Наукова електронна бібліотека
періодичних видань НАН України
періодичних видань НАН України
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
Репозиторій DSpace/Manakin
- Домашня сторінка
- →
- Фізико-технічні та математичні науки
- →
- Відділення математики
- →
- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
- →
- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10
- →
- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10, випуск за цей рік
- →
- Переглянути статтю
JavaScript is disabled for your browser. Some features of this site may not work without it.
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One
Інший ID:
2010 Mathematics Subject Classification: 22E46; 33C47
DOI: http://dx.doi.org/10.3842/SIGMA.2014.113
DOI: http://dx.doi.org/10.3842/SIGMA.2014.113
Посилання:
Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One / Maarten van Pruijssen , P. Román // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 40 назв. — англ.
Підтримка:
The research for this paper was partly conducted when the first author visited the University of
Cordoba in August and September 2012. We would like to thank the Mathematics Departments
of the Universities of Nijmegen and Cordoba for their generous supports that made this visit
possible. Finally we would like to thank to the anonymous referees, whose comments and
suggestions have helped us to improve the paper.
Дата:
2014
Переглядів:
584
Завантажень:
262
Анотація:
We present a method to obtain infinitely many examples of pairs (W,D) consisting of a matrix weight W in one variable and a symmetric second-order differential operator D. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs (G,K) of rank one and a suitable irreducible K-representation. The heart of the construction is the existence of a suitable base change Ψ₀. We analyze the base change and derive several properties. The most important one is that Ψ₀ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group G as soon as we have an explicit expression for Ψ0. The weight W is also determined by Ψ₀. We provide an algorithm to calculate Ψ₀ explicitly. For the pair (USp(2n),USp(2n−2)×USp(2)) we have implemented the algorithm in GAP so that individual pairs (W,D) can be calculated explicitly. Finally we classify the Gelfand pairs (G,K) and the K-representations that yield pairs (W,D) of size 2×2 and we provide explicit expressions for most of these cases.
