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- W3198491186 abstract "This thesis aims to study several aspects of Hall algebras associated with curves and quivers. These algebras encode the extensions between the objects of the category of representations of a quiver or of coherent sheaves on a smooth projective curve. For quivers, they appeared in Ringel's work in his geometric construction of quantum groups. The Hall algebra of curves was first considered by Kapranov to understand the functional equations between Eisenstein series. In the first chapter, we explicitly describe the cuspidal functions of affine quivers. This work is based on Auslander-Reiten. In a second chapter, we are interested in the behavior of Kac polynomials when we increase the number of arrows. Asymptotic and convergence properties emerge. The third chapter deals with more geometric aspects: we prove that the simple perverse sheaves of the Lusztig category (a subcategory of the category of perverse sheaves on the stack of quiver representations) are characterized by the nilpotency of their singular support. Finally, the last chapter studies the analog of this question for elliptic curves. We show that the characteristic cycle gives an isomorphism between a completion of the Grothendieck group of the category of spherical Eisenstein sheaves on the stack of coherent sheaves on an elliptic curve and the group of integer-valued functions on the set irreducible components of the global elliptic nilpotent cone." @default.
- W3198491186 created "2021-09-13" @default.
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- W3198491186 date "2021-07-13" @default.
- W3198491186 modified "2023-09-28" @default.
- W3198491186 title "The Hall algebra of curves and quivers : cuspidal functions, perverse sheaves and Kac polynomials" @default.
- W3198491186 hasPublicationYear "2021" @default.
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