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Zonal spherical function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, a zonal spherical function or often just spherical function is a function on a <a href="Locally_compact_group" title="Locally compact group">locally compact group</a> G with compact subgroup K (often a <a href="Maximal_compact_subgroup" title="Maximal compact subgroup">maximal compact subgroup</a>) that arises as the <a href="Matrix_coefficient" title="Matrix coefficient">matrix coefficient</a> of a K-invariant vector in an <a href="Irreducible_representation" title="Irreducible representation">irreducible representation</a> of G. The key examples are the matrix coefficients of the <a href="Principal_series_representation" title="Principal series representation">spherical principal series</a>, the irreducible representations appearing in the decomposition of the <a href="Unitary_representation" title="Unitary representation">unitary representation</a> of G on L2(G/K). In this case the <a href="Commutant" class="mw-redirect" title="Commutant">commutant</a> of G is generated by the algebra of biinvariant functions on G with respect to K acting by right <a href="Convolution" title="Convolution">convolution</a>. It is <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> if in addition G/K is a <a href="Symmetric_space" title="Symmetric space">symmetric space</a>, 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 <a href="Spectrum" title="Spectrum">spectrum</a> of the corresponding

<a href="C*_algebra" class="mw-redirect" title="C* algebra">C* algebra</a> generated by the biinvariant functions of <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a>, often called a <a href="Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra</a>. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the <a href="Complementary_series" class="mw-redirect" title="Complementary series">complementary series</a>, obtained by analytic continuation of the spherical principal series.

Zonal spherical functions have been explicitly determined for real semisimple groups by <a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a>. For <a href="Special_linear_group" title="Special linear group">special linear groups</a>, they were independently discovered by <a href="Israel_Gelfand" title="Israel Gelfand">Israel Gelfand</a> and <a href="Mark_Naimark" title="Mark Naimark">Mark Naimark</a>. For complex groups, the theory simplifies significantly, because G is the <a href="Complexification" title="Complexification">complexification</a> of K, and the formulas are related to analytic continuations of the <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a> on K. The abstract <a href="Functional_analysis" title="Functional analysis">functional analytic</a> theory of zonal spherical functions was first developed by <a href="Roger_Godement" title="Roger Godement">Roger Godement</a>. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous <a href="Eigenfunction" title="Eigenfunction">eigenfunctions</a> for the natural action of the centre of the <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of G on L2(G/K), as <a href="Differential_operator" title="Differential operator">differential operators</a> on the symmetric space G/K. For semisimple <a href="P-adic" class="mw-redirect" title="P-adic">p-adic</a> Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and <a href="Ian_G._Macdonald" title="Ian G. Macdonald">Ian G. Macdonald</a>. The analogues of the <a href="Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> and <a href="Fourier_inversion_formula" class="mw-redirect" title="Fourier inversion formula">Fourier inversion formula</a> in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for <a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">singular ordinary differential equations</a>: they were obtained in full generality in the 1960s in terms of <a href="Harish-Chandra's_c-function" title="Harish-Chandra's c-function">Harish-Chandra's c-function</a>.

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.

<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra of a locally compact group</a></div>

Let G be a <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="Unimodular_group" class="mw-redirect" title="Unimodular group">unimodular</a> <a href="Topological_group" title="Topological group">topological group</a> and K a <a href="Compact_space" title="Compact space">compact</a> <a href="Subgroup" title="Subgroup">subgroup</a> and let H1 = L2(G/K). Thus, H1 admits a <a href="Unitary_representation" title="Unitary representation">unitary representation</a> π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2(G) with left and right <a href="Regular_representation" title="Regular representation">regular representations</a> λ and ρ of G and P is the <a href="Orthogonal_projection" class="mw-redirect" title="Orthogonal projection">orthogonal projection</a>

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<annotation encoding="application/x-tex">{\displaystyle P=\int _{K}\rho (k)\,dk}</annotation>

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</math><img src="./ab3b0848b526e7b730d246d1fc63a6095dd5febf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.253ex; height:5.676ex;" alt="{\displaystyle P=\int _{K}\rho (k)\,dk}" loading="lazy"></dd></dl>

from H to H1 then H1 can naturally be identified with PH with the action of G given by the restriction of λ.

On the other hand, by <a href="Commutation_theorem" class="mw-redirect" title="Commutation theorem">von Neumann's commutation theorem</a><a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \lambda (G)^{\prime }=\rho (G)^{\prime \prime },}</annotation>

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</math><img src="./f8e3d0e6f626867ecb7d88e8c35e5628ab618955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.396ex; height:3.009ex;" alt="{\displaystyle \lambda (G)^{\prime }=\rho (G)^{\prime \prime },}" loading="lazy"></dd></dl>

where S' denotes the <a href="Commutant" class="mw-redirect" title="Commutant">commutant</a> of a set of operators S, so that

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<annotation encoding="application/x-tex">{\displaystyle \pi (G)^{\prime }=P\rho (G)^{\prime \prime }P.}</annotation>

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</math><img src="./9317d3dd6c341a5ba7df0f7aa8c68db350731e6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.864ex; height:3.009ex;" alt="{\displaystyle \pi (G)^{\prime }=P\rho (G)^{\prime \prime }P.}" loading="lazy"></dd></dl>

Thus the commutant of π is generated as a <a href="Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebra</a> by operators

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<annotation encoding="application/x-tex">{\displaystyle P\rho (f)P=\int _{G}f(g)(P\rho (g)P)\,dg}</annotation>

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</math><img src="./183bf6ab1627b85c404337cf87d66f82ac02c8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.433ex; height:5.676ex;" alt="{\displaystyle P\rho (f)P=\int _{G}f(g)(P\rho (g)P)\,dg}" loading="lazy"></dd></dl>

where f is a continuous function of compact support on G.<a href="#cite_note-2">[a]</a>

However Pρ(f) P is just the restriction of ρ(F) to H1, where

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<annotation encoding="application/x-tex">{\displaystyle F(g)=\int _{K}\int _{K}f(kgk^{\prime })\,dk\,dk^{\prime }}</annotation>

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</math><img src="./853de3adaff9e5cd39ae3fd71cb7aee8a76632ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.134ex; height:5.676ex;" alt="{\displaystyle F(g)=\int _{K}\int _{K}f(kgk^{\prime })\,dk\,dk^{\prime }}" loading="lazy"></dd></dl>

is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides.

Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in

Cc(K\G/K), the K-biinvariant continuous functions of compact support on G.

These functions form a <a href="*_algebra" class="mw-redirect" title="* algebra">* algebra</a> under <a href="Convolution" title="Convolution">convolution</a> with involution

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<annotation encoding="application/x-tex">{\displaystyle F^{*}(g)={\overline {F(g^{-1})}},}</annotation>

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</math><img src="./8182db06f5561cc40b13ce5f1504ef3c144e9551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.656ex; height:3.843ex;" alt="{\displaystyle F^{*}(g)={\overline {F(g^{-1})}},}" loading="lazy"></dd></dl>

often called the <a href="Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra</a> for the pair (G, K).

Let A(K\G/K) denote the <a href="C*_algebra" class="mw-redirect" title="C* algebra">C* algebra</a> generated by the operators ρ(F) on H1.

The pair (G, K)

is said to be a <a href="Gelfand_pair" title="Gelfand pair">Gelfand pair</a><a href="#cite_note-FOOTNOTEDieudonné1978-3">[2]</a> if one, and hence all, of the following algebras are <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>:

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<annotation encoding="application/x-tex">{\displaystyle \pi (G)^{\prime }}</annotation>

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</math><img src="./ef0aa977a6037ddba7b1d289d0a4c7d09df72f95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.653ex; height:3.009ex;" alt="{\displaystyle \pi (G)^{\prime }}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle C_{c}(K\backslash G/K)}</annotation>

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</math><img src="./ccb05cf7a8516071fb8f81751be3694ad9b42ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.699ex; height:2.843ex;" alt="{\displaystyle C_{c}(K\backslash G/K)}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle A(K\backslash G/K).}</annotation>

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</math><img src="./7a757d750f713903acf1c233493d922f7ff791b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle A(K\backslash G/K).}" loading="lazy"></li></ul>

Since A(K\G/K) is a commutative <a href="C*_algebra" class="mw-redirect" title="C* algebra">C* algebra</a>, by the <a href="Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a> it has the form C0(X),

where X is the locally compact space of norm continuous * <a href="Homomorphism" title="Homomorphism">homomorphisms</a> of A(K\G/K) into C.

A concrete realization of the * homomorphisms in X as K-biinvariant <a href="Uniformly_bounded" class="mw-redirect" title="Uniformly bounded">uniformly bounded</a> functions on G is obtained as follows.<a href="#cite_note-FOOTNOTEDieudonné1978-3">[2]</a><a href="#cite_note-FOOTNOTEGodement1952-4">[3]</a><a href="#cite_note-FOOTNOTEHelgason2001-5">[4]</a><a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a><a href="#cite_note-FOOTNOTELang1985-7">[6]</a>

Because of the estimate

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<annotation encoding="application/x-tex">{\displaystyle \|\pi (F)\|\leq \int _{G}|F(g)|\,dg\equiv \|F\|_{1},}</annotation>

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</math><img src="./41cf20d9fe4ce90efff6029d98b8c0c06959687e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.053ex; height:5.676ex;" alt="{\displaystyle \|\pi (F)\|\leq \int _{G}|F(g)|\,dg\equiv \|F\|_{1},}" loading="lazy"></dd></dl>

the representation π of Cc(K\G/K) in A(K\G/K) extends by continuity

to L1(K\G/K), the <a href="*_algebra" class="mw-redirect" title="* algebra">* algebra</a> 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

also continuous for the norm ||·||1. Since the <a href="Banach_space" title="Banach space">Banach space dual</a> of L1 is L∞,

it follows that

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<annotation encoding="application/x-tex">{\displaystyle \chi (\pi (F))=\int _{G}F(g)h(g)\,dg,}</annotation>

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</math><img src="./fb95ee1c1ee4d4939d0d99e2653ea41bee98a5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.745ex; height:5.676ex;" alt="{\displaystyle \chi (\pi (F))=\int _{G}F(g)h(g)\,dg,}" loading="lazy"></dd></dl>

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).

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

A zonal spherical function h has the following properties:<a href="#cite_note-FOOTNOTEDieudonné1978-3">[2]</a>

<ol><li>h is uniformly continuous on G</li>

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<annotation encoding="application/x-tex">{\displaystyle h(x)h(y)=\int _{K}h(xky)\,dk\,\,(x,y\in G).}</annotation>

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</math><img src="./e21dc68254f5c289e6180c925662392c2c7ace71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.329ex; height:5.676ex;" alt="{\displaystyle h(x)h(y)=\int _{K}h(xky)\,dk\,\,(x,y\in G).}" loading="lazy"></li>

<li>h(1) =1 (normalisation)</li>

<li>h is a <a href="Positive_definite_function_on_a_group" class="mw-redirect" title="Positive definite function on a group">positive definite function</a> on G</li>

<li>f * h is proportional to h for all f in Cc(K\G/K).</li></ol>

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

with <a href="Unitary_representation" title="Unitary representation">unitary representations</a>. For semisimple Lie groups, there is a further characterization as eigenfunctions of

<a href="Invariant_differential_operator" title="Invariant differential operator">invariant differential operators</a> on G/K (see below).

In fact, as a special case of the <a href="Gelfand%E2%80%93Naimark%E2%80%93Segal_construction" title="Gelfand–Naimark–Segal construction">Gelfand–Naimark–Segal construction</a>, there is one-one correspondence between

irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions

h given by

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<annotation encoding="application/x-tex">{\displaystyle h(g)=(\sigma (g)v,v).}</annotation>

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</math><img src="./2ce95dea83df8d73dc6b89ea8bd2af00d81217bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.363ex; height:2.843ex;" alt="{\displaystyle h(g)=(\sigma (g)v,v).}" loading="lazy"></dd></dl>

Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the <a href="Induced_representation" title="Induced representation">induced representation</a> π on H1. Each representation σ extends uniquely by continuity

to A(K\G/K), so that each zonal spherical function satisfies

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<annotation encoding="application/x-tex">{\displaystyle \left|\int _{G}f(g)h(g)\,dg\right|\leq \|\pi (f)\|}</annotation>

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</math><img src="./2506680fb4dd5143579f055b5d100701a8a0b1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.528ex; height:5.843ex;" alt="{\displaystyle \left|\int _{G}f(g)h(g)\,dg\right|\leq \|\pi (f)\|}" loading="lazy"></dd></dl>

for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative,

there is a unique probability measure μ on the space of * homomorphisms X such that

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<annotation encoding="application/x-tex">{\displaystyle \int _{G}|f(g)|^{2}\,dg=\int _{X}|\chi (\pi (f))|^{2}\,d\mu (\chi ).}</annotation>

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</math><img src="./5c15daf0f453d5712986cba8c7d941ef297b0058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.833ex; height:5.676ex;" alt="{\displaystyle \int _{G}|f(g)|^{2}\,dg=\int _{X}|\chi (\pi (f))|^{2}\,d\mu (\chi ).}" loading="lazy"></dd></dl>

μ is called the <a href="Plancherel_measure" title="Plancherel measure">Plancherel measure</a>. Since π(G)' is the <a href="Von_Neumann_algebra" title="Von Neumann algebra">centre</a> of the von Neumann algebra generated by G, it also gives the measure associated with the <a href="Direct_integral" title="Direct integral">direct integral</a> decomposition of H1 in terms of the irreducible representations σχ.

<div class="mw-heading mw-heading2"><h2 id="Gelfand_pairs">Gelfand pairs</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Gelfand_pair" title="Gelfand pair">Gelfand pair</a></div>

If G is a <a href="Connectedness" title="Connectedness">connected</a> <a href="Lie_group" title="Lie group">Lie group</a>, then, thanks to the work of <a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a>, <a href="Anatoly_Maltsev" title="Anatoly Maltsev">Malcev</a>, <a href="Kenkichi_Iwasawa" title="Kenkichi Iwasawa">Iwasawa</a> and <a href="Claude_Chevalley" title="Claude Chevalley">Chevalley</a>, G has a <a href="Maximal_compact_subgroup" title="Maximal compact subgroup">maximal compact subgroup</a>, unique up to conjugation.<a href="#cite_note-FOOTNOTECartier1954–1955-8">[7]</a><a href="#cite_note-FOOTNOTEHochschild1965-9">[8]</a> In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition <a href="Semisimple_Lie_group" class="mw-redirect" title="Semisimple Lie group">semisimple</a>, this can be seen directly using the <a href="Cartan_decomposition" title="Cartan decomposition">Cartan decomposition</a> associated to the <a href="Symmetric_space" title="Symmetric space">symmetric space</a> G/K, a generalisation of the <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> of invertible matrices. Indeed, if τ is the associated period two automorphism of G with fixed point subgroup K, then

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<annotation encoding="application/x-tex">{\displaystyle G=P\cdot K,}</annotation>

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</math><img src="./6d49fd7231bfef12f1e5f28da8532849bfdd73ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.062ex; height:2.509ex;" alt="{\displaystyle G=P\cdot K,}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle P=\{g\in G|\tau (g)=g^{-1}\}.}</annotation>

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</math><img src="./848f4c64b029f315e6b744430edbf7ae7291e83a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.923ex; height:3.176ex;" alt="{\displaystyle P=\{g\in G|\tau (g)=g^{-1}\}.}" loading="lazy"></dd></dl>

Under the <a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a>, P is diffeomorphic to the -1 eigenspace of τ in the <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> of G.

Since τ preserves K, it induces an automorphism of the Hecke algebra Cc(K\G/K). On the

other hand, if F lies in Cc(K\G/K), then

so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,

<ul><li>the Hecke algebra is commutative</li>

<li>(G,K) is a Gelfand pair.</li></ul>

More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:<a href="#cite_note-FOOTNOTEDieudonné197855–57-10">[9]</a>

<ul><li>G is a unimodular locally compact group;</li>

<li>K is a compact subgroup arising as the fixed points of a period two automorphism τ of G;</li>

<li>G =K·P (not necessarily a direct product), where P is defined as above.</li></ul>

The two most important examples covered by this are when:

<ul><li>G is a compact connected semisimple Lie group with τ a period two automorphism;<a href="#cite_note-FOOTNOTEDieudonné1977-11">[10]</a><a href="#cite_note-FOOTNOTEHelgason1978249-12">[11]</a></li>

<li>G is a semidirect product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rtimes K}">

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<annotation encoding="application/x-tex">{\displaystyle A\rtimes K}</annotation>

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</math><img src="./3a0b977ef8a22b04c258d695d9790e414c024924.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.649ex; height:2.176ex;" alt="{\displaystyle A\rtimes K}" loading="lazy">, with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a−1 for a in A and k in K.</li></ul>

The three cases cover the three types of <a href="Symmetric_space" title="Symmetric space">symmetric spaces</a> G/K:<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a>

<ol><li>Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;</li>

<li>Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;</li>

<li>Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Cartan–Helgason_theorem">Cartan–Helgason theorem</h2></div>

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 = Gτ. In this case K is a connected compact Lie group.<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a> In addition let T be a <a href="Maximal_torus" title="Maximal torus">maximal torus</a> of G invariant under τ, such that T <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }">

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<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>

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</math><img src="./9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"> P is a maximal torus in P, and set<a href="#cite_note-FOOTNOTEHelgason1978257–264-13">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle S=K\cap T=T^{\tau }.}</annotation>

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</math><img src="./be23e021d2bbbbc8f1b5193299e9595bf952431f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.43ex; height:2.343ex;" alt="{\displaystyle S=K\cap T=T^{\tau }.}" loading="lazy"></dd></dl>

S is the direct product of a torus and an <a href="Elementary_abelian_group" title="Elementary abelian group">elementary abelian 2-group</a>.

In 1929 <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> found a rule to determine the decomposition of L2(G/K) into the direct sum of finite-dimensional <a href="Irreducible_representation" title="Irreducible representation">irreducible representations</a> of G, which was proved rigorously only in 1970 by <a href="Sigurdur_Helgason_(mathematician)" class="mw-redirect" title="Sigurdur Helgason (mathematician)">Sigurdur Helgason</a>. Because the commutant of G on L2(G/K) is commutative, each irreducible representation appears with multiplicity one. By <a href="Frobenius_reciprocity" title="Frobenius reciprocity">Frobenius reciprocity</a> for compact groups, the irreducible representations V that occur are precisely those admitting a non-zero vector fixed by K.

From the <a href="Weyl_character_formula" title="Weyl character formula">representation theory of compact semisimple groups</a>, irreducible representations of G are classified by their <a href="Root_system" title="Root system">highest weight</a>. This is specified by a homomorphism of the maximal torus T into T.

The Cartan–Helgason theorem<a href="#cite_note-FOOTNOTEHelgason1984534–538-14">[13]</a><a href="#cite_note-FOOTNOTEGoodmanWallach1998549–550-15">[14]</a> states that

<td>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.

</td></tr></tbody></table></dd></dl>

The corresponding irreducible representations are called spherical representations.

The theorem can be proved<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a> using the <a href="Iwasawa_decomposition" title="Iwasawa decomposition">Iwasawa decomposition</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}},}</annotation>

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</math><img src="./932432da9d9175791a2c0519ae2eddbfb888b20a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.89ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

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</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

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</math><img src="./16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}" loading="lazy"> are the complexifications of the <a href="Lie_algebra" title="Lie algebra">Lie algebras</a> of G, K, A = T <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }">

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<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>

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</math><img src="./9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"> P and

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}=\bigoplus {\mathfrak {g}}_{\alpha },}</annotation>

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</math><img src="./50fca34a68f29655216fd17594aa3686e84f73ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.324ex; height:3.843ex;" alt="{\displaystyle {\mathfrak {n}}=\bigoplus {\mathfrak {g}}_{\alpha },}" loading="lazy"></dd></dl>

summed over all eigenspaces for T in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> corresponding to <a href="Root_system" title="Root system">positive roots</a> α not fixed by τ.

Let V be a spherical representation with highest weight vector v0 and K-fixed vector vK. Since v0 is an eigenvector of the solvable Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\oplus {\mathfrak {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\oplus {\mathfrak {n}}}</annotation>

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</math><img src="./de39fce269c3289e8e15b3fb16e2d118c63f3d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.228ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}\oplus {\mathfrak {n}}}" loading="lazy">, the <a href="Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem" title="Poincaré–Birkhoff–Witt theorem">Poincaré–Birkhoff–Witt theorem</a>

implies that the K-module generated by v0 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 <a href="Haar_measure" title="Haar measure">Haar measure</a>, it follows that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {v_{K}=cQv_{0}}}</annotation>

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</math><img src="./7b83f206e013bf39fb97cbe79658052e5ca70e8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.946ex; height:2.509ex;" alt="{\displaystyle \displaystyle {v_{K}=cQv_{0}}}" loading="lazy"></dd></dl>

for some non-zero constant c. Because vK is fixed by S and v0 is an eigenvector for S, the subgroup S must actually fix v0, an equivalent form of the triviality condition on S.

Conversely if v0 is fixed by S, then it can be shown<a href="#cite_note-FOOTNOTEGoodmanWallach1998550-16">[15]</a> that the matrix coefficient

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle {f(g)=(gv_{0},v_{0})}}</annotation>

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</math><img src="./9dee10f7d34273b51a7835cd9fc0d7dbbdff46d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.625ex; height:2.843ex;" alt="{\displaystyle \displaystyle {f(g)=(gv_{0},v_{0})}}" loading="lazy"></dd></dl>

is non-negative on K. Since f(1) > 0, it follows that (Qv0, v0) > 0 and hence that Qv0 is a non-zero vector fixed by K.

<div class="mw-heading mw-heading2"><h2 id="Harish-Chandra's_formula">Harish-Chandra's formula</h2></div>

If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the <a href="Cartan_decomposition" title="Cartan decomposition">Cartan decomposition</a>. If A is a maximal Abelian subgroup of G contained in P, then A is diffeomorphic to its Lie algebra under the <a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">exponential map</a> and, as a <a href="Lie_group_decompositions" class="mw-redirect" title="Lie group decompositions">further generalisation</a> of the <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> of matrices, every element of P is conjugate under K to an element of A, so that<a href="#cite_note-17">[16]</a>

There is also an associated <a href="Iwasawa_decomposition" title="Iwasawa decomposition">Iwasawa decomposition</a>

where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus

S=AN is a closed <a href="Solvable_group" title="Solvable group">solvable subgroup</a> of G, the <a href="Semidirect_product" title="Semidirect product">semidirect product</a> of N by A, and G = KS.

If α in Hom(A,T) is a <a href="Character_(mathematics)" title="Character (mathematics)">character</a> of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding <a href="Unitary_representation" title="Unitary representation">unitary</a> <a href="Induced_representation" title="Induced representation">induced representation</a> σ of G on L2(G/S) = L2(K),<a href="#cite_note-FOOTNOTEHarish-Chandra1954a251-18">[17]</a> a so-called <a href="Principal_series_representation" title="Principal series representation">(spherical) principal series representation</a>.

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 ΔS the <a href="Haar_measure#The_modular_function" title="Haar measure">modular function</a> of S. Then<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle \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.}</annotation>

</semantics>

</math><img src="./ae9f62153e096fde6a199e5ddd07c9e74d9b13b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:62.136ex; height:5.676ex;" alt="{\displaystyle \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.}" loading="lazy"></dd></dl>

The principal series representation σ is realised on L2(K) as<a href="#cite_note-FOOTNOTEWallach1973-19">[18]</a>

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<annotation encoding="application/x-tex">{\displaystyle (\sigma (g)\xi )(k)=\alpha ^{\prime }(g^{-1}k)^{-1}\,\xi (U(g^{-1}k)),}</annotation>

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</math><img src="./85589765988174cfec1d2e9f84c9f1b3cb569882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.317ex; height:3.176ex;" alt="{\displaystyle (\sigma (g)\xi )(k)=\alpha ^{\prime }(g^{-1}k)^{-1}\,\xi (U(g^{-1}k)),}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle g=U(g)\cdot X(g)}</annotation>

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</math><img src="./ebb6f1e6cb21207bac5748ddcc6651fcc311e24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.507ex; height:2.843ex;" alt="{\displaystyle g=U(g)\cdot X(g)}" loading="lazy"></dd></dl>

is the Iwasawa decomposition of g with U(g) in K and X(g) in S and

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{\prime }(kx)=\Delta _{S}(x)^{1/2}\alpha (x)}</annotation>

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</math><img src="./bf1bd5ae46c4f5a9285b92bce8de70f4b55c7daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.313ex; height:3.343ex;" alt="{\displaystyle \alpha ^{\prime }(kx)=\Delta _{S}(x)^{1/2}\alpha (x)}" loading="lazy"></dd></dl>

for k in K and x in S.

The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K,

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(g)=(\sigma (g)v,v)}</annotation>

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</math><img src="./4b5357e1048d7b8ca6de479ce24eeb349ceb9afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.182ex; height:2.843ex;" alt="{\displaystyle \varphi _{\alpha }(g)=(\sigma (g)v,v)}" loading="lazy"></dd></dl>

defines a zonal spherical function of G.

Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\alpha }(g)=\int _{K}\alpha ^{\prime }(gk)^{-1}\,dk}</annotation>

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</math><img src="./d4677e174db757d1c04bd06506ab179b69714302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.657ex; height:5.676ex;" alt="{\displaystyle \varphi _{\alpha }(g)=\int _{K}\alpha ^{\prime }(gk)^{-1}\,dk}" loading="lazy">

</td></tr></tbody></table></dd></dl>

as an integral over K.

Harish-Chandra proved that these zonal spherical functions exhaust the characters of the <a href="C*_algebra" class="mw-redirect" title="C* algebra">C* algebra</a> generated by the Cc(K \ G / K) acting by right convolution on L2(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 <a href="Weyl_group" title="Weyl group">Weyl group</a> of A

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<annotation encoding="application/x-tex">{\displaystyle W(A)=N_{K}(A)/C_{K}(A),}</annotation>

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</math><img src="./e9791d8d71eab7eda9aa408af0e8b5395cdcf150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.914ex; height:2.843ex;" alt="{\displaystyle W(A)=N_{K}(A)/C_{K}(A),}" loading="lazy"></dd></dl>

the quotient of the <a href="Normaliser" class="mw-redirect" title="Normaliser">normaliser</a> of A in K by its <a href="Centraliser" class="mw-redirect" title="Centraliser">centraliser</a>, a <a href="Finite_reflection_group" class="mw-redirect" title="Finite reflection group">finite reflection group</a>.

It can also be verified directly<a href="#cite_note-FOOTNOTEDieudonné1978-3">[2]</a> 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-<a href="Invariant_differential_operator" title="Invariant differential operator">invariant differential operators</a> on G/K and their simultaneous <a href="Eigenfunctions" class="mw-redirect" title="Eigenfunctions">eigenfunctions</a> (see below).<a href="#cite_note-FOOTNOTEHelgason2001-5">[4]</a><a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a> In the case of complex semisimple groups, Harish-Chandra and <a href="Felix_Berezin" title="Felix Berezin">Felix Berezin</a> realised independently that the formula simplified considerably and could be proved more directly.<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a><a href="#cite_note-FOOTNOTEBerezin1956a-20">[19]</a><a href="#cite_note-FOOTNOTEBerezin1956b-21">[20]</a><a href="#cite_note-FOOTNOTEHarish-Chandra1954b-22">[21]</a><a href="#cite_note-FOOTNOTEHarish-Chandra1954c-23">[22]</a>

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 "<a href="Complementary_series" class="mw-redirect" title="Complementary series">complementary series</a>" was first studied by <a href="#CITEREFBargmann1947">Bargmann (1947)</a> for G = SL(2,R) and by <a href="#CITEREFHarish-Chandra1947">Harish-Chandra (1947)</a> and <a href="#CITEREFGelfandNaimark1947">Gelfand & Naimark (1947)</a> for G = SL(2,C).

Subsequently in the 1960s, the construction of a <a href="Complementary_series" class="mw-redirect" title="Complementary series">complementary series</a> by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, <a href="Elias_Stein" class="mw-redirect" title="Elias Stein">Elias Stein</a> and <a href="Bertram_Kostant" title="Bertram Kostant">Bertram Kostant</a>.<a href="#cite_note-FOOTNOTEKunzeStein1961-24">[23]</a><a href="#cite_note-FOOTNOTEStein1970-25">[24]</a><a href="#cite_note-FOOTNOTEKostant1969-26">[25]</a> Since these irreducible representations are not <a href="Tempered_representation" title="Tempered representation">tempered</a>, they are not usually required for harmonic analysis on G (or G / K).

<div class="mw-heading mw-heading2"><h2 id="Eigenfunctions">Eigenfunctions</h2></div>

Harish-Chandra proved<a href="#cite_note-FOOTNOTEHelgason2001-5">[4]</a><a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a> 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 <a href="Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of G fixed under the <a href="Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint action</a> of K. As for the commutant of G on L2(G/K) and the corresponding Hecke algebra, this algebra of operators is <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>; indeed it is a subalgebra of the <a href="Affiliated_operator" title="Affiliated operator">algebra of measurable operators</a> 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 <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> by the <a href="Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem" title="Chevalley–Shephard–Todd theorem">Chevalley–Shephard–Todd theorem</a> on polynomial invariants of <a href="Finite_reflection_group" class="mw-redirect" title="Finite reflection group">finite reflection groups</a>. The simplest invariant differential operator on G/K is the <a href="Laplacian_operator" class="mw-redirect" title="Laplacian operator">Laplacian operator</a>; up to a sign this operator is just the image under π of the <a href="Casimir_operator" class="mw-redirect" title="Casimir operator">Casimir operator</a> 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 λD such that

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \pi (D)f=\lambda _{D}f,}</annotation>

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</math><img src="./195182b701119d0ba832a73a08cab70846df2006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.316ex; height:2.843ex;" alt="{\displaystyle \displaystyle \pi (D)f=\lambda _{D}f,}" loading="lazy"></dd></dl>

i.e. f is a simultaneous <a href="Eigenfunction" title="Eigenfunction">eigenfunction</a> 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 <a href="Elliptic_differential_operator" class="mw-redirect" title="Elliptic differential operator">elliptic differential operator</a> with <a href="Real_analytic" class="mw-redirect" title="Real analytic">real analytic</a> coefficients. By <a href="FBI_transform" class="mw-redirect" title="FBI transform">analytic elliptic regularity</a>,

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.<a href="#cite_note-FOOTNOTEHarish-Chandra1958-27">[26]</a><a href="#cite_note-28">[27]</a><a href="#cite_note-FOOTNOTEHelgason1984418-29">[28]</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Example:_SL(2,C)">Example: SL(2,C)</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="SL(2%2CC)" class="mw-redirect" title="SL(2,C)">SL(2,C)</a>; <a href="Representations_of_the_Lorentz_group" class="mw-redirect" title="Representations of the Lorentz group">Representations of the Lorentz group</a>; and <a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></div>

The group G = SL(2,C) is the <a href="Complexification" title="Complexification">complexification</a> of the <a href="Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact Lie group</a> K = SU(2) and the <a href="Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a> of the <a href="Lorentz_group" title="Lorentz group">Lorentz group</a>. The infinite-dimensional representations of the Lorentz group were first studied by <a href="Paul_Dirac" title="Paul Dirac">Dirac</a> in 1945, who considered the <a href="Discrete_series" class="mw-redirect" title="Discrete series">discrete series</a> 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 <a href="Laplacian_operator" class="mw-redirect" title="Laplacian operator">Laplacian operator</a>.<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a>

Indeed, any unimodular complex 2×2 matrix g admits a unique <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> 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

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a={\begin{pmatrix}e^{r/2}&0\\0&e^{-r/2}\end{pmatrix}},}</annotation>

</semantics>

</math><img src="./e50682d3e1d28f7e5240b2caacc6f1e0881e9008.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.151ex; height:6.509ex;" alt="{\displaystyle a={\begin{pmatrix}e^{r/2}&0\\0&e^{-r/2}\end{pmatrix}},}" loading="lazy"></dd></dl>

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<a href="#cite_note-FOOTNOTEDavies1990-30">[29]</a>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=-\partial _{r}^{2}-2\coth r\partial _{r}.}</annotation>

</semantics>

</math><img src="./d838c9ded4565dd48bb8af3dba5df6ea723009e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.96ex; height:2.843ex;" alt="{\displaystyle L=-\partial _{r}^{2}-2\coth r\partial _{r}.}" loading="lazy"></dd></dl>

Setting f(r) = sinh (r)·ψ(r), it follows that f is an <a href="Odd_function" class="mw-redirect" title="Odd function">odd function</a> of r and an eigenfunction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{r}^{2}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{r}^{2}}</annotation>

</semantics>

</math><img src="./f2412b52965a23a21a2d65943c0f43d0c25e4b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.398ex; height:2.843ex;" alt="{\displaystyle \partial _{r}^{2}}" loading="lazy">.

Hence

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)={\sin(\ell r) \over \ell \sinh r}}</annotation>

</semantics>

</math><img src="./325ba930dc0d59f705858ad1ef6213fd701b792e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.253ex; height:5.843ex;" alt="{\displaystyle \varphi (r)={\sin(\ell r) \over \ell \sinh r}}" loading="lazy">

</td></tr></tbody></table></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> is real.

There is a similar elementary treatment for the <a href="Lorentz_group#Generalization_to_higher_dimensions" title="Lorentz group">generalized Lorentz groups</a> SO(N,1) in <a href="#CITEREFTakahashi1963">Takahashi (1963)</a> and <a href="#CITEREFFarautKorányi1994">Faraut & Korányi (1994)</a> (recall that SO0(3,1) = SL(2,C) / ±I).

<div class="mw-heading mw-heading2"><h2 id="Complex_case">Complex case</h2></div>

If G is a complex semisimple Lie group, it is the <a href="Complexification" title="Complexification">complexification</a> of its maximal compact subgroup K. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation>

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

</semantics>

</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy"> are their Lie algebras, then

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus i{\mathfrak {k}}.}</annotation>

</semantics>

</math><img src="./72521a9fabde5c08b8608a35f9247e6c5d5ca3ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.369ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus i{\mathfrak {k}}.}" loading="lazy"></dd></dl>

Let T be a <a href="Maximal_torus" title="Maximal torus">maximal torus</a> in K with Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy">. Then

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=\exp i{\mathfrak {t}},\,\,P=\exp i{\mathfrak {k}}.}</annotation>

</semantics>

</math><img src="./498d3e52ba53cc1ce02b37c0d77098413ddfc451.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.338ex; height:2.509ex;" alt="{\displaystyle A=\exp i{\mathfrak {t}},\,\,P=\exp i{\mathfrak {k}}.}" loading="lazy"></dd></dl>

Let

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=N_{K}(T)/T}</annotation>

</semantics>

</math><img src="./de83ba2aebb8ee7ff19d21c8902c201489b2e1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.337ex; height:2.843ex;" alt="{\displaystyle W=N_{K}(T)/T}" loading="lazy"></dd></dl>

be the <a href="Weyl_group" title="Weyl group">Weyl group</a> of T in K. Recall characters in Hom(T,T) are called <a href="Weight_(representation_theory)" title="Weight (representation theory)">weights</a> and can be identified with elements of the <a href="Weight_lattice" class="mw-redirect" title="Weight lattice">weight lattice</a> Λ in

Hom(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy">, R) = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}^{*}}</annotation>

</semantics>

</math><img src="./9c5b648f9ba6c51bb6a8bbe364503e70ada85f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.874ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {t}}^{*}}" loading="lazy">. There is a natural ordering on weights and every finite-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the <a href="Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint representation</a> of K on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}\ominus {\mathfrak {t}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}\ominus {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./852fdb7c12d1e2371128d276e964e8cf0a4db10d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.554ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {k}}\ominus {\mathfrak {t}}}" loading="lazy"> are called roots and ρ is used to denote half the sum of the <a href="Positive_root" class="mw-redirect" title="Positive root">positive roots</a> α, <a href="Weyl's_character_formula" class="mw-redirect" title="Weyl's character formula">Weyl's character formula</a> asserts that for z = exp X in T

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle \chi _{\lambda }(e^{X})\equiv {\rm {Tr}}\,\pi (z)=A_{\lambda +\rho }(e^{X})/A_{\rho }(e^{X}),}</annotation>

</semantics>

</math><img src="./a4936216c771d52d4a9c200f9a8600a6535b08f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.321ex; height:3.343ex;" alt="{\displaystyle \displaystyle \chi _{\lambda }(e^{X})\equiv {\rm {Tr}}\,\pi (z)=A_{\lambda +\rho }(e^{X})/A_{\rho }(e^{X}),}" loading="lazy"></dd></dl>

where, for μ in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}^{*}}</annotation>

</semantics>

</math><img src="./9c5b648f9ba6c51bb6a8bbe364503e70ada85f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.874ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {t}}^{*}}" loading="lazy">, Aμ denotes the antisymmetrisation

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle A_{\mu }(e^{X})=\sum _{s\in W}\varepsilon (s)e^{i\mu (sX)},}</annotation>

</semantics>

</math><img src="./aa8485bc0159ede66c68b594eebd0adfe75a6362.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.521ex; height:5.676ex;" alt="{\displaystyle \displaystyle A_{\mu }(e^{X})=\sum _{s\in W}\varepsilon (s)e^{i\mu (sX)},}" loading="lazy"></dd></dl>

and ε denotes the sign character of the <a href="Finite_reflection_group" class="mw-redirect" title="Finite reflection group">finite reflection group</a> W.

<a href="Weyl's_denominator_formula" class="mw-redirect" title="Weyl's denominator formula">Weyl's denominator formula</a> expresses the denominator Aρ as a product:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),}</annotation>

</semantics>

</math><img src="./ff5ee23caadc666532db09a57391d25b433f157e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.753ex; height:5.509ex;" alt="{\displaystyle \displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),}" loading="lazy"></dd></dl>

where the product is over the positive roots.

<a href="Weyl's_dimension_formula" class="mw-redirect" title="Weyl's dimension formula">Weyl's dimension formula</a> asserts that

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.}">

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

</mrow>

</munder>

</mrow>

<mrow>

</mrow>

</munder>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.}</annotation>

</semantics>

</math><img src="./f65921a86401376f583a17ae8d9d70bfae26e99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.654ex; height:6.843ex;" alt="{\displaystyle \displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.}" loading="lazy"></dd></dl>

where the <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}^{*}}</annotation>

</semantics>

</math><img src="./9c5b648f9ba6c51bb6a8bbe364503e70ada85f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.874ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {t}}^{*}}" loading="lazy"> is that associated with the <a href="Killing_form" title="Killing form">Killing form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

</semantics>

</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy">.

Now

<ul><li>every irreducible representation of K extends holomorphically to the complexification G</li>

<li>every irreducible character χλ(k) of K extends holomorphically to the complexification of K and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}^{*}}</annotation>

</semantics>

</math><img src="./9c5b648f9ba6c51bb6a8bbe364503e70ada85f5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.874ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {t}}^{*}}" loading="lazy">.</li>

<li>for every λ in Hom(A,T) = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\mathfrak {t}}^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\mathfrak {t}}^{*}}</annotation>

</semantics>

</math><img src="./70844e86f9119f4c015bf39badef3405072ac992.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.677ex; height:2.343ex;" alt="{\displaystyle i{\mathfrak {t}}^{*}}" loading="lazy">, there is a zonal spherical function φλ.</li></ul>

The Berezin–Harish–Chandra formula<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a> asserts that for X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./973b107fe0b633e49a428e636085907e6c42778c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.612ex; height:2.176ex;" alt="{\displaystyle i{\mathfrak {t}}}" loading="lazy">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over \chi _{\lambda }(1)}.}</annotation>

</semantics>

</math><img src="./ed7e37bc66241d354bab00f535e170348b0909e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.988ex; height:6.676ex;" alt="{\displaystyle \varphi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over \chi _{\lambda }(1)}.}" loading="lazy">

</td></tr></tbody></table></dd></dl>

In other words:

<ul><li>the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.</li></ul>

One of the simplest proofs<a href="#cite_note-FOOTNOTEHelgason1984432–433-31">[30]</a> of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of <a href="Freudenthal" title="Freudenthal">Freudenthal</a>'s classical proof of the <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a>, using the radial component on T of the Laplacian on K.<a href="#cite_note-FOOTNOTEHelgason1984501–502-32">[31]</a>

In the latter case the <a href="Class_function" title="Class function">class functions</a> on K can be identified with W-invariant functions on T. The

radial component of ΔK on T is just the expression for the restriction of ΔK to W-invariant functions on T, where

it is given by the formula

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle \Delta _{K}=h^{-1}\circ \Delta _{T}\circ h+\|\rho \|^{2},}</annotation>

</semantics>

</math><img src="./1b30bfa30818cb435d8848d9195ea317fa43f86a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.521ex; height:3.176ex;" alt="{\displaystyle \displaystyle \Delta _{K}=h^{-1}\circ \Delta _{T}\circ h+\|\rho \|^{2},}" loading="lazy"></dd></dl>

where

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \displaystyle h(e^{X})=A_{\rho }(e^{X})}</annotation>

</semantics>

</math><img src="./d4acb6d45745e0073790e7386c6c43cc8407a689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.313ex; height:3.343ex;" alt="{\displaystyle \displaystyle h(e^{X})=A_{\rho }(e^{X})}" loading="lazy"></dd></dl>

for X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy">. If χ is a character with highest weight λ, it follows that φ = h·χ satisfies

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{T}\varphi =(\|\lambda +\rho \|^{2}-\|\rho \|^{2})\varphi .}</annotation>

</semantics>

</math><img src="./6b893f97c7104785733bfd8211f603a0e2aeafab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.118ex; height:3.176ex;" alt="{\displaystyle \Delta _{T}\varphi =(\|\lambda +\rho \|^{2}-\|\rho \|^{2})\varphi .}" loading="lazy"></dd></dl>

Thus for every weight μ with non-zero <a href="Fourier_coefficient" class="mw-redirect" title="Fourier coefficient">Fourier coefficient</a> in φ,

<msup>

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</msup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \|\lambda +\rho \|^{2}=\|\mu +\rho \|^{2}.}</annotation>

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</math><img src="./d1277292c7031e0f44d3a1d537f83172cde7c2fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.345ex; height:3.176ex;" alt="{\displaystyle \displaystyle \|\lambda +\rho \|^{2}=\|\mu +\rho \|^{2}.}" loading="lazy"></dd></dl>

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 ΔG on A is just the expression for the restriction of ΔG to W(A)-invariant functions on A.

It is given by the formula

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \Delta _{G}=H^{-1}\circ \Delta _{A}\circ H-\|\rho \|^{2},}</annotation>

</semantics>

</math><img src="./9af56aae4270a305973bb6830cc0198830cf47da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.917ex; height:3.176ex;" alt="{\displaystyle \displaystyle \Delta _{G}=H^{-1}\circ \Delta _{A}\circ H-\|\rho \|^{2},}" loading="lazy"></dd></dl>

where

<msup>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle H(e^{X})=A_{\rho }(e^{X})}</annotation>

</semantics>

</math><img src="./025975b1e607d1dba9ecf5a8fed8d3271415b642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.038ex; height:3.343ex;" alt="{\displaystyle \displaystyle H(e^{X})=A_{\rho }(e^{X})}" loading="lazy"></dd></dl>

for X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./973b107fe0b633e49a428e636085907e6c42778c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.612ex; height:2.176ex;" alt="{\displaystyle i{\mathfrak {t}}}" loading="lazy">.

The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle f=H\cdot \varphi ,}</annotation>

</semantics>

</math><img src="./d87c2451109ee890a1b03ea2cef28d166d1659d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.287ex; height:2.676ex;" alt="{\displaystyle \displaystyle f=H\cdot \varphi ,}" loading="lazy"></dd></dl>

which is an eigenfunction of the Laplacian ΔA. Since K is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">simple roots</a>, 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 <a href="Periodic_function" title="Periodic function">periodic function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./973b107fe0b633e49a428e636085907e6c42778c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.612ex; height:2.176ex;" alt="{\displaystyle i{\mathfrak {t}}}" loading="lazy"> with respect to some <a href="Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">sublattice</a>. 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.

<div class="mw-heading mw-heading2"><h2 id="Example:_SL(2,R)">Example: SL(2,R)</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="SL(2%2CR)" class="mw-redirect" title="SL(2,R)">SL(2,R)</a>; <a href="Representation_theory_of_SL2(R)" title="Representation theory of SL2(R)">Representation theory of SL2(R)</a>; and <a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></div>

The theory of zonal spherical functions for <a href="SL(2%2CR)" class="mw-redirect" title="SL(2,R)">SL(2,R)</a> originated in the work of <a href="Gustav_Ferdinand_Mehler" title="Gustav Ferdinand Mehler">Mehler</a> 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 <a href="Mehler%E2%80%93Fock_transform" title="Mehler–Fock transform">Mehler–Fock transform</a>. It was already put on a firm footing in 1910 by <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>'s important work on the <a href="Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">spectral theory of ordinary differential equations</a>. The radial part of the Laplacian in this case leads to a <a href="Hypergeometric_differential_equation" class="mw-redirect" title="Hypergeometric differential equation">hypergeometric differential equation</a>, 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 semisimple 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 = <a href="SL(2%2CR)" class="mw-redirect" title="SL(2,R)">SL(2,R)</a> is a <a href="Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a> of the 3-dimensional <a href="Lorentz_group" title="Lorentz group">Lorentz group</a> SO(2,1), the <a href="Symmetry_group" title="Symmetry group">symmetry group</a> of the <a href="Hyperbolic_space" title="Hyperbolic space">hyperbolic plane</a> with its <a href="Poincar%C3%A9_metric" title="Poincaré metric">Poincaré metric</a>. It acts by <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>. The upper half-plane can be identified with the unit disc by the <a href="Cayley_transform" title="Cayley transform">Cayley transform</a>. Under this identification G becomes identified with the group <a href="SU(1%2C1)" class="mw-redirect" title="SU(1,1)">SU(1,1)</a>, also acting by Möbius transformations. Because the action is <a href="Transitive_action" class="mw-redirect" title="Transitive action">transitive</a>, both spaces can be identified with G/K, where K = <a href="SO(2)" class="mw-redirect" title="SO(2)">SO(2)</a>. The metric is invariant under G and the associated Laplacian is G-invariant, coinciding with the image of the <a href="Casimir_operator" class="mw-redirect" title="Casimir operator">Casimir operator</a>. In the upper half-plane model the Laplacian is given by the formula<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a><a href="#cite_note-FOOTNOTELang1985-7">[6]</a>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \Delta =-4y^{2}(\partial _{x}^{2}+\partial _{y}^{2}).}</annotation>

</semantics>

</math><img src="./73797459676c63b768ec0699a62e7c69d9ee7580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.321ex; height:3.343ex;" alt="{\displaystyle \displaystyle \Delta =-4y^{2}(\partial _{x}^{2}+\partial _{y}^{2}).}" loading="lazy"></dd></dl>

If s is a complex number and z = x + i y with y > 0, the function

<msub>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot \log y),}</annotation>

</semantics>

</math><img src="./b535654a9fe77e2710f06b02c45218b6bf1cc62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.693ex; height:2.843ex;" alt="{\displaystyle \displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot \log y),}" loading="lazy"></dd></dl>

is an eigenfunction of Δ:

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \Delta f_{s}=4s(1-s)f_{s}.}</annotation>

</semantics>

</math><img src="./3a743865b8f1c5447c777c3080298e8ebded06c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.122ex; height:2.843ex;" alt="{\displaystyle \displaystyle \Delta f_{s}=4s(1-s)f_{s}.}" loading="lazy"></dd></dl>

Since Δ commutes with G, any left translate of fs is also an eigenfunction with the same eigenvalue. In particular, averaging over K, the function

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{s}(z)=\int _{K}f_{s}(k\cdot z)\,dk}</annotation>

</semantics>

</math><img src="./494ed2919521cbb85cd30af78b442da5b88fe0f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.637ex; height:5.676ex;" alt="{\displaystyle \varphi _{s}(z)=\int _{K}f_{s}(k\cdot z)\,dk}" loading="lazy">

</td></tr></tbody></table></dd></dl>

is a K-invariant eigenfunction of Δ on G/K. When

<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle s={1 \over 2}+i\tau ,}</annotation>

</semantics>

</math><img src="./2151c9e0d101bf3d5dee1db5fb44d0f22cd4d709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.679ex; height:5.176ex;" alt="{\displaystyle \displaystyle s={1 \over 2}+i\tau ,}" loading="lazy"></dd></dl>

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, φs is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by K in the <a href="Principal_series_representation" title="Principal series representation">principal series</a>. Various arguments are available to prove that there are no others. One of the simplest classical <a href="Lie_algebra" title="Lie algebra">Lie algebraic</a> arguments<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a><a href="#cite_note-FOOTNOTELang1985-7">[6]</a><a href="#cite_note-FOOTNOTEBargmann1947-33">[32]</a><a href="#cite_note-FOOTNOTEHoweTan1992-34">[33]</a><a href="#cite_note-FOOTNOTEWallach1988-35">[34]</a> 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 to the matrix coefficient for a vector v and representation σ, the vector v is an <a href="Unitary_representation" title="Unitary representation">analytic vector</a> for G and

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</munderover>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle (\sigma (e^{X})v,v)=\sum _{n=0}^{\infty }(\sigma (X)^{n}v,v)/n!}</annotation>

</semantics>

</math><img src="./e6ae1c8a8ba7bbff514bd466e0751208764a8ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.047ex; height:6.843ex;" alt="{\displaystyle \displaystyle (\sigma (e^{X})v,v)=\sum _{n=0}^{\infty }(\sigma (X)^{n}v,v)/n!}" loading="lazy"></dd></dl>

for X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i{\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i{\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./973b107fe0b633e49a428e636085907e6c42778c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.612ex; height:2.176ex;" alt="{\displaystyle i{\mathfrak {t}}}" loading="lazy">. The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann.<a href="#cite_note-FOOTNOTEBargmann1947-33">[32]</a><a href="#cite_note-FOOTNOTEHoweTan1992-34">[33]</a> 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<a href="#cite_note-FOOTNOTEHelgason2001405-36">[35]</a> proceeds by showing that on radial functions the Laplacian has the form

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \Delta =-\partial _{r}^{2}-\coth(r)\cdot \partial _{r},}</annotation>

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</math><img src="./d317c43a8d30b61dceba9e49b8464e0696e87e20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.865ex; height:3.009ex;" alt="{\displaystyle \displaystyle \Delta =-\partial _{r}^{2}-\coth(r)\cdot \partial _{r},}" loading="lazy"></dd></dl>

so that, as a function of r, the zonal spherical function φ(r) must satisfy the <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \displaystyle \varphi ^{\prime \prime }+\coth r\,\varphi ^{\prime }=\alpha \,\varphi }</annotation>

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</math><img src="./c7519b748d53f53fdb9a39a66404822f5baff25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.411ex; height:3.009ex;" alt="{\displaystyle \displaystyle \varphi ^{\prime \prime }+\coth r\,\varphi ^{\prime }=\alpha \,\varphi }" loading="lazy"></dd></dl>

for some constant α. The change of variables t = sinh r transforms this equation into the <a href="Hypergeometric_differential_equation" class="mw-redirect" title="Hypergeometric differential equation">hypergeometric differential equation</a>. The general solution in terms of <a href="Legendre_functions" class="mw-redirect" title="Legendre functions">Legendre functions</a> of complex index is given by<a href="#cite_note-FOOTNOTEDieudonné1978-3">[2]</a><a href="#cite_note-FOOTNOTEBatemanErdélyi1953156-37">[36]</a>

<msub>

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</msub>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \varphi (r)=P_{\rho }(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }(\cosh r+\sinh r\,\cos \theta )^{\rho }\,d\theta ,}</annotation>

</semantics>

</math><img src="./9362acc40f2cf43f3dc9537d4883680d67aa301a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.834ex; height:6.176ex;" alt="{\displaystyle \varphi (r)=P_{\rho }(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }(\cosh r+\sinh r\,\cos \theta )^{\rho }\,d\theta ,}" loading="lazy">

</td></tr></tbody></table></dd></dl>

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.<a href="#cite_note-FOOTNOTEFlensted-Jensen1978-38">[37]</a><a href="#cite_note-FOOTNOTEHelgason1984489–491-39">[38]</a>

<div class="mw-heading mw-heading2"><h2 id="Further_directions">Further directions</h2></div>

<ul><li>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.<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a></li>

<li><a href="Plancherel_theorem_for_spherical_functions" title="Plancherel theorem for spherical functions">Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions</a>.<a href="#cite_note-FOOTNOTEHelgason1984434–458-40">[39]</a> This is an important special case of his <a href="Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a> for real semisimple Lie groups.</li>

<li>The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between Cc∞(K \ G / K ) and Cc∞(A)W, the subalgebra invariant under the Weyl group.<a href="#cite_note-FOOTNOTEGodement1952-4">[3]</a> This is straightforward to establish for SL(2,R).<a href="#cite_note-FOOTNOTELang1985-7">[6]</a></li>

<li>Spherical functions for <a href="Euclidean_group" title="Euclidean group">Euclidean motion groups</a> and <a href="Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact Lie groups</a>.<a href="#cite_note-FOOTNOTEHelgason1984-6">[5]</a></li>

<li>Spherical functions for <a href="P-adic" class="mw-redirect" title="P-adic">p-adic</a> Lie groups. These were studied in depth by Satake and <a href="Ian_G._Macdonald" title="Ian G. Macdonald">Macdonald</a>.<a href="#cite_note-FOOTNOTESatake1963-41">[40]</a><a href="#cite_note-FOOTNOTEMacdonald1971-42">[41]</a> 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 <a href="Langlands_program" title="Langlands program">Langlands program</a>.</li></ul>

<ul><li><a href="Plancherel_theorem_for_spherical_functions" title="Plancherel theorem for spherical functions">Plancherel theorem for spherical functions</a></li>

<li><a href="Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra of a locally compact group</a></li>

<li><a href="Representations_of_Lie_groups" class="mw-redirect" title="Representations of Lie groups">Representations of Lie groups</a></li>

<li><a href="Non-commutative_harmonic_analysis" class="mw-redirect" title="Non-commutative harmonic analysis">Non-commutative harmonic analysis</a></li>

<li><a href="Tempered_representation" title="Tempered representation">Tempered representation</a></li>

<li><a href="Positive_definite_function_on_a_group" class="mw-redirect" title="Positive definite function on a group">Positive definite function on a group</a></li>

<li><a href="Symmetric_space" title="Symmetric space">Symmetric space</a></li>

<li><a href="Gelfand_pair" title="Gelfand pair">Gelfand pair</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> If σ is a unitary representation of G, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (f)=\int _{G}f(g)\sigma (g)\,dg}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sigma (f)=\int _{G}f(g)\sigma (g)\,dg}</annotation>

</semantics>

</math><img src="./6002e5dcddf144ac8b378d3a518e2d63d3b6fd1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.897ex; height:5.676ex;" alt="{\displaystyle \sigma (f)=\int _{G}f(g)\sigma (g)\,dg}" loading="lazy">.

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFDixmier1996">Dixmier 1996</a>, Algèbres hilbertiennes.

</li>

<li id="cite_note-FOOTNOTEDieudonné1978-3">^ <a href="#cite_ref-FOOTNOTEDieudonné1978_3-0">a</a> <a href="#cite_ref-FOOTNOTEDieudonné1978_3-1">b</a> <a href="#cite_ref-FOOTNOTEDieudonné1978_3-2">c</a> <a href="#cite_ref-FOOTNOTEDieudonné1978_3-3">d</a> <a href="#cite_ref-FOOTNOTEDieudonné1978_3-4">e</a> <a href="#CITEREFDieudonné1978">Dieudonné 1978</a>.

</li>

<li id="cite_note-FOOTNOTEGodement1952-4">^ <a href="#cite_ref-FOOTNOTEGodement1952_4-0">a</a> <a href="#cite_ref-FOOTNOTEGodement1952_4-1">b</a> <a href="#CITEREFGodement1952">Godement 1952</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason2001-5">^ <a href="#cite_ref-FOOTNOTEHelgason2001_5-0">a</a> <a href="#cite_ref-FOOTNOTEHelgason2001_5-1">b</a> <a href="#cite_ref-FOOTNOTEHelgason2001_5-2">c</a> <a href="#CITEREFHelgason2001">Helgason 2001</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason1984-6">^ <a href="#cite_ref-FOOTNOTEHelgason1984_6-0">a</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-1">b</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-2">c</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-3">d</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-4">e</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-5">f</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-6">g</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-7">h</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-8">i</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-9">j</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-10">k</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-11">l</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-12">m</a> <a href="#cite_ref-FOOTNOTEHelgason1984_6-13">n</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>.

</li>

<li id="cite_note-FOOTNOTELang1985-7">^ <a href="#cite_ref-FOOTNOTELang1985_7-0">a</a> <a href="#cite_ref-FOOTNOTELang1985_7-1">b</a> <a href="#cite_ref-FOOTNOTELang1985_7-2">c</a> <a href="#cite_ref-FOOTNOTELang1985_7-3">d</a> <a href="#CITEREFLang1985">Lang 1985</a>.

</li>

<li id="cite_note-FOOTNOTECartier1954–1955-8"><a href="#cite_ref-FOOTNOTECartier1954–1955_8-0">^</a> <a href="#CITEREFCartier1954–1955">Cartier 1954–1955</a>.

</li>

<li id="cite_note-FOOTNOTEHochschild1965-9"><a href="#cite_ref-FOOTNOTEHochschild1965_9-0">^</a> <a href="#CITEREFHochschild1965">Hochschild 1965</a>.

</li>

<li id="cite_note-FOOTNOTEDieudonné197855–57-10"><a href="#cite_ref-FOOTNOTEDieudonné197855–57_10-0">^</a> <a href="#CITEREFDieudonné1978">Dieudonné 1978</a>, pp. 55–57.

</li>

<li id="cite_note-FOOTNOTEDieudonné1977-11"><a href="#cite_ref-FOOTNOTEDieudonné1977_11-0">^</a> <a href="#CITEREFDieudonné1977">Dieudonné 1977</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason1978249-12"><a href="#cite_ref-FOOTNOTEHelgason1978249_12-0">^</a> <a href="#CITEREFHelgason1978">Helgason 1978</a>, p. 249.

</li>

<li id="cite_note-FOOTNOTEHelgason1978257–264-13"><a href="#cite_ref-FOOTNOTEHelgason1978257–264_13-0">^</a> <a href="#CITEREFHelgason1978">Helgason 1978</a>, pp. 257–264.

</li>

<li id="cite_note-FOOTNOTEHelgason1984534–538-14"><a href="#cite_ref-FOOTNOTEHelgason1984534–538_14-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, pp. 534–538.

</li>

<li id="cite_note-FOOTNOTEGoodmanWallach1998549–550-15"><a href="#cite_ref-FOOTNOTEGoodmanWallach1998549–550_15-0">^</a> <a href="#CITEREFGoodmanWallach1998">Goodman & Wallach 1998</a>, pp. 549–550.

</li>

<li id="cite_note-FOOTNOTEGoodmanWallach1998550-16"><a href="#cite_ref-FOOTNOTEGoodmanWallach1998550_16-0">^</a> <a href="#CITEREFGoodmanWallach1998">Goodman & Wallach 1998</a>, p. 550.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFHelgason1978">Helgason 1978</a>, Chapter IX.

</li>

<li id="cite_note-FOOTNOTEHarish-Chandra1954a251-18"><a href="#cite_ref-FOOTNOTEHarish-Chandra1954a251_18-0">^</a> <a href="#CITEREFHarish-Chandra1954a">Harish-Chandra 1954a</a>, p. 251.

</li>

<li id="cite_note-FOOTNOTEWallach1973-19"><a href="#cite_ref-FOOTNOTEWallach1973_19-0">^</a> <a href="#CITEREFWallach1973">Wallach 1973</a>.

</li>

<li id="cite_note-FOOTNOTEBerezin1956a-20"><a href="#cite_ref-FOOTNOTEBerezin1956a_20-0">^</a> <a href="#CITEREFBerezin1956a">Berezin 1956a</a>.

</li>

<li id="cite_note-FOOTNOTEBerezin1956b-21"><a href="#cite_ref-FOOTNOTEBerezin1956b_21-0">^</a> <a href="#CITEREFBerezin1956b">Berezin 1956b</a>.

</li>

<li id="cite_note-FOOTNOTEHarish-Chandra1954b-22"><a href="#cite_ref-FOOTNOTEHarish-Chandra1954b_22-0">^</a> <a href="#CITEREFHarish-Chandra1954b">Harish-Chandra 1954b</a>.

</li>

<li id="cite_note-FOOTNOTEHarish-Chandra1954c-23"><a href="#cite_ref-FOOTNOTEHarish-Chandra1954c_23-0">^</a> <a href="#CITEREFHarish-Chandra1954c">Harish-Chandra 1954c</a>.

</li>

<li id="cite_note-FOOTNOTEKunzeStein1961-24"><a href="#cite_ref-FOOTNOTEKunzeStein1961_24-0">^</a> <a href="#CITEREFKunzeStein1961">Kunze & Stein 1961</a>.

</li>

<li id="cite_note-FOOTNOTEStein1970-25"><a href="#cite_ref-FOOTNOTEStein1970_25-0">^</a> <a href="#CITEREFStein1970">Stein 1970</a>.

</li>

<li id="cite_note-FOOTNOTEKostant1969-26"><a href="#cite_ref-FOOTNOTEKostant1969_26-0">^</a> <a href="#CITEREFKostant1969">Kostant 1969</a>.

</li>

<li id="cite_note-FOOTNOTEHarish-Chandra1958-27"><a href="#cite_ref-FOOTNOTEHarish-Chandra1958_27-0">^</a> <a href="#CITEREFHarish-Chandra1958">Harish-Chandra 1958</a>.

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFHelgason2001">Helgason 2001</a>, pages 418–422, 427-434

</li>

<li id="cite_note-FOOTNOTEHelgason1984418-29"><a href="#cite_ref-FOOTNOTEHelgason1984418_29-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, p. 418.

</li>

<li id="cite_note-FOOTNOTEDavies1990-30"><a href="#cite_ref-FOOTNOTEDavies1990_30-0">^</a> <a href="#CITEREFDavies1990">Davies 1990</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason1984432–433-31"><a href="#cite_ref-FOOTNOTEHelgason1984432–433_31-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, pp. 432–433.

</li>

<li id="cite_note-FOOTNOTEHelgason1984501–502-32"><a href="#cite_ref-FOOTNOTEHelgason1984501–502_32-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, pp. 501–502.

</li>

<li id="cite_note-FOOTNOTEBargmann1947-33">^ <a href="#cite_ref-FOOTNOTEBargmann1947_33-0">a</a> <a href="#cite_ref-FOOTNOTEBargmann1947_33-1">b</a> <a href="#CITEREFBargmann1947">Bargmann 1947</a>.

</li>

<li id="cite_note-FOOTNOTEHoweTan1992-34">^ <a href="#cite_ref-FOOTNOTEHoweTan1992_34-0">a</a> <a href="#cite_ref-FOOTNOTEHoweTan1992_34-1">b</a> <a href="#CITEREFHoweTan1992">Howe & Tan 1992</a>.

</li>

<li id="cite_note-FOOTNOTEWallach1988-35"><a href="#cite_ref-FOOTNOTEWallach1988_35-0">^</a> <a href="#CITEREFWallach1988">Wallach 1988</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason2001405-36"><a href="#cite_ref-FOOTNOTEHelgason2001405_36-0">^</a> <a href="#CITEREFHelgason2001">Helgason 2001</a>, p. 405.

</li>

<li id="cite_note-FOOTNOTEBatemanErdélyi1953156-37"><a href="#cite_ref-FOOTNOTEBatemanErdélyi1953156_37-0">^</a> <a href="#CITEREFBatemanErdélyi1953">Bateman & Erdélyi 1953</a>, p. 156.

</li>

<li id="cite_note-FOOTNOTEFlensted-Jensen1978-38"><a href="#cite_ref-FOOTNOTEFlensted-Jensen1978_38-0">^</a> <a href="#CITEREFFlensted-Jensen1978">Flensted-Jensen 1978</a>.

</li>

<li id="cite_note-FOOTNOTEHelgason1984489–491-39"><a href="#cite_ref-FOOTNOTEHelgason1984489–491_39-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, pp. 489–491.

</li>

<li id="cite_note-FOOTNOTEHelgason1984434–458-40"><a href="#cite_ref-FOOTNOTEHelgason1984434–458_40-0">^</a> <a href="#CITEREFHelgason1984">Helgason 1984</a>, pp. 434–458.

</li>

<li id="cite_note-FOOTNOTESatake1963-41"><a href="#cite_ref-FOOTNOTESatake1963_41-0">^</a> <a href="#CITEREFSatake1963">Satake 1963</a>.

</li>

<li id="cite_note-FOOTNOTEMacdonald1971-42"><a href="#cite_ref-FOOTNOTEMacdonald1971_42-0">^</a> <a href="#CITEREFMacdonald1971">Macdonald 1971</a>.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>

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<li><cite id="CITEREFBatemanErdélyi1953" class="citation cs2"><a href="Harry_Bateman" title="Harry Bateman">Bateman, Harry</a>; <a href="Arthur_Erd%C3%A9lyi" title="Arthur Erdélyi">Erdélyi, Arthur</a> (1953), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf">Higher transcendental functions, Vol. I</a> (PDF), McGraw–Hill, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-019546-3</bdi>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/bateman/Vol1.pdf">the original</a> (PDF) on 2011-08-11, retrieved 2011-07-30</cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)</li>

<li><cite id="CITEREFBerezin1956a" class="citation cs2"><a href="Felix_Berezin" title="Felix Berezin">Berezin, F. A.</a> (1956a), "Операторы Лапласа на полупростых группах" [Laplace operators on semisimple groups], <a href="Doklady_Akademii_Nauk_SSSR" class="mw-redirect" title="Doklady Akademii Nauk SSSR">Doklady Akademii Nauk SSSR</a>, 107: 9–12.</cite></li>

<li><cite id="CITEREFBerezin1956b" class="citation cs2">Berezin, F. A. (1956b), "Representation of complex semisimple Lie groups in Banach space", Doklady Akademii Nauk SSSR, 110: 897–900</cite></li>

<li><cite id="CITEREFBrychkovPrudnikov2001" class="citation cs2">Brychkov, Yu A.; Prudnikov, A.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Spherical_functions">"Spherical functions"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFCartier1954–1955" class="citation cs2"><a href="Pierre_Cartier_(mathematician)" title="Pierre Cartier (mathematician)">Cartier, Pierre</a> (1954–1955), <a rel="nofollow" class="external text" href="http://archive.numdam.org/article/SSL_1954-1955__1__A24_0.pdf">Structure topologique des groupes de Lie généraux, Exposé No. 22</a> (PDF), Séminaire "Sophus Lie", vol. 1</cite>.</li>

<li><cite id="CITEREFDavies1990" class="citation cs2">Davies, E. B. (1990), Heat Kernels and Spectral Theory, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-40997-7</bdi></cite></li>

<li><cite id="CITEREFDieudonné1977" class="citation cs2"><a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1977), Treatise on Analysis, Vol. V, Academic Press</cite></li>

<li><cite id="CITEREFDieudonné1978" class="citation cs2"><a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1978), Treatise on Analysis, Vol. VI, Academic Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-215506-8</bdi></cite></li>

<li><cite id="CITEREFDirac1945" class="citation cs2"><a href="P._A._M._Dirac" class="mw-redirect" title="P. A. M. Dirac">Dirac, P. A. M.</a> (1945), "Unitary representations of the Lorentz group", <a href="Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 183 (994): 284–295, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1945RSPSA.183..284D">1945RSPSA.183..284D</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1945.0003">10.1098/rspa.1945.0003</a></cite></li>

<li><cite id="CITEREFDixmier1996" class="citation cs2"><a href="Jacques_Dixmier" title="Jacques Dixmier">Dixmier, Jacques</a> (1996), Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Les Grands Classiques Gauthier-Villars., Éditions Jacques Gabay, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>2-87647-012-8</bdi></cite></li>

<li><cite id="CITEREFFarautKorányi1994" class="citation cs2">Faraut, Jacques; <a href="%C3%81d%C3%A1m_Kor%C3%A1nyi" title="Ádám Korányi">Korányi, Adam</a> (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-853477-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1446489">1446489</a></cite>, Chapter XIV.</li>

<li><cite id="CITEREFFlensted-Jensen1978" class="citation cs2">Flensted-Jensen, Mogens (1978), "Spherical functions of a real semisimple Lie group. A method of reduction to the complex case", J. Funct. Anal., 30: 106–146, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-1236%2878%2990058-7">10.1016/0022-1236(78)90058-7</a></cite></li>

<li><cite id="CITEREFGelfandNaimark1947" class="citation cs2"><a href="Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="Mark_Naimark" title="Mark Naimark">Naimark, M.A.</a> (1947), "Unitary representations of the Lorentz group", Izv. Akad. Nauk SSSR, Ser. Mat., 37: 411–504</cite></li>

<li><cite id="CITEREFGelfandNaimark1948" class="citation cs2"><a href="Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="Mark_Naimark" title="Mark Naimark">Naimark, M.A.</a> (1948), "An analogue of Plancherel's theorem for the complex unimodular group", Doklady Akademii Nauk SSSR, 63: 609–612</cite></li>

<li><cite id="CITEREFGelfandNaimark1952" class="citation cs2"><a href="Israel_Gelfand" title="Israel Gelfand">Gelfand, I.M.</a>; <a href="Mark_Naimark" title="Mark Naimark">Naimark, M.A.</a> (1952), "Unitary representations of the unimodular group containing the identity representation of the unitary subgroup", Trudy Moscov. Mat. Obšč., 1: 423–475</cite></li>

<li><cite id="CITEREFGodement1952" class="citation cs2"><a href="Roger_Godement" title="Roger Godement">Godement, Roger</a> (1952), "A theory of spherical functions. I", Transactions of the American Mathematical Society, 73 (3): 496–556, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990805">10.2307/1990805</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990805">1990805</a></cite></li>

<li><cite id="CITEREFGoodmanWallach1998" class="citation cs2">Goodman, Roe; Wallach, Nolan (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-66348-2</bdi></cite></li>

<li><cite id="CITEREFHarish-Chandra1947" class="citation cs2"><a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a> (1947), "Infinite irreducible representations of the Lorentz group", <a href="Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 189 (1018): 372–401, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1947RSPSA.189..372H">1947RSPSA.189..372H</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1947.0047">10.1098/rspa.1947.0047</a></cite></li>

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<li><cite id="CITEREFHarish-Chandra1954c" class="citation cs2"><a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a> (1954c), "The Plancherel formula for complex semisimple Lie groups", Trans. Amer. Math. Soc., 76 (3): 485–528, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990793">10.2307/1990793</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990793">1990793</a>, <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063477">1063477</a>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16589034">16589034</a></cite> (Second proof of formula for complex semisimple Lie groups)</li>

<li><cite id="CITEREFHarish-Chandra1958" class="citation cs2"><a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a> (1958), "Spherical functions on a semisimple Lie group I, II", Amer. J. Math., 80 (2): 241–310, 553–613, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372786">10.2307/2372786</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372786">2372786</a>, <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC528464">528464</a>, <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16590028">16590028</a></cite> (Determination of Plancherel measure)</li>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFCasselman" class="citation cs2">Casselman, William, <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/research/pdf/Macdonald.pdf">Notes on spherical functions</a> (PDF)</cite></li></ul></div><div><div class="zim-footer">

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