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Theses

Caractères de représentations unitaires de plus haut poids via la correspondance de Howe et la formule de Rossmann-Duflo-Vergne

Abstract : The topic of this thesis is the character of an irreducible unitary highest weight (IUHW) representation of the group U(p,q,C). As shown by Harish-Chandra, an irreducible unitary representation of a real reductive group has a character, which is a locally integrable function on the group. Integrating that function against a test function gives the trace of the operator obtained by applying the representation to the test function. Also, there is a bijective correspondence between the equivalence classes of irreducible unitary representations and their characters. If the group is compact then the representation is finite dimensional and the the character is equal to the trace of the representation. Weyl classified all irreducible unitary representations of compact connected Lie groups and described their characters. By taking derivatives, one obtains a representation of the complexified Lie algebra. A Borel subalgebra has a one-dimensional eigenspace. The corresponding eigencharacter is the highest weight of the representation. So, there is a bijection between equivalence classes of irreducible unitary representations and some highest weights. The equivalence classes of irreducible unitary representations of a real reductive group are far from being understood. However Harish-Chandra introduced the notion of an irreducible admissible highest weight representation, which is not necessarily unitary. Each of them is determined by an eigencharacter of a Borel subalgebra, also called the highest weight. By the works of Jacobsen, Enright-Howe-Wallach it is clear now which highest weights in Harish-Chandra's construction correspond to unitary representations. If an IUHW representation of a real reductive group occurs as a subrepresentation of the space of the square integrable functions on the group, then a formula of Harish-Chandra describes the restriction of the character to the subset of the regular elliptic elements, i.e. the union of conjugacy classes passing through the regular points of a maximal compact Cartan subgroup. Later Enright gave a formula for the restriction of the character to the subset of the regular elliptic elements of an arbitrary IUHW representation of a classical group. Enright's formula is explicit, but contains many hidden cancellations. Also, the proof is based on homological algebra rather than a direct analytic method as in Harish-Chandra's construction. In this thesis we compute the restriction of the character of an IUHW representation to the subset of the regular elliptic elements of U(p,q,C). The result is more explicit than Enright's and we check in a few cases that the two formulas are equivalent. Checking the equivalence in general is not easy and we do not do it. We use two independent methods. The first one is to compute an integral involving the character of Weil representation via the Residue Theorem. It is straightforward, but the result lacks any pleasing structure. The second method is to realize the representation in Howe's correspondence, i.e. as a subrepresentation of the Weil representation restricted to a dual pair with one member compact. Then we combine the known explicit description of the Weil representation, adopted to this context in a work of McKee-Pasquale-Przebinda, with a formula of Rossmann-Duflo-Vergne (also based on a study of the Weil representation) for the Fourier transform of an elliptic orbital integral. The resulting formula is compatible with the orbit correspondence and has close resemblance to Harish-Chandra's formula.
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https://hal.archives-ouvertes.fr/tel-03351060
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Submitted on : Tuesday, September 21, 2021 - 9:26:58 PM
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Allan Merino. Caractères de représentations unitaires de plus haut poids via la correspondance de Howe et la formule de Rossmann-Duflo-Vergne. Mathématiques [math]. Université de Lorraine, 2017. Français. ⟨tel-03351060⟩

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