STABILISATION OF THE LHS SPECTRAL SEQUENCE FOR ALGEBRAIC GROUPS

ALISON E. PARKER AND DAVID I. STEWART

Abstract. In this note, we consider the Lyndon–Hochschild–Serre spectral sequence corresponding to the first Frobenius kernel of an algebraic group G and computing the extensions between simple G-modules. We state and discuss a conjecture that E_{2} = E_{∞} and provide general conditions for low-dimensional terms on the E_{2}-page to be the same as the corresponding terms on the E_{∞}-page, i.e. its abutment.

1. Introduction

Let G be a reductive algebraic group over an algebraically closed field 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 kernel G_{1}; we refer to [Jan03] for notation. It is the purpose of this short note to state and provide some evidence towards the following conjecture.

Conjecture. Suppose all G_{1}-injective hulls have the structure of G-modules, for instance if p ≥ 2h − 2. Then the Lyndon–Hochschild–Serre spectral sequence

(*)  E_{2}^{ij} = Ext_{G/G_{1}}^{i}(k, Ext_{G_{1}}^{j}(L(λ), L(μ))) ⇒ Ext_{G}^{i+j}(L(λ), L(μ))

stabilises (i.e. reaches its abutment) at the E_{2}-page. That is, E_{2}^{ij} ≅ E_{∞}^{ij} for all i, j.

Hence

Ext_{G}^{n}(L(λ), L(μ)) ≅ ⨁_{i+j=n} Ext_{G/G_{1}}^{i}(k, Ext_{G_{1}}^{j}(L(λ), L(μ))).

Note that it is an open conjecture of Humphreys and Verma that all G_{1}-injective hulls do indeed have the structure of G-modules, possibly making the first hypothesis trivially satisfied.

Let us underline the fact that we are unaware of any occasion where any differential in the spectral sequence (*) is known to be non-zero—even after replacing G with an arbitrary connected algebraic group and replacing L(λ) and L(μ) by arbitrary G-modules. Showing that certain differentials in the spectral sequence are zero has some history; we pick out a few cases. For a large class of naturally occurring modules V and W, it was shown in [Par07] that when G = SL_{2} the spectral sequence does stabilise at the E_{2}-page. In particular the conjecture is confirmed for the case G = SL_{2}, with no condition on p. It was shown by Donkin in [Don82] that the differentials d_{m,1} : E_{2}^{m,1} → E_{2}^{m+2,0} are zero, also with no condition on p. Some other special cases involving maps needed to compute second cohomology were considered in work of McNinch [McN02], the second author [Ste10,Ste12], and Ibraev [Ibr11, Ibr12].

Another case in which the conjecture is true is if λ and μ are p-regular restricted weights, p ≥ 2h − 2 and p is large enough that the Lusztig Character Formula holds. Then [PS13, Theorem 5.3] shows that the G-module Ext_{G_{1}}^{n}(L(λ), L(μ))^{[−1]} has a good filtration for each n. Under these circumstances the spectral sequence moreover degenerates to a line; in particular the conjecture is true.

Note that the conjecture is not true if G is replaced by an arbitrary group. See [BF94, §6], [Lea93] and [Sie00] for examples of non-zero differentials.

The main theorem of this paper is a confirmation of the conjecture in a generic sense. Here, the vanishing of differentials of degree much lower than p is guaranteed.

Theorem. Suppose p ≥ (r + 1)(h − 1). Then the differentials d_{n}^{ij} in the spectral sequence (*) satisfying i ≤ r − 1 and n ≥ 2 or j = 0 and n ≥ 2 or j = 1 and n ≥ 2 are all zero.

In particular,

Ext_{G}^{i}(L(μ), L(λ)) ≅ ⨁_{j=0}^{i} E_{2}^{i−j,j}

for i ≤ r + 1.

We prove the above theorem by applying techniques from [Par07]. First, we show, in a proposition, that part of a minimal G_{1}-injective resolution has a compatible G-structure. We then reconstruct the spectral sequence (*) in such a way that the bottom-most complex in the double complex giving the E_{0}-page contains this part of a minimal G_{1}-injective resolution. It follows that many maps in the E_{0}-page are zero. Then some derived couple arguments prove the theorem.

2. Proposition and proof of the theorem

In the proposition below, note that the case r = 0 would be a special case of the Humphreys–Verma conjecture. (It is not known if the bound p ≥ 2h − 2 could be reduced to p ≥ h − 1 for G_{1}-injective hulls to lift to G-modules.)

Proposition. Let r ≥ 1 and let μ ∈ X_{1}. Provided p ≥ (r + 1)(h − 1), there is a minimal G_{1}- resolution

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-modules 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, let us treat the case r = 1. Set I_{0} = Q_{1}(μ). The hypotheses imply that p ≥ 2h − 2; thus we know that Q_{1}(μ) has the structure of a G-module. The injection L(μ) → Q_{1}(μ) is then a map of G-modules.

Let M := Q_{1}(μ)/L(μ). We may write Soc_{G_{1}} M = ⨁_{ν} L(ν_{0}) ⊗ M_{ν}^{F} where ν_{0} ∈ X_{1} and M_{ν} is some G-module. Set I_{1} = ⨁_{ν} Q_{1}(ν_{0}) ⊗ M_{ν}^{F}. So Soc_{G_{1}} I_{1} = Soc_{G_{1}} Q_{1}(μ)/L(μ). (It is worth noting that the condition on the weights here is enough to ensure that Soc_{G_{1}} M = Soc_{G} M but we do not need this fact explicitly.) Thus I_{1} is the G_{1}-injective hull of M, hence if there is a G-map I_{0} → I_{1}, this will be part of a minimal resolution. It remains to show that there is indeed a map I_{0} → I_{1} of G-modules whose kernel is L(μ), i.e. a map I_{0}/L(μ) → I_{1}. Note that we do have a map Soc_{G_{1}} M → I_{1} by construction, so consider the exact sequence

(*)  Hom_{G}(M, I_{1}) → Hom_{G}(Soc_{G_{1}} M, I_{1}) → Ext_{G}^{1}(M/ Soc_{G_{1}} M, I_{1}).

If we could show that the third term in this sequence is zero then we would have that the first map were surjective, hence the G-map Soc_{G_{1}} M → I_{1} would lift to a map M = I_{0}/L(μ) → I_{1} and we would be done.

Now Ext_{G}^{1}(M/ Soc_{G_{1}} M, I_{1}) has a filtration by spaces E = Ext_{G}^{1}(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 spectral sequence, of which part is

Ext_{G/G_{1}}^{1}(k, Hom_{G_{1}}(M/ Soc_{G_{1}} M, Q_{1}(ν_{0}) ⊗ L(ν_{1})^{F})) → E
→ Hom_{G/G_{1}}(k, Ext_{G_{1}}^{1}(M/ Soc_{G_{1}} 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 particular, 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}^{∨}) ≤ 2h−2 and our condition on p implies that they are both in the closure of the lowest alcove, C̄_{ℤ}. So let L(ξ_{0}) ⊗ L(ξ_{1})^{F} be a composition factor of M/ Soc_{G_{1}} M. We compute:

Ext_{G/G_{1}}^{1}(k, Hom_{G_{1}}(L(ξ_{0}) ⊗ L(ξ_{1})^{F} ,Q_{1}(ν_{0}) ⊗ L(ν_{1})^{F}))
≅ Ext_{G}^{1}(L(ξ_{1}), Hom_{G_{1}}(L(ξ_{0}), Q_{1}(ν_{0}))^{[−1]} ⊗ L(ν_{1}))

Now Hom_{G_{1}}(L(ξ_{0}), Q_{1}(ν_{0})) is non-zero, thence equal to k, if an only ξ_{0} = ν_{0}; in that case, the term on the right becomes Ext_{G}^{1}(L(ξ_{1}), L(ν_{1})), and since ξ_{1}, ν_{1} ∈ C̄_{ℤ}, this vanishes by the linkage principle. This concludes the proof in case r = 1.

Now by induction we may assume that we have a sequence of G-modules

0 → I_{0} → ⋯ → I_{r−2} →^{π} I_{r−1},

which is minimal as an injective G_{1}-resolution, 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 similar way to before: set I_{r} = ⨁_{ν} Q_{1}(ν_{0}) ⊗ M_{ν}^{F}, where the sum is over the G-composition factors of Soc_{G_{1}} I_{r−1}/πI_{r−2}, 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 hull of I_{r−1}/ im π so if we can show there is a G-module map I_{r−1} → I_{r} with kernel im π, we will be done.

Of course, it is equivalent to produce a map from M := I_{r−1}/ im π to I_{r}. By construction we do have a map from Soc_{G_{1}} M → I_{r}. Now the same argument as before shows that the third term in the sequence (*) (with I_{r} replacing I_{1}) is zero. This completes 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-resolution which is also a G_{1}-injective resolution:

0 → L(μ_{0}) → I_{0} → I_{1} → ⋯ → I_{r} → ⋯,

where, up to I_{r}, the resolution is minimal for G_{1}.

We denote the differentials by δ_{i} : I_{i} → I_{i+1} and the kernels by K_{i} := ker δ_{i}. Dimension shifting gives us Ext_{G_{1}}^{i}(L(λ_{0}), L(μ_{0})) ≅ Ext_{G_{1}}^{1}(L(λ_{0}), K_{i−1}). Minimality gives us for μ_{1} ∈ X_{1} that Ext_{G_{1}}^{i}(L(λ_{0}), L(μ_{0})) ≅ Hom_{G_{1}}(L(λ_{0}), K_{i}) ≅ Hom_{G_{1}}(L(λ_{0}), I_{i}) for i ≤ r.

We now have a G-resolution:

0 → L(μ) → I_{0} ⊗ L(μ_{1})^{F} →^{∂_{0}} I_{1} ⊗ L(μ_{1})^{F} →^{∂_{1}} ⋯

where ∂_{i} = δ_{i} ⊗ id, as tensoring is exact. Also note that such a resolution stays injective as a G_{1}-resolution as L(μ_{1})^{F} is trivial as a G_{1}-module.

Now consider the E_{0}-page of the LHS spectral sequence that converges to Ext_{G}^{∗}(L(λ), L(μ)) as constructed in [Par07, §2]

E_{0}^{mn} = Hom_{G/G_{1}}(k, Hom_{G_{1}}(L(λ), I_{n} ⊗ L(μ_{1})^{F}) ⊗ J_{m}^{F})

where we have a G-injective resolution of the trivial module:

0 → k → J_{0} → J_{1} → ⋯

and this spectral sequence has E_{1} and E_{2} page

E_{1}^{mn} = Hom_{G/G_{1}}(k, Ext_{G_{1}}^{n}(L(λ), L(μ)) ⊗ J_{m}^{F})
E_{2}^{mn} = H^{m}(G/G_{1}, Ext_{G_{1}}^{n}(L(λ), L(μ))).

Consider the induced maps ∂_{m}^{∗} in the following complex, which has homology Ext_{G_{1}}^{∗}(L(λ), L(μ)):

Hom_{G_{1}}(L(λ), I_{0} ⊗ L(λ_{1})^{F}) ⟶^{∂_{0}^{∗}} Hom_{G_{1}}(L(λ), I_{1} ⊗ L(μ_{1})^{F}) ⟶^{∂_{1}^{∗}} · · · .

Now

Ext_{G_{1}}^{m}(L(λ), L(μ)) ≅ Ext_{G_{1}}^{m}(L(λ_{0}), L(μ_{0})) ⊗ L(μ_{1})^{F} ⊗ L(λ_{1}^{∗})^{F}
≅ Hom_{G_{1}}(L(λ_{0}), I_{m}) ⊗ L(μ_{1})^{F} ⊗ L(λ_{1}^{∗})^{F} ≅ Hom_{G_{1}}(L(λ), I_{m} ⊗ L(μ_{1})^{F})

for m ≤ r. Thus all the differentials ∂_{m}^{∗} for m ≤ r must be zero.

Now by [Ben98, §3.2, §3.4] we know that the spectral sequence can be constructed using derived couples. We have

D_{0}^{mn} = ⨁_{m+n=e+f, e≥m} E_{0}^{ef}

E_{1}^{mn} = H(E_{0}^{mn}, d_{0})

D_{1}^{mn} = H(E_{0}^{mn} ⊕ E_{0}^{m+1,n−1} ⊕ ⋯, d_{0} + d_{1})

We define the higher derived couples by taking the derived couple of the previous one. We have an exact diagram of doubly graded k-modules

\xymatrix{D_{l} \ar[rr]^{i_{l}} &  & D_{l} \ar[dl]^{j_{l}} \\  & E_{l} \ar[ul]^{k_{l}}}

The derived couple (for l ≥ 1) is defined by

D_{l+1}^{mn} = im i_{l}^{m+1,n−1} ⊆ D_{l}^{mn}  E_{l+1}^{mn} = H(E_{l}^{mn}, d_{l})
i_{l+1}^{mn} = i_{l}^{mn}|_{D_{l+1}}  j_{l+1}^{mn}(i_{l}^{m+1,n−1}(x)) = j_{l}^{m+1,n−1}(x) + im(d_{l})
k_{l+1}^{mn}(z + im(d_{l})) = k_{l}^{mn}(z)  d_{l+1} = j_{l+1} ∘ k_{l+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 2.1] we have that d_{0}^{mn} = 0 implies that k_{2}^{m−1,n+1} = 0. Thus since d_{0}^{mn} = 0 for m ≤ r we have k_{2}^{mn} = 0 for m ≤ r − 1. Thus all k_{l}^{mn} = 0 for all l ≥ 2 and m ≤ r − 1. As d_{l}^{mn} = j_{l}^{m+1,n} ∘ k_{l}^{mn} we also get d_{l}^{mn} = 0 for all l ≥ 2 and m ≤ r − 1.

In other words, as all these differentials are zero on the E_{2} page and remain zero, the terms E_{2}^{mn} with m ≤ r − 1 must already be the stable value. That is, E_{∞}^{mn} = E_{2}^{mn} for m ≤ r − 1.

This easily gives us that

Ext_{G}^{i}(L(λ), L(μ)) = ⨁_{j=0}^{i} E_{2}^{i−j,j}

for i < r. To get the result for i = r, we note that all the terms in the sum

⨁_{j=0}^{r} E_{∞}^{r−j,j}

stabilise at the E_{2} page by the above, except, possibly the term E_{∞}^{r0}. But here clearly E_{∞}^{r0} = E_{2}^{r0} as all incoming differentials are zero by the above, and the leaving differential d_{l}^{r,0} is always zero as our spectral sequence is first quadrant.

We may similarly argue for r + 1. We consider

⨁_{j=0}^{r+1} E_{∞}^{r+1−j,j}.

As before all terms except possibly E_{∞}^{r,1} and E_{∞}^{r+1,0} stabilise at the E_{2} page. The same argument as in the previous case gives E_{∞}^{r+1,0} = E_{2}^{r+1,0}.

Now note that all incoming differentials to E_{l}^{r,1} are zero for l ≥ 2 by the above. We also have that d_{l}^{r,1} = 0 for l ≥ 3, again since the spectral sequence is first quadrant. So we need only check that d_{2}^{r1} = 0, but this is true using [Don82, Main Theorem]. Thus we also get the result for r + 1. □

Acknowledgements. The authors wish to thank Len Scott and Dan Nakano for comments on a previous version of this paper.

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School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK,

E-mail address: a.e.parker@leeds.ac.uk (Parker)

New College, Oxford, Oxford, UK

E-mail address: david.stewart@new.ox.ac.uk (Stewart)