Zonal spherical function

Last updated

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 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/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.

Contents

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 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 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

from 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 [1]

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. [lower-alpha 1]

However Pρ(f) P is just the restriction of ρ(F) 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(F) with F in Cc(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 H1.

The pair (G, K) is said to be a Gelfand pair [2] 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 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. [2] [3] [4] [5] [6]

Because of the estimate

the representation π of Cc(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: [2]

  1. h is uniformly continuous on G
  2. h(1) =1 (normalisation)
  3. h is a positive definite function on G
  4. f * h is proportional to h for all f in Cc(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 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 H1. 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 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. [7] [8] 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 Cc(K\G/K). On the other hand, if F lies in Cc(K\G/K), then

Fg) = F(g1),

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: [9]

The two most important examples covered by this are when:

The three cases cover the three types of symmetric spaces G/K: [5]

  1. Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;
  2. Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;
  3. 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 = Gτ. In this case K is a connected compact Lie group. [5] In addition let T be a maximal torus of G invariant under τ, such that TP is a maximal torus in P, and set [12]

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 [13] [14] states that

the irreducible representations of G admitting a non-zero vector fixed by K are precisely those with highest weights corresponding to homomorphisms trivial on S.

The corresponding irreducible representations are called spherical representations.

The theorem can be proved [5] using the Iwasawa decomposition:

where , , are the complexifications of the Lie algebras of G, K, A = TP 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 [15] that the matrix coefficient

is non-negative on K. Since f(1) > 0, it follows that (Qv0, v0) > 0 and hence that Qv0 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 [16]

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), [17] 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 ΔS the modular function of S. Then [5]

The principal series representation σ is realised on L2(K) as [18]

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 Cc(K \ G / K) acting by right convolution on L2(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 [2] 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). [4] [5] 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. [5] [19] [20] [21] [22]

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 Bargmann (1947) for G = SL(2,R) and by Harish-Chandra (1947) and Gelfand & Naimark (1947) 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. [23] [24] [25] Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K).

Eigenfunctions

Harish-Chandra proved [4] [5] 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 λD 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. [26] [27] [28] 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. [5]

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 [29]

Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of .

Hence

where is real.

There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Takahashi (1963) and Faraut & Korányi (1994) (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 and are their Lie algebras, then

Let T be a maximal torus in K with Lie algebra . 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(, R) = . 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 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 , Aμ denotes the antisymmetrisation

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

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

where the product is over the positive roots.

Weyl's dimension formula asserts that

where the inner product on is that associated with the Killing form on .

Now

The Berezin–Harish–Chandra formula [5] asserts that for X in

In other words:

One of the simplest proofs [30] 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. [31]

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

where

for X in . 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 ΔG on A is just the expression for the restriction of ΔG to W(A)-invariant functions on A. It is given by the formula

where

for X in .

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 ΔA. 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 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 [5] [6]

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 fs 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, φs 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 [5] [6] [32] [33] [34] 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 . The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann. [32] [33] 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 [35] 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 [2] [36]

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. [37] [38]

Further directions

See also

Notes

  1. If σ is a unitary representation of G, then .

Citations

Sources

Related Research Articles

<span class="mw-page-title-main">Representation of a Lie group</span> Group representation

In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

<span class="mw-page-title-main">Lie algebra representation</span>

In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

In mathematics, a Casimir element is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

<span class="mw-page-title-main">Compact group</span> Topological group with compact topology

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect.

<span class="mw-page-title-main">Representation theory of the Lorentz group</span> Representation of the symmetry group of spacetime in special relativity

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly established, and the conjunction of these two theories is the study of the infinite-dimensional unitary representations of the Lorentz group. These have both historical importance in mainstream physics, as well as connections to more speculative present-day theories.

<span class="mw-page-title-main">Hermitian symmetric space</span> Manifold with inversion symmetry

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl. There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra , is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center of the universal enveloping algebra of a reductive Lie algebra to the elements of the symmetric algebra of a Cartan subalgebra that are invariant under the Weyl group .

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.

In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space

In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

<span class="mw-page-title-main">Representation theory</span> Branch of mathematics that studies abstract algebraic structures

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and Harish-Chandra introduced a more general c-function called Harish-Chandra's (generalized) C-function. Gindikin and Karpelevich introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.

<span class="mw-page-title-main">Borel–de Siebenthal theory</span>

In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.

<span class="mw-page-title-main">Complexification (Lie group)</span> Universal construction of a complex Lie group from a real Lie group

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

This is a glossary of representation theory in mathematics.

In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra . There is a closely related theorem classifying the irreducible representations of a connected compact Lie group . The theorem states that there is a bijection

<span class="mw-page-title-main">Glossary of Lie groups and Lie algebras</span>

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

<span class="mw-page-title-main">Representation theory of semisimple Lie algebras</span>

In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.