Proper p-harmonic functions and harmonic morphisms on the classical non-compact semi-Riemannian Lie groups
aa r X i v : . [ m a t h . DG ] F e b PROPER p -HARMONIC FUNCTIONS AND HARMONICMORPHISMS ON THE CLASSICAL NON-COMPACTSEMI-RIEMANNIAN LIE GROUPS ELSA GHANDOUR AND SIGMUNDUR GUDMUNDSSON
Abstract.
We apply the method of eigenfamilies to construct new ex-plicit complex-valued p -harmonic functions on the non-compact classicalLie groups, equipped with their natural semi-Riemannian metrics. Wethen employ this same approach to manufacture explicit complex-valuedharmonic morphisms on these groups. Introduction
For a positive integer p , the complex-valued p -harmonic functions aresolutions to a partial differential equation of order 2 p . This equation arisesin various contexts, see for example the extensive analysis in [5] and a historicaccount in [15]. The best known applications are in physics e.g. for p = 2in the areas of continuum mechanics, including elasticity theory and thesolution of Stokes flows. The literature on 2-harmonic functions is vast, butuntil quiet recently, the domains were either surfaces or open subsets of flatEuclidean space, with only very few exceptions. For this see the regularlyupdated online bibliography [8] maintained by the second author.In their recent article [11], the authors produce p -harmonic functions onthe classical Lie groups equipped with their standard Riemannian metrics.The primary goal of this work is to extend the study to the semi-Riemanniansituation. By Theorem 3.2 we show how the problem can be reduced tofinding an eigenfamily , i.e. a collection of complex-valued functions whichare eigen both with respect to the Laplace-Beltrami operator τ and the conformality operator κ , on the semi-Riemannian manifolds involved. Themain part of this paper is devoted to the construction of such families on thefollowing classical Lie groups equipped with their natural semi-Riemannianmetrics GL n ( C ) , GL n ( R ) , GL n ( H ) , SL n ( C ) , SL n ( R ) , SL n ( H ) , SO ( n, C ) , Sp ( n, C ) , Sp ( n, R ) , SO ∗ (2 n ) , SU ( p, q ) , SO ( p, q ) , Sp ( p, q ) . Our eigenfamilies can also be used to manufacture complex-valued har-monic morphisms on these manifolds as explained in Theorem 4.3. They
Mathematics Subject Classification.
Key words and phrases. p -harmonic functions, harmonic morphisms, semi-Riemannianclassical groups. can therefore be seen as an interesting byproduct of the process presentedhere.For semi-Riemannian geometry we recommend O’Neill’s standard text[16]. Readers not familiar with harmonic morphisms are advised to consultthe standard text [2] by Baird and Wood, [3], [4], [13] and the regularlyupdated online bibliography [7]. For the details from Lie theory, used inthis paper, we refer the reader to [12] and [14].2. Eigenfunctions and Eigenfamilies
In this paper we manufacture explicit complex-valued proper p -harmonicfunctions and harmonic morphisms on semi-Riemannian manifolds. For thiswe apply two different construction techniques which are presented in The-orem 3.2 and in Theorem 4.3, respectively. The main ingredients for boththese recipes are the common eigenfunctions for the tension field τ and theconformality operator κ which we now describe.Let ( M, g ) be an m -dimensional semi-Riemannian manifold and T C M bethe complexification of the tangent bundle T M of M . We extend the metric g to a complex-bilinear form on T C M . Then the gradient ∇ φ of a complex-valued function φ : ( M, g ) → C is a section of T C M . In this situation, thewell-known complex linear Laplace-Beltrami operator (alt. tension field ) τ on ( M, g ) acts locally on φ as follows τ ( φ ) = div( ∇ φ ) = m X i,j =1 p | g | ∂∂x j (cid:16) g ij p | g | ∂φ∂x i (cid:17) . For two complex-valued functions φ, ψ : (
M, g ) → C we have the followingwell-known fundamental relation τ ( φ · ψ ) = τ ( φ ) · ψ + 2 · κ ( φ, ψ ) + φ · τ ( ψ ) , where the complex bilinear conformality operator κ is given by κ ( φ, ψ ) = g ( ∇ φ, ∇ ψ ) . Locally this satisfies κ ( φ, ψ ) = m X i,j =1 g ij · ∂φ∂x i ∂ψ∂x j . Definition 2.1. [10] Let (
M, g ) be a semi-Riemannian manifold. Thena complex-valued function φ : M → C is said to be an eigenfunction ifit is eigen both with respect to the Laplace-Beltrami operator τ and theconformality operator κ i.e. there exist complex numbers λ, µ ∈ C such that τ ( φ ) = λ · φ and κ ( φ, φ ) = µ · φ . A set E = { φ i : M → C | i ∈ I } of complex-valued functions is said to be an eigenfamily on M if there exist complex numbers λ, µ ∈ C such that for all -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 3 φ, ψ ∈ E we have τ ( φ ) = λ · φ and κ ( φ, ψ ) = µ · φ · ψ. The following Theorem 2.2 shows that, given an eigenfamily E , one canemploy this to produce an extensive collection H d E of further such objects.The result is a semi-Riemannian version of Theorem 2.2 proven in [6] forthe Riemannian case. Theorem 2.2.
Let ( M, g ) be a semi-Riemannian manifold and the set ofcomplex-valued functions E = { φ k : M → C | k = 1 , , . . . , n } be an eigenfamily on M i.e. there exist complex numbers λ, µ ∈ C such thatfor all φ, ψ ∈ E τ ( φ ) = λ · φ and κ ( φ, ψ ) = µ · φ · ψ. Then the set of complex homogeneous polynomials of degree d H d E = { P : M → C | P ∈ C [ φ , φ , . . . , φ n ] , P ( α · φ ) = α d · P ( φ ) , α ∈ C } is an eigenfamily on M such that for all P, Q ∈ H d E we have τ ( P ) = ( d · λ + d ( d − · µ ) · P and κ ( P, Q ) = d · µ · P · Q. Proof.
The statement can be proven with exactly the same arguments as itsRiemannian counterpart, see Theorem 2.2 in [6]. (cid:3) Proper p -Harmonic Functions In this section we describe a method for manufacturing complex-valuedproper p -harmonic functions on semi-Riemannian manifolds. This methodwas recently introduced for the Riemannian case in [11]. Definition 3.1.
Let (
M, g ) be a semi-Riemannian manifold. For a positiveinteger p , the iterated Laplace-Beltrami operator τ p is given by τ ( φ ) = φ and τ p ( φ ) = τ ( τ ( p − ( φ )) . We say that a complex-valued function φ : ( M, g ) → C is(i) p -harmonic if τ p ( φ ) = 0 and(ii) proper p -harmonic if τ p ( φ ) = 0 and τ ( p − ( φ ) does not vanish iden-tically.The following Theorem 3.2 can be proven in exactly the same way as itsRiemannian counterpart found as Theorem 3.1 in [11]. Theorem 3.2.
Let φ : ( M, g ) → C be a complex-valued function on a semi-Riemannian manifold and ( λ, µ ) ∈ C \ { } be such that the tension field τ and the conformality operator κ satisfy τ ( φ ) = λ · φ and κ ( φ, φ ) = µ · φ . ELSA GHANDOUR AND SIGMUNDUR GUDMUNDSSON
Then for any positive integer p ∈ Z + the non-vanishing function Φ p : W = { x ∈ M | φ ( x ) ( −∞ , } → C with Φ p ( x ) = c · log( φ ( x )) p − , if µ = 0 , λ = 0 c · log( φ ( x )) p − + c · log( φ ( x )) p − , if µ = 0 , λ = µc · φ ( x ) − λµ log( φ ( x )) p − + c · log( φ ( x )) p − , if µ = 0 , λ = µ is a proper p -harmonic function. Here c , c are complex coefficients notboth zero. Complex-Valued Harmonic Morphisms
In this section we describe a method for constructing complex-valued har-monic morphisms φ : ( M, g ) → C from semi-Riemannian manifolds. This isa special case of the much studied harmonic morphisms φ : ( M, g ) → ( N, h )between semi-Riemannian manifolds. They are maps which pull back localreal-valued harmonic functions on (
N, h ) to harmonic functions on (
M, g ).The standard reference for the extensive theory of harmonic morphisms isthe book [2], but we also recommend the updated online bibliography [7].The following result is a direct consequence of Theorem 3 of the paper [4]by B. Fuglede.
Proposition 4.1.
A function φ : ( M, g ) → C from a semi-Riemannianmanifold to the standard Euclidean complex plane, is a harmonic morphismif and only if it is harmonic and horizontally conformal i.e. τ ( φ ) = 0 and κ ( φ, φ ) = 0 . The following Theorem 4.2 is a semi-Riemannian version of Theorem 5.2of [1], see also [2]. It gives the theory of complex-valued harmonic morphismsa strong geometric flavour and provides a useful tool for the construction ofminimal submanifolds of codimension two. This is our main motivation forstudying these maps.
Theorem 4.2.
Let φ : ( M, g ) → C be a horizontally conformal map from asemi-Riemannian manifold to the standard Euclidean complex plane. Then φ is harmonic if and only if its fibres are minimal at regular points of φ . The next result shows that eigenfamilies can be utilised to manufacturea variety of harmonic morphisms.
Theorem 4.3. [10]
Let ( M, g ) be a semi-Riemannian manifold and E = { φ k : M → C | k = 1 , , . . . , n } be an eigenfamily of complex-valued functions on M . If P, Q : C n → C arelinearily independent homogeneous polynomials of the same positive degree -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 5 then the quotient P ( φ , . . . , φ n ) Q ( φ , . . . , φ n ) is a non-constant harmonic morphism on the open and dense subset { p ∈ M | Q ( φ ( p ) , . . . , φ n ( p )) = 0 } . The Semi-Riemannian Lie Group GL n ( C )The complex general linear group GL n ( C ) of invertible n × n matrices isgiven by GL n ( C ) = { z ∈ C n × n | det z = 0 } . Its Lie algebra gl n ( C ) of left-invariant vector fields on GL n ( C ) can be iden-tified with C n × n i.e. the linear space of complex n × n matrices. We equip GL n ( C ) with its natural semi-Riemannian metric g induced by the semi-Euclidean inner product gl n ( C ) × gl n ( C ) → R on gl n ( C ) satisfying g ( Z, W )
7→ − Re trace( Z · W ) . For gl n ( C ) we then have the orthogonal decomposition gl n ( C ) = gl + n ( C ) ⊕ gl − n ( C ) , where gl + n ( C ) = u ( n ) = { Z ∈ gl n ( C ) | Z + ¯ Z t = 0 } is the set of skew-Hermitian matrices and gl − n ( C ) = i · u ( n ) = { Z ∈ gl n ( C ) | Z − ¯ Z t = 0 } the set of the Hermitian ones. Here u ( n ) is the Lie algebra of the unitarygroup U ( n ) which is the maximal compact subgroup of GL n ( C ) satisfying U ( n ) = { z ∈ GL n ( C ) | z ¯ z t = I n } . For 1 ≤ r, s ≤ n , we shall by E rs ∈ R n × n denote the matrix given by( E rs ) αβ = δ rα δ sβ and for r < s let X rs , Y rs be the symmetric and skew-symmetric matrices X rs = 1 √ E rs + E sr ) , Y rs = 1 √ E rs − E sr ) , respectively. Further, let D r be the diagonal elements with D t = E tt . By B + we denote the orthonormal basis for the Lie algebra u ( n ) satisfying B + = { Y rs , i X rs | ≤ r < s ≤ n } ∪ { i D t | t = 1 , , . . . , n } and B − = i B + . Then B = B + ∪ B − is an orthonormal basis for gl n ( C ) suchthat g ( Z, Z ) = 1 if Z ∈ B + and g ( Z, Z ) = − Z ∈ B − . For later use wedefine the two standard matrices J n and I pq by J n = (cid:20) I n − I n (cid:21) and I pq = (cid:20) − I p I q (cid:21) . ELSA GHANDOUR AND SIGMUNDUR GUDMUNDSSON
Let G be a subgroup of GL n ( C ) with Lie algebra g inheriting the inducedleft-invariant semi-Riemannian metric from g . Then employing the Koszulformula for the Levi-Civita connection ∇ on ( G, g ), we see that for all
Z, W ∈ g we have g ( ∇ ZZ, W ) = g ([ W, Z ] , Z )= − Re trace( W Z − ZW ) Z = − Re trace W ( ZZ − ZZ )= 0 . If Z ∈ g is a left-invariant vector field on G and φ : U → C is a localcomplex-valued function on G then the first and second order derivativessatisfy Z ( φ )( p ) = dds (cid:0) φ ( p · exp( sZ )) (cid:1)(cid:12)(cid:12)(cid:12) s =0 ,Z ( φ )( p ) = d ds (cid:0) φ ( p · exp( sZ )) (cid:1)(cid:12)(cid:12)(cid:12) s =0 . This implies that the tension field τ and the conformality operator κ on G fulfill τ ( φ ) = X Z ∈B g g ( Z, Z ) · (cid:0) Z ( φ ) − ∇ ZZ ( φ ) (cid:1) = X Z ∈B g g ( Z, Z ) · Z ( φ ) κ ( φ, ψ ) = X Z ∈B g g ( Z, Z ) · Z ( φ ) · Z ( ψ ) , where B g is any orthonormal basis for the Lie algebra g .The restriction of the semi-Riemannian metric g on GL n ( C ) to its max-imal compact subgroup U ( n ) is its standard Riemannian metric. For thiswe have the following result, see Lemma 5.1 of [10]. Lemma 5.1.
Let z jα : U ( n ) → C be the complex-valued matrix elementsof the standard representation of the unitary group U ( n ) . Then the ten-sion field τ and the conformality operator κ on U ( n ) satisfy the followingrelations τ ( z jα ) = − n · z jα ,κ ( z jα , z kβ ) = − z kα · z jβ . With this at hand we yield the following statement.
Proposition 5.2.
Let z jα : GL n ( C ) → C be the complex-valued matrixelements of the standard representation of the general linear group GL n ( C ) .Then the tension field τ and the conformality operator κ on GL n ( C ) fulfillthe following relations τ ( z jα ) = − n · z jα , -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 7 κ ( z jα , z kβ ) = − · z kα · z jβ . Proof.
This is an immediate consequence of Lemma 5.1 and how the semi-Riemannian metric is defined on the complex Lie algebra gl n ( C ) = u ( n ) ⊕ i · u ( n ) . (cid:3) Theorem 5.3.
Let v be a non-zero element of C n . Then the complex n -dimensional vector space E v = { φ a : GL n ( C ) → C | φ a ( z ) = trace( v t az t ) , a ∈ C n } is an eigenfamily on GL n ( C ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − n · φ and κ ( φ, ψ ) = − · φ · ψ. Proof.
This result can be proven in exactly the same way as Theorem 5.2presented in [10]. (cid:3) The Semi-Riemannian Lie Group GL n ( R )The real general linear group GL n ( R ) of invertible n × n matrices is givenby GL n ( R ) = { x ∈ R n × n | det x = 0 } . The Lie algebra gl n ( C ) of the complex general linear group GL n ( C ) has anatural orthogonal decomposition gl n ( C ) = gl n ( R ) ⊕ i · gl n ( R ) , where gl n ( R ) is the Lie algebra of GL n ( R ) consisting of the real n × n matrices. Proposition 6.1.
Let x jα : GL n ( R ) → R be the real-valued matrix elementsof the standard representation of the general linear group GL n ( R ) . Thenthe tension field τ and the conformality operator κ on GL n ( R ) satisfy thefollowing relations τ ( x jα ) = − n · x jα ,κ ( x jα , x kβ ) = − x kα · x jβ . Proof.
This is an immediate consequence of Proposition 5.2 and how thesemi-Riemannian metric is defined on the complex Lie algebra gl n ( C ) = gl n ( R ) ⊕ i · gl n ( R ) . (cid:3) As an immediate consequence of Proposition 6.1 we have the followingresult.
ELSA GHANDOUR AND SIGMUNDUR GUDMUNDSSON
Theorem 6.2.
Let v be a non-zero element of C n . Then the complex n -dimensional vector space E v = { φ a : GL n ( R ) → C | φ a ( x ) = trace( v t ax t ) , a ∈ C n } is an eigenfamily on GL n ( R ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − n · φ and κ ( φ, ψ ) = − φ · ψ. Proof.
This is a direct consequence of Proposition 6.1 and Theorem 5.3. (cid:3) The Semi-Riemannian Lie Group GL n ( H )In this section we consider the quaternionic general linear group GL n ( H ).Its standard complex representation π : GL n ( H ) → GL n ( C ) is given by π : ( z + jw ) g = z . . . z n w . . . w n ... . . . ... ... . . . ... z n . . . z nn w n . . . w nn − ¯ w . . . − ¯ w n ¯ z . . . ¯ z n ... . . . ... ... . . . ... − ¯ w n . . . − ¯ w nn ¯ z n . . . ¯ z nn . The Lie algebra gl n ( H ) of GL n ( H ) clearly satisfies gl n ( H ) = n (cid:20) Z W − ¯ W ¯ Z (cid:21) (cid:12)(cid:12)(cid:12) Z, W ∈ gl n ( C ) o . As a subgroup of GL n ( C ) the quaternionic general linear group GL n ( H )inherits its natural semi-Riemannian metric g induced by the semi-Euclideaninner product gl n ( H ) × gl n ( H ) → gl n ( H ) on gl n ( H ) given by g ( Z, W ) = − Re trace( Z · W ) . For the Lie algebra gl n ( H ) we have the orthogonal splitting gl n ( H ) = gl + n ( H ) ⊕ gl − n ( H ) , where gl + n ( H ) = sp ( n ) = n (cid:20) Z W − ¯ W ¯ Z (cid:21) ∈ C n × n (cid:12)(cid:12)(cid:12) Z + ¯ Z t = 0 , W − W t = 0 o . By B + we denote the following orthonormal basis for the Lie algebra sp ( n )of the quaternionic unitary group Sp ( n ) which is the maximal compactsubgroup of GL n ( H ). This satisfies B + = n √ (cid:20) iX rs iX rs (cid:21) , √ (cid:20) X rs − X rs (cid:21) , √ (cid:20) iX rs − iX rs (cid:21) , √ (cid:20) Y rs Y rs (cid:21) , √ (cid:20) D t − D t (cid:21) , √ (cid:20) iD t − iD t (cid:21) , √ (cid:20) iD t iD t (cid:21) (cid:12)(cid:12)(cid:12) ≤ r < s ≤ n, ≤ t ≤ n o . -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 9 For the orthogonal complement gl − n ( H ) of sp ( n ) in gl n ( H ) we have the or-thonormal basis B − = n √ (cid:20) Y rs − Y rs (cid:21) , √ (cid:20) iY rs iY rs (cid:21) , √ (cid:20) iY rs − iY rs (cid:21) , √ (cid:20) X rs X rs (cid:21) , √ (cid:20) D t D t (cid:21) (cid:12)(cid:12)(cid:12) ≤ r, s ≤ n, ≤ t ≤ n o . Then B = B + ∪ B − is an orthonormal basis for gl n ( H ) such that g ( Z, Z ) = 1if Z ∈ B + and g ( Z, Z ) = − Z ∈ B − . With this at our disposal we can now prove the following statement.
Proposition 7.1.
Let z jα , w jα : GL n ( H ) → C be the complex-valued matrixelements of the standard representation of the quaternionic general lineargroup GL n ( H ) . Then the tension field τ and the conformality operator κ on GL n ( H ) satisfy the following relations τ ( z jα ) = − n · z jα , τ ( w jα ) = − n · w jα ,κ ( z jα , z kβ ) = − z kα · z jβ , κ ( w jα , w kβ ) = − w kα · w jβ ,κ ( z jα , w kβ ) = − z kα · w jβ . Proof.
For the tension field τ on GL n ( H ) we have τ ( z jα ) = X Z ∈B g ( Z, Z ) · Z ( z jα )= X Z ∈B + Z ( z jα ) − X Z ∈B − Z ( z jα )= 12 e j z n X r
00 0 (cid:21) Z t − X Z ∈B + Z (cid:20) E αβ
00 0 (cid:21) Z t o z t e tk = e j z n − X r
00 0 (cid:21) + X r
00 0 (cid:21) − n X t =1 (cid:20) D t E αβ D t
00 0 (cid:21) o z t e tk = − e j z ( E βα ) z t e tk = − z kα · z jβ . The other identities can be proven in exactly the same way. (cid:3)
Theorem 7.2.
Let u, v be a non-zero elements of C n . Then the complex n -dimensional vector space E uv = { φ ab : GL n ( H ) → C | φ ab ( g ) = trace( u t az t + v t bw t ) , a, b ∈ C n } is an eigenfamily on GL n ( H ) such that for all φ, ψ ∈ E uv we have τ ( φ ) = − n · φ and κ ( φ, ψ ) = − φ · ψ. The Semi-Riemannian Lie group SL n ( C )In this section we construct eigenfamilies on the semisimple non-compactcomplex special linear group SL n ( C ) = { z ∈ GL n ( C ) | det z = 1 } equippedwith its semi-Riemannian metric inherited from GL n ( C ). For the Lie alge-bra sl n ( C ) = { Z ∈ gl n ( C ) | trace Z = 0 } we have the orthogonal decomposition sl n ( C ) = su ( n ) ⊕ i · su ( n ). Here su ( n ) = { Z ∈ u ( n ) | trace Z = 0 } is the Lie algebra of the special unitary group SU ( n ) = { z ∈ U ( n ) | det z = 1 } which is the maximal compact subgroup of SL n ( C ). Lemma 8.1.
Let z jα : SU ( n ) → C be the complex-valued matrix elementsof the standard representation of the special unitary group SU ( n ) . Thenthe tension field τ and the conformality operator κ on SU ( n ) satisfy thefollowing relations τ ( z jα ) = − ( n − n · z jα , -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 11 κ ( z jα , z kβ ) = − ( z kα · z jβ − n · z jα · z kβ ) . Proof.
For the Lie algebra u ( n ) of the unitary group U ( n ) we have theorthogonal splitting u ( n ) = su ( n ) ⊕ l , where l is the real line generated by the unit vector E n = i I n / √ n . Hencethe tension field ˆ τ on the unitary group U ( n ) satisfiesˆ τ ( φ ) = τ ( φ ) + E n ( φ ) , so we have τ ( z jα ) = ˆ τ ( z jα ) − E n ( z jα ) = − ( n · z jα − n · z jα ) = − ( n − n · z jα . For the conformality operator ˆ κ on U ( n ) we similarily yieldˆ κ ( φ, ψ ) = κ ( φ, ψ ) + E n ( φ ) · E n ( ψ ) . Hence κ ( z jα , z kβ ) = ˆ κ ( z jα , z kβ ) − E n ( z jα ) · E n ( z kβ )= − ( z kα · z jβ − n · z jα · z kβ ) . (cid:3) For the special linear group SL n ( C ) we have the following statement. Proposition 8.2.
Let z jα : SL n ( C ) → C be the complex-valued matrixelements of the standard representation of the special linear group SL n ( C ) .Then the tension field τ and the conformality operator κ on SL n ( C ) satisfythe following relations τ ( z jα ) = − n − n · z jα ,κ ( z jα , z kβ ) = − z kα · z jβ − n · z jα · z kβ ) . Proof.
This is an immediate consequence of Lemma 8.1 and how the semi-Riemannian metric is defined on the complex Lie algebra sl n ( C ) = su ( n ) ⊕ i · su ( n ) . (cid:3) Let
P, Q : SL n ( C ) → C be homogeneous polynomials of the matrix ele-mens z jα : SL n ( C ) → C of degree one i.e. of the form P ( z ) = trace( A · z t ) = n X j,α =1 a jα z jα , Q ( z ) = trace( B · z t ) = n X k,β b kβ z kβ for some A, B ∈ C n × n . As a direct consequence of Proposition 8.2 we seethat τ ( P ) = − n − n · P, τ ( Q ) = − n − n · Q and κ ( P, Q ) + 2 ( n − n · P Q = n X j,α,k,β =1 a jα b kβ κ ( z jα , z kβ ) + 2 ( n − n n X j,α,k,β =1 a jα b kβ z jα z kβ = − n X j,α,k,β =1 a jα b kβ z jβ z kα + 2 n n X j,α,k,β =1 a jα b kβ z jα z kβ + 2 ( n − n n X j,α,k,β =1 a jα b kβ z jα z kβ = 2 n X j,α,k,β =1 ( a jα b kβ z jα z kβ − a jα b kβ z jβ z kα )= 2 n X j,α,k,β =1 ( a jα b kβ − a kα b jβ ) z jα z kβ . Theorem 8.3.
Let v be a non-zero element of C n . Then the complex n -dimensional vector space E v = { φ a : SL n ( C ) → C | φ a ( z ) = trace( v t az t ) , a ∈ C n } is an eigenfamily on SL n ( C ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − n − n · φ, κ ( φ, ψ ) = − n − n · φ · ψ. Proof.
Assume that a, b ∈ C n and define A = v t a and B = v t b . By con-struction any two columns of the matrices A and B are linearly dependent.This means that for all 1 ≤ j, α, k, β ≤ n det (cid:20) a jα b jβ a kα b kβ (cid:21) = a jα b kβ − a kα b jβ = 0 . The statement now follows from the calculation above. (cid:3) The Semi-Riemannian Lie Group SL n ( R )In this section we construct eigenfamilies on the semisimple non-compactspecial linear group SL n ( R ) equipped with its semi-Riemannian metric in-herited from GL n ( C ). The special linear group SL n ( R ) is the subgroup of GL n ( R ) satisfying SL n ( R ) = { x ∈ GL n ( R ) | det x = 1 } with Lie algebra sl n ( R ) = { X ∈ gl n ( R ) | trace X = 0 } . -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 13 Proposition 9.1.
Let x jα : SL n ( R ) → R be the real-valued matrix elementsof the standard representation of the special linear group SL n ( R ) . Thenthe tension field τ and the conformality operator κ on SL n ( R ) satisfy thefollowing relations τ ( x jα ) = − ( n − n · x jα ,κ ( x jα , x kβ ) = − ( x jβ · x kα − n · x jα · x kβ ) . Proof.
The Lie algebra sl n ( C ) of the complex special linear group SL n ( C )is the complexification of sl n ( R ) and we have the orthogonal decomposition sl n ( C ) = sl n ( R ) ⊕ i · sl n ( R ) . Hence the statement is an immediate consequence of Lemma 8.2. (cid:3)
The next result is a direct consequence of Theorem 8.3, Propositions 8.2and 9.1.
Theorem 9.2.
Let v be a non-zero element of C n . Then the complex n -dimensional vector space E v = { φ a : SL n ( R ) → C | φ a ( x ) = trace( v t ax t ) , a ∈ C n } is an eigenfamily on SL n ( R ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − ( n − n · φ, κ ( φ, ψ ) = − ( n − n · φ · ψ . The Semi-Riemannian Lie Group SL n ( H ) ∼ = SU ∗ (2 n )In this section we construct eigenfamilies on the semisimple non-compactquaternionic special linear group SL n ( H ). This can be realised as SU ∗ (2 n ) = n (cid:20) z w − ¯ w ¯ z (cid:21) ∈ GL n ( C ) (cid:12)(cid:12)(cid:12) ( z + j w ) ∈ SL n ( H ) o , with Lie algebra su ∗ (2 n ) = n (cid:20) Z W − ¯ W ¯ Z (cid:21) ∈ gl n ( H ) (cid:12)(cid:12)(cid:12) Re trace Z = 0 o . For the Lie algebra gl n ( H ) of GL n ( H ) we have the orthogonal decomposition gl n ( H ) = su ∗ (2 n ) ⊕ l where l is the real line in gl n ( H ) generated by the unit vector E n = I n / √ n . Proposition 10.1.
Let z jα , w kβ : SU ∗ (2 n ) → C be the complex-valuedmatrix elements of the standard representation of the Lie group SU ∗ (2 n ) .Then the tension field τ and the conformality operator κ on SU ∗ (2 n ) satisfythe following relations τ ( z jα ) = − (4 n − n · z jα , τ ( w jα ) = − (4 n − n · w jα , Lie group Eigenfunctions φ λ µ
Conditions GL n ( C ) trace( v t az t ) − n − a ∈ C n GL n ( R ) trace( v t ax t ) − n − a ∈ C n GL n ( H ) trace( u t az t + v t bw t ) − n − a, b ∈ C n SL n ( C ) trace( v t az t ) − n − n − n − n a ∈ C n SL n ( R ) trace( v t ax t ) − ( n − n − ( n − n a ∈ C n SL n ( H ) trace( u t az t + v t bw t ) − (4 n − n − (2 n − n a, b ∈ C n Table 1.
Eigenfunctions on classical non-compact Lie groups. κ ( z jα , z kβ ) = − ( z kα · z jβ − n · z jα · z kβ ) ,κ ( z jα , w kβ ) = − ( z kα · w jβ − n · z jα · w kβ ) ,κ ( w jα , w kβ ) = − ( w kα · w jβ − n · w jα · w kβ ) . Proof.
Let ˆ τ and ˆ κ denote the tension field and the conformality operatoron GL n ( H ), respectively. Then it follows from Proposition 7.1 and theorthogonal decomposition gl n ( H ) = su ∗ (2 n ) ⊕ l that τ ( z jα ) = ˆ τ ( z jα ) + E n ( z jα )= − n · z jα + 12 n · z jα = − (4 n − n · z jα . Similarly, we have κ ( z jα , z kβ ) = ˆ κ ( z jα , z kβ ) + E n ( z jα ) · E n ( z kβ )= − ( z kα · z jβ − n · z jα · z kβ ) . (cid:3) -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 15 Theorem 10.2.
Let u, v be a non-zero elements of C n . Then the complex n -dimensional vector space E uv = { φ ab : SU ∗ (2 n ) → C | φ ab ( g ) = trace( u t az t + v t bw t ) , a, b ∈ C n } is an eigenfamily on SU ∗ (2 n ) such that for all φ, ψ ∈ E uv we have τ ( φ ) = − (4 n − n · φ and κ ( φ, ψ ) = − (2 n − n · φ · ψ. Proof.
Here the statement can be proven in the exactly the same way asthat of Theorem 8.3. (cid:3)
The Semi-Riemannian Lie Group SO ( n, C )The semisimple complex special orthogonal group SO ( n, C ) is the sub-group of GL n ( C ) defined by SO ( n, C ) = { z ∈ SL n ( C ) | z · z t = I n } . Its Lie algebra so ( n, C ) = { Z ∈ gl n ( C ) | Z + Z t = 0 } has the orthogonal decomposition so ( n, C ) = so ( n ) ⊕ i · so ( n ) , where so ( n ) is the Lie algebra of the special orthogonal group SO ( n ) con-sisting of the real skew-symmetric n × n matrices. The restriction of thesemi-Riemannian metric g on SO ( n, C ) to its maximal compact subgroup SO ( n ) is its standard Riemannian metric. For this we have the followingresult, see Lemma 4.1 of [10]. Lemma 11.1.
Let x jα : SO ( n ) → R be the real-valued matrix elements ofthe standard representation of the special orthogonal group SO ( n ) . Thenthe tension field τ and the conformality operator κ on SO ( n ) satisfy thefollowing relations τ ( x jα ) = − ( n − · x jα ,κ ( x jα , x kβ ) = − · ( x jβ · x kα − δ jk · δ αβ ) . Proposition 11.2.
Let z jα : SO ( n, C ) → C be the complex-valued matrixelements of the standard representation of the complex special orthogonalgroup SO ( n, C ) . Then the tension field τ and the conformality operator κ on SO ( n, C ) satisfy the following relations τ ( z jα ) = − ( n − · z jα ,κ ( z jα , z kβ ) = − ( z jβ · z kα − δ jk · δ αβ ) . Proof.
This is an immediate consequence of Lemma 11.1 and the fact howthe semi-Riemannian metric is defined on the complex Lie algebra so ( n, C ) = so ( n ) ⊕ i · so ( n ) . (cid:3) Theorem 11.3.
Let v ∈ C n be a non-zero isotropic element i.e. ( v, v ) = 0 ,then the complex n -dimensional vector space E v = { φ a : SO ( n, C ) → C | φ a ( z ) = trace( v t az t ) , a ∈ C n } is an eigenfamily on SO ( n, C ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − ( n − · φ, κ ( φ, ψ ) = − φ · ψ. Proof.
The Lie algebra so ( n, C ) of the complex special linear group SO ( n, C )is the complexification of so ( n ) and we have the orthogonal decomposition so ( n, C ) = so ( n ) ⊕ i · so ( n ) . Hence the statement is an immediate consequence of Theorem 4.3 of [10]. (cid:3)
The Semi-Riemannian Lie Group Sp ( n, C )The semisimple complex symplectic group Sp ( n, C ) is the subgroup of SL n ( C ) with Sp ( n, C ) = { z ∈ SL n ( C ) | z J n z t = J n } and Lie algebra sp ( n, C ) = { Z ∈ sl n ( C ) | Z J n + J n Z t = 0 } . The maximal compact subgroup of Sp ( n, C ) is the quaternionic unitarygroup Sp ( n ) = { z ∈ Sp ( n, C ) | z ¯ z t = I n } with Lie algebra sp ( n ) = { Z ∈ sp ( n, C ) | Z + ¯ Z t = 0 } . For sp ( n, C ) we have the orthogonal decomposition sp ( n, C ) = sp ( n ) ⊕ i · sp ( n ) . The restriction of the semi-Riemannian metric g on Sp ( n, C ) to Sp ( n ) isits standard Riemannian metric. For this we have the following result, seeLemma 6.1 of [10] and Lemma 6.1 of [9]. Lemma 12.1.
Let z jα , w jα : Sp ( n ) → C be the complex-valued matrixelements of the standard representation of the quaternionic unitary group Sp ( n ) . Then the tension field τ and the conformality operator κ on Sp ( n ) satisfy the following relations τ ( z jα ) = − n + 12 · z jα , τ ( w kβ ) = − n + 12 · w kβ ,κ ( z jα , z kβ ) = − · z kα · z jβ , κ ( w jα , w kβ ) = − · w kα · w jβ , -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 17 κ ( z jα , w kβ ) = − · z kα · w jβ . With this at hand we then yield the following result.
Proposition 12.2.
Let z jα , w jα : Sp ( n, C ) → C be the complex-valuedmatrix elements of the standard representation of the quaternionic unitarygroup Sp ( n ) . Then the tension field τ and the conformality operator κ on Sp ( n ) satisfy the following relations τ ( z jα ) = − (2 n + 1) · z jα , τ ( w kβ ) = − (2 n + 1) · w kβ ,κ ( z jα , z kβ ) = − z kα · z jβ , κ ( w jα , w kβ ) = − w kα · w jβ ,κ ( z jα , w kβ ) = − z kα · w jβ . Proof.
This is an immediate consequence of Lemma 12.1 and how the semi-Riemannian metric is defined on the complex Lie algebra sp ( n, C ) = sp ( n ) ⊕ i · sp ( n ) . (cid:3) Theorem 12.3.
Let u, v ∈ C n be a non-zero elements of C n , then thecomplex n -dimensional vector space E uv = { φ ab : Sp ( n, C ) → C | φ ab ( g ) = trace( u t az t + v t bw t ) , a, b ∈ C n } is an eigenfamily on Sp ( n, C ) such that for all φ, ψ ∈ E uv we have τ ( φ ) = − (2 n + 1) · φ, κ ( φ, ψ ) = − φ · ψ. Proof.
The statement is an immediate consequence of Proposition 12.2. (cid:3)
The Semi-Riemannian Lie Group Sp ( n, R )The semisimple real symplectic group Sp ( n, R ) is the subgroup of thecomplex symplectic group Sp ( n, C ) given by Sp ( n, R ) = { x ∈ SL n ( R ) | x J n x t = J n } with Lie algebra sp ( n, R ) = { X ∈ gl n ( R ) | X J n + J n X t = 0 } . For the Lie algebra sp ( n, C ) we have the orthogonal decomposition sp ( n, C ) = sp ( n, R ) ⊕ i sp ( n, R ) . Theorem 13.1.
Let v be a non-zero element of C n , then the complex n -dimensional vector space E v = { φ a : Sp ( n, R ) → C | φ a ( x ) = trace( v t ax t ) , a ∈ C n } is an eigenfamily on Sp ( n, R ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − n + 12 · φ, κ ( φ, ψ ) = − · φ · ψ. Proof.
The result follows directly from Theorem 12.3 and the fact that sp ( n, C ) = sp ( n, R ) ⊕ i sp ( n, R ). (cid:3) The Semi-Riemannian Lie Group SO ∗ (2 n )In this section we construct eigenfamilies of complex-valued functions onthe semisimple non-compact Lie group SO ∗ (2 n ) = { g ∈ SU ( n, n ) | g · I nn · J n · g t = I nn · J n } , where SU ( n, n ) = { z ∈ SL n ( C ) | z · I n,n · z ∗ = I n,n } . For the Lie algebra so ∗ (2 n ) = n (cid:20) Z W − ¯ W ¯ Z (cid:21) ∈ C n × n (cid:12)(cid:12)(cid:12) Z + Z ∗ = 0 and W + W t = 0 o of SO ∗ (2 n ) we have the orthogonal splitting so ∗ + (2 n ) ⊕ so ∗− (2 n ), where thesubspaces so ∗ + (2 n ) and so ∗− (2 n ) have the orthonormal basis B + and B − ,respectively, with B + = n √ (cid:20) Y rs Y rs (cid:21) , √ (cid:20) iX rs − iX rs (cid:21) , √ (cid:20) iD t − iD t (cid:21)(cid:12)(cid:12)(cid:12) ≤ r < s ≤ n and 1 ≤ t ≤ n o and B − = (cid:26) √ (cid:20) Y rs − Y rs (cid:21) , √ (cid:20) iY rs iY rs (cid:21) (cid:12)(cid:12)(cid:12) ≤ r < s ≤ n (cid:27) . Here g ( Z, Z ) = 1 for all Z ∈ B + and g ( Z, Z ) = − Z ∈ B − . Proposition 14.1.
Let z jα , w kβ : SO ∗ (2 n ) → C be the complex-valued ma-trix coefficients of the standard representation of SO ∗ (2 n ) . Then the tensionfield τ and the conformality operator κ on SO ∗ (2 n ) satisfy the following re-lations τ ( z jα ) = − n − · z jα , τ ( w jα ) = − n − · w jα ,κ ( z jα , z kβ ) = − · z kα · z jβ , κ ( w jα , w kβ ) = − · w kα · w jβ ,κ ( z jα , w kβ ) = − · z kα · w jβ . Proof.
Here we can apply exactly the same strategy as for the proof ofProposition 7.1. (cid:3)
Theorem 14.2.
Let u, v ∈ C n be a non-zero elements of C n , then thecomplex n -dimensional vector space E uv = { φ ab : SO ∗ (2 n ) → C | φ ab ( g ) = trace( u t az t + v t bw t ) , a, b ∈ C n } , is an eigenfamily on SO ∗ (2 n ) such that for all φ, ψ ∈ E uv we have τ ( φ ) = − n − · φ, κ ( φ, ψ ) = − · φ · ψ. Proof.
The statement is an immediate consequence of Proposition 14.1. (cid:3) -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 19
The Semi-Riemannian Lie Group SU ( p, q )In this section we construct eigenfamilies on the non-compact semisimpleLie group SU ( p, q ) = { z ∈ SL p + q ( C ) | z · I p,q · z ∗ = I p,q } . For its Lie algebra su ( p, q ) = { Z ∈ sl p + q ( C ) | Z · I p,q + I p,q · Z ∗ = 0 } we have a natural orthogonal splitting su ( p, q ) = s ( u ( p ) + u ( q )) ⊕ i · m such that su ( p + q ) = s ( u ( p ) + u ( q )) ⊕ m is the Lie algebra of the special orthogonal group SU ( p + q ). Proposition 15.1.
Let z jα : SU ( p, q ) → C be the complex-valued matrixelements of the standard representation of the special unitary group SU ( p, q ) .Then the tension field τ and the conformality operator κ on SU ( p, q ) satisfythe following relations τ ( z jα ) = − ( p + q ) − p + q ) · z jα ,κ ( z jα , z kβ ) = − ( z kα · z jβ − p + q ) · z jα · z kβ ) . Proof.
This is an immediate consequence of Lemma 8.1, the above rela-tionship between the Lie algebras su ( p + q ) and su ( p, q ) and how the semi-Riemannian metric g is defined on the complex Lie algebra gl p + q ( C ). (cid:3) Theorem 15.2.
Let v be a non-zero element of C p + q . Then the complex ( p + q ) -dimensional vector space E v = { φ a : SU ( p, q ) → C | φ a ( z ) = trace( v t az t ) , a ∈ C p + q } is an eigenfamily on SU ( p, q ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − ( p + q ) − p + q ) · φ, κ ( φ, ψ ) = − ( p + q − p + q ) · φ · ψ. Proof.
The statement follows directly from Proposition 15.1. (cid:3)
The Semi-Riemannian Lie Group SO ( p, q )In this section we construct eigenfamilies on the non-compact semisimpleLie group SO ( p, q ) = { x ∈ SL p + q ( R ) | x · I p,q · x t = I p,q } . For its Lie algebra so ( p, q ) = { X ∈ sl p + q ( R ) | X · I p,q + I p,q · X t = 0 } we have a natural orthogonal spilling so ( p, q ) ∼ = ( so ( p ) ⊕ so ( q )) ⊕ i · m such that so ( p + q ) = ( so ( p ) ⊕ so ( q )) ⊕ m is the Lie algebra of the special orthogonal group SO ( p + q ). Proposition 16.1.
Let x jα : SO ( p, q ) → R be the real-valued matrix ele-ments of the standard representation of the special orthogonal group SO ( p, q ) .Then the tension field τ and the conformality operator κ on SO ( p, q ) satisfythe following relations τ ( x jα ) = − ( p + q − · x jα ,κ ( x jα , x kβ ) = − · ( x jβ · x kα − δ jk · δ αβ ) . Proof.
This is an immediate consequence of Lemma 11.1, the above rela-tionship between the Lie algebras so ( p + q ) and so ( p, q ) and how the semi-Riemannian metric g is defined on the complex Lie algebra gl p + q ( R ). (cid:3) Theorem 16.2.
Let v ∈ C p + q be a non-zero isotropic element i.e. ( v, v ) =0 , then the complex ( p + q ) -dimensional vector space E v = { φ a : SO ( p, q ) → C | φ a ( z ) = trace( v t az t ) , a ∈ C p + q } is an eigenfamily on SO ( p, q ) such that for all φ, ψ ∈ E v we have τ ( φ ) = − ( p + q − · φ, κ ( φ, ψ ) = − · φ · ψ. Proof.
The statement follows directly from Proposition 16.1. (cid:3)
The Semi-Riemannian Lie Group Sp ( p, q )In this section we construct eigenfamilies on the non-compact semisimpleLie group Sp ( p, q ) = { g ∈ SL p + q ( H ) | g · I p,q · g ∗ = I p,q } . For its Lie algebra sp ( p, q ) = { Z ∈ sl p + q ( H ) | Z · I p,q + I p,q · Z ∗ = 0 } we have a natural orthogonal decomposition sp ( p, q ) = ( sp ( p ) ⊕ sp ( q )) ⊕ i · m such that sp ( p + q ) = ( sp ( p ) ⊕ sp ( q )) ⊕ m is the Lie algebra of the quaternionic unitary group Sp ( p + q ). Proposition 17.1.
Let z jα , w jα : Sp ( p, q ) → C be the complex-valued ma-trix elements of the standard representation of the quaternionic unitarygroup Sp ( p, q ) . Then the tension field τ and the conformality operator κ on Sp ( p, q ) satisfy the following relations τ ( z jα ) = − p + q ) + 12 · z jα , τ ( w kβ ) = − p + q ) + 12 · w kβ , -HARMONIC FUNCTIONS AND HARMONIC MORPHISMS ON LIE GROUPS 21 Lie group Eigenfunctions φ λ µ
Conditions SO ( n, C ) trace( v t az t ) − ( n − − a ∈ C n , ( v, v ) = 0 Sp ( n, C ) trace( u t az t + v t bw t ) − (2 n + 1) − a, b ∈ C n Sp ( n, R ) trace( v t ax t ) − n +12 − a ∈ C n SO ∗ (2 n ) trace( u t az t + v t bw t ) − n − − a, b ∈ C n SU ( p, q ) trace( v t az t ) − ( p + q ) − p + q ) − ( p + q − p + q ) a ∈ C n SO ( p, q ) trace( v t ax t ) − ( p + q − − a ∈ C n Sp ( p, q ) trace( u t az t + v t bw t ) − p + q )+12 − a, b ∈ C n Table 2.
Eigenfunctions on classical non-compact Lie groups. κ ( z jα , z kβ ) = − · z kα · z jβ , κ ( w jα , w kβ ) = − · w kα · w jβ ,κ ( z jα , w kβ ) = − · z kα · w jβ . Proof.
This is an immediate consequence of Lemma 12.1, the above rela-tionship between the Lie algebras sp ( p + q ) and sp ( p, q ) and how the semi-Riemannian metric g is defined on the complex Lie algebra gl p + q ( H ). (cid:3) Theorem 17.2.
Let u, v ∈ C p + q be a non-zero element of C p + q , then thecomplex ( p + q ) -dimensional vector space E uv = { φ ab : Sp ( p, q ) → C | φ ab ( g ) = trace( u t az t + v t bw t ) , a, b ∈ C p + q } is an eigenfamily on Sp ( p, q ) such that for all φ, ψ ∈ E uv we have τ ( φ ) = − p + q ) + 12 · φ, κ ( φ, ψ ) = − · φ · ψ. Proof.
The statement follows directly from Proposition 17.1. (cid:3)
Acknowledgements
The first author would like to thank the Department of Mathematics atLund University for its great hospitality during her time there as a postdoc.
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Mathematics, Faculty of Science, University of Lund, Box 118, Lund 22100, Sweden
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Mathematics, Faculty of Science, University of Lund, Box 118, Lund 22100, Sweden
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