We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.

Suppose $G$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is‎ ‎the Lie algebra of the unipotent radical $N.$ Under the adjoint‎ ‎action of its stabilizer in $M,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in [2]. Here we study the minimal free resolution of the invariant ring. Recently it was shown that if G is a finite linearly reductive group, then the ring of invariants is generated in degree ≤ |G| (see [5, 6, 3]). This extends the classical result of Noether who proved the bound in c...

We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as t...

We give an explicit description of a set of irreducible representations of a Weyl group which parametrizes the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group.

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over field $C$ characteristic different from $p$. When is algebraically closed, for many groups $G$, list $C$-types $(J,\lambda)$ has been produced satisfying exhaustion, sometimes restricted kind representations, and often unicity. verify that those lists Aut($C$)-stability we produce similar when no ...

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of Langlands program. In this article, we study its derived version. It was shown by second author that category smooth G $G$ -representations over k $k$ , p $p$ -adic reductive group and field characteristic equivalent to certain differential graded -algebra H • $H_G^\bullet$ whose zeroth cohom...

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over C. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH ...