Zonal spherical function

From Wikipedia, the free encyclopedia

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with maximal compact subgroup K 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 L²(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 L²(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 Ian G. Macdonald.

1 Definitions
2 Properties
3 Gelfand pairs
4 Harish-Chandra's formula
5 Eigenfunctions
6 Example: SL(2,C)
7 Complex case
8 Example: SL(2,R)
9 Real case
10 Plancherel measure
11 Notes
12 References
13 See also

[edit] Definitions

See also: Hecke algebra

Let G be a locally compact unimodular topological group and K a compact subgroup and let H₁ = L²(G/K). Thus H₁ admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L²(G) with left and right regular representations λ and ρ of G and P is the orthogonal projection

$P =\int_K \rho(k) \, dk$

then H₁ 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^[1]

$\lambda(G)^\prime= \rho(G)^{\prime\prime},$

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

$\pi(G)^\prime = P \rho(G)^{\prime\prime}P.$

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

$P\rho(f)P = \int_G f(g) (P \rho(g)P) \, dg$

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

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

The pair (G, K) is said to be a Gelfand pair ^[2] if one, and hence all, of the following algebras are commutative:

$\pi(G)^\prime$

$C_c(K\backslash G /K)$

$A(K\backslash G /K)$

Since A(K\G/K) is a commutative C* algebra, by the Gelfand-Naimark theorem it has the form C₀(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.^[3]^[4]^[5]^[6]^[7]

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 continuity to L¹(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is 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).

[edit] Properties

A zonal spherical function h has the following properties:^[8]

h is uniformly continuous on G
$h(x) h(y) = \int_K h(xky) \,dk \,\,(x,y\in G).$
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 connection with 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 between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by

h (g) = (σ(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 H₁. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies

$\left|\int_G f(g) h(g)\, dg\right| \le \|\pi(f)\|$

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

$\int_G |f(g)|^2 \, dg = \int_X |\chi(\pi(f))|^2 \, d\mu(\chi).$

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 H₁ in terms of the irreducible representations σ_χ.

[edit] Gelfand pairs

See also: Gelfand pair

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.^[9]^[10] 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

$G=P\cdot K,$

where

$P= \{g\in G| \tau(g)=g^{-1}\}.$

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

F(τg) = 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.

[edit] Harish-Chandra's formula

If G is a semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is isomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that^[11]

G =KAK.

There is also an associated Iwasawa decomposition

G =KAN,

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 solvable subgroup of G, the semidirect product of N by A, and G = KS.

If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L²(G/S) = L²(K), ^[12] a so-called a principal series representation.

This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and Δ_S the modular function of S. Then^[13]

$\int_G f(g) \,dg = \int_S\int_K f(x\cdot k) \, dx\, dk = \int_S\int_K f(k\cdot x) \Delta_S(x)\,dx\, dk.$

The principal series representation σ is realised on L²(K) as^[14]

$(\sigma(g) \xi)(k) = \alpha^\prime(g^{-1}x)^{-1} \, \xi(U(g^{-1}k)),$

where

$g = U(g)\cdot X(g)$

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

$\alpha^\prime(kx) = \Delta_S(x)^{1/2} \alpha(x)$

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,

ψ α (g) = (σ(g) v, v)

defines a zonal spherical function of G.

Harish-Chandra proved that

every zonal spherical function arises in this way.

He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A

W (A) = N K (A) / C K (A),

the quotient of the normaliser of A in K by its centraliser, a finite reflection group.

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

$\psi_\alpha(g) = \int_K \alpha^\prime(gk)^{-1}\, dk$

as an integral over K.

It can also be verified directly^[15] that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below).^[16]^[17] In the case of complex semisimple groups, Harish-Chandra and Berezin realised independently that the formula simplified considerably and could be proved more directly.^[18]^[19] ^[20] ^[21] ^[22]

[edit] Eigenfunctions

Harish-Chandra proved^[23]^[24] that zonal spherical functions can be characterised as those normalised positive definite K-invariant functions on G/K that are eigenfunctions of D(G/K), the algebra of invariant differential operators on G. This algebra acts on G/K and commutes with the natural action of G by left translation. It can be identified with the subalgebra of the universal enveloping algebra of G fixed under the adjoint action of K. As for the commutant of G on L²(G/K) and the corresponding Hecke algebra, this algebra of operators is commutative; indeed it is a subalgebra of the algebra of mesaurable operators affiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of W(A)-invariant polynomials on the Lie algebra of A, which itself is a polynomial ring by the Chevalley-Shephard-Todd theorem on polynomial invariants of finite reflection groups. The simplest invariant differential operator on G/K is the Laplacian operator; up to a sign this operator is just the image under π of the Casimir operator in the centre of the universal enveloping algebra of G.

Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K) there is a constant λ_D such that

π(D) f = λ D f,

i.e. f is a simultaneous eigenfunction of the operators π(D).

If ψ is a zonal spherical function, then, regarded as a function on G/K, it is an eigenfunction of the Laplacian there, an elliptic differential operator with real analytic coefficients. By analytic elliptic regularity, ψ is a real analytic function on G/K, and hence G.

[edit] Example: SL(2,C)

See also: SL(2,C) and Spectral theory of ordinary differential equations

[edit] Complex case

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

$\mathfrak{g} = \mathfrak{k} \oplus i\mathfrak{k}.$

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-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the adjoint representation of K on $\mathfrak{k}\ominus \mathfrak{t}$ are called roots and 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

$\chi_\lambda(e^X)\equiv {\rm Tr} \, \pi(z) = A_{\lambda+\rho}(e^X)/A_{\rho}(e^X),$

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

$A_\mu(e^X) =\sum_{s\in W} \varepsilon(s) e^{i\mu(sX)},$

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

Weyl's denominator formula expresses the denominator A_ρ as a product:

A_ρ(e^X) = e^iρ(X)	∏	(1 − e ^{− iα(X)}),
	α > 0

where the product is over the positive roots.

Weyl's dimension formula asserts that

$\chi_\lambda(1) \equiv {\rm dim}\, V = {\prod_{\alpha>0} (\lambda + \rho,\alpha)\over \prod_{\alpha>0} (\rho,\alpha)}.$

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) = $i\mathfrak{t}^*$ , there is a zonal spherical function ψ_λ.

The Berezin-Harish-Chandra formula^[25] asserts that for X in $i\mathfrak{t}$

ψ λ (e X) = χ λ (e X) / χ λ (1).

In other words:

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

One of the simplest proofs^[26] of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of Freudenthal's classical proof of the Weyl character formula, using the radial component on T of the Laplacian on K.^[27]

In the latter case the class functions on K can be identified with W-invariant functions on T. The radial component of Δ_K on T is just the expression for the restriction of Δ_K to W-invariant functions on T, where it is given by the formula

$\Delta_K= h^{-1}\circ \Delta_T \circ h + \|\rho\|^2,$

where

h (e X) = A ρ (e X)

for X in $\mathfrak{t}$ . If χ is a character with highest weight λ, it follows that φ = h·χ satisfies

$\Delta_T \varphi= (\|\lambda + \rho\|^2 -\|\rho\|^2) \varphi.$

Thus for every weight μ with non-zero Fourier coefficient in φ,

$\|\lambda +\rho\|^2 = \|\mu+\rho\|^2.$

The classical argument of Freudenthal shows that μ + ρ must have the form s(λ + ρ) for some s in W, so the character formula follows from the antisymmetry of φ.

Similarly K-biinvariant functions on G can be identified with W(A)-invariant functions on A. The radial component of Δ_G on A is just the expression for the restriction of Δ_G to W(A)-invariant functions on A. It is given by the formula

$\Delta_G= H^{-1}\circ \Delta_A\circ H - \|\rho\|^2,$

where

H (e X) = A ρ (e X)

for X in $i\mathfrak{t}$ .

[edit] Example: SL(2,R)

See also: SL(2,R) and Spectral theory of ordinary differential equations

[edit] Real case

[edit] Plancherel measure

[edit] Notes

^ Dixmier (1996), Algèbres hilbertiennes.
^ Dieudonné (1978)
^ Godement (1954)
^ Dieudonné (1978)
^ Helgason (2001)
^ Helgason (1984)
^ Lang (1985)
^ Dieudonné (1978).
^ Cartier (1954-1955).
^ Hochschild (1965).
^ Helgason (1978), Chapter IX.
^ Harish-Chandra (1954a), page 251.
^ Helgason (1984)
^ Wallach (1973)
^ Dieudonné (1978)
^ Helgason (2001)
^ Helgason (1984)
^ Berezin (1956a)
^ Berezin (1956b)
^ Harish-Chandra (1954b)
^ Harish-Chandra (1954c)
^ Helgason (1984)
^ Helgason (2001)
^ Helgason (1984)
^ Helgason (1984)
^ Helgason (1984), pages 432-433
^ Helgason (1984), pages 501-502

[edit] References

Barnett, Adam & Smart, Nigel P. (2003), “Mental Poker revisited, in Cryptography and Coding”, Lecture Notes in Comput. Sci. (Springer-Verlag) 2898: 370-383
Berezin, F. A. (1956), “Laplace operators on semisimple Lie groups”, Dokl. Akad. Nauk SSSR 107: 9-12
Berezin, F. A. (1956), “Representation of complex semisimple Lie groups in Banach space”, Dokl. Akad. Nauk SSSR 110: 897-900
Cartier, Pierre (1954-1955), Structure topologique des groupes de Lie généraux, Exposé No. 22, vol. 1, Séminaire "Sophus Lie" . Online version
Dieudonné, Jean (1978), Treatise on Analysis, Vol. VI, Academic Press, ISBN 0-12-215506-8
Dixmier, Jacques (1996), Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Les Grands Classiques Gauthier-Villars., Éditions Jacques Gabay, ISBN 2-87647-012-8
Gelfand, I.M. & Naimark, M.A. (1948), “An analogue of Plancherel's theorem for the complex unimodular group”, Dokl. Akad. Nauk. USSR 63: 609-612
Gelfand, I.M. & Naimark, M.A. (1952), Unitary representations of the unimodular group containing the identity representation of the unitary subgroup, vol. 1, pp. 423-475
Godement, Roger (1952), “A theory of spherical functions. I”, Transactions of the American Mathematical Society 73: 496-556
Harish-Chandra (1954a), “Representations of Semisimple Lie Groups. III”, Trans. Amer. Math. Soc. 76: 26-65 (Formula for zonal spherical functions on a semisimple Lie group)
Harish-Chandra (1954b), “Representations of Semisimple Lie Groups. III”, Trans. Amer. Math. Soc. 76: 234-253 (Simplification of formula for complex semisimple Lie groups)
Harish-Chandra (1954c), “The Plancherel formula for complex semisimple Lie groups”, Trans. Amer. Math. Soc. 76: 485-528 (Second proof of formula for complex semisimple Lie groups)
Harish-Chandra (1958), “Spherical functions on a semisimple Lie group I, II”, Amer. J. Math. 80: 241-310, 553-613 (Determination of Plancherel measure)
Helgason, Sigurdur (2001), Differential geometry and symmetric spaces (reprint of 1962 edition), American Mathematical Society, ISBN 0821827359
Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3
Hochschild, Gerhard P. (1965), The structure of Lie groups, Holden-Day
Lang, Serge (1985), SL(2,R), vol. 105, Graduate Texts in Mathematics, Springer-Verlag, ISBN 0387961984
Macdonald, Ian G. (1971), Spherical Functions on a Group of p-adic Type, vol. 2, Publ. Ramanujan Institute, University of Madras
Wallach, Nolan (1973), Harmonic Analysis on Homogeneous Spaces, Marcel Decker, ISBN 0824760107

Zonal spherical function

From Wikipedia, the free encyclopedia

Contents

[edit] Definitions

[edit] Properties

[edit] Gelfand pairs

[edit] Harish-Chandra's formula

[edit] Eigenfunctions

[edit] Example: SL(2,C)

[edit] Complex case

[edit] Example: SL(2,R)

[edit] Real case

[edit] Plancherel measure

[edit] Notes

[edit] References

[edit] See also

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