Download Lie Theory: Harmonic Analysis on Symmetric Spaces—General by Erik P. van den Ban (auth.), Jean-Philippe Anker, Bent PDF
By Erik P. van den Ban (auth.), Jean-Philippe Anker, Bent Orsted (eds.)
Semisimple Lie teams, and their algebraic analogues over fields except the reals, are of primary significance in geometry, research, and mathematical physics. 3 self sufficient, self-contained volumes, below the final name Lie Theory, function survey paintings and unique effects by way of well-established researchers in key parts of semisimple Lie theory.
Harmonic research on Symmetric Spaces—General Plancherel Theorems provides huge surveys via E.P. van den Ban, H. Schlichtkrull, and P. Delorme of the brilliant development during the last decade in deriving the Plancherel theorem on reductive symmetric spaces.
Van den Ban’s introductory bankruptcy explains the fundamental setup of a reductive symmetric house besides a cautious examine of the constitution thought, relatively for the hoop of invariant differential operators for the correct category of parabolic subgroups. complicated themes for the formula and realizing of the evidence are lined, together with Eisenstein integrals, regularity theorems, Maass–Selberg family, and residue calculus for root platforms. Schlichtkrull presents a cogent account of the elemental materials within the harmonic research on a symmetric house during the clarification and definition of the Paley–Wiener theorem. drawing close the Plancherel theorem via another perspective, the Schwartz area, Delorme bases his dialogue and evidence on asymptotic expansions of eigenfunctions and the speculation of intertwining integrals.
Well suited to either graduate scholars and researchers in semisimple Lie idea and neighboring fields, in all probability even mathematical cosmology, Harmonic research on Symmetric Spaces—General Plancherel Theorems presents a large, in actual fact concentrated exam of semisimple Lie teams and their crucial value and purposes to analyze in lots of branches of arithmetic and physics. wisdom of easy illustration idea of Lie teams in addition to familiarity with semisimple Lie teams, symmetric areas, and parabolic subgroups is required.
Read or Download Lie Theory: Harmonic Analysis on Symmetric Spaces—General Plancherel Theorems PDF
Similar symmetry and group books
Derived Equivalences for Group Rings
A self-contained advent is given to J. Rickard's Morita conception for derived module different types and its fresh purposes in illustration concept of finite teams. specifically, Broué's conjecture is mentioned, giving a structural reason behind relatives among the p-modular personality desk of a finite crew and that of its "p-local structure".
This new version of utilizing teams to aid humans has been written with the pursuits, wishes, and issues of team therapists and crew employees in brain. it really is designed to aid practitioners to devise and behavior healing teams of various varieties, and it offers frameworks to help practitioners to appreciate and choose how you can reply to the original events which come up in the course of staff classes.
- Profinite groups
- Semigroups and Near-Rings of Continuous Functions
- Groups Involving a Small Number of Conjugates
- The Galaxies of the Local Group
Extra info for Lie Theory: Harmonic Analysis on Symmetric Spaces—General Plancherel Theorems
Example text
Moreover, d P = 1 on M P and d P = eρ P on A P . Multiplication by the function d P induces a topological linear isomorphism from C ∞ (X1P ) onto itself; moreover, if m ∈ M1P , then L −1 m ◦ d P ◦ L m = d P (m) d P . It follows that conjugation by d P induces a linear automorphism of D(X1P ). Accordingly, for D ∈ D(X) we define the differential operator µ P (D) := d P−1 ◦ µ P (D) ◦ d P ∈ D(X1P ). Let b be a θ-stable maximal abelian subalgebra of q, containing aq . Let γ : D(X) → I (b) be the Harish-Chandra isomorphism introduced in Section 4 and let γ X1P : D(X1P ) → I P (b) be the similar isomorphism for the space X1P ; here I P (b) denotes the subalgebra of W (m1P C , b)-invariants in S(b).
The endomorphism by which the operator µ P,v (D : λ) acts on this module is denoted by µ P,v (D : ξ : λ). Finally, the direct sum of these endomorphisms, for v ∈ P W, is an endomorphism of V (P, ξ ), denoted by µ P (D : ξ : λ) := ⊕v∈ P W µ P,v (D : ξ : λ). 8. 3), with respect to which every endomorphism µ P (D : ξ : λ) diagonalizes, for D ∈ D(X) and λ ∈ a∗PqC . Action on Generalized Vectors We can now finally describe the action of the algebra of invariant differential operators on the H -fixed generalized vectors introduced in the previous section.
On the open H -orbits one expects the elements of C −∞ (P : ξ : λ) H to be just functions, which may be evaluated in points. Let ϕ ∈ C −∞ (P : ξ : λ) H and let v ∈ P W. Then one expects that ϕ(v) is a vector in Hξ−∞ which is fixed for ξ ⊗ (λ + ρ P ) ⊗ 1 P∩v H v −1 , because of the formal identity, for p ∈ P ∩ v H v −1 , [ξ ⊗ (λ + ρ p ) ⊗ 1]( p) ϕ(v) = ϕ( pv) = ϕ(vv −1 pv) = [πξ,λ (v −1 pv)ϕ](v) = ϕ(v), −1 since v −1 pv ∈ H . This implies that ϕ(v) ∈ (Hξ−∞ ) M P ∩v H v and (λ + ρ P )|aP ∩h = 0. The latter condition is equivalent to λ|aP ∩h = 0, in view of the following lemma.