If you want to know more about my research, you can take a look at my Research Statement. You can check my preprints on ArXiV and/or my profile on ResearchGate - ORCID.

For every irreducible reductive dual pair (G,G′) in the real symplectic group Sp(W), R. Howe proved the existence of an isomorphism between the spaces R(G) and R(G′), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of G ̃ (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the previous duality have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the previous duality (or Howe duality).

Publications

• Characters of Unitary Highest Weight Representations via Howe Correspondence and Rossmann-Duflo-Vergne formula, PhD Thesis (HAL).

• Characters of some unitary highest weight representations via the theta correspondence, Journal of Functional Analysis, Volume 279, Issue 8, November 2020 (ArXiV, Link to published article).

• Transfer of characters in the theta correspondence with one compact member, Journal of Lie Theory, 2020, no. 4, 997-1026 (ArXiV, Link to published article).

• Characters of irreducible unitary representations of U(n, n+1) via double lifting from U(1), Representation Theory, 26 (2022), 325-369 (ArXiV, Link to published article).

• 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, Link to published article).

Work in Progress

• Distribution character of the theta lift of a discrete series representations.

In his wonderful paper "Remarks on classical invariant theory", Roger Howe suggested that his classical duality should be extendable to superalgebras/supergroups. Roughly speaking, by restricting the spinor-oscillator representation (w, H) of the complex orthosymplectic Lie superalgebra spo(V) to some particular dual pairs (g, g'), he proved that the joint action of g and g' on every g-isotypic component is irreducible. A similar phenomenon has been obtained for other pairs by the works of other mathematicians (Wang, Cheng, Zhang, Nishiyama, etc ..) but a general theory for the real or complex orthosymplectic Lie superalgebra (or supergroup) is not known yet. This is precisely the motivation behind this large project. Since 30 years, the classical Howe's duality theorem has proven its crucial role in the study of irreducible representations of real classical groups, and the hope is that this "expected" duality will help in the study of irreducible representations of Lie supergroups.

Publications

• 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, Link to published article).

• Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup (joint work with Hadi Salmasian), to appear in Transformation Groups (ArXiV).

• Howe duality for the dual pair (SpO(2n|1), osp(2k|2l)) (joint work with Roman Lavicka), submitted for publication (ArXiV).

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)

• 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.

• Dual pairs in an orthosymplectic Lie supergroup, double commutant theorem, and duality (Video)

• Howe duality and Lie supergroups (Video, Beamer)