On Lusztig Sheaves for Tame Quivers
Abstract
Let Q be a connected quiver of tame type, and let Uv- be the positive part of the corresponding quantized affine algebra. In Lusztig's construction of the canonical basis of Uv-, the simple perverse sheaves occurring, up to shift, as direct summands of the flag sheaf complexes are given by intersection cohomology complexes IC(X(c,m),ℒχ). We study the restrictions of these intersection cohomology complexes to the locally closed subsets Y(c,m) obtained by fixing the preprojective, non-homogeneous regular, and preinjective summands in the category rep Q. We prove pointwise purity on the loci where the homogeneous regular part is a direct sum of pairwise non-isomorphic quasi-simple modules, and establish a semisimple decomposition of the restrictions to the corresponding larger strata. The graded multiplicities of the local systems associated with irreducible characters of symmetric groups determine the transition coefficients from the canonical basis to the PBW basis constructed via quiver representations. This gives a geometric explanation of the positivity of these coefficients and the unitriangularity of the transition matrix.
Related
- Lusztig sheaves, local systems, and the extended Springer resolution
- Hybrid Kazhdan-Lusztig basis and parabolic induction
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
- Base change and the Mackey formula in Deligne--Lusztig theory
- Relative Rouquier linearity and inverse hybrid Kazhdan-Lusztig positivity
BibTeX
@misc{wu2026lusztig,
title = {{On Lusztig Sheaves for Tame Quivers}},
author = {Yumeng Wu and Jie Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29750},
eprint = {2610.29750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29750},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Lusztig Sheaves for Tame Quivers
AU - Yumeng Wu
AU - Jie Xiao
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29750
UR - https://arxiv.org/abs/2610.29750
AB - Let Q be a connected quiver of tame type, and let U_v^- be the positive part of the corresponding quantized affine algebra. In Lusztig's construction of the canonical basis of U_v^-, the simple perverse sheaves occurring, up to shift, as direct summands of the flag sheaf complexes are given by intersection cohomology complexes IC(X(c,m),ℒ_χ). We study the restrictions of these intersection cohomology complexes to the locally closed subsets Y(c,m) obtained by fixing the preprojective, non-homogeneous regular, and preinjective summands in the category rep Q. We prove pointwise purity on the loci where the homogeneous regular part is a direct sum of pairwise non-isomorphic quasi-simple modules, and establish a semisimple decomposition of the restrictions to the corresponding larger strata. The graded multiplicities of the local systems associated with irreducible characters of symmetric groups determine the transition coefficients from the canonical basis to the PBW basis constructed via quiver representations. This gives a geometric explanation of the positivity of these coefficients and the unitriangularity of the transition matrix.
ER -