arXiv subjects

Marcelo De Martino

Publications and source records attributed to Marcelo De Martino.

12 recordsLinked to original sources

arXiv2026

The double dihedral Dunkl total angular momentum algebra

The Dunkl total angular momentum algebra (TAMA) is realised as the dual partner of the orthosymplectic Lie superalgebra containing the Dunkl deformation of the Dirac operator. In this paper, we consider the case when the reflection group associated with the Dunkl operators is a product of two dihedral groups acting on a four-dimensional Euclidean space. We show that in this case there is a subalgebra of the total angular momentum algebra that admits a triangular decomposition. In analogy to the celebrated theory of semisimple Lie algebras, we use this triangular subalgebra to give precise necessary conditions that a finite-dimensional irreducible representation must obey, in terms of weights. In specific cases, which includes unitary representations, we construct a basis of weight vectors with explicit actions of all TAMA elements. Examples of these modules occur in the kernel of the Dunkl--Dirac operator in the context of deformations of Howe dual pairs.

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arXiv2025

A remark on the Langlands correspondence for tori

For an algebraic torus defined over a local (or global) field $F$, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to-one) homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. Finally, we compare the group of unramified characters with the Galois (co)invariants of the dual torus.

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arXiv2025

On angular momentum algebras and their relations

In this paper, we study the centraliser of $\mathfrak{osp}(1|2)$, denoted the total angular momentum algebra (TAMA), in the Weyl Clifford algebra. The TAMA extends the angular momentum algebra (AMA), which arises as the centraliser of $\mathfrak{sl}(2)$ and admits a diagrammatic presentation via the crossing relation described by Feigin and Hakobyan. Using Young symmetrisers we construct an analogue relation for the even subalgebra of the TAMA. We prove that for rank $4$ and $5$ these relations generate a presentation for the even subalgebra of the TAMA.

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arXiv2024

On the Quantum Metaplectic Howe Duality

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups $(\mathcal U_{q^2}(\mathfrak{sl}_2),\mathcal U_{q}(\mathfrak{sl}_2))$ realized in a suitable algebra of $q$-differential operators acting on the space of symplectic polynomial spinors. We obtain $q$-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss $q$-analogues of generalized symmetries of the $q$-symplectic Dirac operator raising the homogeneous polynomial degree.

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arXiv2022

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum algebra $O_{t,c}(V,W)$ arises as the centraliser algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ containing a Dunkl deformation of the Dirac operator, inside the tensor product of $\mathsf{H}_{t,c}(V,W)$ and the Clifford algebra generated by $V$. We show that, for every value of the parameter $c$, the centre of $O_{t,c}(V,W)$ is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not $(-1)_V$ is an element of the group $W$. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into $O_{t,c}(V,W)$, we establish results analogous to ``Vogan's conjecture'' for a family of operators depending on suitable elements of the double cover $\tilde{W}$.

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arXiv2022

On the unramified spherical automorphic spectrum

For an unramified connected reductive group $G$ defined over a number field $F$, consider the part of the spherical automorphic spectrum with cuspidal support $[T,\mathcal{O}(χ)]$, where $T$ is a maximal torus and $χ$ is an unramified automorphic character. We define a normalization of the Eisenstein series and we give the precise spectral decomposition of the closure of the subspace spanned by the normalized pseudo-Eiseinstein series. The proof uses residue distributions which were introduced by the third author (in joint work with G. Heckman) in the study of graded affine Hecke algebras, which is an ingredient of a purely local nature. In the case when $G$ is split and $χ$ is the trivial character, we show that the normalized spectrum is in fact the whole spherical automorphic spectrum. The necessary argument to conclude the result in the split case are based on combinatorial results proved in [DMHO].

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arXiv2022

Residue distributions, iterated residues, and the spherical automorphic spectrum

Let $G$ be a split reductive group over a number field $F$. We consider the computation of the inner product of two $K$-spherical pseudo Eisenstein series of $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following a classical approach initiated by Langlands. We show that only the singularities of the intertwining operators due to the poles of the completed Dedekind zeta function $Λ_F$ contribute to the spectrum, while the singularities caused by the zeroes of $Λ_F$ do not contribute to any of the iterated residues which arise as a result of the necessary contour shifts. In the companion paper [DMHO] we use this result to explicitly determine the spectral measure of $L^2(G(F)\backslash G(\mathbb{A}_F),ξ)^K_{[T,\mathcal{O}(1)]}$ by a comparison of the iterated residues with the residue distributions of [HO1].

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arXiv2022

Dirac Operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

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arXiv2021

Deformations of unitary Howe dual pairs

We study deformations of the Howe dual pairs $(\mathrm{U}(n),\mathfrak{u}(1,1))$ and $(\mathrm{U}(n),\mathfrak{u}(2|1))$ to the context of a rational Cherednik algebra $H_{1,c}(G,E)$ associated with a real reflection group $G$ acting on a real vector space $E$ of even dimension. For each pair, we show that the Lie (super)algebra structure of one partner is preserved under the deformation, which leads to a multiplicity-free decomposition of the standard module or its tensor product with a spinor space. For the case where $E$ is two-dimensional and $G$ is a dihedral group, we provide complete descriptions for the deformed pair and the relevant joint-decomposition.

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arXiv2020

The Dunkl-Cherednik Deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{W}\otimes\mathcal{C}$ deforms to a Lie superalgebra inside $H_c\otimes\mathcal{C}$ and compute its centraliser algebra. For a generic parameter $c$ such that the standard $H_c$-module is unitary, we compute the joint $((H_c)^{\mathfrak{s}},\mathfrak{s})$- and $((H_c\otimes\mathcal{C})^{\mathfrak{g}},\mathfrak{g})$-decompositions of the relevant modules.

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arXiv2020

Symplectic Dirac operators for Lie algebras and graded Hecke algebras

We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic Lie algebras $\mathfrak g=\mathfrak k+\mathfrak p$ and of graded affine Hecke algebras $\mathbb H$.

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arXiv2017

Dirac induction for rational Cherednik algebras

We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler-Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.

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