Adapted pairs in type A and regular nilpotent elements

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• 作者：Florence Fauquant-Millet^a ; ^{florence.millet@univ-st-etienne.fr" class="auth_mail" title="E-mail the corresponding author} ; Anthony Joseph^b ; ^{anthony.joseph@weizmann.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：17B35
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：453
• 期：Complete
• 页码：1-67
• 全文大小：1363 K

文摘

Let g$\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field k of characteristic zero and G its adjoint group. A biparabolic subalgebra q$\mathfrak{q}$ of g$\mathfrak{g}$ is the intersection of two parabolic subalgebras whose sum is g$\mathfrak{g}$. The algebra Sy(q)$\mathit{Sy}(\mathfrak{q})$ of semi-invariants on q^⁎${\mathfrak{q}}^{⁎}$ of a proper biparabolic subalgebra q$\mathfrak{q}$ of g$\mathfrak{g}$ is polynomial in most cases, in particular when g$\mathfrak{g}$ is simple of type A or C  . On the other hand q$\mathfrak{q}$ admits a canonical truncation q_Λ${\mathfrak{q}}_{\mathrm{\Lambda }}$ such that Sy(q)=Sy(q_Λ)=Y(q_Λ)$\mathit{Sy}(\mathfrak{q})=\mathit{Sy}({\mathfrak{q}}_{\mathrm{\Lambda }})=Y({\mathfrak{q}}_{\mathrm{\Lambda }})$ where Y(q_Λ)$Y({\mathfrak{q}}_{\mathrm{\Lambda }})$ denotes the algebra of invariant functions on (q_Λ)^⁎${({\mathfrak{q}}_{\mathrm{\Lambda }})}^{⁎}$. An adapted pair for q_Λ${\mathfrak{q}}_{\mathrm{\Lambda }}$ is a pair $(h,\eta )\in {\mathfrak{q}}_{\mathrm{\Lambda }}\times {({\mathfrak{q}}_{\mathrm{\Lambda }})}^{⁎}$ such that η   is regular in (q_Λ)^⁎${({\mathfrak{q}}_{\mathrm{\Lambda }})}^{⁎}$ and $(adh)(\eta )=-\eta$. In Joseph (2008) [9] adapted pairs for every truncated biparabolic subalgebra q_Λ${\mathfrak{q}}_{\mathrm{\Lambda }}$ of a simple Lie algebra g$\mathfrak{g}$ of type A   were constructed and then provide Weierstrass sections for Y(q_Λ)$Y({\mathfrak{q}}_{\mathrm{\Lambda }})$ in (q_Λ)^⁎${({\mathfrak{q}}_{\mathrm{\Lambda }})}^{⁎}$. These Weierstrass sections are linear subvarieties η+V$\eta +V$ of (q_Λ)^⁎${({\mathfrak{q}}_{\mathrm{\Lambda }})}^{⁎}$ such that the restriction map induces an algebra isomorphism of Y(q_Λ)$Y({\mathfrak{q}}_{\mathrm{\Lambda }})$ onto the algebra of regular functions on η+V$\eta +V$. The main result of the present work is to show that for each of the adapted pairs $(h,\eta )$ constructed in Joseph (2008) [9] one can express η (not quite uniquely) as the image of a regular nilpotent element y   of g^⁎${\mathfrak{g}}^{⁎}$ under the restriction map g^⁎→q^⁎${\mathfrak{g}}^{⁎}\to {\mathfrak{q}}^{⁎}$. This is a significant extension of Joseph and Fauquant-Millet (2011) [12], which obtains this result in the rather special case of a truncated biparabolic of index one. Observe that y must be a G translate of the standard regular nilpotent element defined in terms of the already chosen set π of simple roots. Consequently one may attach to y a unique element of the Weyl group W   of g$\mathfrak{g}$. Ultimately one can then hope to be able to describe adapted pairs (in general, that is not only for g$\mathfrak{g}$ of type A) through the Weyl group.