Related papers: Affine Harish-Chandra bimodules and Steinberg--Whi…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…
Let G=GL(N), K=GL(p)xGL(q), where p+q=N, and n be a positive integer. We construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type…
This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination…
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the…
In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe…
We generalize the Hernandez-Jimbo category O of representations of Borel subalgebras of quantum affine algebras to the case of quantum loop algebras for arbitrary Kac-Moody g (as well as related algebras, such as quantum toroidal gl_1).…
The equivariant $K$-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a $q$-Heisenberg algebra. The action of the…
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional…
A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine…
We study the Whittaker category $\mathcal N(\zeta)$ of the Lie superalgebra $\mathfrak g$ for an arbitrary character $\zeta$ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of $\mathfrak g$. We…
We quantize parabolic flag manifolds and describe categories of equivariant quantum $\D$-modules on them at a singular central character. We compute global sections at any $q \in \C^*$ and we also prove a singular version of…
We use the Kazhdan-Lusztig tensoring to define affine translation functors, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA, and to sketch a functorial approach to ``affine…
We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on…
The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining…
We introduce the notion of a spectral character for finite-dimensional representations of affine algebras. These can be viewed as a suitable q=1 limit of the elliptic characters defined by Etingof and Moura for quantum affine algebras. We…
This paper addresses Question 1 posed by Dipendra Prasad in his recent problem list: classify all irreducible smooth representations of an unramified reductive p-adic group such that the space of vectors fixed by the pro-unipotent radical…
A well-known theorem of Mathieu's states that a Harish-chandra module over the Virasoro algebra is either a highest weight module, a lowest weight module or a module of the intermediate series. It is proved in this paper that an analogous…
In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type $A_n$ to the category of finite-dimensional modules of the graded affine Hecke algebra of type $A_\ell$.…
We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev (\cite{Losev2011}). We explore the applications of this theory to unipotent representations of real reductive groups. For complex groups,…
We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we…