Spherical function on G/K

Intuition [Beginner]

A spherical function is a special function on a symmetric space $G/K$ that is simultaneously invariant under the compact subgroup $K$ and an eigenfunction of every rotationally-invariant differential operator. Think of it as the higher-dimensional analogue of a Legendre polynomial.

On the sphere $S^2=SO(3)/SO(2)$, the spherical functions are the Legendre polynomials $P_{\ell }(\cos \theta )$. They depend only on the polar angle $\theta$ (not the azimuthal angle) — that is the $K$-invariance. And they satisfy the Laplace eigenvalue equation — that is the differential equation.

For a general symmetric space $G/K$, the spherical functions ${\varphi }_{\lambda }$ are indexed by a spectral parameter $\lambda$ living in a dual space. The spherical transform (a generalised Fourier transform) decomposes bi-$K$-invariant functions on $G/K$ into spherical functions, just as the ordinary Fourier transform decomposes periodic functions into exponentials.

Visual [Beginner]

A symmetric space $G/K$ drawn as a curved surface with the compact subgroup $K$ fixing a base point. The spherical function ${\varphi }_{\lambda }$ takes the same value at all points on each $K$-orbit (concentric shells around the base point). Different values of $\lambda$ give different radial profiles.

Spherical functions: the radial building blocks for harmonic analysis on symmetric spaces.

Worked example [Beginner]

On the hyperbolic plane ${\mathbb{H}}^2=SL(2,\mathbb{R})/SO(2)$, the spherical functions are given by the Legendre function of the first kind:

$${\varphi }_s(r)=\frac{\sin (sr)}{s\sinh (r)}$$

where $r$ is the hyperbolic distance from the base point and $s$ is the spectral parameter.

At $s=0$: ${\varphi }_0(r)=r/\sinh (r)$, which decays exponentially as $r$ grows. For real $s$, the functions oscillate and then decay. The spherical transform on ${\mathbb{H}}^2$ adds up $f(r)$ weighted by the spherical function ${\varphi }_s(r)$ and the volume element $\sinh (r)$. This is the Helgason-Fourier transform on the hyperbolic plane. The integral adds up $f(r)$ weighted by the spherical function ${\varphi }_s(r)$ and the volume element $\sinh (r)$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Zonal spherical function). Let $G$ be a semisimple Lie group with maximal compact subgroup $K$. A zonal spherical function is a continuous function $\varphi :G\to \mathbb{C}$ satisfying:

1. $\varphi (e)=1$ (normalisation).
2. $\varphi (k_{1}g{k}_2)=\varphi (g)$ for all $k_1,k_2\in K$ and $g\in G$ (bi-$K$-invariance).
3. $D\varphi =\chi (D)\varphi$ for every $G$-invariant differential operator $D$ on $G/K$ (eigenfunction property).

Theorem (Harish-Chandra integral formula). The zonal spherical function with spectral parameter $\lambda \in {\mathfrak{a}}_{\mathbb{C}}^{\ast }$ is:

$${\varphi }_{\lambda }(g)={\int }_K{e}^{(i\lambda +\rho )(A(gk))}dk$$

where $A(gk)$ is the $A$-component in the Iwasawa decomposition $gk=k^{\prime }\cdot \exp (A(gk))\cdot n$, and $\rho =\frac{1}{2}{\sum }_{\alpha >0}\alpha$ is the Weyl vector.

Key theorem with proof [Intermediate+]

Theorem (Harish-Chandra, 1958). The spherical functions ${\varphi }_{\lambda }$ for $\lambda \in \mathfrak{a}^$ satisfy:*

1. ${\varphi }_{\lambda }$ is bi-$K$-invariant and a joint eigenfunction of $D(G/K)$.
2. ${\varphi }_{\lambda }={\varphi }_{\mu }$ if and only if $\lambda =w\cdot \mu$ for some $w\in W$.
3. The spherical transform $\hat{f}(\lambda )={\int }_{G}f(g){\varphi }_{-\lambda }(g)dg$ inverts on $L^2(G/K{)}^K$:

$$f(g)=\frac{1}{|W|}{\int }_{{\mathfrak{a}}^{\ast }}\hat{f}(\lambda ){\varphi }_{\lambda }(g)|c(\lambda ){|}^{-2}d\lambda$$

where $c(\lambda )$ is the Harish-Chandra $c$-function.

Proof sketch. (1) The bi-$K$-invariance follows from the integration over $K$. The eigenfunction property follows from the fact that invariant differential operators act on $A(gk)$ by the Harish-Chandra homomorphism, which sends $D$ to a polynomial $p_D(\lambda )$, and $D{\varphi }_{\lambda }=p_D(i\lambda ){\varphi }_{\lambda }$.

(2) The Weyl group invariance ${\varphi }_{w\lambda }={\varphi }_{\lambda }$ follows from the Weyl group acting on $A(gk)$. The converse uses the asymptotic expansion ${\varphi }_{\lambda }(\exp tH)\sim c(\lambda )e^{(i\lambda -\rho )(tH)}+c(-\lambda )e^{(-i\lambda -\rho )(tH)}$ as $t\to \infty$, which determines $\lambda$ up to the $W$-action.

(3) The Plancherel formula follows from the spectral theory of the commuting algebra of invariant differential operators on $G/K$, combined with the Harish-Chandra Plancherel theorem for the group. $\square$

Bridge. The spherical transform generalises the Fourier transform on Euclidean space to symmetric spaces; the foundational reason it works is that bi-$K$-invariant functions form a commutative algebra under convolution (the Gelfand property). This pattern appears again in 07.04.14 where Hermitian symmetric spaces carry additional holomorphic structure. The bridge is that spherical functions are the joint eigenfunctions of the invariant differential operators, and the spherical transform diagonalises the convolution algebra.

Exercises [Intermediate+]

Advanced results [Master]

The Plancherel measure. The spherical transform is an isometry from $L^2(G/K{)}^K$ to $L^2({\mathfrak{a}}^{\ast }/W,|c(\lambda ){|}^{-2}d\lambda )$. The density $|c(\lambda ){|}^{-2}$ is the Plancherel measure. For $SL(2,\mathbb{R})/SO(2)$: $|c(\lambda ){|}^{-2}=\frac{\lambda \tanh (\pi \lambda )}{2\pi }$.

The Paley-Wiener theorem. A function $f$ on $G/K$ is smooth and compactly supported if and only if its spherical transform $\hat{f}(\lambda )$ extends to a holomorphic function of exponential type on ${\mathfrak{a}}_{\mathbb{C}}^{\ast }$ satisfying appropriate symmetry conditions. This generalises the classical Paley-Wiener theorem for the Fourier transform.

Applications to special functions. For rank-one symmetric spaces, the spherical functions reduce to classical special functions: Legendre polynomials ($S^n$), Jacobi functions (${\mathbb{H}}^n$), Bessel functions (Euclidean spaces). The general theory unifies these into a single framework parametrised by root system data.

Synthesis. Spherical functions are the eigenfunction basis for harmonic analysis on symmetric spaces; the central insight is that the Gelfand property (commutativity of the bi-$K$-invariant convolution algebra) forces the spherical transform to be a diagonalisation. This pattern appears again in the Selberg trace formula where spherical functions appear as the kernel of the trace. The bridge is that spherical functions generalise classical orthogonal polynomials to the setting of Lie-theoretic symmetric spaces, identifying harmonic analysis with representation theory via the spherical principal series.

Full proof set [Master]

Proposition (Gelfand property). The algebra $C_c(K\backslash G/K)$ of compactly supported bi-$K$-invariant functions is commutative under convolution.

Proof. The involution $\theta (g)=g^{-1}$ preserves $K$-double cosets (since $\theta (k_{1}g{k}_2)=k_2^{-1}{g}^{-1}{k}_1^{-1}$ is again a $K$-double coset). For $f_1,f_2\in C_c(K\backslash G/K)$: $(f_1\ast f_2)(g)={\int }_G{f}_1(g{h}^{-1})f_2(h)dh$. Applying $\theta$: $(f_1\ast f_2)(g^{-1})=(f_2\ast f_1)(g)$. Since $f_1\ast f_2$ is bi-$K$-invariant: $(f_1\ast f_2)(g^{-1})=(f_1\ast f_2)(g)$. So $f_1\ast f_2=f_2\ast f_1$. $\square$

Connections [Master]

The Iwasawa decomposition 07.04.09 provides the $A$-component $A(gk)$ used in the Harish-Chandra integral formula for spherical functions.

Riemannian symmetric spaces 07.04.07 are the geometric setting; the spherical functions are harmonic-analytic invariants of the symmetric space structure.

Hermitian symmetric spaces 07.04.14 carry additional complex structure that refines the spherical functions into holomorphic discrete series representations.

The Weyl integration formula 07.07.04 is the special case for compact groups where the "symmetric space" is $G$ itself with the adjoint action.

Bibliography [Master]

@article{harish-chandra1958,
  author = {Harish-Chandra},
  title = {Spherical functions on a semisimple {L}ie group {I}, {II}},
  journal = {Amer. J. Math.},
  volume = {80},
  pages = {241--310, 553--613},
  year = {1958}
}

@book{helgason-gga,
  author = {Helgason, Sigurdur},
  title = {Groups and Geometric Analysis},
  publisher = {Academic Press},
  year = {1984}
}

@book{terras-harmonic,
  author = {Terras, Audrey},
  title = {Harmonic Analysis on Symmetric Spaces and Applications},
  publisher = {Springer},
  year = {1985}
}

@book{sugiura-unitary,
  author = {Sugiura, Mitsuo},
  title = {Unitary Representations and Harmonic Analysis},
  edition = {2},
  publisher = {North-Holland},
  year = {1990}
}

Historical & philosophical context [Master]

Harish-Chandra developed the theory of spherical functions in his monumental papers of 1958 ^{[Harish-Chandra 1958]}, completing the program begun by Gel'fand and Naimark. The spherical transform provides the non-abelian generalisation of the Fourier transform, applicable whenever a commutative convolution algebra exists (the Gelfand pair condition).

The theory has profound applications: the Selberg trace formula in number theory, the resolution of the Helgason conjecture on the surjectivity of the horospherical transform, and the explicit computation of Plancherel measures for all symmetric spaces. In physics, spherical functions appear as the radial wave functions for the quantum mechanical hydrogen atom and for scattering on hyperbolic spaces.