How tempered representations of a semisimple Lie group contract to its Cartan motion group

Alexandre Afgoustidis (1, 2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

George W. Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact semisimple Lie group G and those of its Cartan motion group − the semidirect product G 0 of a maximal compact subgroup of G and a vector space. In these notes, I focus on the carrier spaces for these representations and try to give a precise meaning to some of Mackey's remarks. I first describe a bijection, based on Mackey's suggestions, between the tempered dual of G − the set of equivalence classes of irreducible unitary representations which are weakly contained in L 2 (G) − and the unitary dual of G 0. I then examine the relationship between the individual representations paired by this bijection : there is a natural continuous family of groups interpolating between G and G 0 , and starting from the Hilbert space H for an irreducible representation of G, I prove that there is an essentially unique way of following a vector through the contraction from G to G 0 within a fixed Fréchet space that contains H. It then turns out that there is a limit to this contraction process on vectors, and that the subspace of our Fréchet space thus obtained naturally carries an irreducible representation of G 0 whose equivalence class is that predicted by Mackey's analogy.

Mots clés: Representations of semisimple Lie groups; Lie group contractions; Mackey analogy; Tempered dual

Domaines: Mathématiques [math]; Théorie des représentations [math.RT]; Théorie des groupes [math.GR]

Alexandre Afgoustidis. How tempered representations of a semisimple Lie group contract to its Cartan motion group. 2015. ⟨hal-01214358v1⟩