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Irreducibility and K-types of the discrete series

Statement

Assume the Axiom of Choice (The Axiom of Choice). For every integer $n \geq 2$ , the representations $D_{n}^{-} = (π_{n}, H_{n}^{+})$ and $D_{n}^{+} = (π_{n}^{-}, H_{n}^{-})$ of $G = {S L}_{2} (R)$ are irreducible strongly continuous unitary representations. Their K-type decompositions are $H_{n}^{+} = \overline{⨁_{j \geq 0} C f_{n, j}}, π_{n} (k_{θ}) f_{n, j} = e^{- i (n + 2 j) θ} f_{n, j}, H_{n}^{-} = \overline{⨁_{j \geq 0} C {\tilde{f}}_{n, j}}, π_{n}^{-} (k_{θ}) {\tilde{f}}_{n, j} = e^{i (n + 2 j) θ} {\tilde{f}}_{n, j},$ each character occurring with multiplicity one. The displayed algebraic K-finite subspaces $V_{n}^{-}, V_{n}^{+}$ of Holomorphic and antiholomorphic discrete-series models are isomorphic as $(g, K)$ -modules to $M_{- n}^{+}$ and $M_{n}^{-}$ , the extremal submodules of Highest- and lowest-weight submodules at the exceptional parameters(b) at $ν = n - 1$ ; in particular, the Casimir $Ω$ of Smooth and K-finite vectors for SL2(R), and the (g,K)-module acts on these algebraic modules by $\frac{1}{8} ((n - 1)^{2} - 1)$ .

Facts & Assumptions

Given: AC; the holomorphic and antiholomorphic models and their smooth displayed K-finite vectors; and the Hilbert, K-type, and invariant norm results of the preceding weighted-space items.

[F1]

The displayed vectors are smooth and have derived actions $L_{E_{+}} f_{n, j} = - j f_{n, j - 1}$ (zero for $j = 0$ ), $L_{E_{-}} f_{n, j} = (n + j) f_{n, j + 1}$ ; on conjugates, $L_{E_{+}} {\tilde{f}}_{n, j} = (n + j) {\tilde{f}}_{n, j + 1}$ and $L_{E_{-}} {\tilde{f}}_{n, j} = - j {\tilde{f}}_{n, j - 1}$ (Holomorphic and antiholomorphic discrete-series models).

[F2]

The $f_{n, j}$ form a complete orthogonal K-eigenbasis of $H_{n}^{+}$ , the projections $P_{j}$ have range $C f_{n, j}$ , and the conjugate family has the analogous properties (The weighted discrete-series space is a Hilbert space with K-type basis).

[F3]

The model actions are strongly continuous unitary representations on the Hilbert spaces (The weighted area form is SL2(R)-invariant).

[F4]

At $ν = n - 1$ , the extremal modules $M_{- n}^{+}$ and $M_{n}^{-}$ are irreducible with the same K-weights and ladder actions; the Casimir acts on them by $\frac{1}{8} ((n - 1)^{2} - 1)$ (Highest- and lowest-weight submodules at the exceptional parameters(b)). The model's algebraic K-finite spans identify with these modules (Holomorphic and antiholomorphic discrete-series models).

Proof

technique · direct

Given: The assumptions and notation of the Statement.

1.1F2F3algebra

Let $W \subseteq H_{n}^{+}$ be a nonzero closed $G$ -invariant subspace and choose $0 \neq v \in W$ . It is K-invariant. By [F2], $v = \sum_{j \geq 0} P_{j} v$ in Hilbert norm, so some $P_{j} v$ is nonzero; each $P_{j} v$ belongs to $W$ because its defining K-orbit integral is a norm limit of sums of vectors in $W$ . Thus $f_{n, j} \in W$ for some $j$ .

2.1F1F2step 1.1algebra

Since $W$ is closed and G-invariant, for every real $X \in g$ and smooth $w \in W$ the difference quotients $(π_{n} (\exp (t X)) w - w) / t$ lie in $W$ and converge in norm to $L_{X} w$ ; hence $L_{X} w \in W$ . Applying the complex-linear combinations $E_{+}$ and $E_{-}$ to the smooth K-finite vectors in [F1] shows that $W$ contains $f_{n, j - 1}$ whenever $j > 0$ , and contains $f_{n, j + 1}$ for every $j \geq 0$ , since the coefficients $- j$ and $n + j$ are nonzero in those cases. Iteration gives every $f_{n, k} \in W$ . Their span is dense by [F2], so closedness yields $W = H_{n}^{+}$ .

3.1F1F2F3step 1.1step 2.1

The same argument applies to $H_{n}^{-}$ : from any nonzero closed invariant subspace, a nonzero character projection gives some ${\tilde{f}}_{n, j}$ ; the nonzero coefficients $n + j$ upward and $- j$ downward generate all ${\tilde{f}}_{n, k}$ , whose span is dense. Thus both representations are irreducible. Their strong continuity and unitarity are [F3].

4.1F4algebra∎

The module identifications in [F4] match the K-weights and both ladder operators, and therefore identify the algebraic K-finite modules of $D_{n}^{-}$ and $D_{n}^{+}$ with $M_{- n}^{+}$ and $M_{n}^{-}$ . The same supplier gives the stated Casimir scalar on these modules, with the normalization of Smooth and K-finite vectors for SL2(R), and the (g,K)-module.

Depends on

Used by

Dependency tree · two levels

48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emmanuel Kowalski, An Introduction to the Representation Theory of Groups (AMS GSM 155; author's PDF) (standard reference, not scraped)
Matt Kerr, Notes on the Representation Theory of SL2(R) (CBMS workshop writeup) (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups, MIT 18.757 Lecture 9 (standard reference, not scraped)