Structure of the degenerate principal series on symmetric R -spaces and small representations

Abstract

Let G be a simple real Lie group with maximal parabolic subgroup P whose nilradical is abelian. Then X = G/P is called a symmetric R-space.We study the degenerate principal series representations of G on C ∞ (X) in the case where P is not conjugate to its opposite parabolic. We find the points of reducibility, the composition series and all unitarizable constituents. Among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent K C -orbit in p * C , where K C is the complexification of a maximal compact subgroup K ⊆ G and g = k + p the corresponding Cartan decomposition.

Bessel operators on Jordan pairs and small representations of semisimple Lie groups

“…For P and P conjugate this is shown in [11,Corollary 2.32]. For P and P not conjugate we showed in [21,Theorem 5.3 (2)] that the associated variety of J(ν 1 ) is the minimal nilpotent K C -orbit. Since the Joseph ideal is the unique completely prime ideal in U (g) with associated variety the minimal nilpotent K C -orbit, it remains to show that the annihilator ideal of J(ν 1 ) is completely prime.…”

“…In many cases these operators can be obtained from standard families of intertwining operators such as the Knapp-Stein intertwiners. However, as observed in [21] in the case where P and P are not conjugate such families do not exist. Still, unitarizable quotients can occur in this setting, and hence the corresponding intertwiners cannot be obtained from standard families by regularization.…”

“…The rank r of (n, n) agrees with the rank of the compact symmetric space G/P . For ν ∈ a * C we form the degenerate principal series (smooth normalized parabolic induction) π ν = Ind G P (1 ⊗ e ν ⊗ 1) and realize it on a space I(ν) ⊆ C ∞ (n) of smooth functions on n. The structure of these representations has been completely determined by Sahi [24] for G Hermitian (see also Johnson [12,13] and Ørsted-Zhang [22]), by Sahi [25] and Zhang [28] for G non-Hermitian and P and P conjugate, and by the authors [21] for the remaining cases. The precise results differ from case to case, but they all have a common feature: There exist 0 < ν r−1 < .…”

Branching laws for small unitary representations ofGL(n, ℂ)

“…[11,Corollary 2.4.3 & 2.5.8]). In [24,Theorem 5.3] we showed that the unitary principal series representations π iλ,k attain the minimal possible Gelfand-Kirillov dimension among all infinite-dimensional unitary irreducible representations of G. (In fact, we only showed the statement for k = 0, but the proof carries over to the case k ∈ Z. )…”

“…Let π t,k := Ind G P (χ t,k ) denote the corresponding unitary principal series representations (normalized parabolic induction). The structure of these representations was studied by Dooley-Zhang [4] (see also [11,24]). For t = iλ ∈ iR and k ∈ Z the representation π t,k is unitary and irreducible and called unitary principal series representation.…”