Marko Tadić

Construction of discrete series for classical p- adic groups

In this paper the authors complete the classification of irreducible square integrable representations of classical

On reducibility of parabolic induction

p

An external approach to unitary representations

-adic groups, assuming a natural assumption which is expected to hold in general. This classification implies a parameterization of irreducible tempered representations of these groups and it implies a classification of the non-unitary duals of these groups (modulo cuspidal data).

On characters of irreducible unitary representations of general linear groups

Jacquet modules of a reducible parabolically induced representation of a reductivep-adic group reduce in a way consistent with the transitivity of Jacquet modules. This fact can be used for proving irreducibility of parabolically induced representations. Classical groups are particularly convenient for application of this method, since we have very good information about part of the representation theory of their Levi subgroups (general linear groups are factors of Levi subgroups, and therefore we can apply the Bernstein-Zelevinsky theory). In the paper, we apply this type of approach to the problem of determining reducibility of parabolically induced representations ofp-adic Sp(n) and SO(2n+1). We present also a method for getting Langlands parameters of irreducible subquotients. In general, we describe reducibility of certain generalized principal series (and some other interesting parabolically induced representations) in terms of the reducibility in the cuspidal case. When the cuspidal reducibility is known, we get explicit answers (for example, for representations supported in the minimal parabolic subgroups, the cuspidal reducibility is well-known rank one reducibility).

On tempered and square integrable representations of classical p-adic groups

The principal ideas of harmonic analysis on a locally compact group G which is not necessarily compact or commutative, were developed in the 1940’s and early 1950’s. In this theory, the role of the classical fundamental harmonics is played by the irreducible unitary representations of G. The set of all equivalence classes of such representations is denoted by Ĝ and is called the dual object of G or the unitary dual of G. Since the 1940’s, an intensive study of the foundations of harmonic analysis on complex and real reductive groups has been in progress (for a definition of reductive groups, the reader may consult the appendix at the end of the second section). The motivation for this development came from mathematical physics, differential equations, differential geometry, number theory, etc. Through the 1960’s, progress in the direction of the Plancherel formula for real reductive groups was great, due mainly to Harish-Chandra’s monumental work, while, at the same time, the unitary duals of only a few groups had been parametrized. With F. Mautner’s work [Ma], a study of harmonic analysis on reductive groups over other locally compact non-discrete fields was started. We shall first describe such fields. In the sequel, a locally compact non-discrete field will be called a local field. If we have a non-discrete absolute value on the field Q of rational numbers, then it is equivalent either to the standard absolute value (and the completion is the field R of real numbers), or to a p-adic absolute value for some prime number p. For r 2 Q× write r = pa/b where α, a and b are integers and neither a nor b are divisible by p. Then the p-adic absolute value of r is jrjp = p−α. A completion of Q with respect to the p-adic absolute value is denoted by Qp. It is called a field of p-adic numbers. Each finite dimensional extension F of Qp has a natural topology of a vector space over Qp. With this topology, F becomes a local field. The topology of F can be also introduced with an absolute value which is denoted by j jF (in the fifth section we shall fix a natural absolute value). The fields of real and complex numbers, together with the finite extensions of p-adic numbers, exhaust all local fields of characteristic zero up to isomorphisms ([We]). Let F be a finite field. Denote by F((X)) the field of formal power series over F. Elements of this field are series of the form f = P∞ n=k anX , an 2 F, for some integer k. Fix q > 1.

On Jacquet Modules of Induced Representations of p—Adic Symplectic 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

Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case)

This paper has two aims. The first is to give a description of irreducible tempered representations of classical p-adic groups which follows naturally the classification of irreducible square integrable representations modulo cuspidal data obtained by Moeglin and the author of this article (2002). The second aim of the paper is to give a description of an invariant (partially defined function) of irreducible square integrable representation of a classical p-adic group (defined by Moeglin using embeddings) in terms of subquotients of Jacquet modules. As an application, we describe behavior of partially defined function in one construction of square integrable representations of a bigger group from such representations of a smaller group (which is related to deformation of Jordan blocks of representations).

Structure Arising from Induction and Jacquet Modules of Representations of Classical p-Adic Groups

We fix a reductive p-adic group G. One very useful tool in the representation theory of reductive p-adic groups is the Jacquet module. Let us recall the definition of the Jacquet module. Let (π, V) be a smooth representation of G and let P be a parabolic subgroup of G with a Levi decomposition P = MN. The Jacquet module of V with respect to N is \( {V_N} = V/spa{n_{\mathbb{C}}}\left\{ {\pi (n)v - v;n \in N,v \in V} \right\} \).