Characters and related distributions on reductive p-adic groups; distinguished representations

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Our proposed research involves projects in the representation theory of reductive p-adic groups. Suppose that G is a group, r is a representation of G, where, for each g in G, r(g) is an invertible operator on a vector space V. If H is a subgroup of G, we say that the representation r is H-distinguished if the space of H-invariant linear functionals on V is nonzero. Reductive p-adic groups are matrix groups that are important in number theory. We are interested in studying H-distinguished representations of complex representations of reductive p-adic groups G, in the special case where H is the fixed points of an involution of G. Here, an involution of G is an automorphism of order 2. One aspect of our proposed research involves finding various ways to construct H-distinguished representations. Another aspect of our research involves the study of the behaviour of spherical characters of H-distinguished representations. These are functions on G whose definition is quite technical. Roughly speaking, these functions are determined by the various H-invariant linear functionals on the spaces of H-distinguished representations of G.