Lie Group Of Real Rank Research Articles

On an analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type

Let X be a symmetric space and G the connected component of the group of isometries of X. If / G Lι (X), we consider conditions under which Sp{*/: g e G} is dense in Lι(X) in terms of the of /. This continues earlier work on this kind of problem by L. Ehrenpreis and F. I. Mautner, R. Krier and the author. 1. Introduction. Let /be an integrable function on R. Then we have the famous Wiener-Tauberian theorem: If the Fourier transform / is a nowhere vanishing function on i?, then the ideal generated by / is dense in Lι(R). In [EMI] Ehrenpreis and Mautner observed that the exact analogue of the above theorem is no longer true if one considers the commutative Banach algebra of ^-bi-invarian t Lι-functions on a non-compact semi-simple Lie group G, where K is a maximal compact subgroup of G—i.e. Let I (G) denote the commutative Banach algebra of Λ^-bi-invaria nt Lι-functions on G. For / e I\(G) let / denote its spherical Fourier transform (see §2). Then there exist functions / G I\(G) such that / is nowhere vanishing on the maximal ideal space M of I (G) and yet the algebra generated by / is not dense in h{G). However when G = SL(2, i?) they were able to show that a modified version of Wiener's theorem is true i.e. / nowhere vanishing on M together with the condition that it does not go to zero too fast at oo would indeed imply that the ideal generated by / is dense in I\(G). (Theorems 6 and 7 of [EMI].) This kind of result has been generalized by R. Krier [Kl] when G is a non-compact connected semisimple Lie group of real rank 1 and by the author for G of arbitrary rank [Si].

Local rigidity of discrete groups acting on complex hyperbolic space

The superrigidity theorem of Margulis, see Zimmer [17], classifies finite dimensional representations of lattices in semi-simple Lie groups of real rank strictly larger than 1. It is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank I semi-simple Lie groups. That this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or, in other words, the existence of non-trivial deformations. In Johnson-Millson [6] and Kourouniotis [8] such deformations were constructed for certain representations of lattices in SO(n, 1) based on a construction of Thurston called bending. The deformation space of the representation of a lattice F=SO(n, 1) obtained by restricting an inclusion SO(n, 1)-,SO(n+ l, 1) to F is of particular interest. If n > 2 the space of infinitesimal deformations is Hi(F, R "+1) where F acts on R "+1 by the restriction of the standard action of SO(n, 1). The space of infinitesimal deformations is non-zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable (in [6] it is also shown that some are not). In this paper, we study the complex analogue of the above example. We let F be a cocompact torsion free lattice in SU(n, 1) and consider the deformation space of the representation of F obtained by restricting an inclusion SU(n, 1)--*SU(n+I, 1). If n > l the space of infinitesimal deformations is H~(F, R)@H~(F, C,+ 1). In the first summand F acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming F in U(n, 1) by a curve of homomorphisms into the center of U(n, 1) (observe that the above inclusion factors as SU(n, 1)~U(n , 1)~SU(n+I, 1)). In the second summand F acts by the restriction of the standard action of SU(n, 1). This summand is non-zero for the standard arithmetic examples,

The contraction of K to [formula omitted

By studying the contraction of K to NM inside the Iwasawa decomposition of a semisimple Lie group of real rank 1, we realize the representations of NM as limits of representations of K, obtaining in particular, limiting formulas for matrix entries.

The Embeddings of the Discrete Series in the Principal Series for Semisimple Lie Groups of Real Rank One

The Embeddings of the Discrete Series in the Principal Series for Semisimple Lie Groups of Real Rank One

The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one

We consider the problem of finding all the “embeddings” of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is Spin ⁡ ( 2 n , 1 ) , SU ( n , 1 ) , SP ( n , 1 ) or F 4 ( n ⩾ 2 ) \operatorname {Spin} (2n,\,1),{\text {SU}}(n,\,1),\,{\text {SP}}(n,\,1)\,{\text {or}}\,{F_4}\,(n\, \geqslant \,2) . The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.

The Plancherel Formula for parabolic subgroups

We prove an explicit Plancherel Formula for the parabolic subgroups of the simple Lie groups of real rank one. The key point of the formula is that the operator which compensates lack of unimodularity is given, not as a family of implicitly defined operators on the representation spaces, but rather as an explicit pseudo-differential operator on the group itself. That operator is a fractional power of the Laplacian of the center of the unipotent radical, and the proof of our formula is based on the study of its analytic properties and its interaction with the group operations.

Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.

Fourier inversion for semisimple Lie groups of real rank one

Fourier inversion for semisimple Lie groups of real rank one

Singular Integrals and the Principal Series, II

A further study is made of the intertwining operators for the representations of the simple Lie groups of real rank one. We extend and apply this to the higher rank case, obtaining various results dealing with reducibility of principal series and existence of complementary series.

SINGULAR INTEGRALS AND THE PRINCIPAL SERIES

The L(2) theory of singular integral operators on nilpotent Lie groups is studied, extending known results for IR(n). The intertwining operators for the representations of the simple Lie groups of real rank one turn out to be of this type. As a result we determine which representations of the principal series of these groups are irreducible.