Translation Functors and Equivalences of Derived Categories for Blocks of Algebraic Groups

Jeremy Rickard

School of Mathematics University of Bristol University Walk Bristol BS8 1TW England j.rickard@bristol.ac.uk

ABSTRACT.

We prove the existence of many self-equivalences of the derived categories of blocks of reductive groups in prime characteristic that are not induced by self-equivalences of the module categories of the blocks. We conjecture that our examples are just the simplest special case of a much more general phenomenon. This is analogous to conjectures about derived categories of blocks of symmetric groups.

1. Introduction

In (3), Happel took the machinery of derived categories from algebraic geometry and introduced it into the representation theory of finite-dimensional algebras, showing that the tilting theory of algebras has an extremely elegant interpretation in terms of equivalences of derived categories of module categories. Since then, more general equivalences of derived categories have been studied in some detail. Perhaps the most intriguing situation where such equivalences have been discovered is in the modular representation theory of finite groups. Broué, in (1), made some deep conjectures about conditions that would ensure that two blocks have equivalent derived categories. If proved, these conjectures would have important consequences for the study of block invariants, such as the number of characters in a block, since most such invariants are actually invariants of the derived category.

One consequence of an equivalence of derived categories for blocks is that it induces a ‘perfect isometry’ between the groups of characters of the two blocks. One important class of examples of perfect isometries was found by Enguehard (2); he proved that any two blocks of (possibly different) symmetric groups in characteristic p are ‘perfectly isometric’ if they have the same defect group. This gives a strong indication that such blocks should have equivalent derived categories. At about the same time, Scopes (6) proved that many blocks of symmetric groups are actually Morita equivalent, and that in fact there are only finitely many Morita equivalence classes for a given defect group. The pairs of blocks for which Enguehard’s perfect isometry can be taken to be a bijection between irreducible characters without signs are precisely the pairs of blocks which Scopes proves to be Morita equivalent.

Several people seem to have independently noticed an analogy between Scopes’ equivalences and Jantzen’s translation functors, which give many equivalences of categories between blocks of rational representations of reductive groups in positive characteristic. During an only partially successful attempt to generalize the ideas of Scopes to find equivalences of derived categories for symmetric groups, I realized that the analogy seemed to extend to these derived equivalences. In neither the case of symmetric groups nor the case of reductive groups have I so far been able to prove all that I would like, but in this paper I will describe one of the simplest cases for reductive groups.

Let us now recall some basic facts about blocks of reductive groups. Jantzen’s book (4) is a convenient reference for all the facts we shall use. First we shall fix some notation.

− Let k be an algebraically closed field of prime characteristic p. − Let G be a connected reductive group defined over k. − Let T be a maximal torus of G. − Let B be a Borel subgroup of G containing T. − Let X(T) be the group of characters of T. − Let X(T)_{+} be the set of dominant weights in X(T). − Let mod-G be the category of finite-dimensional rational G-modules.

1. The simple rational G-modules are parametrized by the set X(T)_{+} of dominant weights: for each λ ∈ X(T)_{+} there is a unique simple module L(λ) with highest weight λ (4, II.2).

2. Each λ ∈ X(T) determines a line bundle on G/B and the cohomology groups H^{i}(λ) of these line bundles are rational G-modules (4, I.5). By Kempf’s Vanishing Theorem (4, II.4), H^{i}(λ) = 0 for i > 0 if λ is dominant.

3. Each M in mod-G has a formal character ch(M) in the group ring Z[X(T)]; the function taking M to ch(M) is additive on short exact sequences. The set of formal characters ch(L(λ)) of the simple modules form a basis for the group generated by formal characters of all modules (i.e., ch(M) determines the composition factors of M). The formal characters ch(H^{0}(λ)) for λ ∈ X(T)_{+} form another basis.

4. For any λ ∈ X(T), not necessarily dominant, we define an element χ(λ) of Z[X(T)] by

χ(λ) = ∑_{i≥0}(−1)^{i}ch(H^{i}(λ)),

so if λ is dominant, then χ(λ) = ch(H^{0}(λ)).

5. The Weyl group W of G acts as a reflection group on X(T), and this action extends to an action of the affine Weyl group W_{p} which is generated by W together with all translations by pβ for roots β. By shifting the origin of this action, we get a new action, the dot action, of W_{p} on X(T), given by

w · λ = w(λ + ρ) − ρ,

where ρ is half the sum of the positive roots (4, II.6.1).

6. If w ∈ W, then there is a very simple formula (4, II.5.9) for the effect of w on

the characters χ(λ):

χ(w · λ) = det(w)χ(λ).

7. Weyl’s Character Formula (4, II.5.10) implies that χ(λ) is independent of the characteristic of k, and so (4, II.5.5) applied in characteristic zero implies that χ(λ) = 0 unless λ is conjugate to a dominant weight under the dot action of W on X(T). Hence χ(λ) is either zero or is ±ch(H^{0}(μ)) for some μ ∈ X(T)_{+}.

8. If we define

\overline{C}Z = \{λ ∈ X(T) : 0 ≤ ⟨λ + ρ, β^{∨}⟩ ≤ p for all positive roots β\},

then \overline{C}Z is a fundamental domain for the action of W_{p} on X(T).

9. For each λ ∈ X(T), we can define the full subcategory M_{λ} of mod-G to consist of those modules whose simple composition factors all have highest weights in W_{p} ·λ. The Linkage Principle (4, II.6.17) implies that every indecomposable rational G-module lies in M_{λ} for some λ. In other words, each M_{λ} is a sum of blocks (usually a single block) of G.

2. The Main Theorem

Let us first recall the definition of Jantzen’s translation functors (4, III.7).

Let λ and μ be two weights in \overline{C}Z, and let pr_{λ} be the functor from mod-G to M_{λ} that takes a module M to the largest submodule of M that lies in M_{λ} (i.e., pr_{λ} is ‘projection onto the block M_{λ}’), and let pr_{μ} be the similarly defined functor from mod-G to M_{μ}.

Since X(T)_{+} is a fundamental domain for the action of W on X(T), there is a unique dominant weight ν_{1} in W (μ − λ). We now define the translation functor

T_{λ}^{μ} : M_{λ} ⟶ M_{μ}

by

T_{λ}^{μ}(V) = pr_{μ}(L(ν_{1}) ⊗ V ).

This definition gives an exact functor, and one of the basic properties of these functors is that T_{λ}^{μ} is both left and right adjoint to T_{μ}^{λ}.

If λ and μ are in \overline{C}Z, then the effect of the translation functor T_{λ}^{μ} on formal characters is easy to describe (4, II.7.8). If V is in M_{λ}, then

ch(V) = ∑_{w∈W_{p}} a_{w}χ(w · λ)

for some coefficients a_{w} ∈ Z with almost all a_{w} zero. Then

ch(T_{λ}^{μ}(V)) = ∑_{w∈W_{p}} a_{w} ∑_{w_{1}} χ(ww_{1} · μ),

where w_{1} runs through a set of coset representatives for

Stab_{W_{p}}(λ)/(Stab_{W_{p}}(λ) ∩ Stab_{W_{p}}(μ)).

In particular, if λ and μ have the same stabilizer in W_{p}, then

ch(T_{λ}^{μ}(V)) = ∑_{w∈W_{p}} a_{w}χ(w · μ),

from which the ‘translation principle’, saying that in this case T_{λ}^{μ} is an equivalence of categories from M_{λ} to M_{μ}, follows quite quickly (4, II.7.9).

More generally, if λ and μ are arbitrary elements of X(T) with the same stabilizer in W_{p}, then

θ : ∑_{w∈W_{p}} a_{w}χ(w · λ) ⟼ ∑_{w∈W_{p}} a_{w}χ(w · μ)

is still a well-defined isomorphism between the groups of characters of M_{λ} and of M_{μ}. However, when λ, μ ∉ \overline{C}Z this map sends the character of a module to a virtual character that need not be the character of a module, so in general the map cannot be realized by an equivalence of module categories.

For those familiar with Broué’s ‘perfect isometries’ (1) for characters of finite groups, however, the map θ has some suggestive properties. The characters χ(w · λ) for dominant w · λ are the analogue for reductive groups of the ordinary irreducible characters for blocks of finite groups, so θ takes an ‘ordinary irreducible’ character to another ‘ordinary irreducible’ character multiplied by a sign ±1 — just like a perfect isometry.

The philosophy of perfect isometries is that they frequently occur for blocks of finite groups because they are the character-theoretic ‘shadow’ of a structural phenomenon: an equivalence of triangulated categories between the derived categories of the blocks. So it is natural to ask whether the map θ comes from an equivalence between the derived categories of M_{λ} and M_{μ}. We shall show that this is so in at least some cases.

THEOREM 2.1. Let G be a connected reductive group defined over k. Suppose that G has a simply-connected derived subgroup and that the root system of G has a Coxeter number h ≤ p. Let T be a maximal torus of G, and let λ, μ ∈ X(T) be weights with trivial stabilizer in W_{p}. There is an equivalence of triangulated categories

D^{b}(M_{λ}) ⟶ D^{b}(M_{μ})

that induces the map

χ(w · λ) ⟼ χ(w · μ)

on characters.

This theorem will follow easily from the following.

PROPOSITION 2.2. Let G be a connected reductive group defined over k whose derived subgroup is simply-connected and whose root system has Coxeter number h ≤ p. Let λ ∈ \overline{C}Z be a weight with trivial stabilizer in W_{p}, let s ∈ W_{p} be reflection in a wall of \overline{C}Z, and let μ = s · λ. Then there is an equivalence of triangulated categories

D^{b}(M_{λ}) ⟶ D^{b}(M_{μ})

that induces the map

χ(w · λ) ⟼ χ(w · μ)

on characters.

The next section will be devoted to the proof of this proposition. Let us first show how it may be used to deduce Theorem 2.1.

Proof of Theorem 2.1: Since \overline{C}Z is a fundamental domain for the action of W_{p} on X(T), μ ∈ W_{p}·λ′ for some λ′ ∈ \overline{C}Z. By the translation principle (4, II.7.9) there is an equivalence of categories

M_{λ} ⟶ M_{λ′}

that induces the map

χ(w · λ) ⟼ χ(w · λ′)

on characters, so without loss of generality we may and shall assume that λ = λ′. The set of reflections in walls of \overline{C}Z generates W_{p} as a group (4, II.6.3), so

μ = s_{1}s_{2} … s_{n} · λ

for some sequence s_{1},… s_{n} of reflections in walls of \overline{C}Z. By Proposition 2.2, for each reflection s_{i} there is an equivalence of triangulated categories

D^{b}(M_{λ}) ⟶ D^{b}(M_{λ})

inducing the map

χ(w · λ) ⟼ χ(ws_{i} · λ)

on characters. The theorem follows by taking a composition of such equivalences. □

We should remark that with λ and μ as in Theorem 2.1, both of these weights are conjugate under the dot action of W_{p} to weights in \overline{C}Z which also have trivial stabilizer. Applying the translation principle for these weights, we immediately deduce that M_{λ} and M_{μ} are equivalent categories, and are both equivalent to the principal block M_{0} of G. So the fact that the corresponding derived categories are equivalent is not in itself surprising. The content of Theorem 2.1 is rather that there are many non-trivial self-equivalences of these derived categories that realize various natural automorphisms of the group of characters. If, as we suggested before we stated the main theorem, the obvious generalization of the theorem to weights with non-trivial stabilizer is also true, then this would provide equivalences of derived categories for pairs of blocks for which the translation principle does not give an equivalence of module categories.

3. Proof of Proposition 2.2

Since the derived subgroup of G is simply-connected and h ≤ p, it follows from (4, II.6.3) that there is some ν ∈ X(T) on the wall in which s reflects.

Consider the bounded complex of functors

… ⟶ 0 ⟶ T_{ν}^{λ}T_{λ}^{ν} ⟶ id ⟶ 0 ⟶ …

from M_{λ} to itself, where T_{ν}^{λ}T_{λ}^{ν} is in degree zero and the map from T_{ν}^{λ}T_{λ}^{ν} to the identity functor is the counit of the adjunction. Since the functors involved are all exact, this complex defines an exact functor

F : D^{b}(M_{λ}) ⟶ D^{b}(M_{λ})

by applying the complex of functors to a complex of modules and then taking the total complex of the resulting double complex. We shall show that F is an equivalence of categories.

There is another bounded complex

… ⟶ 0 ⟶ id ⟶ T_{ν}^{λ}T_{λ}^{ν} ⟶ 0 ⟶ …

of functors, giving a functor F^{∨} from D^{b}(M_{λ}) to itself, where again T_{ν}^{λ}T_{λ}^{ν} is in degree zero, and this time the map from the identity functor to T_{ν}^{λ}T_{λ}^{ν} is the unit of the adjunction. The first thing we want to show is that F^{∨} is both left and right adjoint to F.

Let C and D be arbitrary categories and let

L, L′ : C ⟶ D and R, R′ : D ⟶ C

be functors and suppose we are given adjunctions making R right adjoint to L and R′ right adjoint to L′. Then by (5, IV.7, Theorem 2), for every map of functors L ⟶ L′ there is a unique map of functors R′ ⟶ R (the conjugate map) such that the diagram

Hom_{D}(L′−, −) ≅ Hom_{C}(−, R′−)
↓ ↓
Hom_{D}(L−, −) ≅ Hom_{C}(−, R−)

commutes.

It follows easily from the definition of conjugate maps that the conjugate of a composition of maps of functors is the composition of the conjugates of the individual maps, and that if C and D are additive categories then the conjugate of the zero map between additive functors is also the zero map. Hence, if

L^{∗} = … ⟶ L^{n} ⟶ L^{n+1} ⟶ …

is a bounded complex of functors, where each L_{n} has a right adjoint R^{n}, then we get a bounded complex of functors

R^{∗} = … ⟶ R^{n+1} ⟶ R^{n} ⟶ …

with degree i term R^{−i} by taking the conjugates of all the differentials of L^{∗}. If X^{∗} and Y^{∗} are bounded complexes over C and D respectively, then the definition of conjugacy immediately gives a natural isomorphism

Hom_{C}(X^{∗}, R^{∗}Y^{∗}) ≅ Hom_{D}(L^{∗}X^{∗}, Y^{∗})

of triple complexes. Since the degree zero homology groups of the total complexes of these triple complexes are the groups of chain homotopy classes of maps from X^{∗} to the total complex of R^{∗}Y^{∗} and from the total complex of L^{∗}X^{∗} to Y^{∗} respectively, we deduce that the functor given by L^{∗} from the homotopy category K^{b}(C) of complexes over C to K^{b}(D) is left adjoint to the functor given by R^{∗} from K^{b}(D) to K^{b}(C).

If, furthermore, C and D are abelian categories and all the functors L^{i} and R^{i} are exact, then this adjunction of functors between homotopy categories provides an adjunction of functors between the derived categories D^{b}(C) and D^{b}(D). Returning now to our functors F and F^{∨}, we see that to prove that F^{∨} is right adjoint to F, we just need to show that the unit

id ⟶ T_{ν}^{λ}T_{λ}^{ν}

is conjugate to the counit

T_{ν}^{λ}T_{λ}^{ν} ⟶ id.

This is true because if we regard T_{ν}^{λ}T_{λ}^{ν} as a functor from mod-G to mod-G, vanishing on modules with no summand in M_{λ} and sending every module to an object of M_{λ}, then the counit map can be expressed as the composition of three maps: the inclusion of the direct summand

T_{ν}^{λ}T_{λ}^{ν} ⟶ − ⊗ L(α_{1}) ⊗ L(α_{1})^{∗},

where α_{1} is the unique dominant weight in W (ν − λ), the map

− ⊗ L(α_{1}) ⊗ L(α_{1})^{∗} ⟶ id_{G} ≅ − ⊗ k

induced by the natural map L(α_{1}) ⊗ L(α_{1})^{∗} ⟶ k, and the projection

id_{G} ⟶ pr_{λ}

onto a direct summand. But it is clear that the conjugate of the inclusion of a direct summand of a functor is just projection onto the corresponding direct summand of the adjoint, and that the conjugate of the natural map

− ⊗ L(α_{1}) ⊗ L(α_{1})^{∗} ⟶ − ⊗ k

is just the natural map

− ⊗ k ⟶ − ⊗ L(α_{1}) ⊗ L(α_{1})^{∗}.

The claim about the conjugate of the counit map now follows by composing the three conjugates. Hence F^{∨} is right adjoint to F, and similarly F^{∨} is also left adjoint to F.

Let us now consider the composite F^{∨}F. This is induced by the complex of functors we get by taking the total complex of the double complex whose only non-zero terms are

T_{ν}^{λ}T_{λ}^{ν} ⟶ id
↓ ↓
T_{ν}^{λ}T_{λ}^{ν}T_{ν}^{λ}T_{λ}^{ν} ⟶ T_{ν}^{λ}T_{λ}^{ν},

where the left hand vertical map is the map

ϵ ∘ T_{ν}^{λ}T_{λ}^{ν} : id ∘ T_{ν}^{λ}T_{λ}^{ν} ⟶ T_{ν}^{λ}T_{λ}^{ν}T_{ν}^{λ}T_{λ}^{ν}

induced by the unit

ϵ : id ⟶ T_{ν}^{λ}T_{λ}^{ν}

of the adjunction, and the bottom horizontal map is the map

T_{λ}^{ν}T_{ν}^{λ} ∘ η

induced by the counit η. But for any functor L with a right adjoint R, the composition

L ⟶^{Lϵ} LRL ⟶^{ηL} L

is the identity (5, IV.1, Theorem 1), where ϵ and η are respectively the unit and counit of the adjunction. Hence the left hand vertical map is a split monomorphism, and similarly the bottom horizontal map is a split epimorphism, so the total complex is split, and so is chain homotopy equivalent to a complex of functors concentrated in degree zero. Thus the functor F^{∨}F is isomorphic to a functor induced by an exact functor from the abelian category M_{λ} to itself. In particular, if we regard M_{λ} as a full subcategory of its derived category D^{b}(M_{λ}) in the usual way then F^{∨}F (V) is in M_{λ} for every V in M_{λ}.

Let us next consider the maps

K_{0}(M_{λ}) ⟶ K_{0}(M_{λ})

induced by F and F^{∨}. Let us call these maps ϕ and ϕ^{∨} respectively. These maps are easy to evaluate on characters of the form χ(w · λ), since we know the effect of translation functors on characters of this form. So

ϕ(χ(w · λ)) = [χ(w · λ) + χ(ws · λ)] − χ(w · λ)
= χ(ws · λ)
= χ(w · μ)

and

ϕ^{∨}(χ(w · λ)) = [χ(w · λ) + χ(ws · λ)] − χ(w · λ)
= χ(ws · λ)
= χ(w · μ),

since T_{λ}^{ν} maps χ(w · λ) to χ(w · ν) and T_{ν}^{λ} maps χ(w · ν) to χ(w · λ) + χ(ws · λ). Therefore, since s^{2} is the identity,

ϕ^{∨}ϕ = id = ϕϕ^{∨}.

Since we already know that F^{∨}F maps modules to modules, and we now know that it preserves characters, it must send a simple module L(w · λ) to an isomorphic simple module. Since the unit id ⟶ F^{∨}F of the adjunction evaluates as a non-zero map on every object of D^{b}(M_{λ}), it must therefore evaluate as an isomorphism on all simple modules. Since the simple modules generate D^{b}(M_{λ}) as a triangulated category, the unit of the adjunction is an isomorphism of functors, so F^{∨}F ≅ id. Similarly F F^{∨} ≅ id, and so F and F^{∨} are equivalences of categories, completing the proof of the Proposition. □

4. Concluding Remarks

Our main theorem has variants that apply in slightly different situations. Indeed, an almost identical proof gives the obvious generalization of Theorem 2.1 to representations of blocks of G_{r}T or G_{r}B, the infinitesimal thickenings of a maximal torus or Borel subgroup of a reductive group G, since all the properties that we use for translation functors for G also hold for G_{r}T and G_{r}B (4, II.9).

As we indicated in Section 2, character calculations suggest that Theorem 2.1 generalizes to more general blocks. We make the following conjecture.

CONJECTURE 4.1. Let G be a reductive group over k that has a simply-connected derived group, and whose Coxeter number is less than p. Let λ and μ be any two weights in X(T) with the same stabilizer under the dot action of W_{p}. Then there is an equivalence of triangulated categories

D^{b}(M_{λ}) ⟶ D^{b}(M_{μ})

that induces the map

χ(w · λ) ⟼ χ(w · μ)

on characters.

It is not clear to us whether or not the conditions on the derived subgroup and the Coxeter number are strictly necessary.

References

[1] M. Broué. Isométries parfaites, types de blocs, catégories dérivées. Astérisque, 181–182:61–92, [2] M. Enguehard. Isométries parfaites entre blocs de groupes symétriques. Astérisque, 181–

182:157–171, 1990.

[3] D. Happel. Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge, 1987.

[4] J. C. Jantzen. Representations of Algebraic Groups. Academic Press, Orlando, 1987.

[5] S. MacLane. Categories for the Working Mathematician. Springer-Verlag, New York, 1971.

[6] J. Scopes. Cartan matrices and Morita equivalence for blocks of the symmetric groups. J. Algebra, 142:441–455, 1991.