[Paper Review] Localization of modules for a semisimple Lie algebra in prime characteristic

Roman Bezrukavnikov, Ivan Mirković+1 more|ArXiv.org|May 14, 2002

Algebraic structures and combinatorial modelsMathematics35 references132 citations

TL;DR

This paper computes the coherent sheaves on the Springer resolution of the nilpotent cone corresponding to irreducible and projective modules for the small quantum group of $\mathfrak{sl}(3)$ in prime characteristic $p$. Using the equivalence $\Upsilon$ between derived categories of coherent sheaves and modules over the small quantum group, it identifies sheaves via Frobenius pushforward and cohomological computations, establishing that the irreducible module $L((p-2)\rho)$ corresponds to a cone of a nontrivial morphism between twisted structure sheaves, and projective covers are realized as line bundles or extensions on $\widetilde{\mathcal{N}}$. The key contribution is a complete explicit classification of these sheaves for $\mathfrak{sl}(3)$.

ABSTRACT

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.

Motivation & Objective

• To explicitly compute the coherent sheaves on $\widetilde{\mathcal{N}}^{(1)}$ corresponding to irreducible $U_{\hat{0}}^{0}$-modules under the derived equivalence $\Upsilon$.
• To determine the projective covers of these irreducible modules in terms of coherent sheaves on $\widetilde{\mathcal{N}}^{(1)}$.
• To verify the $\Upsilon$-equivalence by checking Ext-groups and constructing sheaf-theoretic models for all irreducible and projective modules in the $\mathfrak{sl}(3)$ case.
• To establish that the sheaf corresponding to $L((p-2)\rho)$ is the cone of a nontrivial morphism between $i_*\mathcal{O}_{\mathcal{B}}$ and $i_*\mathcal{O}_{\mathcal{B}}(-\rho)[3]$.
• To show that projective covers are realized as vector bundles or extensions on $\widetilde{\mathcal{N}}$, including $\mathcal{O}_{\widetilde{\mathcal{N}}}(\rho)$ as a nontrivial extension of $\mathcal{O}_{\widetilde{\mathcal{N}}}$ by $\mathcal{O}_{\widetilde{\mathcal{N}}}(\rho)$.

Proposed method

• The authors use the derived equivalence $\Upsilon: \mathcal{D}^b(\mathrm{Coh}_{\mathcal{B}^{(1)}}(\widetilde{\mathcal{N}}^{(1)})) \to \mathcal{D}^b(U_{\hat{0}}^{0}\text{-}{\rm Mod}^{fg})$ with normalization such that $\Upsilon(i_*\mathcal{F}) = R\Gamma(\mathcal{B}, {\rm Fr}_{\mathcal{B}}^*\mathcal{F})$.
• They compute cohomology of twisted sheaves on $\mathcal{B}$ using the Borel-Weil-Bott theorem and the Koszul resolution of $\mathcal{O}_{\mathcal{B}}$ over $\mathrm{S}(\mathcal{T}_{\mathcal{B}})$.
• For irreducible modules, they use the exact sequence $0 \to \Omega^1_{\mathbb{P}^2} \to \mathcal{O}_{\mathbb{P}^2}(-1)^{\oplus 3} \to \mathcal{O}_{\mathbb{P}^2} \to 0$ to compute $R\Gamma$ of Frobenius pullbacks.
• They identify $L((p-2)\rho)$ as the quotient of the Weyl module $[H^0((p-2)\rho)]^*$, and use Serre duality to show $\Upsilon(i_*\mathcal{O}_{\mathcal{B}}(-\rho)[-3]) = [H^0((p-2)\rho)]^*[-3]$, leading to the cone construction.
• For projective covers, they construct sheaves as vector bundles or extensions on $\widetilde{\mathcal{N}}$, such as $\mathcal{O}_{\widetilde{\mathcal{N}}}(\rho)$ as a nontrivial extension in $H^1(\mathcal{T}_{\mathcal{B}}(-\rho))$, and verify Ext vanishing via spectral sequences and cohomological computations.
• They verify that the Ext groups between projective covers and irreducibles are one-dimensional only in the correct degree, using the structure of the derived category and the $\Upsilon$-equivalence.

Experimental results

Research questions

• RQ1 What are the coherent sheaves on $\widetilde{\mathcal{N}}^{(1)}$ corresponding to the irreducible $U_{\hat{0}}^{0}$-modules for $\mathfrak{sl}(3)$ in prime characteristic?
• RQ2 How are the projective covers of these irreducible modules realized as coherent sheaves under the $\Upsilon$-equivalence?
• RQ3 What is the geometric realization of the irreducible module $L((p-2)\rho)$, and how does it arise as a cone of a morphism between twisted structure sheaves?
• RQ4 How can the Ext groups between projective covers and irreducible modules be computed explicitly using sheaf-theoretic methods?
• RQ5 What is the role of $H^1(\mathcal{T}_{\mathcal{B}}(-\rho))$ in constructing the projective cover of the trivial module?

Key findings

• The irreducible module $L(0) = \Bbbk$ corresponds to $i_*\mathcal{O}_{\mathcal{B}}$, and $L((p-3)\omega_j)$ for $j=1,2$ corresponds to $i_*\mathcal{O}_{\mathcal{B}}(-\omega_j)[2]$, computed via $R\Gamma(\mathcal{B}, \mathcal{O}_{\mathcal{B}}(-p\omega_j))$.
• The module $L((p-2)\omega_1 + \omega_2)$ corresponds to $i_*\pi_1^*\Omega^1_{\mathbb{P}^2}(1)[1]$, arising from the cohomology of the Koszul resolution of the exact sequence on $\mathbb{P}^2$.
• The irreducible module $L((p-2)\rho)$ is realized as the cone of the unique nonzero morphism $i_*\mathcal{O}_{\mathcal{B}} \to i_*\mathcal{O}_{\mathcal{B}}(-\rho)[3]$, with $\dim \mathrm{Ext}^3(i_*\mathcal{O}_{\mathcal{B}}, i_*\mathcal{O}_{\mathcal{B}}(-\rho)) = 1$.
• The projective cover of $L(0)$ is $\mathcal{P} = \mathcal{O}_{\widetilde{\mathcal{N}}}(\rho)$, a nontrivial extension of $\mathcal{O}_{\widetilde{\mathcal{N}}}$ by $\mathcal{O}_{\widetilde{\mathcal{N}}}(\rho)$, arising from $H^1(\mathcal{T}_{\mathcal{B}}(-\rho)) \cong \Bbbk$.
• The projective cover of $L((p-2)\omega_1 + \omega_2)$ is $\mathcal{O}_{\widetilde{\mathcal{N}}}(\omega_1)$, and the projective cover of $L(\omega_1 + (p-2)\omega_2)$ is $\mathcal{O}_{\widetilde{\mathcal{N}}}(\omega_2)$, realized as line bundles on $\widetilde{\mathcal{N}}$.
• The projective covers of $L((p-3)\omega_1)$ and $L((p-3)\omega_2)$ are $p^*((\pi_2^*\Omega^1_{\mathbb{P}^2})(\omega_1 + 2\omega_2))$ and $p^*((\pi_1^*\Omega^1_{\mathbb{P}^2})(2\omega_1 + \omega_2))$, respectively, pulled back from $\mathcal{B}$ via the projection $p: \widetilde{\mathcal{N}} \to \mathcal{B}$.