On restriction to the maximal compact subgroup GL(3, R), an unramified principal series representation of the p-adic group GL(3, F ) decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents.

Let Λ be a minimal Kac–Moody group of rank 2 defined over the finite field Fq , where q = pa with p prime. Let G be the topological Kac–Moody group obtained by completing Λ. An example is G = SL2(K), where K is the field of formal Laurent series over Fq . The group G acts on its Bruhat–Tits building X, a tree, with quotient a single edge. We construct new examples of cocompact lattices in G, ma...

Let G be a split adjoint semisimple group over Q and K∞ ⊂ G(R) a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K∞. This proves a conjecture of Sarnak for Q-split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula.

A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent n ≫ 1 is given in the following form. Let Q be a group of odd exponent n > 2 and G be an HNN-extension of Q. If A ∈ G then let F(A) denote the maximal subgroup of Q which is normalized by A. By τA denote the automorphism of F(A) which is induced by conjugation by A. Suppose that for every A ∈ G, wh...

Let F be a local complete field with discrete valuation, and let G be a quasisplit group over F which splits over some unramified extension of F . Let P be a parahoric subgroup of the group G(F ) of F -points of G; the open normal pronilpotent subgroups of P can be classified using the standard normal filtration subgroups of Prasad and Raghanathan. More precisely, we show that if G is quasisimp...

Given a finite set, X, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is “as big as possible” inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals o...

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann....

1.1. Let K be a nonarchimedean local field and let K be a maximal unramified field extension of K. Let O (resp. O) be the ring of integers of K (resp. K) and let p (resp. p) be the maximal ideal of O (resp. O). Let K = K −{0}. We write O/p = Fq, a finite field with q elements, of characteristic p. Let G be a semisimple almost simple algebraic group defined and split over K with a given O-struct...

Let GR denote the real points of a connected complex reductive algebraic group defined over R). (For the basic results on cells below, this class of groups is unnecessarily restrictive. But we shall need the assumption when we discuss unipotent representations below.) Let gR denote the Lie algebra of GR and write g for the complexification of gR. Let KR denote a maximal compact subgroup in G, a...

There are several approaches addressing the problem of understanding the mechanism of color confinement in non-Abelian gauge theories. The most popular share the idea that only a subset of the degrees of freedom are relevant for confinement. In the Abelian projection approach, one takes into account the maximal Abelian subgroup U(1) of the gauge group SU(N) and monopoles are the effective degre...