BLOCKS OF THE CYCLOTOMIC HECKE ALGEBRA.

I. Grojnowski

January 1999

In this note we determine the blocks of the cyclotomic Hecke algebra. We show that the obvious central elements already distinguish distinct blocks. This was conjectured by Graham and Lehrer [GL] in an equivalent but different form.

We will freely use notation from [G], but we recall the most essential definitions. We write H_{n}^{aff} for the affine Hecke algebra over a field R, λ = (q_{1}, …, q_{r}) ∈ (R^{∗})^{r}, I_{λ} = H_{n}^{aff} · ∏_{1≤i≤r}(X_{1} − q_{i}) · H_{n}^{aff}, and H_{n}^{λ} = H_{n}^{aff}/I_{λ} for the cyclotomic Hecke algebra. (In [G] we studied the most interesting case, when q_{i} is a power of q, but there is no need for such a restriction here).

Given any algebra A, Rep A is the category of finite dimensional representations of A. As in [G], ev^{∗} : Rep H_{n}^{λ} ⟶ Rep H_{n}^{aff} and pr : Rep H_{n}^{aff} ⟶ Rep H_{n}^{λ} are the obvious pair of adjoint functors; pr M = M/I_{λ}M is right exact. Denote the i’th cohomology of the left derived functor of pr by L^{i} pr.

Finally, for s ∈ (R^{∗})^{n} we will write Rep_{s} H_{n}^{λ} for the full subcategory of modules on which the obvious central algebra R[X_{1}^{±1}, …, X_{n}^{±1}]^{S_{n}} acts with only one irreducible constituent, the character s. It is clear that Rep_{s} H_{n}^{λ} is a sum of blocks; the theorem we prove below is it is in fact indecomposable (or zero).

1.1 Recall that if A is an algebra, and M and N are irreducible A-modules, then M and N are in the same block if there is a sequence M = V_{0}, V_{1}, …, V_{n} = N of irreducible A-modules such that, for all i, either Ext^{1}(V_{i}, V_{i+1}) ≠ 0 or Ext^{1}(V_{i+1}, V_{i}) ≠ 0.

Theorem. Let M and N be irreducible H_{n}^{λ}-modules. Then M and N are in the same block of Rep H_{n}^{λ} if and only if ev^{∗} M and ev^{∗} N are in the same block of H_{n}^{aff}.

Corollary. The blocks of Rep H_{n}^{λ} are the subcategories Rep_{s} H_{n}^{λ}, s ∈ (R^{∗})^{n}/S_{n}.

The blocks of an algebra are the blocks of its center, so the corollary follows immediately from Bernstein’s theorem describing the center of H_{n}^{aff}.

So we must prove the theorem.

As H_{n}^{aff} surjects onto H_{n}^{λ}, it is clear that if M and N are in the same block of Rep H_{n}^{λ} then ev^{∗} M and ev^{∗} N are in the same block of H_{n}^{aff}. So the theorem will follow from the following proposition, whose proof we break up into two lemmas.

Proposition. Let M and N be irreducible H_{n}^{λ}-modules, and suppose that M ≠ N. Let

0 ⟶ M ⟶ X ⟶ N ⟶ 0

Typeset by AMS-TEX

be an exact sequence of H_{n}^{aff}-modules. Then pr X = X, and the sequence is an exact sequence of H_{n}^{λ}-modules.

We write Ind and Res for the functors of induction and restriction between H_{n−1,1}^{aff} = H_{n−1}^{aff} ⊗ H_{1}^{aff} and H_{n}^{aff}.

Lemma 1. Let M be a H_{n}^{λ} module. There is a surjective map

pr Ind Res M ↠ L^{1} pr(M).

In fact, this map is an isomorphism, but we do not need this here.

Proof. We have pr(M) = (H_{n}^{aff}/I_{λ}) ⊗_{H_{n}^{aff}} M. We may compute the left derived functor of tensor product by resolving H_{n}^{aff}/I_{λ} by H_{n}^{aff}-bimodules which are free as right H_{n}^{aff}-modules. The first part of such a resolution is

(H_{n}^{aff} ⊗_{H_{1,n−1}^{aff}} H_{n}^{aff} ⟶^{d_{1}} H_{n}^{aff}) ⟶^{d_{0}} H_{n}^{aff}/I_{λ}

where d_{1}(a ⊗ b) = a · ∏(X_{1} − q_{i}) · b and d_{0} = ev is the natural projection. Set Z_{i} = ker(d_{i}), so Z_{0} = I_{λ}. Then we have short exact sequences

0 ⟶ Z_{0} ⟶ H_{n}^{aff} ⟶ H_{n}^{aff}/I_{λ} ⟶ 0, 0 ⟶ Z_{1} ⟶ H_{n}^{aff} ⊗_{H_{1,n−1}^{aff}} H_{n}^{aff} ⟶ Z_{0} ⟶ 0

which give long exact sequences on tensoring with M. These give L^{i+1} pr(M) = Tor_{H_{n}^{aff}}^{i+1}(H_{n}^{aff}/I_{λ}, M) = Tor_{H_{n}^{aff}}^{i}(Z_{0}, M) for i ≥ 0, and

⟶ Tor_{H_{n}^{aff}}^{0}(Z_{1}, M) ⟶ Tor_{H_{n}^{aff}}^{0}(H_{n}^{aff} ⊗_{H_{1,n−1}^{aff}} H_{n}^{aff}, M) ⟶ Tor_{H_{n}^{aff}}^{0}(Z_{0}, M) ⟶ 0,

and hence a surjection H_{n}^{aff}⊗_{H_{1,n−1}^{aff}}M ↠ L^{1} pr(M), hence a surjection Ind Res(M) ↠ L^{1} pr(M) as claimed.

Lemma 2. If M and N are irreducible H_{n}^{aff} modules in the same H_{n}^{aff}-block, and M ≠ N, then

Hom_{H_{1,n−1}^{aff}}(Res N, Res M) = 0.

Proof. We will prove instead that Hom_{H_{n−1,1}^{aff}}(Res N, Res M) = 0. This is equivalent to the statement of the lemma by applying the diagram automorphism of H_{n}^{aff}, and it allows us to use the results of [G] without needing to change notation.

The forgetful map H_{n−1}^{aff} ↪ H_{n−1,1}^{aff} induces an injection

Hom_{H_{n−1,1}^{aff}}(Res N, Res M) ↪ ⊕_{i} Hom_{H_{n−1}^{aff}}(e_{i}^{∗}N, e_{i}^{∗}M),

so it suffices to show the stronger statement Hom(e_{i}^{∗}N, e_{i}^{∗}M) = 0 for all i ∈ R^{∗}.

Write N = f̃_{i}^{ε}N′, where ε_{i}(N′) = 0. As Ind(N′ ⊠ K_{ε}) ↠ N, it is enough to show that Hom_{i} =: Hom(e_{i}^{∗} Ind(N′ ⊠ K_{ε}), e_{i}^{∗}M) is zero. But e_{i}^{∗} Ind(N′ ⊠ K_{ε}) = Ind(N′ ⊠ e_{i}^{∗}K_{ε}) by [G], so

Hom_{i} = Hom(Ind(N′ ⊠ e_{i}^{∗}K_{ε} ⊠ J_{m}), M)

for m ≫ 0. Now Ind(N′⊠e_{i}^{∗}K_{ε}⊠J_{m}) admits a filtration with subquotients Ind(N′⊠ K_{ε−1} ⊠ J_{1}) = Ind(N′ ⊠ K_{ε}), so if ϕ ∈ Hom_{i} is non-zero, it must induce a non-zero map Ind(N′ ⊠ K_{ε}) ⟶ M, hence a non-zero map cosoc Ind(N′ ⊠ K_{ε}) ⟶ M. But cosoc Ind(N′ ⊠ K_{ε}) = f̃_{i}^{ε}N′ = N, and N ≠ M. It follows that Hom_{i} = 0.

We now prove the proposition. Suppose given such an exact sequence 0 ⟶ M ⟶ X ⟶ N ⟶ 0. Applying pr, we get an exact sequence

⟶ L^{1} pr N ⟶^{α} M ⟶ pr X ⟶ N ⟶ 0.

We must show that α = 0. By lemma 1, it suffices to show that Hom(pr Ind Res N, M) = 0. But Hom(pr Ind Res N, M) = Hom(Ind Res N, M) = Hom(Res N, Res M) = 0, by lemma 2. We are done.

References

[GL] J. Graham and G. Lehrer, Cellular algebras, Invent. Math 123 (1996), 1–34.

[G] I. Grojnowski, Affine 𝔰̂𝔩_{p} controls the modular representation theory of the symmetric group and related Hecke algebras, preprint.

DPMMS, 16 Mill Lane, Cambridge, CB2 1SB