Timeline for Strictly contracting elements in the center of a Levi subgroup

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Mar 13, 2014 at 5:33	comment	added	user76758		Typo in above comment: in the final sum indexed by characters $\chi \in \Delta - I$, should be summing the associated (isogeny category) "cocharacters" $\chi^{\ast}$ in the dual basis $\Delta^{\ast}$.
Mar 12, 2014 at 13:09	comment	added	user76758		@Arkandias: You seek $\lambda$ without specifying $M$ (so with $P_G(\lambda)=P$ you could choose $Z_G(\lambda)$ to be $M$). Let $B$ be a minimal parabolic $k$-subgroup of $G$ contained in $P$ and containing $S$, so $\Phi(B,S)$ is a positive system of roots in $\Phi(G,S)$ contained in the parabolic set of roots $\Phi(P,S)$. Let $\Delta$ be the base of $\Phi(B,S)$, $\Delta^{\ast}$ the dual basis of X$_{\ast}(S\cap\mathscr{D}(G))_{\mathbf{Q}}$. Then $P=P_I$ with $I\subset\Delta$ as usual. Use $\lambda=N\cdot \sum_{\chi \not\in I}\chi$ for $N>0$ divisible to make $\lambda \in{\rm{X}}_{\ast}(S)$.
Mar 12, 2014 at 12:58	vote	accept	Arkandias
Mar 12, 2014 at 12:58	comment	added	Arkandias		Thank you. I was looking for such $\lambda$. Is there an explicit description (given $G$, $P$ and a maximal split torus $S \subset G$) ?
Mar 11, 2014 at 13:12	history	edited	user76758	CC BY-SA 3.0	Simplified discussion be removing the use of a root system base adapted to P.
Mar 11, 2014 at 3:50	history	edited	user76758	CC BY-SA 3.0	edited body
Mar 11, 2014 at 2:01	history	answered	user76758	CC BY-SA 3.0