In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by Harish-Chandra. It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group.
Prerequisites
Length functions Let G {\displaystyle G} be a topological group. A length on G {\displaystyle G} is a continuous function l : G ↦ [ 0 , + ∞ [ {\displaystyle l:G\mapsto [0,+\infty [} , such that for all g , g ′ ∈ G {\displaystyle g,g'\in G} , l ( g g ′ ) ≤ l ( g ) + l ( g ′ ) {\displaystyle l(gg')\leq l(g)+l(g')} . These functions are used to define spaces with rapidly decreasing functions on locally compact topological groups, since they are used to define polynomial weights to ensure rapid decay.
The σ {\displaystyle \sigma } function Let G {\displaystyle G} be a semisimple connected Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , and let K {\displaystyle K} be its maximal compact subgroup. Then, the Cartan decomposition of G {\displaystyle G} allows one to state that the mapping
( k , X ) ↦ k exp ( X ) {\displaystyle (k,X)\mapsto k\exp(X)}
, k ∈ K {\displaystyle k\in K} and X ∈ p {\displaystyle X\in {\mathfrak {p}}} is an analytic diffeomorphism of K × p {\displaystyle K\times {\mathfrak {p}}} onto G {\displaystyle G} . Let b {\displaystyle b} be the Killing form of g {\displaystyle {\mathfrak {g}}} . Its restriction on p {\displaystyle {\mathfrak {p}}} provides an euclidean norm ‖ X ‖ {\displaystyle \|X\|} , K {\displaystyle K} -invariant, and such that ⟨ Z , ( a d Z ) 2 Y ⟩ = ⟨ ( a d Z ) 2 X , Y ⟩ {\displaystyle \langle Z,(adZ)^{2}Y\rangle =\langle (adZ)^{2}X,Y\rangle } for all X , Y , Z ∈ p {\displaystyle X,Y,Z\in {\mathfrak {p}}} , where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product corresponding to ‖ ⋅ ‖ {\displaystyle \|\cdot \|} . The σ {\displaystyle \sigma } function of G {\displaystyle G} is then defined as σ ( x ) = ‖ X ‖ {\displaystyle \sigma (x)=\|X\|} for x = k exp ( X ) {\displaystyle x=k\exp(X)} . It is proved that σ {\displaystyle \sigma } is a length function. Of course, σ {\displaystyle \sigma } is equal to 0 {\displaystyle 0} on K {\displaystyle K} , and for all x ∈ G {\displaystyle x\in G} , σ ( x − 1 ) = σ ( x ) {\displaystyle \sigma (x^{-1})=\sigma (x)} .
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