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Spyridon Afentoulidis-Almpanis

Publications and source records attributed to Spyridon Afentoulidis-Almpanis.

7 recordsLinked to original sources
arXiv2026

Dirac operators for Cartan motion groups

We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.

math.RT↗
arXiv2026

Dirac operators for algebraic families

We introduce algebraic families of Dirac operators for the deformation family (and other related families) associated with a real reductive Lie group that interpolates the reductive group and the corresponding Cartan motion group. We prove Vogan's conjecture in this setting, relating the infinitesimal character of an algebraic family of Harish-Chandra modules and its Dirac cohomology.

math.RT↗
arXiv2026

Structure and Representation Theory of basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded color Lie algebras

We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded (color) Lie algebras, which we call basic. As an application, assuming that the Cartan subalgebra is self-centralizing, we classify all finite-dimensional representations of these algebras by proving a highest weight theorem and a complete reducibility theorem.

math.RT↗
arXiv2023

Dirac cohomology for the BGG category $\mathcal{O}$

We study Dirac cohomology $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ for modules belonging to category $\mathcal{O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ while we show that in the case of a Hermitian symmetric pair $(\mathfrak{g},\mathfrak{k})$ and an irreducible unitary module $M\in\mathcal{O}$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for $M\in\mathcal{O}$.

math.RT↗
arXiv2023

Dirac cohomology and $Θ$-correspondence for complex dual pairs

We study the behavior of Dirac cohomology under Howe's $Θ$-correspondence in the case of complex reductive dual pairs. More precisely, if $(G_1,G_2)$ is a complex reductive dual pair with $G_1$ and $G_2$ viewed as real groups, we describe those Harish-Chandra modules $π_1$ of $G_1$ with nonzero Dirac cohomology whose $Θ$-liftings $Θ(π_1)$ still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $Θ(π_1)$.

math.RT↗
arXiv2022

Noncubic Dirac operators for finite dimensional modules

We study the decomposition into irreducibles of the kernel of noncubic Dirac operators attached to finite-dimensional modules. We compare this decomposition with features of Kostant's cubic Dirac operator. In particular, we show that the kernel of noncubic Dirac operators need not contain full isotypic components. The cases of classical and exceptional complex Lie algebras are studied in detail. As a by-product, we deduce some information on the kernel of noncubic geometric Dirac operators acting on sections over compact manifolds studied by Slebarski.

math.RT↗