In this paper we prove an analogue of Jantzen's sum formula for the $q$-Weyl modules of the $q$-Schur algebra and, as a consequence, derive the analogue of Schaper's theorem for the $q$-Specht modules of the Hecke algebras of type $\bf A$. We apply these results to classify the irreducible $q$-Weyl modules and the irreducible ($e$-regular) $q$-Specht modules, defined over any field. In turn, this allows us to identify all of the ordinary irreducible representations of the finite general linear group $GL_n(q)$ which remain irreducible modulo a prime $p$ not dividing~$q$.

1991 Mathematics Subject Classification: 20C32.

Let $(W,S)$ be a finite Coxeter system and $B=B(W)$ the corresponding Artin-Tits braid group. The natural map $B \rightarrow W$ has a canonical section $r:W \rightarrow B$ defined by the condition that if $w \in W$ is written as a reduced expression in the generators in ~$S$ then $r(w)$ is the corresponding product taken in $B$. The main result of the present paper is as follows. Let $C$ be a conjugacy class in $W$ whose elements have order $d$ say. Then there exists an element $w \in C$ of minimal length in $C$ such that $r(w)^d$ is a product of terms of the form $r(w_I)^{d_I}$ where the following hold: $d_I$ is a non-negative even integer, $I$ runs over a sequence of subsets of $S$ which decreases (which implies that the terms commute), and $w_I$ is the longest element in the corresponding parabolic subgroup of $W$. Such an element will be called a ‘good’ element in $C$. The result is proved case by case, using the classification of irreducible finite Coxeter groups and the knowledge of representatives of minimal length from the article by Geck and Pfeiffer in Advances in Math. 102 (1993) 79-94.The main application of this result concerns the problem of calculating character values of Iwahori-Hecke algebras. The generic Iwahori-Hecke algebra $H$ associated with $(W,S)$ is a quotient of the group algebra of $B$ by the ideal generated by quadratic relations of the form $(s-q)(s+1)$ where $s \in S$ and $q$ is an indeterminate. Thus, $H$ is an algebra over a suitable field of rational functions in the variable ~$q$. The above result implies that if $w$ is a good element in the class $C$ of $W$, then the eigen values of the standard basis element $T_w$ of $H$ in an irreducible representation of $H$ are roots of unity times fractional powers of ~$q$, and the fractional powers occurring can be explicitly determined from the ordinary character table of $W$. This result is used to compute the character table of the Iwahori-Hecke algebra of type $E_8$. To determine the roots of unity, we use additional relations coming from the modular representation theory of ~$H$. This completes the program of determining the character tables of Iwahori-Hecke algebras.

1991 Mathematics Subject Classification: 20C20, 20F36.

We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity

$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$

This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.

1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.

We study the subgroup growth of profinite groups. We obtain a structure theorem for profinite groups of polynomial subgroup growth (PSG groups, for short), which essentially reduces their characterization to the case where the group is a cartesian product of finite simple groups. Analysing the growth behaviour of such cartesian products, we construct, for any real number $\alpha \ge 1$, a PSG profinite group whose degree is exactly $\alpha$. Applications to the behaviour of the abscissa of convergence of the associated zeta function $\sum a_n(G)n^{-s}$ are drawn. We also show that there is no gap between polynomial and non-polynomial subgroup growth by constructing non-PSG groups whose subgroup growth is arbitrarily slow. Our arguments rely heavily on the use of sieve methods in number theory. In particular, a Bombieri-type short intervals theorem and the so-called Fundamental lemma in sieve theory play an essential role in this paper.

1991 Mathematics Subject Classification: 20E07, 11N36.

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field.

When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity.

1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

The aim of the paper is to construct a calculus of pseudodifferential operators (PDOs) on a smooth manifold $M$ without using local coordinate systems. Instead we deal with linear connections $\Gamma$ of $M$.

The fact that a linear connection $\Gamma$ is a global object enables one to associate with a PDO its full symbol, which is a function on the cotangent bundle $T^*M$ (depending on the choice of $\Gamma\,$). This idea was put forward by H. Widom. He has suggested a method of defining full symbols of PDOs on a manifold with a linear connection and constructed a version of symbolic calculus. However, H. Widom defined the classes of PDOs in local coordinates, using standard local phase functions.

On the contrary, in the paper the PDOs are defined in a coordinate-free way, with use of invariant oscillatory integrals over $T^*M\,$. The invariant approach allows us to define $\tau$-symbols of pseudodifferential operators acting on a manifold; in particular, we introduce the Weyl symbols. In fact, it is a kind of quantization on a manifold provided with a linear connection.

The results are applied to study of semi-elliptic differential operators and functions of the Laplacian on a closed Riemannian manifold.

1991 Mathematical Subject Classification: 35S05, 58G30.

We introduce the notion of continuous twisted partial actions of a locally compact group on a C*-algebra. With such, we construct an associated C*-algebraic bundle called the semidirect product bundle. Our main theorem shows that, given any C*-algebraic bundle which is second countable and which satisfies a certain regularity condition (automatically satisfied if the unit fibre algebra is stable), there is a continuous twisted partial action of the base group on the unit fibre algebra, whose associated semidirect product bundle is isomorphic to the given one.

1991 Mathematics Subject Classification: 46L05, 46L40, 46L55.

We construct a $C^*$-algebra version of the theory of distributions. For a $C^*$-algebra $A$ with an action $\alpha$ of $R^n$, the spaces $\cal D_{\alpha}^{\prime}(A)$ and $\cal S_{\alpha}^{\prime}(A)$ are introduced as an analogy of the classical spaces $\cal D^{\prime}(R^n)$ and $\cal S^{\prime}(R^n)$ of distributions on $R^n$. The original algebra $A$ is densely embedded into them as an $A^{\infty}$ (=the subalgebra of $A$ of all smooth elements)-bimodule.

Derivations coming from the action $\alpha$ are extended on them so that all elements of $A$ can be differentiable. Fourier transforms behave well on them. We prove that any member of $\cal D_{\alpha}^{\prime}(A)$ can be expressed as a finite linear combination of finite order derivatives of elements of the weak closure $A^{\prime\prime}$ of $A$. As a result, for any von Neumann algebra $M$ with an action $\alpha$ of $R^n$, the smallest completion of $M$ under differential calculus is given by our distribution space $\cal D_{\alpha}^{\prime}(A_M)$ for an associated $C^*$-algebra $A_M$ with action $\alpha$. The spectrum of an element of $\cal S_{\alpha}^{\prime}(A)$ is defined as support of its Fourier transform, which is a generalization of Arveson's spectrum. We then state the $C^*$-algebra version of the classical Paley-Wiener-Schwartz theorem which characterizes the condition that $\text{Sp}_{\alpha}(x)$ is compact in terms of the possibility of extension of the function $\alpha_t(x), t \in R^n$ to entire function of exponential type.

1991 Mathematics Subject Classification: primary 46L55, 46L89; secondary 46F25.