New Developments in Lie theory and their applications I edited by Juan Tirao, Nolan Wallach. p. c... more New Developments in Lie theory and their applications I edited by Juan Tirao, Nolan Wallach. p. cm.-(Progress in mathematics: 105) Papers from the Third Workshop on Representation Theory of Lie Groups and Their Applications, held Aug.-Sept. 1989 in C6rdoba, Argentina. Includes bibliographical references.
The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of... more The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions ${}_{p+1}F_p$.
We took due care of all of referee’s suggestions. Namely: (1) (a) We do not study the weight in t... more We took due care of all of referee’s suggestions. Namely: (1) (a) We do not study the weight in the most absolute deep, since we are focused mainly in the spherical functions and how to obtain sequences of matrix orthogonal polynomials that are eigenfunctions of a second order differential operator. However, we add Proposition 10.9 to state a criteria of reducibility. In [1] the authors affirm that there is no further reduction, but that is not clear to us. (b) We add the Subsection 9.2 giving the explicit values of the coefficients a j in terms of Racah polynomials. (c) In page 42 of the new version we add the fact of that U∗U is diagonal and the Remark 10.7 which mention the LDU-decomposition of the weight, both indicated by the referee. (d) At the end of the paper we add Remark 10.11 in order to mention more precisely some connections and relations with [2]. (2) At the beginning of Subsection 2.3 we add the Cartan involution and say that the Lie algebra k is spanned by Y1, Y2 and Y3. (3) We change the old proof of Lemma 3.3 by the more natural one suggested by the referee. (4) We add an Appendix with the proofs of Propositions 3.4 and 3.5 (5) We suppress the words ”such” and ”And” in pages 13 and 20, respectively, and replace ”lemma.2” by ”Lemma 5.1” as the referee pointed out.
Abstract. In this paper, we deal with a linear control system E denned on a Lie group G with Lie ... more Abstract. In this paper, we deal with a linear control system E denned on a Lie group G with Lie algebra fl. The dynamic of E is determined by a drift vector field, which is an element in the normalizer of 0 in the Lie algebra of all smooth vector fields on G, and by the control ...
In this paper, we describe the irreducible spherical functions of fundamental $K$-types associate... more In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=(\mathrm{SO}(n+1),\mathrm{SO}(n))$ in terms of matrix hypergeometric functions. The output of these description is that the irreducible spherical functions of the same $K$-type are encoded news examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$. Moreover, we show that $W$ admits a second order symmetric hypergeometric operator $D$.
In this paper, we shall deal with a linear control system (Sigma) defined on a Lie group G with L... more In this paper, we shall deal with a linear control system (Sigma) defined on a Lie group G with Lie algebra g. The dynamic of (Sigma) is determined by the drift vector field which is an element in the normalizer of g in the Lie algebra of all smooth vector field on G and by the ...
Page 300. Contemporary Mathematics Contemporary Mathematics Volume 537, 2011 Contemporary Mathema... more Page 300. Contemporary Mathematics Contemporary Mathematics Volume 537, 2011 Contemporary Mathematics Volume 537, 2011 The algebra of differential operators associated to a weight matrix: a first example Juan Tirao Abstract. ...
... III. Wolf, Joseph Albert, i936-. IV. ... In the case of GL (n) over fields of large residual ... more ... III. Wolf, Joseph Albert, i936-. IV. ... In the case of GL (n) over fields of large residual characteristic, Howe and Moy found a way to study arbitrary repre-sentations, replacing J and the t rival representation of J by smaller compact open subgroups and representations of them. ...
The main purpose of this paper is to compute all irreducible spherical functions on G = SU(3) of ... more The main purpose of this paper is to compute all irreducible spherical functions on G = SU(3) of arbitrary type δ ∈K, where K = S(U(2) × U(1)) ≃ U(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the complex projective plane P2(C) = G/K. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair (G, K). But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of H are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions p+1Fp.
Abstract. In this paper, we describe the irreducible spherical functions of fundamental K-types a... more Abstract. In this paper, we describe the irreducible spherical functions of fundamental K-types associated with the pair (G,K) = (SO(n+ 1),SO(n)) in terms of matrix hypergeo-metric functions. The output of this description is that the irreducible spherical functions of the same K-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size 2 and 3, with respect to a matrix-weight W supported on [0, 1]. Moreover, we show that W has a second order symmetric hypergeometric operator D. Key words: matrix-valued spherical functions; matrix orthogonal polynomials; the matrix hypergeometric operator; n-dimensional sphere 2010 Mathematics Subject Classification: 22E45; 33C45; 33C47 1
The aim of this paper is to determine all irreducible spherical functions of the pair (G, K) = (S... more The aim of this paper is to determine all irreducible spherical functions of the pair (G, K) = (SU(n + 1), U(n)), where the highest weight of their K-types π are of the form (m + ℓ,. .. , m + ℓ, m,. .. , m). Instead of looking at a spherical function Φ of type π we look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of Pn(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type π turn out to be parameterized by S = {(w, r) ∈ Z × Z : 0 ≤ w, 0 ≤ r ≤ ℓ, 0 ≤ m + w + r}. A key result to characterize the associated function Hw,r is the existence of a matrix-valued polynomial function Ψ of degree ℓ such that Fw,r(t) = Ψ(t) −1 Hw,r (t) becomes an eigenfunction of a matrix hypergeometric operator with eigenvalue λ(w, r), explicitly given. In the last section we assume that m ≥ 0 and define the matrix polynomial Pw as the (ℓ + 1) × (ℓ + 1) matrix whose r-row is the polynomial Fw,r. This leads to interesting families of matrix-valued orthogonal Jacobi polynomials P α,β w for α, β > −1.
In this paper we establish a close relationship between the spherical functions of the n-dimensio... more In this paper we establish a close relationship between the spherical functions of the n-dimensional sphere S n ≃ SO(n + 1)/SO(n) and the spherical functions of the n-dimensional real projective space P n (R) ≃ SO(n + 1)/O(n). In fact, for n odd a function on SO(n + 1) is an irreducible spherical function of some type π ∈ŜO(n) if and only if it is an irreducible spherical function of some type γ ∈Ô(n). When n is even this is also true for certain types, and in the other cases we exhibit a clear correspondence between the irreducible spherical functions of both pairs (SO(n + 1), SO(n)) and (SO(n + 1), O(n)). Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.
We describe families of polynomials arising in the study of the universal central extensions of L... more We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa [DJKM83] in their work on the Landau-Lifshitz equations. We show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials.
The fundamental properties of spherical functions have been establis-hed by R. Godement in a well... more The fundamental properties of spherical functions have been establis-hed by R. Godement in a well known paper [1} in 1952. There he defines
ABSTRACT. Let Go bea non compact real semisimple Lie group with finite center, and let U(g)K deno... more ABSTRACT. Let Go bea non compact real semisimple Lie group with finite center, and let U(g)K denote the centralizer in U(g) of a maximal compact subgroup Ko of Go. By the fundamental work of Harish-Chandra it is known that many deep questions concerning the infinite dimensional representation theory of Go reduce to questions about the structure and finite dimensional representation theory of the algebra U(g)!\, called the classifying ring of Go. To study the algebra U(g)](, B. Kostant suggested to consider the projection map P: U(g)-> U(t)<2>U(a), associated to an Iwasawa decomposition Go = KoAoNo of Go, adapted to Ko. When P is restricted to U(g)K P becomes an injective anti-homomorphism of algebras. In this paper we use the characterization of the image of U(g)](, when Go =SO(n,l) or SU(n,l) obtained in Tirao [11], to prove that U(g)] ( ~ Z(g) <2> Z(£), where Z(g) and Z(t) denote respectively the centers of U(g) and of U(t). By a well known theorem of Harish-Chandra...
Proceedings of the National Academy of Sciences, 2003
The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1 ... more The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1 , 11–13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123–162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39–83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127–172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1–24]. The hypergeometric function known also as Gauss' function is the unique solution of the hypergeometric equation analytic at z = 0 and with value 1 at z = 0. This function, because of its remarkable properties, has been used for centuries in the whole subject of special functions. In this article we give a matrix-valued analog of the hypergeometric differential equation and of Gauss' function. One can only speculate that many of the connections that made Gauss' function a vital part of mathematics at the end of the 20th century will be shared by its matrix-valued version, discussed here.
New Developments in Lie theory and their applications I edited by Juan Tirao, Nolan Wallach. p. c... more New Developments in Lie theory and their applications I edited by Juan Tirao, Nolan Wallach. p. cm.-(Progress in mathematics: 105) Papers from the Third Workshop on Representation Theory of Lie Groups and Their Applications, held Aug.-Sept. 1989 in C6rdoba, Argentina. Includes bibliographical references.
The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of... more The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by associating to a spherical function $\Phi$ on $G$ a matrix valued function $H$ on the complex projective plane $P_2(\mathbb{C})=G/K$. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair $(G,K)$. But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of $H$ are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions ${}_{p+1}F_p$.
We took due care of all of referee’s suggestions. Namely: (1) (a) We do not study the weight in t... more We took due care of all of referee’s suggestions. Namely: (1) (a) We do not study the weight in the most absolute deep, since we are focused mainly in the spherical functions and how to obtain sequences of matrix orthogonal polynomials that are eigenfunctions of a second order differential operator. However, we add Proposition 10.9 to state a criteria of reducibility. In [1] the authors affirm that there is no further reduction, but that is not clear to us. (b) We add the Subsection 9.2 giving the explicit values of the coefficients a j in terms of Racah polynomials. (c) In page 42 of the new version we add the fact of that U∗U is diagonal and the Remark 10.7 which mention the LDU-decomposition of the weight, both indicated by the referee. (d) At the end of the paper we add Remark 10.11 in order to mention more precisely some connections and relations with [2]. (2) At the beginning of Subsection 2.3 we add the Cartan involution and say that the Lie algebra k is spanned by Y1, Y2 and Y3. (3) We change the old proof of Lemma 3.3 by the more natural one suggested by the referee. (4) We add an Appendix with the proofs of Propositions 3.4 and 3.5 (5) We suppress the words ”such” and ”And” in pages 13 and 20, respectively, and replace ”lemma.2” by ”Lemma 5.1” as the referee pointed out.
Abstract. In this paper, we deal with a linear control system E denned on a Lie group G with Lie ... more Abstract. In this paper, we deal with a linear control system E denned on a Lie group G with Lie algebra fl. The dynamic of E is determined by a drift vector field, which is an element in the normalizer of 0 in the Lie algebra of all smooth vector fields on G, and by the control ...
In this paper, we describe the irreducible spherical functions of fundamental $K$-types associate... more In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=(\mathrm{SO}(n+1),\mathrm{SO}(n))$ in terms of matrix hypergeometric functions. The output of these description is that the irreducible spherical functions of the same $K$-type are encoded news examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$. Moreover, we show that $W$ admits a second order symmetric hypergeometric operator $D$.
In this paper, we shall deal with a linear control system (Sigma) defined on a Lie group G with L... more In this paper, we shall deal with a linear control system (Sigma) defined on a Lie group G with Lie algebra g. The dynamic of (Sigma) is determined by the drift vector field which is an element in the normalizer of g in the Lie algebra of all smooth vector field on G and by the ...
Page 300. Contemporary Mathematics Contemporary Mathematics Volume 537, 2011 Contemporary Mathema... more Page 300. Contemporary Mathematics Contemporary Mathematics Volume 537, 2011 Contemporary Mathematics Volume 537, 2011 The algebra of differential operators associated to a weight matrix: a first example Juan Tirao Abstract. ...
... III. Wolf, Joseph Albert, i936-. IV. ... In the case of GL (n) over fields of large residual ... more ... III. Wolf, Joseph Albert, i936-. IV. ... In the case of GL (n) over fields of large residual characteristic, Howe and Moy found a way to study arbitrary repre-sentations, replacing J and the t rival representation of J by smaller compact open subgroups and representations of them. ...
The main purpose of this paper is to compute all irreducible spherical functions on G = SU(3) of ... more The main purpose of this paper is to compute all irreducible spherical functions on G = SU(3) of arbitrary type δ ∈K, where K = S(U(2) × U(1)) ≃ U(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the complex projective plane P2(C) = G/K. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair (G, K). But the relation of spherical functions of types of dimension bigger than one with classical analysis, has not been worked out even in the case of an example of a rank one pair. The entries of H are solutions of two systems of ordinary differential equations. There is no ready made approach to such a pair of systems, or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions p+1Fp.
Abstract. In this paper, we describe the irreducible spherical functions of fundamental K-types a... more Abstract. In this paper, we describe the irreducible spherical functions of fundamental K-types associated with the pair (G,K) = (SO(n+ 1),SO(n)) in terms of matrix hypergeo-metric functions. The output of this description is that the irreducible spherical functions of the same K-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size 2 and 3, with respect to a matrix-weight W supported on [0, 1]. Moreover, we show that W has a second order symmetric hypergeometric operator D. Key words: matrix-valued spherical functions; matrix orthogonal polynomials; the matrix hypergeometric operator; n-dimensional sphere 2010 Mathematics Subject Classification: 22E45; 33C45; 33C47 1
The aim of this paper is to determine all irreducible spherical functions of the pair (G, K) = (S... more The aim of this paper is to determine all irreducible spherical functions of the pair (G, K) = (SU(n + 1), U(n)), where the highest weight of their K-types π are of the form (m + ℓ,. .. , m + ℓ, m,. .. , m). Instead of looking at a spherical function Φ of type π we look at a matrix-valued function H defined on a section of the K-orbits in an affine subvariety of Pn(C). The function H diagonalizes, hence it can be identified with a column vector-valued function. The irreducible spherical functions of type π turn out to be parameterized by S = {(w, r) ∈ Z × Z : 0 ≤ w, 0 ≤ r ≤ ℓ, 0 ≤ m + w + r}. A key result to characterize the associated function Hw,r is the existence of a matrix-valued polynomial function Ψ of degree ℓ such that Fw,r(t) = Ψ(t) −1 Hw,r (t) becomes an eigenfunction of a matrix hypergeometric operator with eigenvalue λ(w, r), explicitly given. In the last section we assume that m ≥ 0 and define the matrix polynomial Pw as the (ℓ + 1) × (ℓ + 1) matrix whose r-row is the polynomial Fw,r. This leads to interesting families of matrix-valued orthogonal Jacobi polynomials P α,β w for α, β > −1.
In this paper we establish a close relationship between the spherical functions of the n-dimensio... more In this paper we establish a close relationship between the spherical functions of the n-dimensional sphere S n ≃ SO(n + 1)/SO(n) and the spherical functions of the n-dimensional real projective space P n (R) ≃ SO(n + 1)/O(n). In fact, for n odd a function on SO(n + 1) is an irreducible spherical function of some type π ∈ŜO(n) if and only if it is an irreducible spherical function of some type γ ∈Ô(n). When n is even this is also true for certain types, and in the other cases we exhibit a clear correspondence between the irreducible spherical functions of both pairs (SO(n + 1), SO(n)) and (SO(n + 1), O(n)). Summarizing, to find all spherical functions of one pair is equivalent to do so for the other pair.
We describe families of polynomials arising in the study of the universal central extensions of L... more We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa [DJKM83] in their work on the Landau-Lifshitz equations. We show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials.
The fundamental properties of spherical functions have been establis-hed by R. Godement in a well... more The fundamental properties of spherical functions have been establis-hed by R. Godement in a well known paper [1} in 1952. There he defines
ABSTRACT. Let Go bea non compact real semisimple Lie group with finite center, and let U(g)K deno... more ABSTRACT. Let Go bea non compact real semisimple Lie group with finite center, and let U(g)K denote the centralizer in U(g) of a maximal compact subgroup Ko of Go. By the fundamental work of Harish-Chandra it is known that many deep questions concerning the infinite dimensional representation theory of Go reduce to questions about the structure and finite dimensional representation theory of the algebra U(g)!\, called the classifying ring of Go. To study the algebra U(g)](, B. Kostant suggested to consider the projection map P: U(g)-> U(t)<2>U(a), associated to an Iwasawa decomposition Go = KoAoNo of Go, adapted to Ko. When P is restricted to U(g)K P becomes an injective anti-homomorphism of algebras. In this paper we use the characterization of the image of U(g)](, when Go =SO(n,l) or SU(n,l) obtained in Tirao [11], to prove that U(g)] ( ~ Z(g) <2> Z(£), where Z(g) and Z(t) denote respectively the centers of U(g) and of U(t). By a well known theorem of Harish-Chandra...
Proceedings of the National Academy of Sciences, 2003
The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1 ... more The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1 , 11–13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123–162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39–83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127–172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1–24]. The hypergeometric function known also as Gauss' function is the unique solution of the hypergeometric equation analytic at z = 0 and with value 1 at z = 0. This function, because of its remarkable properties, has been used for centuries in the whole subject of special functions. In this article we give a matrix-valued analog of the hypergeometric differential equation and of Gauss' function. One can only speculate that many of the connections that made Gauss' function a vital part of mathematics at the end of the 20th century will be shared by its matrix-valued version, discussed here.
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