Mackey analogy via 𝒟-modules for SL(2, ℝ)

Abstract

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of SL(2, R) by considering families of D-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.

The algebraic Mackey-Higson bijections

“…temp that correspond to each other under the Mackey bijection there is a family of representations {π t } t∈[0,∞) that extends π 0 and π 1 and, in a certain sense, deform the underlying Hilbert spaces of π 0 and π 1 . This description is in agreement with [TYY17] where the bijection is constructed via a family of D-modules on the flag variety of SL 2 (R). We shall now write these families of representations in terms of parabolically induced representations and reinterpret the formulas for the families in [Afg15] and [TYY17] using our language.…”

“…Let I 0 (ǫ, λ) be the corresponding representation of SO(2) ⋉ R 2 . Starting with I(ǫ, λ) with λ = 0 the recipe of [Afg15] (which is very similar to that of [TYY17]) is to look on the family I t := I(ǫ, λ t ) with t > 0 and then to show that in a certain sense I t converges to I 0 (ǫ, λ) as t goes to zero. This construction seems a bit ad hoc (one can try to take other functions of t) and the infinitesimal character of I t that is given by λ t −1 as well as the action of the Casimir that is given by λ 2 r 2 − 1, blow up as we approach 0.…”

“…Note that the choice of sign for the square root is irrelevant since I(ǫ, λ) ≃ I(ǫ, −λ). In our coordinate system the Cartan motion group is obtained at r = ∞ which is the same as R = 0, in [Afg15,TYY17] it is obtained at t = 0. Furthermore, from the way the Lie brackets scales with the parameter we must have r = R −1 = ct −1 for some nonzero real constant c. We shall choose c = 1 which corresponds to η R=1 , the other choices lead to all other Mackey-Higson bijections.…”

“…The Mackey bijection for the tempered duals in the case of SL 2 (R) was studied before at [Geo09]. Recently it was reformulated in terms of twisted D-modules on the flag variety of SL 2 (R) [TYY17]. Most of the formalism of algebraic groups that we use here was developed in [BHS16,BHS17].…”

Orbital integrals and K-theory classes

“…It turns out, however, that the combination of the isomorphism class of K 0 (C * r G), the topological space T and the maps τ g : K 0 (C * r G) → C, for g ∈ T , together determine the Cartan motion group K ⋉ p and vice versa. The tempered representation theory of K ⋉ p is closely related to that of G; this is the Mackey analogy [3,20,21,33,38,43]. Also, the analytic assembly map for G can be defined in terms of a continuous deformation from K ⋉ p to G, see pp.…”