Symmetry Breaking Operators between Generalized Principal Series of GL(n, R) and GL(n−1, R)
In representation theory typical problems are decomposition problems, i.e. the question how a given representation decomposes into smallest possible parts. A subclass of that are branching problems, asking for the decomposition of a representation that is irreducible in terms of the whole algebraic structure, when restricted to a substructure. For real reductive Lie groups these problems have recently been studied in terms of symmetry breaking operators which are intertwining operators between a restricted group representation and a subgroup representation, intertwining with respect to the subgroup. We will address this problem in the category of Harish–Chandra modules, which are both representations of the Lie algebra and of the maximal compact subgroup of a Lie group, realized on the spaces of K-finite vectors of smooth representations. This way we will give a full classification of intertwining operators between Harish–Chandra modules of generalized principal series representations of the groups (GL(n, R), GL(n−1, R)), intertwining with respect to the Lie algebra and the maximal compact subgroup of GL(n−1, R).