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| In [[mathematics]], a '''zonal spherical function''' or often just '''spherical function''' is a function on a [[locally compact group]] ''G'' with compact subgroup ''K'' (often a [[maximal compact subgroup]]) that arises as the [[matrix coefficient]] of a ''K''-invariant vector in an [[irreducible representation]] of ''G''. The key examples are the matrix coefficients of the ''[[principal series|spherical principal series]]'', the irreducible representations appearing in the decomposition of the [[unitary representation]] of ''G'' on ''L''<sup>2</sup>(''G''/''K''). In this case the [[commutant]] of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right [[convolution]]. It is [[commutative]] if in addition ''G''/''K'' is a [[symmetric space]], for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the [[spectrum]] of the corresponding
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| [[C* algebra]] generated by the biinvariant functions of [[compact support]], often called a [[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra]]. The spectrum of the commutative Banach *-algebra of biinvariant ''L''<sup>1</sup> functions is larger; when ''G'' is a semisimple Lie group with maximal compact subgroup ''K'', additional characters come from matrix coefficients of the [[complementary series]], obtained by analytic continuation of the spherical principal series.
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| Zonal spherical functions have been explicitly determined for real semisimple groups by [[Harish-Chandra]]. For [[special linear group]]s, they were independently discovered by [[Israel Gelfand]] and [[Mark Naimark]]. For complex groups, the theory simplifies significantly, because ''G'' is the [[complexification]] of ''K'', and the formulas are related to analytic continuations of the [[Weyl character formula]] on ''K''. The abstract [[functional analysis|functional analytic]] theory of zonal spherical functions was first developed by [[Roger Godement]]. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group ''G'' also provide a set of simultaneous [[eigenfunction]]s for the natural action of the centre of the [[universal enveloping algebra]] of ''G'' on ''L''<sup>2</sup>(''G''/''K''), as [[differential operator]]s on the symmetric space ''G''/''K''. For semisimple [[p-adic]] Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and [[Ian G. Macdonald]]. The analogues of the [[Plancherel theorem]] and [[Fourier inversion formula]] in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for [[Spectral theory of ordinary differential equations|singular ordinary differential equations]]: they were obtained in full generality in the 1960s in terms of [[Harish-Chandra's c-function]].
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| The name "zonal spherical function" comes from the case when ''G'' is SO(3,'''R''') acting on a 2-sphere and ''K'' is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.
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| ==Definitions==
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| {{See also|Hecke algebra}}
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| Let ''G'' be a [[locally compact]] [[unimodular group|unimodular]] [[topological group]] and ''K'' a [[compact space|compact]] [[subgroup]] and let ''H''<sub>1</sub> = ''L''<sup>2</sup>(''G''/''K''). Thus ''H''<sub>1</sub> admits a [[unitary representation]] π of ''G'' by left translation. This is a subrepresentation of the regular representation, since if ''H''= ''L''<sup>2</sup>(''G'') with left and right [[regular representation]]s λ and ρ of ''G'' and ''P'' is the [[orthogonal projection]]
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| :<math> P =\int_K \rho(k) \, dk</math>
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| from ''H'' to ''H''<sub>1</sub> then ''H''<sub>1</sub> can naturally be identified with ''PH'' with the action of ''G'' given by the restriction of λ.
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| On the other hand by [[commutation theorem|von Neumann's commutation theorem]]<ref>{{harvnb|Dixmier|1996}}, Algèbres hilbertiennes.</ref>
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| :<math> \lambda(G)^\prime= \rho(G)^{\prime\prime},</math>
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| where ''S''' denotes the [[commutant]] of a set of operators ''S'', so that
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| :<math> \pi(G)^\prime = P \rho(G)^{\prime\prime}P. </math>
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| Thus the commutant of π is generated as a [[von Neumann algebra]] by operators
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| :<math> P\rho(f)P = \int_G f(g) (P \rho(g)P) \, dg</math>
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| where ''f'' is a continuous function of compact support on ''G''.<ref>If σ is a unitary representation of ''G'', then <math>\sigma(f)=\int_G f(g)\sigma(g)\, dg</math>.</ref>
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| However ''P''ρ(''f'') ''P'' is just the restriction of ρ(''F'') to ''H''<sub>1</sub>, where
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| :<math> F(g) =\int_K \int_K f(kgk^\prime) \, dk\, dk^\prime </math>
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| is the ''K''-biinvariant continuous function of compact support obtained by averaging ''f'' by ''K'' on both sides.
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| Thus the commutant of π is generated by the restriction of the operators ρ(''F'') with ''F'' in
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| ''C''<sub>c</sub>(''K''\''G''/''K''), the ''K''-biinvariant continuous functions of compact support on ''G''.
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| These functions form a [[* algebra]] under [[convolution]] with involution
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| :<math> F^*(g) =\overline{F(g^{-1})}, </math>
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| often called the [[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra]] for the pair (''G'', ''K'').
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| Let ''A''(''K''\''G''/''K'') denote the [[C* algebra]] generated by the operators ρ(''F'') on ''H''<sub>1</sub>.
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| The pair (''G'', ''K'')
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| is said to be a [[Gelfand pair]] <ref name="harvnb|Dieudonné |1978">{{harvnb|Dieudonné |1978}}</ref> if one, and hence all, of the following algebras are [[commutative]]:
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| * <math> \pi(G)^\prime</math>
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| *<math> C_c(K\backslash G /K) </math>
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| *<math> A(K\backslash G /K).</math>
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| Since ''A''(''K''\''G''/''K'') is a commutative [[C* algebra]], by the [[Gelfand–Naimark theorem]] it has the form ''C''<sub>0</sub>(''X''),
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| where ''X'' is the locally compact space of norm continuous * [[homomorphism]]s of ''A''(''K''\''G''/''K'') into '''C'''.
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| A concrete realization of the * homomorphisms in ''X'' as ''K''-biinvariant [[uniformly bounded]] functions on ''G'' is obtained as follows.<ref name="harvnb|Dieudonné |1978"/><ref name="harvnb|Godement|1952">{{harvnb|Godement|1952}}</ref><ref name="Helgason2001">{{harvnb|Helgason|2001}}</ref><ref name="harvnb|Helgason|1984">{{harvnb|Helgason|1984}}</ref><ref name="Lang1985">{{harvnb|Lang|1985}}</ref>
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| Because of the estimate
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| :<math> \|\pi(F)\|\le \int_G |F(g)| \, dg \equiv \|F\|_1,</math>
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| the representation π of ''C''<sub>c</sub>(''K''\''G''/''K'') in ''A''(''K''\''G''/''K'') extends by continuity
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| to L<sup>1</sup>(''K''\''G''/''K''), the [[* algebra]] of ''K''-biinvariant integrable functions. The image forms | |
| a dense * subalgebra of ''A''(''K''\''G''/''K''). The restriction of a * homomorphism χ continuous for the operator norm is
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| also continuous for the norm ||·||<sub>1</sub>. Since the [[Banach space|Banach space dual]] of L<sup>1</sup> is L<sup>∞</sup>,
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| it follows that
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| :<math> \chi(\pi(F)) =\int_G F(g) h(g) \, dg,</math>
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| for some unique uniformly bounded ''K''-biinvariant function ''h'' on ''G''. These functions ''h'' are exactly the '''zonal spherical functions''' for the pair (''G'', ''K'').
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| ==Properties==
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| A zonal spherical function ''h'' has the following properties:<ref name="harvnb|Dieudonné |1978"/>
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| # ''h'' is uniformly continuous on ''G''
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| # <math> h(x) h(y) = \int_K h(xky) \,dk \,\,(x,y\in G).</math>
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| # ''h''(1) =1 (normalisation)
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| # ''h'' is a [[positive definite function on a group|positive definite function]] on ''G''
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| # ''f'' * ''h'' is proportional to ''h'' for all ''f'' in ''C''<sub>c</sub>(''K''\''G''/''K'').
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| These are easy consequences of the fact that the bounded linear functional χ defined by ''h'' is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection
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| with [[unitary representation]]s. For semisimple Lie groups, there is a further characterization as eigenfunctions of
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| [[invariant differential operator]]s on ''G''/''K'' (see below).
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| In fact, as a special case of the [[Gelfand–Naimark–Segal construction]], there is one-one correspondence between
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| irreducible representations σ of ''G'' having a unit vector ''v'' fixed by ''K'' and zonal spherical functions
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| ''h'' given by
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| :<math> h(g) = (\sigma(g) v,v).</math>
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| Such irreducible representations are often described as having '''class one'''. They are precisely the irreducible representations required to decompose the [[induced representation]] π on ''H''<sub>1</sub>. Each representation σ extends uniquely by continuity
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| to ''A''(''K''\''G''/''K''), so that each zonal spherical function satisfies
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| :<math> \left|\int_G f(g) h(g)\, dg\right| \le \|\pi(f)\|</math>
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| for ''f'' in ''A''(''K''\''G''/''K''). Moreover, since the commutant π(''G'')' is commutative,
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| there is a unique probability measure μ on the space of * homomorphisms ''X'' such that
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| :<math> \int_G |f(g)|^2 \, dg = \int_X |\chi(\pi(f))|^2 \, d\mu(\chi).</math>
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| μ is called the '''[[Plancherel measure]]'''. Since π(''G'')' is the [[von Neumann algebra|centre]] of the von Neumann algebra generated by ''G'', it also gives the measure associated with the [[direct integral]] decomposition of ''H''<sub>1</sub> in terms of the irreducible representations σ<sub>χ</sub>.
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| ==Gelfand pairs==
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| {{See also|Gelfand pair}}
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| If ''G'' is a [[Connectedness|connected]] [[Lie group]], then, thanks to the work of [[Élie Cartan|Cartan]], [[Anatoly Maltsev|Malcev]], [[Kenkichi Iwasawa|Iwasawa]] and [[Claude Chevalley|Chevalley]], ''G'' has a [[maximal compact subgroup]], unique up to conjugation.<ref>{{harvnb|Cartier|1954-1955}}</ref><ref>{{harvnb|Hochschild|1965}}</ref> In this case ''K'' is connected and the quotient ''G''/''K'' is diffeomorphic to a Euclidean space. When ''G'' is in addition [[Semisimple Lie group|semisimple]], this can be seen directly using the [[Cartan decomposition]] associated to the [[symmetric space]] ''G''/''K'', a generalisation of the [[polar decomposition]] of invertible matrices. Indeed if τ is the associated period two automorphism of ''G'' with fixed point subgroup ''K'', then
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| :<math>G=P\cdot K,</math>
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| where
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| :<math> P= \{g\in G| \tau(g)=g^{-1}\}.</math>
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| Under the [[exponential map]], ''P'' is diffeomorphic to the -1 eigenspace of τ in the [[Lie algebra]] of ''G''.
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| Since τ preserves ''K'', it induces an automorphism of the Hecke algebra ''C''<sub>c</sub>(''K''\''G''/''K''). On the
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| other hand, if ''F'' lies in ''C''<sub>c</sub>(''K''\''G''/''K''), then
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| :''F''(τ''g'') = ''F''(''g''<sup>−1</sup>),
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| so that τ induces an anti-automorphism, because inversion does. Hence, when ''G'' is semisimple,
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| *the Hecke algebra is commutative
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| *(''G'',''K'') is a Gelfand pair.
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| More generally the same argument gives the following criterion of Gelfand for (''G'',''K'') to be a Gelfand pair:<ref>{{harvnb|Dieudonné|1978|pp=55–57}}</ref>
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| *''G'' is a unimodular locally compact group;
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| *''K'' is a compact subgroup arising as the fixed points of a period two automorphism τ of ''G'';
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| *''G'' =''K''·''P'' (not necessarily a direct product), where ''P'' is defined as above.
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| The two most important examples covered by this are when:
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| * ''G'' is a compact connected semisimple Lie group with τ a period two automorphism;<ref>{{harvnb|Dieudonné|1977}}</ref><ref>{{harvnb|Helgason|1978|p=249}}</ref>
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| * ''G'' is a semidirect product <math>A\rtimes K</math>, with ''A'' a locally compact Abelian group without 2-torsion and τ(''a''· ''k'')= ''k''·''a''<sup>−1</sup> for ''a'' in ''A'' and ''k'' in ''K''.
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| The three cases cover the three types of [[symmetric space]]s ''G''/''K'':<ref name="harvnb|Helgason|1984"/>
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| # ''Non-compact type'', when ''K'' is a maximal compact subgroup of a non-compact real semisimple Lie group ''G'';
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| # ''Compact type'', when ''K'' is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group ''G'';
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| # ''Euclidean type'', when ''A'' is a finite-dimensional Euclidean space with an orthogonal action of ''K''.
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| ==Cartan–Helgason theorem==
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| Let ''G'' be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a ''G'' with fixed point subgroup ''K'' = ''G''<sup>τ</sup>. In this case ''K'' is a connected compact Lie group.<ref name="harvnb|Helgason|1984" /> In addition let ''T'' be a [[maximal torus]] of ''G'' invariant under τ, such that ''T'' <math>\cap</math> ''P'' is a maximal torus in ''P'', and set<ref>{{harvnb|Helgason|1978|pp=257–264}}</ref>
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| :<math>S= K\cap T = T^\tau.</math>
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| ''S'' is the direct product of a torus and an [[elementary abelian group|elementary abelian 2-group]].
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| In 1929 [[Élie Cartan]] found a rule to determine the decomposition of L<sup>2</sup>(''G''/''K'') into the direct sum of finite-dimensional [[irreducible representation]]s of ''G'', which was proved rigorously only in 1970 by [[Sigurdur Helgason (mathematician)|Sigurdur Helgason]]. Because the commutant of ''G'' on L<sup>2</sup>(''G''/''K'') is commutative, each irreducible representation appears with multiplicity one. By [[Frobenius reciprocity]] for compact groups, the irreducible representations ''V'' that occur are precisely those admitting a non-zero vector fixed by ''K''.
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| From the [[Weyl character formula|representation theory of compact semisimple groups]], irreducible representations of ''G'' are classified by their [[root system|highest weight]]. This is specified by a homomorphism of the maximal torus ''T'' into '''T'''.
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| The '''Cartan–Helgason theorem'''<ref>{{harvnb|Helgason|1984|pp=534–538}}</ref><ref>{{harvnb|Goodman|Wallach|1998|pp=549–550}}</ref> states that
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| :{| border="1" cellspacing="0" cellpadding="5" |
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| |the irreducible representations of ''G'' admitting a non-zero vector fixed by ''K'' are precisely those with highest weights corresponding to homomorphisms trivial on ''S''.
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| |}
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| The corresponding irreducible representations are called ''spherical representations''.
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| The theorem can be proved<ref name="harvnb|Helgason|1984"/> using the [[Iwasawa decomposition]]:
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| :<math> \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n},</math>
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| where <math>\mathfrak{g}</math>, <math>\mathfrak{k}</math>, <math>\mathfrak{a}</math> are the complexifications of the [[Lie algebra]]s of ''G'', ''K'', ''A'' = ''T'' <math>\cap</math> ''P'' and
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| :<math>\mathfrak{n}=\bigoplus \mathfrak{g}_\alpha,</math>
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| summed over all eigenspaces for ''T'' in <math>\mathfrak{g}</math> corresponding to [[root system|positive roots]] α not fixed by τ.
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| Let ''V'' be a spherical representation with highest weight vector ''v''<sub>0</sub> and ''K''-fixed vector ''v''<sub>''K''</sub>. Since ''v''<sub>0</sub> is an eigenvector of the solvable Lie algebra <math>\mathfrak{a}\oplus\mathfrak{p}</math>, the [[Poincaré–Birkhoff–Witt theorem]]
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| implies that the ''K''-module generated by ''v''<sub>0</sub> is the whole of ''V''. If ''Q'' is the orthogonal projection onto the fixed points of ''K'' in ''V'' obtained by averaging over ''G'' with respect to [[Haar measure]], it follows that
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| :<math>\displaystyle{ v_K = c Qv_0}</math> | |
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| for some non-zero constant ''c''. Because ''v''<sub>''K''</sub> is fixed by ''S'' and ''v''<sub>0</sub> is an eigenvector for ''S'', the subgroup ''S'' must actually fix ''v''<sub>0</sub>, an equivalent form of the triviality condition on ''S''.
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| Conversely if ''v''<sub>0</sub> is fixed by ''S'', then it can be shown<ref>{{harvnb|Goodman|Wallach|1998|p=550}}</ref> that the matrix coefficient
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| :<math> \displaystyle{f(g) =(gv_0,v_0)}</math>
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| is non-negative on ''K''. Since ''f''(1) > 0, it follows that (''Qv''<sub>0</sub>, ''v''<sub>0</sub>) > 0 and hence that ''Qv''<sub>0</sub> is a non-zero vector fixed by ''K''.
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| ==Harish-Chandra's formula==
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| If ''G'' is a non-compact semisimple Lie group, its maximal compact subgroup ''K'' acts by conjugation on the component ''P'' in the [[Cartan decomposition]]. If ''A'' is a maximal Abelian subgroup of ''G'' contained in ''P'', then ''A'' is isomorphic to its Lie algebra under the exponential map and, as a [[Lie group decompositions|further generalisation]] of the [[polar decomposition]] of matrices, every element of ''P'' is conjugate under ''K'' to an element of ''A'', so that<ref>{{harvnb|Helgason|1978}}, Chapter IX.</ref>
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| :''G'' =''KAK''.
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| There is also an associated [[Iwasawa decomposition]]
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| :''G'' =''KAN'',
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| where ''N'' is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by ''A''. Thus
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| ''S''=''AN'' is a closed [[solvable group|solvable subgroup]] of ''G'', the [[semidirect product]] of ''N'' by ''A'', and ''G'' = ''KS''.
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| If α in Hom(''A'','''T''') is a [[character (mathematics)|character]] of ''A'', then α extends to a character of ''S'', by defining it to be trivial on ''N''. There is a corresponding [[unitary representation|unitary]] [[induced representation]] σ of ''G'' on L<sup>2</sup>(''G''/''S'') = L<sup>2</sup>(''K''),<ref>{{harvnb|Harish-Chandra|1954a|p= 251}}</ref> a so-called [[principal series representation|(spherical) principal series representation]].
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| This representation can be described explicitly as follows. Unlike ''G'' and ''K'', the solvable Lie group ''S'' is not unimodular. Let ''dx'' denote left invariant Haar measure on ''S'' and Δ<sub>''S''</sub> the [[Haar measure#The modular function|modular function]] of ''S''. Then<ref name="harvnb|Helgason|1984" />
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| :<math> \int_G f(g) \,dg = \int_S\int_K f(x\cdot k) \, dx\, dk = \int_S\int_K f(k\cdot x) \Delta_S(x)\,dx\, dk.</math>
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| The principal series representation σ is realised on L<sup>2</sup>(''K'') as<ref>{{harvnb|Wallach|1973}}</ref>
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| :<math> (\sigma(g) \xi)(k) = \alpha^\prime(g^{-1}k)^{-1} \, \xi(U(g^{-1}k)), </math>
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| where
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| :<math>g = U(g)\cdot X(g)</math>
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| is the Iwasawa decomposition of ''g'' with ''U''(''g'') in ''K'' and ''X''(''g'') in ''S'' and | |
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| :<math>\alpha^\prime(kx) = \Delta_S(x)^{1/2} \alpha(x)</math>
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| for ''k'' in ''K'' and ''x'' in ''S''.
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| The representation σ is irreducible, so that if ''v'' denotes the constant function 1 on ''K'', fixed by ''K'',
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| :<math> \varphi_\alpha(g)=(\sigma(g)v,v)</math>
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| defines a zonal spherical function of ''G''.
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| Computing the inner product above leads to '''Harish-Chandra's formula''' for the zonal spherical function
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |<math>\varphi_\alpha(g) = \int_K \alpha^\prime(gk)^{-1}\, dk </math>
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| |}
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| as an integral over ''K''. | |
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| Harish-Chandra proved that these zonal spherical functions exhaust the characters of the [[C* algebra]] generated by the ''C''<sub>''c''</sub>(''K'' \ ''G'' / ''K'') acting by right convolution on ''L''<sup>2</sup>(''G'' / ''K''). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·''s'', where ''s'' is in the [[Weyl group]] of ''A''
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| :<math> W(A)=N_K(A)/C_K(A),</math>
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| the quotient of the [[normaliser]] of ''A'' in ''K'' by its [[centraliser]], a [[finite reflection group]].
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| It can also be verified directly<ref name="harvnb|Dieudonné |1978"/> that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of ''G''-[[invariant differential operator]]s on ''G''/''K'' and their simultaneous [[eigenfunctions]] (see below).<ref name="Helgason2001" /><ref name="harvnb|Helgason|1984"/> In the case of complex semisimple groups, Harish-Chandra and [[Felix Berezin]] realised independently that the formula simplified considerably and could be proved more directly.<ref name="harvnb|Helgason|1984" /><ref>{{harvnb|Berezin|1956a}}</ref><ref>{{harvnb|Berezin|1956b}}</ref><ref>{{harvnb|Harish-Chandra|1954b}}</ref><ref>{{harvnb|Harish-Chandra|1954c}}</ref>
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| The remaining positive-definite zonal spherical functions are given
| |
| by Harish-Chandra's formula with α in Hom(''A'','''C'''*) instead of Hom(''A'','''T'''). Only certain α are permitted and the corresponding irreducible
| |
| representations arise as analytic continuations of the spherical principal series. This so-called "[[complementary series]]" was first studied by {{harvtxt|Bargmann|1947}} for ''G'' = SL(2,'''R''') and by {{harvtxt|Harish-Chandra|1947}} and {{harvtxt|Gelfand|Naimark|1947}} for ''G'' = SL(2,'''C''').
| |
| Subsequently in the 1960s, the construction of a [[complementary series]] by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, [[Elias Stein]] and [[Bertram Kostant]].<ref>{{harvnb|Kunze|Stein|1961}}</ref><ref>{{harvnb|Stein|1970}}</ref><ref>{{harvnb|Kostant|1969}}</ref> Since these irreducible representations are not [[tempered representation|tempered]], they are not usually required for harmonic analysis on ''G'' (or ''G'' / ''K'').
| |
| | |
| ==Eigenfunctions==
| |
| Harish-Chandra proved<ref name="Helgason2001" /><ref name="harvnb|Helgason|1984"/> that zonal spherical functions can be characterised as those normalised positive definite ''K''-invariant functions on ''G''/''K'' that are eigenfunctions of ''D''(''G''/''K''), the algebra of invariant differential operators on ''G''. This algebra acts on ''G''/''K'' and commutes with the natural action of ''G'' by left translation. It can be identified with the subalgebra of the [[universal enveloping algebra]] of ''G'' fixed under the [[Adjoint representation of a Lie group|adjoint action]] of ''K''. As for the commutant of ''G'' on L<sup>2</sup>(''G''/''K'') and the corresponding Hecke algebra, this algebra of operators is [[commutative]]; indeed it is a subalgebra of the [[affiliated operator|algebra of mesurable operators]] affiliated with the commutant π(''G'')', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of ''W''(''A'')-invariant polynomials on the Lie algebra of ''A'', which itself is a [[polynomial ring]] by the [[Chevalley–Shephard–Todd theorem]] on polynomial invariants of [[finite reflection group]]s. The simplest invariant differential operator on ''G''/''K'' is the [[Laplacian operator]]; up to a sign this operator is just the image under π of the [[Casimir operator]] in the centre of the universal enveloping algebra of ''G''.
| |
| | |
| Thus a normalised positive definite ''K''-biinvariant function ''f'' on ''G'' is a zonal spherical function if and only if for each ''D'' in ''D''(''G''/''K'') there is a constant λ<sub>''D''</sub> such that
| |
| | |
| :<math>\displaystyle\pi(D)f =\lambda_D f,</math>
| |
| | |
| i.e. ''f'' is a simultaneous [[eigenfunction]] of the operators π(''D'').
| |
| | |
| If ψ is a zonal spherical function, then, regarded as a function on ''G''/''K'', it is an eigenfunction of the Laplacian
| |
| there, an [[elliptic differential operator]] with [[real analytic]] coefficients. By [[FBI transform#Holmgren's uniqueness theorem|analytic elliptic regularity]],
| |
| ψ is a real analytic function on ''G''/''K'', and hence ''G''.
| |
| | |
| Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.<ref>{{harvnb|Harish-Chandra|1958}}</ref><ref>{{harvnb| Helgason|2001}}, pages 418–422, 427-434</ref><ref>{{harvnb|Helgason|1984|p=418}}</ref> Indeed the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
| |
| | |
| ==Example: SL(2,C)==
| |
| {{See also|SL(2,C)|Representations of the Lorentz group|Spectral theory of ordinary differential equations}}
| |
| The group ''G'' = SL(2,'''C''') is the [[complexification]] of the [[compact Lie group]] ''K'' = SU(2) and the [[Double covering group|double cover]] of the [[Lorentz group]]. The infinite-dimensional representations of the Lorentz group were first studied by [[Paul Dirac|Dirac]] in 1945, who considered the [[discrete series]] representations, which he termed ''expansors''. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and
| |
| Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial
| |
| component of the [[Laplacian operator]].<ref name="harvnb|Helgason|1984" />
| |
| | |
| Indeed any unimodular complex 2×2 matrix ''g'' admits a unique [[polar decomposition]] ''g'' = ''pv'' with ''v'' unitary and ''p'' positive. In turn
| |
| ''p'' = ''uau''*, with ''u'' unitary and ''a'' a diagonal matrix with positive entries. Thus ''g'' = ''uaw'' with ''w'' = ''u''* ''v'', so that any ''K''-biinvariant function on ''G'' corresponds to a function of the diagonal matrix
| |
| | |
| :<math>a = \begin{pmatrix} e^{r/2} & 0 \\ 0 & e^{-r/2} \end{pmatrix},</math>
| |
| | |
| invariant under the Weyl group. Identifying ''G''/''K'' with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate ''r'', the Laplacian is given by<ref>{{harvnb|Davies|1990}}</ref>
| |
| | |
| :<math>L= -\partial_r^2 - 2 \coth r \partial_r.</math>
| |
| | |
| Setting ''f''(''r'') = sinh (''r'')·ψ(''r''), it follows that ''f'' is an [[odd function]] of ''r'' and an eigenfunction of <math>\partial_r^2</math>.
| |
| | |
| Hence
| |
| | |
| :{| border="1" cellspacing="0" cellpadding="5"
| |
| |<math>\varphi(r) = {\sin (\ell r)\over \ell \sinh r}</math>
| |
| |}
| |
| | |
| where <math>\ell</math> is real.
| |
| | |
| There is a similar elementary treatment for the [[Lorentz group#General dimensions|generalized Lorentz group]]s SO(''N'',1) in {{harvtxt|Takahashi|1963}} and {{harvtxt|Faraut|Korányi|1994}} (recall that SO<sup>0</sup>(3,1) = SL(2,'''C''') / ±I).
| |
| | |
| ==Complex case==
| |
| If ''G'' is a complex semisimple Lie group, it is the [[complexification]] of its maximal compact subgroup ''K''. If <math>{\mathfrak
| |
| g}</math> and <math>\mathfrak{k}</math> are their Lie algebras, then
| |
| | |
| :<math> \mathfrak{g} = \mathfrak{k} \oplus i\mathfrak{k}.</math>
| |
| | |
| Let ''T'' be a [[maximal torus]] in ''K'' with Lie algebra <math>\mathfrak{t}</math>. Then
| |
| | |
| :<math>A= \exp i \mathfrak{t}, \,\, P= \exp i \mathfrak{k}.</math>
| |
| | |
| Let
| |
| | |
| :<math> W= N_K(T)/T</math>
| |
| | |
| be the [[Weyl group]] of ''T'' in ''K''. Recall characters in Hom(''T'','''T''') are called [[weight (representation theory)|weights]] and can be identified with elements of the [[weight lattice]] Λ in
| |
| Hom(<math>\mathfrak{t}</math>, '''R''') = <math>\mathfrak{t}^*</math>. There is a natural ordering on weights and very finite-dimensional irreducible representation (π, ''V'') of ''K'' has a unique highest weight λ. The weights of the [[Adjoint representation of a Lie group|adjoint representation]] of ''K'' on <math>\mathfrak{k}\ominus \mathfrak{t}</math> are called roots and ρ is used to denote half the sum of the [[positive root]]s α, [[Weyl's character formula]] asserts that for ''z'' = exp ''X'' in ''T''
| |
| | |
| :<math> \displaystyle\chi_\lambda(e^X)\equiv {\rm Tr} \, \pi(z) = A_{\lambda+\rho}(e^X)/A_{\rho}(e^X),</math>
| |
| | |
| where, for μ in <math>\mathfrak{t}^*</math>, ''A''<sub>μ</sub> denotes the antisymmetrisation
| |
| | |
| :<math>\displaystyle A_\mu(e^X) =\sum_{s\in W} \varepsilon(s) e^{i\mu(sX)},</math>
| |
| | |
| and ε denotes the ''sign character'' of the [[finite reflection group]] ''W''.
| |
| | |
| [[Weyl's denominator formula]] expresses the denominator ''A''<sub>ρ</sub> as a product:
| |
| | |
| :<math>\displaystyle A_\rho(e^X) = e^{i\rho(X)} \prod_{\alpha>0}(1 - e^{-i\alpha(X)}),</math>
| |
| | |
| where the product is over the positive roots.
| |
| | |
| [[Weyl's dimension formula]] asserts that
| |
| | |
| :<math>\displaystyle\chi_\lambda(1) \equiv {\rm dim}\, V = {\prod_{\alpha>0} (\lambda + \rho,\alpha)\over \prod_{\alpha>0} (\rho,\alpha)}.</math>
| |
| | |
| where the [[inner product]] on <math>\mathfrak{t}^*</math> is that associated with the [[Killing form]] on <math>\mathfrak{k}</math>.
| |
| | |
| Now
| |
| | |
| *every irreducible representation of ''K'' extends holomorphically to the complexification ''G''
| |
| | |
| *every irreducible character χ<sub>λ</sub>(''k'') of ''K'' extends holomorphically to the complexification of ''K'' and <math>\mathfrak{t}^*</math>.
| |
| | |
| * for every λ in Hom(''A'','''T''') = <math> i\mathfrak{t}^*</math>, there is a zonal spherical function φ<sub>λ</sub>.
| |
| | |
| The '''Berezin–Harish–Chandra formula'''<ref name="harvnb|Helgason|1984"/> asserts that for ''X'' in <math>i\mathfrak{t}</math>
| |
| | |
| :{| border="1" cellspacing="0" cellpadding="5"
| |
| |<math> \varphi_\lambda(e^X) = {\chi_\lambda(e^X)\over\chi_\lambda(1)}.</math>
| |
| |}
| |
| | |
| In other words:
| |
| | |
| *'''''the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.'''''
| |
| | |
| One of the simplest proofs<ref>{{harvnb|Helgason|1984|pp=432–433}}</ref> of this formula involves the ''radial component'' on ''A'' of the Laplacian on ''G'', a proof formally parallel to Helgason's reworking of [[Freudenthal]]'s classical proof of the [[Weyl character formula]], using the radial component on ''T'' of the Laplacian on ''K''.<ref>{{harvnb|Helgason|1984|pp= 501–502}}</ref>
| |
| | |
| In the latter case the [[class function]]s on ''K'' can be identified with ''W''-invariant functions on ''T''. The
| |
| radial component of Δ<sub>''K''</sub> on ''T'' is just the expression for the restriction of Δ<sub>''K''</sub> to ''W''-invariant functions on ''T'', where
| |
| it is given by the formula
| |
| | |
| :<math>\displaystyle \Delta_K= h^{-1}\circ \Delta_T \circ h + \|\rho\|^2,</math>
| |
| | |
| where
| |
| | |
| :<math>\displaystyle h(e^X) = A_\rho(e^X)</math>
| |
| | |
| for ''X'' in <math>\mathfrak{t}</math>. If χ is a character with highest weight λ, it follows that φ = ''h''·χ satisfies
| |
| | |
| :<math> \Delta_T \varphi= (\|\lambda + \rho\|^2 -\|\rho\|^2) \varphi.</math>
| |
| | |
| Thus for every weight μ with non-zero [[Fourier coefficient]] in φ,
| |
| | |
| :<math>\displaystyle \|\lambda +\rho\|^2 = \|\mu+\rho\|^2.</math>
| |
| | |
| The classical argument of Freudenthal shows that μ + ρ must have the form ''s''(λ + ρ) for some ''s'' in ''W'', so the character formula
| |
| follows from the antisymmetry of φ.
| |
| | |
| Similarly ''K''-biinvariant functions on ''G'' can be identified with ''W''(''A'')-invariant functions on ''A''. The
| |
| radial component of Δ<sub>''G''</sub> on ''A'' is just the expression for the restriction of Δ<sub>''G''</sub> to ''W''(''A'')-invariant functions on ''A''.
| |
| It is given by the formula
| |
| | |
| :<math>\displaystyle \Delta_G= H^{-1}\circ \Delta_A\circ H - \|\rho\|^2,</math>
| |
| | |
| where
| |
| | |
| :<math>\displaystyle H(e^X) = A_\rho(e^X)</math>
| |
| | |
| for ''X'' in <math>i\mathfrak{t}</math>.
| |
| | |
| The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
| |
| | |
| :<math>\displaystyle f= H\cdot\varphi,</math>
| |
| | |
| which is an eigenfunction of the Laplacian Δ<sub>''A''</sub>. Since ''K'' is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to [[root of a polynomial|simple roots]], its complexification ''G'' is generated by the corresponding homomorphic images of SL(2,'''C'''). The formula for zonal spherical functions of SL(2,'''C''') implies that ''f'' is a [[periodic function]] on <math>i\mathfrak{t}</math> with respect to some [[lattice (discrete subgroup)|sublattice]]. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
| |
| | |
| ==Example: SL(2,R)==
| |
| {{See also|SL(2,R)|Representation theory of SL2(R)|Spectral theory of ordinary differential equations}}
| |
| The theory of zonal spherical functions for [[SL(2,R)|SL(2,'''R''')]] originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by [[Hermann Weyl]]'s important work on the [[spectral theory of ordinary differential equations]]. The radial part of the Laplacian in this case leads to a [[hypergeometric differential equation]], the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semimisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,'''R'''), the general theory of unitary irreducible representations of SL(2,'''R''') was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.
| |
| | |
| The group ''G'' = [[SL(2,R)|SL(2,'''R''')]] is a [[Double covering group|double cover]] of the 3-dimensional [[Lorentz group]] SO(2,1), the [[symmetry group]] of the [[Hyperbolic space|hyperbolic plane]] with its [[Poincaré metric]]. It acts by [[Moebius transformation]]s. The upper half-plane can be identified with the unit disc by the [[Cayley transform]]. Under this identification ''G'' becomes identified with the group SU(1,1), also acting by Moebius transformations. Because the action is [[transitive action|transitive]], both spaces can be identified with ''G''/''K'', where ''K'' = [[SO(2)]]. The metric is invariant under ''G'' and the associated Laplacian is ''G''-invariant, coinciding with the image of the [[Casimir operator]]. In the upper half-plane model the Laplacian is given by the formula<ref name="harvnb|Helgason|1984" /><ref name="Lang1985" />
| |
| | |
| :<math>\displaystyle\Delta=-4y^{2}(\partial_x^2 +\partial_y^2).</math>
| |
| | |
| If ''s'' is a complex number and ''z'' = ''x + i y'' with ''y'' > 0, the function
| |
| | |
| :<math>\displaystyle f_s(z) =y^{s}= \exp({s}\cdot\log y),</math>
| |
| | |
| is an eigenfunction of Δ:
| |
| | |
| :<math>\displaystyle \Delta f_s = 4s(1-s) f_s.</math>
| |
| | |
| Since Δ commutes with ''G'', any left translate of ''f''<sub>''s''</sub> is also an eigenfunction with the same eigenvalue. In particular, averaging over ''K'', the function
| |
| | |
| :{| border="1" cellspacing="0" cellpadding="5"
| |
| |<math> \varphi_s(z) =\int_K f_s(k\cdot z)\, dk</math>
| |
| |}
| |
| | |
| is a ''K''-invariant eigenfunction of Δ on ''G''/''K''. When
| |
| | |
| :<math>\displaystyle s={1\over 2} + i\tau,</math>
| |
| | |
| with τ real, these functions give all the zonal spherical functions on ''G''. As with Harish-Chandra's more general formula for semisimple Lie groups, φ<sub>''s''</sub> is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by ''K'' in the [[principal series]]. Various arguments are available to prove that there are no others. One of the simplest classical [[Lie algebra]]ic arguments<ref name="harvnb|Helgason|1984"/><ref name="Lang1985" /><ref name="Bargmann 1947">{{harvnb|Bargmann|1947}}</ref><ref name="Howe 1992">{{harvnb|Howe | Tan |1992}}</ref><ref>{{harvnb|Wallach|1988}}</ref> is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds is the matrix coefficient for a vector ''v'' and representation σ, the vector ''v'' is an [[unitary representation|analytic vector]] for ''G'' and
| |
| | |
| :<math>\displaystyle(\sigma(e^{X})v,v)= \sum_{n=0}^\infty (\sigma(X)^n v,v)/n!</math>
| |
| | |
| for ''X'' in <math>i\mathfrak{t}</math>. The infinitesimal form of the irreducible unitary representations with a vector fixed by ''K'' were worked out classically by Bargmann.<ref name="Bargmann 1947"/><ref name="Howe 1992"/> They correspond precisely to the principal series of SL(2,'''R'''). It follows that the zonal spherical function corresponds to a principal series representation.
| |
| | |
| Another classical argument<ref>{{harvnb|Helgason|2001|p= 405}}</ref> proceeds by showing that on radial functions the Laplacian has the form
| |
| | |
| :<math>\displaystyle\Delta=-\partial_r^2 - \coth(r)\cdot \partial_r,</math>
| |
| | |
| so that, as a function of ''r'', the zonal spherical function φ(''r'') must satisfy the [[ordinary differential equation]]
| |
| | |
| :<math>\displaystyle\varphi^{\prime\prime} + \coth r\, \varphi^\prime = \alpha \, \varphi</math>
| |
| | |
| for some constant α. The change of variables ''t'' = sinh ''r'' transforms this equation into the [[hypergeometric differential equation]]. The general solution in terms of [[Legendre functions]] of complex index is given by<ref name="harvnb|Dieudonné |1978"/><ref>{{harvnb|Bateman|Erdelyi|1953|p=156}}</ref>
| |
| | |
| :{| border="1" cellspacing="0" cellpadding="5"
| |
| |<math>\varphi(r)=P_\rho(\cosh r) = {1\over 2\pi} \int_0^{2\pi} (\cosh r + \sinh r \, \cos \theta)^\rho \, d\theta,</math>
| |
| |}
| |
| | |
| where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on ''G''.
| |
| | |
| There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,'''R'''), including the Plancherel formula, from the corresponding results for SL(2,'''C'''), which are simple consequences of the Plancherel formula and Fourier inversion formula for '''R'''. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.<ref>{{harvnb|Flensted-Jensen|1978}}</ref><ref>{{harvnb|Helgason|1984|pages=489–491}}</ref>
| |
| | |
| ==Further directions==
| |
| *''The theory of zonal functions that are not necessarily positive-definite.'' These are given by the same formulas as above, but without restrictions on the complex parameter ''s'' or ρ. They correspond to non-unitary representations.<ref name="harvnb|Helgason|1984" />
| |
| *[[Plancherel theorem for spherical functions|''Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions]].''<ref>{{harvnb|Helgason|1984|pp= 434–458}}</ref> This is an important special case of his [[Plancherel theorem]] for real semisimple Lie groups.
| |
| *''The structure of the Hecke algebra''. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between C<sub>c</sub><sup>∞</sup>(''K'' \ ''G'' / ''K'' ) and C<sub>c</sub><sup>∞</sup>(''A'')<sup>''W''</sup>, the subalgebra invariant under the Weyl group.<ref name="harvnb|Godement|1952"/> This is straightforward to establish for SL(2,'''R''').<ref name="Lang1985" />
| |
| *''Spherical functions for [[Euclidean group|Euclidean motion groups]] and [[compact Lie group]]s''.<ref name="harvnb|Helgason|1984"/>
| |
| *''Spherical functions for [[p-adic]] Lie groups''. These were studied in depth by Satake and [[Ian G. Macdonald|Macdonald]].<ref>{{harvnb|Satake|1963}}</ref><ref>{{harvnb|Macdonald|1971}}</ref> Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the [[Langlands program]].
| |
| | |
| ==See also==
| |
| *[[Plancherel theorem for spherical functions]]
| |
| *[[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra of a locally compact group]]
| |
| *[[Representations of Lie groups]]
| |
| *[[Non-commutative harmonic analysis]]
| |
| *[[Tempered representation]]
| |
| *[[Positive definite function on a group]]
| |
| *[[Symmetric space]]
| |
| *[[Gelfand pair]]
| |
| | |
| ==Notes==
| |
| {{reflist|3}}
| |
| | |
| ==References==
| |
| *{{citation|first=V.|last= Bargmann|authorlink=Valentine Bargmann|title=Irreducible Unitary Representations of the Lorentz Group|journal= Annals of Mathematics|volume= 48|pages= 568–640|year=1947|doi=10.2307/1969129|issue=3|publisher=Annals of Mathematics|jstor=1969129}}
| |
| *{{citation|last=Barnett|first=Adam|last2= Smart|first2=Nigel P.|title=Mental Poker revisited, in Cryptography and Coding|pages= 370–383|year=2003|journal= [[Lecture Notes in Computer Science]]|volume= 2898|publisher= Springer-Verlag}}
| |
| *{{citation|title=Higher transcendental functions, Vol.I|first= Harry|last= Bateman|authorlink1=Harry Bateman|first2= Arthur|last2= Erdélyi|authorlink2=Arthur Erdélyi|year= 1953
| |
| |publisher=McGraw–Hill|ISBN= 0-07-019546-3|url=http://apps.nrbook.com/bateman/Vol1.pdf}}
| |
| *{{Citation|last=Berezin|first= F. A.|authorlink=Felix Berezin|title=Операторы Лапласа на полупростых группах|trans_title= Laplace operators on semisimple groups|journal=[[Doklady Akademii Nauk SSSR]] |volume= 107 |year=1956|pages =9–12|postscript=.}}
| |
| *{{citation|last=Berezin|first= F. A.|title= Representation of complex semisimple Lie groups in Banach space|journal=Doklady Akademii Nauk SSSR |volume= 110 |year=1956b|pages =897–900}}
| |
| *{{springer|id=s/s086670|title=Spherical functions|first1=Yu.A.|last1=Brychkov|first2=A.P.|last2=Prudnikov|year=2001}}
| |
| *{{citation|first=Pierre|last=Cartier|authorlink=Pierre Cartier (mathematician)|title=Structure topologique des groupes de Lie généraux, Exposé No. 22|series=Séminaire "Sophus Lie"|volume=1|year=1954–1955 |url=http://archive.numdam.org/article/SSL_1954-1955__1__A24_0.pdf}}.
| |
| *{{citation|title=Heat Kernels and Spectral Theory|first=E. B.|last= Davies|year=1990|publisher=Cambridge University Press|ISBN= 0-521-40997-7}}
| |
| *{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. V|publisher=Academic Press|year=1977}}
| |
| *{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. VI|publisher=Academic Press|year=1978|isbn=0-12-215506-8}}
| |
| *{{citation|last=Dirac|first= P. A. M.|authorlink=P. A. M. Dirac|title=Unitary representations of the Lorentz group|journal=[[Proceedings of the Royal Society A]]| volume=183|year=1945|pages= 284–295|doi=10.1098/rspa.1945.0003|issue=994}}
| |
| *{{citation|last=Dixmier| first=Jacques| authorlink=Jacques Dixmier|title=Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann) |series=Les Grands Classiques Gauthier-Villars.|publisher= Éditions Jacques Gabay|year= 1996|ISBN= 2-87647-012-8}}
| |
| *{{citation|last=Flensted-Jensen|first=Mogens|title=Spherical functions of a real semisimple Lie group. A method of reduction to the complex case
| |
| |journal=J. Funct. Anal.|volume= 30|year=1978|pages=106–146|doi=10.1016/0022-1236(78)90058-7}}
| |
| *{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=Unitary representations of the Lorentz group|journal=Izv. Akad. Nauk SSSR, Ser. Mat.|year=1947|volume=37|pages=411–504}}
| |
| *{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=An analogue of Plancherel's theorem for the complex unimodular group|journal=Doklady Akademii Nauk SSSR| year =1948|volume=63|pages=609–612}}
| |
| *{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=Unitary representations of the unimodular group containing the identity representation of the unitary subgroup|year=1952|volume =1|pages=423–475|journal=Trudy Moscov. Mat. Obšč.}}
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| *{{citation|first=Roger|last= Godement|authorlink=Roger Godement|title=A theory of spherical functions. I|journal=Transactions of the American Mathematical Society|year=1952|volume=73|pages=496–556|doi=10.2307/1990805|issue=3|publisher=American Mathematical Society|jstor=1990805}}
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| *{{citation|last2=Wallach|first2=Nolan|last=Goodman|first=Roe|title=Representations and Invariants of the Classical Groups|publisher=Cambridge University Press|year=1998|ISBN= 0-521-66348-2}}
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| *{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Infinite irreducible representations of the Lorentz group|journal=[[Proceedings of the Royal Society A]]|volume= 189| year=1947|pages= 372–401|doi=10.1098/rspa.1947.0047|issue=1018}}
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| *{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Representations of Semisimple Lie Groups. II|journal=Trans. Amer. Math. Soc.|year=1954a|volume=76|pages=26–65|doi=10.2307/1990743}}
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| *{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Representations of Semisimple Lie Groups. III|journal=Trans. Amer. Math. Soc.|year=1954b|volume=76|pages=234–253|doi=10.2307/1990767|issue=2|publisher=American Mathematical Society|jstor=1990767}} (Simplification of formula for complex semisimple Lie groups)
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| *{{citation|last=Hochschild|first = Gerhard P.|title= The structure of Lie groups|publisher=Holden–Day|year=1965}}
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| year=1992|publisher=Springer-Verlag|series=Universitext|ISBN= 0-387-97768-6}}
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| *{{citation|last=Kostant|first= Bertram|authorlink=Bertram Kostant|title=On the existence and irreducibility of certain series of representations|journal=Bull. Amer. Math. Soc.|volume= 75 |year=1969|pages =627–642|doi=10.1090/S0002-9904-1969-12235-4|issue=4}}
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| *{{citation|last=Kunze|first=Raymond A.|last2 =Stein|first2= Elias M.|authorlink2=Elias Stein|title=Analytic continuation of the principal series |volume=67|year= 1961|pages=593–596| journal=[[Bull. Amer. Math. Soc.]]|doi=10.1090/S0002-9904-1961-10705-2|issue=6}}
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| *{{citation|first=Serge|last=Lang|authorlink=Serge Lang|title=SL(2,'''R''')|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=105|
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| year=1985|ISBN= 0-387-96198-4}}
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| *{{citation|first=Nolan|last=Wallach|title=Harmonic Analysis on Homogeneous Spaces|year=1973|publisher=Marcel Decker|ISBN= 0-8247-6010-7}}
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| *{{citation|first=Nolan|last=Wallach|title=Real Reductive Groups I|year=1988|publisher=Academic Press|ISBN= 0-12-732960-9}}
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| *{{citation|last=Takahashi|first= R.|title=Sur les représentations unitaires des groupes de Lorentz généralisés|journal=Bull. Soc. Math. France|
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| volume= 91| year=1963|pages= 289–433}}
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| ==External links==
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| *{{citation|last=Casselman|first=William|title=Notes on spherical functions|url=http://www.math.ubc.ca/~cass/research.html}}
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| {{DEFAULTSORT:Zonal Spherical Function}}
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Harmonic analysis]]
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| [[Category:Functional analysis]]
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| [[Category:Types of functions]]
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