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On reductive algebraic semigroups. (English) Zbl 0840.20041
Gindikin, S. G. (ed.) et al., Lie groups and Lie algebras: E. B. Dynkin’s seminar. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 169, 145-182 (1995).
An algebraic semigroup $$S$$ is an affine variety with an associative multiplication, which is a morphism of varieties. The case where $$S$$ is a monoid which is irreducible as an algebraic variety over an algebraically closed field $$k$$ of characteristic zero is considered. $$S$$ is called reductive if the unit group of $$S$$ is a reductive group. The theory of reductive monoids was developed by Putcha and Renner. In the present paper, a new approach to the classification problem of reductive monoids is proposed. The starting point is the action of $$G\times G$$ on $$S$$ given by: $$(g,h)s=gsh^{-1}$$. Then $$k[S]$$ is defined as the subalgebra of $$G\times G$$-invariants in $$k[G]$$. Fix a Borel subgroup $$B$$ of $$G$$ and a Cartan subgroup $$T\subseteq B$$. It is known that $$k[S]= \bigoplus_{\Lambda\in {\mathcal L}} k[G]_\Lambda$$, where $${\mathcal L}$$ is a subset of the semigroup of dominant characters of $$T$$ with respect to $$B$$. Here $$k[G]_\Lambda$$ is the space of the matrix entries of the irreducible linear representation of $$G$$ with highest weight $$\Lambda$$.
The first of the main results of the paper identifies the sets $${\mathcal L}$$ which define algebraic monoids with unit group $$G$$. Next, the case where the underlying algebraic variety is normal is described. If $$G_0$$ is the commutator group of $$G$$, then for the quotient $$A= S/ (G_0\times G_0)$$ we let $$k[A]= k[S]^{G_0\times G_0}$$. The embedding $$k[A]\subseteq k[S]$$ defines a morphism $$\pi: S\to A$$, called the abelianization of $$S$$. The paper continues with a study of $$\pi$$, in particular with a determination of the cases in which $$\pi$$ is flat, that is when $$k[S]$$ is a flat $$k[A]$$-module. In this case the $$G_0 \times G_0$$-orbits are just the intersections of $$G\times G$$-orbits with the fibers of $$\pi$$. On the other hand, it is shown that, if $$G_0$$ is a fixed connected semisimple group, then there exists a distinguished (unique up to isomorphism) flat reductive semigroup $$Env(G_0)$$, whose commutator of the unit group is isomorphic to $$G_0$$, which has a certain universal property. It is called the enveloping semigroup of $$G_0$$. Moreover, if the group $$G_0$$ acts on an affine variety $$X_0$$, then there exists a variety $$E$$ containing $$X_0$$ with an action of $$S$$ extending the action of $$G_0$$ on $$X_0$$. Next, the $$G\times G$$-orbit structure on $$Env (G_0)$$ is described. Finally, stabilizers of $$G\times G$$-action on $$k[S]$$ are studied. It is known that every orbit contains a unique up to conjugation idempotent $$e$$. The author identifies the stabilizer of $$e$$ as a certain subgroup of $$P\times P^-$$, where $$P$$ is an appropriate parabolic subgroup of $$G$$ and $$P^-$$ is its opposite.
For the entire collection see [Zbl 0828.00011].

##### MSC:
 20G15 Linear algebraic groups over arbitrary fields 20M20 Semigroups of transformations, relations, partitions, etc. 14L30 Group actions on varieties or schemes (quotients)