Research

I am researching matrix spherical functions for symmetric pairs (of Lie groups) and quantum symmetric pairs (of a quantum group and a coideal subalgebra).

Matrix Spherical Functions for Symmetric Pairs

More concretely, if \(G\) is a Lie group and \(K\le G\) is a subgroup such that there is an involutive automorphism \(\Theta\) of \(G\) with \[ (G^\Theta)_0 \le K\le G^\Theta, \] then \((G,K)\) is a symmetric pair. Given two finite-dimensional \(K\)-modules \(V,W\), we can then consider functions \(f: G\to\operatorname{Hom}(V,W)\) satisfying \[ \forall g\in G, k,k’\in K:\quad f(kgk’) = \pi_W(k) f(g) \pi_V(k’). \] These functions are called matrix spherical functions and we can equivalently view them as equivariant sections of the associated vector bundle for the principal \(K\)-bundle \(G\to G/K\) over the symmetric space \(G/K\) and the \(K\)-representation \(\operatorname{Hom}(V,W)\) (only considering the right action).

In the case of \(V,W\) being the trivial representation, these functions are known¹ to be Heckman–Opdam hypergeometric functions (and in particular Jacobi polynomials for compact symmetric pairs), and similar statements hold true for other one-dimensional representations². For higher-dimensional representations, however, it is unclear how and if matrix spherical functions are connected to Heckman–Opdam theory.

In my upcoming preprint² with Mikhail Isachenkov we undertake steps towards this question by devising a radial part decomposition for invariant differential operators even for cases where \(K\) is non-compact.

Matrix Spherical Functions for Quantum Symmetric Pairs

It can be shown that on the Lie algebra side there is a classification of all possible symmetric pairs in terms of Satake diagrams. Given such a Satake diagram, it is possible³ to construct a Drin’feld–Jimbo quantum group \(U\) and a parameter-dependent coideal subalgebra \(B\) that in the \(q\to1\) limit specialise to \(U(\mathfrak{g}),U(\mathfrak{k})\), respectively. Such a pair \((U,B)\) is called a quantum symmetric pair.

We can follow the same definition as for symmetric pairs to define matrix spherical functions for quantum symmetric pairs (now using two finite-dimensional \(B\)-modules). For the case of \(V,W\) being trivial representations, it is known⁴ that the matrix spherical functions are symmetric Macdonald polynomials, and similarly, for other one-dimensional representations, my colleague Stein Meereboer recently showed the same.

Furthermore, there are five examples for which I managed to establish a connection between matrix spherical functions and intermediate Macdonald polynomials. —-