Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N= Norm (T), and W= N/T is the Weyl group. If w W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.

We study the Ginzburg dg algebra Γ_T associated with the quiver with potential arising from a triangulation T of a decorated marked surface S_△⁠, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories D_{fd}(Γ_T)⁠, denoted by D_{fd}(S_△)⁠. As an application, we show that the spherical twist group ST(S_△) associated with D_{fd}(S_△) acts faithfully on its space of stability conditions.

We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lusztig theory for Lie algebras of type B/C.

We study the oriented exchange graph EG(Gamma_N Q) ofreachable hearts in the finite-dimensional derived category D(Gamma_N Q) of the CY-N Ginzburg algebra Gamma_N Q associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D(Q) via some ‘Lagrangian immersion’ L :D(Q)->D(Gamma_N Q). We build on this to show that the quotient of EG(Gamma_N Q) by the Seidel–Thomas braid group is the exchange graph CEG_N−1(Q)of cluster tilting sets in the (higher) cluster category C_N−1(Q). As an application, we interpret Buan–Thomas’ coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D(Gamma_N Q).

No abstract uploaded!