Algebraic Group Of Adjoint Type Research Articles

A parametrization of sheets of conjugacy classes in bad characteristic

AbstractLet G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

Rigidity of Bott–Samelson–Demazure–Hansen variety for $${\varvec{F}}_{\varvec{4}}$$ and $${\varvec{G}}_{\varvec{2}}$$

Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced expressions of $w_{0}$ in terms of a Coxeter element such that $Z(w_{0}, \underline{i})$ is rigid (see Theorem 8.1). Further, if $G$ is of type $G_{2},$ there is no reduced expression $\underline{i}$ of $w_{0}$ for which $Z(w_{0}, \underline{i})$ is rigid (see Theorem 8.2).

-TYPE SUBGROUPS CONTAINING REGULAR UNIPOTENT ELEMENTS

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X=\text{PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_{1}$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p)=(E_{6},13)$ or $(E_{7},19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the embedding of $X$ in $G$. We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.

Parabolic subgroups and automorphism groups of Schubert varieties

Let G be a simple algebraic group of adjoint type over the field C of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and X(w) be the Schubert variety in G/B corresponding to w. In this article we show that given any parabolic subgroup P of G containing B properly, there is an element w∈W such that P is the connected component, containing the identity element of the group of all algebraic automorphisms of X(w).

On the Automorphism Group of a Smooth Schubert Variety

Let G be a simple algebraic group of adjoint type over the field \(\mathbb {C}\) of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and let X(w) be the Schubert variety in G/B corresponding to w. Let α0 denote the highest root of G with respect to T and B. Let P be the stabiliser of X(w) in G. In this paper, we prove that if G is simply laced and X(w) is smooth, then the connected component of the automorphism group of X(w) containing the identity automorphism equals P if and only if w−1(α0) is a negative root (see Theorem 4.2). We prove a partial result in the non simply laced case (see Theorem 6.6).

AUTOMORPHISM GROUP OF A BOTT–SAMELSON–DEMAZURE–HANSEN VARIETY

Let G be a simple algebraic group of adjoint type over the field of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G, w be an element of the Weyl group W and X(w) be the Schubert variety in G/B corresponding to w. Let Z(w; i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a reduced expression i of w. In this article, we compute inductively, the connected component Aut0(Z(w; i)) of the automorphism group of Z(w; i) containing the identity automorphism. We show that Aut0(Z(w; i)) contains a closed subgroup isomorphic to B if and only if w -1(α0) < 0, where α0 is the highest root. If w 0 denotes the longest element of W, then we prove that Aut0(Z(w 0; i)) is a parabolic subgroup of G. It is also shown that this parabolic subgroup depends very much on the chosen reduced expression i of w 0 and we describe all parabolic subgroups of G that occur as Aut0(Z(w 0; i)). If G is simply laced, then we show that for every w ϵ W, and for every reduced expression i of w, Aut0(Z(w; i)) is a quotient of the parabolic subgroup Aut0(Z(w 0; j)) of G for a suitable choice of a reduced expression j of w 0 (see Theorem 7.3).

On equivariant principal bundles over wonderful compactifications

Let G be a simple algebraic group of adjoint type over C, and let M be the wonderful compactification of a symmetric space G/H. Take a G˜-equivariant principal R-bundle E on M, where R is a complex reductive algebraic group and G˜ is the universal cover of G. If the action of the isotropy group H˜ on the fiber of E at the identity coset is irreducible, then we prove that E is polystable with respect to any polarization on M. Further, for wonderful compactification of the quotient of PSL(n,C), n≠4 (respectively, PSL(2n,C), n>1) by the normalizer of the projective orthogonal group (respectively, the projective symplectic group), we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification.

Hecke modifications, wonderful compactifications and moduli of principal bundles

In this paper we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of universal Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the wonderful, or De Concini-Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.

The H-polynomial of an irreducible representation

Let G be a simple algebraic group. Associated with the finite-dimensional rational representation ρ : G → End ( V ) of G there is the monoid M ρ = K ⁎ ρ ( G ) ¯ ⊆ End ( V ) and the projective G × G -embedding P ρ = [ M ρ ∖ { 0 } ] / K ⁎ . One can identify the cases where P ρ is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of P ρ . In this paper we consider the situation where ρ is irreducible. We then determine H explicitly in terms of combinatorial invariants of ρ. Indeed, there is a canonical cellular decomposition for P ρ . These cells are defined in terms of idempotents, B × B -orbits and other natural quantities obtained from M ρ . Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of M ρ . As a special case we reacquire the well-known formula for the Poincaré polynomial of a “wonderful embedding” of a simple algebraic group of adjoint type.

On Intersections of Certain Partitions of a Group Compactification

Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let be the wonderful compactification of G. For a fixed pair (B,B − ) of opposite Borel subgroups of G, we look at intersections of Lusztig’s G-stable pieces and the B − ×B-orbits in , as well as intersections of B×B-orbits and B − ×B − -orbits in . We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of G.

Fixed point spaces in primitive actions of simple algebraic groups

Let G be a simple algebraic group of adjoint type acting primitively on an algebraic variety Ω. We study the dimensions of the subvarieties of fixed points of involutions in G. In particular, we obtain a close to best possible function f( h), where h is the Coxeter number of G, with the property that with the exception of a small finite number of cases, there exists an involution t in G such that the dimension of the fixed point space of t is at least f(h) dimΩ .

Remarks on the wonderful compactification of semisimple algebraic groups

We prove that ifG is a semisimple algebraic group of adjoint type over the field of complex numbers,H is the subgroup of all fixed points of an involution σ ofG that is induced by an involution σ of the simply connected coveringĜ ofG, then the wonderful compactification $$\overline {G/H} $$ of the homogeneous spaceG/H is isomorphic to the G.I.T quotientG ss (L)//H of the wonderful compactificationG ofG for a suitable choice of a line bundleL onG. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.

THE SMALLEST FIELD OF DEFINITION OF A SUBGROUP OF THE GROUP

As previously proved by the author, for each semisimple algebraic group of adjoint type that is dense in the Zariski topology there exists a smallest field of definition which is an invariant of the class of commensurable subgroups. In the present paper an algorithm is given for finding the smallest field of definition of a dense finitely generated subgroup of the group . A criterion for arithmeticity of a lattice in or in terms of this field is presented. Bibliography: 7 titles.

Equivariant betti numbers for symmetric varieties

The purpose of this paper is to give an explicit procedure for the computation of the equivariant Poincart series of a complete symmetric variety, in terms of associated combinatorial objects. We recall that given G, a semisimple algebraic group of adjoint type over the complex numbers, 0: G + G an order two automorphism, and letting H = G” = (g E G 1 g(g) = g), by a wonderful symmetric variety we mean a pointed smooth G-variety (Y, p) such that