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 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 L2. 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 L1 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 L2, 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 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 H1 = L2. Thus, H1 admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2 with left and right regular representations λ and ρ of G and P is the orthogonal projectionfrom 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 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 Pρ P is just the restriction of ρ to H1, 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 with F in
Cc, 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.
Let A denote the C* algebra generated by the operators ρ on H1.
The pair
is said to be a Gelfand pair if one, and hence all, of the following algebras are commutative:
where X is the locally compact space of norm continuous * homomorphisms of A 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 Cc in A extends by continuity
to L1, the * algebra of K-biinvariant integrable functions. The image forms
a dense * subalgebra of A. 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.
Properties
A zonal spherical function h has the following properties:- h is uniformly continuous on G
- h =1
- h is a positive definite function on G
- f * h is proportional to h for all f in Cc.
with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of
invariant differential operators on G/''K.
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 H1. Each representation σ extends uniquely by continuity
to A, so that each zonal spherical function satisfies
for f in A. Moreover, since the commutant π' 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' 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 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, thenwhere
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 Cc. On the
other hand, if F lies in Cc, then
so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,
- the Hecke algebra is commutative
- is a Gelfand pair.
- 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, where P'' is defined as above.
- G is a compact connected semisimple Lie group with τ a period two automorphism;
- G is a semidirect product, with A a locally compact Abelian group without 2-torsion and τ= k·''a−1 for a'' in A and k in K.
Cartan–Helgason theorem
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 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 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,, are the complexifications of the Lie algebras of G, K, A = T ''P and
summed over all eigenspaces for T'' in corresponding to positive roots 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, the Poincaré–Birkhoff–Witt theorem
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 Haar measure, it follows that
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 that the matrix coefficient
is non-negative on K. Since f > 0, it follows that > 0 and hence that Qv0 is a non-zero vector fixed by K.