For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, the decomposition of $\mathcal{C}$ is determined by the decomposition of $\mathcal{T}$ satisfying "vanishing condition" of negative extension groups, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $\mbox{Hom} _{\mathcal{C}}(\mathcal{T} _i[t],\mathcal{T} _j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that for any two cluster tilting objects $T, T'$ in a $2-$Calabi-Yau triangulated category $\mathcal{C}$, the Gabriel quivers of endomorphism algebra End$_{\mathcal{C}}T$ of $T$ is connected if and only if so is End$_{\mathcal{C}}T'$. As an application, we prove that indecomposable $2-$Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs.

A general framework for constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms, which is called the Principle of Endoscopy Correspondences, extending the well-known Howe Principle of Theta Correspondences. This suggests another principle, called the (τ, b)-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski- Shapiro, Rallis, Kudla and others on standard L-functions of classical groups and theta correspondence.

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmuller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.

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.

Let G be a connected reductive complex algebraic group, and E a complex elliptic curve. Let GE denote the connected component of the trivial bundle in the stack of semistable Gbundles on E. We introduce a complex analytic uniformization of G_E by adjoint quotients of reductive subgroups of the loop group of G. This can be viewed as a nonabelian version of the classical complex analytic uniformization E=C^∗/q^Z. We similarly construct a complex analytic uniformization of G itself via the exponential map, providing a nonabelian version of the standard isomorphism C^∗= C/Z, and a complex analytic uniformization of G_E generalizing the standard presentation E = C/(Z ⊕ Zτ). Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from Sh_N(G_E) (the semistable part of the automorphic category) to IndCoh_{Nˇ}(Locsys_{Gˇ}(E)) (the spectral category).