PhD research topic - Paolo Bravi
spherical varieties, symmetric spaces
Spherical varieties are special algebraic varieties with an action of a linear algebraic group G. Toric varieties, flag varieties, symmetric spaces are well-known examples of spherical varieties. The classification of spherical varieties is now complete.
The techniques developed to achieve the classification can be applied to study the multiplication of sections of globally generated line bundles on special spherical varieties called wonderful, see [BGM].
Special cases of such multiplication is just the tensor product of simple G-modules, for which one can see the so-called saturation conjecture [Ku]. That conjecture can be seen as a particular case of more general questions naturally arising while looking at the multiplication on different wonderful varieties which share the same combinatorial invariants.
The multiplication of Harish-Chandra spherical functions on symmetric spaces is yet another example. This can be seen from the point of view of Heckmann-Opdam hypergeometric functions, eigenfunctions of a system of differential operators in a Cherednik algebra (double affine Hecke algebra), multivariate orthogonal polynomials, such as Jack, Jacobi, Koornwinder polynomials, see [Ma,He].
Some open questions/conjectures on Jacobi polynomials can be reformulated in terms of the combinatorial invariants of wonderful varieties.
[BGM] Bravi, Gandini, Maffei, Repr.Theory 2016
[He] Heckmann, Sem.Bourbaki 1997
[Ku] Kumar, ICM 2010
[Ma] Macdonald, Sem.Bourbaki 1996