Distinguished representations of reductive p-adic groups

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My research forms part of the Langlands program, a branch of mathematics involving deep connections between number theory, automorphic forms, representation theory, and algebraic geometry. I study the properties of representations of certain matrix groups known as reductive p-adic groups. These matrix groups are defined in terms of number theory - the matrix entries belong to p-adic fields - these fields are nonarchimedean completions of number fields (finite extensions of the rational numbers). In recent years, there has been considerable interest in harmonic analysis on p-adic symmetric varieties and connections with the Langlands program. My work involves the study and construction of distinguished representations of reductive p-adic groups. These distinguished representations exhibit specific symmetry properties relative to the involution that defines the p-adic symmetric space. The basic building blocks in the theory of distinguished representations are the so-called relatively supercuspidal representations. My main focus is on construction of relatively supercuspidal representations and the study of their properties, particularly those that are relevant to the Langlands program.