2025/10/28 by Chatterjee, Tamanna
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2510.24506
We develop a theory of primitive pairs for 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 G0 of a fixed cocharacter χ in a connected, reductive, algebraic group G and its action on the eigenspaces \mathfrakgn of χ. Building on the framework of parity sheaves and the Fourier transform established in \citeCh,Ch1, we show that every indecomposable parity sheaf on \mathfrakgn 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 \citeCh. This confirms the organizing role of primitive pairs in the block decomposition of the category of G0-equivariant parity sheaves on \mathfrakgn. 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.