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The infinitesimal cone of a totally positive semigroup. (English) Zbl 0894.22011

Given a pinning \(\{T,B^+, B^-,x_i, y_i; i\in I\}\) of a complex reductive linear algebraic group \(G\) that is split over \(\mathbb{R}\) the author (following G. Lusztig) considers the semigroup \(G_{\geq 0}\) of \(G\) generated by \[ \bigl\{x_i (a),y_i(a), \chi(b) \mid a\geq 0,\;b>0;\;a, b\in \mathbb{R};\;i\in I,\;\chi\in X^\vee (T)\bigr\} \] and the convex cone \[ {\mathfrak g}_{\geq 0}=\text{Lie} (T)+ \sum_{i\in I} (\mathbb{R}^+ dx_i+ \mathbb{R}^+dy_i). \] The purpose of the paper is to show the equality \[ {\mathfrak g}_{\geq 0} =\bigl\{X\in {\mathfrak g} \mid\exp (\mathbb{R}^+X) \subseteq G_{\geq 0} \bigr\}. \] The inclusion “\(\subset\)” has been shown by G. Lusztig, so it only remained to show the reverse inclusion.
The basic idea of the proof given is to use a canonical basis for the adjoint representation in the sense of Lusztig together with the fact (proven by G. Lusztig) that the matrix elements of \(G_{\geq 0}\) with respect to such a basis are nonnegative. An explicit calculation in simply laced roots systems then leads to the desired restrictions on \(X\in {\mathfrak g}\) resulting from \(\exp (\mathbb{R}^+X) \subseteq G_{\geq 0}\).

MSC:

22E46 Semisimple Lie groups and their representations
22A15 Structure of topological semigroups
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References:

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