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Theorem 5. On \( {C}_{c}^{\infty }\left( {G//K}\right) \) we have a commutative diagram \ni.e. \( \mathbf{S} = \mathbf{{MH}} \), and all the arrows are isomorphisms. | Proof.\n\n\[ \mathbf{S}f\left( s\right) = {\int }_{G}f\left( x\right) {\varphi }_{s}\left( x\right) {dx} \]\n\n\[ = {\int }_{K}{\int }_{G}f\left( x\right) \rho {\left( kx\right) }^{s + 1}{dxdk} \]\n\n\[ = {\int }_{A}{\int }_{N}{\int }_{K}f\left( {an}\right) \rho {\left( ka\right) }^{s + 1}{dadndk} \]\n\n\[ = \mathbf{{M... | Yes |
Corollary 1. \( {\varphi }_{s} = {\varphi }_{-s} \) | Proof. For every \( f \in {C}_{c}^{\infty }\left( {G//K}\right) \) we get \( \mathbf{S}f\left( s\right) = \mathbf{S}f\left( {-s}\right) \), so the integrals of \( {\varphi }_{s} \) and \( {\varphi }_{-s} \) against \( f \) give the same value. Hence \( {\varphi }_{s} = {\varphi }_{-s} \). | Yes |
Corollary 2. \( {\varphi }_{s} \) is bounded by 1 for \( - 1 \leq \operatorname{Re}s \leq 1 \) . | Proof. We shall give two proofs, each one illustrating a useful technique. The first is that of Helgason-Johnson [He, Jo], who proved the result in general. Let \( f \in {\mathcal{L}}^{1}\left( G\right) \) be bi-invariant. Then\n\n\[ \int \left| {f\left( x\right) }\right| {dx} = \int \left| {f\left( {kan}\right) }\righ... | Yes |
Theorem 6. Let \( f \in {C}_{c}^{\infty }\left( {G//K}\right) \) . Then\n\n\[ f\left( 1\right) = {\int }_{-\infty }^{\infty }\mathrm{S}f\left( {i\tau }\right) \tau \tanh \left( {\pi \tau }\right) \frac{d\tau }{2\pi }.\] | Proof. We keep our old notation, letting\n\n\[ {h}_{a} = \left( \begin{matrix} a & 0 \\ 0 & {a}^{-1} \end{matrix}\right) ,\;v = \frac{{a}^{2} + {a}^{-2}}{2},\;a = {e}^{t/2} \]\n\nand\n\n\[ \mathbf{H}f\left( {h}_{a}\right) = F\left( v\right) \]\n\nBy Mellin inversion we find\n\n\[ F\left( \frac{{a}^{2} + {a}^{-2}}{2}\ri... | Yes |
Theorem 7. Let\n\n\[ \nP\left( s\right) {ds} = \frac{s}{i}\tanh \left( \frac{\pi s}{i}\right) \frac{ds}{2\pi i}. \]\n\nIf \( g \) is in the Paley-Wiener space and is even, then\n\n\[ \n{\mathrm{S}}^{-1}g\left( x\right) = {\int }_{\operatorname{Re}s = 0}g\left( s\right) \varphi \left( {x, s}\right) P\left( s\right) {ds}... | Proof. From the inversion formula for \( f\left( 1\right) \), we shall get the general inversion for \( f\left( x\right) \) by a reduction, using the formalism of spherical functions. For any \( f \in {C}_{c}^{\infty }\left( {G//K}\right) \), let \( {f}_{x} \) be defined by\n\n\[ \n{f}_{x}\left( y\right) = {\int }_{K}f... | Yes |
Theorem 8. If \( \operatorname{Re}\left( s\right) \) does not lie in the interval \( \left\lbrack {-1,1}\right\rbrack \), then the spherical function \( {\varphi }_{s} \) is not bounded. | The proof we have given is that in Helgason-Johnson [He, Jo], for arbitrary semisimple Lie groups, but collapsing to advanced calculus in the case of \( S{L}_{2}\left( \mathbf{R}\right) \) . | No |
Theorem 9. We have the asymptotic behavior for \( \epsilon \rightarrow 0 \) :\n\n\[ \n{\int }_{0}^{\infty }\frac{1}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + {it}}\right) }{\left( 1 + {\epsilon }^{2}{v}^{2}\right) }^{\frac{1}{2}\left( {1 - {it}}\right) }}{dv} = c\left( {it}\right) + c\left( {-{it}}\right) {\... | Proof. We need some lemmas. | No |
Lemma 1. There exists a function \( c \), of \( t \) alone such that for \( N \rightarrow \infty \) we have \[ {\int }_{0}^{N}\frac{dv}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + {it}}\right) }} = - \frac{{N}^{-{it}}}{it} + {c}_{t} + O\left( {N}^{-1}\right) . | Proof. Write \[ {\int }_{0}^{N} = {\int }_{0}^{a} + {\int }_{a}^{N} \] where \( a > 0 \) is a fixed constant. We have first \[ {\int }_{a}^{N}\frac{1}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + {it}}\right) }}{dv} = {\int }_{a}^{N}\left\lbrack {\frac{1}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + {it... | Yes |
Lemma 2. The value \( {c}_{t} \) is equal to \( c\left( {it}\right) \) . | Proof. We consider\n\n\[ \n{\int }_{0}^{a}\frac{1}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + s}\right) }}{dv} + {\int }_{a}^{\infty }\left\lbrack {\frac{1}{{\left( 1 + {v}^{2}\right) }^{\frac{1}{2}\left( {1 + s}\right) }} - \frac{1}{{v}^{1 + s}}}\right\rbrack {dv} + \frac{{a}^{-s}}{s}, \n\] \n\nwhich is clea... | Yes |
Lemma 1. Let \( X \) be a measured space with positive measure \( \mu \) . Let \( U \) be an open subset of \( {\mathbf{R}}^{n} \) . Let \( f : X \times U \rightarrow E \) be a mapping into a Banach space. Assume:\n\ni) For each \( y \in U \) the map \( x \mapsto f\left( {x, y}\right) \) is in \( {\mathcal{L}}^{1}\left... | Proof. It suffices to prove that for any sequence \( \left\{ {y}_{k}\right\} \) converging to \( y \) ,\n\n\[ {\int }_{X}f\left( {x,{y}_{k}}\right) {d\mu }\left( x\right) \;\text{ converges to }\;{\int }_{X}f\left( {x, y}\right) {d\mu }\left( x\right) . \]\n\nLet \( {f}_{k}\left( x\right) = f\left( {x,{y}_{k}}\right) \... | Yes |
Lemma 2. Let \( X \) be a measured space with positive measure \( \mu \) . Let \( U \) be an open subset of \( {\mathbf{R}}^{n} \) . Let \( f : X \times U \rightarrow E \) be a mapping into a Banach space. Assume:\n\ni) For each \( y \in U \) the map \( x \mapsto f\left( {x, y}\right) \) is in \( {\mathcal{L}}^{1}\left... | Proof. We have\n\n\[ \frac{\Phi \left( {y + h{e}_{j}}\right) - \Phi \left( y\right) }{h} = {\int }_{X}\frac{1}{h}\left\lbrack {f\left( {x, y + h{e}_{j}}\right) - f\left( {x, y}\right) }\right\rbrack {d\mu }\left( x\right) .\n\]\n\nUsing the mean value theorem and (iii), together with the dominated convergence theorem, ... | Yes |
Lemma 3. Let \( G \) be a Lie group and \( f \in {\mathcal{L}}^{1}\left( {G, H}\right) \) where \( H \) is a Banach space. Let \( \varphi \in {C}_{c}^{\infty }\left( {G,\mathbf{C}}\right) \) and let \( X \in {\mathfrak{g}}_{\mathbf{C}} \) . Then \( f * \varphi \) is \( {C}^{\infty } \), and\n\n\[{\mathcal{L}}_{X}\left(... | Proof. Exactly the same as above for \( {\mathbf{R}}^{n} \) . By definition,\n\n\[f * \varphi \left( y\right) = {\int }_{G}f\left( {x}^{-1}\right) \varphi \left( {xy}\right) {dx}.\n\nIn the neighborhood of a point \( y \) we can choose local coordinates identifying this neighborhood with an open set in Euclidean space.... | No |
Lemma 4. Let \( s \) be a real number, and put \( {g}_{s} = \exp \left( {sY}\right) \) . Let\n\n\[ \varphi \left( X\right) = f\left( {\exp X}\right) \]\n\nThen\n\n\[ {\left. \frac{d}{ds}\varphi \left( {g}_{s}^{-1}X{g}_{s}\right) \right| }_{s = 0} = {\varphi }^{\prime }\left( X\right) \left( {{XY} - {YX}}\right) . \] | Proof. We have\n\n\[ {g}_{s}^{-1}X{g}_{s} = \left( {I - {sY} + O\left( {s}^{2}\right) }\right) X\left( {I + {sY} + O\left( {s}^{2}\right) }\right) \]\n\n\[ = X + s\left( {{XY} - {YX}}\right) + O\left( {s}^{2}\right) . \]\n\nHence\n\n\[ \varphi \left( {{g}_{s}^{-1}X{g}_{s}}\right) = \varphi \left( X\right) + {\varphi }^... | Yes |
Lemma 5. Let \( y = \exp \left( {uY}\right) \), where \( u \) is a fixed real number. Let\n\n\[ J\left( {u, s, t}\right) = f\left( {y\exp \left( {t{g}_{s}^{-1}X{g}_{s}}\right) }\right) .\n\]\n\nThen\n\n\[ {D}_{3}{D}_{2}J\left( {u,0,0}\right) = {\varphi }^{\prime }\left( O\right) \left( {{XY} - {YX}}\right) . \] | Proof. The expression on the left is equal to\n\n\[ {\left. \frac{d}{dt}{D}_{2}J\left( u,0, t\right) \right| }_{t = 0} = {\left. \frac{d}{dt}\left\lbrack {\varphi }^{\prime }\left( tX\right) t\left( XY - YX\right) \right\rbrack \right| }_{t = 0}. \]\n\nThe lemma follows from the rule for the derivative of a product. | No |
Theorem 1. Let \( G \) be a connected Lie group and let\n\n\[ \n\pi : G \rightarrow {GL}\left( H\right) \n\]\n\nbe a representation on a Banach space. Let \( V \) be an algebraic subspace of \( H \) consisting of analytic vectors, and invariant under \( {d\pi }\left( \mathrm{g}\right) \) . Then the closure of \( V \) i... | Proof. Let \( v \in V \) . For \( X \in \mathfrak{g} \) and \( \left| X\right| \) small, we apply the corollary to the analytic map \( f\left( x\right) = \pi \left( x\right) v \), and get\n\n\[ \n\pi \left( {\exp \left( X\right) }\right) v = \sum \frac{1}{n!}{d\pi }{\left( X\right) }^{n}v \in V. \n\]\n\nBut exp of a sm... | Yes |
Lemma 1. Let \( \pi : G \rightarrow {GL}\left( H\right) \) be a representation. Let \( X \in \mathfrak{g} \) . Let \( v \in {H}_{\pi }^{\infty } \) be an eigenvector for \( {d\pi }\left( X\right) \) with eigenvalue \( \lambda \), i.e. assume that\n\n\[ \n{d\pi }\left( X\right) v = {\lambda v} \n\]\n\nThen\n\n\[ \n\pi \... | Proof. Let \( f\left( t\right) = \pi \left( {\exp \left( {tX}\right) }\right) d \) . Then \( f \) is differentiable by assumption, and\n\n\[ \n\frac{f\left( {t + h}\right) - f\left( t\right) }{h} = \frac{\pi \left( {\exp \left( {t + h}\right) X}\right) v - \pi \left( {\exp \left( {tX}\right) }\right) v}{h} \n\]\n\n\[ \... | Yes |
Theorem 2. Let \( \pi : G \rightarrow {GL}\left( H\right) \) be an irreducible admissible representation of \( G = S{L}_{2}\left( \mathbf{R}\right) \) in a Banach space. Let \( n \) be an integer such that \( {H}_{n} \neq \{ 0\} \) . Then \( {H}_{n} \) has dimension 1, any element of \( {H}_{n} \) is a \( {C}^{\infty }... | Proof. We know from II, \( §1 \), Th. 2 that \( {H}_{n} \) has dimension 1 . Since \( {\pi }^{1}\left( {S}_{n, n}^{\infty }\right) {H}_{n} \neq 0 \), if \( \left\{ {v}_{n}\right\} \) is a basis of \( {H}_{n} \) over \( \mathbf{C} \), then \[ {\pi }^{1}\left( {S}_{n, n}^{\infty }\right) {v}_{n} = \mathbb{C}{v}_{n} \] Th... | Yes |
Theorem 3. Let \( \pi : G \rightarrow {GL}\left( H\right) \) be an irreducible representation of \( G \) in a Banach space. Let \( m \) be an integer. Then the sum \[ \mathop{\sum }\limits_{{n \equiv m\left( {\;\operatorname{mod}\;2}\right) }}{H}_{n} \] is stable under \( {d\pi }\left( \mathfrak{g}\right) \) . Furtherm... | Proof. Let \( v \in {H}_{n} \) . Then DER 3 shows \[ {d\pi }\left( W\right) {d\pi }\left( {E}^{ + }\right) v = {d\pi }\left\lbrack {W,{E}^{ + }}\right\rbrack v + {d\pi }\left( {E}^{ + }\right) {d\pi }\left( W\right) v \] \[ = {2id\pi }\left( {E}^{ + }\right) v + {ind\pi }\left( {E}^{ + }\right) v \] \[ = i\left( {n + 2... | Yes |
Lemma 2. If \( \psi \in {C}_{c}^{\infty }\left( G\right) \) is odd (resp. even), then \( {\pi }^{1}\left( \psi \right) \) maps \( H \) into \( {H}^{ - } \) (resp. \( {\mathrm{H}}^{ + } \) ) and annihilates \( {\mathrm{H}}^{ + } \) (resp. \( {\mathrm{H}}^{ - } \) ). | Proof. This is immediate from the definitions of \( {\pi }^{1}\left( \psi \right) \) . | No |
Theorem 4. Let \( \pi ,{\pi }^{\prime } \) be irreducible admissible representations of \( G \) on Banach spaces \( H,{H}^{\prime } \) . Assume that there exists a postive integer \( m \) such that \( \pi ,{\pi }^{\prime } \) have a lowest weight vector of weight \( m \), say \( {u}_{m},{u}_{m}^{\prime } \) respectivel... | Proof. For simplicity write \( {Xv} \) instead of \( {d\pi }\left( X\right) v \), and similarly for \( X{v}^{\prime } \) , \( {v}^{\prime } \in {H}^{\prime } \) . Let\n\n\[ {E}_{ + }^{r}{u}_{m} = {u}_{m + {2r}},\;{E}_{ + }^{r}{u}_{m}^{\prime } = {u}_{m + {2r}}^{\prime }.\]\n\nBy irreducibility, \( {u}_{m + {2r}} \) and... | Yes |
Lemma 1. If \( \pi \) is a unitary representation of \( G \), and \( X \in \mathfrak{g} \), then \( {d\pi }\left( X\right) \) is skew symmetric on \( {H}_{\pi }^{\infty } \) . | Proof. Obvious from the definition of the derived representation. | No |
Lemma 2. If \( \pi \) is irreducible admissible on the Banach space \( H \), and if \( \mathcal{Q} \) is the algebra of operators on \( H\left( K\right) \) generated over \( \mathbf{C} \) by the elements of \( {d\pi }\left( \mathrm{g}\right) \), then given any non-zero element \( u \in {H}_{m} \) for some \( m \), we h... | Proof. This is obvious from Theorem 3, §2, and Theorem 1, §1. | No |
Lemma 3. Let \( V = \sum {H}_{n} \) be a vector space over \( \mathbf{C} \), expressed as a direct sum of subspaces \( {H}_{n} \) . Let \( \mathcal{Q} \) be an algebra of linear endomorphisms of \( V \) generated by elements \( {X}_{1},\ldots ,{X}_{d} \) . Let \( m \) be an integer such that \( {H}_{m} \) has dimension... | Proof. After multiplying one scalar product by a constant, we may assume that the two scalar products are equal on \( {H}_{m} \) . What we have to show is that a scalar product satisfying the conditions of the theorem is determined by its values on \( {H}_{m} \) . It suffices to consider the scalar product of elements ... | Yes |
Lemma 4. Let \( \pi \) be an irreducible admissible representation of \( G \) on a Banach space \( H \) . Any two positive definite hermitian products on \( H\left( K\right) \) for which the elements of \( {d\pi }\left( \mathrm{g}\right) \) are skew symmetric are positive scalar multiples of each other. | Proof. We take the algebra \( \mathcal{Q} \) to be the one generated by \( {d\pi }\left( X\right), X \in \mathfrak{g} \) , and apply Lemmas 2, 3. | No |
Theorem 5. Let \( {\pi }_{1},{\pi }_{2} \) be irreducible unitary representations of \( G \), and let \( L : {H}_{1}\left( K\right) \rightarrow {H}_{2}\left( K\right) \) be an infinitesimal isomorphism. Then there exists \( c > 0 \) such that \( {cL} \) is unitary, and its unitary extension \( {H}_{1} \rightarrow {H}_{... | Proof. The first statement is immediate from Lemma 4. After multiplying \( L \) by a positive number, let us assume that \( L \) is unitary. For \( X \) sufficiently small, we have Taylor's formula,\n\n\[ \langle \pi \left( {\exp X}\right) v, w\rangle = \sum \frac{1}{n!}\left\langle {{d\pi }{\left( X\right) }^{n}v, w}\... | No |
Theorem 6. If two representations of \( G \) which are unitary on \( K \) are infinitesimally isomorphic, and the n-th eigenspace for the representations has dimension 1, then the n-th coefficient functions of the two representations are equal. | Proof. Let \( H\left( K\right) \) be the space of \( K \) -finite vectors for the first representation, let \( u \) be a unit vector in \( {H}_{n} \), and let\n\n\[ f\left( x\right) = \langle \pi \left( x\right) u, u\rangle . \]\n\nThen the function \( f \) is analytic on \( G \) (see below), and for all small \( X \in... | Yes |
Lemma 5. Let \( \pi \) be the representation by right translation on \( {L}^{2}\left( G\right) \), i.e. \( \pi \left( x\right) f\left( y\right) = f\left( {yx}\right) \). Let \( f \in {L}^{2}\left( G\right) \) and \( \varphi \in {C}_{c}^{\infty }\left( G\right) \). Then \( f * \varphi \) is a \( {C}^{\infty } \) vector ... | Proof. It suffices to prove that the map \( x \mapsto \pi \left( x\right) f \) is \( {C}^{\infty } \) near the origin in \( G \). Let \[ {\pi }_{tX}f\left( y\right) = f\left( {y\exp \left( {tX}\right) }\right) . \] We first prove the formula, and we have to show that \[ \frac{{\pi }_{tX}\left( {f * \varphi }\right) - \... | Yes |
Theorem 7. Let \( \pi \) be an irreducible admissible representation of \( G \) in a Banach space E. Let \( \lambda \in {E}^{\prime },\lambda \neq 0 \), and let\n\n\[ \n{f}_{v}\left( x\right) = \lambda \left( {\pi \left( x\right) v}\right) \n\]\n\n\( v \in E\left( K\right) \)\n\nbe the corresponding coordinate function... | Proof. By irreducibility, \( {\sum \pi }\left( {S}_{m, n}^{\infty }\right) \) acts transitively on \( E\left( K\right) \) . By formula (1), we conclude that \( {f}_{v} \in {L}^{2}\left( G\right) \) for all \( v \in E\left( K\right) \) . Lemma 5 and formulas (2),\n\n(3),(4) immediately show that the map \( v \mapsto {f}... | Yes |
Lemma 1. Let \( \pi \) be the representation by right translation of \( G \) on \( H\left( s\right) \) . Let \( X \in \mathfrak{g} \), let \( f \in H\left( s\right) \), and assume that \( f \) is \( {C}^{\infty } \) on \( G \) . Then \( f \) is a \( {C}^{\infty } \) vector as element of \( {L}^{2}\left( K\right) \), an... | Proof. Let\n\n\[ \n{\pi }_{tX}f\left( g\right) = f\left( {g\exp \left( {tX}\right) }\right) . \n\]\n\nWe have to verify that\n\n\[ \n\frac{{\pi }_{tX}f - f}{t} \rightarrow {\mathcal{L}}_{X}f \n\]\n\nin \( {L}^{2}\left( K\right) \), i.e. that\n\n\[ \n{\int }_{K}{\left| \frac{f\left( {k\exp \left( {tX}\right) }\right) - ... | Yes |
Lemma 2. Let \( f \) be an analytic function on \( G \) . Let \( \pi \left( x\right) f \) be the right translation, \[ \pi \left( x\right) f\left( y\right) = f\left( {yx}\right) . \] Then the map \[ x \mapsto \pi \left( x\right) f \mid K \] is an analytic map of \( G \) into \( {L}^{2}\left( K\right) \) . | Proof. Let \( {x}_{0} \in G \) . If we have proved that \( x \mapsto \pi \left( x\right) f \mid K \) is analytic in a neighborhood of the origin \( e \) on \( G \), then \[ x \mapsto \pi \left( {{x}_{0}^{-1}x}\right) f = \pi \left( {x}_{0}^{-1}\right) \pi \left( x\right) f \mid K \] is seen to be analytic because \( \p... | Yes |
Theorem 9. Given \( m \) as above, there exists \( \psi \in {C}_{c}^{\infty }\left( G\right) \) such that:\n\ni) \( {\pi }^{1}\left( \psi \right) \mid {H}_{m}^{\left( m\right) } = \) identity on \( {H}_{m}^{\left( m\right) } \).\n\nii) \( {\pi }^{1}\left( \psi \right) \) annihilates \( {H}_{n}^{\left( m\right) } \) if ... | Proof. Say \( m \geq 1 \) . For any \( f \in {S}_{m, m}^{\infty } \) we have\n\n\[ \n{\pi }^{1}\left( f\right) {H}_{m}^{\left( m\right) }\; = {H}_{m}^{\left( m\right) }\;\text{ or }\;0, \n\]\n\n\[ \n{\pi }^{1}\left( f\right) {H}_{n}^{\left( q\right) }\; = 0\;\text{ if }\;n \neq m. \n\]\n\nWe visualize the weighted seri... | Yes |
Theorem 1. Let \( q \) be a \( {C}^{\infty } \) function on \( K \times K \) . Then the integral operator \( Q \) defined by \( q \) is of trace class. | Proof. We have to express \( Q \) as a product of two Hilbert-Schmidt operators. Let \( d \) be a large positive integer. Let \( {P}_{m, n} \) be the integral operator defined by the kernel \( {\varphi }_{m} \otimes {\bar{\varphi }}_{n} \), so that\n\n\[ \n{P}_{m, n}{\varphi }_{j} = 0\text{ if }j \neq n, \n\]\n\n\[ \n{... | Yes |
Theorem 2. Let \( \pi \) be a strictly admissible representation of \( G \) on a Hilbert space \( H \) . If \( f \in {C}_{c}^{\infty }\left( G\right) \), then \( {\pi }^{1}\left( f\right) \) is of trace class. | Proof. The idea is similar to the idea used in the proof of Theorem 1. Write \( G = {ANK} \) and let \( B = {AN} \) . Then by definition,\n\n\[ \n{\pi }^{1}\left( f\right) = {\int }_{G}f\left( x\right) \pi \left( x\right) {dx} = {\int }_{B}{\int }_{K}f\left( {bk}\right) \pi \left( b\right) \pi \left( k\right) {dkdb} \n... | Yes |
Lemma 1. Let \( G = {ANK} \) . Then\n\n\[ \n{G}^{\prime } = \pm {A}^{\prime G} \cup {K}^{\prime G}.\n\]\n\nThe complement of \( {G}^{\prime } \) in \( G \) has measure 0 . | Proof. If an element \( g \in S{L}_{2}\left( \mathbf{R}\right) \) has a single eigenvalue of multiplicity 2, then by the Jordan normal form it is conjugate to an element in \( \pm N \) . Since \( {AN} \) normalizes \( N \), the image of \( N \) under conjugation by \( G \) has dimension 2, and therefore has measure 0 .... | Yes |
Lemma 2. Let \( {\pi }_{1} \) be a representation of \( G \) on a Hilbert space \( {H}_{1} \), and let \( {H}_{2} \) be the same space as \( {H}_{1} \) but with an equivalent scalar product. Let \( {\pi }_{2} \) be the same map as \( {\pi }_{1} \), but viewed as a representation on \( {H}_{2} \). Then\n\n\[ \operatorna... | Proof. Let \( T : {H}_{1} \rightarrow {H}_{2} \) be the identity map, which is bicontinuous. Then\n\n\[ {\pi }_{2}\left( x\right) = T{\pi }_{1}\left( x\right) {T}^{-1} \]\n\nand the equality between the traces follows by the general theory of traces (cf. the appendix to this chapter, last theorem). | Yes |
Theorem 3. Let \( {\pi }_{1},{\pi }_{2} \) be representations of \( G \) on Hilbert spaces. Assume that the \( K \) -eigenspaces in each have dimension 0 or 1, and that \( {\pi }_{1},{\pi }_{2} \) are infinitesimally isomorphic. Then for every \( \psi \in {C}_{c}^{\infty }\left( G\right) \) we have\n\n\[ \operatorname{... | Proof. By the preceding remarks, we may assume that \( {\pi }_{1},{\pi }_{2} \) are unitary on \( K \) . We can then apply VI, \( §3 \), Th. 5, where we proved that the coefficient functions are equal. A fortiori, their sums are equal, to\n\n\[ \mathop{\sum }\limits_{i}{\int }_{G}\psi \left( x\right) \left\langle {\pi ... | Yes |
Theorem 4. The trace \( \operatorname{tr}{\pi }_{\mu } \) is a distribution, which can be represented by the function \( {T}_{\mu } \), defined by:\n\n\[ \n{T}_{\mu }\left( a\right) = 2\frac{\mu \left( a\right) + \mu \left( {a}^{-1}\right) }{\left| D\left( a\right) \right| }\;\text{ if }\;a \in {A}^{\prime }, \n\]\n\n\... | Proof. According to the integral formula INT 3, we have\n\n\[ \n{\int }_{{A}^{\prime G}}\psi \left( x\right) {dx} = \frac{1}{4}{\int }_{A}{\int }_{A \smallsetminus G}\psi \left( {{x}^{-1}{ax}}\right) {\left| D\left( a\right) \right| }^{2}{dadx}. \n\]\n\n\[ \n= \frac{1}{4}{\int }_{A}{\mathbf{H}}^{A}\psi \left( a\right) ... | Yes |
Lemma 1. Let \( n \) be an integer \( \geq 1 \), let \( m = n + 1 \), and let\n\n\[ \epsilon \left( {-1}\right) = {\left( -1\right) }^{m}. \]\n\nThe trace of the representation in \( H\left( {n,\epsilon }\right) \) is a distribution represented by the function invariant under conjugation such that\n\n\[ T\left( {z{h}_{... | Proof. This is a special case of the trace found in the last section for an arbitrary induced representation. | No |
Lemma 2. For each pair of integers \( p, q \geq 0 \) such that \( p + q = n - 1 \), let \( {f}_{p, q} \) be the function of \[ x = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \] such that \( {f}_{p, q}\left( x\right) = {c}^{p}{d}^{q} \) . Then \( {f}_{p, q} \in H\left( {-n,\epsilon }\right) \) . The func... | Proof. The functions \( {f}_{p, q} \) obviously lie in \( H\left( {-n}\right) \), they are linearly independent, and matrix multiplication shows that the space they generate is stable under right translation by \( G \) . Each function is an eigenvector of \( \rho \left( {h}_{t}\right) \) , [VII, §4] and in fact \[ \rho... | Yes |
Lemma 3. Let \( V \) be the finite dimensional irreducible representation of dimension \( n = m - 1 \), and let \( \rho \) be the representation in \( V \) . Then\n\n\[ \operatorname{tr}\rho \left( {k}_{\theta }\right) = \frac{{e}^{in\theta } - {e}^{-{in\theta }}}{{e}^{i\theta } - {e}^{-{i\theta }}}.\] | Proof. We know that \( V \) is generated by the \( K \) -eigenvectors having eigenvalues\n\n\[ {e}^{i\left( {-m + 2}\right) \theta },{e}^{i\left( {-m + 4}\right) \theta },\ldots ,{e}^{i\left( {m - 2}\right) \theta }.\]\n\nWe find the trace on elements of \( K \) by summing these eigenvalues, i.e. summing part of a geom... | No |
Lemma 4. Let \( n \) be an integer \( \geq 1 \) and let \( m = n + 1 \) . Let\n\n\[ \epsilon \left( {-1}\right) = {\left( -1\right) }^{m}. \]\n\nThen the trace of the representation on \( {H}^{\left( m\right) } + {H}^{\left( -m\right) } \) is a distribution represented by the function:\n\n\[ T\left( {z{h}_{t}}\right) =... | Proof. This is obvious from Lemmas 1, 2, 3, subtracting the values found in Lemmas 2, 3 from the values found in Lemma 1. | No |
Theorem 5. Let \( n \) be an integer \( \neq 0 \) and let \( m = \left| n\right| + 1 \) . Let \( z = \pm 1 \), and \( \epsilon \left( {-1}\right) = {\left( -1\right) }^{m} \) . Let \( {\sigma }_{n} \) be the representation in \( {H}^{\left( n + 1\right) } \) if \( n > 0 \) and \( {H}^{\left( -n - 1\right) } \) if \( n ... | Proof. The proof is a little elaborate and is due to Harish-Chandra. The main difficulty is to eliminate the possibility that there is a contribution from the singular set (the complement of \( {G}^{\prime } \) in \( G \) ) which may cancel in the sum of the discrete series with positive and negative weight. I shall om... | No |
Theorem 6. We have the relation\n\n\[ \n{S}_{n}\left( \psi \right) = - \left( {\operatorname{sign}n}\right) {\int }_{0}^{2\pi }{\mathbf{H}}^{K}\psi \left( {k}_{\theta }\right) {e}^{in\theta }{d\theta } \n\]\n\n\[ \n+ \frac{1}{2}{\int }_{0}^{\infty }\left\lbrack {{\mathbf{H}}^{A}\psi \left( {h}_{t}\right) + {\left( -1\r... | Proof. We compute \( {S}_{n}\left( \psi \right) \) by integrating over \( {K}^{\prime G} \) and \( \pm {A}^{\prime G} \) respectively. Let us start with \( {K}^{\prime G} \) . We have\n\n\[ \n{\int }_{{K}^{\prime G}}\psi \left( x\right) \left( {-\operatorname{sign}n}\right) \frac{{e}^{{in\theta }\left( x\right) }}{{e}^... | Yes |
Theorem 7. Let \( \left\{ {T}_{n}\right\} \) be a sequence of operators on \( H \), converging weakly to an operator \( T \) . In other words, for each \( v, w \in H,\left\langle {{T}_{n}v, w}\right\rangle \rightarrow \langle {Tv}, w\rangle \) . Let \( A \) be of trace class. Then\n\n\[ \operatorname{tr}\left( {TA}\rig... | Proof. Assume first that \( A = P \) is positive symmetric. Since \( A \) is compact (because \( A \) is Hilbert-Schmidt and HS 3), there is an orthonormal basis of \( H \) consisting of eigenvectors, say \( \left\{ {u}_{i}\right\} \), with \( A{u}_{i} = {c}_{i}{u}_{i} \) . For fixed \( v, w \in H \) the set \( \left\{... | Yes |
Theorem 1. Let \( f \in {C}_{c}^{\infty }\left( {G, K}\right) \) . Then\n\n\[{\left. {\mathbf{H}}^{K}f\right\rbrack }_{0 - }^{0 + } = {\mathbf{H}}^{K}f\left( {0 + }\right) - {\mathbf{H}}^{K}f\left( {0 - }\right) = \frac{i}{2}{\mathbf{H}}^{A}f\left( 1\right) ,\]\n\n\[{\left. {\mathbf{H}}^{K}f\right\rbrack }_{\pi - }^{\p... | Proof. By the calculus lemma and \( w{n}_{u}{w}^{-1} = {\bar{n}}_{-u} \) we find\n\n\[{\mathbf{H}}^{K}f\left( {0 + }\right) = \frac{i}{2}{\int }_{0}^{\infty }f\left( \begin{array}{ll} 1 & u \\ 0 & 1 \end{array}\right) {du}\]\n\nand\n\n\[{\mathbf{H}}^{K}f\left( {0 - }\right) = - \frac{i}{2}{\int }_{-\infty }^{0}f\left( ... | Yes |
Theorem 2. For \( f \in {C}_{c}^{\infty }\left( G\right) \), we have \( {\left( {\mathbf{H}}^{K}f\right) }^{\prime }\left( 0\right) = - {if}\left( 1\right) \) . | Proof. Immediate from (iii) of the calculus lemma. | No |
Theorem 3. The Fourier series of \( {\left( {\mathbf{H}}^{K}f\right) }^{\prime } \) converges to the function at 0 . | Proof. From (iv) of the calculus lemma, we know that\n\n\[ \n{\left( {\mathbf{H}}^{K}f\right) }^{\prime }\left( \theta \right) = \text{ constant } + O\left( {\left| {\theta \log }\right| \theta \parallel }\right) .\n\]\n\nThe Dirichlet kernel from which the Fourier series is obtained by convolution is equal to (up to a... | Yes |
Lemma 1. We have the two identities:\n\n\[ \mathop{\sum }\limits_{{n\text{ odd }}}\sin \left( {\left| n\right| \theta }\right) {e}^{-\left| n\right| t} = \frac{2\sin \theta \cosh t}{\cosh {2t} - \cos {2\theta }}, \]\n\n\[ \mathop{\sum }\limits_{{n\text{ even }}}\sin \left( {\left| n\right| \theta }\right) {e}^{-\left| ... | Proof. Consider \( n \) positive, odd, \( n = {2d} + 1 \) . We have\n\n\[ \mathop{\sum }\limits_{{d = 0}}^{\infty }{e}^{i\left( {{2d} + 1}\right) \theta }{e}^{-\left( {{2d} + 1}\right) t} = {e}^{{i\theta } - t}\mathop{\sum }\limits_{{d = 0}}^{\infty }{e}^{{2d}\left( {{i\theta } - t}\right) } \]\n\n\[ = \frac{1}{{e}^{-\... | Yes |
Lemma 2. For \( 0 < \theta < \pi \) and \( \lambda > 0 \) we have\n\n\[ \mathop{\lim }\limits_{{c \rightarrow \infty }}{\int }_{-c}^{c}\frac{{e}^{i\lambda t}}{\cosh {2t} - \cos {2\theta }}{dt} = \frac{\pi }{\sin {2\theta }}\frac{\sinh \left( {\frac{\pi }{2} - \theta }\right) \lambda }{\sinh \frac{\pi \lambda }{2}}. \] | Proof. We integrate around the rectangle as shown.\n\n\n\nThe integral\n\n\[ \oint \frac{{e}^{i\lambda z}}{\cosh {2z} - \cos {2\theta }}{dz} \]\n\nis equal to \( {2\pi i} \) times the sum of the residues. We have\n\n... | Yes |
Lemma 3. For \( 0 < \theta < \pi \) and \( \lambda > 0 \) we have\n\n\[ \mathop{\lim }\limits_{{c \rightarrow \infty }}{\int }_{-c}^{c}\frac{{e}^{i\lambda t}\cosh t}{\cosh {2t} - \cos {2\theta }}{dt} = \frac{\pi }{2\sin \theta }\frac{\cosh \left( {\frac{\pi }{2} - \theta }\right) \lambda }{\cosh \frac{\pi \lambda }{2}}... | Proof. We integrate\n\n\[ \oint \frac{{e}^{i\lambda z}\cosh z}{\cosh {2z} - \cos {2\theta }}{dz} \]\n\nover the same contour as before. The residue at \( {z}_{n} = \left( {{n\pi } \pm \theta }\right) i \) is\n\n\[ \frac{{e}^{-\lambda \left( {{n\pi } \pm \theta }\right) }\cosh \left( {{n\pi } \pm \theta }\right) i}{{2i}... | Yes |
Theorem 4. Let \( {S}_{n} \) be the function defined in VII,§5, Th. 6. Let \( f \in {C}_{c}^{\infty }\left( {G, K}\right) \) . For \( 0 < \theta < \pi \) we have\n\n\[ \frac{2\pi }{i}{J}^{K}f\left( \theta \right) = - \mathop{\sum }\limits_{{n \neq 0}}{S}_{n}\left( f\right) \sin \left| n\right| \theta \]\n\n\[ + \frac{1... | Proof. When subtracting the Fourier series above, the constant terms cancel, and we find\n\n(3)\n\n\[ \frac{2\pi }{i}{J}^{K}f\left( \theta \right) = - \mathop{\sum }\limits_{{n \neq 0}}\sin {n\theta }{\int }_{0}^{2\pi }{\mathbf{H}}^{K}f\left( \varphi \right) {e}^{in\varphi }{d\varphi }. \]\n\nSubstituting the value fou... | Yes |
Lemma 1. Let \( m \) be an integer \( \geq 2 \) . Then the function \( {\alpha }^{-m} \) is in \( {L}^{2}\left( G\right) \) . | Proof. By the integral formula INT 2, we know that\n\n\[ \n{\int }_{G}{\left| \alpha \right| }^{-{2m}}{dx} = {\int }_{0}^{\infty }{\left( \cosh t\right) }^{-{2m}}\sinh {2tdt} \n\]\n\nup to a constant factor. Since \( \sinh {2t} = \left( {{e}^{t} + {e}^{-t}}\right) \left( {{e}^{t} - {e}^{-t}}\right) /2 \), the integral ... | Yes |
Theorem 1. Let \( {\varphi }_{m + {2r}} = {\alpha }^{-m - r}{\beta }^{r} \) for \( r = 0,1,2,\ldots \) . Then the functions \( {\varphi }_{m + {2r}} \) are eigenvectors of \( K \) with eigenvalue \( {e}^{i\left( {m + {2r}}\right) \theta } \) . The closed subspace of \( {L}^{2}\left( G\right) \) which they generate is i... | Proof. The eigenvalue property is immediate from (5) and (6), i.e. essentially directly from the definition of \( \alpha \) and \( \beta \) . Observe that \( \beta /\alpha \) is a function of absolute value \( < 1 \), and that for any given \( y,\left| {\beta \left( y\right) /\alpha \left( y\right) }\right| < 1 \) . Th... | Yes |
Lemma 1. If a sequence of holomorphic functions \( \left\{ {f}_{n}\right\} \) is \( {L}^{2} \) -convergent in an open set in the complex plane, then it is uniformly convergent to a holomorphic function on any compact set. In fact, locally, we have domination of norms: | Proof. We work in the neighborhood of a point, which we may assume to be the origin without loss of generality. Our estimates will depend on some disc of fixed radius \( \delta \) around any point, and again it suffices to bound \( \left| {f\left( 0\right) }\right| \) in terms of the \( {L}^{1} \) and \( {L}^{2} \) nor... | Yes |
Theorem 2. \( {\pi }_{m} \) is a unitary representation. | Proof. We first verify the unitary property. Let \( w = {\sigma }^{-1}z \) . Recall that\n\n\[ \operatorname{Im}{\sigma }^{-1}z = \frac{y}{{\left| cz + d\right| }^{2}}. \]\n\nWe have\n\n\[ {\begin{Vmatrix}{\pi }_{m}\left( \sigma \right) f\end{Vmatrix}}_{2}^{2} = {\int }_{0}^{\infty }{\int }_{-\infty }^{\infty }{\left| ... | Yes |
Lemma 2. Let \( n \) be an integer \( \geq 0 \) and let\n\n\[ \n{\psi }_{n}\left( z\right) = {\left( \frac{z - i}{z + i}\right) }^{n}{\left( z + i\right) }^{-m}.\n\]\n\nThen \( {\psi }_{n} \in H \) . | Proof. As \( \left| z\right| \rightarrow \infty ,{\psi }_{n}\left( z\right) \rightarrow 0 \), and we have\n\n\[ \n\left| \frac{z - i}{z + i}\right| \leq 1\n\]\n\nfor all \( z \) in the upper half plane. Let \( \delta > 0 \) . The proof consists in proving that the \( {L}^{2} \) -integrals above \( \delta \) and below \... | Yes |
Theorem 3. The representation \( {\pi }_{m} \) on \( H = {L}_{\mathrm{{hol}}}^{2}\left( {\mathfrak{S},{\mu }_{m}}\right) \) is irreducible. Let \( {H}_{m + {2n}} \) be the one-dimensional subspace generated by the function \( {\psi }_{n} \) . Then \( {H}_{m + {2n}} \) is an eigenspace of \( K \), with character \( m + ... | The proof of Theorem 3 is best carried out by changing the model for the representation under the analytic isomorphism between the upper half plane and the unit disc. We shall do this in the next section. | No |
Lemma 1. The map\n\n\[ \n{T}_{m} : {L}_{\mathrm{{hol}}}^{2}\left( {\mathfrak{H},{\mu }_{m}}\right) \rightarrow {L}_{\mathrm{{hol}}}^{2}\left( {D,{v}_{m}}\right) \n\]\n\nis an isometry. | Proof. We have \( {dw} \land d\bar{w} = - {2du} \land {dv} \) and \( {dz} \land d\bar{z} = - {2idx} \land {dy} \) . We get\n\n\[ \n\frac{4dudv}{{\left( 1 - {\left| w\right| }^{2}\right) }^{2}} = \frac{dxdy}{{y}^{2}} \n\]\n\nThe isometry amounts to\n\n\[ \n\iint {\left| f\left( -i\frac{w + 1}{w - 1}\right) {\left( \frac... | Yes |
Theorem 4. The functions \( \left\{ {1, w,{w}^{2},\ldots }\right\} \) form a complete orthogonal basis for \( {L}_{\mathrm{{hol}}}^{2}\left( {D,{v}_{m}}\right) \) . | Proof. Let \( f \in {L}_{\text{hol }}^{2}\left( {D,{v}_{m}}\right) \) . Then \( f \) has a power series expansion\n\n\[ f\left( w\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{w}^{n} \]\n\nIt suffices to prove that the series converges in \( {L}^{2}\left( {\nu }_{m}\right) \) . Let \( 0 < {r}^{\prime } < 1 ... | Yes |
Lemma 2. The elements \( 1, w,{w}^{2},\ldots \) in \( \widetilde{H} \) are analytic vectors. | Proof. The proof is entirely similar to the proof of the analogous statement for the continuous series, VI, §5, Lemma 2, and will be left to the reader. | No |
Theorem 1. The map from \( \mathcal{U}\left( \mathfrak{g}\right) \) into the algebra of differential operators on \( {C}^{\infty }\left( G\right) \) is injective. If we identify \( \mathcal{U}\left( \mathfrak{g}\right) \) with its image, then \( \mathcal{A}\left( \mathfrak{g}\right) \) has a basis consisting of the ele... | Proof. Without loss of generality it suffices to prove the assertion for one particular basis, say \( {E}_{ + },{E}_{ - }, W \) . In any associative algebra \( A \), given \( x \in A \) , the mapping\n\n\[ \ny \mapsto \left\lbrack {x, y}\right\rbrack = {xy} - {yx} \]\n\n\nis a derivation, i.e. satisfies \( D\left( {yz}... | Yes |
The centralizer of \( \mathbf{W} \) in \( \mathcal{U}\left( \mathfrak{g}\right) \) is the set of all linear combinations of monomials \[ {E}_{ + }^{p}{W}^{q}{E}_{ - }^{p} \] \[ p, q \geq 0 \] | Proof. The rule for the derivative of a product shows that \[ \left\lbrack {W,{E}_{ + }^{p}{W}^{q}{E}_{ - }^{r}}\right\rbrack = {2i}\left( {p - r}\right) {E}_{ + }^{p}{W}^{q}{E}_{ - }^{r}. \] This shows that all the above monomials commute with \( W \) . Conversely, suppose that a linear combination \[ \sum {c}_{pqr}{E... | Yes |
Lemma 1. The map \( h \) is a multiplicative homomorphism on \( \mathfrak{Z}\left( W\right) \) . | Proof. Let \( {\mathbf{Y}}_{1} \equiv h\left( {\mathbf{Y}}_{1}\right) {\;\operatorname{mod}\;\mathcal{U}}{\mathbf{E}}_{ - } \) and let \( {\mathbf{Y}}_{2} \equiv h\left( {\mathbf{Y}}_{2}\right) {\;\operatorname{mod}\;\mathcal{U}}{\mathbf{E}}_{ - } \) . Then\n\n\[ \n{\mathbf{Y}}_{2}{\mathbf{Y}}_{1} \equiv h\left( {\math... | Yes |
Lemma 2. The map \( h \) is injective on \( \mathcal{Z}\left( \mathcal{U}\right) \) . | Proof. Suppose\n\n\[ Y = \mathop{\sum }\limits_{q}{c}_{q}{E}_{ + }^{r}{W}^{q}{E}_{ - }^{r} + \mathop{\sum }\limits_{{p > r}}{c}_{pq}{E}_{ + }^{p}{W}^{q}{E}_{ - }^{p}, \]\n\nsome coefficient \( {c}_{rq} \neq 0 \) and \( r \geq 1 \) . We show that \( \mathbf{Y} \) cannot commute with \( {E}_{ - } \) . Consider the irredu... | Yes |
Theorem 3. The center of \( \mathcal{U}\left( \mathfrak{g}\right) \) is the polynomial algebra in one variable \( \mathbb{C}\left\lbrack \omega \right\rbrack \) . Its image under \( h \) is \( \mathbb{C}\left\lbrack {\left( W - i\right) }^{2}\right\rbrack \) . | Proof. Since \( h \) is injective on \( \mathfrak{T}\left( \mathcal{U}\right) \) and since we have already seen that \( {\left( W - i\right) }^{2} \) already occurs in the image of the center under \( h \), it will suffice to prove that only polynomials in even powers of \( \left( {W - i}\right) \) can occur. This amou... | Yes |
Theorem 4. The centralizer of \( \mathfrak{k} \) (i.e. of \( \mathbf{W} \) ) in \( \mathcal{A} \) is the commutative polynomial algebra \( \mathbb{C}\left\lbrack {\omega, W}\right\rbrack \) . | Proof. The expression of \( \omega \) in terms of \( \mathbf{W},{\mathbf{E}}_{ + },{\mathbf{E}}_{ - } \) shows that the above commutative algebra contains \( {\mathbf{E}}_{ + }{\mathbf{E}}_{ - } \), whence \( {\left( {\mathbf{E}}_{ + }{\mathbf{E}}_{ - }\right) }^{p} \) for every integer \( p \geq 0 \) . We then prove b... | Yes |
Lemma 3. For \( X \in \mathfrak{g} \) we have\n\n\[ \n{\operatorname{Ad}}_{{q}_{1}}\left( {\exp X}\right) = \exp \left( {\operatorname{ad}X}\right) \n\]\n\nwhere \( \left( {\operatorname{ad}X}\right) \left( Y\right) = {XY} - {YX} \). | Proof. Entirely similar to the proof of the lemma in Chapter VII, §2. We leave it to the reader. | No |
Theorem 5. Let \( \mathbf{Y} \in \mathcal{U}\left( \mathfrak{g}\right) \) and assume that \( \mathbf{Y} \) commutes with some \( \mathbf{X} \) with\n\n\( X \in \mathfrak{g} \) . Then \( \mathbf{Y} \) commutes with translation on the right and left by \( \exp X \) (as\n\nan operator on \( {C}^{\infty }\left( {G, H}\righ... | Proof. By definition, \( \mathbf{Y} \) is a left invariant differential operator, and we have only to prove that \( \mathbf{Y} \) is also right invariant. Let \( g = \exp X \) . Then\n\n\[ f\left( {x\exp \left( {tY}\right) g}\right) = f\left( {{xg}\exp \left( {t{g}^{-1}{Yg}}\right) }\right) ,\]\n\nand therefore, if \( ... | Yes |
Theorem 7. Let \( \pi \) be a representation of \( G \) in a Banach space \( H \) . Assume that for some integer \( n \), the space \( {H}_{n} \) has dimension 1, and let \( \{ v\} \) be a basis of \( {H}_{n} \) . Then \( v \) is an analytic vector. | Proof. Let \( {f}_{v}\left( g\right) = \pi \left( g\right) v \) . Then\n\n\[ \n{f}_{v}\left( {g{k}_{\theta }}\right) = {e}^{in\theta }{f}_{v}\left( g\right) \n\]\n\nIt follows that \( {f}_{v} \) is an eigenvector of \( W \) with eigenvalue in. Since \( \omega \) commutes with \( \pi \left( k\right) \) for all \( k \in ... | Yes |
Theorem 8. Let \( f \) be a \( {C}^{\infty } \) function on \( G \), bi-invariant under \( K \) and taking the value 1 at e. Then \( f \) is a spherical function if and only if there exists \( \lambda \in \mathbb{C} \) such that \( {\omega f} = {\lambda f} \) . | Proof. Assume first that \( f \) is a spherical function. Let \( \varphi \in {C}_{c}^{\infty }\left( {G//K}\right) \) . By definition,\n\n\[ f * \varphi \left( x\right) = {\int }_{G}f\left( {x{y}^{-1}}\right) \varphi \left( y\right) {dy} = {\int }_{G}f\left( {y}^{-1}\right) \varphi \left( {yx}\right) {dy}. \]\n\nSince ... | Yes |
Theorem 9. Let \( \lambda : \mathcal{Z}\left( \mathfrak{k}\right) \rightarrow \mathbf{C} \) be a character, i.e. an algebra homomorphism. Let \( f \) be an analytic function on \( G \) such that \( f \) is invariant under conjugation by \( K \), i.e.\n\n\[ f\left( {{kx}{k}^{-1}}\right) = f\left( x\right) ,\;x \in G, k ... | Proof. This is an immediate consequence of Taylor's formula\n\n\[ f\left( {\exp X}\right) = \sum \frac{{X}^{m}f\left( e\right) }{m!}, \]\n\nand formula (2) above, together with the assumption that \( f \) is an eigenvector for \( \mathcal{Z}\left( f\right) \) with eigencharacter \( \lambda \) . | Yes |
Theorem 10. (i) If two functions on \( G \) are conjugate invariant under \( K \) and are eigenfunctions of \( \mathfrak{X}\left( \mathfrak{k}\right) \) with the same character, normalized to have the value 1 at the origin, then they are equal. | Proof. The first part has already been proved. | No |
Theorem 11. The only spherical functions are the ones which we have already exhibited, i.e. the functions \( {f}_{s} \) above, \( s \in \mathbf{C} \) . | Since \( f \) is bi-invariant, we can view \( f \) as a function of the \( A \) -variable only in the Cartan decomposition \( G = K{A}^{ + }K \) . The above differential equation is a second order linear differential equation, and it has two linearly independent solutions. Given an eigenvalue \( \lambda \) and a spheri... | Yes |
Theorem 1. The natural homomorphism of the above free group onto \( S{L}_{2}\left( F\right) \) is an isomorphism. | Proof. Any group with generators \( u\left( b\right) \left( {b \in F}\right), w \), and \( s\left( a\right) \) defined by (*), satisfying our stated relations, whether it is the free group or not, consists of all elements of the form\n\n\[ u\left( b\right) s\left( a\right) \;\text{ or }\;u\left( b\right) s\left( a\righ... | Yes |
Theorem 2. Let \( S = S\left( {\mathbf{R}}^{2}\right) \) be the Schwartz space of \( {\mathbf{R}}^{2} \) . There exists a unique algebraic representation \( r \) of \( S{L}_{2}\left( \mathbf{R}\right) \) on \( S \) (so we impose no continuity condition) satisfying the following properties:\n\n(1)\n\n\[ r\left( w\right)... | Proof. We have to check that the above operations defined in terms of \( a, b, w \) satisfy the relations of \( \$ 2 \) . Note that\n\n\[ r\left( {w}^{2}\right) f = - {f}^{ - }. \]\n\nThe easiest way to proceed is to verify first that if we define \( r\left( {s\left( a\right) }\right) \) by formula (3), then the relati... | No |
Lemma 1. \( S\left( {\mathbf{C},\chi }\right) \) is stable under the representation \( r \) . | Proof. Let \( {\alpha }^{\prime } \) denote the complex conjugate of \( \alpha \) . Then\n\n\[ \n{h}_{b}\left( {\alpha z}\right) = {h}_{b}\left( z\right) \n\]\n\nif \( \alpha {\alpha }^{\prime } = 1 \), and so \( r\left( {u\left( b\right) }\right) f \) lies in \( S\left( {\mathbf{C},\chi }\right) \) if \( f \) does. Al... | Yes |
Theorem 3. Let \( {L}^{2}\left( {\mathbf{C},\omega }\right) \) be the completion of \( S\left( {\mathbf{C},\omega }\right) \) with respect to the \( {L}^{2} \)-norm. Then \( {L}^{2}\left( {\mathbf{C},\omega }\right) \) is irreducible for \( {r}_{\omega } \) . | Proof. The algebraic subspace \( S\left( {\mathbf{C},\omega }\right) \) is dense in \( {L}^{2}\left( {\mathbf{C},\omega }\right) \) . Map the space \( S\left( {\mathbf{C},\omega }\right) \) into a function space on \( \mathbf{C} \) by the mapping \( T \) such that (5) \[ {Tf}\left( z\right) = \left| z\right| \omega \le... | Yes |
Lemma 2. The association\n\n\[ \nf \mapsto {\left( Tf\right) }^{ * }\n\]\n\nis a unitary isomorphism with respect to the hermitian product taken with\n\nLebesgue measure on \( \mathbf{C} \) and \( {\pi dt}/t \) on \( {\mathbf{R}}^{ + } \) . | Proof. Immediate, and left to the reader. | No |
Lemma 3. Let \( \\left( {X,\\mu }\\right) \) be a \( \\sigma \) -finite measure space. Let\n\n\[ A : {L}^{2}\\left( {X,\\mu }\\right) \\rightarrow {L}^{2}\\left( {X,\\mu }\\right) \]\n\nbe a continuous linear map which commutes with multiplication by all functions in \( {\\mathcal{L}}^{\\infty }\\left( {X,\\mu }\\right... | Proof. Let \( \\varphi \\in {\\mathcal{L}}^{2}\\left( {X,\\mu }\\right) \) be an essentially positive function. For instance, decompose\n\n\[ X = \\cup {X}_{n} \]\n\nas a disjoint union of sets of finite measure, and let \( \\varphi \) on \( {X}_{n} \) be the constant function\n\n\[ \\frac{1}{{n}^{2}\\mu {\\left( {X}_{... | Yes |
Lemma 4. Let \( {\mathbf{R}}^{ + } \) be taken with its Haar measure \( {dt}/t \) . Let\n\n\[ A : {L}^{2}\left( {\mathbf{R}}^{ + }\right) \rightarrow {L}^{2}\left( {\mathbf{R}}^{ + }\right) \]\n\nbe a bounded linear map, which commutes with multiplication by all\n\nfunctions \( {e}^{ibt} \) for all \( b \in \mathbf{R} ... | Proof. We first prove the assertion of the theorem when \( \varphi \in {C}_{c}^{\infty }\left( {\mathbf{R}}^{ + }\right) \) . Let \( N \) be a large integer, and let \( {\psi }_{N} \) be the extension of \( \varphi \) to \( {\mathbf{R}}^{ + } \) by periodicity over the interval \( (0, N\rbrack \), so that the graph of ... | Yes |
Theorem 1. All the one-parameter unipotent subgroups of \( S{L}_{2}\left( \mathbf{R}\right) \) are conjugate. Let \( N,{N}^{\prime } \) be two unipotent subgroups. We have \( N = {N}^{\prime } \) if and only if \( \operatorname{Norm}\left( N\right) = \operatorname{Norm}\left( {N}^{\prime }\right) \) . | Proof. Given \( X \) nilpotent, there exists a matrix \( M \in G{L}_{2}\left( \mathbf{R}\right) \) such that\n\n\[ \n{MX}{M}^{-1} = \left( \begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)\n\]\n\nby the Jordan normal form theorem. We can change \( M \) by\n\n\[ \n\left( \begin{array}{rr} 1 & 0 \\ 0 & - 1 \end{array}\... | Yes |
Theorem 2. A subgroup \( {N}_{x} \) is cuspidal for \( S{L}_{2}\left( \mathbb{Z}\right) \) if and only if \( x \) is rational, or \( \infty \) . | Proof. This is immediately seen from the explicit description of matrices in \( {N}_{x} \) given in (1) above. If the intersection of \( {N}_{x} \) and \( S{L}_{2}\left( \mathbf{Z}\right) \) contains an element other than 1, then \( u = n \) is an integer and \( 1 - {ux} \) is also an integer, so \( x \) is rational. T... | Yes |
Theorem 3. Let \( \Gamma \) be a discrete subgroup of \( S{L}_{2}\left( \mathbf{R}\right) \) such that the quotient \( \Gamma \smallsetminus S{L}_{2}\left( \mathbf{R}\right) \) is compact. Then there is no cuspidal group for \( \Gamma \) . | Proof. We have to show that \( \Gamma \) contains no unipotent matrix other than 1. We first prove that the conjugacy class of any element \( \gamma \) under \( S{L}_{2}\left( \mathbf{R}\right) \) is closed in \( S{L}_{2}\left( \mathbf{R}\right) \) . Indeed, let \( W \) be a compact set such that \[ S{L}_{2}\left( \mat... | Yes |
Theorem 4. We have \( {\mathcal{Q}}^{0}\left( {\Gamma \smallsetminus G, m}\right) \subset {}^{0}{L}^{2}\left( {\Gamma \smallsetminus G}\right) \) . | Proof. Let \( \varphi \) be the function on \( \mathfrak{H} \) corresponding to a lifted function \( \Phi \) in \( {\mathcal{Q}}^{0}\left( {\Gamma \smallsetminus G, m}\right) \) . Since \( {a}_{0} = 0 \), we see that the power series in \( q \) starts with a term\n\n\[ \n{a}_{1}{e}^{-{2\pi y}}{e}^{2\pi ix} \n\]\n\nand ... | Yes |
Theorem 5. Let \( f \) be a holomorphic function on \( \mathfrak{H} \), satisfying\n\n\[ f\parallel {\left\lbrack \gamma \right\rbrack }_{m} = f \]\n\nall \( \gamma \in \Gamma \) .\n\nAssume that its 0 -th coefficient in the q-expansion is 0, i.e.\n\n\[ {f}^{ * }\left( q\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty... | Proof. The function \( h\left( z\right) = \left| {f\left( z\right) }\right| {y}^{m/2} \) is invariant under \( \Gamma \), as is seen at once from the definitions of the operation \( {\left\lbrack \gamma \right\rbrack }_{m},\gamma \in \Gamma \) . Also, \( h\left( z\right) \rightarrow 0 \) as \( y \rightarrow \infty \) .... | Yes |
Theorem 6. Let \( X \) be a locally compact space with a finite positive measure \( \mu \) . Let \( H \) be a closed subspace of \( {L}^{2}\left( {X,\mu }\right) = {L}^{2}\left( X\right) \), and let \( T \) be a linear map of \( H \) into the vector space of bounded continuous functions on \( X \) . Assume that there e... | Proof. We know that an operator which can be represented by a kernel in \( {L}^{2} \) is compact, cf. Chapter I,§3. To get hold of the kernel, we use the hypothesis that for each \( x \in X \), the map\n\n\[ \nf \mapsto {Tf}\left( x\right)\n\]\n\n\( f \in H \)\n\nis continuous linear on \( H \) . Hence there exists a f... | Yes |
Lemma 2. Let \( H \) be a Hilbert space with countable base, and let \( X \) be a measured space. If \( f, g : X \rightarrow H \) are weakly measurable maps, then the map \( x \mapsto \langle f\left( x\right), g\left( x\right) \rangle \), i.e. the map \( \langle f, g\rangle \), is measurable. | Proof. Let \( \left\{ {u}_{\mathrm{i}}\right\} \) be a Hilbert basis of \( H \), and let\n\n\[ f\left( x\right) = \sum {f}_{i}\left( x\right) {u}_{i} \]\n\n\[ g\left( x\right) = \sum {g}_{i}\left( x\right) {u}_{i} \]\n\nbe the Fourier expansions of \( f \) and \( g \) . Then by hypothesis, each \( {f}_{i},{g}_{i} \) is... | Yes |
Theorem 7. If \( \varphi \in {C}_{c}^{\infty }\left( G\right) \), then there exists a number \( {C}_{\varphi } \) such that for all \( f \in {}^{0}{L}^{2}\left( {\Gamma \smallsetminus G}\right) \) we have\n\n\[ \parallel \pi \left( \varphi \right) f\parallel \leq {C}_{\varphi }\parallel f{\parallel }_{2} \]\n\nwhere \(... | In view of Theorem 6 in the last section, we obtain\n\nCorollary. The operator \( \pi \left( \varphi \right) \) is compact. | No |
Theorem 8. The representation \( \pi \) by right translation on \( {}^{0}{L}^{2}\left( {\Gamma \smallsetminus G}\right) \) is completely reducible, and each irreducible component occurs only a finite number of times in it. | As usual, \( G = {NAK} \) and \( {\Gamma }_{N} = \Gamma \cap N \) . We proceed with the proof of Theorem 7. We have\n\n\[ \pi \left( \varphi \right) f\left( x\right) = {\int }_{G}f\left( y\right) \varphi \left( {{x}^{-1}y}\right) {dy} \]\n\n\[ = {\int }_{{\Gamma }_{N} \smallsetminus G}\mathop{\sum }\limits_{{\eta \in {... | Yes |
For each positive integer \( d \) there exists a constant \( C\left( {\varphi, d,{F}_{c}}\right) \) such that\n\n\[ \left| {{\widehat{\varphi }}_{x, y}\left( \lambda \right) }\right| \leq C\left( {\varphi, d,{F}_{c}}\right) {v}_{x}^{1 - d}{\left| \lambda \right| }^{-d}. \] | By definition,\n\n\[ {\widehat{\varphi }}_{x, y}\left( \lambda \right) = {\int }_{\mathbb{R}}\varphi \left( {{x}^{-1}\left( \begin{array}{ll} 1 & u \\ 0 & 1 \end{array}\right) y}\right) {e}^{-{2\pi i\lambda u}}{du} \]\n\n\[ = {\int }_{\mathbf{R}}\varphi \left( {{x}^{-1}{a}_{x} \cdot {a}_{x}^{-1}\left( \begin{array}{ll}... | Yes |
iv) Let \( \widehat{\varphi } \) be the Fourier transform of \( \varphi \), viewed as a function on \( {\mathbf{R}}^{2} \) . We have \( \widehat{\varphi }\left\lbrack 0\right\rbrack = 0 \) if and only if \( {T\varphi } \) is orthogonal to the constant 1 on \( \Gamma \smallsetminus G \) . | Proof. All of these statements are immediate from the adjointness relation. The Fourier transform of a function on \( {\mathbf{R}}^{2} \) is normalized to be, in the present case,\n\n\[ \n\widehat{\varphi }\left\lbrack z\right\rbrack = {\int }_{{\mathbb{R}}^{2}}\varphi \left\lbrack \xi \right\rbrack {e}^{-{2\pi i\xi } ... | Yes |
Lemma 1. If \( \varphi \in {C}_{c}\left( {N \smallsetminus G}\right) \), then \( Z\left( {\varphi, y,{2s}}\right) \) is entire in \( s \) . | This is essentially obvious since nothing horrible occurs either near 0 or near \( \infty \) . We had already encountered this situation in Chapter V,§3. | No |
For \( \varphi \in S\left( {N \smallsetminus G}\right) \), the Eisenstein series\n\n\[ E\left( {\varphi, y, s}\right) = \mathop{\sum }\limits_{{{\Gamma }_{N} \smallsetminus \Gamma }}Z\left( {\varphi ,{\gamma y},{2s}}\right) = {TZ}\left( {\varphi, y,{2s}}\right) \]\n\nconverges absolutely for \( \operatorname{Re}s > 1 \... | Proof. Replacing \( \varphi \) by its \( y \) -translate on the right, it suffices to prove the convergence with \( y = e \), i.e. deal with the sum\n\n\[ \mathop{\sum }\limits_{{{\Gamma }_{N} \smallsetminus \Gamma }}Z\left( {\varphi ,\gamma ,{2s}}\right) \]\n\nBut if\n\n\[ \gamma = \left( \begin{array}{ll} * & * \\ c ... | Yes |
Theorem 2. Let \( \varphi \in S\left( {N \smallsetminus G}\right) \) be an even function, and assume that\n\n\[ \varphi \left\lbrack 0\right\rbrack = \widehat{\varphi }\left\lbrack 0\right\rbrack = 0 \]\n\nDefine\n\n\[ {E}^{ * }\left( {\varphi, y, s}\right) = \zeta \left( {2s}\right) E\left( {\varphi, y, s}\right) . \]... | Proof. By definition\n\n\[ Z\left( {\varphi, y,{2s}}\right) = {\int }_{0}^{\infty }\varphi \left( {{h}_{a}y}\right) {a}^{-{2s}}\frac{da}{a}. \]\n\nLet\n\n\[ \gamma = \left( \begin{array}{ll} * & * \\ c & d \end{array}\right) \]\n\nThen\n\n\[ Z\left( {\varphi ,{\gamma y},{2s}}\right) = {\int }_{0}^{\infty }\varphi \left... | Yes |
Lemma 1. If \( f \in S\\left( \\mathbf{R}\\right) \) and \( f\\left( 0\\right) = 0 \) then \( {Mf}\\left( {2s}\\right) \) is a meromorphic function of \( s \) with at most simple poles at\n\n\[ s = - \\frac{1}{2}, - 1, - \\frac{3}{2},\\cdots \\text{.} \]\n | Proof. Assuming \( f\\left( 0\\right) = 0 \) implies that \( f\\left( a\\right) = a{f}_{1}\\left( a\\right) \) where \( {f}_{1} \) is \( {C}^{\\infty } \).\n\nIntegrating by parts the integral\n\n\[ {\\int }_{0}^{1}f\\left( a\\right) {a}^{2s}\\frac{da}{a} = {\\int }_{0}^{1}f\\left( a\\right) {a}^{{2s} - 1}{da} \]\n\ngi... | No |
Lemma 2. Let \( f \in S\left( \mathbf{R}\right) \) and assume \( f\left( 0\right) = 0 \) . Then \( {Mf}\left( {2s}\right) \) is rapidly decreasing on every vertical line uniformly in each strip \[ {\sigma }_{0} \leq \sigma \leq {\sigma }_{1} \] outside a neighborhood of the poles. | Proof. Suppose first \( \operatorname{Re}s > 0 \) . For any integer \( n > 0 \) we have, after integrating by parts, \[ \left( {{2s} + 1}\right) \left( {{2s} + 2}\right) \cdots \left( {{2s} + n}\right) {Mf}\left( {2s}\right) = M{f}^{\left( n\right) }\left( {{2s} + n}\right) , \] where \( {f}^{\left( n\right) } \) is th... | Yes |
Lemma 3. Let \( \sigma = \operatorname{Re}s > 1 \) . If the functions \( f, g \) on \( {\mathbf{R}}^{ + } \) (with respect to Haar measure \( {da}/a \) ) are such that \( f\left( a\right) {a}^{{2s} - 2} \) and \( g\left( a\right) {a}^{-2\bar{s}} \) are in \( {\mathcal{L}}^{1} \cap {\mathcal{L}}^{2}\left( {\mathbb{R}}^{... | Proof. Put \( {a}^{2} = {e}^{b} \), so that \( a = {e}^{b/2} \) . Then\n\n\[ \n\frac{da}{a} = \frac{1}{2}{db} \n\]\n\nLet \( s = \sigma + {i\tau } \) . Let\n\n\[ \n{f}_{1}\left( b\right) = f\left( {e}^{b/2}\right) {e}^{b\left( {\sigma - 1}\right) }\;\text{ and }\;{g}_{1}\left( b\right) = g\left( {e}^{b/2}\right) {e}^{-... | Yes |
Lemma 1. If \( \theta \in {C}_{c}\left( {\Gamma \smallsetminus G}\right) \), then \( {T}^{0}\theta \left( {{h}_{a}y}\right) = 0 \) for large \( a \) . | Proof. By definition,\n\n\[ \n{T}^{0}\theta \left( {{h}_{a}y}\right) = {\int }_{{\Gamma }_{N} \smallsetminus N}\theta \left( {n{h}_{a}y}\right) {dn}.\n\]\n\nAs \( a \rightarrow \infty, n{h}_{a} \) tends to infinity (i.e. lies outside a given compact set). Hence the expression under the integral sign is 0 for large \( a... | Yes |
Lemma 2. The integral\n\n\[ \nZ\left( {{T}^{0}\theta, y,2 - {2s}}\right) = {\int }_{0}^{\infty }{T}^{0}\theta \left( {{h}_{a}y}\right) {a}^{{2s} - 2}\frac{da}{a} \n\]\n\nconverges absolutely for \( \operatorname{Re}s > 1 \) . | Proof. Using Lemma 1 to cut off at infinity, our integral is estimated by\n\n\[ \n{\int }_{0}^{B}{a}^{{2\sigma } - 2}\frac{da}{a} = {\left. \frac{{a}^{{2\sigma } - 2}}{{2\sigma } - 2}\right| }_{0}^{B} \n\]\n\nwhich exists, as desired. | Yes |
Theorem 3. Let \( \varphi \in {C}_{c}^{\infty }\left( {N \smallsetminus G}\right) \) be even, and assume that \( \varphi \left\lbrack 0\right\rbrack = \widehat{\varphi }\left\lbrack 0\right\rbrack = 0 \) . Then for \( \operatorname{Re}s > 1 \) , \[ \frac{1}{2}Z\left( {{T}^{0}{T\varphi }, y,2 - {2s}}\right) = Z\left( {\... | The right-hand side is a meromorphic function in the whole plane, giving the analytic continuation of the left-hand side. It is holomorphic for \( \operatorname{Re}s \geq \frac{1}{2} \) . The proof of the above theorem will result from a number of computations, and transformations, giving rise to relations (1), (2), an... | Yes |
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