Related papers: Affine Jacquet functors and Harish-Chandra categor…

Let $\mathcal{V}^c(\mathfrak{gl}_N)$ be Etingof--Kazhdan's quantum affine vertex algebra associated with the trigonometric $R$-matrix. We establish a connection between suitably generalized deformed $\phi$-coordinated…

We develop a theory of tdos and twisted $\mathcal D$-modules over general base schemes with a focus on functorial aspects. In particular, we introduce a flat base change functor and establish its compatibility with globalization and direct…

For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form…

In this paper we prove a formula for fusion coefficients of affine Kac-Moody algebras first conjectured by Walton [Wal2], and rediscovered in [Fe]. It is a reformulation of the Frenkel-Zhu affine fusion rule theorem [FZ], written so that it…

Let ${\mathsf F}$ be the Schur functor from the category of finite dimensional ${\mathcal H}_{\vartriangle}(r)_\mathbb C$-modules to the category of finite dimensional ${\mathcal S}_{\vartriangle}(n,r)_{\mathbb{C}}$-modules, where…

We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms…

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules…

In this paper we study the indecomposable module categories over $\mathcal{C}(\mathfrak{sl}_N, k)$, the category of integrable level-$k$ respresentations of affine Kac-Moody $\mathfrak{sl}_N$. Our first main result classifies these module…

We investigate the structure and representation theory of finite-dimensional $\mathbb{Z}$-graded Lie algebras, including the corresponding root systems and Verma, irreducible, and Harish-Chandra modules. This extends the familiar theory for…

For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with a fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which…

For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of…

Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In…

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak{h}$. We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for…

The classical Gindikin-Karpelevich formula appears in Langlands' calculation of the constant terms of Eisenstein series on reductive groups and in Macdonald's work on p-adic groups and affine Hecke algebras. The formula has been generalized…

We introduce the $h$-adic quantum vertex algebras associated with the trigonometric $R$-matrices in types $B$, $C$ and $D$, thus generalizing the well-known Etingof-Kazhdan construction in type $A$. We show that restricted modules for…

We study the Gaberdiel-Goddard spaces of systems of correlation functions attached to an affine Kac-Moody Lie algebra $\gh$. We prove that these spaces are isomorphic to the spaces of coinvariants with respect to certain subalgebras of…

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an…

Let g be a finite-dimensional complex simple Lie algebra. Fix a non-negative integer l, we consider the set of dominant weights {\lambda} of g such that l{\Lambda}_0+{\lambda} is a dominant weight for the corresponding untwisted affine…

The goal of this paper is to relate the quantum category $\mathcal{O}$ (known also as the category of modules over the mixed quantum group) at an odd root of unity to the affine Hecke category. Namely, we prove equivalences of highest…

We give a classification of the Harish-Chandra modules generated by the pullback to~$\SL{2}(\RR)$ of \emph{poly}harmonic Maa\ss{} forms for congruence subgroups of~$\SL{2}(\ZZ)$ with exponential growth allowed at the cusps. This extends…