Related papers: On exponentiation and infinitesimal one-parameter …

Let $U$ be the quantum group with divided powers in $l-$th root of unity and let $u\subset U$ be the Frobenius kernel. V.Ginzburg and S.Kumar proved that the cohomology algebra of $u$ with trivial coefficients is isomorphic to the functions…

Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positive root system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let…

Let $F$ be a finite unramified extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_F$, and let $\mathbf{G}$ denote a split, connected reductive group over $\mathcal{O}_F$. We fix a Borel subgroup $\mathbf{B} =…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As…

Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that…

Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup…

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary…

Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case…

Let $U(KG)$ be the group of units of the group ring $KG$ of the group $G$ over a commutative ring $K$. The anti-automorphism $g\mapsto g\m1$ of $G$ can be extended linearly to an anti-automorphism $a\mapsto a^*$ of $KG$. Let $S_*(KG)=\{x\in…

Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by…

We study reductive subgroups $H$ of a reductive linear algebraic group $G$ -- possibly non-connected -- such that $H$ contains a regular unipotent element of $G$. We show that under suitable hypotheses, such subgroups are $G$-irreducible in…

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq…

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth…

Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = {\l}\oplus \u$, where $\u$ is…

We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a…

A rational group of hermitian type is an algebraic group over the rational numbers whose symmetric space is a hermitian symmetric space. We assume such a group $G$ to be given, which we assume is isotropic. Then, for any rational parabolic…

Let $F$ be a locally compact non-archimedean field of residue characteristic $p$, $\textbf{G}$ a connected reductive group over $F$, and $R$ a field of characteristic $p$. When $R$ is algebraically closed, the irreducible admissible…

Let $G$ be a group and $H \le K \le G$. We say that $H$ is $c$-embedded in $G$ with respect to $K$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H \cap B \le Z(K)$. Given a finite group $G$, a prime number $p$ and a Sylow…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…