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". The key examples are the matrix coefficients of the "spherical principal series", the irreducible representations appearing in the decomposition of the unitary representation of "G" 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"/"K" is a symmetric space, for example when "G" is a connected semisimple Lie group with finite centre and "K" is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding
C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant "L"1 functions is larger; when "G" is a semisimple Lie group with maximal compact subgroup "K", additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
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 ordinary differential 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
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
where "S"' denotes the commutant of a set of operators "S", so that
Thus the commutant of π is generated as a von Neumann algebra by operators
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
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
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 if one, and hence all, of the following algebras are commutative:
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.
Because of the estimate
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
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:
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 eigenfunctions 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
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 continuity
to "A"("K"\"G"/"K"), so that each zonal spherical function satisfies
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
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 σχ.
Gelfand pairs.
If "G" is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, "G" has a maximal compact subgroup, unique up to conjugation. 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
where
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
so that τ induces an anti-automorphism, because inversion does. Hence, when "G" is semisimple,
More generally the same argument gives the following criterion of Gelfand for ("G","K") to be a Gelfand pair:
The two most important examples covered by this are when:
The three cases cover the three types of symmetric spaces "G"/"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. In addition let "T" be a maximal torus of "G" invariant under τ, such that "T" formula_19 "P" is a maximal torus in "P", and set
"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 proved rigorously only 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 states that
The corresponding irreducible representations are called "spherical representations".
The theorem can be proved using the Iwasawa decomposition:
where formula_22, formula_23, formula_24 are the complexifications of the Lie algebras of "G", "K", "A" = "T" formula_19 "P" and
summed over all eigenspaces for "T" in formula_22 corresponding to positive roots α not fixed by τ.
Let "V" be a spherical representation with highest weight vector "v"0 and "K"-fixed vector "v""K". Since "v"0 is an eigenvector of the solvable Lie algebra formula_28, the Poincaré–Birkhoff–Witt theorem
implies that the "K"-module generated by "v"0 is the whole of "V". If "Q" is the orthogonal projection onto the fixed points of "K" in "V" obtained by averaging over "G" with respect to Haar measure, it follows that
for some non-zero constant "c". Because "v""K" is fixed by "S" and "v"0 is an eigenvector for "S", the subgroup "S" must actually fix "v"0, an equivalent form of the triviality condition on "S".
Conversely if "v"0 is fixed by "S", then it can be shown that the matrix coefficient
is non-negative on "K". Since "f"(1) > 0, it follows that ("Qv"0, "v"0) > 0 and hence that "Qv"0 is a non-zero vector fixed by "K".
Harish-Chandra's formula.
If "G" is a non-compact semisimple Lie group, its maximal compact subgroup "K" acts by conjugation on the component "P" in the Cartan decomposition. If "A" is a maximal Abelian subgroup of "G" contained in "P", then "A" is diffeomorphic 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
There is also an associated Iwasawa decomposition
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"), a so-called (spherical) 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
The principal series representation σ is realised on L2("K") as
where
is the Iwasawa decomposition of "g" with "U"("g") in "K" and "X"("g") in "S" and
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",
defines a zonal spherical function of "G".
Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function
as an integral over "K".
Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the "C""c"("K" \ "G" / "K") acting by right convolution on "L"2("G" / "K"). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·"s", where "s" is in the Weyl group of "A"
the quotient of the normaliser of "A" in "K" by its centraliser, a finite reflection group.
It can also be verified directly 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). In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.
The remaining positive-definite zonal spherical functions are given
by Harish-Chandra's formula with α in Hom("A",C*) instead of Hom("A",T). Only certain α are permitted and the corresponding irreducible
representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by for "G" = SL(2,R) and by and for "G" = SL(2,C).
Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant. Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on "G" (or "G" / "K").
Eigenfunctions.
Harish-Chandra proved 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 mesurable operators affiliated with the commutant π("G")', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of "W"("A")-invariant polynomials on the Lie algebra of "A", which itself is a polynomial ring by the Chevalley–Shephard–Todd theorem on polynomial invariants of finite reflection 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
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".
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. Indeed, the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
Example: SL(2,C).
The group "G" = SL(2,C) is the complexification of the compact Lie group "K" = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed "expansors". A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and
Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial
component of the Laplacian operator.
Indeed, any unimodular complex 2×2 matrix "g" admits a unique polar decomposition "g" = "pv" with "v" unitary and "p" positive. In turn
"p" = "uau"*, with "u" unitary and "a" a diagonal matrix with positive entries. Thus "g" = "uaw" with "w" = "u"* "v", so that any "K"-biinvariant function on "G" corresponds to a function of the diagonal matrix
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
Setting "f"("r") = sinh ("r")·ψ("r"), it follows that "f" is an odd function of "r" and an eigenfunction of formula_40.
Hence
where formula_41 is real.
There is a similar elementary treatment for the generalized Lorentz groups SO("N",1) in and (recall that SO0(3,1) = SL(2,C) / ±I).
Complex case.
If "G" is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup "K". If formula_42 and formula_23 are their Lie algebras, then
Let "T" be a maximal torus in "K" with Lie algebra formula_45. Then
Let
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(formula_45, R) = formula_49. 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 adjoint representation of "K" on formula_50 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"
where, for μ in formula_49, "A"μ denotes the antisymmetrisation
and ε denotes the "sign character" of the finite reflection group "W".
Weyl's denominator formula expresses the denominator "A"ρ as a product:
where the product is over the positive roots.
Weyl's dimension formula asserts that
where the inner product on formula_49 is that associated with the Killing form on formula_23.
Now
The Berezin–Harish–Chandra formula asserts that for "X" in formula_60
In other words:
One of the simplest proofs 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".
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
where
for "X" in formula_45. If χ is a character with highest weight λ, it follows that φ = "h"·χ satisfies
Thus for every weight μ with non-zero Fourier coefficient in φ,
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
where
for "X" in formula_60.
The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
which is an eigenfunction of the Laplacian Δ"A". Since "K" is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to simple roots, its complexification "G" is generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies that "f" is a periodic function on formula_60 with respect to some sublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
Example: SL(2,R).
The theory of zonal spherical functions for SL(2,R) originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by Hermann Weyl's important work on the spectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general 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" = SL(2,R) is a double cover of the 3-dimensional Lorentz group SO(2,1), the symmetry group of the hyperbolic plane with its Poincaré metric. It acts by Möbius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under this identification "G" becomes identified with the group SU(1,1), also acting by Möbius transformations. Because the action is transitive, both spaces can be identified with "G"/"K", where "K" = SO(2). The metric is invariant under "G" and the associated Laplacian is "G"-invariant, coinciding with the image of the Casimir operator. In the upper half-plane model the Laplacian is given by the formula
If "s" is a complex number and "z" = "x + i y" with "y" > 0, the function
is an eigenfunction of Δ:
Since Δ commutes with "G", any left translate of "f""s" is also an eigenfunction with the same eigenvalue. In particular, averaging over "K", the function
is a "K"-invariant eigenfunction of Δ on "G"/"K". When
with τ real, these functions give all the zonal spherical functions on "G". As with Harish-Chandra's more general formula for semisimple Lie groups, φ"s" is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by "K" in the principal series. Various arguments are available to prove that there are no others. One of the simplest classical Lie algebraic arguments 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 analytic vector for "G" and
for "X" in formula_60. The infinitesimal form of the irreducible unitary representations with a vector fixed by "K" were worked out classically by Bargmann. 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 proceeds by showing that on radial functions the Laplacian has the form
so that, as a function of "r", the zonal spherical function φ("r") must satisfy the ordinary differential equation
for some constant α. The change of variables "t" = sinh "r" transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by
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.