = χ N ,c ( τ ) ( z p ) · (cid:18) − z p − z (cid:19) n . If G is a reflection group for which the algebra of invariants ( S h ∗ ) G is a polynomial algebra with homo-geneous generators of degrees d , . . . d n , then the characters of baby Verma modules are: χ N ,c ( τ ) ( z ) = χ M ,c ( τ ) ( z )(1 − z d )(1 − z d ) . . . (1 − z d n ) ,χ N ,c ( τ ) ( z ) = χ M ,c ( τ ) ( z )(1 − z pd )(1 − z pd ) . . . (1 − z pd n ) . There is no known general formula for the characters of the irreducible modules L t,c ( τ ), even in char-acteristic zero. The main focus of the second half of this paper is describing these modules for particularseries of groups G , in terms of their characters, or through describing the generators for the maximal propersubmodules J t,c ( τ ), or through describing the composition series of baby Verma modules and Verma modules.It is clear from the definition that Hilb L t,c ( τ ) ( z ) = ∞ X i =0 rank( B i ) z i . his is useful because matrices B i and their ranks can be calculated in many examples using algebra software.We used MAGMA [2] to do these calculations for small examples in order to form conjectures which becamesections 4 and 5 of this paper. Some unused computational data of this kind can be found in the AppendixA.3.2. Characters of L t,c ( τ ) at generic value of the parameter c . By definition, the i -th graded pieceof L t,c ( τ ) is, as a representation of G , equal to the quotient of S i h ∗ ⊗ τ by the kernel of B i . Let us fix t andconsider c = ( c s ) s as variables; B i depends on them polynomially. Let k | conj | be the space of functions fromthe finite set of conjugacy classes in G to k , and think of it as the space of all possible parameters c .Let d be the dimension of S i h ∗ ⊗ τ and let r be the rank of B i , seen as an operator over k [ c ]. For c outside of finitely many hypersurfaces in k | conj | , the rank of B i evaluated at c is equal to r , and the kernelof B i is some ( d − r )-dimensional representation of G , depending on c . All these representations have thesame composition series. (To see this, let V ( c ) be a flat family of G -representations, for example Ker B i forgeneric c . Let { σ i } i be a complete set of pairwise non-isomorphic irreducible k G − modules, and for all i let π i be a projective cover of σ i . Then the number [ V ( c ) : σ i ] of times σ i appears as a composition factor in V ( c ) is equal to the dimension of Hom( π i , V ( c )). So, for generic c it is the same, and for special c it might bebigger. But P i [ V ( c ) : σ i ] dim( σ i ) = dim V ( c ) is constant, so [ V ( c ) : σ i ] does not depend on c , and all V ( c )have the same factors in their composition multiplicities. They might however not be isomorphic, becausethey might be different extensions of their irreducible composition factors.)The map c Ker( B i ) = J t,c ( τ ) i , defined on the open complement of hypersurfaces in k | conj | , can bethought of as a rational function from k | conj | to the Grassmannian of ( d − r )-dimensional subspaces of S i h ∗ ⊗ τ .For c in some finite family of hypersurfaces in the parameter space k | conj | , the rank of B i evaluated at c is smaller than r , and the dimension of the kernel J t,c ( τ ) i is larger than d − r . We will now use the aboverational function to define a subspace J t, ( τ ) ′ i ⊆ J t, ( τ ) i at c = 0, which has similar properties to those J t, ( τ ) i would have if c = 0 was a generic point.If c = 0 is generic and rank of B i at c = 0 is d , let J t, ( τ ) ′ i = J t, ( τ ) i . Otherwise, pick a line inthe parameter space k | conj | which does not completely lie in one of the hypersurfaces, and which passesthrough 0. The composition of the inclusion of this line to k | conj | and the rational map from k | conj | to theGrassmannian is then a rational map from the punctured line to a projective space, and such a map canalways be extended to a regular map on the whole line. This associates to c = 0 a vector space J t, ( τ ) ′ i . Itgenerally depends on the choice of a line in the parameter space, and it always has the following properties: • dim( J t, ( τ ) ′ ) i = r − d ; • J t, ( τ ) ′ i ⊆ Ker B i = J t, ( τ ) i ; • J t, ( τ ) ′ i is G -invariant; • J t, ( τ ) ′ i has the same composition series as J t,c ( τ ) i for generic c .By making consistent choices for all i (for example, by choosing the same line in the parameter space for all i ), one can ensure an extra property: • J t, ( τ ) ′ = ⊕ i J t, ( τ ) ′ i is a H t, ( G, h ) subrepresentation of M t, ( τ ).This produces a subrepresentation J ′ t, ( τ ) at c = 0 such that the quotient M t, ( τ ) /J t, ( τ ) ′ behaves like L t,c ( τ ) at generic c , even when c = 0 is not generic. In particular, M t, ( τ ) /J t, ( τ ) ′ and L t,c ( τ ) at generic c have the same character. Example 3.2.
For G = GL ( F ), τ = triv, the form B restricted to M ,c (triv) ∼ = S h ∗ has a matrix,written here in the ordered basis ( x , x x , x x , x x , x ): B = c ( c + 1) c ( c + 1) c ( c + 1) c ( c + 1) 0 c ( c + 1) c ( c + 1) 0 0 c ( c + 1) c ( c + 1) 0 0 0 c ( c + 1) c ( c + 1) 0 0 c ( c + 1) c ( c + 1)0 c ( c + 1) c ( c + 1) c ( c + 1) c ( c + 1) . When c = 0 ,
1, this matrix has rank 4, and a one-dimensional kernel J ,c (triv) spanned by x + x x + x .For c = 0, the matrix is zero and J , (triv) is the whole S h ∗ . The above procedure defines J , (triv) ′ tobe k ( x + x x + x ). e will now draw conclusions about the character of L t,c ( τ ) for generic c using information about M t, ( τ ) /J ′ t, ( τ ). Lemma 3.3.
Let M be a free finitely-generated graded S h ∗ -module with free generators b , . . . , b m , and N a graded submodule of M . For f ∈ S h ∗ , y ∈ h , define ∂ y f b i = ( ∂ y f ) b i . If N is stable under ∂ y for all y ∈ h ,then it is generated by elements of the form P f pi b i for some f i ∈ S h ∗ .Proof. First, assume there is only one generator, so M ∼ = S h ∗ as left S h ∗ modules. Let N ′ = { f p | f ∈ S h ∗ } ∩ N . We claim that S h ∗ N ′ = N .Clearly, S h ∗ N ′ ⊂ N . To show that N ⊂ S h ∗ N ′ , we need to show that any f ∈ N can be written as asum of elements of the form h ( x , . . . , x n ) f ′ ( x p , . . . , x pn ), for some h ∈ S h ∗ and f ′ ( x p , . . . , x pn ) ∈ N .As N is graded, assume f is homogeneous of degree d . Write it as f = p − X i =0 x i f i ( x p , x , . . . , x n ) . The space N is stable under all partial derivatives, so for each j = 0 , . . . , p − x j ∂ j f = p − X i =1 i ( i − . . . ( i − j + 1) x i f i ( x p , x , . . . , x n ) ∈ N. The coefficient i ( i − . . . ( i − j + 1) is zero for i < j and is nonzero for i = j , so the matrix [ i ( i − . . . ( i − j + 1)] i,j is invertible, implying that x i f i ( x p , x , . . . , x n ) is in N for all i , and therefore (after applying ∂ i ),also f i ( x p , x , . . . , x n ) ∈ N .Applying the same argument on each f i for x , . . . , x n , it follows that f is of desired form. The claim for M ∼ = ⊕ S h ∗ b i follows directly from the one for S h ∗ . (cid:3) Let S ( p ) h ∗ be the quotient of S h ∗ by the ideal generated by x p , . . . x pn . Proposition 3.4.
The character of L ,c ( τ ) , for generic value of c , is of the form χ L ,c ( τ ) ( z ) = χ S ( p ) h ∗ ( z ) H ( z p ) for H ∈ K [ z ] the character of some graded G -representation. In particular, the Hilbert series of L ,c ( τ ) isof the form Hilb L ,c ( τ ) ( z ) = (cid:18) − z p − z (cid:19) n · h ( z p ) , for h a polynomial with nonnegative integer coefficients.Proof. As commented above, the character of L ,c ( τ ) = M ,c ( τ ) /J ,c ( τ ) is the same for all c outside of finitelymany hypersurfaces, and it is equal to the character of M , ( τ ) /J , ( τ ) ′ . At these values of parameter, t = 1and c = 0, Dunkl operators are particularly simple, and equal to partial derivatives: D y = ∂ y . By theprevious lemma, J , ( τ ) ′ is generated by p -th powers. Let f i ( x p , . . . x pn ) ⊗ v i , for some f i ∈ S h ∗ , v i ∈ τ , bethese generators.Define J ∗ to be the ( S h ∗ ) p -module generated by f i ( x p , . . . x pn ) ⊗ v i . Let the reduced module R t,c ( τ ) be the k [ G ] ⋉ S h ∗ -module defined as the quotient of S h ∗ ⊗ τ by the ideal generated by f i ( x , . . . x n ) ⊗ v i . Call itscharacter (in the sense of Definition 3.1) the reduced character of L t,c ( τ ), and denote it by H ( z ) ∈ K ( G )[ z ].Consider the multiplication map µ : S ( p ) h ∗ ⊗ (( S h ∗ ) p ⊗ τ ) /J ∗ → S h ∗ ⊗ τ /J , ( τ ) ′ . It is an isomorphism of graded G -representations, so it preserves characters. From this it follows that forgeneric c , χ L ,c ( τ ) ( z ) = χ M , ( τ ) /J , ( τ ) ′ ( z ) = χ S ( p ) h ∗ ( z ) H ( z p ) . (cid:3) By inspecting the proof and using that c is non-generic on a union of finitely many hypersurfaces, onecan strengthen the claim of the proposition as follows: for any hyperplane P passing through the origin inthe space of functions from the conjugacy classes of G to k , there exists a function H P ( z ) ∈ K ( G )[ z ] suchthat, for c generic in P , the character of L t,c ( τ ) is of the form χ S ( p ) h ∗ ( z ) H P ( z p ). .3. A dimension estimate for L ,c ( τ ) .Lemma 3.5. Any irreducible H ,c ( G, h ) -representation has dimension less than or equal to p n | G | .Proof. We begin with a definition, which will only be used in this proof. Let A be an algebra. A polynomialidentity is a nonzero polynomial f ( x , . . . , x r ) in non-commuting variables x , . . . x r , with a property that f ( a , . . . , a r ) = 0 for all a , . . . , a r ∈ A . Given an algebraically closed field k , a polynomial identity algebra ,or PI algebra is a k -algebra A that satisfies a polynomial identity. We say a PI algebra has degree r if itsatisfies the polynomial identity s r = P σ ∈ S r sgn( σ ) Q ri =1 x σ ( i ) .Our first claim is that H ,c ( G, h ) is a PI algebra. By Proposition V.5.4 in [1], A is a PI algebra if and only ifevery localization of A is also a PI algebra. By the localization lemma (Lemma 2.6), H loc1 ,c ( G, h ) = H loc1 , ( G, h ).Thus, it suffices to show that H , ( G, h ) is a PI algebra.Let Z be the center of H , ( G, h ). It is easy to see that Z = ( S ( h ⊕ h ∗ ) p ) G . Z is commutative, so wecan consider A ′ = Frac( Z ) ⊗ Z H , ( G, h ), which is an algebra over the field Frac( Z ). By Theorem V.8.1 in[1], this is a central simple algebra , i.e. an algebra that is finite-dimensional, simple, and whose center isexactly its field of coefficients. By the Artin-Wedderburn theorem, a central simple algebra is isomorphic toa matrix algebra over a division ring. To calculate the size of these matrices, we would like to determine thedimension of A ′ . We need to calculatedim Frac( Z ) A ′ = dim Frac(( S ( h ⊕ h ∗ ) p ) G ) Frac(( S ( h ⊕ h ∗ ) p ) G ) ⊗ ( S ( h ⊕ h ∗ ) p ) G H , ( G, h )Using the PBW theorem and the fact that for all algebras B and C , B ∼ = C ⊗ C B , we get that this is equalto dim Frac(( S ( h ⊕ h ∗ ) p ) G ) Frac(( S ( h ⊕ h ∗ ) p ) G ) ⊗ ( S ( h ⊕ h ∗ ) p ) G S ( h ⊕ h ∗ ) p ⊗ S ( h ⊕ h ∗ ) p S ( h ⊕ h ∗ ) ⊗ k G. Next, we use the following fact: if G acts on some vector space V (in our case, V = ( h ⊕ h ∗ ) p ) in such away that there exists a vector v ∈ V with a trivial stabilizer, then the G − orbit of v is a Frac( SV G ) − basisfor Frac( SV G ) ⊗ SV G SV , and dim Frac( SV G ) Frac( SV G ) ⊗ SV G SV = | G | .We get that the dimension above is equal todim Frac(( S ( h ⊕ h ∗ ) p ) G ) Frac(( S ( h ⊕ h ∗ ) p ) G ) ⊗ ( S ( h ⊕ h ∗ ) p ) G S ( h ⊕ h ∗ ) p ⊗ S ( h ⊕ h ∗ ) p S ( h ⊕ h ∗ ) ⊗ k G == | G | · p dim S h ⊕ h ∗ · | G | = | G | p n . So, A ′ is isomorphic to the matrix algebra over Frac(( S ( h ⊕ h ∗ ) p ) G ) of size | G | p n . By Corollary V.8.4 in[1], an r × r matrix algebra satisfies s r , so A ′ is a PI algebra of degree p n | G | . Consequently, H , ( G, h ) is a PIalgebra of degree p n | G | , and so is H ,c ( G, h ). By Proposition V.6.1(ii) in [1], any irreducible representationof a PI algebra of degree r has dimension less than or equal to r , and the result follows. (cid:3) Corollary 3.6.
Let h be the reduced Hilbert series of L ,c ( τ ) for generic c . Then L ,c ( τ ) has dimension h (1) p n , and ≤ h (1) ≤ | G | . Some observations, questions and remarks.Remark 3.7.
In many examples we considered, in particular whenever G = GL n ( F q ) or G = SL n ( F q ) and τ = triv, h (1) is equal to 1 or to | G | . In many other cases, it divides | G | . However, this is not always true.For G = GL ( F p ), τ = S p − h , the order of the group is ( p − p − p ), and the reduced Hilbert series is p + ( p − z + pz . So, h (1) = 3 p −
2, which does not always divide ( p − p − p ) (for example, when p = 3). Question 3.8.
For h ( z ) = P i a (1) i z i the reduced Hilbert series of L ,c ( τ ) and h ( z ) = P i a (0) i z i the Hilbertseries of L ,c ( τ ) , does the inequality a (0) i ≤ a (1) i hold? There is computational data supporting the positive answer. In many examples, particularly for G = GL n ( F q ) and SL n ( F p ), the equality h = h holds. An example when strict inequality is achieved is G = SL ( F ), τ = triv: the reduced Hilbert series is h ( z ) = (1 + z + z + z )(1 + z + z + z + z + z ),and the Hilbert series of L ,c ( τ ) is h ( z ) = 1. ecall that a finite dimensional Z + graded algebra A = ⊕ i A i is Frobenius if the top degree A d is onedimensional, and multiplication A i ⊗ A d − i → A d is a non-degenerate pairing. As a consequence, the Hilbertseries of A is a palindromic polynomial.The irreducible module L t,c (triv) is a quotient of M t,c (triv) ∼ = S h ∗ by the H t,c ( G, h ) submodule J t,c (triv),which is in particular an S h ∗ submodule. So, we can consider it as a quotient of the algebra S h ∗ by the leftideal J t,c (triv), and therefore as a finite dimensional graded commutative algebra. Proposition 3.9.
Assume that t, c, τ are such that the top graded piece of L t,c (triv) is one dimensional.Then L t,c (triv) is Frobenius.Proof. Let us first prove: a finite dimensional graded commutative algebra A = ⊕ di =0 A i is Frobenius if andonly if the kernel on A of the multiplication by A + = ⊕ i> A i is one dimensional. One implication is clear:if A is Frobenius, the kernel is the one dimensional space A d . For the other, assume the kernel on A of themultiplication by A + is one dimensional. The top nontrivial graded piece A d is always contained in it, so A d is one dimensional and equal to the kernel. Now assume there exists a nonzero element a n ∈ A n suchthat multiplication by a n , seen as a map A d − n → A d , is zero, and let 0 < n < d be the maximal index forwhich such an a n exists. As a n is not in the kernel of the multiplication by A + , there exists some b ∈ A + such that a n b = 0. We can assume without loss of generality that b is homogeneous, b ∈ A m , 0 < m < d − n .Then a n b ∈ A n + m , with n < n + m < d , is a nonzero element such that multiplication by it, seen as a map A d − n − m → A d , is zero, contrary to the choice of n as the largest such index.Now assume that A = L t,c (triv) has one dimensional top degree. Let 0 = f be in the kernel of multi-plication by A + . As the kernel is graded, assume without loss of generality that f is homogeneous. Then xf = 0 ∈ L t,c (triv) for all x ∈ h ∗ , so x is a highest weight vector . Under the action of H t,c ( G, h ), f generatesa subrepresentation of L t,c (triv) for which the highest graded piece consists of G -translates of x . As L t,c (triv)is irreducible, this subrepresentation has to be the entire L t,c (triv), and f is in the top degree, which is byassumption one dimensional. (cid:3) Remark 3.10.
In many instances we observed, the algebra L t,c (triv) is Frobenius for generic c and haspalindromic Hilbert series. However, this is not true in general: let k = F , G = S the symmetric groupon five letters, h the four dimensional reflection representation { ( z , . . . z ) ∈ k | z + . . . + z = 0 } with thepermutation action, and τ = triv. Then the character of L ,c (triv) is(1 + t )(1 + t + t )(1 + 2 t + 3 t + 4 t ) . We thank Sheela Devadas and Steven Sam for pointing out this counterexample to us.3.5.
Invariants and characters of baby Verma modules for G = GL n ( F q ) and G = SL n ( F q ) . Remember that N t,c ( τ ) is a quotient of M t,c ( τ ) ∼ = S h ∗ ⊗ τ by ( S h ∗ ) G + ⊗ τ if t = 0 and by (( S h ∗ ) G ) p + ⊗ τ if t = 0. For all groups G for which we know the Hilbert series of the space of invariants ( S h ∗ ) G , we cancalculate the character of baby Verma modules easily. An especially nice case is when ( S h ∗ ) G is a polynomialalgebra generated by algebraically independent elements of homogeneous degrees d , . . . d n . In that case, suchelements are called fundamental invariants , and d i are called degrees of G .In [3], Dickson shows that GL n ( F q ) is such a group; the result for SL n ( F q ) follows easily and is explainedin [7]. Let us recall the construction of invariants and the calculation of their degrees before calculating thecharacters of baby Verma modules.Remember that for G = GL n ( F q ) and G = SL n ( F q ), the reflection representation h is the vector repre-sentation k n , and so h F = F nq , h ∗ F = F nq , h ∗ = k n . Let x , . . . , x n be the tautological (coordinate) basis for h ∗ F . For an ordered n -tuple of non-negative integers e , . . . , e n , let [ e , . . . , e n ] ∈ S h ∗ be the determinant ofthe matrix whose entry in the i -th row and j − th column is x q ei j . The action of G on h ∗ is dual to thetautological action, so the matrix of g ∈ GL n ( F q ) in the basis ( x i ) i is ( g − ) t . Taking determinants is amultiplicative map, so a direct calculation gives that for g ∈ GL n ( F q ), g . [ e , . . . , e n ] = (det( g )) − [ e , . . . , e n ] . Define L n := [ n − , n − , . . . , , ,Q i := [ n, n − , . . . , i + 1 , i − , . . . , , /L n , i = 1 , . . . n − , nd Q = L q − n . The paper [3] shows that [ n, n − , . . . , i + 1 , i − , . . . , ,
0] is divisible by L n , and so Q i are indeed in S h ∗ .From the observation that all [ e , . . . , e n ] transform as (det g ) − under the action of g ∈ GL n ( F q ), it followsthat Q i , i = 0 , . . . , n − S h ∗ . The main theorem in [3] states a stronger claim: Theorem 3.11.
Polynomials Q i , i = 0 , . . . , n − , form a fundamental system of invariants for GL n ( F q ) in S h ∗ , i.e. they are algebraically independent and generate the subalgebra of invariants. A comment in Section 3 of [7] gives us the following corollary.
Corollary 3.12.
Polynomials L n and Q i for i = 1 , . . . , n − form a fundamental system of invariants for SL n ( F q ) . The degrees of these invariants are: deg L n = 1 + q + . . . + q n − deg Q = ( q −
1) deg L n = ( q − q + . . . + q n − ) = q n − Q i = (1 + q + . . . + q n − q i ) − (1 + q + . . . + q n − ) = q n − q i . From this, we can calculate the characters of the baby Verma modules for these groups.
Corollary 3.13.
For G = GL n ( F q ) , the characters of the baby Verma modules are χ N ,c ( τ ) ( z ) = χ M ,c ( τ ) ( z ) n − Y n =0 (1 − z q n − q i ) ,χ N t,c ( τ ) ( z ) = χ M t,c ( τ ) ( z ) n − Y n =0 (1 − z p ( q n − q i ) ) , t = 0 . For G = SL n ( F q ) , the characters of the baby Verma modules are χ N ,c ( τ ) ( z ) = χ M ,c ( τ ) ( z )(1 − z q + ... + q n − ) n − Y n =1 (1 − z q n − q i ) ,χ N t,c ( τ ) ( z ) = χ M t,c ( τ ) ( z )(1 − z p (1+ q + ... + q n − ) ) n − Y n =1 (1 − z p ( q n − q i ) ) , t = 0 . Description of L t,c (triv) for GL n ( F p r )4.1. GL n ( F p r ) as a reflection group. For this section, let us fix G = GL n ( F q ), for q = p r a prime powerand n >
1. It acts tautologically on h F = F nq , which has a coordinate basis y , . . . , y n , and by the dualrepresentation on h ∗ F , which has a dual basis x , . . . , x n . The underlying field is k = F q , and the reflectionrepresentation is h = h F ⊗ F q k = k n . We will study L t,c ( τ ) for all t, c and the trivial representation τ = k ,where every element of G acts as identity.The group GL n ( F q ) is indeed a reflection group, generated by all the conjugates of elements d λ for λ ∈ F × q (for definition of d λ , see Section 2.1). The order of GL n ( F q ) is Q n − i =0 ( q n − q i ), which is divisible bythe characteristic p of k . It is an example of a reflection group in characteristic p which has no counterpartin characteristic zero. Lemma 4.1.
Reflections in GL n ( F q ) are elements that are conjugate in GL n ( F q ) to one of the d λ , λ ∈ F × q .There are q − conjugacy classes of reflections in GL n ( F q ) , with representatives d λ . Each semisimpleconjugacy class consists of ( q n − q n − q − reflections. The unipotent conjugacy class (elements conjugate to d )consists of ( q n − q n − − q − reflections. roof. Every semisimple reflection in GL n ( F q ) is diagonalizable, with eigenvalue 1 of multiplicity n − λ − of multiplicity 1, for some λ = 0 , = 1. Such a reflection is conjugate in GL n ( F q ) to d λ . For λ = µ , d λ is not conjugate to d µ . Thus, to every λ ∈ F q , λ = 0 , = 1, we associate a conjugacy class C λ containing it. The centralizer of d λ is GL ( F q ) × GL n − ( F q ) ⊆ GL n ( F q ), so the number of reflections in theconjugacy class C λ is | GL n ( F q ) | / | GL ( F q ) × GL n − ( F q ) | = ( q n − q n − q − .The unipotent conjugacy class C is the orbit of d , which is centralized by any element of the form: a b · · · c a · · · ∗ ∗ f ∗ ∗ Here, a = 0, the rest of the variables are arbitrary, and the bottom right ( n − × ( n −
2) submatrix isinvertible. The order of the centralizer is q n − ( q − | GL n − ( F p ) | , and the order of the conjugacy class is ( q n − q n − − q − . (cid:3) Let C λ be the conjugacy class containing d λ , let c λ be the value of c on C λ . Reflection s ∈ C λ has(generalized, if λ = 1) eigenvalues λ, , . . . , h ∗ and λ − , , . . . , h . Example 4.2. If n = 2, the parametrization of C λ by α ⊗ α ∨ ∈ h ∗ ⊗ h described in Lemma 2.3 is as follows: λ = 1 : C λ ↔ { (cid:20) b (cid:21) ⊗ (cid:20) − λ − bdd (cid:21) | b, d ∈ F q } ∪ { (cid:20) (cid:21) ⊗ (cid:20) a − λ (cid:21) | a ∈ F q } λ = 1 : C ↔ { (cid:20) b (cid:21) ⊗ (cid:20) − bdd (cid:21) | b, d ∈ F q , d = 0 } ∪ { (cid:20) (cid:21) ⊗ (cid:20) a (cid:21) | a ∈ F q , a = 0 } . Irreducible modules with trivial lowest weight for GL n ( F q ) at t = 0 .Theorem 4.3. The characters of the irreducible modules L ,c (triv) for the rational Cherednik algebra H ,c ( GL n ( F q ) , h ) are: ( q, n ) c χ L ,c (triv) ( z ) Hilb L ,c (triv) ( z )( q, n ) = (2 , any [triv] 1(2 ,
2) 0 [triv] 1(2 , c = 0 [triv] + [ h ∗ ] z + ([ S h ∗ ] − [triv]) z + 1 + 2 z + 2 z + z +([ S h ∗ ] − [ h ∗ ] − [triv]) z Proof.
We claim that when ( n, q ) = (2 , x ∈ h ∗ ⊗ triv ∼ = M ,c (triv) are singular. To seethat, remember that the Dunkl operator associated to y ∈ h is D y = t∂ y ⊗ − X s ∈ S c s ( y, α s ) α s (1 − s ) ⊗ s, which for t = 0 and τ = triv becomes the operator D y = − X s ∈ S c s ( y, α s ) α s (1 − s )on M ,c (triv) ∼ = S h ∗ . We claim that for all values of the parameter c , for all x ∈ h ∗ and for all y ∈ h ,the value D y ( x ) = 0. To see that, calculate the coefficient of c λ in D y ( x ). Using the parametrization ofconjugacy classes from Proposition 2.3, this coefficient is equal to − X s ∈ C λ ( y, α s ) α s (1 − s ) . x = − X α ⊗ α ∨ =0( α,α ∨ )=1 − λ ( y, α )( α ∨ , x ) = − X α ( α, y ) x, X ( α,α ∨ )=1 − λ α ∨ . We claim that for fixed α ∈ h ∗ , the sum P ( α,α ∨ )=1 − λ α ∨ is zero. Fix α and let us change the coordinatesso that α is the first element of some new ordered basis. Write the sum P ( α,α ∨ )=1 − λ α ∨ in the dual of thisbasis. The set of all nonzero α ∨ such that ( α, α ∨ ) = 1 − λ written in these new coordinates then consists of all ∨ = (1 − λ, a , . . . , a n ) = 0, for a i ∈ F q . If λ = 1, the sum of all such α ∨ is the sum over all a , . . . , a n ∈ F q ,so the sum is zero. If λ = 1, the first coordinate is always zero, so the sum is over all a i ∈ F q which are notall simultaneously 0. However, adding (0 , . . . ,
0) doesn’t change the sum, which is then equal to X a ,...,a n ∈ F q (0 , a , . . . a n ) = (0 , q n − X a ∈ F q a, . . . , q n − X a ∈ F q a ) . This is equal to 0, as claimed, unless n = 2 and q = 2.In case ( n, q ) = (2 , B implies the claim. Moreprecisely, if c = 0 then all D y = 0, the form B is zero in degree one, and the module L , (triv) is onedimensional. If c = 0, then the only vectors in the kernels of matrices B i are the invariants in degrees 2 and3 and all their S h ∗ multiples. This implies that L ,c (triv) = N ,c (triv), and gives the character formula fromthe statement of the theorem. (cid:3) Irreducible modules with trivial lowest weight for GL n ( F q ) at t = 0 . For the rest of this sectionwe assume that t = 1. As the parameters t and c can be simultaneously rescaled, the results we obtain for t = 1 hold (after rescaling c by 1 /t ) for all t = 0. Proposition 4.4.
Suppose ( q, n ) = (2 , . For any x ∈ h ∗ , the vector x p is singular in M ,c (triv) .Proof. Method of proof is an explicit calculation analogous to the proof of Theorem 4.3. By definition, D y ( x p ) = ∂ y ( x p ) − X s ∈ S c s ( α s , y ) 1 α s (1 − s ) . x p = 0 − X s ∈ S c s ( α s , y ) 1 α s ((1 − s ) . x ) p We will show that for every conjugacy class C λ of reflections, the sum X s ∈ C λ ( α s , y ) 1 α s ((1 − s ) . x ) p vanishes. By Proposition 2.3, this is equal to X α ⊗ α ∨ =0( α,α ∨ )=1 − λ ( α, y ) 1 α (( α ∨ , x ) α ) p = X α =0 ( α, y ) α p − X α ∨ =0( α,α ∨ )=1 − λ ( α ∨ , x ) p It is enough to fix α and show that the inner sum over α ∨ is zero. After fixing α , let us change the basisso that α is the first element of the new ordered basis. The set of all α ∨ = 0 such that ( α, α ∨ ) = 1 − λ ,written in the dual of this new basis, is A := { ((1 − λ ) , a , . . . , a n ) = 0 | a , . . . , a n ∈ F q } . For a fixed x , theexpression X α ∨ ∈ A ( x, α ∨ s ) p is a sum over all possible values of n − a , . . . , a n of a polynomial of degree p . By Lemma 2.26,this is zero if p < ( n − q − n = 2, p = q . In that case ( α ∨ s , x ) p = ( α ∨ s , x ) forall x ∈ h ∗ F , so the expression is equal to the sum over all possible values of one variable a of a polynomialof degree 1; this sum is again by Lemma 2.26 equal to zero whenever 1 < p −
1. So, the expression is equalto zero, as desired, whenever ( n, q ) = (2 , (cid:3) When ( n, q ) = (2 , n, q ) = (2 ,
2) separately by explicit calculations.As explained before, studying L ,c (triv) is the same as studying the contravariant form B on M ,c (triv).The following proposition tells us that the set of singular vectors from the previous proposition is large, inthe sense that the quotient of M ,c (triv) by the submodule generated by them is already finite dimensional. Corollary 4.5.
Suppose ( q, n ) = (2 , . Then, the form B k = 0 and L ,c (triv) k = 0 for all k ≥ np − n + 1 . roof. Any degree n ( p −
1) + 1 monomial must have one of the n basis elements, say x i , raised to at leastthe p -th power by the pigeonhole principle. Then, B ( x a . . . x a n n , y ) = B ( x pi , y ′ ) = 0, where y ′ is the result ofthe Dunkl operator being applied to y successively. (cid:3) We write the matrices of the form B k in the monomial basis in x i for S h ∗ and y i for S h , both inlexicographical order. In the case of GL n ( F q ) these matrices are surprisingly simple. Proposition 4.6.
Suppose ( q, n ) = (2 , . Then the matrices of B k are diagonal for all k .Proof. We will use invariance of B with respect to G to show that for ( a , . . . , a n ), ( b , . . . , b n ) ∈ Z n ≥ , suchthat P a i = P b i = k , if B ( x a . . . x a n n , y b . . . y b n n ) = 0 then ( a , . . . , a n ) = ( b , . . . , b n ). This means that thematrices B k , written in monomial basis in lexicographical order, are diagonal.Let g ∈ G be a diagonal matrix with entries λ , . . . , λ n ∈ F × q , so that g . y i = λ i y i and g . x i = λ − i x i . Thenfor any ( a , . . . , a n ), ( b , . . . , b n ) ∈ Z n ≥ , such that P a i = P b i = k , we have B ( x a . . . x a n n , y b . . . y b n n ) = B ( g . ( x a . . . x a n n ) , g . ( y b . . . y b n n ))= B (( λ − x ) a . . . ( λ − n x n ) a n , ( λ y ) b . . . ( λ n y n ) b n )= λ b − a . . . λ b n − a n n B ( x a . . . x a n n , y b . . . y b n n ) . So, whenever B ( x a . . . x a n n , y b . . . y b n n ) = 0, it follows that λ b − a . . . λ b n − a n n = 1 for all λ , . . . , λ n ∈ F × q .Fix i . Set all λ j where j = i to be equal to 1, and set λ i to be a multiplicative generator of F × q . Then,necessarily, b i − a i ≡ q − q > p and b i − a i ≡ q − a i = b i for all i , or there exists an index i for which b i − a i ≡ q − a i = b i . In the second case, either a i or b i is greater than or equal to p , so by theprevious proposition, B ( x a . . . x a n n , y b . . . y b n n ) = 0. This finishes the proof if q > p .Now assume q = p , and B ( x a . . . x a n n , y b . . . y b n n ) = 0, so a i ≡ b i (mod p −
1) for all i , and a i , b i < p forall i . Then either all a i = b i as claimed, or, there exists an index i such that { a i , b i } = { , p − } . Assumewithout loss of generality that i = 1, a = 0, b = p −
1. Using that P a j = P b j = k , there exists anotherindex, which we can assume without loss of generality to be 2, such that a = p − b = 0. Now we areclaiming that for any f monomial in x , . . . , x n , any f ′ monomial in y , . . . , y n , B ( x p − f, y p − f ′ ) = 0 . We will be working only with indices 1 and 2 and choosing g ∈ G which fixes all others, so assume withoutloss of generality that n = 2, f = f ′ = 1. We use the invariance of B with respect to d ∈ G : B ( x p − , y p − ) = B ( d . ( x p − ) , d . ( y p − )) = B (( x − x ) p − , y p − )= B (( x p − + . . . + x p − ) , y p − )= B ( x p − , y p − ) + 0 + B ( x p − , y p − )where the terms in the middle are all zero, as their exponents do not differ by a multiple of p −
1. Thus, B ( x p − , y p − ) = 0 as desired. (cid:3) Elements of G ⊆ H t,c ( G, h ) have degree 0 and preserve the graded pieces. So, every graded piece is afinite dimensional representation of G . This makes the following lemma useful. Lemma 4.7.
For every i , the space S i h ∗ / ( h x p , . . . , x pn i ∩ S i h ∗ ) ∼ = span k { x a . . . x a n n | a , . . . , a n < p } is eitherzero or an irreducible G -representation.Proof. This can be found in [8], Section 2, Coraollary 2 .
6. The result there describes certain irreduciblemodules W λ . In the notation of [8] and for λ = ( d, , , . . . ) , the the Corollary says that the modulesspan k { x a . . . x a n n | a , . . . , a n < p } , when they are nonzero, are the irreducible modules W ( d, , ,... ) . (cid:3) Proposition 4.8.
Suppose ( q, n ) = (2 , . Then, in each degree i , the diagonal elements of the matrix of B i are constant multiples of the same polynomial in c s . roof. By previous lemmas, every B i is a diagonal matrix, with diagonal entries polynomials f m in c parametrized by n -tuples of integers m = ( m , . . . , m n ) such that P m j = i . The kernel of B at spe-cific c is spanned by all monomials x m for which f m ( c ) = 0. As the kernel is a submodule of S i h ∗ containing < x p , . . . , x pn > ∩ S i h ∗ , by the previous lemma it can either be < x p , . . . , x pn > ∩ S i h ∗ or the whole S i h ∗ . Inother words, all polynomials f m where one of m j is ≥ p are identically zero, and all others have the sameroots, so they are constant multiples of the same polynomial in c s . (cid:3) Next, we will find these polynomials f m in each degree and calculate their zeros. Proposition 4.9.
Suppose ( q, n ) = (2 , . Then, a) If n = 2 and q = p , then B i depends on the c s for p − ≤ i ≤ p − . The diagonal entries of B i are k -multiples of c + . . . + c p − − for i ≥ p − and k -multiples of ( c + . . . + c p − − c + 2 c + . . . + ( p − c p − + 1) for i ≥ p . b) If n = 3 and q = 2 , then B i depends on the c s for i = 2 , , and the diagonal entries are k -multiplesof c + 1 . c) Otherwise, the form B doesn’t depend on c .Proof. The matrices B i are diagonal, with all diagonal entries being constant multiples of the same polyno-mial. Our strategy is to compute one nonzero diagonal entry.First, we will show claim c). It is sufficient to show that the Dunkl operators on the quotient M ,c ( τ ) / Ker B are independent of c . As in the proof of Proposition 4.4, we compute the part of the Dunkl operator asso-ciated to the conjugacy class C λ with eigenvalue λ , and claim that for any monomial f ∈ S i h ∗ , the part of D y ( f ) which is the coefficient of c λ , − X s ∈ C λ ( y, α s ) 1 α s (1 − s ) . f, is in Ker B. Using Proposition 2.3, we can write this sum over nonzero α ∈ h ∗ and α ∨ ∈ h , such that ( α, α ∨ ) = 1 − λ .Writing it as consecutive sums over α and then over α ∨ , it is enough to show that for any choice of α , theinner sum, over all α ∨ such that ( α, α ∨ ) = 1 − λ , is contained in h x p , . . . , x pn i . As in the previous calculations,we fix α , and change basis of h ∗ to x ′ , . . . , x ′ n so that x ′ = α . Let the dual basis of h be y ′ , . . . , y ′ n . The innersum, with vectors written in the basis y ′ i , is then over α ∨ ∈ A λ := { ((1 − λ ) , a , . . . , a n ) = 0 | a , . . . , a n ∈ F q } .By Proposition 2.3, the reflection s corresponding to α ⊗ α ∨ , α = (1 , , . . . , α ∨ = ((1 − λ ) , a , . . . , a n ),acts on h ∗ as s . x ′ = λx ′ s . x ′ i = x ′ i − a i x ′ , i > α , let us also factor out the constant − ( y, α ). The inner sum for fixed α is then P α ∨ ∈ A λ α (1 − s ) . f . The set A λ is parametrized by ( a , . . . , a n ) ∈ F n − q if λ = 1, and by ( a , . . . , a n ) = 0 ∈ F n − q if λ = 1. However, if λ = 1, the summand corresponding to ( a , . . . , a n ) = 0 is 0, so we can assume thesum is over all ( a , . . . , a n ) ∈ F n − q in both cases.The inner sum we are calculating is equal to( ⋆ ) X ( a ,...,a n ) ∈ F n − q x ′ (cid:16) f ( x ′ , . . . , x ′ n ) − f ( λx ′ , x ′ − a x ′ , . . . , x ′ n − a n x ′ ) (cid:17) . It is enough to calculate it for f of the form f = x ′ b . . . x ′ b n n , b i < p . For such f ,( ⋆ ) = X ( a ,...,a n ) ∈ F n − q − x ′ (cid:16) ( λx ′ ) b ( x ′ − a x ′ ) b . . . ( x ′ n − a n x ′ ) b n − x ′ b . . . x ′ b n n (cid:17) == X ( a ,...,a n ) ∈ F n − q X i ,...i n − (cid:18) b i (cid:19) . . . (cid:18) b n i n (cid:19) λ b ( − a ) i . . . ( − a n ) i n x ′ b + i + ...i n − x ′ b − i . . . x ′ b n − i n n the last sum being over all 0 ≤ i j ≤ b j such that not all i j are 0 at the same time. The coefficient of eachmonomial in x i can be seen as a monomial of degree i + . . . i n in variables a i , so when we sum it over all( a , . . . a n ) ∈ F n − q to get the sum ( ⋆ ), we can use Lemma 2.26 to conclude ( ⋆ ) is 0 whenever the degree of allpolynomials appearing is small enough, more precisely, whenever there exists an index j such that i j < q − . s i j ≤ b j < p for all j , this only fails when p = q and b = b = . . . = b n = p −
1. In other words, thisproves c) whenever q = p .Now assume q = p . By the above argument, the only monomials f for which ( ⋆ ) is not yet known to bezero are the ones of the form f = x ′ b x ′ p − · · · x ′ p − n . For such f , the sum ( ⋆ ) is by the above argument equalto ( − ( p − n − X ( a ,...,a n ) ∈ F n − q λ b ( a ) p − . . . ( a n ) p − x ′ b +( p − n − − While this is not 0, the monomial x ′ b +( p − n − is by Lemma 4.4 in Ker B whenever it has degree at least p , meaning whenever b + ( p − n − − ≥ p. If n = 2, this condition is b ≥
2, which is not satisfied only when b = 0 ,
1. Thus, for n = 2, q = p , thediagonal matrices B i do not depend on c in degrees i < p −
1, their entries are multiples of some polynomialin c in degree p −
1, and some other polynomial (divisible by the first one) in degrees p and higher. Finally,by corollary 4.5, all matrices B i become zero at degrees 2 p − n = 3, this condition is b + 2( p − − ≥ p , which is equivalent to p + b ≥
3. This will not be satisfiedonly if p = 2 and b = 0. So, for n = 3, p = 2, the matrices B and B do not depend on c , the diagonalentries of B i are multiples of the same polynomial in c for i = 2 ,
3, and B = 0. For n = 3 and all other p ,the matrices B i do not depend on c .For n >
3, the inequality b + ( p − n − − ≥ p is always satisfied, so there is never a dependence ofmatrices B i on the parameters c .This finishes the proof of (c) and describes the cases in a) and b) for which there might be dependence onparameters c . To finish the proof, it remains to find specific polynomials in cases: (a) n = 2 , q = p , degrees p − p and; (b) n = 3 , q = 2, degrees 2 and 3.Next, we prove (a). Let n = 2 and q = p . We need to compute one nonzero entry of the matrix B p − andone nonzero entry of B p .To compute the polynomial in degree p −
1, by Proposition 4.8, it suffices to compute B ( x p − , y p − ) = B ( D y ( x p − ) , y p − ). For that, by Proposition 4.6, it suffices to show that the coefficient of x p − in D y ( x p − )is a constant multiple of c + . . . + c p − −
1. Compute D y ( x p − ) = ∂ y ( x p − ) − X C c s X s ∈ C ( α s , y ) 1 α s ( x p − − ( s . x ) p − ) . The coefficient of x p − in ∂ y ( x p − ) = ( p − x p − is p − −
1, so it suffices to show that for each conjugacyclass C the coefficient of x p − in X s ∈ C ( α s , y ) 1 α s (( s . x ) p − − x p − )is 1. Using (cid:0) p − i (cid:1) = ( − i , we can write this as X s ∈ C ( α, y ) 1 α s (( x − ( x , α ∨ s ) α s ) p − − x p − ) == X s ∈ C ( α s , y ) p − X i =1 ( α ∨ s , x ) i α i − s x p − − i . The bases x i and y i are dual, so α s = ( α s , y ) x + ( α s , y ) x , and the coefficient of x i − in α i − s is ( α s , y ) i − .Thus, the coefficient of x p − in the above sum is X s ∈ C ( y , α s ) p − X i =1 ( α ∨ s , x ) i ( y , α s ) i − which can be written as X s ∈ C (cid:0) (( α ∨ s , x )( α s , y ) − p − − (cid:1) . Each term (( α ∨ s , x )( α s , y ) − p − − α ∨ s , x )( α s , y ) = 1, in which case it is − p − α s , α ∨ s that make their product 1. The product of the second oordinate must now be ( α, α ∨ ) −
1. This is nonzero, so there are p − p − ( −
1) = −
1, as desired. Note that this term will appear as a multiplicative factorin higher degrees, since the matrices of B are defined inductively. This proves the claim for degree p − p , with n = 2 and q = p . We will calculate the value of B ( x p − x , y p − y ) = B ( D y ( x p − x ) , y p − ), equal to the product of the coefficient of x p − in D y ( x p − x ) with B ( x p − , y p − ).We proved that B ( x p − , y p − ) is a constant multiple of c + . . . + c p − −
1, so we are now calculating thecoefficient of x p − in D y ( x p − x ) = ∂ y ( x p − x ) − X λ c λ X s ∈ Cλ ( α s , y ) 1 α s (1 − s ) . ( x p − x )= x p − − X λ c λ X s ∈ C λ ( α s , y ) 1 α s ( x p − x − ( s . x ) p − ( s . x )) . We now use the parametrization of conjugacy classes C λ by α ⊗ α ∨ from Lemma 2.3 and Example 4.2: λ = 1 : C λ ↔ { (cid:20) b (cid:21) ⊗ (cid:20) − λ − bdd (cid:21) | b, d ∈ F p } ∪ { (cid:20) (cid:21) ⊗ (cid:20) a − λ (cid:21) | a ∈ F p } λ = 1 : C ↔ { (cid:20) b (cid:21) ⊗ (cid:20) − bdd (cid:21) | b, d ∈ F p , d = 0 } ∪ { (cid:20) (cid:21) ⊗ (cid:20) a (cid:21) | a ∈ F p , a = 0 } . We are calculating the coefficient of x p − in x p − − X λ c λ X α ⊗ α ∨ ↔ C λ ( α, y ) 1 α (cid:16) x p − x − ( x − ( α ∨ , x ) α ) p − ( x − ( α ∨ , x ) α ) (cid:17) == x p − − X λ c λ X α ⊗ α ∨ ↔ C λ ( α, y ) 1 α (cid:16) x p − x − ( x − ( α ∨ , x ) α ) p − x (cid:17) −− X λ c λ X α ⊗ α ∨ ↔ C λ ( α, y ) 1 α (cid:0) ( x − ( α ∨ , x ) α ) p − ( α ∨ , x ) α (cid:1) . The term x p − − ( x − ( α ∨ , x ) α ) p − is divisible by α , so1 α (cid:16) x p − x − ( x − ( α ∨ , x ) α ) p − ( x − ( α ∨ , x ) α ) (cid:17) , written in a monomial basis in x and x , is divisible by x . These terms can be disregarded when calculatingthe coefficient of x p − in the above sum.Let α b = x + bx . We are calculating the coefficient of x p − in x p − − X λ c λ X α ⊗ α ∨ ↔ C λ ( α, y ) 1 α (cid:0) ( x − ( α ∨ , x ) α ) p − ( α ∨ , x ) α (cid:1) == x p − − X λ c λ (cid:16) X b,d b α b (cid:0) ( x − (1 − λ − bd ) α b ) p − dα b (cid:1) + X a x (cid:0) ( x − ax ) p − (1 − λ ) x (cid:1) (cid:17) = x p − − X λ c λ (cid:16) X b,d bd p − X i =0 (1 − λ − bd ) i x p − − i ( x + bx ) i + (1 − λ ) X a ( x − ax ) p − (cid:17) Here, the sum over is over all b, d ∈ F p and over all a ∈ F p if λ = 1, and over a, b, d ∈ F p , d, a = 0 if λ = 1.The coefficient of x p − is: 1 − X λ c λ (cid:16) X b,d bd p − X i =0 (1 − λ − bd ) i + (1 − λ ) X a (cid:17) =1 − X λ =1 c λ (cid:16) X b,d ( bd ) p − ( − λ ) (cid:17) − c (cid:16) X b,d ( bd ) p − (cid:17) = 1 + X λ λc λ . This ends the proof of (a).To prove (b), n = 3, q = 2, we computed the matrices B i explicitly. (cid:3) he combination of these results and the explicit computations in case ( n, q ) = (2 ,
2) gives us the maintheorem of this section:
Theorem 4.10.
Let k be an algebraically closed field of characteristic p . Let G = GL n ( F q ) for q = p r and n ≥ . The following is a complete classification of characters of L ,c (triv) for all values of c : ( q, n ) c χ L ,c (triv) ( z ) Hilb L ,c (triv) ( z )( q, n ) = (2 , generic P i ≥ ( (cid:2) S i h ∗ / h x p , . . . , x pn i ∩ S i h ∗ (cid:3) ) z i (cid:18) − z p − z (cid:19) n (2 , generic (cid:16)P i ≥ (cid:2) S i h ∗ (cid:3) z i (cid:17) (1 − z )(1 − z ) (1 − z )(1 − z )(1 − z ) (2 , c = 1 [triv] + [ h ∗ ] z z ( p, , p = 2 p − X i =1 c i = 1 p − X i =0 (cid:2) S i h ∗ (cid:3) z i p − X i =0 ( i + 1) z i ( p, , p = 2 p − X i =1 c i = 1 , p − X i =1 ic i = − p − X i =0 (cid:2) S i h ∗ (cid:3) z i p − X i =0 ( i + 1) z i (2 , c = 1 [triv] 1(2 , c = 0 [triv] + [ h ∗ ] z + [triv] z z + z Proof.
For ( n, q ) = (2 , B i of the form B can be computed explicitly, and one can see thatthey are 0 starting in degree 1 when c = 1 and starting in degree 3 when c = 0. For all other c , they arefull rank on N ,c (triv), so L ,c (triv) = N ,c (triv) is the quotient of the Verma module by squares of theinvariants, which are in degrees 4 and 6.For ( q, n ) = (2 ,
2) and generic c , we saw in Proposition 4.4 that J ′ (triv) contains h x p , . . . , x pn i , so byProposition 3.4 the reduced module R ,c (triv) is a quotient of the trivial module. From this it follows thatfor generic c , J ,c (triv) = h x p , . . . , x pn i and that the character of L ,c (triv) for generic c and ( q, n ) = (2 ,
2) isas stated above.The characters for special c are computed by looking at the roots of polynomials on the diagonal in B i ,and are computed directly from Proposition 4.9. (cid:3) Remark 4.11.
Notice that for n, p, r large enough, the character does not depend on c at all. This neverhappens in characteristic zero. Remark 4.12.
Notice that the claims from Remarks 3.7 and 3.10 and Question 3.8 hold in the case of G = GL n ( F q ). Namely, by observing the characters one can see that all L t,c (triv) for generic c have onedimensional top degree and are thus Frobenius; that for h the reduced Hilbert series of L ,c (triv) at generic c , h (1) is either | G | (in the case of ( q, n ) = (2 , h (1) = 1 (in all other cases),and that for h the Hilbert series of L ,c (triv) at generic c , the equality h = h always holds.5. Description of L t,c (triv) for SL n ( F p r )In this section we explore category O for the rational Cherednik algebra associated to the special lineargroup over a finite field. We start with some preliminary facts about relations between rational Cherednikalgebras associated to some group and to its normal subgroup, and by looking more carefully into conjugacyclasses of reflections in SL n ( F p r ).5.1. Normal subgroups of reflection groups.
Let G ⊆ GL ( h ) be any reflection group, and assume N ⊂ G is a normal subgroup with a property that two reflections in N are conjugate in N if and only ifthey are conjugate in G . Let c be a k -valued conjugation invariant function on reflections of N , and extend c to all reflections in G by defining it to be zero on reflections which are not in N . Then one can considerthe rational Cherednik algebra H t,c ( N, h ) as a subalgebra of H t,c ( G, h ); it has fewer generators and the samerelations.Let τ be an irreducible representation of G , and assume it is irreducible as a representation τ | N of N .Consider two representations of H t,c ( N, h ): the irreducible representation L t,c ( τ | N ) = L t,c ( N, h , τ | N ), andthe irreducible representation L t,c ( τ ) = L ,c ( G, h , τ ) of H t,c ( G, h ) restricted to H t,c ( N, h ). emma 5.1. As representations of H ,c ( N, h ) , L ,c ( τ | N ) ∼ = L ,c ( τ ) | H ,c ( N, h ) .Proof. The reflections in N are a subset of reflections in G . Because N is normal in G , every conjugacy classin G is either contained in N or does not intersect it. By the assumption, two reflections in N which areconjugate in G are also conjugate in N , so conjugacy classes in N are a subset of conjugacy classes in G .The Verma modules M t,c ( G, h , τ ) and M t,c ( N, h , τ | N ) are both naturally isomorphic to S h ∗ ⊗ τ as vectorspaces, and this induces their natural isomorphism as H t,c ( N, h ) representations. The modules L t,c ( τ ) and L t,c ( τ | N ) are their quotients by the kernel of the contravariant form, which is controlled by Dunkl operators.Because of the discussion of conjugacy classes in N and G and because of the definition of c , the Dunkloperators are the same for H t,c ( N, h ) and H t,c ( G, h ). (cid:3) One could define Verma modules, baby Verma modules and their quotients by the kernel of the con-travariant form (chosen so that it is non-degenerate on the lowest weights) even in cases when the lowestweight is not irreducible as a representation of the reflection group. In that case, an analogous lemma isthat the composition series of τ as a representation of N is the same as the composition series of L t,c ( τ ) asa representation of H t,c ( N, h ). We will not need this here.5.2. Conjugacy classes of reflections in SL n ( F q ) . In this section we will study G = SL n ( F q ) for q = p r a prime power and n >
1. As before, let h = k n , k = F p , and τ be the trivial representation. Further, let Q be the set of nonzero squares in F q and R be the set of non-squares.All reflections in SL n ( F q ) are unipotent and conjugate in GL n ( F q ) to d . It is easy to see that SL n ( F q ) isgenerated by them. The group SL n ( F q ) is a normal subgroup of GL n ( F q ), and it contains all the reflectionsfrom the unipotent conjugacy class in GL n ( F q ). However, the second condition from the above discussion,that two reflections are conjugate in GL n ( F q ) if and only if they are conjugate in SL n ( F q ), is not satisfiedfor all n, q . For example, (cid:20) −
10 1 (cid:21) = (cid:20) − (cid:21) (cid:20) (cid:21) (cid:20) − (cid:21) is not conjugate in SL ( F ) to (cid:20) (cid:21) . Proposition 5.2.
Let q = p r be a prime power. If n ≥ , or p = 2 , then two reflections are conjugatein SL n ( F q ) if and only if they are conjugate in GL n ( F q ) , and there is one conjugacy class of reflections in SL n ( F q ) . Otherwise (for n = 2 and q = p r , p = 2 ), there are two conjugacy classes of unipotent reflectionsin SL n ( F q ) : C Q = { gd g − | g ∈ GL ( F q ) , det( g ) is a square } and C R = { gd g − | g ∈ GL ( F q ) , det( g ) is not a square } . Proof.
Direct computation, using the description of conjugacy classes in GL n ( F q ) and the fact that tworeflections are conjugate in SL n ( F q ) if and only if they are conjugate in GL n ( F q ) by some element whosedeterminant is in Q (the set of squares in F q ). (cid:3) If n ≥ p = 2, then the only conjugacy class in SL n ( F q ) is equal to the conjugacy class C of GL n ( F q ).Let us call this class C , and the constant associated to it c . If n = 2 and p = 2, to the two conjugacyclasses C R and C Q we will associate parameters c R and c Q . Note that C R ∪ C Q = C . In the case of onlyone conjugacy class, we will use Lemma 5.1 to transfer character formulas for rational Cherednik algebrasassociated to GL n ( F q ) to character formulas for rational Cherednik algebras associated to SL n ( F q ). In thecase of two conjugacy classes, we will have to do more computations to get character formulas. Let us firstlook more closely into the case of two conjugacy classes. Lemma 5.3.
Let n = 2 , q = p r , and p = 2 . Let γ ∈ R be an arbitrary non-square in F q . Let s be a reflectionin SL ( F q ) . Then, s and γs − ( γ − are in different conjugacy classes.Proof. The proof follows from Proposition 5.2. The map F γ : s γs − ( γ −
1) maps reflections to reflections,and its inverse is F γ − . So, it is enough to show it maps s ∈ C Q to an element of C R .Remember that d γ , γ = 0 , γ − ,
1, and d is a matrixwith all generalized eigenvalues equal to 1 and one Jordan block of size 2. For s = d , we have F γ ( d ) = d − γ d d γ ∈ C R . If s is a conjugate in SL ( F q ) to d , say s = hd h − , then F γ ( s ) = hd − γ d d γ h − =( hd − γ h − ) s ( hd − γ h − ) − ∈ C R . (cid:3) he following lemma is useful in computations, and is a stronger version of Lemma 2.26. Lemma 5.4. If d ≡ q − , then P i ∈ Q i d = P i ∈ R i d = q − . If d ≡ q − (mod q − , then P i ∈ Q i d = − P i ∈ R i d = q − . Otherwise, P i ∈ Q i d = P i ∈ R i d = 0 .Proof. For this proof, let S Q = P i ∈ Q i d and S R = P i ∈ R i d . Suppose d ≡ q − i d = 1 forall nonzero i ∈ F q , and S R = S Q = | Q | = | R | = q − . Suppose d ≡ q − (mod q − i ∈ Q , i d = 1, and if i ∈ R , then i d = −
1. Thus, S Q = − S R = q − . Suppose neither holds. For any a ∈ Q it is easyto see that aR = R and aQ = Q , so S R = a d S R and S Q = a d S Q . If a is a multiplicative generator of thecyclic multiplicative group F × q , then 1 and a d are different elements of Q , so (1 − a d ) S Q = (1 − a d ) S R = 0implies that S R = S Q = 0. (cid:3) Next, we parametrize reflections in each conjugacy class. Remember the notation from Proposition 2.3:unipotent reflections are identified with all α ⊗ α ∨ ∈ h ∗ ⊗ h such that ( α, α ∨ ) = 0 , in such a way that theaction of a reflection s on x ∈ h ∗ and y ∈ h is s.x = x − ( α ∨ , x ) αs.y = y + ( α, y ) α ∨ . Lemma 5.5.
The conjugacy classes C Q and C R of reflections in SL ( F p r ) , p > , are parametrized through α ⊗ α ∨ as C Q = (cid:26) γ (cid:20) a (cid:21) ⊗ (cid:20) a − (cid:21) | a ∈ F q , γ ∈ Q (cid:27) [ (cid:26) γ (cid:20) (cid:21) ⊗ (cid:20) (cid:21) | γ ∈ Q (cid:27) ,C R = (cid:26) γ (cid:20) a (cid:21) ⊗ (cid:20) a − (cid:21) | a ∈ F q , γ ∈ R (cid:27) [ (cid:26) γ (cid:20) (cid:21) ⊗ (cid:20) (cid:21) | γ ∈ R (cid:27) . Proof.
The proof is straightforward. The reflection d is identified with (cid:20) (cid:21) ⊗ (cid:20) (cid:21) . Conjugating it by g = (cid:20) a bc d (cid:21) ∈ GL ( F q ) gives us a reflection gd g − identified with1det g (cid:20) − ca (cid:21) ⊗ (cid:20) ac (cid:21) . As g can always be scaled by an element of the centralizer of d so that either c = − c = 0 and a = 1,and that gd g − is in C Q or C R depending whether det g is in Q or R , the description follows. (cid:3) Description of L ,c (triv) for SL n ( F p r ) .Theorem 5.6. Characters of the irreducible modules L ,c (triv) for the rational Cherednik algebra H ,c ( SL n ( F q , h )) are: ( q, n ) c χ L ,c (triv) ( z ) Hilb L ,c (triv) ( z )( q, n ) = (2 , any [triv] 1(2 ,
2) 0 [triv] 1(2 , c = 0 [triv] + [ h ∗ ] z + ([ S h ∗ ] − [triv]) z + 1 + 2 z + 2 z + z +([ S h ∗ ] − [ h ∗ ] − [triv]) z Proof.
The group SL n ( F q ) is a normal subgroup of GL n ( F q ), and for n ≥ p = 2, by Proposition 5.2 itsatisfies the conditions of Lemma 5.1. Thus, in those cases, irreducible representations of H t,c ( SL n ( F q ) , h )have the same characters as irreducible representations of H t,c ( GL n ( F q ) , h ), where c is extended to conjugacyclasses of reflections in GL n ( F q ) − SL n ( F q ) by zero. So, in those cases we can deduce character formulas for L ,c (triv) from Theorem 4.3.The remaining case is SL ( F q ), p = 2, for which we claim that D y ( x ) = 0 for all x and y , and so thecharacter is trivial. We have D y ( x ) = − c R X α ⊗ α ∨ ∈ C R ( y, α )( x, α ∨ ) − c Q X α ⊗ α ∨ ∈ C Q ( y, α )( x, α ∨ ) , o it is enough to show that for T = Q or T = R , X α ⊗ α ∨ ∈ C T α ⊗ α ∨ is zero.We know from the proof of Theorem 4.3 that P α ⊗ α ∨ ∈ C R ∪ Q α ⊗ α ∨ = 0, so it is enough to prove that thesum over C Q is zero. For this, let us calculate, using parametrization from Lemma 5.5: X α ⊗ α ∨ ∈ C Q α ⊗ α ∨ = X γ ∈ Q X a ∈ F q γ ( x + ax ) ⊗ ( ay − y ) + X γ ∈ Q γx ⊗ y By Lemma 5.4, if q = 3, then P γ ∈ Q γ = 0 and the whole sum is zero, as claimed. For q = 3, Q = { } , sothe sum is equal to X a ∈ F (cid:0) − x ⊗ y − ax ⊗ y + ax ⊗ y + a x ⊗ y (cid:1) + x ⊗ y == − x ⊗ y + x ⊗ y = 0 . (cid:3) Description of L ,c (triv) for SL n ( F p r ) if n ≥ or p = 2 . As explained above and demonstrated in thecase of t = 0, we can get character formulas for H ,c ( SL n ( F q ) , h ) directly from the ones for H ,c ( GL n ( F q ) , h )when n ≥ p = 2. The following is a corollary of Lemma 5.1, Proposition 5.2, and results from Section4, most notably 4.10. Corollary 5.7.
Let n ≥ or p = 2 . Consider the rational Cherednik algebra H ,c ( SL n ( F q ) , h ) , its repre-sentations M ,c (triv) , the contravariant form B on it, and the irreducible quotient L ,c (triv) . Then all theresults proved previously for the group GL n ( F q ) hold also for SL n ( F q ) . Specifically, a) D y ( x p ) = 0 in M ,c (triv) . b) The form B on M ,c (triv) is zero in degrees np − n + 1 and higher. c) The matrices of the form B on M ,c (triv) in lexicographically ordered monomial bases are diagonalin all degrees. d) All diagonal elements of the matrix of the form B i on any graded piece M ,c (triv) i are k -multiplesof the same polynomial in c . e) If ( q, n ) = (2 , , ( q, n ) = (2 , the matrices B i of the form on M ,c (triv) i do not depend on c . f) If ( q, n ) = (2 , , only the matrices B and B depend on c . Their nonzero diagonal coefficients areconstant multiples of c + 1 . g) If ( q, n ) = (2 , , then GL n ( F q ) = SL n ( F q ) so the character formulas are the same. h) The character formulas for L ,c (triv) for the rational Cherednik algebra H ,c ( SL n ( F q ) , h ) are thesame as for the rational Cherednik algebra H ,c ( GL n ( F q ) , h ) , with the parameter function c extendedto all classes of reflections in GL n ( F q ) − SL n ( F q ) by zero. Description of L ,c (triv) for SL n ( F p r ) if n = 2 and p > . As in the case of t = 0, we need to studythe case n = 2 and p >
2, when there are two conjugacy classes of reflections in SL n ( F q ), separately. Thecase q = 3 is the most complicated and we solve it by calculating the matrices of the form B explicitly. Thefollowing results address the remaining cases. Proposition 5.8.
Let n = 2 , q = p r for p an odd prime, and q = 3 . In the Verma module M ,c (triv) for H ,c ( SL ( F q )) , all the vectors x p , x ∈ h ∗ , are singular.Proof. We need to show that for every conjugacy class C T , for T = R or T = Q , and any x, y , the coefficientin D y ( x p ) of c T is zero. This coefficient is equal to X α ⊗ α ∨ ∈ C T ( y, α )( x, α ∨ ) p α p − . Again, as C R ∪ C Q = C is a conjugacy class of reflections in GL ( F q ), and the result holds there, it isenough to show this for C Q . s in the proof of Proposition 4.4, we claim that after writing it as a double sum, with the outer sumbeing over α and the inner over α ∨ , the inner sum is zero for any α . Fix α and change coordinates, so thatwe assume without loss of generality that α = x . Then the inner sum is X α ∨ = γy γ ∈ Q ( x, α ∨ ) p = ( y , x ) p X γ ∈ Q γ p . Using Lemma 5.4, this is zero unless p ≡ , q − (mod q − q = 2 and q = 3. (cid:3) Next, we prove that acting by Dunkl operator produces elements of M ,c (triv) of a specific form. Lemma 5.9.
Let n = 2 , q = p r for p an odd prime, and q = 3 , and consider the Verma module M ,c (triv) for H ,c ( SL n ( F q )) . For any f ∈ M ,c (triv) / h x pi i ∼ = S h ∗ / h x pi i and any y ∈ h , there exists h ∈ S h ∗ such that,as elements of M ,c (triv) / h x pi i , D y ( f ) = ∂ y f + c Q · h + c R · h Proof.
The Dunkl operator action in the case of SL ( F q ) is D y ( f ) = ∂ y ( f ) − X T ∈{ Q,R } c T X s ∈ C T ( α s , y ) α s (1 − s ) .f The strategy is to compute the sum P s ∈ C T ( α s ,y ) α s (1 − s ) . ( f ) parallelly for T = Q, R , disregarding all termsthat do not depend on the choice of T (these contribute equally to the coefficient of c Q and c R ), and anyelements of the ideal h x pi i . We will use: X a ∈ F q a m = 0 unless m ≡ q − , m = 0(1) X γ ∈ Q γ m = X γ ∈ R γ m unless m ≡ q −
12 (mod q − x pi = 0 in M ,c (triv) / h x pi i (3)and the parametrization of conjugacy classes from 5.5:(4) C T = (cid:26) γ (cid:20) a (cid:21) ⊗ (cid:20) a − (cid:21) | a ∈ F q , γ ∈ T (cid:27) [ (cid:26) γ (cid:20) (cid:21) ⊗ (cid:20) (cid:21) | γ ∈ T (cid:27) . The rest of the proof is this computation.We will do it for D y ( f ), y = y , f = x u x v . The general statement follows from this case by symmetryand linearity. We can assume u, v ≤ p − ⋆ ) X s ∈ C T ( α s , y ) α s ( x u x v − ( s.x ) u ( s.x ) v )does not depend on T .Reflections s corresponding to elements of the form γ (cid:20) (cid:21) ⊗ (cid:20) (cid:21) satisfy ( α s , y ) = ( γx , y ) = 0, sothey do not contribute to the sum.For s of the form γ (cid:20) a (cid:21) ⊗ (cid:20) a − (cid:21) , let us write the action on x i ∈ h ∗ explicitly, using the notation α ′ a = x + ax . The explicit action is s.x = x − aγα ′ a s.x = x + γα ′ a . ubstituting this into ( ⋆ ), we get( ⋆ ) = X γ ∈ T X a ∈ F q ( γα ′ a , y ) γα ′ a ( x u x v − ( x − aγα ′ a ) u ( x + γα ′ a ) v ) == X γ ∈ T X a ∈ F q u X i =0 v X j =0( i,j ) =(0 , α ′ a ( − (cid:18) ui (cid:19)(cid:18) vj (cid:19) x u − i ( − i a i γ i α ′ ia x v − j γ j α ′ ja == X γ ∈ T X a ∈ F q u X i =0 v X j =0( i,j ) =(0 , (cid:18) ui (cid:19)(cid:18) vj (cid:19) ( − i +1 a i γ i + j x u − i x v − j ( x + ax ) i + j − == X γ ∈ T X a ∈ F q u X i =0 v X j =0( i,j ) =(0 , i + j − X k =0 (cid:18) ui (cid:19)(cid:18) vj (cid:19)(cid:18) i + j − k (cid:19) ( − i +1 a i + k γ i + j x u + j − − k x v − j + k Let us evaluate the sum P a ∈ F q a i + k . By (1), this is only nonzero if i + k is divisible by q −
1. We knowthat i + k ≤ i + j − ≤ u + v − ≤ p − − < p − ≤ q − , so let us consider three different cases: i + k = 0, i + k = q − i + k = 2( q − CASE 1: i + k = 0 . In that case, P a ∈ F q a = 0, so this does not contribute either. CASE 2: i + k = q − . After substituting P a ∈ F q a q − = − k = q − − i , and after that m = i + j ,we get that part of ( ⋆ ) corresponding to this case equals X γ ∈ T u X i =0 v X j =0( i,j ) =(0 , (cid:18) ui (cid:19)(cid:18) vj (cid:19)(cid:18) i + j − q − − i (cid:19) ( − i γ i + j x u + j + i − q x v − j + q − − i == X γ ∈ T u + v X m =1 v X j =0 (cid:18) um − j (cid:19)(cid:18) vj (cid:19)(cid:18) m − q − − m + j (cid:19) ( − m − j γ m x u + m − q x v − m + q − Now we use (2) to describe P γ ∈ T γ m and disregard all terms except m ≡ q − (mod q −
1) (these terms wedisregarded contribute to the coefficient h ). There are again few cases, as m ≤ u + v ≤ p − <
52 ( p − ≤ q − , so we consider separately m = q − and m = q − . CASE 2.1: m = q − . One of the binomial coefficients in the expression is (cid:0) m − q − − m + j (cid:1) , and we claim itis always zero for this choice of m . This is because q − − m + j = q −
12 + 2 j ≥ q − > q − − m − . In other words, case 2 . . never actually appears in the sum. CASE 2.2: m = q − . We will show that this part of the sum is zero. First,3( q − m ≤ u + v ≤ p − p = q . Next, because we x p = 0 in the quotient, the only terms thatcan be nonzero are the ones with the power of x less than p , so u + m − q ≤ p − u ≤ p − q − m = p + 12 . ext, the term (cid:0) um − j (cid:1) is zero unless j ≥ m − u ≥ p − − p + 12 = p − . Since j ≤ v ≤ p − , it follows that j ∈ { p − , p − } . In both those cases, the binomial coefficient (cid:0) m − q − − m + j (cid:1) is 0, as the numeratorhas a factor p and the denominator does not. CASE 3: i + k = 2( q − . The part of the sum ( ⋆ ) corresponding to this case is X γ ∈ T u X i =0 v X j =0( i,j ) =(0 , (cid:18) ui (cid:19)(cid:18) vj (cid:19)(cid:18) i + j − q − − i (cid:19) ( − i γ i + j x u + j − − q +2+ i x v − j +2 q − − i = X γ ∈ T u + v X m =1 v X j =0 (cid:18) um − j (cid:19)(cid:18) vj (cid:19)(cid:18) m − q − − m + j (cid:19) ( − m − j γ m x u + m − − q +21 x v − m +2 q − The powers of x in this sum and the original power of x are both ≤ p −
1, so p − ≥ v − m + 2( q − p − ≥ u ≥ m − v ≥ q − − ( p − ≥ p − . From the last string of inequalities, u = p − m = u + v and p = q . The above sum then becomes X γ ∈ T v X j =0 (cid:18) p − p − v − j (cid:19)(cid:18) vj (cid:19)(cid:18) p + v − p − − v + j (cid:19) ( − p − v − j γ p − v x v − x p − As j ≤ v , the first binomial coefficient in this sum is zero unless j = v , producing X γ ∈ T (cid:18) p + v − p − (cid:19) γ p − v x v − x p − . The sum P γ ∈ T γ p − v = P γ ∈ T γ v only depends on T if v ≡ p − (mod p − , which only happens if v = p − . In that case, (cid:0) p + v − p − (cid:1) = (cid:0) p + v − p − (cid:1) = 0, as the numerator is divisible by p . (cid:3) We can use the previous proposition to transfer the results we had about GL ( F q ) to SL ( F q ), as in theprevious chapter. Namely, the structure of irreducible modules for H ,c ( SL ( F q ) , h ), where c takes values c Q on C Q and c R on C R , is determined by Dunkl operators. By the previous proposition, X s ∈ C Q ( α s , y ) α s (1 − s ) = X s ∈ C R ( α s , y ) α s (1 − s ) , so the Dunkl operator is equal to D y = ∂ y − X s ∈ C Q c Q ( α s , y ) α s (1 − s ) − X s ∈ C R c R ( α s , y ) α s (1 − s )= ∂ y − c R + c Q X s ∈ C R ∪ C Q ( α s , y ) α s (1 − s )In GL ( F q ), the union C Q ∪ C R is one conjugacy class C (unipotent reflections). Define the function c on allreflections in GL ( F q ) by letting it be c = c R + c Q on all unipotent reflections, and c λ = 0 on all semisimplereflections. Then the Dunkl operators controlling the structure of L ,c (triv) for H ,c ( GL ( F q ) , h ) are D y = ∂ y − q X λ =1 X s ∈ C λ c λ ( α s , y ) α s (1 − s )= ∂ y − X s ∈ C c ( α s , y ) α s (1 − s ) , hich is exactly the same as the Dunkl operator for H ,c ( SL ( F q ) , h ). From this we get: Corollary 5.10.
Let n = 2 , q = p r for p an odd prime, and q = 3 , and consider the Verma module L ,c (triv) for H ,c ( SL n ( F q )) . All the results proved previously for the rational Cherednik algebra associated to GL n ( F q ) hold for SL n ( F q ) . Namely, a) D y ( x p ) = 0 in M ,c (triv) . b) The form B on M ,c (triv) is zero in degrees p − and higher. c) The form B on M ,c (triv) is diagonal in all degrees. d) All diagonal elements of the matrix of the form B i on any graded piece M ,c (triv) i are k -multiplesof a single polynomial in c . e) If q = p r with r > , then B does not depend on c . f) If q = p , the matrices of B i on M ,c (triv) i are constant for i = 0 , . . . , p − , constant multiples of c Q + c R − for i = p − , and constant multiples of ( c Q + c R − c Q + c R + 2) for i = p, . . . , p − . Putting together the previous Corollary, Corollary 5.7, explicit computations for the rational Cherednikalgebra associated to SL ( F ), and noticing that SL ( F ) = GL ( F ), we get the main theorem of thissection. Theorem 5.11.
Let p be a prime, q = p r and n ≥ . The characters of L ,c (triv) for the rational Cherednikalgebra H ,c ( SL n ( F q ) , h ) over an algebraically closed field of characteristic p are as follows: ( q, n ) c χ L ,c (triv) ( z ) Hilb L ,c (triv) ( z )( q, n ) = (2 , , (3 , generic P i ≥ ( (cid:2) S i h ∗ / h x p , . . . , x pn i ∩ S i h ∗ (cid:3) ) z i (cid:18) − z p − z (cid:19) n (3 , generic (cid:16)P i ≥ (cid:2) S i h ∗ (cid:3) z i (cid:17) (1 − z )(1 − z ) (1 − z )(1 − z )(1 − z ) (2 , generic (cid:16)P i ≥ (cid:2) S i h ∗ (cid:3) z i (cid:17) (1 − z )(1 − z ) (1 − z )(1 − z )(1 − z ) (2 , c = 1 [triv] 1(2 , c = 0 [triv] + [ h ∗ ] z + [triv] z z + z (2 , c = 1 [triv] + [ h ∗ ] z z ( p, , p = 2 , c Q + c R = 2 p − X i =0 (cid:2) S i h ∗ (cid:3) z i p − X i =0 ( i + 1) z i ( p, , p = 2 , c Q + c R = − p − X i =0 (cid:2) S i h ∗ (cid:3) z i p − X i =0 ( i + 1) z i We omit the characters for ( q, n ) = (3 , and special c as there are too many cases to concisely list. Remark 5.12.
All L t,c (triv) for generic c have one dimensional top degree and are Frobenius. For h thereduced Hilbert series of L ,c (triv) at generic c , h (1) is either | G | or 1. For h the Hilbert series of L ,c (triv)at generic c , the inequality h ≤ h term by term always holds, but not the equality: for SL ( F ), h ( z ) = 1,and h ( z ) = (1 − z )(1 − z )(1 − z ) . ppendix A. A conjecture about orthogonal groups
In the preliminary stages of this project, we used MAGMA [2] to gather data about characters of irreduciblerepresentations of a larger class of groups than we ended up studying. Here we list some conjectures aboutHilbert series of irreducible representations with trivial lowest weight for orthogonal groups over a finite field.The groups O n ( A, F q ) are defined as subgroups of GL n ( F q ) preserving a quadratic form F nq → F q given byan invertible matrix A as x x t Ax . For odd n there is only one orthogonal group up to isomorphism, so wechoose A = I and denote such a group by O n ( F q ). For even n , there are two possible nonisomorphic groups: O + n ( F q ), preserving the form given by A a diagonal matrix with diagonal entries 1 , − , , − , . . . , −
1, (sothat the total space k n is an orthogonal sum of hyperbolic lines (( x , x ) , ( y , y )) x y − x y ) and O − n ( F q ), preserving the form given by A a diagonal matrix with diagonal entries 1 , − , , − , . . . , − , , − r ,for r an arbitrary quadratic non-residue (in which case the total space k n is an orthogonal sum of severalhyperbolic lines and a two dimensional anisotropic space). Conjecture A.1.
The reduced Hilbert series for irreducible representation with trivial lowest weight ofrational Cherednik algebras associated to orthogonal groups of low rank are as follows:
Hilb O +2 ( F q ) ( z ) = (1 + z )(1 + z + z + . . . z q − )Hilb O − ( F q ) ( z ) = (1 + z )(1 + z + z + . . . z q )Hilb O ( F q ) ( z ) = 1 . The conjecture was checked computationally for pairs ( n, q ) = (2 , , , ,
2) and (3 , Acknowledgments
We are very grateful to Pavel Etingof for suggesting the problem and devoting his time to it through nu-merous helpful conversations. This project was initiated as part of the Summer Program for UndergraduateResearch (SPUR) at the Department of Mathematics at MIT, and partially funded by SPUR and Undergrad-uate Research Opportunities Program (UROP) at MIT. The work of H.C. was partially supported by theLord Foundation through UROP. The work of M.B. was partially supported by the NSF grant DMS-0758262.
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Transactions of the American Mathematical Society , 12(1):75–98, 1911.[4] Pavel Etingof and Xiaoguang Ma. Lecture notes on Cherednik algebras. arXiv , 1001.0432v4, 2010.[5] Iain Gordon. Baby Verma modules for rational Cherednik algebras.
Bulletin of the London Mathematics Society , 35(3):321–336, 2003.[6] Stephen Griffeth. Towards a combinatorial representation theory for the rational Cherednik algebra of type G ( r, p, n ). Proceedings of the Edinburgh Mathematical Society , 53(2):419–445, 2010.[7] Gregor Kemper and Gunter Malle. The finite irreducible linear groups with polynomial ring of invariants, transformationgroups 2.
Transformation Groups , (2):57–89, 1997.[8] Leonid Krop. On the representations of the full matrix semigroup on homogeneous polynomials.
Journal of Algebra , 99(2):370– 421, 1986.[9] Larry Smith.
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M.B: Department of Mathematics, University of York, York, YO10 5DD, UK, and, Department of Mathemat-ics, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia, H.C: Department of Mathematics, University ofCalifornia, 852 Evans Hall, Berkeley, CA 94720 USA
E-mail address : [email protected], [email protected]@york.ac.uk, [email protected]