John Brundan and the centers of blocks of category O

So, those of you who like the categorical approach to representation theory probably know about the center of a category:

Definition. The center of an abelian category is the rings of endomorphisms of the identity functor. That is, an element of the center is a system of maps \phi_C: C\to C for all objects C in your category such that for ANY morphism \xi:C\to D, we have \phi_D\xi=\xi\phi_C.

For example, if A is a finite dimensional algebra, then the action of any element of the center of A gives an element of the center of the catgeory \mathrm{Rep}(A) (ANY element of A satisfies the commutation condition by definition, but it will only give a morphism in \mathrm{Rep}(A) if it is central). In this case, the center of A will actually be the whole center of the category, by silly universal algebra considerations (you can recover a finite-dimensional algebra as the endomorphism algebra of the forgetful functor from \mathrm{Rep}(A) to vector spaces).

But in more general cases, it can be hard to be sure you’ve got the whole center of the category (just as it’s hard to be sure you’ve got the whole center of an algebra you don’t know very well).

Once you have your hands on the center of a category \mathcal C, you can use it for block decomposition by central characters. Let’s make our lives easier, and suppose our category is linear over an algebraically closed field k. Then, for any simple object C, and any \phi is the center of \mathcal C, then the morphism \phi_C must be a multiple of the identity, which we might call \chi_C(\phi). This defines a character \chi_C:Z(\mathcal{C})\to\mathbb{C}, which we call the central character of C.

Now, imagine C and D are simple objects with different central characters \chi_C\neq \chi_D. Then, if one looks at the space \mathrm Ext^i(C,D) we can see that the center must act on it by \chi_C by composition on the left, and by \chi_D by composition on the right. Since these are different, the only way this is possible is for all these spaces to be trivial. Thus

Theorem. Let \mathcal{C}_\chi be the Serre subcategory of \mathcal{C} generated by simples with central character \chi_C, and \mathcal{C}'_\chi be the subcategory generated by all other simples. Then \mathcal{C}\cong \mathcal{C}_\chi\oplus \mathcal{C'}_\chi. More generally, we have a decomposition of \mathcal{C} as \mathcal{C}\cong\bigoplus_{\chi}\mathcal{C}_\chi.

(A Serre subcategory of an abelian category where if latex A\subset B$ and two of A,B and B/A are in the subcategory, the third one is as well).

On the other hand, if one has a direct sum decomposition of your category \mathcal{C}\cong \mathcal{C}'\oplus\mathcal{C}'', then then projection to the unique summand of C lying in \mathcal{C}' is an element of the center which distinguishes these. Thus, the central characters separate blocks.

However, if you have an infinite dimensional algebra, and pick out some nice category of representations for it, then there’s no guarantee that the center of the algebra surjects onto the center of the category.

One interesting example of this is so-called category \mathcal O. This is the Serre subcategory of the category of all representations of a semi-simple complex Lie algebra $\mathfak{g}$ generated by Verma modules (alternatively by simple modules with a weight decomposition and highest weight vector). More generally, we’ll want to consider the parabolic category \mathcal O^{\mathfrak p} for some parabolic $\mathfrak p\subset \mathfrak g$, which is the subcategory of \mathcal O consisting of modules which are a direct sum of finite dimensional representations for the action of $\mathfrak l$, the Levi subgroup of $\mathfrak p$.

Let’s restrict, for ease, to the case where they have integral weights as well. In this case, the blocks are in bijection with orbits of the shifted Weyl group action w\bullet\lambda=w(\lambda+\rho)-\rho where \rho is the sum of the fundamental weights, as usual. The action of the center of the universal enveloping algebra of \mathfrak g gets exactly this far: it’s canonically identified with the functions invariant under this action of the Weyl group by the Harish-Chandra homomorphism. One need only look at the Verma modules of elements with dominant (or almost dominant weight) to check that each of these subcategories is indecomposible.

But does the universal enveloping algebra get the whole center of each block? How could you check?

To see this, you should realize your category as representations of a finite dimensional algebra. Luckily, this is possible for any category with finitely many simple objects and enough projectives. If you take a projective generator P of your block (the direct sum of the projective cover of each simple module in the block), and look at it’s endomorphism algebra E, then you’ll see that \mathrm{Hom}(P,-) is an equivalence of categories to \mathrm{Rep}(E).

So, now we just need to find the center of E. But this is not a particularly easy task. Soergel worked out the the universal enveloping algebra does surject in usual category \mathcal O using the following trick: let P_a be the unique indecomposible module of your block which is both projective and injective. Then, the functor \mathrm{Hom}(P_a,-) to representations of C=\mathrm{End}(P_a) is full and faithful on projectives (it kills lots of other modules, but not projectives). Thus, one can calculate the endomorphisms of a projective generator just as well after applying this functor. It turns out that C is commutative, and the obvious map into $\mathrm E$ is an isomorphism onto the center. One can explicitly check that the center of U(\mathfrak{g}) surjects onto C, and thus onto the center of the block.

This result is simply not true in the parabolic case, though. There are explicit examples in type $B$ where the map to the center is not sujective. However, in type $A$, it’s actually true, and John Brundan gave a tlak here in Denmark illuminating his proof of this fact. I don’t really have the energy to say much about it, other than that it relies on a very explicit description of a projective generator of \mathcal{O}^{\mathfrak p}, which most interestingly shows that the center of a block of \mathcal{O}^{\mathfrak p} is isomorphic to the cohomology of a Spaltenstein variety in G/P. There’s no geometric proof of this fact, a problem begging to be rectified. But we’ll save talking about that for another day.

8 thoughts on “John Brundan and the centers of blocks of category O

  1. I’d be grateful to learn if somebody ever ran into prominent examples of the following slight generalization of the concept of the center of a category:
    The center Z(\mathcal{C}) of \mathcal{C} is an abelian monoid since it is constructed as a one-object one-morphism 2-category, namely
    Z(\mathcal{C}) :=\mathrm{End}_{\mathrm{Cat}}(\mathrm{Id}_{\mathcal{C}})).
    But now suppose we have a finite group G acting strictly on \mathcal{C}
    R : \Sigma G : \mathrm{Aut}_{\mathrm{Cat}}(\mathcal{C})
    by functors R_g : \mathcal{C} \to \mathcal{C} for all g \in G.
    Then we can consider the G-graded monoid
    \mathrm{Hom}_{\mathrm{Cat}}(\mathrm{Id}_{\mathcal{C}},-) := \bigcup_{g \in G} \mathrm{Hom}_{\mathrm{Cat}}(\mathrm{Id}_{\mathcal{C}},R_g).
    This is no longer abelian. But almost so: we may pass an element b of this monoid past an element a in degree g up to a twist
    a \cdot b = \mathrm{Ad}_g(b) \cdot a.
    This game may be played not just with 1-categories, but also for instance with 2-categories. There it turns out to reproduce a concept introduced by Turaev and Kirillov, that of G-equivariant fusion categories: these generalize braided monoidal categories of reps of vertex operator algebras to something like G-orbifold versions.
    But here I am looking for places where this “G-twisted center” has been considered for 1-categories. Has anyone seen this?

  2. Sorry for the typesetting mess. Without a comment preview this is hard to avoid. The lesson I learned this time: don’t use line breaks in LaTeX environment…

  3. Sounds vaguely reminiscent of some of this stuff Noah’s been doing with braided monoidal categories fibered over a group, though he has a very specific example in mind, rather than a general construction.

  4. stuff Noah’s been doing with braided monoidal categories fibered over a group

    Is that the one referred to as

    Braiding for quantum groups at roots of unity (in preparation)

    on Noah’s website?

    I’d be interested in seeing that. I did think about looking into categories fibered over something in this context, but there are some reasons for me to look at it from another point of view.

    I now have a detailed description of what I have in mind here:

    On G-equivariant fusion categories.

    I happen to be interested in that mostly as a way to understand rational SCFT a little more conceptually. I am talking about that relation in Supercategories.

  5. That is what Ben was refering to. It’s definitely a subject with a similar flavor to what you’re dicussing, but also a bit different. You can see the basic outline in Kashaev and Reshetikhin’s papers (especially this one which I think still isn’t on the arxiv).

    The paper Kolya and I are writing will hopefully be done sometime early in the fall. I wish it were going to be done sooner but it got delayed due to Kolya’s being in Denmark, and now I have Mathcamp and so won’t have a chance to write it. However, in terms of the general structure it doesn’t add much to what’s already in Kashaev and Reshetikhin’s papers. It mostly concerns certain technical issues in generalizing from sl_2 to arbitrary type.

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