Unitary Representations of Nilpotent Super Lie Groups

Abstract

Abstract. We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.

Harmonic analysis on Heisenberg–Clifford Lie supergroups

“…Indeed, we expect that it will generalize to arbitrary Lie supergroups (though, of course, these may not have any unitary representations in general). The irreducible unitary representations of the Heisenberg-Clifford Lie supergroup have been classified by Salmasian [17], and our approach depends heavily on his classification. The Fourier transform F that we introduce takes values in a certain endomorphism algebra H. We define a convolution product in analogy with (f * g)(x) = G f (y)g(y −1 x) dy, which behaves well with respect to the Fourier transform in the sense that the Fourier transform F (F * G) agrees with the pointwise product F (F )F (G) in H. This convolution product seems to be new.…”

Superunitary Representations of Heisenberg Supergroups

“…Building on [16], we introduce a new definition of Hilbert superspaces, which generalizes the standard one [9,7,10,11]. Their basic properties are derived by using the theory of Krein spaces [15].…”

“…We provide independently a definition of SUR for super Harish-Chandra pairs and for Lie supergroups and prove that they are equivalent. Our definition of SUR generalizes the one in [10,11].…”

“…the standard superunitary dual of H 2m|k, essentially coincides with the unitary dual of H 2m . On the contrary, if k = 0, the standard superunitary theory of H 2m|k, is empty, there is no standard SUR except the trivial one [11].…”

Q-representations and unitary representations of the super-Heisenberg group and harmonic superanalysis

“…To make contact with our previous considerations, let us illustrate the super Harish-Chandra pair notion on the example of the super Heisenberg group [15], and has been used in mathematical approach to supersymmetry [16,17,6]. In such an approach the super-Hilbert space is defined as a Z 2 -graded complex Hilbert space H = H 0 ⊕ H 1 such that…”