On Characters of Irreducible Unitary Representations of General Linear Groups

Founding harmonic analysis on classical simple complex groups, I.M. Gelfand and M.A. Naimark in their classical book [GN] posed three basic questions: unitary duals, characters of irreducible unitary representations and Plancherel measures. In the case of reductive p-adic groups, the only series of reductive groups where unitary duals are known are general linear groups. In this paper we reduce characters of irreducible unitary representations of GL(n) over a non-archimedean local field F , to characters of irreducible square-integrable representations of GL(m), with m ≤ n (we get an explicit expression for characters of irreducible unitary representations in terms of characters of irreducible square-integrable representations). In other words, we express characters of irreducible unitary representations in terms of the standard characters. We get also a formula expressing the characters of irreducible unitary representations in terms of characters of segment representations of Zelevinsky (the formula for the Steinberg character of GL(n) is a very special case of this formula). The classification of irreducible square-integrable representations of GL(m,F )’s has recently been completed ([Z], [BuK], [Co]). The characters of these representations are not yet known in the full generality, although there exists a lot of information about them ([Ca2], [CoMoSl], [K], [Sl]). Zelevinsky’s segment representations supported by minimal parabolic subgroups are one dimensional, so their characters are obvious. Therefore, we get the complete formulas for characters of irreducible unitary representations supported by minimal parabolic subgroups. By the classification theorem for general linear groups over any locally compact nondiscrete field, any irreducible unitary representation is parabolically induced by a tensor product of representations u(δ, n) where δ is an irreducible essentially square integrable representation of some general linear group and n a positive integer (see the second section for precise statements). Since there exists a simple formula for characters of parabolically induced representations in terms of the characters of inducing representations ([D]), it is enough to know the characters of u(δ, n)’s. Our idea in getting the formula for characters of irreducible unitary representations was to use the fact that unitary duals in archimedean and non-archimedean case can be expressed in the same way. Using the fact that there also exists a strong similarity of behavior of ends of complementary series, we relate in these two cases the formulas that