If you would like to learn more about my research, you can read my Research Statement or browse my preprints on ArXiV. You can also find a complete list of my publications on my ResearchGate and ORCID profiles.
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Howe Duality and Characters
For every irreducible reductive dual pair (G,G') in the real symplectic group Mp(W), R. Howe established a remarkable correspondence between irreducible admissible representations of the metaplectic covers of G and G', known today as the Howe correspondence or theta correspondence. More precisely, he proved the existence of a natural bijection between the spaces R(G) and R(G'), where R(G) denotes the set of infinitesimal equivalence classes of irreducible admissible representations of the metaplectic cover of G that occur as quotients of the Weil (or metaplectic) representation. Since every representation in this correspondence is characterized by its distribution character, a natural problem is to understand how characters behave under the Howe correspondence.
Publications
Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral, International Mathematics Research Notices, Volume 2023, Issue 8, April 2023, Pages 6845-6900 (ArXiV, Article).
Characters of irreducible unitary representations of U(n, n+1) via double lifting from U(1), Representation Theory, 26 (2022), 325-369 (ArXiV, Article).
Transfer of characters in the theta correspondence with one compact member, Journal of Lie Theory, 2020, no. 4, 997-1026 (ArXiV, Article).
Characters of some unitary highest weight representations via the theta correspondence, Journal of Functional Analysis, Volume 279, Issue 8, November 2020 (ArXiV, Article).
Characters of Unitary Highest Weight Representations via Howe Correspondence and Rossmann-Duflo-Vergne formula, PhD Thesis (HAL).
Work in Progress
Distribution character of the theta lift of a discrete series representation.
Extension of Howe duality to Lie supergroups
In his seminal paper Remarks on Classical Invariant Theory, Roger Howe suggested that his classical duality should admit an extension to Lie superalgebras and Lie supergroups. Roughly speaking, he showed that, for certain dual pairs, the spinor-oscillator representation decomposes into irreducible components under the joint action of the two members of the pair. Since then, similar results have been obtained for several other dual pairs by Cheng, Wang, Zhang, Nishiyama, and others. However, a general theory for real or complex orthosymplectic Lie superalgebras (or Lie supergroups) is still unknown. This question has been one of the main motivations behind my work in this area. For more than thirty years, Howe's classical duality has played a fundamental role in the representation theory of real classical groups, and it is expected that an analogous theory for Lie supergroups will provide equally powerful tools for understanding their irreducible representations.
Publications
Howe duality for the dual pair (SpO(2n|1), osp(2k|2l)) (joint work with Roman Lavicka), to appear in Journal of Lie Theory, 2026 (ArXiV).
Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup (joint work with Hadi Salmasian), Transformation Groups 30 (2025), no. 4, 1915–1977 (ArXiV, Article).
Dual pairs in the Pin-group and duality for the corresponding spinorial representation (joint work with Clement Guerin and Gang Liu), Algebras and Representation Theory, Volume 24, Issue 6, December 2021 (ArXiV, Article).
Work in Progress
Howe duality for dual pair (gl(1|1), gl(p+q)): the non-semisimple case (joint wotk with Hadi Salmasian)
Pieri rule for the orthosymplectic Lie superalgebra spo(2n|1) (joint work with Roman Lavicka)
Matrix Analysis
Matrix analysis studies matrices through the combined perspectives of linear algebra, operator theory, geometry, and functional analysis. It has applications in optimization, quantum information theory, mathematical physics, statistics, and numerical analysis. My recent research has focused on the theory of matrix means, with particular emphasis on Kubo–Ando means and alternative means. I am interested in their algebraic, analytic, and geometric properties, the development of explicit formulas and decomposition results, and the characterization of when these means admit simple linear representations. More recently, I have also been studying positive definite matrices over the real division algebras and the interplay between matrix analysis, symmetric cones, and representation theory.
Publications
On the linearization of alternative means (with Raluca Dumitru and Jose Franco), submitted (ArXiV).
Spectral decomposition and linearization of Kubo-Ando means (with Raluca Dumitru and Jose Franco), submitted (ArXiV).
Correspondence of Kubo-Ando means over real division algebras and linearization of means (with Jose Franco), submitted (ArXiV).
The cone of J-hermitian matrices and a geometric mean (with Jose Franco), submitted (ArXiV).
Short notes
Transfer on characters for discrete series representation in the equal rank case Pdf (here is a shortest handwriting note in French Pdf).
Transfer of characters in the theta correspondence with one compact member Pdf.
From Weyl-Schur duality to Pin-duality Pdf.
Polynomial Invariants and Character Varieties of Classical Groups Pdf.
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