A Note on primitive pairs for graded Lie algebras
Tamanna Chatterjee
TL;DR
The paper develops a graded analogue of the generalized Springer framework in positive characteristic by introducing primitive pairs for $\mathbb{Z}$-graded Lie algebras and studying parity sheaves on graded pieces $frak g_n$. It proves that every indecomposable parity sheaf on $frak g_n$ is a direct summand of an induced complex from primitive data on a Levi, extending cuspidal-induced decompositions to the graded setting. It also shows that primitive pairs on the nilpotent cone induce graded primitive data and that primitivity is preserved under the Fourier--Sato transform, highlighting strong compatibility between the graded geometry and representation theory. Finally, it introduces quasi-monomial and good objects and establishes a decomposition theorem that expresses parity sheaves as summands of objects induced from primitive data, laying the groundwork for a graded generalized Springer correspondence in positive characteristic.
Abstract
We develop a theory of primitive pairs for $\mathbb{Z}$-graded Lie algebras when the sheaves have coefficients in a field $\Bbbk$ of positive characteristic, providing a graded analogue of the role played by cuspidal pairs in the generalized Springer correspondence. We consider the centralizer $G_0$ of a fixed cocharacter $χ$ in a connected, reductive, algebraic group $G$ and its action on the eigenspaces $\mathfrak{g}_n$ of $χ$. Building on the framework of parity sheaves and the Fourier transform established in \cite{Ch,Ch1}, we show that every indecomposable parity sheaf on $\mathfrak{g}_n$ can be expressed as a direct summand of a complex induced from primitive data on a Levi subgroup. This result extends the fact that, in the graded setting, any indecomposable parity sheaf is direct summand of an induced cuspidal datum \cite{Ch}. This confirms the organizing role of primitive pairs in the block decomposition of the category of $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$. We further establish that primitive pairs on the nilpotent cone induce primitive pairs in the graded setting, and we prove that primitivity is preserved under the Fourier--Sato transform. These results reveal a deep compatibility between the geometry of graded Lie algebras and their representation-theoretic structures, forming the foundation for a graded version of the generalized Springer correspondence in positive characteristic.