On monomial representations of finitely generated nilpotent groups

Introduction. In applying symmetry considerations to quantum mechanics, one is often forced to consider representations T of a symmetry group G by means of Hilbert space operators which may be either unitary or conjugate-unitary. Indeed, in some situations there is a well-determined subgroup N of G (necessarily normal and of index 2 in G) such that T, must be unitary for x E N and conjugate-unitary for x E G N. Such a representation T will be called a conjugating representation of G (relative to N). More generally one can consider projective conjugating representations T, in which the homomorphism relation TT, = T, is replaced by Tj, = A(x, y)Txy (the A(x, y) being complex scalars). These also are required in the applications to quantum mechanics. In ?1, by a small modification of Mackey's analysis, we shall classify the equivalence classes of irreducible projective conjugating representations of G in terms of the irreducible projective representations of N (provided N is of Type I). As an immediate application of this, we obtain in ?2 a classification of all the irreducible unitary representations of any Type I group acting in real or quaternionic Hilbert space, in terms of those acting in complex Hilbert space. In working out this classification for a given group G, the essential step is to subdivide the self-conjugate irreducible complex representations T of G into two classes-those of real type and those of quaternionic type-according as the conjugate-linear map setting up the equivalence of T with its conjugate has positive or negative square. Exactly the same results are obtained if, instead of unitary (not necessarily finite-dimensional) representations, we consider finite-dimensional (not necessarily unitary) representations. The remaining sections of this paper concern finite-dimensional representations of semisimple groups. Fix a connected semisimple Lie group G. The family G of all finite-dimensional (not necessarily unitary) irreducible complex representations of G is parametrized by the well-known Cartan-Weyl method of dominant weights. Which of the self-conjugate elements of Gi are of real type and which are of quaternionic type? In ??3 and 4 this question is answered in two steps: First, Theorem 5 reduces the general case to the case that G is compact. It turns out that, if G is not compact, each self-conjugate element T of G gives rise to a self-conjugate element

On Schur multiplier and projective representations of Heisenberg groups

Let G be a connected semi-simple Lie group with finite center, say — then it is not necessarily the case that G possesses enough finite dimensional representations to separate points (cf. 3.1.1). In fact G will, in general, admit no non-trivial finite dimensional unitary representations (cf. number 4.3.2). Nevertheless, despite these apparently discouraging facts, the finite dimensional representations of G are important — this will be elaborated on below.

In an earlier paper [P1]; we studied self-dual complex representations of a finite group of Lie type. In this paper; we make an analogous study in thep-adic case. We begin by recalling the main result of that paper. LetG(F) be the group of F rational points of a connected reductive algebraic group G over a finite field F. Fix a Borel subgroup B of G; defined over F; which always exists by Lang’s theorem. LetBDTU be a Levi decomposition of the Borel subgroupB. Suppose that there is an element t02 T(F) that operates byi1 on all the simple roots in U with respect to the maximal split torus inT. It can be seen thatt 2 belongs to the center ofG. We proved in [P1] thatt 2 operates by 1 on a self-dual irreducible complex representation …; which has a Whittaker model; if and only if… carries a symmetricG-invariant bilinear form. If … carries a symmetric G-invariant bilinear form; then we call … an orthogonal representation. If … is an irreducible admissible self-dual representation of a p-adic group G; then there exists a nondegenerateG-invariant bilinear formB: …£…! C. This form is unique up to scalars by a simple application of Schur’s lemma; and it is either symmetric or skew-symmetric. The aim of this paper is to provide for a criterion to decide which of the two possibilities holds in the context of p-adic groups similar to our work in the finite field case in [P1]. This partly answers a question raised by Serre (see [P1]). We are able to say nothing about representations not admitting a Whittaker model. Our method; however; works in some cases even when there is no element inT which operates on each simple root byi1. This is the case; for instance; in some cases of SLn andSpn; where we work instead with the similitude group where such an element exists; then we deduce information on SLn andSpn.

The paper studies the structure of J-unitary representations of connected nilpotent groups on \(\Pi _{k}\)-spaces, that is, the representations on a Hilbert space preserving a quadratic form “with a finite number of negative squares”. Apart from some comparatively simple cases, such representations can be realized as double extensions of finite-dimensional representations by unitary ones. So their study is based on some special cohomological technique. We concentrate mostly on the problems of the decomposition of these representations and the classification of “non-decomposable” ones.

Journal Article The space of degenerate Whittaker models for general linear groups over a finite field Get access Dipendra Prasad Dipendra Prasad Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2000, Issue 11, 2000, Pages 579–595, https://doi.org/10.1155/S1073792800000313 Published: 01 January 2000 Article history Received: 20 October 1999 Published: 01 January 2000 Revision received: 25 January 2000

Complex Conjugation of Group Representations by Inner Automorphisms

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

In an earlier work [M. Havlı́ček et al., J. Math. Phys. 40, 2135 (1999)] we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation Uq′(so3) of the Lie algebra so3 where q is not a root of unity [for each dimension only one of them (called classical) admits limit q→1]. In the first part of this paper we show that any finite-dimensional irreducible representation is equivalent to some of these representations. In the case qn=1 we derive new Casimir elements of Uq′(so3) and show that a dimension of any irreducible representation is not higher than n. These elements are Casimir elements of Uq′(som) for all m and even of Uq′(isom+1) due to Inönu–Wigner contraction. According to the spectrum of one of the generators, the representations are found to belong to two main disjoint sets. We give full classification and explicit formulas for all representations from the first set (we call them nonsingular representations). If n is odd, we have full classification also for the remaining singular case with the exception of a finite number of representations.

In the note, a 30-year old result concerning the von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected and the smallest von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional (not necessarily continuous) complex unitary representations of the group) is described.

The concepts of semi-direct product, quasi semi-direct product and the method of constructing quasisymmetry (P-symmetry)groups [Krishnamurty, Prasad & Rama Mohana Rao (1978). J. Phys. A, 11, 805-811; (1980). J. Phys. A, 13, 1947-1956] have been explored and a general method of constructing quasisymmetry (P-symmetry) groups with the crystallographic space groups as generators is suggested. The study is restricted to only the cubic system for the chosen boundary condition T 2 = T 2 = T 2 = E. The minor quasisymmetry cubic space groups so obtained are associated with the one-dimensional complex and two-dimensional real irreducible representations of the generator groups using the ideas of little groups and their one-dimensional allowable irreducible representations. The symmorphic cubic space groups F23, F432 and the non-symmorphic cubic space groups Fd3, Fd3m are exemplified. For the rest of the cubic space groups the results obtained are tabulated. Some suggestions have been made as to the possible studies in which the groups obtained here can be applied.

Smith theory and Hecke operators

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.