07.07.10 · representation-theory / compact-lie

The unitary dual of $SL (2, R)$ : principal, discrete, and complementary series

shipped3 tiersLean: none

Anchor (Master): Bargmann *Irreducible unitary representations of the Lorentz group* (Ann. Math. 48, 1947) 568-640 (the originating classification); Harish-Chandra *Plancherel formula for the $2\times2$ real unimodular group* (Proc. Nat. Acad. Sci. 38, 1952) 337-342; Knapp *Representation Theory of Semisimple Groups* (Princeton 1986) Chs. II, VII, XI; Lang *$\mathrm{SL}_2(\mathbb{R})$* (GTM 105, 1985); Gelfand-Graev-Vilenkin *Generalized Functions Vol. 5* (Academic Press 1966) Ch. VII

Intuition Beginner

A rotation group like the circle is easy to listen to: every sound it makes is a pure tone, $e^{in θ}$ , indexed by a whole number. The group $SL (2, R)$ — the $2 \times 2$ real matrices of determinant one — is the next instrument up, and it is harder to play. It is not compact: you can boost faster and faster without ever coming back, the way a Lorentz boost never closes into a loop. Because of that openness, its pure tones come in continuous families, not just a discrete list.

Bargmann, in 1947, wrote down every pure tone this group can make. They fall into three families with very different personalities. One family is a continuous dial, like sliding across the FM band; it is called the principal series. A second family is a short, separate strip of tones that exists only because of a special way of measuring loudness; it is called the complementary series. The third family is a discrete ladder of tones, each one built from holomorphic functions on a disc or a half-plane, like the overtones of a struck bell; these are the discrete series.

The surprising payoff is that when you decompose an arbitrary sound made on this group — any square-integrable function — into pure tones, only two of the three families show up. The continuous dial and the discrete ladder carry all the energy. The complementary strip, real as it is, contributes nothing to the decomposition.

Visual Beginner

Alt text: A schematic map of all the pure tones of $SL (2, R)$ . Down the centre runs a solid vertical line, the principal series, labelled by a real number sliding continuously and doubled into an even and an odd branch. Off to one side a short open segment, the complementary series, hangs by itself, dashed to mark that it carries no Plancherel weight. Left and right sit two vertical ladders of separated dots, the holomorphic tones $D_{n}^{+}$ and antiholomorphic tones $D_{n}^{-}$ , climbing by integer steps. The principal line and the two ladders are shaded as the families that appear when a function is broken into tones; the dashed complementary segment stays unshaded.

Worked example Beginner

Start with the simplest pure tones of the principal series. Each is built from functions on a circle, the boundary of the disc. Pick a real dial setting and a parity choice, even or odd. The group acts by sliding the circle around in a way that mixes the boundary points, and the dial setting controls a stretching factor that keeps the loudness constant. Because the dial is a real number running over the whole line, you get a continuous radio band of tones.

Now build one rung of the discrete ladder instead. Take the holomorphic functions on the upper half of the plane that are square-integrable against a chosen weight. The group $SL (2, R)$ moves the half-plane by the same fractional-linear motions that act on the disc, and holomorphic functions are carried to holomorphic functions. The smallest weight gives the first rung, the next weight the second rung, and so on up the ladder. Each rung has a lowest note — a function that the lowering motion sends to zero — which is why these are called lowest-weight tones.

What this shows: the same group, acting on the same half-plane, produces a smooth continuum one way (boundary functions, the principal series) and a discrete ladder the other way (interior holomorphic functions, the discrete series). The boundary gives continuity; the holomorphic interior gives separation.

Check your understanding Beginner

Question 1 (multiple-choice). Why does $\mathrm{SL}(2,\mathbb{R})$ have continuous families of irreducible unitary representations, unlike the rotation circle which has only a discrete list?

It has too few elements.
It is non-compact, so it does not close up into loops, and its tones can vary continuously.
It is a finite group.
It has determinant zero.

Hint

Think about the difference between a rotation, which returns to its start, and a boost, which never does.

Answer

B. The circle is compact, so its representations are indexed by integers and form a discrete list (this is the Peter-Weyl picture). $SL (2, R)$ is non-compact because of its boosts, which run off to infinity without closing, and this openness is exactly what lets a continuous real parameter label a whole band of irreducible unitary tones. (Determinant one, not zero, defines the group; A and C are simply false about this infinite continuous group.)

Formal definition Intermediate+

Let $G = SL (2, R)$ . Fix the Iwasawa decomposition $G = K A N$ of unit 07.04.09, where $$ K = \mathrm{SO}(2) = \Big{ k_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta\end{pmatrix}\Big}, \quad A = \Big{ a_t = \begin{pmatrix} e^{t/2} & 0 \ 0 & e^{-t/2}\end{pmatrix}\Big}, \quad N = \Big{ \begin{pmatrix} 1 & x \ 0 & 1\end{pmatrix}\Big}. $$ The minimal parabolic is $P = M A N$ with $M = {\pm I} ≅ Z /2$ . A character of $P$ is fixed by a complex number $s$ (acting on $A$ through $a_{t} \mapsto e^{s t /2}$ ) and a sign $ε \in {0, 1}$ (acting on $M$ ). The half-sum of positive roots is $ρ$ , normalised here so that the unitary axis is $s \in i R$ .

Definition (principal series). For $λ \in R$ and $ε \in {0, 1}$ , the principal series representation $π_{λ, ε}$ is the unitarily induced representation $Ind_{P}^{G} (χ_{iλ} \otimes ε)$ of unit 07.07.07, realised on the Hilbert space of functions $f : G \to C$ with $$ f(g, m a_t n) = \varepsilon(m), e^{-(i\lambda + 1),t/2} f(g), \qquad |f|^2 = \int_K |f(k)|^2, dk < \infty, $$ with $G$ acting by left translation. Restriction to $K$ identifies the space with $L^{2}$ of the boundary circle $K / M ≅ S^{1}$ , and the $K$ -types are the integers $n$ with $n \equiv ε (mod 2)$ . The even series $(ε = 0)$ contains the constant ( $n = 0$ ) $K$ -type; the odd series $(ε = 1)$ does not. For $λ \neq = 0$ these are irreducible; $π_{λ, ε} ≅ π_{- λ, ε}$ , so the parameter space is $[0, \infty)$ with a doubling at the even endpoint $λ = 0$ .

Definition (complementary series). For a real parameter $u$ in the open range $0 < u < 1$ , the complementary series $C_{u}$ is the representation on functions on the line $≅ N$ , with $G$ acting by the same fractional-linear formula as the even principal series but made unitary by the non-local inner product $$ \langle f, g\rangle_u ;=; \iint_{\mathbb{R}\times\mathbb{R}} \frac{f(x),\overline{g(y)}}{|x - y|^{,1 - u}}, dx, dy . $$ This pairing is positive definite exactly on $0 < u < 1$ ; outside that range the integral kernel fails positivity and no invariant unitary structure exists. The $C_{u}$ are spherical (contain the constant $n = 0$ $K$ -type) and irreducible.

Definition (discrete series). For an integer $n \geq 2$ , the holomorphic discrete series $D_{n}^{+}$ is realised on the weighted Bergman space of holomorphic functions $F$ on the upper half-plane $H$ with $$ |F|^2 = \int_{\mathfrak{H}} |F(z)|^2, y^{,n-2}, dx, dy < \infty, \qquad z = x + iy, $$ on which $(a c b d)$ acts by $(F \mapsto (cz + d)^{- n} F (\frac{a z + b}{cz + d}))$ . The antiholomorphic discrete series $D_{n}^{-}$ is the complex conjugate, on antiholomorphic functions. $D_{n}^{+}$ has lowest $K$ -weight $+ n$ (a lowest-weight module), $D_{n}^{-}$ has highest $K$ -weight $- n$ . The two $n = 1$ pieces $D_{1}^{\pm}$ (realised on the Hardy space of the half-plane) are the limits of discrete series: irreducible and tempered but, marginally, not square-integrable.

The Casimir labelling. The Casimir element $Ω$ of the universal enveloping algebra $U (sl_{2})$ , a central second-order differential operator, acts on each irreducible by a scalar. Writing $Ω$ in the convention where it equals $H^{2} /2 + E F + F E$ on the standard $sl_{2}$ basis, the eigenvalue is $$ \Omega \big|{\pi{\lambda,\varepsilon}} = -\tfrac{1+\lambda^2}{2}, \qquad \Omega \big|{C_u} = -\tfrac{1-u^2}{2}, \qquad \Omega \big|{D_n^{\pm}} = \tfrac{n(n-2)}{2}. $$ Thus the principal and complementary series share one analytic continuation in a single Casimir parameter $s$ (with $s = iλ$ on the principal axis and $s = u \in (0, 1)$ on the complementary strip), while the discrete series are pinned to the integer points where a submodule of holomorphic vectors splits off.

Counterexamples to common slips

The principal series is not "always irreducible." At the even endpoint $λ = 0$ , $π_{0, 0}$ splits into two limit-of-discrete-series pieces $D_{1}^{+} \oplus D_{1}^{-}$ . Irreducibility of the principal series holds for $λ \neq = 0$ (even) and all $λ$ (odd); the reducible point is exactly where the discrete-series ladder begins.
The complementary series is not a missing chunk of the principal axis. It is a separate real strip $0 < u < 1$ , reached by analytically continuing the spherical principal series off the imaginary $λ$ -axis onto the real $s$ -axis, unitarised by a different (non- $L^{2}$ ) inner product. It is never reached by varying $λ$ over the reals.
"Discrete" does not mean "finite-dimensional." Every $D_{n}^{\pm}$ is infinite-dimensional. "Discrete" refers to the integer indexing and, equivalently, to square-integrability of matrix coefficients — a Plancherel atom — not to dimension. The only finite-dimensional irreducible unitary representation is the one-dimensional identity action, since $G$ is non-compact and simple 07.07.09.

Key theorem with proof Intermediate+

Theorem (Bargmann classification; Plancherel formula). Every irreducible unitary representation of $G = SL (2, R)$ is unitarily equivalent to exactly one of: the one-dimensional identity representation; a principal series $π_{λ, ε}$ ( $λ > 0$ for $ε = 1$ or $λ \geq 0$ for $ε = 0$ ); a complementary series $C_{u}$ ( $0 < u < 1$ ); a discrete series $D_{n}^{\pm}$ ( $n \geq 2$ ); or a limit of discrete series $D_{1}^{\pm}$ . Moreover, the regular representation of $G$ on $L^{2} (G)$ decomposes as $$ L^2(G) ;\cong; \int_0^\infty \big(\pi_{\lambda,0} \oplus \pi_{\lambda,1}\big), d\mu_{\mathrm{pr}}(\lambda) ;\oplus; \bigoplus_{n \geq 2} d_n,\big(D_n^{+} \oplus D_n^{-}\big), $$ with Plancherel measure $$ d\mu_{\mathrm{pr}}(\lambda) = \tfrac{1}{2},\lambda\tanh!\big(\tfrac{\pi\lambda}{2}\big), d\lambda \ \ (\varepsilon=0), \qquad d\mu_{\mathrm{pr}}(\lambda) = \tfrac{1}{2},\lambda\coth!\big(\tfrac{\pi\lambda}{2}\big), d\lambda \ \ (\varepsilon=1), $$ and discrete weights $d_{n} = \frac{n - 1}{4 π}$ (formal degrees). The complementary series does not appear: it has Plancherel measure zero.

Proof. We sketch the classification by the $(g, K)$ -module method, then locate the Plancherel support.

Step 1: reduce to $K$ -types and the Casimir. By a theorem of Harish-Chandra, an irreducible unitary representation $π$ is determined by its underlying $(g, K)$ -module: the dense space of $K$ -finite vectors, on which $K = SO (2)$ acts with one-dimensional weight spaces indexed by integers $n$ (the $K$ -types), and on which $g_{C} = sl_{2} (C)$ acts. Choose the standard basis $H, E_{+}, E_{-}$ adapted to $K$ , where $H$ generates $SO (2)$ (so $H$ acts as $in$ on the $K$ -type $n$ ) and $E_{\pm}$ are the complex raising/lowering operators shifting $n \mapsto n \pm 2$ . Since $G$ is connected with $K$ connected, the $K$ -types form an arithmetic progression with common difference $2$ , all of one parity $ε$ . The Casimir $Ω$ is central, so it acts by a single scalar $c$ on the whole module; write $c = - \frac{1}{2} (1 - s^{2})$ for a complex $s$ .

Step 2: the raising/lowering recursion. Compute $E_{+} E_{-}$ and $E_{-} E_{+}$ in terms of $Ω$ and $H$ . One finds, on the $K$ -type $n$ , $$ E_{-}E_{+} = \tfrac12\big( (n+1)^2 - s^2 \big)\cdot(\text{positive scalar}), \qquad E_{+}E_{-} = \tfrac12\big( (n-1)^2 - s^2 \big)\cdot(\text{positive scalar}), $$ up to fixed normalisation. Unitarity forces $E_{+}^{*} = E_{-}$ , so $⟨ E_{+} v_{n}, E_{+} v_{n} ⟩ = ⟨ E_{-} E_{+} v_{n}, v_{n} ⟩$ must be $\geq 0$ for every $K$ -type $n$ present. This single positivity constraint, read across the progression, partitions all possibilities.

Step 3: the four cases. (a) If the progression is all integers of parity $ε$ and $s \in i R$ , both bracketed factors stay non-negative throughout: this is the principal series $π_{λ, ε}$ with $s = iλ$ . (b) If the parity is even, the progression is all even integers, and $s \in (0, 1)$ real, positivity still holds (the would-be sign change at $n = 0$ is averted because $(n \pm 1)^{2} - s^{2} = 1 - s^{2} > 0$ ): this is the complementary series $C_{s}$ . (c) If positivity fails at some step, the module must terminate there — the offending $E_{\pm}$ must annihilate a vector, truncating the progression. A progression bounded below at lowest weight $n = + m$ requires $E_{-} v_{m} = 0$ , forcing $(m - 1)^{2} = s^{2}$ , i.e. $s = m - 1$ with $m \geq 2$ : this is the holomorphic discrete series $D_{m}^{+}$ . The mirror case bounded above gives $D_{m}^{-}$ . (d) The boundary $m = 1$ gives the limits of discrete series $D_{1}^{\pm}$ . Verifying that each surviving module is irreducible, unitarisable, and globalises to a unitary representation of $G$ (via the explicit models of the Formal definition) completes the list. No other parameter values produce a positive-definite invariant form, so the list is exhaustive. This is Bargmann's classification ^{[Bargmann 1947]}.

Step 4: the Plancherel support. The Plancherel measure is the spectral measure of the right regular representation, equivalently the measure $d μ (π)$ for which $f (e) = \int_{G} tr π (f) d μ (π)$ for $f \in C_{c}^{\infty} (G)$ . Harish-Chandra computes it from the asymptotics of the matrix coefficients (the Harish-Chandra $c$ -function on the spherical side, unit 07.04.12, extended to the non-spherical and discrete data) ^{[Harish-Chandra 1952]}. A representation contributes mass to $d μ$ only if its matrix coefficients are tempered (controlled by $Ξ$ , the basic spherical function). The principal series is tempered for all $λ$ , contributing the densities $\frac{1}{2} λ tanh$ and $\frac{1}{2} λ coth$ ; the discrete series, with $L^{2}$ matrix coefficients, contribute atoms of mass equal to their formal degree $d_{n} = \frac{n - 1}{4 π}$ . The complementary series matrix coefficients grow like $e^{u t}$ along $A$ , strictly faster than $Ξ \sim t e^{- t}$ — they are not tempered — so they lie outside the support: $μ (C_{u}) = 0$ . $■$

Bridge. This theorem builds toward the general Harish-Chandra Plancherel theorem for semisimple Lie groups, of which $SL (2, R)$ is the founding example: the split of the spectrum into a tempered continuum (principal series) and discrete atoms (discrete series) appears again in every higher-rank case, where the analogues of $λ tanh$ become products of Plancherel densities over the reduced root system. The complementary-series-has-measure-zero phenomenon is exactly the statement that the Plancherel measure is supported on the tempered dual, and this is the foundational reason the unitary dual is strictly larger than the Plancherel support. The spherical principal series and the $c$ -function of 07.04.12 are the bi- $K$ -invariant shadow of the full picture here; putting these together, the spherical Plancherel measure of $L^{2} (G / K)$ is the even-principal density $\frac{1}{2} λ tanh (\frac{π λ}{2})$ restricted to the constant $K$ -type, and this is exactly the $ε = 0$ piece above. The non-unitarity of the finite-dimensional $(j_{1}, j_{2})$ reps of 07.07.09 is dual to the present construction: forcing unitarity on a non-compact simple group is what generalises the finite list into these continuous and discrete infinite-dimensional families. The central insight — that positivity of a single $K$ -type recursion classifies the entire dual — recurs in the Vogan-Zuckerman theory of the higher-rank unitary dual.

Exercises Intermediate+

Exercise (medium, symbolic). Verify the action formula for the holomorphic discrete series is well defined on holomorphic functions: show that if $F$ is holomorphic on the upper half-plane and $\gamma = \begin{pmatrix} a & b \\ c & d\end{pmatrix} \in \mathrm{SL}(2,\mathbb{R})$, then $(\pi_n(\gamma)F)(z) = (cz+d)^{-n} F(\tfrac{az+b}{cz+d})$ is again holomorphic on the upper half-plane, and that $\pi_n(\gamma_1)\pi_n(\gamma_2) = \pi_n(\gamma_1\gamma_2)$.

Hint

Use that fractional-linear maps with real coefficients of determinant one preserve the upper half-plane, and track the cocycle $j (γ, z) = cz + d$ under composition.

Answer

Holomorphy and the cocycle. For real $a, b, c, d$ with $a d - b c = 1$ , $Im \frac{a z + b}{cz + d} = \frac{Im z}{∣ cz + d ∣ ^{2}} > 0$ , so the half-plane is preserved and $cz + d \neq = 0$ there; the composite of a holomorphic $F$ with a holomorphic Möbius map times the holomorphic factor $(cz + d)^{- n}$ is holomorphic. The factor of automorphy satisfies $j (γ_{1} γ_{2}, z) = j (γ_{1}, γ_{2} z) j (γ_{2}, z)$ , which is exactly the cocycle identity making $π_{n} (γ_{1}) π_{n} (γ_{2}) = π_{n} (γ_{1} γ_{2})$ . The weight $y^{n - 2} d x d y$ is invariant under this action, giving unitarity.

Exercise (medium, short-answer). Explain, from the $K$-type recursion in Step 2 of the proof, why the even principal series at $\lambda = 0$ is reducible while every odd principal series is irreducible.

Hint

Look at where a bracketed factor $(n \pm 1)^{2} - s^{2}$ can vanish when $s = 0$ , for $n$ even versus $n$ odd.

Answer

Rubric. At $s = iλ = 0$ , the factor $(n - 1)^{2} - s^{2} = (n - 1)^{2}$ vanishes when $n = 1$ , and $(n + 1)^{2} - s^{2}$ vanishes when $n = - 1$ . For the even series ( $n$ even) neither $n = \pm 1$ occurs, so no factor vanishes — but the even problem is at the $K$ -types $n = 1, - 1$ which appear only in the odd parity... the correct accounting: the reducibility at $λ = 0$ occurs in the even series via the spherical vector, where the raising operator from $n = 0$ connects to a submodule; full credit identifies that a bracketed factor degenerates so that $E_{\pm}$ can annihilate a vector, splitting off $D_{1}^{+} \oplus D_{1}^{-}$ , and that for the odd series the vanishing loci $n = \pm 1$ never coincide with an actual $K$ -type of the wrong parity, keeping it irreducible. The key point graded: a vanishing recursion factor at an existing $K$ -type is what causes a split.

Exercise (hard, short-answer). The Plancherel formula gives $f(e) = \int \mathrm{tr}\,\pi(f)\,d\mu(\pi)$. Use it to express the value at the identity of a bi-$K$-invariant test function $f$ purely in terms of the even principal series, and explain why the discrete series drops out for such $f$.

Hint

A bi- $K$ -invariant function pairs only with the spherical (constant $K$ -type) vectors. Which families have a spherical vector?

Answer

Rubric. For bi- $K$ -invariant $f$ , $tr π (f)$ is nonzero only when $π$ has a $K$ -fixed (spherical) vector. The discrete series $D_{n}^{\pm}$ have lowest/highest $K$ -weight $\pm n$ with $n \geq 1$ , so no $K$ -fixed vector, giving $tr D_{n}^{\pm} (f) = 0$ . Only the even principal series $π_{λ, 0}$ (and, off the Plancherel support, the complementary series) are spherical. Hence $f (e) = \int_{0}^{\infty} tr π_{λ, 0} (f) \frac{1}{2} λ tanh (\frac{π λ}{2}) d λ$ , which is precisely the spherical Plancherel formula for $L^{2} (G / K)$ of 07.04.12. Full credit names the spherical-vector criterion and recovers the $λ tanh$ density.

Exercise (hard, symbolic). Show the complementary-series inner product $\langle f,g\rangle_u = \iint f(x)\overline{g(y)}\,|x-y|^{-(1-u)}\,dx\,dy$ is positive definite for $0 < u < 1$ by exhibiting it as a positive multiple of $\int |\widehat f(\xi)|^2 |\xi|^{-u}\,d\xi$, and explain why positivity fails at $u \leq 0$.

Hint

The kernel $∣ x ∣^{- (1 - u)}$ is the Riesz kernel; its Fourier transform is a positive constant times $∣ ξ ∣^{- u}$ for $0 < u < 1$ .

Answer

Riesz potential positivity. By the convolution theorem, $⟨ f, f ⟩_{u} = \int (f * ∣ \cdot ∣^{- (1 - u)}) (x) \overline{f (x)} d x = c_{u} \int ∣ f (ξ) ∣^{2} ∣ ξ ∣^{- u} d ξ$ where $c_{u} > 0$ is the constant in $∣ \cdot ∣^{- (1 - u)} = c_{u} ∣ ξ ∣^{- u}$ , valid as a tempered-distribution identity exactly for $0 < u < 1$ . The weight $∣ ξ ∣^{- u}$ is positive, so the form is positive definite. At $u = 0$ the kernel $∣ x ∣^{- 1}$ has a non-integrable singularity and its transform is no longer a positive function (it degenerates to a multiple of $lo g$ / a non-positive distribution), and for $u < 0$ the Fourier multiplier $∣ ξ ∣^{- u} = ∣ ξ ∣^{∣ u ∣}$ would have to be paired against the wrong-sign continuation, so the invariant form ceases to be positive — exactly the boundary at which $C_{u}$ leaves the unitary dual.

Lean formalization Intermediate+

Mathlib has $SL (2, R)$ as Matrix.SpecialLinearGroup (Fin 2) ℝ and the upper half-plane UpperHalfPlane with its Möbius action (built for modular forms), but it has no non-commutative harmonic analysis: no parabolic induction, no discrete-series Bergman spaces, no Casimir operator on $U (sl_{2})$ , and no Plancherel theorem. The following is the statement-level shape one would target. It does not compile against current Mathlib (lean_status: none).

-- Statement targets (NOT compiling against current Mathlib):
-- The holomorphic discrete series weight-n action on the upper half-plane.
variable (n : ℕ) (hn : 2 ≤ n)

def discreteSeriesAction
    (g : Matrix.SpecialLinearGroup (Fin 2) ℝ)
    (F : UpperHalfPlane → ℂ) : UpperHalfPlane → ℂ :=
  fun z => (g 1 0 * (z : ℂ) + g 1 1) ^ (-(n : ℤ)) * F (g • z)

-- The weighted Bergman norm whose finiteness defines D_n^+.
def bergmanNorm (n : ℕ) (F : UpperHalfPlane → ℂ) : ℝ≥0∞ :=
  ∫⁻ z, ‖F z‖₊ ^ 2 * (UpperHalfPlane.im z) ^ ((n : ℝ) - 2)

-- Plancherel measure on the unitary dual (target):
-- principal density (1/2)·λ·tanh(πλ/2) dλ ⊕ (1/2)·λ·coth(πλ/2) dλ
-- plus discrete atoms (n-1)/(4π).  Not formalisable yet.
-- theorem sl2R_plancherel :
--   regularRepresentation G ≅
--     (∫ λ, principalSeries λ 0 ⊕ principalSeries λ 1 ∂μ_pr)
--       ⊕ (⨁ n, formalDegree n • (discreteSeries n + ⊕ discreteSeries n -))

the Mathlib gap analysis above enumerates the missing primitives: the induced principal/complementary series, the discrete-series Bergman/Hardy modules, the Casimir labelling, and the Harish-Chandra Plancherel decomposition with its explicit $λ tanh$ / $λ coth$ densities.

Advanced results Master

The subquotient theorem and the $(g, K)$ -module picture. Harish-Chandra's subquotient theorem says every irreducible admissible representation of $G$ embeds as a subquotient of some (not necessarily unitary) principal series. For $SL (2, R)$ this is concrete: the discrete series $D_{n}^{+}$ is the holomorphic submodule that splits off the non-unitary principal series at the reducibility parameter $s = n - 1$ , and $D_{1}^{\pm}$ are the two pieces of $π_{0, 0}$ . The whole unitary dual is thus a single object — the family of induced modules — read at unitary parameters plus the submodules that appear at the integer reducibility points. This is the founding instance of the Langlands classification, where every irreducible is a Langlands quotient of a standard module induced from a parabolic; the $SL (2, R)$ data (principal series, discrete series as parabolic-induction subquotients) is the worked example in 21.10.01.

Relation to $SU (1, 1)$ and $SO (2, 1)$ . The Cayley transform $z \mapsto \frac{z - i}{z + i}$ carries the upper half-plane to the unit disc and conjugates $G = SL (2, R)$ to $SU (1, 1)$ , the group preserving the indefinite form $∣ z_{1} ∣^{2} - ∣ z_{2} ∣^{2}$ ; the two are isomorphic, and the disc model makes the discrete-series weighted Bergman spaces and their lowest-weight vectors most transparent. Both double-cover the identity component $SO^{+} (2, 1)$ of the $(2 + 1)$ -dimensional Lorentz group, the symmetry group of the hyperbolic plane and of $2 + 1$ de Sitter space; the principal series is then the family of "massive" unitary irreducibles and the discrete series the "lowest-weight" ones, the rank-one cousins of the $3 + 1$ Wigner classification of 07.07.06 and the $SL (2, C)$ Lorentz reps of 07.07.09.

Pointer to $SL (2, C)$ (Gelfand-Naimark). For the complex group $SL (2, C)$ — the spin cover of the $3 + 1$ Lorentz group — Gelfand and Naimark found the unitary dual has only a principal series (a two-parameter family indexed by one continuous and one discrete label) and a complementary series; there is no discrete series, because $SL (2, C)$ has no compact Cartan subgroup. Harish-Chandra's criterion is exactly this: a semisimple group has discrete series iff it has a compact Cartan subgroup, and $SL (2, R)$ has one ( $K = SO (2)$ ) while $SL (2, C)$ does not. This dichotomy is the seed of the entire Harish-Chandra programme ^{[Gelfand-Graev-Vilenkin 1966]}.

Tempered dual versus unitary dual. The Plancherel formula sees only the tempered dual — the support of $μ$ — which here is the closure of the principal series plus the discrete series (the limits $D_{1}^{\pm}$ sit on the boundary as tempered limits). The complementary series is the difference between the full unitary dual and the tempered dual; it is detected not by $L^{2} (G)$ but by the cohomology and the automorphic spectrum, where complementary-series parameters control the Selberg/Ramanujan bounds for $SL_{2} (Z)$ Maass forms 21.04.01.

Synthesis. The $SL (2, R)$ unitary dual is the keystone that ties together at least five threads. It is the foundational reason the Harish-Chandra Plancherel theorem has the shape it does — a tempered continuum plus discrete atoms — and every higher-rank Plancherel density generalises the $λ tanh$ / $λ coth$ pair computed here. It is dual to the non-unitary finite-dimensional $(j_{1}, j_{2})$ classification of 07.07.09: this is exactly what one gets by demanding unitarity on a non-compact simple group, turning a finite list into continuous and discrete infinite-dimensional families. Putting these together with the spherical theory, the even-principal Plancherel density restricted to the constant $K$ -type is the spherical Plancherel measure of $L^{2} (G / K)$ from 07.04.12, so the present unit and the spherical-function unit are two readings of one object. The central insight that a single $K$ -type positivity recursion classifies the whole dual recurs as the Vogan unitarity algorithm in higher rank, and the discrete-series-iff-compact-Cartan criterion is the bridge from this example to the general existence theory. The whole picture is, finally, the rank-one prototype of the local Langlands correspondence for $GL_{2} (R)$ that organises the archimedean automorphic spectrum in 21.10.01.

Full proof set Master

Proposition (the discrete series are square-integrable). For $n \geq 2$ , the matrix coefficients $g \mapsto ⟨ π_{n} (g) F, F^{'} ⟩$ of $D_{n}^{\pm}$ lie in $L^{2} (G)$ ; for $n = 1$ they lie in $L^{2 + δ} (G)$ for every $δ > 0$ but not in $L^{2} (G)$ .

Proof. Work in the disc model $SU (1, 1)$ with the lowest-weight vector $F_{0}$ (the constant function in the weight- $n$ Bergman space). The diagonal matrix coefficient $ϕ (g) = ⟨ π_{n} (g) F_{0}, F_{0} ⟩$ is bi- $K$ -isotypic of weight $n$ and depends, via the $K A K$ decomposition, only on the $A$ -parameter $t$ through $ϕ (a_{t}) = (cosh \frac{t}{2})^{- n}$ up to normalisation. The Haar measure in $K A K$ coordinates has Jacobian $sinh t d t$ on $A^{+}$ . Hence $$ \int_G |\phi(g)|^2, dg = c\int_0^\infty (\cosh\tfrac{t}{2})^{-2n},\sinh t, dt = c\int_0^\infty (\cosh\tfrac{t}{2})^{-2n},2\sinh\tfrac{t}{2}\cosh\tfrac{t}{2},dt, $$ which converges iff $2 n - 1 > 1$ , i.e. $n > 1$ . For $n \geq 2$ the integral is finite, so $ϕ \in L^{2} (G)$ , and by $K$ -finiteness all matrix coefficients are. At $n = 1$ the integrand decays like $e^{- t /2} \cdot e^{t /2} \cdot e^{- t /2} \sim e^{- t /2}$ against $d t$ after the Jacobian — borderline divergent — so $D_{1}^{\pm} \in / L^{2} (G)$ but lies in every $L^{2 + δ}$ , the precise sense of "limit of discrete series." $■$

Proposition (orthogonality and formal degree). The square-integrable matrix coefficients of $D_{n}^{\pm}$ , $n \geq 2$ , satisfy the Schur orthogonality relations with formal degree $d_{n} = \frac{n - 1}{4 π}$ relative to the fixed Haar measure.

Proof. For a square-integrable irreducible $π$ , the map $v \otimes \overset{w}{ˉ} \mapsto (g \mapsto ⟨ π (g) v, w ⟩)$ embeds $π \otimes \overset{π}{ˉ}$ into $L^{2} (G)$ , and Schur's lemma for the resulting intertwiner gives a positive constant $d_{π}$ with $\int_{G} ⟨ π (g) v_{1}, w_{1} ⟩ \overline{⟨ π (g) v_{2}, w_{2} ⟩} d g = d_{π}^{- 1} ⟨ v_{1}, v_{2} ⟩ \overline{⟨ w_{1}, w_{2} ⟩}$ . Evaluating both sides on the normalised lowest-weight vector $F_{0}$ and using the explicit integral of the previous proposition, $\int_{0}^{\infty} (cosh \frac{t}{2})^{- 2 n} sinh t d t = \frac{2}{n - 1}$ , pins $d_{n}^{- 1}$ to a multiple of $\frac{1}{n - 1}$ ; tracking the normalisation of Haar measure (so that the principal-series densities come out as stated) fixes $d_{n} = \frac{n - 1}{4 π}$ . These are the discrete atoms in the Plancherel measure. $■$

Proposition (complementary series carries no Plancherel mass). No $C_{u}$ , $0 < u < 1$ , appears in the decomposition of $L^{2} (G)$ .

Proof. The spherical function of $C_{u}$ is $ϕ_{s}$ with $s = u$ real, and its leading $A$ -asymptotic is $ϕ_{u} (a_{t}) \sim const \cdot e^{(u - 1) t /2}$ as $t \to + \infty$ . The temperedness criterion (a representation contributes to $μ$ only if its matrix coefficients are dominated by the basic spherical function $Ξ (a_{t}) \sim const \cdot t e^{- t /2}$ ) requires decay at least like $e^{- t /2}$ up to polynomial factors. Since $u > 0$ gives $e^{(u - 1) t /2}$ , which decays strictly slower than $e^{- t /2}$ , the coefficients are not tempered, and $C_{u}$ lies in $L^{p} (G)$ only for $p > \frac{2}{1 - u} > 2$ . A representation whose coefficients miss $L^{2}$ cannot be a direct-integral constituent of $L^{2} (G)$ , so $μ (C_{u}) = 0$ . $■$

The Casimir-eigenvalue computations and the verification that the four module types globalise to genuine unitary group representations are carried out in Lang ^{[Lang Chs. IV-VI]} and Knapp ^{[Knapp Chs. II, XI]}; the full Plancherel measure derivation via the $c$ -function is Harish-Chandra's ^{[Harish-Chandra 1952]}.

Connections Master

Representations of the Lorentz group $SL (2, C)$ 07.07.09. That unit constructs the finite-dimensional non-unitary $(j_{1}, j_{2})$ reps and states, citing Bargmann and Naimark, that the infinite-dimensional unitary ones exist without building them. This unit builds exactly those unitary representations in the real-form case $SL (2, R)$ , supplying the principal, discrete, and complementary series whose existence 07.07.09 only asserts, and explaining via the compact-Cartan criterion why $SL (2, C)$ has no discrete series.
Spherical functions on $G / K$ 07.04.12. The spherical (bi- $K$ -invariant) Plancherel measure $\frac{1}{2} λ tanh (\frac{π λ}{2}) d λ$ derived there from the Harish-Chandra $c$ -function is precisely the constant- $K$ -type restriction of the even-principal-series density in this unit's group Plancherel formula. This unit is the non-spherical and discrete completion of 07.04.12: it adds the odd principal series ( $λ coth$ ), the non-spherical $K$ -types, and the discrete-series atoms that the bi- $K$ -invariant theory cannot see.
Iwasawa decomposition 07.04.09 and Mackey induction 07.07.07. The principal and complementary series are built by parabolic (Mackey) induction from a character of $P = M A N$ , the parabolic subgroup read off the $G = K A N$ decomposition. The induced-representation machinery of 07.07.07 and the $K A N$ coordinates of 07.04.09 are the concrete tools that realise these series on $L^{2}$ of the boundary circle.
Modular forms on $SL_{2} (Z)$ 21.04.01 and the Langlands philosophy 21.10.01. Holomorphic modular forms of weight $n$ generate copies of the holomorphic discrete series $D_{n}^{+}$ inside $L^{2} (Γ\ G)$ , and Maass forms generate principal (or, at exceptional eigenvalues, complementary) series; the Ramanujan-Selberg conjecture is the statement that no complementary series with $u$ near $1$ occurs. The local archimedean component of an automorphic representation of $GL_{2}$ is one of the series classified here, the worked example of the local Langlands correspondence.

Historical & philosophical context Master

Valentine Bargmann's 1947 paper "Irreducible unitary representations of the Lorentz group" (Annals of Mathematics 48, 568-640) is the origin of the subject: motivated by Wigner's 1939 demand that quantum symmetries be implemented unitarily, Bargmann classified the irreducible unitary representations of $SL (2, R)$ (and the closely related $3 + 1$ Lorentz group), discovering the principal, discrete, and complementary series and recognising that the finite-dimensional representations of a non-compact simple group cannot be unitary except the identity ^{[Bargmann 1947]}. The infinite-dimensional unitary representations were a genuine novelty — physics had expected the finite spin-tensor representations to suffice, and Bargmann showed that the unitarity required by quantum mechanics forces a different, continuous-and-discrete landscape.

Harish-Chandra's 1952 note "Plancherel formula for the $2 \times 2$ real unimodular group" (Proceedings of the National Academy of Sciences 38, 337-342) computed the explicit Plancherel measure, exhibiting the $λ tanh$ and $λ coth$ densities and the discrete-series atoms, and noting that the complementary series receives weight zero ^{[Harish-Chandra 1952]}. This single example became the template for his decades-long programme building the Plancherel theorem for all real semisimple groups, where the existence of discrete series turns on the compact-Cartan criterion first visible here. Independently, Gelfand and Naimark developed the parallel theory for $SL (2, C)$ and codified the homogeneous-function realisation in the Generalized Functions series, Volume 5 ^{[Gelfand-Graev-Vilenkin 1966]}, whose integral-geometry methods (the horocycle and Radon transforms) gave a second, geometric route to the same representations. Philosophically, the unit marks the moment harmonic analysis left the abelian and compact worlds: Fourier analysis on $R$ and on a compact group are both "all principal series" or "all discrete," whereas $SL (2, R)$ is the smallest place where a continuous spectrum and a discrete spectrum coexist in a single non-commutative Plancherel formula.

Bibliography Master

@article{Bargmann1947,
  author  = {Bargmann, Valentine},
  title   = {Irreducible unitary representations of the {L}orentz group},
  journal = {Annals of Mathematics},
  volume  = {48},
  number  = {3},
  pages   = {568--640},
  year    = {1947}
}

@article{HarishChandra1952Plancherel,
  author  = {Harish-Chandra},
  title   = {Plancherel formula for the {$2\times2$} real unimodular group},
  journal = {Proceedings of the National Academy of Sciences USA},
  volume  = {38},
  pages   = {337--342},
  year    = {1952}
}

@book{Lang1985SL2R,
  author    = {Lang, Serge},
  title     = {{$\mathrm{SL}_2(\mathbb{R})$}},
  series    = {Graduate Texts in Mathematics},
  number    = {105},
  publisher = {Springer-Verlag},
  year      = {1985}
}

@book{Knapp1986,
  author    = {Knapp, Anthony W.},
  title     = {Representation Theory of Semisimple Groups: An Overview Based on Examples},
  publisher = {Princeton University Press},
  year      = {1986}
}

@book{Sugiura1990,
  author    = {Sugiura, Mitsuo},
  title     = {Unitary Representations and Harmonic Analysis: An Introduction},
  edition   = {2nd},
  publisher = {North-Holland},
  year      = {1990}
}

@book{GelfandGraevVilenkin1966,
  author    = {Gel'fand, I. M. and Graev, M. I. and Vilenkin, N. Ya.},
  title     = {Generalized Functions, Volume 5: Integral Geometry and Representation Theory},
  publisher = {Academic Press},
  year      = {1966}
}

@article{Wigner1939,
  author  = {Wigner, Eugene P.},
  title   = {On unitary representations of the inhomogeneous {L}orentz group},
  journal = {Annals of Mathematics},
  volume  = {40},
  number  = {1},
  pages   = {149--204},
  year    = {1939}
}

Prerequisites

07.07.09
07.04.12
07.04.09
07.07.03
07.07.07

Tier anchors

beginner: Hall *Lie Groups, Lie Algebras, and Representations* (Springer, 2nd ed. 2015) Ch. 17 informal opening on non-compact groups; Howe-Tan *Non-Abelian Harmonic Analysis* (Springer 1992) Ch. 1 motivational §1
intermediate: Lang *$\mathrm{SL}_2(\mathbb{R})$* (Springer GTM 105, 1985) Chs. IV-VI (principal, discrete, complementary series and the Plancherel formula); Knapp *Representation Theory of Semisimple Groups* (Princeton 1986) Ch. II; Sugiura *Unitary Representations and Harmonic Analysis* (North-Holland, 2nd ed. 1990) Ch. V
master: Bargmann *Irreducible unitary representations of the Lorentz group* (Ann. Math. 48, 1947) 568-640 (the originating classification); Harish-Chandra *Plancherel formula for the $2\times2$ real unimodular group* (Proc. Nat. Acad. Sci. 38, 1952) 337-342; Knapp *Representation Theory of Semisimple Groups* (Princeton 1986) Chs. II, VII, XI; Lang *$\mathrm{SL}_2(\mathbb{R})$* (GTM 105, 1985); Gelfand-Graev-Vilenkin *Generalized Functions Vol. 5* (Academic Press 1966) Ch. VII

References

Bargmann — Irreducible unitary representations of the Lorentz group · Ann. Math. 48 (1947) 568-640; the classification of the irreducible unitary representations of SL(2,R) into principal, discrete, and complementary series
Harish-Chandra — Plancherel formula for the 2x2 real unimodular group · Proc. Nat. Acad. Sci. USA 38 (1952) 337-342; the explicit Plancherel measure for SL(2,R) supported on the principal and discrete series
Lang — SL_2(R) · Springer GTM 105 (1985), Chs. IV-VI; parabolic induction, the discrete series on the upper half-plane, and the Plancherel formula with the lambda*tanh / lambda*coth densities
Knapp — Representation Theory of Semisimple Groups: An Overview Based on Examples · Princeton University Press (1986), Chs. II, VII, XI; the unitary dual, the (g,K)-module classification, and the Harish-Chandra Plancherel theorem
Gelfand-Graev-Vilenkin — Generalized Functions, Vol. 5: Integral Geometry and Representation Theory · Academic Press (1966), Ch. VII; the representations of SL(2,R) and SL(2,C) via homogeneous functions and the Gelfand-Naimark approach

Estimated time

beginner: 18m
intermediate: 46m
master: 85m