STABILISATION OF THE LHS SPECTRAL SEQUENCE FOR ALGEBRAIC GROUPS. 1. Introduction

Transcription

1 STABILISATION OF THE LHS SPECTRAL SEQUENCE FOR ALGEBRAIC GROUPS ALISON E. PARKER AND DAVID I. STEWART arxiv: v1 [math.rt] 19 Feb 014 Abstract. In this note, we consider the Lyndon Hochschid Serre spectra sequence corresponding to the first Frobenius kerne of an agebraic group G and computing the extensions between simpe G-modues. We state and discuss a conjecture that E = E and provide genera conditions for ow-dimensiona terms on the E -page to be the same as the corresponding terms on the E -page, i.e. its abutment. 1. Introduction Let G be a reductive agebraic group over an agebraicay cosed fied k of characteristic p > 0. Let λ and µ be two dominant weights for G. This paper concerns the representation theory of G and its first Frobenius kerne G 1 ; we refer to [Jan03] for notation. It is the purpose of this short note to state and provide some evidence towards the foowing conjecture. Conjecture. Suppose a G 1 -injective hus have the structure of G-modues, for instance if p h. Then the Lyndon Hochschid Serre spectra sequence (*) E ij = Exti G/G 1 (k,ext j G 1 (L(λ),L(µ))) Ext i+j G (L(λ),L(µ)) stabiises (i.e. reaches its abutment) at the E -page. That is, E ij = E ij for a i,j. Hence Ext n G(L(λ),L(µ)) = i+j=n Ext i G/G 1 (k,ext j G 1 (L(λ),L(µ))). Note that it is an open conjecture of Humphreys and Verma that a G 1 -injective hus do indeed have the structure of G-modues, possiby making the first hypothesis triviay satisfied. Let us underine the fact that we are unaware of any occasion where any differentia in the spectra sequence (*) is known to be non-zero even after repacing G with an arbitrary connected agebraic group and repacing L(λ) and L(µ) by arbitrary G-modues. Showing that certain differentias in the spectra sequence are zero has some history; we pick out a few cases. For a arge cass of naturay occurring modues V and W, it was shown in [Par07] that when G = SL the spectra sequence does stabiise at the E -page. In particuar the conjecture is confirmed for the case G = SL, with no condition on p. It was shown by Donkin in [Don8] that the differentias d m,1 : E m,1 E m+,0 are zero, aso with no condition on p. Some other specia cases invoving maps needed to compute second cohomoogy were considered in work of McNinch [McN0], the second author [Ste10,Ste1], and Ibraev [Ibr11, Ibr1]. Another case in which the conjecture is true is if λ and µ are p-reguar restricted weights, p h and p is arge enough that the Lusztig Character Formua hods. Then [PS13, Theorem 5.3] shows that the G-modue Ext n G 1 (L(λ),L(µ)) [ 1] has a good fitration for each n. Under these 1

2 circumstances the spectra sequence moreover degenerates to a ine; in particuar the conjecture is true. Note that the conjecture is not true if G is repaced by an arbitrary group. See [BF94, 6], [Lea93] and [Sie00] for exampes of non-zero differentias. The main theorem of this paper is a confirmation of the conjecture in a generic sense. Here, the vanishing of differentias of degree much ower than p is guaranteed. Theorem. Suppose p (r + 1)(h 1). Then the differentias d ij n in the spectra sequence (*) satisfying i r 1 and n or j = 0 and n or j = 1 and n are a zero. In particuar, for i r+1. Ext i G (L(µ),L(λ)) = i E i j,j We prove the above theorem by appying techniques from [Par07]. First, we show, in a proposition, that part of a minima G 1 -injective resoution has a compatibe G-structure. We then reconstruct the spectra sequence (*) in such a way that the bottom-most compex in the doube compex giving the E 0 -page contains this part of a minima G 1 -injective resoution. It foows that many maps in the E 0 -page are zero. Then some derived coupe arguments prove the theorem.. Proposition and proof of the theorem In the proposition beow, note that the case r = 0 woud be a specia case of the Humphreys Verma conjecture. (It is not known if the bound p h coud be reduced to p h 1 for G 1 -injective hus to ift to G-modues.) Proposition. Let r 1 and et µ X 1. Provided p (r + 1)(h 1), there is a minima G 1 - resoution 0 L(µ) I 0 I 1 I r such that the sequence up to term I r has a G-structure. Proof. We prove a fortiori that there is such a sequence of G-modues with I r having weights λ = λ 0 +pλ 1 with λ 0 X 1, which satisfy (λ 1,α 0 ) (r +1)(h 1). First, et us treat the case r = 1. Set I 0 = Q 1 (µ). The hypotheses impy that p h ; thus we know that Q 1 (µ) has the structure of a G-modue. The injection L(µ) Q 1 (µ) is then a map of G-modues. Let M := Q 1 (µ)/l(µ). We may write Soc G1 M = ν L(ν 0) M F ν where ν 0 X 1 and M ν is some G-modue. Set I 1 = ν Q 1(ν 0 ) M F ν. So Soc G1 I 1 = Soc G1 Q 1 (µ)/l(µ). (It is worth noting that the condition on the weights here is enough to ensure that Soc G1 M = Soc G M but we do not need this fact expicity.) Thus I 1 is the G 1 -injective hu of M, hence if there is a G-map I 0 I 1, this wi be part of a minima resoution. It remains to show that there is indeed a map I 0 I 1 of G-modues whose kerne is L(µ), i.e. a map I 0 /L(µ) I 1. Note that we do have a map Soc G1 M I 1 by construction, so consider the exact sequence (*) Hom G (M,I 1 ) Hom G (Soc G1 M,I 1 ) Ext 1 G(M/Soc G1 M,I 1 ).

3 If we coud show that the third term in this sequence is zero then we woud have that the first map were surjective, hence the G-map Soc G1 M I 1 woud ift to a map M = I 0 /L(µ) I 1 and we woud be done. Now Ext 1 G (M/Soc G 1 M,I 1 ) has a fitration by spaces E = Ext 1 G (M/Soc G 1 M,Q 1 (ν 0 ) L(ν 1 ) F ) over certain weights ν = ν 0 +pν 1. And E can be computed via the 5-term exact sequence of the LHS spectra sequence, of which part is Ext 1 G/G 1 (k,hom G1 (M/Soc G1 M,Q 1 (ν 0 ) L(ν 1 ) F )) E Hom G/G1 (k,ext 1 G 1 (M/Soc G1 M,Q 1 (ν 0 ) L(ν 1 ) F )). Now the third term here is zero, as Q 1 (ν 0 ) is injective for G 1, hence, to show E = 0, it suffices to show that the first term is zero. Now I 0 = Q 1 (µ), being a direct G 1 -summand of St 1 L((p 1)ρ µ), has weights satisfying ξ = ξ 0 + pξ 1 with (ξ 1,α 0 ) h 1. Thus the composition factors of I 0 (hence of M) are of the form L(ξ 0 ) L(ξ 1 ) F. In particuar, we have that the weights ν 1 satisfy (ν 1,α 0 ) h 1. Thus (ξ 1 +ρ,α 0 ),(ν 1+ρ,α 0 ) h and our condition on p impies that they are both in the cosure of the owest acove, CZ. So et L(ξ 0 ) L(ξ 1 ) F be a composition factor of M/Soc G1 M. We compute: Ext 1 G/G 1 (k,hom G1 (L(ξ 0 ) L(ξ 1 ) F,Q 1 (ν 0 ) L(ν 1 ) F )) = Ext 1 G (L(ξ 1),Hom G1 (L(ξ 0 ),Q 1 (ν 0 )) [ 1] L(ν 1 )) Now Hom G1 (L(ξ 0 ),Q 1 (ν 0 )) is non-zero, thence equa to k, if an ony ξ 0 = ν 0 ; in that case, the term on the right becomes Ext 1 G (L(ξ 1),L(ν 1 )), and since ξ 1,ν 1 C Z, this vanishes by the inkage principe. This concudes the proof in case r = 1. Now by induction we may assume that we have a sequence of G-modues 0 I 0 I r π Ir 1, which is minima as an injective G 1 -resoution, such that the composition factors of I r 1 have high weights λ satisfying λ = λ 0 + pλ 1 with λ 0 X 1 and (λ 1,α 0 ) r(h 1). We construct I r in a simiar way to before: set I r = ν Q 1(ν 0 ) Mν F, where the sum is over the G-composition factors of Soc G1 I r 1 /πi r, where ν 0 X 1 and a weight ν 1 of M ν satisfies (ν 1,α 0 ) r(h 1). Thus a weight ξ of I r, say ξ 0 + pξ 1 with ξ 0 X 1 satisfies (ξ 1,α 0 ) (ν 1,α 0 ) + (ρ,α 0 ) = (r + 1)(h 1) as required. Note that I r is again a G 1 -injective hu of I r 1 /imπ so if we can show there is a G-modue map I r 1 I r with kerne imπ, we wi be done. Of course, it is equivaent to produce a map from M := I r 1 /imπ to I r. By construction we do have a map from Soc G1 M I r. Now the same argument as before shows that the third term in the sequence (*) (with I r repacing I 1 ) is zero. This competes the proof. Proof of the theorem. We write L(µ) = L(µ 0 ) L(µ 1 ) F using Steinberg s tensor product theorem where µ 0 X 1 and µ 1 X +. Using the proposition we have a G-resoution which is aso a G 1 -injective resoution: 0 L(µ 0 ) I 0 I 1 I r, where, up to I r, the resoution is minima for G 1. 3

4 We denote the differentias by δ i : I i I i+1 and the kernes by K i := kerδ i. Dimension shifting gives us Ext i G 1 (L(λ 0 ),L(µ 0 )) = Ext 1 G 1 (L(λ 0 ),K i 1 ). Minimaity gives us for µ 1 X 1 that Ext i G 1 (L(λ 0 ),L(µ 0 )) = Hom G1 (L(λ 0 ),K i ) = Hom G1 (L(λ 0 ),I i ) for i r. We now have a G-resoution: 0 L(µ) I 0 L(µ 1 ) F 0 I1 L(µ 1 ) F 1 where i = δ i id, as tensoring is exact. Aso note that such a resoution stays injective as a G 1 -resoution as L(µ 1 ) F is trivia as a G 1 -modue. Now consider the E 0 -page of the LHS spectra sequence that converges to Ext G (L(λ),L(µ)) as constructed in [Par07, ] 0 = Hom G/G1 (k,hom G1 (L(λ),I n L(µ 1 ) F ) J F m) where we have a G-injective resoution of the trivia modue: and this spectra sequence has E 1 and E page 0 k J 0 J 1 1 = Hom G/G1 (k,ext n G 1 (L(λ),L(µ)) J F m ) = H m (G/G 1,Ext n G 1 (L(λ),L(µ))). Consider the induced maps m in the foowing compex, which has homoogy Ext G 1 (L(λ),L(µ)): Now Hom G1 (L(λ),I 0 L(λ 1 ) F ) 0 Hom G1 (L(λ),I 1 L(µ 1 ) F ) 1. Ext m G 1 (L(λ),L(µ)) = Ext m G 1 (L(λ 0 ),L(µ 0 )) L(µ 1 ) F L(λ 1 )F = Hom G1 (L(λ 0 ),I m ) L(µ 1 ) F L(λ 1 )F = HomG1 (L(λ),I m L(µ 1 ) F ) for m r. Thus a the differentias m for m r must be zero. Now by [Ben98, 3., 3.4] we know that the spectra sequence can be constructed using derived coupes. We have D0 mn = m+n=e+f, e m 1 = H( 0,d 0 ) E ef 0 D mn 1 = H( 0 E m+1,n 1 0,d 0 +d 1 ) We define the higher derived coupes by taking the derived coupe of the previous one. We have an exact diagram of douby graded k-modues i D D k j E 4

5 The derived coupe (for 1) is defined by D+1 mn = imim+1,n 1 i mn +1 = imn D+1 D mn j+1 mn (im+1,n 1 +1 = H(Emn,d ) (x)) = j m+1,n 1 (x)+im(d ) k mn +1 (z +im(d )) = k mn (z) d +1 = j +1 k +1 And the degrees of the maps k, j and d are: deg(i n ) = ( 1,1), deg(j n ) = (n 1,n+1), deg(k n ) = (1,0). Now using [Par07, Lemma.1] we have that d mn 0 = 0 impies that k m 1,n+1 = 0. Thus since d mn 0 = 0 for m r we have k mn = 0 for m r 1. Thus a k mn = 0 for a and m r 1. As d mn = j m+1,n k mn we aso get d mn = 0 for a and m r 1. In other words, as a these differentias are zero on the E page and remain zero, the terms E mn with m r 1 must aready be the stabe vaue. That is, E mn = Emn for m r 1. This easiy gives us that Ext i G(L(λ),L(µ)) = i E i j,j for i < r. To get the resut for i = r, we note that a the terms in the sum r E r j,j stabiise at the E page by the above, except, possiby the term E. r0 But here ceary E r0 = E r0 as a incoming differentias are zero by the above, and the eaving differentia d r,0 is aways zero as our spectra sequence is first quadrant. We may simiary argue for r +1. We consider r+1 E r+1 j,j. As before a terms except possiby E r,1 and E r+1,0 as in the previous case gives E r+1,0 = E r+1,0. stabiise at the E page. The same argument Now note that a incoming differentias to E r,1 are zero for by the above. We aso have that d r,1 = 0 for 3, again since the spectra sequence is first quadrant. So we need ony check that d r1 = 0, but this is true using [Don8, Main Theorem]. Thus we aso get the resut for r +1. Acknowedgements. The authors wish to thank Len Scott and Dan Nakano for comments on a previous version of this paper. References [Ben98] D. J. Benson, Representations and cohomoogy. II, second ed., Cambridge Studies in Advanced Mathematics, vo. 31, Cambridge University Press, Cambridge, 1998, Cohomoogy of groups and modues. MR (99f:0001b) 5

6 [BF94] D. J. Benson and M. Feshbach, On the cohomoogy of spit extensions, Proc. Amer. Math. Soc. 11 (1994), no. 3, MR (94i:0096) [Don8] StephenDonkin, On Ext 1 for semisimpe groups and infinitesima subgroups, Math. Proc. Cambridge Phios. Soc. 9 (198), no., MR (84i:0037) [Ibr11] S. S. Ibraev, The second cohomoogy groups of simpe modues for G, Sib. Èektron. Mat. Izv. (011). [Ibr1], The second cohomoogy of simpe modues over Sp 4 (k), Comm. Agebra (01). [Jan03] Jens Carsten Jantzen, Representations of agebraic groups, second ed., Mathematica Surveys and Monographs, vo. 107, American Mathematica Society, Providence, RI, 003. MR MR (004h:0061) [Lea93] Ian Leary, A differentia in the Lyndon-Hochschid-Serre spectra sequence, J. Pure App. Agebra 88 (1993), no. 1-3, MR (94m:010) [McN0] George J. McNinch, The second cohomoogy of sma irreducibe modues for simpe agebraic groups, Pacific J. Math. 04 (00), no., MR (003m:006) [Par07] Aison E. Parker, Higher extensions between modues for SL, Adv. Math. 09 (007), no. 1, MR 947 (008e:0071) [PS13] Brian J. Parsha and Leonard L. Scott, New graded methods in the homoogica agebra of semisimpe agebraic groups. [Sie00] Stephen F. Siege, The cohomoogy of spit extensions of eementary abeian -groups and Totaro s exampe, J. Pure App. Agebra 145 (000), no., MR (000m:0087) [Ste10] David I. Stewart, The second cohomoogy of simpe SL -modues, Proc. Amer. Math. Soc. 138 (010), no., MR (011b:0134) [Ste1], The second cohomoogy of simpe SL 3-modues, Comm. Agebra (01), (to appear). Schoo of Mathematics, University of Leeds, Leeds, LS 9JT, UK, E-mai address: a.e.parker@eeds.ac.uk (Parker) New Coege, Oxford, Oxford, UK E-mai address: david.stewart@new.ox.ac.uk (Stewart) 6