# On the restriction of irreducible representations of the group U_n(k) to the subgroup U_{n&#8722;1}(k)

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Let k be a local &#64257;eld, and let K be a separable quadratic &#64257;eld extension of k. It is known that an irreducible complex representation &#960;_1 of the unitary group G_1 = U_n(k) has a multiplicity free restriction to the subgroup G_2 = U{n&#8722;1}(k) &#64257;xing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if &#960;_2 is an irreducible representation of G_2 , then &#960; = &#960;_1 &#8855; &#960;_2 is an irreducible representation of the product G = G_1 G_2 which we can restrict to the subgroup H = G_2 , diagonally embedded in G. The space of H-invariant linear forms on &#960; has dimension &#8804; 1.

In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of &#960; to predict when the dimension of H-invariant forms is equal to 1, i.e. when the dual of &#960;_2 occurs in the restriction of &#960;_1 . I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.