GRADINGS ON REPRESENTATION CATEGORIES

WOLFGANG SOERGEL UNIVERSITÄT FREIBURG, GERMANY

1. Selfduality of Category O

Let 𝔤 ⊃ 𝔟 ⊃ 𝔥 be a complex semisimple Lie algebra, a Borel and a Cartan. Let O be the category of all finitely generated 𝔤-modules which are locally finite over 𝔟 and semisimple over 𝔥, see [BGG76]. This category is of interest, since it is a close relative of the category of Harish-Chandra modules for the corresponding simply connected complex algebraic group G, considered as real Lie group, see [BG80]. For example, for 𝔤 = 𝔰𝔩(n, ℂ) one takes G = SL(n, ℂ).

It is known [BGG76] that every object of O has finite length and that there are enough projective and injective objects. Let U = U(𝔤) be the universal enveloping algebra of 𝔤. For λ ∈ 𝔥^{∗} consider in O the Verma module M(λ) = U ⊗_{U(𝔟)} ℂ_{λ}, its unique simple quotient L(λ) and an injective hull I(λ) of L(λ) in O, i.e. an injective object with unique simple submodule L(λ). This I(λ) is unique only up to nonunique isomorphism. Every simple object of O is isomorphic to some L(λ) for unique λ ∈ 𝔥^{∗}.

Let Z ⊂ U be the center and Z^{+} = Ann_{Z}ℂ the annihilator of the trivial one-dimensional representation of 𝔤. Let L be the direct sum of all simples L(λ) with Z^{+}L(λ) = 0. There are but finitely many of those, parametrized by the Weyl group. Let I be the direct sum of their respective injective hulls. We are now ready to state the “selfduality theorem for category O”.

Theorem 1. [Soe90, BGS91] There exists an isomorphism of finite dimensional ℂ-algebras End_{𝔤}I ≅ Ext_{O}^{•}(L, L). Furthermore the graded ring on the right is Koszul.

Remarks . 1. Here Ext^{•} stands for the direct sum of all Ext^{i}, made into a ring

via the cup-product.

2. The object I is only unique up to nonunique isomorphism, so we cannot expect the isomorphism of the theorem to be canonical. See however [BGS91], 3.8 for more canonicity.

3. Let (W, S) be the Weyl group and its simple reflections corresponding to 𝔟 ⊂ 𝔤. Let ρ ∈ 𝔥^{∗} be the halfsum of the roots of 𝔟. For x ∈ W, λ ∈ 𝔥^{∗} define x · λ = x(λ + ρ) − ρ. Then L = ⊕_{x∈W}L(x · 0), I = ⊕_{x∈W}I(x · 0). Let w_{∘} ∈ W be the longest element of the Weyl group. The isomorphism of the theorem can be choosen in such a way that the projection onto I(x · 0) on the left corresponds to the projection onto L(x^{−1}w_{∘} · 0) on the right (which lies in End_{𝔤}L = Ext_{O}^{0}(L, L)).

4. A “Koszul ring” is by definition [Pri70, BGS91] a positively ℤ-graded ring A = ⨁_{i≥0} A^{i} such that (a) A^{0} is semisimple and (b) the left A-module A^{0} = A/A^{>0} admits a graded projective resolution ⋯ → P_{2} → P_{1} → P_{0}→ →A^{0} such that P_{i} is generated by its elements that are homogeneous of degree i, i.e. P_{i} = AP_{i}^{i}. For example, the standard Koszul complex shows that the symmetric algebra over a finite dimensional vector space is Koszul.

We can reformulate (b) as follows: Let A-gr denote the abelian category of all ℤ-graded A-modules M = ⨁ M^{i}. (b)′ If M, N ∈ A-gr are concentrated in one degree, say M = M^{m}, N = N^{n}, then Ext_{A-gr}^{i}(M, N) = 0 unless i = n − m. Thus Koszulity is an analogon of semisimplicity for ℤ-graded rings.

5. Put O_{0} = {M ∈ O | (Z^{+})^{n}M = 0 for n ≫ 0}. Since I ∈ O_{0} is an injective generator, the functor

Hom_{𝔤}( , I) : O_{0} → End_{𝔤}I-mof

from O_{0} to the category of finitely generated End_{𝔤}I-modules is an equivalence of categories. The theorem says that End_{𝔤}I admits a ℤ-grading. Thus in some sense (which can be made precise) the category O_{0} admits a ℤ-grading and is even “graded semisimple”.

The same is true for all other blocks of O. This is proven in [BGS91] for blocks with integral central character and then follows for arbitrary blocks with [Soe90], 2.5.

6. Let us explain what is selfdual about this theorem. If A is any positively ℤgraded ring, we may form another positively ℤ-graded ring E(A) = Ext_{A}^{•}(A^{0}, A^{0}). If A is Koszul and A^{1} is finitely generated as a left A^{0}-module, one can prove [Löf86, BGS91] that E(A) is Koszul too and there is a canonical isomorphism E(E(A)) = A. For this reason E(A) is called the Koszul dual of A. But now the Koszul ring A appearing in our theorem is selfdual: Indeed, under the equivalence O_{0} ≅ A-mof the direct sum L of all simple objects of O_{0} corresponds to the direct sum A^{0} of all simple objects of A-mof, and hence

E(A) = Ext_{A}^{•}(A^{0}, A^{0}) ≅ Ext_{O}^{•}(L, L) ≅ End_{𝔤}I = A.

2. Representation of Hecke algebras via Bimodules

To establish the isomorphism of theorem 1, we will describe both sides combinatorially. This section explains the combinatorics involved, which might be of independent interest. Let (W, S) be an arbitrary Coxeter system, ≤ the Bruhat order on W and l : W → ℕ the length function, see [Bou81]. The Hecke algebra H = H(W, S) is a free ℤ[t, t^{−1}]-module H = ⨁_{x∈W} ℤ[t, t^{−1}]T_{x} with basis T_{x} indexed by x ∈ W. Its multiplication is given by the rules T_{x}T_{y} = T_{xy} if l(x) + l(y) = l(xy) and T_{s}^{2} = (t^{2} − 1)T_{s} + t^{2}T_{e} for s ∈ S, where e ∈ W denotes the identity.

Suppose now that S is finite. Consider the geometric representation E of W, its complexification E_{ℂ} = E ⊗_{ℝ} ℂ and the ring R = R(E_{ℂ}) of regular functions on E_{ℂ} alias the symmetric algebra of E_{ℂ}^{∗}. We equip R with the “even” ℤ-grading, defined by the prescription degE_{ℂ}^{∗} = 2.

Let R-mobf_{ℤ}-R be the category of all ℤ-graded R-bimodules which are finitely generated as left modules and as right modules. Let ⟨R-mobf_{ℤ}-R⟩ denote its split Grothendieck group, i.e. the free abelian group generated by the objects modulo the relations ⟨A⟩ = ⟨A′⟩ + ⟨A′′⟩ if A ≅ A′ ⊕ A′′. Certainly this is a ring via ⊗_{R}.

Theorem 2. [Soe92a] There is a ring homomorphism E : H → ⟨R-mobf_{ℤ}-R⟩ such that E(T_{s} + 1) = ⟨R ⊗_{R^{s}} R⟩ for every s ∈ S (here R^{s} are the s-invariants) and E(t) = ⟨R[−1]⟩ (where R[−1] stands for the R-bimodule R with its grading shifted by one in positive direction).

Kazhdan and Lusztig [KL80] defined a new basis C′_{x} of H over ℤ[t, t^{−1}]. It can be characterized as follows. Let i : H → H be the involutive algebra homomorphism with i(t) = t^{−1}, i(T_{x}) = (T_{x^{−1}})^{−1}. Then we have

1. i(C′_{x}) = C′_{x}

2. C′_{x} = t^{−l(x)} ∑_{x≥y} P_{x,y}(t)T_{y} with P_{x,y} ∈ ℤ[t], P_{x,x} = 1 and deg P_{x,y} < l(x) −

l(y) if x ≠ y. Because of property (1) this is often called the selfdual basis. Condition (2) is some kind of minimality condition which ensures uniqueness. The P_{x,y} are called the Kazhdan-Lusztig polynomials. For s ∈ S one checks easily C′_{s} = t^{−1}(T_{s} + 1).

Theorem 3. Suppose W is a Weyl group. Then there are indecomposable bimodules B_{x} ∈ R-mobf_{ℤ}-R such that E(C′_{x}) = ⟨B_{x}⟩.

Remarks . 1. The Krull-Schmid theorem holds in R-mobf_{ℤ}-R, hence the B_{x} are

unique up to isomorphism.

2. Here are some examples. We have B_{e} = R, B_{s} = R ⊗_{R^{s}} R[1] for s ∈ S and B_{w_{∘}} = R ⊗_{R^{W}} R[l(w_{∘})].

3. From the theorem one can deduce the conjectures of Kazhdan and Lusztig concerning composition series of Verma modules in an elementary way [Soe90]. However the proof of the theorem is not elementary and uses the decomposition theorem of [BBD82]. It would be very interesting to have an elementary proof.

4. It would be very interesting to know whether this theorem holds for an arbitrary Coxeter group, or at least for any finite one.

5. It would be very interesting to know for which finite characteristic analogous results hold. An equivalent problem is to determine the intersection cohomology with coefficients in a finite field of complex Schubert varieties. This would determine some composition factor multiplicities for Weyl modules in the same characteristic.

Let me finish this subsection with some indications on how these theorems are proven. For theorem 2 one just has to remark that H is generated over ℤ[t, t^{−1}] by the (T_{s} + 1), s ∈ S subject only to the quadratic relations (T_{s} + 1)^{2} = (T_{s} + 1) + t^{2}(T_{s} + 1) and braid relations. So we only need to establish isomorphisms

R ⊗_{R^{s}} R ⊗_{R^{s}} R ≅ (R ⊗_{R^{s}} R) ⊕ (R ⊗_{R^{s}} R[−2])

for all s ∈ S and

R ⊗_{R^{s}} R ⊗_{R^{t}} R ⊗_{R^{s}} … R ≅ R ⊗_{R^{t}} R ⊗_{R^{s}} R ⊗_{R^{t}} … R

whenever sts … = tst … (the same number of factors on both sides) for s, t ∈ S. This is a bit tricky, but not deep.

The only way I know to prove theorem 3 is to interpret B_{x} as the equivariant intersection cohomology group of some Schubert variety. Namely choose G ⊃ B a linear complex algebraic group with a Borel subgroup such that (W, S) is the corresponding Coxeter system. One knows [Ara86] how to identify the equivariant cohomology ring H_{B}^{•}(G/B) with R ⊗_{R^{W}} R, where R are the regular functions on the Lie algebra of a maximal torus in B. Using this identification we can interpret the equivariant intersection cohomology groups of Schubert varieties as objects of R-mobf_{ℤ}-R. One then shows that these objects satisfy the conditions on the B_{x} required in theorem 3.

3. Combinatorics of Category O

To prove the “selfduality” theorem 1, we will show that both sides admit the same combinatorial description. Let us start describing the left hand side. Choose an exact functor 𝕍 : O_{0} → ℂ-mod which transforms the simple Verma module M(−2ρ) to a one dimensional vector space and annihilates all other simple objects from O_{0}. Such a functor exists up to nonunique isomorphism and can be given as 𝕍 = Hom_{𝔤}( , I(−2ρ)). By functoriality this gives even a functor

𝕍 : O_{0} → Z-mod.

Proposition 1. [Soe90] The functor 𝕍 is fully faithful on injective objects, i.e. for any two injectives I, J ∈ O_{0} it induces an isomorphism

Hom_{𝔤}(I, J) → Hom_{Z}(𝕍I, 𝕍J).

By the way an analogous statement holds for every block of O. Now we describe 𝕍I(x·0). Put S = S(𝔥) and consider the Harish-Chandra homomorphism ξ : Z ↪ S normalized by the condition ξ(z) − z ∈ U𝔫, where 𝔫 ⊂ 𝔟 denotes the nilradical. So in particular ξ^{−1}(S^{+}) = Z^{+}. Via ξ every S-module becomes a Z-module, and in [Soe90] it is proven that 𝕍I(x^{−1} · 0) ≅ B_{x}/B_{x}S^{+}. From this we deduce

End_{𝔤}I = End_{Z}𝕍I
= End_{Z}(⨁_{x} B_{x}/B_{x}S^{+})
= End_{S}(⨁_{x} B_{x}/B_{x}S^{+}),

the last equality since (Z^{+})^{n}I = 0 for n ≫ 0 by definition of O_{0} and ξ induces isomorphisms Z/(Z^{+})^{n} → S/(S^{+})^{n}.

We next describe combinatorially the right hand side. Let G be as before, B ⊃ N connected algebraic subgroups with Lie algebras 𝔟 ⊃ 𝔫. Let D^{b}(G/B) denote the bounded derived category with algebraically constructible cohomology of sheaves of complex vector spaces on G/B. Let D_{N}^{b}(G/B) ⊂ D^{b}(G/B) be the full subcategory of objects which are smooth along N-orbits alias Bruhat cells.

Localization [BB81] along with [Soe86] determines an equivalence of categories (see [Soe89], prop. 6)

D^{b}(O_{0}) →^{∼} D_{N}^{b}(G/B)

mapping L(xw_{∘} ·0) to the intersection cohomology complex IC_{x} = IC(\overline{Nx^{−1}B/B}) of the closure of a Bruhat cell. Recall the ring R = S(𝔥^{∗}) of regular functions on the Lie algebra of a maximal torus. Via the Borel homomorphism R→ →H^{•}(G/B, ℂ) hypercohomology gives us a functor

ℍ^{•} : D^{b}(G/B) ⟶ R-mod.

Proposition 2. [Soe90] The functor ℍ^{•} induces an isomorphism

⨁_{i} Hom_{D}(IC_{x}, IC_{y}[i]) → Hom_{R}(ℍ^{•}IC_{x}, ℍ^{•}IC_{y}).

In addition, we establish there isomorphisms ℍ^{•}IC_{x} ≅ B_{x}/B_{x}R^{+}. Let us put L = ⊕_{x}IC_{x}. From the preceding we now deduce

Ext_{O}^{•}(L, L) = ⨁_{i} Hom_{D}(L, L[i])
= End_{R}(ℍ^{•}L)
= End_{R}(⨁_{x} B_{x}/B_{x}R^{+}).

So indeed the left hand side of the selfduality theorem equals the right hand side, once we identify S with R, i.e. 𝔥 with 𝔥^{∗}. This already suggests that from the very beginning it would have been more natural to take the Langlands dual Lie algebra on one side of our selfduality isomorphism. This point of view allows a partly conjectural generalization of theorem 1, as explained in the next subsection.

4. Real groups

Let X be a complex algebraic variety and H a complex algebraic group acting on X with finitely many orbits. Then we can form a positivly graded ring Ext_{H}^{•}X, the “geometric extension algebra”, as follows: Let D_{H}(X) be the equivariant derived category, see [BL92]. Let L ∈ D_{H}(X) be the direct sum of all simple H-equivariant perverse sheavers on X (i.e. take one from each isomorphism class). Then put Ext_{H}^{•}X = ⨁_{i} Hom_{D_{H}(X)}(L, L[i]). If H ⊂ K is a closed subgroup, then the “induction equivalence” gives Ext_{H}^{•}X = Ext_{K}^{•}(K ×_{H} X).

In our new notation theorem 1 can be rewritten as

O_{0} ≅ Ext_{N^{∨}}^{•}(G^{∨}/B^{∨})-mof

where G^{∨} ⊃ B^{∨} ⊃ N^{∨} are Langlands dual to G ⊃ B ⊃ N. Let Õ_{0} be the category of all finite length 𝔤-modules with all their composition factors from O_{0}. As a variant of the above one proves [Soe92a]

Õ_{0} ≅ Ext_{B^{∨}}^{•}(G^{∨}/B^{∨})-nil

where for a positively ℤ-graded ℂ-Algebra A^{•} we denote by A-nil the category of finite dimensional A-modules annihilated by A^{i} for i ≫ 0. Now by [Soe86] the category Õ_{0} is equivalent to the category H of Harish-Chandra modules for G with trivial central character. On the other hand Ext_{B^{∨}}^{•}(G^{∨}/B^{∨}) = Ext_{G^{∨}}^{•}(G^{∨} ×_{B^{∨}} G^{∨}/B^{∨}), and X = G^{∨} ×_{B^{∨}} G^{∨}/B^{∨} is just the modified Langlands parameter space associated by [ABV92] to H. So we may write

H ≅ Ext_{G^{∨}}^{•}X-nil

and written in this form it is clear how the theorem should generalize to real and p-adic groups. Some special cases are checked in [Soe92b].

5. Positive characteristic

The Koszulity statement in theorem 1 seems to have an analogon in positive characteristic. Let R be a root system, p a prime, G the corresponding simply connected semisimple algebraic group over 𝔽̄_{p}, and 𝔤 = LieG its Lie algebra. This comes equipped with a formal p-power map X ↦ X^{[p]}, X ∈ 𝔤. For example, for R = A_{n} we get 𝔤 = sl(n + 1, 𝔽̄_{p}) and X^{[p]} = X^{p} is just the p-th power in the matrix ring. Let U^{[p]} = U(𝔤)/(X^{p} − X^{[p]}) be the restricted enveloping algebra of 𝔤. (Be careful that here X^{p} stands for the p-th power in U(𝔤)!) This is a finite dimensional 𝔽̄_{p} algebra.

Conjecture 1. [AJS94] Suppose p is at least the Coxeter number. Then U^{[p]} admits a Koszul grading compatible with its natural ℤR-grading.

Remark . By a Koszul grading on a ring we mean a ℤ-grading that makes the ring to a Koszul ring. The conjecture says that the category of G_{1}T -modules should be “graded semisimple”. In [AJS94] we construct a ℤ-grading on U^{[p]} and prove that it gives a Koszul grading on the regular blocks of U^{[p]} if Lusztig’s conjecture holds, i.e. if p ≫ 0 (for R fixed). So the problem is to treat singular blocks and small p.

References

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[Soe89] Wolfgang Soergel, 𝔫-cohomology of simple highest weight modules on walls and purity, Inventiones 98 (1989), 565–580.

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[Soe92b] Wolfgang Soergel, Langlands’ philosophy and Koszul duality, Preprint, 1992.

Mathematisches Institut der Universität Freiburg, Albertstr.23b, 79104 Freiburg, Germany

     E-mail address: soergel@sun1.ruf.uni-freiburg.de