Zonal spherical function

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group "G" with compact subgroup "K" (often a maximal compact subgroup) that arises as the matrix coefficient of a "K"-invariant vector in an irreducible representation of "G".These are precisely the irreducible representations that arise in the decomposition of the unitary representation on "L"²("G"/"K"). In this case the commutant of "G" is generated by the algebra of biinvariant functions on "G" with respect to "K" acting by right convolution. It is commutative if in addition "G" is a semisimple Lie group. The zonal spherical functions describe precisely the spectrum of the commutative
C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because "G" is the complexification of "K", and the formulas are related to analytic continuations of the Weyl character formula on "K". The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group "G" also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of "G" on "L"²("G"/"K"), as differential operators on the symmetric space "G"/"K". For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinarydifferential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function.

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.

Definitions

Let "G" be a locally compact unimodular topological group and "K" a compact subgroup and let "H"₁ = "L"²("G"/"K"). Thus "H"₁ admits a unitary representation π of "G" by left translation. This is a subrepresentation of the regular representation, since if "H"= "L"²("G") with left and right regular representations λ and ρ of "G" and "P" is the orthogonal projection

:$P\; =int\_K\; ho(k)\; ,\; dk$

from "H" to "H"₁ then "H"₁ can naturally be identified with "PH" with the action of "G" given by the restriction of λ.

On the other hand by von Neumann's commutation theorem [ Dixmier (1996), Algèbres hilbertiennes. ]

:$lambda(G)^prime=\; ho(G)^\{primeprime\},$

where "S"' denotes the commutant of a set of operators "S", so that

:$pi(G)^prime\; =\; P\; ho(G)^\{primeprime\}P.$

Thus the commutant of π is generated as a von Neumann algebra by operators

:$P\; ho(f)P\; =\; int\_G\; f(g)\; (P\; ho(g)P)\; ,\; dg$

where "f" is a continuous function of compact support on "G".

However "P"ρ("f") "P" is just the restriction of ρ("F") to "H"₁, where :$F(g)\; =int\_K\; int\_K\; f(kgk^prime)\; ,\; dk,\; dk^prime$

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 "C"_c("K""G"/"K"), the "K"-biinvariant continuous functions of compact support on "G".

These functions form a * algebra under convolution with involution

:$F^*(g)\; =overline\{F(g^\{-1\})\},$

often called the Hecke algebra for the pair ("G", "K").

Let "A"("K""G"/"K") denote the C* algebra generated by the operators ρ("F") on "H"₁.

The pair ("G", "K")is said to be a Gelfand pair [ Dieudonné (1978)] if one, and hence all, of the following algebras are commutative:

* $pi(G)^prime$

*

*

Since "A"("K""G"/"K") is a commutative C* algebra, by the Gelfand-Naimark theorem it has the form "C"₀("X"),where "X" is the locally compact space of norm continuous * homomorphisms of "A"("K""G"/"K") into C.

A concrete realization of the * homomorphisms in "X" as "K"-biinvariant uniformly bounded functions on "G" is obtained as follows. [ Godement (1954)] [Dieudonné (1978)] [Helgason (2001)] [ Helgason (1984)] [Lang (1985)]

Because of the estimate

:$|pi(F)|le\; int\_G\; |F(g)|\; ,\; dg\; equiv\; |F|\_1,$

the representation π of "C"_c("K""G"/"K") in "A"("K""G"/"K") extends by continuityto L¹("K""G"/"K"), the * algebra of "K"-biinvariant integrable functions. The image formsa 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 Banach space dual of L¹ is L^∞, it follows that

:$chi(pi(F))\; =int\_G\; F(g)\; h(g)\; ,\; dg,$

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").

Properties

A zonal spherical function "h" has the following properties: [ Dieudonné (1978).]

# "h" is uniformly continuous on "G"
# $h(x)\; h(y)\; =\; int\_K\; h(xky)\; ,dk\; ,,(x,yin\; G).$
# "h"(1) =1 (normalisation)
# "h" is a positive definite function on "G"
# "f" * "h" is proportional to "h" for all "f" in "C"_c("K""G"/"K").

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 connectionwith unitary representations. For semisimple Lie groups, there is a further characterization as eiegenfunctions of
invariant differential operators on "G"/"K" (see below).

In fact, as a special case of the Gelfand-Naimark-Segal construction, there is one-one correspondence betweenirreducible representations σ of "G" having a unit vector "v" fixed by "K" and zonal spherical functions"h" given by

:$h(g)\; =\; (sigma(g)\; v,v).$

Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on "H"₁. Each representation σ extends uniquely by continuityto "A"("K""G"/"K"), so that each zonal spherical function satisfies

:$left|int\_G\; f(g)\; h(g),\; dg\; ight|\; le\; |pi(f)|$

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

:$int\_G\; |f(g)|^2\; ,\; dg\; =\; int\_X\; |chi(pi(f))|^2\; ,\; dmu(chi).$

μ is called the Plancherel measure. Since π("G")' is the centre of the von Neumann algebra generated by "G", it also gives the measure associated with the direct integral decomposition of "H"₁ in terms of the irreducible representations σ_χ.

Gelfand pairs

If "G" is a connected Lie group, then, thanks to the work of Elie Cartan, Malcev, Iwasawa and Chevalley, "G" has a maximal compact subgroup, unique up to conjugation. [ Cartier (1954-1955).] [Hochschild (1965).] In this case "K" is connected and the quotient "G"/"K" is diffeomorphic to a Euclidean space. When "G" is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space "G"/"K", a generalisation of the polar decomposition of invertible matrices. Indeed if τ is the associated period two automorphism of "G" with fixed point subgroup "K", then

:$G=Pcdot\; K,$

where

:$P=\; \{gin\; G|\; au(g)=g^\{-1\}\}.$

Under the exponential map, "P" is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of "G".Since τ preserves "K", it induces an automorphism of the Hecke algebra "C"_c("K""G"/"K"). On theother hand, if "F" lies in "C"_c("K""G"/"K"), then

:"F"(τ"g") = "F"("g"⁻¹),

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

*the Hecke algebra is commutative

*("G","K") is a Gelfand pair.

More generally the same argument gives the following criterion of Gelfand for ("G","K") to be a Gelfand pair: [harvnb|Dieudonné|1978|p=55-57]

*"G" is a unimodular locally compact group;
*"K" is a compact subgroup arising as the fixed points of a period two automorphism τ of "G";
*"G" ="K"·"P" (not necessarily a direct product), where "P" is defined as above.

The two most important examples covered by this are when:

* "G" is a compact connected semisimple Lie group with τ a period two automorphism; [harvnb|Dieudonné|1977] [ harvnb|Helgason|1978|p=249]
* "G" is a semidirect product $A\; times\; K$, 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".

The three cases cover the three types of symmetric spaces "G"/"K": [harvnb|Helgason|1984]

# "Non-compact type", when "K" is a maximal compact subgroup of a non-compact real semisimple Lie group "G";
# "Compact type", when "K" is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group "G";
# "Euclidean type", when "A" is a finite-dimensional Euclidean space with an orthogonal action of "K".

Cartan–Helgason theorem

Let "G" be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a "G" with fixed point subgroup "K" = "G"^τ. In this case "K" is a connected compact Lie group. [harvnb|Helgason|1984] In addition let "T" be a maximal torus of "G" invariant under τ, such that "T" $cap$ "P" is a maximal torus in "P", and set [harvnb|Helgason|1978|p=257-264]

:$S=\; Kcap\; T\; =\; T^\; au.$

"S" is the direct product of a torus and an elementary abelian 2-group.

In 1929 Élie Cartan found a rule to determine the decomposition of L²("G"/"K") into the direct sum of finite-dimensional irreducible representations of "G", which was only proved rigorously in 1970 by Sigurdur Helgason. Because the commutant of "G" on L²("G"/"K") is commutative, each irreducible representation appears with multiplicity one. By Frobenius reciprocity for compact groups, the irreducible representations "V" that occur are precisely those admitting a non-zero vector fixed by "K".

From the representation theory of compact semisimple groups, irreducible representations of "G" are classified by their highest weight. This is specified by a homomorphism of the maximal torus "T" into T.

The Cartan–Helgason theorem [harvnb|Helgason|1984|p=534-538] [harvnb|Goodman|Wallach|1998|p=549-550] states that

:

where $ell$ is real.

Complex case

If "G" is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup "K". If $\{mathfrakg\}$ and $mathfrak\{k\}$ are their Lie algebras, then

:$mathfrak\{g\}\; =\; mathfrak\{k\}\; oplus\; imathfrak\{k\}.$

Let "T" be a maximal torus in "K" with Lie algebra $mathfrak\{t\}$. Then

:$A=\; exp\; i\; mathfrak\{t\},\; ,,\; P=\; exp\; i\; mathfrak\{k\}.$

Let

:$W=\; N\_K(T)/T$

be the Weyl group of "T" in "K". Recall characters in Hom("T",T) are called weights and can be identified with elements of the weight lattice Λ in Hom($mathfrak\{t\}$, R) = $mathfrak\{t\}^*$.There is a natural ordering on weights and very finite-dimensionalirreducible representation (π, "V") of "K" has a unique highest weightλ. The weights of the adjoint representation of "K" on $mathfrak\{k\}ominus\; mathfrak\{t\}$ are called rootsand ρ is used to denote half the sum of the positive roots α,
Weyl's character formula asserts that for "z" = exp "X" in "T"

:$displaystylechi\_lambda(e^X)equiv\; \{\; m\; Tr\}\; ,\; pi(z)\; =\; A\_\{lambda+\; ho\}(e^X)/A\_\{\; ho\}(e^X),$

where, for μ in $mathfrak\{t\}^*$, "A"_μ denotes the antisymmetrisation

:$displaystyle\; A\_mu(e^X)\; =sum\_\{sin\; W\}\; varepsilon(s)\; e^\{imu(sX)\},$

and ε denotes the "sign character" of the finite reflection group "W".

Weyl's denominator formula expresses the denominator "A"_ρ as a product:

:$displaystyle\; A\_\; ho(e^X)\; =\; e^\{i\; ho(X)\}\; prod\_\{alpha>0\}(1\; -\; e^\{-ialpha(X)\}),$ where the product is over the positive roots.

Weyl's dimension formula asserts that

:$displaystylechi\_lambda(1)\; equiv\; \{\; m\; dim\},\; V\; =\; \{prod\_\{alpha>0\}\; (lambda\; +\; ho,alpha)over\; prod\_\{alpha>0\}\; (\; ho,alpha)\}.$

where the inner product on $mathfrak\{t\}^*$ is that associated with the Killing form on$mathfrak\{k\}$.

Now

*every irreducible representation of "K" extends holomorphically to the complexification "G"

*every irreducible character χ_λ of "K" extends holomorphically to the complexification "G"

* for every λ in Hom("A",T) = $imathfrak\{t\}^*$, there is a zonal spherical function ψ_λ.

The Berezin-Harish-Chandra formula [Helgason (1984)] asserts that for "X" in $imathfrak\{t\}$

:

where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on "G".

Further directions

*"The theory of zonal functions that are not necessarily positive-definite." These are given by the same formulas as above, but without restrictions on the complex parameter "s" or ρ. They correspond to non-unitary representations. [ Helgason (1984)]
*"Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions." [Helgason (1984), pages 434-458] This is an important special case of his Plancherel theorem for real semisimple Lie groups.
*"The structure of the Hecke algebra". Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between C_c^∞("K" "G" / "K" ) and C_c^∞("A")^"W", the subalgebra invariant under the Weyl group. [Godement (1952)] This is straightforward to establish for SL(2,R). [Lang (1985)]
*"Spherical functions for Euclidean motion groups and compact Lie groups". [Helgason (1984)]
*"Spherical functions for p-adic Lie groups". These were studied in depth by Satake and Macdonald. [Satake (1963)] [ Macdonald (1971)] Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program.

ee also

*Hecke algebra of a locally compact group
*Representations of Lie groups
*Non-commutative harmonic analysis
*Tempered representation
*Positive definite function on a group
*Symmetric space
*Gelfand pair

Notes

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