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In this paper, we summarize a general method of transforming DG structures into higher structures on the various complexes related to the reduced bar resolution of a given quiver algebra using algebraic Morse theory. As an application, we…

For every simple Hermitian Lie group $G$, we consider a certain maximal parabolic subgroup whose unipotent radical $N$ is either abelian (if $G$ is of tube type) or two-step nilpotent (if $G$ is of non-tube type). By the generalized…

We provide a convenient formulation of the definition of Cartan graphs and Weyl groupoids introduced by Heckenberger and Schneider, and construct Cartan graphs for regular symmetrizable contragredient Lie superalgebras. For…

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group $S_{n}$. In particular, we show that for any parameter $c \in \mathbb{C}$, the category of Harish-Chandra $H_{c}$-bimodules admits a…

Let B be a weight-$3$ block of an Iwahori--Hecke algebra of type B over any field. We develop the combinatorics of B to prove that the decomposition numbers for B are all 0 or 1.

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $M$. We proved that every ICE-closed subcategory in$\mod A$ can be extended to be some ICE-closed subcategories in$\mod B$.In the same way, every epibrick in $\mod A$ can be…

We explicitly provide minimal Gr\"obner bases for simple, finite-dimensional modules of complex Lie algebras of types A and C, using a homogeneous ordering that is compatible with the PBW filtration on the universal enveloping algebras.

Let $F$ be a non-Archimedean locally compact field and $G$ a connected reductive group defined over $F$. To any unipotent element $u$ in $G(F)$, we have associated in [L] an $F$-stratum $\boldsymbol{\mathfrak{Y}}_{F,u}$ which is a (possibly…

For a field $F$ and a connected reductive group $G$ defined over $F$, we develop a theory of Kempf-Rousseau-Hesselink unipotent $F$-strata in $G(F)$ that should allow us to attack open problems in positive characteristic. As an application,…

We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig's conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type…

In this paper, we explore a possibility to utilize harmonic analysis on $\GL_1$ to understand Langlands automorphic $L$-functions in general, as a vast generalization of the pioneering work of J. Tate. For a split reductive group $G$ over a…

We prove that any holomorphic function $f$ on the Lie ball of even dimension satisfying $\Delta f=0$ is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of…

For any Coxeter system we introduce the concept of singular light leaves, answering a question of Williamson raised in 2008. They provide a combinatorial basis for Hom spaces between singular Soergel bimodules.

Like the RSK correspondence for symmetric groups, Garfinkle defined a domino correspondence for type $\mathrm{B}$ and $\mathrm{D}$ Coxeter groups. Similar to the Knuth relations, Taskin and Pietraho give the plactic relations for the domino…

For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall…

Let F be a non-archimedean local field of odd residual characteristic. Let W be a symplectic vector space over F. It is known that there are different Weil representations of a Meteplectic covering group Mp(W). By some twisted actions, we…

We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete…

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely…

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

We prove a conjecture of Lusztig on a microlocal characterization of his perverse sheaves. For any finite quiver without loops, an equivariant simple perverse sheaf on the variety of quiver representations is a Lusztig's perverse sheaf if…