Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994, [32]: for e a nilpotent element in the Lie algebra of G, the ...

We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group G whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of G in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show th...

We give an explicit formula for the local normal zeta functions of torsion-free, class-2-nilpotent groups at primes p where the associated Pfaffian hypersurface has good reduction mod p and contains no lines. We show how together the functional equations of two types of zeta functions the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by ...

Let B be a p-block of a finite group, and set m = ∑ χ(1), the sum taken over all height zero characters of B. Motivated by a result of M. Isaacs characterising p-nilpotent finite groups in terms of character degrees, we show that B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of |G : P ||P : R|, where P is a defect group of B and where R is the focal subgro...

In the present paper, we prove that if L is a nilpotent Lie algebra whose proper subalge- bras are all nilpotent of class at most n, then the class of L is at most bnd=(d 1)c, where b c denotes the integral part and d is the minimal number of generators of L.

An element $u$ of a ring $R$ is called \textsl{unipotent} if $u-1$ is
 nilpotent. Two elements $a,b\in R$ are \textsl{unipotent equivalent}
 there exist unipotents $p,q\in such that $b=q^{-1}ap$. square
 matrices $A,B$ \textsl{strongly unipotent equivalent} there
 triangular $P,Q$ with $B=Q^{-1}AP$.
 In this paper, over commutative reduced rings, we characterize the mat...

Let F be a p-adic field and let G connected reductive group defined over F. We assume p is large. Denote g the Lie algebra of G. To each vertex s reduced Bruhat–Tits’ building G, we associate as usual gs residual Fq. normalize suitably Fourier-transform f↦fˆ on Cc∞(g(F)). study subspace functions f∈Cc∞(g(F)) such that orbital integrals f fˆ are 0 for element g(F) which not topologically nilpote...

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general...

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...