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.

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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.

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