We prove the following theorem. Let G be a finite group generated by unitary reflections in a complex Hermitian space $$V={\mathbb {C}}^\ell $$ V = C ℓ and let $$G'$$ G ′ be any reflection subgroup of G . Let $${\mathcal {H}}={\mathcal {H}}(G)$$ H = H ( G ) be the space of G -harmonic polynomials on V . There is a degree preserving isomorphism $$\mu :{\mathcal {H}}(G')\otimes {\mathcal {H}}(G)^{G'}\overset{\sim }{{\longrightarrow \;}}{\mathcal {H}}(G)$$ μ : H ( G ′ ) ⊗ H ( G ) G ′ ⟶ ∼ H ( G ) of graded $${\mathcal {N}}$$ N -modules, where $${\mathcal {N}}:=N_{{\text {GL}}(V)}(G)\cap N_{{\text {GL}}(V)}(G')$$ N : = N GL ( V ) ( G ) ∩ N GL ( V ) ( G ′ ) and $${\mathcal {H}}(G)^{G'}$$ H ( G ) G ′ is the space of $$G'$$ G ′ -fixed points of $${\mathcal {H}}(G)$$ H ( G ) . This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups. An application is given to counting rational conjugates of reductive groups over $${\mathbb {F}}_q$$ F q .

中文翻译：

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酉反射群的共变代数的分解定理

我们证明以下定理。设 G 是在复厄米空间 $$V={\mathbb {C}}^\ell $$ V = C ℓ 中由酉反射生成的有限群，并令 $$G'$$ G ′ 是G 。令 $${\mathcal {H}}={\mathcal {H}}(G)$$ H = H ( G ) 是 V 上的 G 调和多项式的空间。有一个度保持同构 $$\mu :{\mathcal {H}}(G')\otimes {\mathcal {H}}(G)^{G'}\overset{\sim }{{\longrightarrow \ ;}}{\mathcal {H}}(G)$$ μ : H ( G ′ ) ⊗ H ( G ) G ′ ⟶ ∼ H ( G ) 的分级 $${\mathcal {N}}$$ N -模块，其中 $${\mathcal {N}}:=N_{{\text {GL}}(V)}(G)\cap N_{{\text {GL}}(V)}(G')$ $ N : = N GL ( V ) ( G ) ∩ N GL ( V ) ( G ′ ) 和 $${\mathcal {H}}(G)^{G'}$$ H ( G ) G ′ 是$$G'$$ G ′ - $${\mathcal {H}}(G)$$ H ( G ) 的不动点的空间。这概括了 Douglass 和 Dyer 对真实反射群的抛物线子群的结果。给出了一个应用来计算 $${\mathbb {F}}_q$$ F q 上的还原群的有理共轭。