An asymptotic cell category for cyclic groups

Abstract

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A Soergel-like category for complex reflection groups of rank one

“…One can cite for instance the 'Spetses' program [4], which provides a sort of generalization of unipotent characters of reductive groups to non-existing reductive groups attached to complex reflection groups. In this framework, some categorification results were obtained for cyclic groups in [1] (building up on [6,7]) and later extended in [14,15]. Note that the ring considered in [1,Theorem 5.5] is related to the ring A W studied in the present paper, as observed in Remark 5.4.…”

“…In this framework, some categorification results were obtained for cyclic groups in [1] (building up on [6,7]) and later extended in [14,15]. Note that the ring considered in [1,Theorem 5.5] is related to the ring A W studied in the present paper, as observed in Remark 5.4. Furthermore, the Artin-Tits groups, which as mentioned above are categorified using complexes of Soergel bimodules, also admit nice generalizations to the complex case [3].…”

“…Here R s ⊆ R denotes the graded subring of s-invariant functions. The classes of the (unshifted) indecomposable objects in the split Grothendieck ring coincide with the elements of the Kazhdan-Lusztig basis of H W (this is Soergel's conjecture, proven in [5]) and the bimodule R ⊗ R s R [1] corresponds to the Kazhdan-Lusztig generator C ′ s . Note that Soergel requires V to be a reflection faithful representation of W ([21, Definition 1.5]), that is, the reflections in W must act by geometric reflections, and distinct reflections have distinct reflecting hyperplanes.…”

“…This paper proposes the construction of the analogue of a category of Soergel bimodules for finite complex reflection groups of rank one. In the Coxeter group case, the category of Soergel bimodules is a graded category monoidally generated by a family of bimodules {B s } over the graded ring R = O(V ), indexed by the simple reflections s ∈ S. Each bimodule B s admits both an algebraic definition as the tensor product R ⊗ R s R [1] (here R s ⊆ R is the graded subring of s-invariant functions and [1] denotes a grading shift) and an equivalent definition as a ring of regular functions on a graph. In the complex case, both definitions can be given, but they produce non-isomorphic bimodules if the reflection does not have order 2.…”

“…As an immediate corollary, we have that the Grothendieck ring Gr st B (D(B)) of the category D(B) − stab B studied in [1, Section 5B] (which is a quotient of the stable category of the module category of the Drinfeld quantum double of the Taft algebra B) is isomorphic to a quotient of A W . More precisely, we have the following isomorphism of Z[v, v −1 ]algebras: Z[v, v −1 ] ⊗ Z Gr st B (D(B)) ∼ = A W /I where I is the ideal generated by C d−1 and C l−1 + s l C d−l−1 for all 1 ≤ l ≤ d − 1.In the notation of[1], this isomorphism sends v stBζ to s and m st B Corollary Let T s be defined as C = T s + v. The quotient algebra A w /(s = 1) is generated by a single element T s and a single relation of the form T d s + a d−1 T d−1 s…”