arXiv subjects
Publications and source records attributed to Pavle Pandžić.
Let $ G $ be a non-compact connected reductive real Lie group and $ P $ its parabolic subgroup. We give a list of equivalent conditions for $ P $ to have an abelian unipotent radical. These conditions are related to skew multiplicity free and multiplicity free actions, and using the theory of Clifford algebra and the cubic Dirac operator we deduce formulas for the de Rham cohomology of compact symmetric spaces $ G/P $.
We consider covariant algebras attached to certain hybrid qudit spaces. We describe these algebras in terms of generators and relations, and we show they are flat deformations of certain explicitly described semigroup algebras.
We extend Kostant's results about $\mathfrak{g}$-invariants in the Clifford algebra $Cl(\mathfrak{g})$ of a complex semisimple Lie algebra $\mathfrak{g}$ to the relative case of $\mathfrak{k}$-invariants in the Clifford algebra $Cl(\mathfrak{p})$, where $(\mathfrak{g},\mathfrak{k})$ is a classical symmetric pair and $\mathfrak{p}$ is the $(-1)$-eigenspace of the corresponding involution. In this setup we prove the Cartan theorem for Clifford algebras, a relative transgression theorem, the Harish--Chandra isomorphism for $Cl(\mathfrak{p})$, and a relative version of Kostant's Clifford algebra conjecture.
We classify unitary highest weight modules with a given integral infinitesimal character for the real Lie algebras $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$. We treat both regular and singular cases. For $\mathfrak{su}(p,q)$ we identify the unitarizable modules in the Hasse diagrams of the highest weight orbit. Analogous results for the other Hermitian Lie algebras were given in our earlier publications.
In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups $SO_{e}(2, n)$, $E_{6(-14)}$ and $E_{7(-25)}$. We also describe unitary highest weight modules with given infinitesimal characters.
In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$. We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.
We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator $D_q$. In the classical case it was done by Alekseev and Meinrenken. We show that the cubic Dirac operator $D_q$ is invariant with respect to the $U_q(\mathfrak{sl}_2)$-action and *-structures on $\mathcal{W}_q(\mathfrak{sl}_2)$, moreover, the square of $D_q$ is central in $\mathcal{W}_q(\mathfrak{sl}_2)$. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of~$U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology.
We propose a definition of a quantised $\mathfrak{sl}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of $\mathfrak{sl}_2$ are natural examples of such algebras.
We show how to use Clifford algebra techniques to describe the de Rham cohomology ring of equal rank compact symmetric spaces $G/K$. In particular, for $G/K=U(n)/U(k)\times U(n-k)$, we obtain a new way of multiplying Schur polynomials, i.e., computing the Littlewood-Richardson coefficients. The corresponding multiplication on the Clifford algebra side is, in a convenient basis given by projections of the spin module, simply the componentwise multiplication of vectors in $\mathbb{C}^N$, also known as the Hadamard product.
Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q(\mathfrak{sl}_n)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q(\mathfrak{sl}_2)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.
We study various kinds of Grassmannians or Lagrangian Grassmannians over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as $\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and $\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotent radical. The same Grassmannians can also be realized as (classical) compact symmetric spaces $G/K$. We give explicit generators and relations for the de Rham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time we describe certain filtered deformations of these rings, related to Clifford algebras and spin modules. While the cohomology rings are of our primary interest, the filtered setting of $K$-invariants in the Clifford algebra actually provides a more conceptual framework for the results we obtain.
Let $G$ be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then $G$ has unitary highest weight representations. The proof of the classification of unitary highest weight representations of $G$ given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.
Let $G$ be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.
We study certain ${\mathfrak s}{\mathfrak l}(2,{\mathbb C})$-actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $({\mathfrak g},{\mathfrak p})$, $({\mathfrak g}',{\mathfrak p}')$ of Lie algebras and their parabolic subalgebras.
Let $G$ be a finite cover of a closed connected transpose-stable subgroup of $GL(n,\bR)$ with complexified Lie algebra $\frg$. Let $K$ be a maximal compact subgroup of $G$, and assume that $G$ and $K$ have equal rank. We prove a translation principle for the Dirac index of virtual $(\frg,K)$-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subalgebra of $\frg$. This ``index polynomial'' generates an irreducible representation of the Weyl group contained in the coherent continuation representation. We show that the index polynomial is the exact analogue on the compact Cartan subgroup of King's character polynomial. The character polynomial was defined in \cite{K1} on the maximally split Cartan subgroup, and it was shown to be equal to the Goldie rank polynomial up to a scalar multiple. In the case of representations of Gelfand-Kirillov dimension at most half the dimension of $G/K$, we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occuring in the associated cycle of the corresponding coherent family.
We modify the definition of Dirac cohomology in such a way that the standard properties of the usual Dirac cohomology, valid for modules with infinitesimal character, become valid also for modules with only generalized infinitesimal character.
In this paper we study the problem of computing the Dirac cohomology of the special unipotent representations of the real groups Sp(2n,R) and U(p,q).