Viernes 28/07/2023, 14.30 hs. Aula 27
Titulo: Matrix valued multivariable polynomials associated to commutative triples
Resumen:
It is well known that there is intimate relation between representations of groups and special functions, and we discuss an instance of this connection. In this case the special functions are matrix valued multivariable orthogonal polynomials. From the group theory point of view, we look at triples (G, K, τ ). Here (G, K) consists of a compact Lie group G and a compact subgroup K. Next τ is a finite-dimensional representation of K acting in the representation V. The triple (G, K, τ )is commutative if a suitable convolution algebra of End(V )-valued functions on G is commutative. Then the special functions arise as characters of this convolution algebra. The general construction will be introduced in some detail. We next focus on an explicit example. Here G = SU(2 + m) and K = S(U(2) × U(m)) with an explicit representation τ coming from the U(2)-part of K is discussed in detail. In this case the orthogonal polynomials are 2-variable polynomials taking values in a matrix algebra which are invariant under the action of the Weyl group of type BC2.
The explicit example is joint work with Jie Liu (currently at Zhejiang University, China).