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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].

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