Projectives in the category of modular representations of Lie algebras

Question

Let $\mathfrak{g}$ be a semi-simple Lie algebra (eg. $\mathfrak{g} = \mathfrak{sl}_n$), defined over an algebraically closed field $\textbf{k}$ with characteristic $p >> 0$. The center $Z(U\mathfrak{g})$ of the universal enveloping algebra is generated by the Frobenius center $Z_{\text{Fr}}$, and the Harish-Chandra center $Z_{\text{HC}}$. Let $\mathcal{C}_{\lambda, e} = \text{Mod}_{\lambda, e}(\text{U} \mathfrak{g})$ be the category of modules where $Z_{\text{Fr}}$ acts via $e \in \mathfrak{g}^*$, and $Z_{\text{HC}}$ acts via $\mu \in \mathfrak{h}^*//W$. I would be happy with an answer for the following question when $e=0$ (this case is closely related to representations of the first Frobenius kernel $G^{(1)}$).

My question is about how one can compute the projective covers of the irreducible objects in $\mathcal{C}_{\lambda, e}$. A parametrization of the irreducibles is well-known (as quotients of baby Verma modules). In category $\mathcal{O}$, the projective covers can be computed by applying translation functors to Verma modules, and taking summands. I'm looking for something roughly along these lines, and I would like to understand the cases $\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3$ very explicitly (including the cases where $\mu$ is a singular weight).

EDIT: Since the general version of the question seems to be out of reach, I'd be happy with an answer in the case of $\mathfrak{g}=\mathfrak{sl}_3$, $e=0$, and a possibly singular Harish-Chandra character.

Answer 1

It's fairly easy to answer your basic question in the second paragraph: at present there is no guaranteed method for computing these projectives, though quite a bit of work has been done in recent decades. But your first paragraph introduces too broad a setting for the case you call $e=0$ (usually written as $\chi =0$ where $\chi$ is a "nilpotent" linear functional on $\mathfrak{g}$). Studying all simple modules (= irreducible representations) of the universal enveloping algebra is a far more complicated problem, for which you can consult my 1998 AMS Bulletin survey and its references. (Note too that "semisimple" has various connotations in prime characteristic. Instead what you really want is the Lie algebra of a connected simple algebraic group such as $\mathrm{SL}_n$.)

As Drupieski indicates in his comment, there is an extensive treatment of representations of these algebraic groups and related schemes involving the Frobenius kernels in Jantzen's 1987 Academic Press book Representations of Algebraic Groups, later republished with many additions and corrections by AMS in 2003. See especially his Chapter II.11, keeping in mind that projective = injective in the finite dimensional algebras attached to Frobenius kernels (e.g., the restricted enveloping algebra of the Lie algebra itself).

The history goes back many decades, with some small rank examples of the sort you indicate treated rather explicitly in papers by me and others. A basic theme is the parallel between the finite dimensional algebras just mentioned and the more complicated group algebras of Chevalley groups or twisted analogues. For example, these share the same simple modules for corresponding powers of $p$. The hope (realized so far for $p \geq 2h-2$ with $h$ the Coxeter number of the Weyl group, thanks to Ballard and Jantzen) has been to lift indecomposable projectives for the Frobenius kernels to the algebraic group and then restrict to finite subgroups followed by further decomposition when needed. My correspondence with Verma in the early 1970s made me overconfident of doing this lifting easily, so our "theorem" became a problem (not quite a conjecture, since it may turn out to be false for small $p$).

One flawed early source is the informal 1976 Springer lecture note volume I wrote, in which useful details are mixed with some errors and oversimplified arguments. A more reliable source is the 2006 LMS lecture notes I wrote to complement Jantzen's book, which has a large number of references; see especially Chapters 9-10 there (and online corrections). Some papers by me and others in the 1970s and 1980s worked out more details for small rank groups and Lie algebras. For all of this see my publication list here. By now there is a lot of related literature, partly because the fundamental questions about simple modules, projective covers, etc., remain unsolved. But examples in low ranks can be useful to study. Everywhere a basic role is played by tensoring with Steinberg modules (for powers of $p$), in the spirit of Jantzen's translation functors. But decomposing such tensor products is typically difficult even after the block decomposition is known (via what I called "linkage").