THE POLYNOMIAL REPRESENTATION OF THE TYPE A n 1 RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n

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1 THE POLYNOMIAL REPRESENTATION OF THE TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n SHEELA DEVADAS AND YI SUN Abstract. We study the polynoial representation of the rational Cherednik algebra of type A n with generic paraeter in characteristic p for p n. We give explicit forulas for generators for the axial proper graded subodule, show that they cut out a coplete intersection, and thus copute the Hilbert series of the irreducible quotient. Our ethods are otivated by taking characteristic p analogues of existing characteristic 0 results. Contents. Introduction 2. An explicit construction of singular vectors 3 3. Coplete intersection properties 7 4. Proof of the ain result 8 References 8. Introduction The present work presents a detailed study of the polynoial representation of the type A n rational Cherednik algebra over a field of characteristic p dividing n. Rational Cherednik algebras were introduced by Etingof-Ginburg in [EG02] as a rational degeneration of the double affine Hecke algebra dependent on two paraeters and c. In characteristic 0, their type A representation theory has been the subject of extensive study. We refer the reader to [EM0] for a survey of these results. In characteristic p and especially in the odular case, uch less is known about the representation theory of the rational Cherednik algebra. In this paper, we consider the odular case p n. For = and generic c, we provide a coplete characteriation of the irreducible quotient of the polynoial representation. We give explicit generators for the unique axial proper graded subodule J c, show that the irreducible quotient is a coplete intersection, and copute its Hilbert series. Our techniques are inspired by taking characteristic p analogues of results about Cherednik algebras in characteristic 0. In particular, our explicit expression for generators of J c was obtained by converting expressions with coplex residues to equivalent expressions dealing only with foral power series which ay be interpreted in characteristic p. While we restrict our study to the polynoial representation in type A, we view it as a test case for this philosophy, which we believe ay adit wider application. We now state our results precisely and explain their relation to other recent work... The rational Cherednik algebra in positive characteristic. We work over an algebraically closed field k of characteristic p > 0 and fix n so that p n. Let S n denote the syetric group on n eleents, V = k n its perutation representation, and s ij S n the transposition peruting i and j. Fix a basis y,..., y n for V and a dual basis x,..., x n for V. Let h and h be the dual n -diensional S n - representations which are subrepresentation and quotient of V and V, respectively given by h = span{y i y j i j} and h = V /x + + x n. The action of S n on h and h is given explicitly by natural perutation of basis vectors. Date: January 25, 206.

2 2 SHEELA DEVADAS AND YI SUN Fix constants and c in k. Denoting the tensor algebra of h h by T h h, the type A n rational Cherednik algebra H,c h is the quotient of k[s n ] T h h by the relations [x i, x j ] = 0, [y i y j, y l y ] = 0, [y i y j, x i ] = cs ij c t i s it, [y i y j, x l ] = cs il cs jl for all i, j, l, n such that i, j, l are distinct and l. There is a Z-grading on H,c h given by setting deg x = for x h, deg y = for y h, and deg g = 0 for g k[s n ]. In addition, H,c h adits a PBW decoposition H,c h = Syh k k[s n ] k Syh. For any α 0, H,c h and H α,αc h are isoorphic as algebras, so only the cases = 0 or = need be considered. In this paper, we restrict our attention to =..2. Polynoial representation of the rational Cherednik algebra. The rational Cherednik algebra H,c h adits a Z 0 -graded representation on the polynoial ring A = Syh, known as the polynoial representation. The actions of Syh and k[s n ] on A are by left ultiplication and the S n action on h, respectively. The action of Syh is ipleented by letting y h act by the Dunkl operator D y = y <l c y, x x l s l x x l, where we note that s l x x l f is a polynoial for f A. Explicitly, we have D yi y j = yi y j c i s i + c s j, x i x x j x j where yi y j is the differential operator satisfying yi y j x = y i y j, x for all x h..3. Maxial proper graded subodule and irreducible quotient of polynoial representation. As described in [BC3, Section 2.5], there is a contravariant for β c : Syh Syh k defined by setting β c, = and iposing for all x h, y h, f Syh, g Syh that β c xf, g = β c f, D x g and β c f, yg = β c D y f, g. where for x h we denote by D x the Dunkl operator ipleenting the action of H,c h on its polynoial representation Syh. The polynoial representation Syh has unique axial graded proper subodule J c = kerβ c. By the definition of β c, J c contains the ideal generated by all hoogeneous vectors f A of positive degree that are killed by all Dunkl operators D y. Such f are known as singular vectors. The quotient L = A/J c is an irreducible representation of H,c h. It inherits a Z 0 -grading fro A, and for L j the degree j subspace of L, we ay define its Hilbert series as h L t = j 0 di L j t j..4. Stateent of the ain result. For a foral power series r, we denote by [ l ]r the coefficient of l in r. Throughout the paper, we will consider foral power series in considered as expansions of rational functions around = 0. For i =,..., n, define the foral power series F i = x i =0 c n x j for c = cc c +!. Denote by f i the coefficients f i = [ p ]F i. Theore 4.. For generic c, f,..., f n are linearly independent and generate the axial proper graded subodule J c of the polynoial representation for H,c h. The irreducible quotient L = A/J c is a coplete intersection with Hilbert series t p n h L t =. t j=

3 POLYNOMIAL REPRESENTATION OF TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n 3 Reark. In Theore 4., by generic c we ean c avoiding finitely any values..5. Connections to previous work. Our study is otivated by previous work on the representation theory of the type A rational Cherednik algebra in both characteristic 0 and p. The type A non-odular case p n was studied in [BFG06], and soe properties of the axial proper graded subodule of the polynoial representation were given in both odular and non-odular cases in [BC3]. In the odular case p n, for p = 2 the polynoial representation associated to the n-diensional perutation representation was studied in [Lia2]. Theore. [Lia2, Theore 5.]. The irreducible quotient of the polynoial representation associated to the n-diensional perutation representation is a coplete intersection with Hilbert series ht = + t n + t 2. The corresponding axial proper graded subodule is generated by n eleents of degree 2 and one eleent of degree 4. It was further conjectured by Lian in [Lia2, Conjecture 5.2] that for all p the corresponding irreducible is a coplete intersection with J c having n generators in degree p and a single generator in degree p 2. Our results are consistent with the restriction of Lian s conjecture to the case when h is the n -diensional quotient. It would be interesting to extend our work to prove Lian s conjecture in full. For general p n, a subodule of the axial proper graded subodule was coputed in [DS4, Proposition 6.]. In characteristic 0, our results parallel the explicit decoposition of the polynoial representation of the type A rational Cherednik algebra given in [BEG03, CE03]. There, the polynoial representation is irreducible unless c = r n for soe integer r, and an explicit set of generators of the axial proper graded subodule is known. Proposition.2 [CE03, Proposition 3.]. If chark = 0 and c = r n, the axial proper graded subodule J c A of the polynoial representation A of H,c h is generated by [ ] d n Res x i c for j =,..., n. x j i= We interpret the characteristic p analogue of Proposition.2 to ean that if r = p and p n, then since p/n is equivalent to 0/0 and thus an indeterinacy in characteristic p, taking c = p/n in characteristic 0 should correspond to taking c generic in characteristic p. While this substitution is of course invalid, Proposition.2 ay be interpreted as a stateent about certain foral power series. By using a power series version of this construction of generators which akes sense in characteristic p, we are able to iic the arguents of [BEG03, CE03] to show that they cut out a coplete intersection and generate the entire ideal. We believe that the philosophy of taking characteristic p analogues of characteristic 0 results for the rational Cherednik algebra should apply ore generally and hope to explore this further in future work..6. Outline of the paper. The reainder of this paper is organied as follows. In Section 2, we check that the generators f i are linearly independent singular vectors. In Section 3, we show that they cut out a coplete intersection. In Section 4, we put these facts together to conclude Theore Acknowledgeents. The authors thank P. Etingof for suggesting the proble and for helpful discussions. Soe exploratory coputations were done using Sage. S. D. was supported by the MIT Undergraduate Research Opportunities Progra UROP. Y. S. was supported by a NSF Graduate Research Fellowship NSF Grant # Both authors were also supported by NSF Grant DMS An explicit construction of singular vectors 2.. Definition of the singular vectors. In A, define the polynoials n c g = x j and F = g. j= In these ters, we have F i = F x i and f i = [ p ] F x i. We will show that f i are singular vectors. =0

4 4 SHEELA DEVADAS AND YI SUN 2.2. Coputation of soe partial derivatives. We begin by coputing soe partial derivatives of F which will be useful for coputing the action of the Dunkl operators. Lea 2.. We have [ 0 ]g = and [ ]g = 0, eaning 2 g. Proof. For eleentary syetric polynoials e 2,..., e n, we have the expansion n g = x j = j= x i + 2 e 2 x,..., x n + + n n e n x,..., x n. i Recalling that i x i = 0 in A, we see that [ ]g = 0 and [ 0 ]g =, so 2 g as desired. Lea 2.2. For soe foral power series V with [ l ]V = 0 for l = 0,..., p, we have n F cx j = V x j F. Proof. We see easily that g = g j c p F = = = j = j = j = j x j x j x j x j g j= F. We now consider. We copute g p 2 c c g g + =0 p 2 x j c c c g + c g x j =0 = x j c c c g c g p x j p =0 cx j x j F + x j c x j j c g p. p Defining the foral power series V = j x j c x j c g p, p cx j we see that F = V n j= x F. It reains only to show that j [l ]V = 0 for l = 0,..., p, which follows by noting that g p V, applying Lea 2., and noting p 2. Lea 2.3. For soe foral power series G with [ l ]G = 0 for l = 0,..., p, we have y2 y F = G x 2 F. Proof. We ay copute y2 y g = g g p c y2 y F = = = x 2 + = x 2 + = x 2 + p = p 2 =0 x + 2 F x y2 y g. Using this, we see that c g g c p 2 c c g + c g + =0 c g p. p

5 POLYNOMIAL REPRESENTATION OF TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n 5 Defining G = x 2 c x p g p, we have shown that x 2 y2 y F = G F It reains only to show that [ l ]G = 0 for l = 0,..., p, which follows by noting that g p G, applying Lea 2., and noting p Proving f i are singular vectors. We now show that the f i are singular vectors suing to 0. Lea 2.4. For c 0, we have n i= f i = 0. Proof. By Lea 2.2, we have that F = V n j= cx j n x j F = V c x j F, where we subtract ncf in the second equality. Notice that [ p ] coefficient of the left side is 0 because [ p ]F = 0 and that [ p ]V = 0. Therefore, we conclude that c i f i = 0, giving the desired. Proposition 2.5. The eleents f i for i =,..., n are singular vectors in A. Proof. We ust show that f i is annihilated by D yj y l for all j l. First, by syetry it suffices to consider f. Because the Dunkl operators D yi y j for all i j are spanned by D yi y for < i n, it suffices to show D yi y f = 0. Finally, because f is syetric in the x i for i >, it suffices to show that D y2 y f = 0. Recall by Lea 2.3 that y2 y F = G F for a power series G with x 2 j= x [ l ]G = 0 for l = 0,..., p. In ters of G, we can calculate y2 y F as In addition, we have that y2 y F = 2 F + y 2 y F s i x x i F = = F + = x 2 x x i x i F + x 2 F = G. Because F is invariant under s ij for i, j >, we now copute D y2 y F = y2 y c s 2 + c s j F x 2 x x j> x j = y2 y F c s 2 F + c x 2 x j> = G + x 2 = G F + j F + G x i F = x i F. F + x j F. s j x x j F x 2 F + j> x j F To show that the p coefficient in D y2 y F vanishes, we ust consider F. By Lea 2.2, we have F = V j cx j x j F,

6 6 SHEELA DEVADAS AND YI SUN where [ l ]V = 0 for l = 0,..., p. Fro this, it follows that F = F F = + x 2 F = V cx j x j F + x 2 F = V j j cx j x j F + x F. We now copute D y2 y F = G F + j = G F + F x j F F + j x j F = G F x 2 F + j = G F 2 F + V 2 = G + 2 V F 2 F, x j c 2 x j F where in the second step we have subtracted nf. We note that [ p ] G+2 V x is a linear cobination of [ l ]G + 2 V for 0 l p, hence a linear cobination of [ l ]G for 0 l p and [ l ]V for 0 l p 2. By Leas 2.2 and 2.3, these coefficients of G and V are all 0, hence [ p ] G+2 V x = 0. We conclude that [ p ]D y2 y F = [ p ] F 2 F. If b = [ p ]F, then [ p ] F = b and [ p ] 2 F = b, which iplies that D y2 y f = [ p ]D y2 y F = [ p ] F 2 F = b + b = Proof of linear independence of f i. Proposition 2.6. For generic c, f,..., f n are linearly independent degree p hoogeneous polynoials. Proof. We have the expansion F i = x i =0 c g = l=0 x l i l =0 c g. Because for any l the coefficient of l in each factor is a hoogeneous polynoial of degree l, we see that [ p ]F i is hoogeneous of degree p. For linear independence, suppose that n i= λ if i = 0 for soe λ i k. Substitute x n =, x j = and x i = 0 for i j, i < n so that g = + = 2 and hence and F i = l=0 0 l l =0 F j = l=0 l =0 c 2 = =0 c 2 c 2 for i < n, i j.

7 POLYNOMIAL REPRESENTATION OF TYPE A n RATIONAL CHEREDNIK ALGEBRA IN CHARACTERISTIC p n 7 If p = 2, we see that [ 2 ]F j = c and [ 2 ]F i = c, so varying j iplies that n λ j = c λ i for all j. i= In particular, all λ i have coon value λ k solving cn λ = 0, which for c and hence for c generic iplies that λ = 0, giving linear independence. If p > 2, we have p /2 c c [ p ]F j = f j = = p /2 =0 and [ p ]F i = f i = 0 for i < n and i j. For c / {, 2,..., p /2} and hence for generic c, we have c p /2 0, eaning that n c λ i f i = λ j = 0, p /2 i= which iplies λ j = 0. Varying j iplies that λ j = 0 for all j, again yielding linear independence. 3. Coplete intersection properties Consider the ideal I c = f,..., f n A generated by the f i. In this section, we will show that A/I c is a coplete intersection. Recall that for an ideal I k[x,..., X ], the quotient ring k[x,..., X ]/I is a coplete intersection if I has a inial set of generators g,..., g l with di k[x,..., X ]/I = l. Proposition 3.. For generic c, the quotient A/I c is a coplete intersection. Proof. By Proposition 2.6, I c has a set of n linearly independent and therefore inial generators {f i } in degree p. We will now show that for generic c, if α,..., α n k satisfy f i α,..., α n = 0 for i =,..., n, then α = = α n = 0. Because f n lies in the span of f,..., f n for c 0 by Lea 2.4, this will iply that suppa/i c = {0,..., 0} and hence A/I c has diension 0, copleting the proof. Suppose that the α i take values {s,..., s r }, where s i occurs with ultiplicity i > 0 so that g = i s i i, i is i = 0, and i i = n. We clai that for generic c this iplies s i = 0 for all i. Let B = k[s,..., s r ]/ i is i = 0; it suffices to check that with I c interpreted as an ideal in B, the quotient B/I c is finite diensional over k. Define the polynoial h = r i= s i, and let the series expansion of F be given by F = i 0 a i s,..., s r, c i, where a i has degree i in s,..., s r and a 0 s,..., s r, c =. Define a degree p polynoial in by p F ; s,..., s r, c := a i s,..., s r, c p i. In this notation, we have that f i = [ p ] s i F = F s i ; s,..., s r, c = 0 i=0 for i =,..., r. Notice that F s i ; s,..., s r, c is a polynoial in s,..., s r of hoogeneous degree p. By the definition of F, we have F ; s,..., s r, 0 = p, eaning that if c = 0, we have s = = s r = 0, hence B/I 0 is finite diensional over k. Now, let B d and Ic d denote the degree d pieces of B and the hoogeneous ideal I c, respectively. Choose a onoial basis {t i } for B d independent of c, and let Ic d be spanned by a finite set of polynoials {g j } with g j = i h jict i. Notice that di Ic d is given by the sie of the axial non-vanishing inor of the atrix H = h ji. On the other hand, we just showed that there is soe degree d > 0 so that I0 d = B d, eaning that di Ic d < di B d exactly when c lies in the ero set of all di B d inors of H. This iplies that for all but finitely any c we have di Ic d = di B d and hence B/I c is finite-diensional over k.

8 8 SHEELA DEVADAS AND YI SUN For each of the finitely any choices of r,,..., r > 0 with i i = n, we conclude that for all but finitely any c, if {α,..., α n } contains s i with ultiplicity i, then α = = α n = 0. Taking the union over excluded possibilities for c, we conclude that f i α,..., α n = 0 iplies α = = α n = 0 unconditionally for generic c, so A/I c is a coplete intersection as desired. Reark. Proposition 3. is a foral power series analogue of [CE03, Theore 3.2]. However, our proof differs fro the residues by parts arguent which appears there, as the crucial [CE03, Lea 3.2] fails in the odular case. It is interesting to note that our proof does not appear to translate to the characteristic 0 case, as no analogue of the polynoial F ; s,..., s r, c exists in that setting. 4. Proof of the ain result We now put everything together to obtain our ain result. Theore 4.. For generic c, f,..., f n are linearly independent and generate the axial proper graded subodule J c of the polynoial representation for H,c h. The irreducible quotient L = A/J c is a coplete intersection with Hilbert series t p n h L t =. t Proof. By [BC3, Proposition 3.4], the Hilbert series of L is t p n h L t = ht p t for a polynoial ht with nonnegative integer coefficients. On the other hand, by Propositions 2.6 and 3., A/I c is a coplete intersection with n linearly independent degree p generators f i. Its Hilbert series is t p n h A/Ic t =. t By Proposition 2.5, the generators f i of I c are singular vectors, so I c J c, iplying that h A/Ic t h A/Jc t coefficient-wise. We conclude that ht =, hence h A/Ic t = h A/Jc t and I c = J c, copleting the proof. Reark. In the proof of Proposition 3., we require that c avoids 0 for Lea 2.4, c avoids {,,..., p /2} for Proposition 2.6, and that c avoids a non-explicit finite set given by vanishing of a deterinantal ideal. These are the only uses of the assuption that c is generic, so Proposition 3. and Theore 4. hold for c avoiding these values. References [BC3] M. Balagović and H. Chen. Representations of rational Cherednik algebras in positive characteristic. J. Pure Appl. Algebra, 274:76 740, 203. [BEG03] Y. Berest, P. Etingof, and V. Ginburg. Finite-diensional representations of rational Cherednik algebras. Int. Math. Res. Not., 9: , [BFG06] R. Berukavnikov, M. Finkelberg, and V. Ginburg. Cherednik algebras and Hilbert schees in characteristic p. Represent. Theory, 0: , With an appendix by P. Etingof. [CE03] T. Chutova and P. Etingof. On soe representations of the rational Cherednik algebra. Represent. Theory, 7:64 650, [DS4] S. Devadas and S. Sa. Representations of rational Cherednik algebras of G, r, n in positive characteristic. J. Cout. Algebra, 64: , 204. [EG02] P. Etingof and V. Ginburg. Syplectic reflection algebras, Calogero-Moser space, and defored Harish-Chandra isoorphis. Invent. Math., 472: , [EM0] P. Etingof and X. Ma. Lecture notes on Cherednik algebras, [Lia2] C. Lian. Representations of Cherednik algebras associated to syetric and dihedral groups in positive characteristic, preprint, E-ail address, S. Devadas: sheelad@stanford.edu E-ail address, Y. Sun: yisun@ath.it.edu