Harmonic Spinors on Reductive Homogeneous Spaces

Mehdi, Salah; Zierau, Roger

doi:10.1007/978-3-319-09934-7_6

Cited by 4 publications

(11 citation statements)

References 11 publications

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“…We now show that the embedding of a globalization of the fundamental series in the kernel of the Dirac operator, as defined by Mehdi and Zierau in [MZ14b], commutes with the Zuckerman translation functor. Let us give a precise statement.…”

Section: The Fundamental Seriesmentioning

confidence: 82%

“…We now choose positive systems as in [MZ14b]. Let ∆ + ⊂ ∆(t + a, g) be defined by a lexicographic order with t h first, then t h ⊥ , and a.…”

Section: The Fundamental Seriesmentioning

confidence: 99%

“…Section 3 is devoted to the construction of various commutative diagrams which are used to prove that a specific embedding of fundamental series representations into the Dirac kernel commutes with translation functors. Finally, in Section 4, we show that the intertwining map defined in [MZ14b], which produces explicit non-zero solutions to Weyl equations D(E µ ) = 0 on G/H, commutes with translations by finite dimensional representations (Theorem 4.1). This provides a systematic geometric process which produces interacting Weyl fermions with a fixed energy level on the homogeneous spacetime G/H as follows:…”

Section: Introductionmentioning

confidence: 95%

“…When H is not compact, the smooth kernel of D(E) contains a principal series representation (now G and H need not be equal rank). This is done in [MZ14b] by constructing an intertwining map P E analogous to the Poisson transform:…”

Section: Introductionmentioning

confidence: 99%

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Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Mehdi¹,

Prudhon²

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We show that the image of the Poisson map, defined by Mehdi and Zierau in [MZ14b], which intertwines principal series representations with a submodule of the kernel of the cubic Dirac operator, commutes with the translation functor. As a byproduct, we obtain a systematic geometric process which produces interacting Weyl fermions with a fixed energy level on homogeneous spacetimes.In honor of Professor Jean LudwigIt is a first order Lorentz-invariant differential operator acting on sections of the spin bundle S over Minkowski spacetime. The spin bundle splits into half-spin bundles S + and S − so that

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Section: The Fundamental Seriesmentioning

confidence: 82%

“…We now choose positive systems as in [MZ14b]. Let ∆ + ⊂ ∆(t + a, g) be defined by a lexicographic order with t h first, then t h ⊥ , and a.…”

Section: The Fundamental Seriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Mehdi¹,

Prudhon²

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…To fix this problem, we will replace Dirac cohomology by the Dirac index. (We note that there is a relationship between Dirac cohomology and translation functors; see [MP], [MPa1], [MPa2], [MPa3]. )…”

Section: Introductionmentioning

confidence: 99%

Translation principle for Dirac index

Mehdi¹,

Pandžić²,

Vogan³

2017

American Journal of Mathematics

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Let G be a finite cover of a closed connected transpose-stable subgroup of GL(n, R) with complexified Lie algebra g. 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 (g, 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 g. 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 [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.2010 Mathematics Subject Classification. Primary 22E47; Secondary 22E46.

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Representation theoretic embedding of twisted Dirac operators

Mehdi

Pandžić

2021

Represent. Theory

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Let G G be a non-compact connected semisimple real Lie group with finite center. Suppose L L is a non-compact connected closed subgroup of G G acting transitively on a symmetric space G / H G/H such that L ∩ H L\cap H is compact. We study the action on L / L ∩ H L/L\cap H of a Dirac operator D G / H ( E ) D_{G/H}(E) acting on sections of an E E -twist of the spin bundle over G / H G/H . As a byproduct, in the case of ( G , H , L ) = ( S L ( 2 , R ) × S L ( 2 , R ) , Δ ( S L ( 2 , R ) × S L ( 2 , R ) ) , S L ( 2 , R ) × S O ( 2 ) ) (G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\mathbb R})\times SL(2,{\mathbb R})),SL(2,{\mathbb R})\times SO(2)) , we identify certain representations of L L which lie in the kernel of D G / H ( E ) D_{G/H}(E) .

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Harmonic Spinors on Reductive Homogeneous Spaces

Cited by 4 publications

References 11 publications

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Translation of Harmonic Spinors and Interacting Weyl Fermions on Homogeneous Spaces

Translation principle for Dirac index

Representation theoretic embedding of twisted Dirac operators

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