Space Of Harmonic Spinors Research Articles

Dirac cohomology and translation functors

We study the relationship between the Dirac cohomology of a (g,K)-module X and the Dirac cohomology of a Jantzen–Zuckerman translate of X. More precisely, we show that if X is unitary, and if some submodule X′ of a translate of X has nonzero Dirac cohomology, then X has nonzero Dirac cohomology. We also show that the space of harmonic spinors (i.e., the kernel of the Dirac operator) related to X′ embeds into a certain product of harmonic spinors for X and harmonic spinors for the finite-dimensional module used to define the translation. This generalizes, with a simpler proof, results of Mehdi and Parthasarathy (2008) [MP1] and Mehdi and Parthasarathy (2010) [MP2].

Representation theoretic harmonic spinors for coherent families

Coherent continuation π2 of a representation π1 of a semisimple Lie alge- bra arises by tensoring π1 with a finite dimensional representation F and projecting it to the eigenspace of a particular infinitesimal character. Some relations exist be- tween the spaces of harmonic spinors (involving Kostant's cubic Dirac operator and the usual Dirac operator) with coefficients in the three modules. For the usual Dirac operator we illustrate with the example of cohomological representations by using their construction as generalized Enright-Varadarajan modules. In (9) we consid- ered only discrete series, which arises as generalized Enright-Varadarajan modules in the particular case when the parabolic subalgebra is a Borel subalgebra. Key wordsSemisimple lie group, Enright-Varadarajan module, Dirac cohomology, Zuckerman translation functor, Coherent family, harmonic spinor.

Harmonic spinors on homogeneous spaces

Let G G be a compact, semi-simple Lie group and H H a maximal rank reductive subgroup. The irreducible representations of G G can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space G / H G/H twisted by bundles associated to the irreducible, possibly projective, representations of H H . Here, we give a quick proof of this result, computing the index and kernel of this twisted Dirac operator using a homogeneous version of the Weyl character formula noted by Gross, Kostant, Ramond, and Sternberg, as well as recent work of Kostant regarding an algebraic version of this Dirac operator.

Fredholm L p-theory for coupled Dirac operators on the Euclidean space

We study coupled Dirac operators on the Euclidean space R d , which are associated with unitary connections which extend to the sphere S d (in a possibly non-trivial bundle). We define Fredholm extensions of these operators to Sobolev type completions, and we prove that the kernel and the cokernel of these extensions can be identified with the corresponding spaces of harmonic spinors on the sphere. For q, p > 1 related by 1/d + 1/q = 1/p result provides bounded L p → L q operators which invert the coupled Dirac operators modulo operators of finite rank. © Académie des Sciences/Elsevier, Paris

Dirac Induction for Unimodular Lie Groups

In this paper, we give a geometric realization of discrete series representations for unimodular Lie groups on the spaces of harmonic spinors by using Connes–Moscovici'sL2-Index theorem. Our work is a continuation of Atiyah–Schmid's geometric realization of discrete series representations for semisimple Lie groups and Connes–Moscovici's realization of square-integrable representations for nilpotent Lie groups.

Harmonic Spinors on Hyperbolic Space

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

Harmonic spinors on Riemann surfaces

We calculate the dimension of the space of harmonic spinors on hyperelliptic Riemann surfaces for all spin structures. Furthermore, we present non-hype relliptic examples of genus 4 and 6 on which the maximal possible number of linearly independent harmonic spinors is achieved.