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The unitary dual of SL2(R) is non-discrete and non-Hausdorff at the stated limits

Statement

Assume the Axiom of Choice (The Axiom of Choice) and use the Fell topology on the unitary dual of $G = {S L}_{2} (R)$ (The Fell topology on the unitary dual, The unitary dual of a locally compact group).

(1) As $s ↓ 0$ through positive values, the classes $[I_{1, i s}]$ converge to both distinct classes $[D_{1}^{-}]$ and $[D_{1}^{+}]$ . Consequently the unitary dual is not Hausdorff.

(2) As $s ↓ 0$ through positive values, the pairwise distinct classes $[I_{0, i s}]$ converge to $[I_{0, 0}]$ . Consequently the unitary dual is not discrete.

(3) For $0 < r < 1$ , the spherical complementary classes $[I_{0, r}]$ are distinct from the trivial class and converge to it as $r ↑ 1$ , as in The spherical complementary series converge to the trivial representation.

Facts & Assumptions

Given: AC; the Fell topology and unitary dual; the compact-picture principal-series family; the limit representations; and the spherical complementary-series convergence result.

[F1]

A basic Fell neighborhood is specified by finitely many diagonal matrix coefficients, compact test sets, and positive tolerances; a class is a point of $\hat{G}$ exactly when its representation is irreducible and strongly continuous unitary (The Fell topology on the unitary dual, The unitary dual of a locally compact group, Matrix coefficient of a unitary representation).

[F2]

For each $ξ \in L_{ε}^{2} (K)$ , the diagonal coefficients of $Π_{ε, s}$ converge uniformly on every compact subset of $G$ to those of $Π_{ε, s_{0}}$ as $s \to s_{0}$ (Fell continuity of the unitary principal series in the parameter).

[F3]

$I_{1, 0} = D_{1}^{-} \oplus D_{1}^{+}$ orthogonally, both limits are irreducible strongly continuous unitary representations, and their K-type supports are the negative and positive odd tails respectively (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series, K-type decomposition of the SL2(R) principal series).

[F4]

$I_{ε, ν}$ is generically irreducible off $W_{ε}$ , and the Casimir scalar on its K-finite module is $(ν^{2} - 1) / 8$ (Generic irreducibility and the exceptional parameter lattice, Smooth and K-finite vectors for SL2(R), and the (g,K)-module).

[F5]

The compact-picture action is strongly continuous and unitary for $ν \in i R$ (The compact picture of the SL2(R) principal series, The normalized principal series I(epsilon, nu)).

[F6]

For $0 < r < 1$ , the class $[I_{0, r}]$ converges to the trivial class in the Fell topology as $r ↑ 1$ (The spherical complementary series converge to the trivial representation).

[F7]

For $0 < r < 1$ , the complementary representation is the positive weighted Hilbert completion of the even Fourier module, with $∥ f_{2} ∥_{r}^{2} = a_{2} (r) = (1 - r) / (1 + r) > 0$ (Unitarity of the complementary series). Its $K$ -action on $f_{2}$ is $π_{r} (k_{θ}) f_{2} = e^{2 i θ} f_{2}$ (K-type decomposition of the SL2(R) principal series, The compact picture of the SL2(R) principal series).

[A1]

AC is assumed and inherited through the compact-picture, unitary-dual, and Fell-topology suppliers (The Axiom of Choice).

Proof

technique · apply compact-uniform coefficient continuity to one positive-parameter net, then distinguish its limit classes by K-types and Casimir

Given: The definitions and supplier claims in the Statement and Facts.

1.1F1F2F3F4F5algebraA1

For every $s > 0$ , [F4] and [F5] make $[I_{1, i s}]$ a unitary-dual point. Fix a basic Fell neighborhood of either $[D_{1}^{+}]$ or $[D_{1}^{-}]$ , testing finitely many diagonal coefficients of vectors in that limit representation on compact subsets of $G$ . By [F3], these vectors embed in $L_{1}^{2} (K)$ , and $Π_{1, 0}$ restricts to the relevant limit on them. Applying [F2] to the finite set of vectors and compact sets gives $δ > 0$ such that every $0 < s < δ$ satisfies all tests. Thus the positive-parameter branch converges to both limits as $s ↓ 0$ . In particular the same sequence $s_{j} = 1 / (j + 1)$ converges to both.

1.2F1F4F5algebra

For $s > 0$ , [F4] and [F5] place $[I_{0, i s}]$ in $\hat{G}$ . If $s, t > 0$ and $[I_{0, i s}] = [I_{0, i t}]$ , a unitary intertwiner maps smooth K-finite vectors to smooth K-finite vectors and intertwines their derived actions by differentiating the group-intertwining identity. It therefore preserves the Casimir scalar. By [F4] these scalars are $- (s^{2} + 1) / 8$ and $- (t^{2} + 1) / 8$ , so $s = t$ . Also $[I_{0, i s}] \neq [I_{0, 0}]$ for $s > 0$ , since their Casimir scalars differ.

2.1F1F3step 1.1algebra

The two limit classes are distinct: their K-type supports in [F3] are disjoint, and a unitary intertwiner must preserve the K-action. A sequence in the unitary dual with two distinct limits contradicts uniqueness of limits in every Hausdorff space. This proves (1).

2.2F1F2F5step 1.2algebra

Fix a basic Fell neighborhood of $[I_{0, 0}]$ . Its finitely many diagonal coefficient tests use vectors in $L_{0}^{2} (K)$ ; [F2] gives compact-uniform convergence of each tested coefficient as $s \to 0$ , so all tests are satisfied by $[I_{0, i s}]$ for sufficiently small positive $s$ . Hence $[I_{0, i s}] \to [I_{0, 0}]$ along the positive branch. By step 1.2 these are distinct classes, so $[I_{0, 0}]$ is not isolated. This proves (2).

3.1F6F7A1algebra∎

For $0 < r < 1$ , [F7] makes $f_{2}$ a nonzero vector in the complementary Hilbert space, with $K$ -character $e^{2 i θ}$ . A unitary intertwiner with the trivial representation would preserve this character; at $θ = π / 2$ , it would send $- f_{2}$ and $f_{2}$ to the same vector, forcing the image of $f_{2}$ to vanish, contrary to injectivity. Hence $[I_{0, r}]$ is distinct from the trivial class. The convergence assertion is [F6]: its supplier proves compact-uniform convergence of the normalized spherical coefficient to $1$ and scales that coefficient to meet every finite test for a Fell neighborhood of the trivial class. This proves (3).

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Sources

Matt Kerr, Notes on the Representation Theory of SL2(R) (NSF/CBMS workshop writeup) (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups (AMS GSM 155; author's PDF) (standard reference, not scraped)