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Parameter identifications in the SL2(R) unitary dual

Statement

Assume the Axiom of Choice (The Axiom of Choice) and use the normalized parameter $I_{ε, ν}$ and exceptional lattice $W_{ε}$ of The normalized principal series I(epsilon, nu). The following records the parameter identifications and reducible endpoints in this normalization.

For $ν \notin W_{ε}$ , the irreducible principal-series representations satisfy $I_{ε, ν} ≅ I_{ε, - ν}$ . For nonzero $ν \in W_{ε}$ , the two reducible full induced modules are not isomorphic. At $ε = 1, ν = 0$ , one has $I_{1, 0} = D_{1}^{-} \oplus D_{1}^{+}$ .

The K-types of $I_{ε, ν}$ are exactly the characters of parity $m \equiv ε (m o d 2)$ , each with multiplicity one. If $n \in W_{ε}$ and $n \geq 1$ , then at $ν = n$ the composition factors are $L_{n - 1}$ and the two extremal modules $M_{n + 1}^{-}, M_{- (n + 1)}^{+}$ , with $L_{n - 1}$ the quotient; at $ν = - n$ the finite-dimensional factor is the submodule and the two tails form the quotient. For $n \geq 2$ , the algebraic K-finite modules of the discrete series $D_{n}^{-}$ and $D_{n}^{+}$ are isomorphic to $M_{- n}^{+}$ and $M_{n}^{-}$ , respectively, at $ν = n - 1$ ; the Hilbert representations are the weighted completions in Holomorphic and antiholomorphic discrete-series models. At $ε = 0, ν = \pm 1$ , the trivial module $L_{0}$ is a subquotient.

The unitary principal-series parameters are $I_{0, i s}$ for $s \geq 0$ and $I_{1, i s}$ for $s > 0$ ; the even point $I_{0, 0}$ is irreducible, while $I_{1, 0}$ is the limit split above. The spherical complementary family has parameters $I_{0, r}$ for $0 < r < 1$ , with the sign labels $r$ and $- r$ identified; $ν = \pm 1$ are degenerate endpoints, not additional irreducible complementary-series points. Within each principal or complementary family, no two different absolute parameter values give isomorphic representations.

Facts & Assumptions

Given: AC; the normalized principal-series conventions; the compact-picture K-type decomposition; the generic irreducibility and exceptional-parameter results; and the discrete and limit-series models.

[F1]

$W_{0}$ is the odd integer lattice and $W_{1}$ is the even integer lattice; the normalized parameter is used throughout (Iwasawa and minimal-parabolic data for SL2(R), The normalized principal series I(epsilon, nu)).

[F2]

In $I_{ε, ν}$ the one-dimensional K-types are precisely $C f_{m}$ , $m \equiv ε (m o d 2)$ , and each occurs once (K-type decomposition of the SL2(R) principal series).

[F3]

Off $W_{ε}$ , $I_{ε, ν}^{K}$ is irreducible and $I_{ε, ν} ≅ I_{ε, - ν}$ . At nonzero exceptional parameters, the finite-dimensional constituent is a quotient at $+ n$ and a submodule at $- n$ (Parameter-sign equivalence and its exceptional failures for SL2(R), Generic irreducibility and the exceptional parameter lattice).

[F4]

At $ν = n \in W_{ε}$ , $n \geq 1$ , the factors are $L_{n - 1}$ and $M_{n + 1}^{-}, M_{- (n + 1)}^{+}$ ; at $- n$ the finite-dimensional submodule and two-tail quotient are reversed. The central element acts by $(ν^{2} - 1) / 8$ (Generic irreducibility and the exceptional parameter lattice, Highest- and lowest-weight submodules at the exceptional parameters).

[F5]

$I_{1, 0}$ splits into $D_{1}^{-}$ and $D_{1}^{+}$ , and for $n \geq 2$ , their algebraic K-finite spans $V_{n}^{-}$ and $V_{n}^{+}$ identify with $M_{- n}^{+}$ and $M_{n}^{-}$ , respectively at $ν = n - 1$ (The two limits of discrete series, Unitarity and irreducibility of the limits of discrete series, Holomorphic and antiholomorphic discrete-series models).

[F6]

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

[A1]

AC is assumed and inherited from the normalized Iwasawa and compact-picture constructions (The Axiom of Choice).

[F7]

For $0 < ∣ ν ∣ < 1$ , the spherical invariant form has positive even Fourier weights $a_{n} (ν)$ , its weighted Hilbert completion is irreducible and strongly continuous unitary, and no real nonzero odd parameter admits a positive-definite full-module invariant form (Unitarity of the complementary series). The weights satisfy $a_{n} (- ν) = a_{n} (ν)^{- 1}$ .

Proof

technique · compile the exact parameter equivalences, K-type data, exceptional subquotients, and unitary ranges; distinguish the parameter values by K-types and the central character

Given: The conventions and claims recorded in the Statement.

1.1F1F3A1

By [F1], $W_{0} = 2 Z + 1$ and $W_{1} = 2 Z$ . The generic parameter set is their complement, so the sign-equivalence claim in the Statement is restricted to irreducible induced modules; nonzero lattice points are reducible.

1.2F3F5F6F7

The compact-picture theorem [F6] supplies the principal unitary axis; [F3] supplies irreducibility at its even zero and [F5] supplies the odd zero split. The positive weighted Hilbert completion and odd-parity invariant-form obstruction in [F7] give precisely the spherical complementary interval and absence of an odd complementary family.

2.1F3F5F7step 1.1algebra

For $ν \notin W_{ε}$ , [F3] gives $I_{ε, ν} ≅ I_{ε, - ν}$ . For nonzero $ν \in W_{ε}$ , [F3] gives the opposite positions of $L_{∣ ν ∣ - 1}$ in the two modules, so they are not isomorphic. For spherical real $0 < ∣ ν ∣ < 1$ , the normalized smooth intertwiner $R_{ν}$ of [F3] has multipliers $a_{n} (ν)$ . By [F7], $B_{- ν} (R_{ν} f, R_{ν} f) = \sum_{n} a_{n} (- ν) ∣ a_{n} (ν) \hat{f} (n) ∣^{2} = B_{ν} (f, f)$ on finite Fourier sums. Its inverse is $R_{- ν}$ , so density extends it to an onto unitary intertwiner of the completed complementary representations. At $ε = 1, ν = 0$ , [F5] gives the direct sum of the two limits.

3.1F2F4F5step 2.1

The K-type parity and multiplicity statement is [F2]. At a positive exceptional integer $n$ , [F4] gives the finite quotient and two extremal submodules; at $- n$ it gives the reversed submodule/quotient orientation. When $n = 1$ and $ε = 0$ , $L_{0}$ is the trivial representation, so the two endpoints $ν = \pm 1$ have the stated trivial subquotient. For $n \geq 2$ , [F5] identifies the extremal modules at $ν = n - 1$ with the algebraic K-finite spans of $D_{n}^{-}$ and $D_{n}^{+}$ , whose weighted Hilbert completions give the group representations. Thus an exceptional full induced module is not a second irreducible class to be counted alongside its irreducible constituents.

4.1F2F3F4step 3.1∎

For two generic parameters of the same parity, [F4] gives the central scalar $c (ν) = (ν^{2} - 1) / 8$ ; equivalent representations must have equal central scalars, hence their parameters differ only by sign. Different parities have disjoint K-type supports by [F2]. On the principal unitary axis $ν = i s$ , the scalar is $- (s^{2} + 1) / 8$ , so it determines $∣ s ∣$ ; on the spherical complementary interval $0 < r < 1$ , it is $(r^{2} - 1) / 8$ , so it determines $r$ . These ranges are disjoint, and the sign equivalence [F3] therefore leaves no further identification between distinct absolute parameter values in either family.

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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)