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Matrix-coefficient formulas and decay for the discrete and principal series

Statement

Assume the Axiom of Choice (The Axiom of Choice).

(a) Discrete series. Let $n \geq 2$ , let $u_{n} = f_{n, 0} / ∥ f_{n, 0} ∥_{n} \in D_{n}^{-}$ be the normalized extremal vector of Holomorphic and antiholomorphic discrete-series models, and let $c_{n} (g) = ⟨ π_{n} (g) u_{n}, u_{n} ⟩$ be its matrix coefficient (Matrix coefficient of a unitary representation). Then for all $τ \in R$ , $c_{n} (a_{τ}) = \cosh (τ / 2)^{- n} .$ Moreover, for every $X, Y \in U (g_{C})$ there is $C = C (n, X, Y) > 0$ such that $∣ ⟨ π_{n} (g) X u_{n}, Y u_{n} ⟩ ∣ \leq C e^{- n ∣ τ ∣ / 2} (g = k_{1} a_{τ} k_{2}) .$

(b) Limits and unitary principal series. In the compact picture of $I_{1, 0}$ (The compact picture of the SL2(R) principal series, Unitarity of the unitary principal series, The two limits of discrete series), let $Π_{0}$ be the unitary action on $L_{1}^{2} (K)$ and $f_{1} (k_{θ}) = e^{i θ}$ the normalized weight-one vector. Then $⟨ Π_{0} (a_{τ}) f_{1}, f_{1} ⟩ = sech (τ / 2) = \frac{2}{e^{τ / 2} + e^{- τ / 2}} .$ Consequently $\int_{0}^{\infty} ∣ ⟨ Π_{0} (a_{τ}) f_{1}, f_{1} ⟩ ∣^{2} \sinh τ d τ = + \infty$ , and this matrix coefficient is not in $L^{2} (G)$ by the KAK formula (KAK integration formula for K-bi-invariant functions on SL2(R)).

More generally, for every odd integer $m$ and every real spectral parameter $s$ , let $c_{m, s} (τ) = ⟨ Π_{i s} (a_{τ}) f_{m}, f_{m} ⟩$ in the unitary compact picture of $I_{1, i s}$ . Then for an absolute constant $C > 0$ , $∣ c_{m, s} (τ) ∣ \leq C (1 + ∣ τ ∣) e^{- ∣ τ ∣ / 2} (τ \in R) .$

Facts & Assumptions

Given: AC; the holomorphic and antiholomorphic models and their K-finite vectors; the weighted disk norm; the odd compact-picture basis; and the unitary principal-series action.

[F1]

The model action is unitary for the weighted inner product, and the vectors $f_{n, j}$ are nonzero, mutually orthogonal $K$ -eigenvectors with characters $e^{- i (n + 2 j) θ}$ (Holomorphic and antiholomorphic discrete-series models, The weighted area form is SL2(R)-invariant, The weighted discrete-series space is a Hilbert space with K-type basis).

[F2]

The model vectors are smooth K-eigenvectors with the displayed raising/lowering actions; every element of $U (g_{C})$ sends $u_{n}$ to a finite sum of these K-types (Smooth and K-finite vectors for SL2(R), and the (g,K)-module, Holomorphic and antiholomorphic discrete-series models).

[F3]

The discrete-series model is unitary and strongly continuous, and its matrix coefficient is defined by the first-variable-linear Hilbert pairing (The weighted area form is SL2(R)-invariant, Matrix coefficient of a unitary representation).

[F4]

In the compact picture, for $k g = a_{t} n_{x} k_{ψ}$ , the action is $(Π_{ν} (g) f) (k) = e^{(1 + ν) t / 2} f (k_{ψ})$ ; the odd Fourier vectors $f_{m} (k_{θ}) = e^{i m θ}$ form an orthonormal basis (The compact picture of the SL2(R) principal series, K-type decomposition of the SL2(R) principal series).

[F5]

For $ν \in i R$ , the compact-picture action is a strongly continuous unitary representation; its matrix coefficients use the same L2 pairing (Unitarity of the unitary principal series, Matrix coefficient of a unitary representation).

[F6]

For continuous nonnegative K-bi-invariant functions, $\int_{G} ψ (g) d g = 2 π \int_{0}^{\infty} ψ (a_{τ}) \sinh τ d τ$ (KAK integration formula for K-bi-invariant functions on SL2(R)).

[F7]

The positive-weight limit vector $f_{1}$ is a unit vector in the $D_{1}^{+}$ summand of $I_{1, 0}$ (The two limits of discrete series).

[A1]

AC is inherited through the weighted and compact-picture constructions (The Axiom of Choice).

Proof

technique · direct

Given: The notation and assumptions of the Statement.

1.1F1F2algebraA1

Let $w = (z - i) / (z + i)$ , $F_{f} (w) = (z + i)^{n} f (z)$ , and $r = \tanh (τ / 2)$ . Solving gives $z = i (1 + w) / (1 - w)$ , so direct substitution gives $F_{f_{n, j}} (w) = (z - i)^{j} (z + i)^{- j} = w^{j}$ . Substituting this inverse coordinate in the inverse-action formula for $a_{τ} = diag (e^{τ / 2}, e^{- τ / 2})$ gives $F_{π_{n} (a_{τ}) f} (w) = \cosh (τ / 2)^{- n} (1 - r w)^{- n} F_{f} ⁣ (\frac{w - r}{1 - r w})$ . For $F_{f_{n, j}} (w) = w^{j}$ , this is $\cosh (τ / 2)^{- n} (w - r)^{j} (1 - r w)^{- n - j}$ . Its Taylor coefficient at $w^{k}$ equals $\cosh (τ / 2)^{- n} Q_{j k} (r)$ , where $Q_{j k} (r) = \sum_{p = 0}^{\min (j, k)} (- 1)^{j - p} (\binom{j}{p}) (\binom{n + j + k - p - 1}{k - p}) r^{j + k - 2 p}$ .

1.2F4F5algebraA1

Write $t = e^{τ}$ and $D (θ, τ) = \cos^{2} θ + e^{2 τ} \sin^{2} θ$ . Multiplying $k_{θ} a_{τ}$ and comparing its bottom row with the $A N K$ form gives $e^{σ / 2} = e^{τ / 2} / \sqrt{D}$ and $e^{i ψ} = (\cos θ + i e^{τ} \sin θ) / \sqrt{D}$ . Hence in $I_{1, i s}$ , $c_{m, s} (τ) = \frac{1}{2 π} \int_{0}^{2 π} {(\frac{e^{τ / 2}}{\sqrt{D}})}^{1 + i s} e^{i m (ψ - θ)} d θ$ . The character factors and the $i s$ -power have modulus one, so $∣ c_{m, s} (τ) ∣ \leq \frac{e^{τ / 2}}{2 π} \int_{0}^{2 π} D^{- 1 / 2} d θ$ . For $τ \geq 1$ , $D = 1 + (e^{2 τ} - 1) \sin^{2} θ \geq 1 + c e^{2 τ} θ^{2}$ on $0 \leq θ \leq π / 2$ for a fixed $c > 0$ , using $\sin θ \geq 2 θ / π$ . Substitution $u = e^{τ} θ$ gives $\int_{0}^{π / 2} D^{- 1 / 2} d θ \leq C (1 + τ) e^{- τ}$ , since $\int_{0}^{R} (1 + c u^{2})^{- 1 / 2} d u \leq C (1 + \log (1 + R))$ . The other three quadrants have the same bound; for $0 \leq τ \leq 1$ it follows after increasing $C$ . Thus $∣ c_{m, s} (τ) ∣ \leq C (1 + τ) e^{- τ / 2}$ for $τ \geq 0$ , uniformly in odd $m$ and real $s$ . For negative $τ$ , unitarity gives $∣ c_{m, s} (- τ) ∣ = ∣ c_{m, s} (τ) ∣$ , proving the stated bound.

2.1F1step 1.1algebra

By [F1] the vectors $f_{n, j}, f_{n, k}$ with $j \neq k$ are orthogonal, so the expansion of step 1.1 gives $⟨ π_{n} (a_{τ}) f_{n, j}, f_{n, k} ⟩ = \cosh (τ / 2)^{- n} Q_{j k} (r) ∥ f_{n, k} ∥_{n}^{2}$ . In particular $Q_{00} = 1$ , so normalizing $f_{n, 0}$ gives $c_{n} (a_{τ}) = \cosh (τ / 2)^{- n}$ . Each fixed polynomial $Q_{j k}$ is bounded for $∣ r ∣ \leq 1$ .

3.1F2F3step 2.1algebra

For fixed $X, Y \in U (g_{C})$ , [F2] writes $X u_{n}$ and $Y u_{n}$ as finite sums of $f_{n, j}$ . In $g = k_{1} a_{τ} k_{2}$ , the left and right K factors multiply each K-type vector by a scalar of modulus one. Thus the matrix coefficient is a fixed finite sum of the radial coefficients in step 2.1, with bounded factors $Q_{j k} (\tanh (τ / 2))$ . Since $\cosh (τ / 2)^{- n} \leq 2^{n} e^{- n ∣ τ ∣ / 2}$ , this proves the derivative-vector bound with a constant depending only on $n, X, Y$ and with polynomial exponent $m = 0$ .

4.1F4F5F6F7step 1.2algebra∎

At $m = 1, s = 0$ , the integrand in step 1.2 has real part $(\cos^{2} θ + t \sin^{2} θ) / D$ and odd imaginary part, so $c_{1, 0} (τ) = \frac{\sqrt{t}}{2 π} (I_{c} + t I_{s})$ , where $I_{c} = \int_{0}^{2 π} \cos^{2} θ / D d θ$ and $I_{s} = \int_{0}^{2 π} \sin^{2} θ / D d θ$ . Substitution $x = \tan θ$ on each quadrant gives $I_{0} : = I_{c} + I_{s} = 2 π / t$ and $I_{s} = 4 \int_{0}^{\infty} \frac{x^{2}}{(1 + x^{2}) (1 + t^{2} x^{2})} d x = 2 π / (t (t + 1))$ ; for $t \neq 1$ the last integral follows from partial fractions, and at $t = 1$ it is $\int_{0}^{2 π} \sin^{2} θ d θ = π$ . Therefore $I_{c} = 2 π / (t + 1)$ and $c_{1, 0} (τ) = 2 \sqrt{t} / (t + 1) = sech (τ / 2)$ . Consequently $∣ c_{1, 0} (τ) ∣^{2} \sinh τ \to 2$ , so the radial integral diverges. Since $∣ c_{1, 0} (k_{1} g k_{2}) ∣ = ∣ c_{1, 0} (g) ∣$ by unitarity and the K-character property of $f_{1}$ , [F6] implies this matrix coefficient is not in $L^{2} (G)$ .

Depends on

Used by

A limit of discrete series is not square-integrable Counterexample
A square-integrable discrete-series matrix coefficient Example
Square integrability of discrete-series matrix coefficients Theorem
The limits of discrete series are not square-integrable Theorem

Dependency tree · two levels

79 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)