Related papers: Super Distributions, Analytic and Algebraic Super …
Over an arbitrary field of characteristic $\ne 2$, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to…
We show basic results on super-manifolds and super Lie groups over a complete field of characteristic $\ne 2$, extensively using Hopf-algebraic techniques. The main results are two theorems. The first main theorem shows a category…
We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi…
We develop a general theory of algebraic group superschemes, which are not necessarily affine. Our key result is a category equivalence between those group superschemes and Harish-Chandra pairs, which generalizes the result known for affine…
It is known that there exists a natural functor $\Phi$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of…
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or…
This expository paper introduces the theory of Harish-Chandra integrals, a family of special functions that express the integral of an exponential function over the adjoint orbits of a compact Lie group. Originally studied in the context of…
The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph…
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 previous supersymmetric generalization of the unitary Harish--Chandra integral prompted the conjecture that the Harish--Chandra formula should extend to all classical supergroups. We prove this conjecture for the unitary orthosymplectic…
These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible Harish-Chandra modules for these quantum groups, following Joseph and Letzter. Along the…
In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.
Recently the correlation functions of the so-called Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors and collaborators) for all classical groups using an integration formula that relates integrals over compact…
Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half…
We review the basic theory of super $G$-spaces. We prove a theorem relating the action of a super Harish-Chandra pair $(G_0, \mathfrak{g})$ on a supermanifold to the action of the corresponding super Lie group $G$. The theorem was stated in…
We study $\mathbb Z_2\times\mathbb Z_2$ bi-graded Lie algebras. We describe their properties in relation to Lie superalgebras with some compatible structures. Then we focus on the approach to the Lie group--algebra correspondence based on…
Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still…
In this paper, we introduce the Harish-Chandra homomorphism for the quantum superalgebra $\mathrm{U}_q(\mathfrak{g})$ associated with a simple basic Lie superalgebra $\mathfrak{g}$ and give an explicit description of its image. We use it to…
We study and classify algebraic families of Harish-Chandra pairs over the complex affine line and over the complex projective line with generic fiber that is isomorphic to the Harish-Chandra pair of $SL_2(\mathbb{R})$.