regular orbits of nilpotent subgroups of solvable linear groups abstract we settle a conjecture of walter carlip 1994 2 conjecture 1 3 suppose that g is a finite solvable group v is a finite faithful f g module over a field of characteristic p and assume o p g 1 let h be a nilpotent subgroup of g assume that h involves no wreath product z r z r for r 2 or r a mersenne prime then h has at least one regular orbit on v

regular orbits on symplectic modules let g be a finite group and let v be a faithful kg module k a field the question is does v contain a regular g orbit the problem has been widely studied in the literature in l berger proves that the answer is positive when a is nilpotent c 1 c free for every p and char k 1 a in 6 hargraves drops the condition of being a c c free for odd primes which are not mersenne primes a simplified proof of the main results of hargraves may be found in fleischmann s paper 3 in s gow shows that the conclusion holds when a is nilpotent char k j j a 1 and both 1 a i and char k are odd in io turull studies the case in which a is supersolvable and v is a completely reducible a module theorem 2 2 of lo extends berger s result in z the author studies a particular situation in which char k 1 1 a it is proven that if h k is a p chief factor of a solvable group g p odd and a is a p subgroup of g acting faithfully on h k then h k contains a regular a orbit the nilpotent case is open in this note we suppose that v is endowed with a g invariant nonsingular symplectic form this allows us to obtain stronger results

blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

large orbits of odd order subgroups of solvable linear groups abstract suppose that g is a finite solvable group and v is a finite faithful and completely reducible g module let h be an odd order subgroup of g then h has at least two regular orbits on v v suppose in addition that v is odd then there exists v v in a regular orbit of f g h such that c h v f 2 g let g be a solvable group h be an odd order hall subgroup of g v be a faithful g module over possibly different finite fields of odd characteristic and assume that v o g is completely reducible then there exists v v such that c h v o g

the existence of regular orbits a well known result of berger see theorem 2 2 of 2 assures that if a is a nilpotent c c free group for all primes p and v is a completely reducible k module acted faithfully by a then v contains a regular a orbit i e there exists v e v with c v 1 this result was improved by hargraves in 7 showing that it suffices to assume that a is c 1 c free for p 2 and p a mersenne prime her proof has been recently simplified by fleischmann in 4 berger s result has been improved in a different direction by turull in lo who extended it to the supersolvable case with extra assumptions if a is nilpotent and both char k and ial are odd then the result holds without any restriction as gow showed in s all these results assume the complete reducibility of v in fact if a is nilpotent they suppose that char k t ial the purpose of this note is to consider the modular case in an important particular situation we prove that if a is a p subgroup of a solvable group g and h k is a p chief factor of g acted faithfully by a then h k contains a regular a orbit provided that a is c2 2 c free when p 2 combining this with the theorems of hargraves and gow we obtain several corollaries one of them is the following in a group of odd order every p subgroup has a regular orbit on each chief factor on which it acts faithfully easily constructed examples show that the analogues of the berger and gow theorems are false when char k 1 jai thus some restriction seems to be necessary we do not know if our theorem is true when p group is replaced by nilpotent we give three applications of our theorem the first is about fixed point free automorphism groups a well known situation is the following suppose ac is a solvable group gaag ai igl l v is a kag module char k 1 ial and c a o then if v is not a primitive kag module we may obtain a nontrivial subgroup of a centralizing a section of g see proposition 4 1 of io nothing seems to be known when ia 1 ici 1 we obtain a weak analogue of turull s result in our proposition 2 1 the main obstruction to obtain a quite similar result in

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

the existence of regular orbits for nilpotent groups consider the following situation suppose r a cr where r is an r group and c is a cyclic group of order qe where r and q are distinct primes let k be a finite field of characteristic p and suppose v is a faithful irreducible k cr module this situation has been widely studied when p q it is similar to the situation occurring in the proof of theorem b of hall and higman 9 when p q it is similar to the situation occurring in the corresponding theorem of shult 131 in this paper a generalization of this situation is considered suppose g 4 ag where g is a solvable group a is nilpotent and ia g 1 again we assume v is a faithful irreducible k ag module two related questions arise in the study of this situation

zeros of brauer characters and linear actions of finite groups small primes abstract we describe the finite groups whose p brauer character table for p 2 or p 3 does not contain any zero this completes the analysis in 6 where we considered the case p 5

p parts of character degrees and the index of the fitting subgroup abstract in a solvable group g if p 2 does not divide 1 for all irr g then we prove that g f g p p 2 this bound is best possible

nilpotent lie algebras of maximal rank and of kac moody type e6 1 let be the kac moody algebra associated to the affine cartan matrix f 4 1 each nilpotent lie algebra of type f 4 1 is isomorphic to a quotient of the positive part of we determine the isomorphism classes of nilpotent lie algebras of type f 4 1

zeros of brauer characters and linear actions of finite groups let g be a finite group and p a prime number greater than 3 it is known that if every irreducible p brauer character of g does not vanish on any p element of g then g is solvable the primary aim of this work is to describe the structure of groups satisfying the above condition among other more specific properties we show that the p length of g is at most 2 the bound being the best possible the structural results are obtained as an application of the main theorem in this paper that deals with particular linear actions of solvable groups on finite vector spaces

nilpotent lie algebras of kac moody affine type abstract the nilpotent lie algebras are grouped into kac moody types simple affine hyperbolic we study the affine type and show that it can be reduced to the simple type concerning the discrete series whereas the continuous families require in addition the study of grassmanians the explicit list for the rank 2 illustrates the whole article

large orbits in coprime actions of solvable groups let g be a solvable group of automorphisms of a finite group k if g and k are coprime then there exists an orbit of g on k of size at least g it is also proved that in a solvable group the largest normal subgroup is the intersection of at most three hall subgroups

finite groups with regular orbits on vector spaces in this article we investigate the following situation which is interesting for example in the theory of character degrees let a be a finite group f q the finite field with q p s elements and v a finite dimensional f q a module during the whole article we assume if c a v denotes the kernel of the representation we ask whether or not a c a v has a regular orbit as a permutation group on v 0 that is we look for an element v e v 0 such that c a v c a v where c a v denotes the centralizer in a of v e v elementary methods using clifford theory lead to a theorem concerning the direct product of relatively prime groups as an application we prove theorems on nilpotent groups which were received with other methods by b bailey hargraves in 1980 j algebra 72 1981

class 2 quotients of solvable linear groups abstract let g be a finite group and let v be a completely reducible faithful g module by a result of glauberman it has been known for a long time that if g is nilpotent of class 2 then g v in this paper we generalize this result as follows assuming g to be solvable we show that the order of the maximal class 2 quotient of g is strictly bounded above by v

regular orbits of finite primitive solvable groups abstract suppose that a finite solvable group g acts faithfully irreducibly and quasi primitively on a finite vector space v then g has a uniquely determined normal subgroup e which is a direct product of extraspecial p groups for various p and we denote e e z e we prove that when e 5 6 7 g will have regular orbits on v this result combined with a previous result of the author will give a complete list of e such that g will have regular orbits on v

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

on the brauer feit bound for abelian defect groups we improve the brauer feit bound on the number of irreducible characters in a p block for abelian defect groups by making use of halasi and podoski every coprime linear group admits a base of size two http arxiv org abs 1212 0199v1 7 and kessar and malle ann math 178 2 321 384 11 we also prove brauer s k b conjecture for 2 blocks with abelian defect groups of rank at most 5 and 3 blocks and 5 blocks with abelian defect groups of rank at most 3

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

orbits of the actions of finite solvable groups abstract suppose v is a completely reducible faithful g module for a solvable group g we show g has a large orbit on v specifically there is v v such that c g v is contained in a normal subgroup of derived length 9 contained in the seventh ascending fitting subgroup of g this is in some ways best possible this is applied to generate many theorems showing that a solvable group must have characters of large degree

invariant eigendistributions on a semisimple lie algebra and homology classes on the conormal variety ii representations of weyl groups abstract we construct representations of w in the homology of certain subvarieties of the cotangent bundle of the flag manifold of g in particular in the homology of the conormal variety and of springer varieties using the integral formula we prove that in top degree the former is isomorphic with the natural representation of w on coherent families of invariant eigendistributions and decomposes into a sum of the latter according to the decomposition of the nilpotent variety into g0 orbits as consequences we obtain springer s theorem on irreducible representations of weyl groups a formula for irreducible characters of g0 as integrals over characteristic varieties of d modules an identification of the harmonic polynomials occuring in the asymptotic expansion at zero of invariant eigendistributions as homology classes on the flag manifold a formula for the fourier transforms of nilpotent g0 orbits and a proof of a conjecture of joseph on the characteristic polynomials of intersections of nilpotent g0 orbits with n0

direct products of groups and regular orbits abstract we prove the following result let p be a prime and let g be a finite p solvable group that is a direct product of two non cyclic subgroups of coprime order and let v be some faithful irreducible module for g over some field in characteristic p then g has a regular orbit that is there exists some v v such that c g v 1

primitive characters of odd order groups abstract let g be a finite group of odd order we show that if is an irreducible primitive character of g then for all primes p dividing the order of g there is a conjugacy class such that the p part of 1 divides the size of that conjugacy class we also show that for some classes of groups the entire degree of an irreducible primitive character divides the size of a conjugacy class

springer s weyl group representations through characteristic classes of cone bundles on realise le groupe symetrique s n comme le groupe des permutations des valeurs propres des matrices diagonales pour l identifier au groupe de weyl w de g sl n c

the regular module problem i in the study of induced representations the following problem arises let h ag be a finite solvable group and k a field with chark a let v be an irreducible faithful primitive k 4g module suppose h contains a normal extraspecial r subgroup r with z r z h and that a acts faith fully on r under what conditions does a have a regular direct summand in vi in this paper we consider this question under the hypotheses that g mr where 1 m is normal abelian in am a is nilpotent a mr m r 1 and r z r is a faithful irreducible am modu e we show that a has at least three regular direct summands in v unless a exp m and r satisfy certain very restrictive conditions let h ag be a finite solvable group where a is nilpotent g is normal and y4 g 1 let k be a field of characteristic not dividing a and v a zc module assume that v is faithful and irreducible the study of induced representations often leads to the following two questions 1 regular orbit problem under what conditions does a have a regular orbit on f