Scalar irreducibility of certain eigenspace representations (original) (raw)

Invariant Differential Operators on Certain Nilpotent Homogeneous Spaces

Monatshefte f�r Mathematik, 2001

Let G exp g be a nilpotent connected and simply connected Lie group, and H exp h an analytic subgroup of G. Let 1 1 f Y f P g à , be a unitary character of H and let (Ind G H 1. Suppose that the multiplicities of all the irreducible components of (are ®nite. Corwin and Greenleaf conjectured that the algebra D (GaH of the differential operators on the Schwartz-space of (which commute with (is isomorphic to the algebra of H-invariant polynomials on the af®ne space f h c. We prove in this paper this conjecture under the condition that there exists a subalgebra which polarizes all generic elements in f h c. We prove also that if h is an ideal of g, then the ®nite multiplicities of (is equivalent to the fact that the algebra D (GaH is commutative.

On the Irreducibility of a Class of Homogeneous Operators

Operator Theory: Advances and Applications

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra A(D) as quotients of certain natural modules over the function algebra A(D 2). These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class B k (D) and that they are mutually inequivalent, irreducible and homogeneous.

Criteria of irreducibility of the Koopman representations for the group GL0(2∞,R)

Journal of Functional Analysis

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group GL 0 (2∞, R) = lim − →n GL(2n − 1, R), the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian noncentered measures. The corresponding G-space X m is a subspace of the space Mat(2∞, R) of infinite in both directions real matrices. In fact, X m is a collection of m infinite in both directions rows. This result was announced in [20]. We give the proof only for m ≤ 2. The general case will be studied later.

Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Ergodic Theory and Dynamical Systems

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

On generalized Cartan subspaces

Transformation Groups, 2011

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2,  an involution of G defined over k, H a k-open subgroup of the fixed point group of  and G k (resp. H k ) the set of k-rational points of G (resp. H ). The variety G k =H k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L 2 .G k =H k / of square integrable functions decomposes into several series, one for each H k -conjugacy class of Cartan subspaces of G k =H k . In this paper we give a characterization of the H k -conjugacy classes of these Cartan subspaces in the case that there exists a splitting extension of order 2 and .G; / is . ; k/-split conjugate (see 3.8). This condition is satisfied for k the the real numbers and several other fields for which the symmetric k-variety has a splitting extension of order 2. For k D R we prove a number of additional results as well.

Notes on group actions, manifolds, lie groups, and lie algebras

2005

CHAPTER 2. REVIEW OF GROUPS AND GROUP ACTIONS 3. Similarly, the sets R of real numbers and C of complex numbers are groups under addition (with identity element 0), and R * = R − {0} and C * = C − {0} are groups under multiplication (with identity element 1). 4. The sets R n and C n of n-tuples of real or complex numbers are groups under componentwise addition: (x 1 ,. .. , x n) + (y 1 , • • • , y n) = (x 1 + y n ,. .. , x n + y n), with identity element (0,. .. , 0). All these groups are abelian. 5. Given any nonempty set S, the set of bijections f : S → S, also called permutations of S, is a group under function composition (i.e., the multiplication of f and g is the composition g • f), with identity element the identity function id S. This group is not abelian as soon as S has more than two elements. 6. The set of n × n matrices with real (or complex) coefficients is a group under addition of matrices, with identity element the null matrix. It is denoted by M n (R) (or M n (C)). 7. The set R[X] of polynomials in one variable with real coefficients is a group under addition of polynomials. 8. The set of n × n invertible matrices with real (or complex) coefficients is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the general linear group and is usually denoted by GL(n, R) (or GL(n, C)). 9. The set of n × n invertible matrices with real (or complex) coefficients and determinant +1 is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the special linear group and is usually denoted by SL(n, R) (or SL(n, C)). 10. The set of n × n invertible matrices with real coefficients such that RR = I n and of determinant +1 is a group called the orthogonal group and is usually denoted by SO(n) (where R is the transpose of the matrix R, i.e., the rows of R are the columns of R). It corresponds to the rotations in R n. 11. Given an open interval ]a, b[, the set C(]a, b[) of continuous functions f : ]a, b[ → R is a group under the operation f + g defined such that (f + g)(x) = f (x) + g(x) for all x ∈]a, b[.

Lie-algebras and linear operators with invariant subspaces

1993

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of some algebra of differential (difference) operators in finite-dimensional representation plus an

Geometric embeddings of operator spaces

Illinois Journal of Mathematics

We denote typically by e4 a C*-algebra with and by G the Banach Lie group of invertible elements of 4. Sometimes we assume that A is represented faithfully in a Hilbert space/C and this representation may change when needed. The general reference for this sort of thing is 15]. Throughout we use reductive homogeneous space of operators (or simply reductive space) for the following type of data: a C*-algebra A is given and the group G or a convenient Lie subgroup of G acts on a C-Banach manifold M in such a way that the isotropy groups of points in M are provided with stable infinitesimal supplements (the "horizontal spaces" of the reductive space) in the sense that the adjoint action of the isotropy groups leave these supplements stable (see [8] for the finite dimensional analogue and [9] for the case considered here). The horizontal spaces provide the canonical connection of the reductive space. Numerous examples of reductive spaces are described in [1], [2], [4], [12]. Given reductive spaces M, M' with groups G, G', a morphism from M to M' is a smooth map P" M-M' together with a Lie group homomorphism p: G G' with the equivariance q (Lge) L(g)P (e) (we denote by L both actions) and also with the infinitesimal condition that the tangent map of p preserves the horizontal spaces. The groups G and G' operate on the space Hom(M, M') of morhpisms from M to M by qg Lgql, g Adg , for g 6 G', ql h ql Lh, 1/r h lp Adh, for h 6 G. It is clear that these left actions commute and that (qlg) h (ffklh)g kI/g(h) (l[rg) h (1]rh) g kllg(h) Denote by Q c A the set of reflections in A, i.e., the invertible elements e of A that satisfy e e-1. This space is studied in detail in [4]. It is a reductive space with group G acting by inner automorphism Age geg-The selfadjoint elements of Q form a smooth submanifold P and the polar decomposition induces a fibration Q-P. More explicitly, if e-/zp with/z > 0 and p unitary then automatically p is in P and zr(e) p. The fibers Qp 7r-l(p) are characterized by Qp {e Q; ep > O} (for