$\mathbb{R}^n$紧扩张的Corwin-Greenleaf重数函数
表示论
2018-07-31 v1
摘要
设G = K ⋉ R n G=K\ltimes\mathbb{R}^n G = K ⋉ R n K K K O ( n ) O(n) O ( n ) R n \mathbb{R}^n R n g ⊃ k \mathfrak{g}\supset\mathfrak{k} g ⊃ k G G G K K K p r : g ∗ ⟶ k ∗ pr: \mathfrak{g}^{*}\longrightarrow\mathfrak{k}^{*} p r : g ∗ ⟶ k ∗ O G ⊂ g ∗ \mathcal{O}^{G}\subset\mathfrak{g}^{*} O G ⊂ g ∗ O K ⊂ k ∗ \mathcal{O}^{K}\subset\mathfrak{k}^{*} O K ⊂ k ∗ n ( O G , O K ) n(\mathcal{O}^{G},\mathcal{O}^{K}) n ( O G , O K ) O G ∩ p r − 1 ( O K ) \mathcal{O}^{G}\cap pr^{-1}(\mathcal{O}^{K}) O G ∩ p r − 1 ( O K ) K K K π ∈ G ^ \pi\in\widehat{G} π ∈ G τ ∈ K ^ \tau\in\widehat{K} τ ∈ K O G \mathcal{O}^G O G O K \mathcal{O}^K O K n ( O G , O K ) n(\mathcal{O}^G,\mathcal{O}^K) n ( O G , O K ) τ \tau τ π \pi π K K K m ( π , τ ) m(\pi,\tau) m ( π , τ ) π \pi π n ( O G , O K ) ≠ 0 n(\mathcal{O}^G,\mathcal{O}^K)\neq 0 n ( O G , O K ) = 0 m ( π , τ ) ≠ 0 m(\pi,\tau)\neq 0 m ( π , τ ) = 0 K = S O ( n ) K=SO(n) K = S O ( n ) n ≥ 3 n\geq 3 n ≥ 3 π \pi π τ \tau τ n ( O G , O K ) = m ( π , τ ) n(\mathcal{O}^G,\mathcal{O}^K)=m(\pi,\tau) n ( O G , O K ) = m ( π , τ )
引用
@article{arxiv.1807.10864,
title = {Corwin-Greenleaf multiplicity function for compact extensions of $\mathbb{R}^n$},
author = {Majdi Ben Halima and Anis Messaoud},
journal= {arXiv preprint arXiv:1807.10864},
year = {2018}
}