Compactly Generated Relative Stable Categories

Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated by the class of finitely generated modules. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. This is also a triangulated category, but no non-trivial examples have been known where it was compactly generated. While the finitely generated modules are compact objects, they do not necessarily generate the category. We show that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.     

Hausdorff dimension in a family of self-similar groups

For every prime p and every monic polynomial f, invertible over p, we define a group G _{p, f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group . We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a characterization of finitely constrained groups in terms of the branching property, and as a corollary conclude that all defined groups are finitely constrained. In addition, we show that all infinite, finitely constrained groups of p-adic automorphisms have positive and rational Hausdorff dimension and we provide a general formula for Hausdorff dimension of finitely constrained groups. Further “finiteness” properties are also discussed (amenability, torsion and intermediate growth). Partially supported by NSF grant DMS-0600975.     

Virtually free factors of pro-p groups

Letp be a prime number,G a pro-p group, andH a closed (topologically) finitely generated subgroup ofG. We give conditions under whichH is virtually a free factor ofG, i.e., that there exists an open subgroupU ofG such thatU is the free pro-p product ofH and some other subgroup ofU. We prove that this happens if eitherG is a free pro-p group of any rank, or ifG is a free pro-p product of finitely generated pro-p groups. Research supported in part by grants from NSERC (Canada) and DGICYT (Spain).     

Injectivity, Projectivity and Supercuspidal Representations

Let G be a reductive p-adic group. Consider the category ofsmooth (complex) representations of G in which a (fixed) closedcocompact subgroup of the centre acts by a (fixed) character.It is well known that the supercuspidal representations in thiscategory are both injective and projective. It is shown that,conversely, an admissible injective or projective object isnecessarily supercuspidal.     

Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties

Let G be a reductive group defined over a p-adic local field L, let P be a parabolic subgroup of G with Levi quotient M, and write G:=G(L), P:=P(L), and M:=M(L). In this paper we construct a functor JP from the category of essentially admissible locally analytic G-representations to the category of essentially admissible locally analytic M-representations, which we call the Jacquet module functor attached to P, and which coincides with the usual Jacquet module functor of [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] on the subcategory of admissible smooth G-representations. We establish several important properties of this functor.     

Representing endomorphisms and principal congruences

For every algebraU there is an algebraU ^* with (up to isomorphism) the same endomorphism, subalgebra and congruence structure as that ofU, for which every finitely generated subalgebra and every finitely generated congruence ofU ^* is singly generated. The theorem is proved in a somewhat more general category theoretic context.Presented by R. W. Quackenbush.This author's research was supported by an OTKA grant from Hungary.This author's research was supported by NSERC, The Natural Sciences and Engineering Research Council of Canada.     

Ramification de certaines extensions de Lie p-adiques

Let G be a p-adic Lie group and let K be a finite extension of the p-adic number field ℚ _p . There are finitely many filtrations of G which could be ramification filtrations of totally ramified Galois extensions of K with Galois group G. Received: 19 October 1998     

Characters,genericity, and multiplicity one for U(3)

Let ψ:U→ℂ be a generic character of the unipotent radicalU of a Borel subgroup of a quasisplitp-adic groupG. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation π ofG was expressed by Rodier in terms of the limit of values of the trace of π at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in [F, p. 47] to provide a local proof of the multiplicity one theorem for U(3). This assert that each discrete spectrum automorphic representation of the quasisplit unitary group U(3) associated with a quadratic extensionE/F of number fields occurs in the discrete spectrum with multiplicity one. It is pointed out in [F, p. 47] that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in [F, p. 48] at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here. Partially supported by a Lady Davis Visiting Professorship at the Hebrew University and the Max-Planck-Institut für Mathematik, Bonn.     

K0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoreticquestions related to the representation theory of the completedgroup algebra kG of G with coefficients in a finite field kof characteristic p. We show that if M is a finitely generatedkG-module with canonical dimension smaller than the dimensionof the centralizer, as a p-adic analytic group, of any p-regularelement of G, then the Euler characteristic of M is trivial.Writing _i for the abelian category consisting of all finitelygenerated kG-modules of dimension at most i, we provide an upperbound for the rank of the natural map from the Grothendieckgroup of _i to that of _d, where d denotes the dimension of G.We show that this upper bound is attained in some special cases,but is not attained in general.     

Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${\cal C}$ Grou ps

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FP_m. This generalises work of J. King for ¹(G) in the case when G is metabelian [9].Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on ¹(G) and on the automorphism group of G.Both authors are partially supported by CNPq, Brazil.     

Limit distributions of expanding translates of certain orbits on homogeneous spaces

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U⁺ ={gεG:a ⁻ⁿ gaⁿ} →e as n → ∞. Let Ω be a non-empty open subset ofU ⁺ andn _i → ∞ be any sequence. It is showed that . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.     

Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.     

Finitely Presented Wreath Products and Double Coset Decompositions

We characterize which permutational wreath products $G \ltimes W^{(X)}We characterize which permutational wreath products are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X ². On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.      

The unitary group over the integers of an unramified local field

Let L be a lattice over the integers of a local field F which has a nontrivial involution. Then U⁺(L) (the subgroup of rotations of the unitary group U(L)) is generated by unitary transvections and quasitransvections contained in U⁺(L) (Theorem 7.8). Let g be a tableau. Then the mixed commutator subgroup of U⁺(L) and U(g) (the congruence subgroup of U⁺(L) corresponding to g) equals E(g) (the subgroup generated by unitary transvections and quasitransvections with orders contained in g) (Theorem 7.7). Finally, let G be a subgroup of U⁺(L) with _o(G) = g, then G is a normal subgroup of U⁺(L) if and only if U(g) G E(g).     

Multivariable Difference Dimension Polynomials

Let K be a difference field with basic set of translations σ and let a partition of the set σ into p disjoint subsets be fixed. Let be the ring of difference operators over K treated as a filtered ring with the natural p-dimensional filtration. We introduce a special type of reduction in a finitely generated free -module and develop the corresponding technique of characteristic sets that allows one to prove that the Hilbert function in p variables associated with a finitely generated difference K-module is polynomial. We also outline a method of calculation of such a function and obtain a generalization of the author's theorem on difference dimension polynomials. In conclusion, we develop a similar technique of characteristic sets for the ring of difference polynomials and prove a theorem on multivariable difference dimension polynomials associated with a finitely generated difference field extension. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Classification des Représentations Tempérées d’un Groupe p-Adique non Connexe

Let G be the group of points defined over a p-adic field of a non-connected reductive group. In this note, we prove that every tempered irreducible representation of G is irreducibly induced from an essential one of a cuspidal Levi subgroup of G.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups*

Let be a class of groups. A subgroup H of a group G is called -s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ H_G belongs to where H_G is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to study some subgroups of Fitting subgroup and generalized Fitting subgroup -s-supplemented and some new criterions of p-nilpotency of finite groups are obtained. *This research is supported by the grant of NSFC and TianYuan Fund of Mathematics of China (Grant #10626047).     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

The Kernel of the Adjoint Representation of a p-Adic Lie Group Need Not Have an Abelian Open Normal Subgroup

Let G be a p-adic Lie group with Lie algebra 𝔤 and Ad: G → Aut(𝔤) be the adjoint representation. It was claimed in the literature that the kernel K?ker(Ad) always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false. It can even happen that K = G, but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.