The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played an important role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc arguments. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set $\mathcal{M}\left(\mathcal{G}\right)$ of each two-sided cell.

We give a necessary condition for Galois covering singularities to be logterminal or $\log$ -canonical singularities, which is also sufficient under acertain restriction on the branch loci of the covering maps. We also give a method constructing explicitly resolutions of 2-dimensional Abel covering singularities.

For $G=S^3\times \text{...}\times S^3$, let $X$ be a space such that the $p$-completion $(X{)}_p^{\wedge }$ is homotopy equivalent to (BG)${)}_p^{\wedge }$ for any prime $p$. We investigate the monoid of rational equivalences of $X$, denoted by ${ϵ}_0\left(X\right)$. This topological question is transformed into a matrix problem over $Q\otimes Z^{\wedge }$, since ${ϵ}_0\left(BG\right)$ is the set of monomial matrices whose nonzero entries are odd squares. It will be shown that a submonoid of ${ϵ}_0\left(X\right)$, denoted by ${\delta }_0\left(X\right)$, determines the decomposability of $X$. Namely, if $N_{odd}$ denotes the monoid of odd natural numbers, Theorem 2 shows that the monoid ${\delta }_0\left(X\right)$ is isomorphic to a direct sum of copies of $N_{odd}$. Moreover the space $X$ splits into $m$ indecomposable spaces if and only if ${\delta }_0\left(X\right)\cong (N_{odd}{)}^m$. When such aspace $X$ is indecomposable, the relationship between $[X,X]$ and $[BG,BG]$ is discussed. Our results indicate that the homotopy set $[X,X]$ contains less maps if $X$ is not homotopy equivalent to the product of quaternionic projective spaces $BG=H{P}^{\infty }\times \text{...}\times H{P}^{\infty }$.

If apolynomial map $f$ : $C^n\to C$ has anice behaviour at infinity (e.g. it is a "good polynomial"), then the Milnor fibration at infinity exists; in particular, one can define the Seifert form at infinity $\Gamma \left(f\right)$ associated with $f$. In this paper we prove a Sebastiani-Thom type formula. Namely, if $f$ : $C^n\to C$ and $g:c^m\to C$ are "good" polynomials, and we define $h=f$$\oplus$$g$ : $C^{n+m}\to C$ by $h(x,y)=f\left(x\right)+g\left(y\right)$, then $\Gamma \left(h\right)=(-I{)}^{mn}\Gamma (f)$$\otimes \Gamma \left(g\right)$. This is the global analogue of the local result, proved independently by K. Sakamoto and P. Deligne for isolated hypersurface singularities.

We prove that if $X$ is aconnected $H$-space with at most three cells of positive dimension, then the self homotopy set of $X$ becomes a group relative to the binary operation induced from any multiplication on $X$, and we determine it's group structure in some cases.

In this paper we give an explicit Fourier expansion of the Eisenstein series on certain quaternion unitary groups of degree 2 by means of Shimura's method. Moreover using an explicit formula of the Fourier coefficients of holomorphic Eisenstein series and Oda's lifted cusp forms, we give some numerical examples.

Let $\mathfrak{g}$ be acomplex semisimple Lie algebra with symmetric decomposition $g=\mathfrak{k}+\mathfrak{p}$. For each irreducible Harish-Chandra $(g,\mathfrak{k})$-module $X$, we construct a family of nilpotent Lie subalgebras $\mathfrak{n}\left(\mathcal{O}\right)$ of $\mathfrak{g}$ whose universal enveloping algebras $U\left(\mathfrak{n}\right(\mathcal{O}\left)\right)$ act on $X$ locally freely. The Lie subalgebras $\mathfrak{n}\left(\mathcal{O}\right)$ are parametrized by the nilpotent orbits $\mathcal{O}$ in the associated variety of $X$, and they are obtained by making use of the Cayley tranformation of $s{I}_2$-triples(Kostant-Sekiguchi correspondence). As aconsequence, it is shown that an irreducible Harish-Chandra module has the possible maximal Gelfand-Kirillov dimension if and only if it admits locally free $U\left({\mathfrak{n}}_m\right)$-action for ${\mathfrak{n}}_m=\mathfrak{n}\left({\mathcal{O}}_{\max }\right)$ attached to aprincipal nilpotent orbit ${\mathcal{O}}_{\max }$ in $\mathfrak{p}$.

In this article, let $k\equiv O$ or 1(mod4) be a fundamental discriminant, and let $\chi \left(n\right)$ be the real even primitive character modulo $k$. The series $L(1,\chi )={\displaystyle \sum }_{n=1}^{\infty }\frac{\chi \left(n\right)}{n}$ can be divided into groups of $k$ consecutive terms. Let $v$ be any nonnegative integer, $j$ an integer, $0\le j\le k-1$, and let $T(v,j,\chi )={\displaystyle \sum }_{n=j+1}^{j+k}\frac{\chi (vk+n)}{vk+n}$ Then $L(1,\chi )={{\displaystyle \sum }}_{v=0}^{\infty }T(v,0,\chi )={{\displaystyle \sum }}_{n=1}^j\chi \left(n\right)/n+{{\displaystyle \sum }}_{v=0}^{\infty }T(v,j,\chi )$. In section 2, Theorems 2.1 and 2.2 reveal asurprising relation between incomplete character sums and partial sums of Dirichlet series. For example, we will prove that for integer $v\ge \max \{1,\sqrt{k}/|M\left|\right\}$ if $M={{\displaystyle \sum }}_{m=1}^{j-1}\chi \left(m\right)+1/2\chi \left(j\right)\ne 0$ and $\left|M\right|\ge 3/2$. In section 3, we will derive algorithm and formula for calculating the class number of a real quadratic field. In section 4, we will attempt to make a connection between two conjectures on real quadratic fields and the sign of $T(0,20,\chi )$.

The classical Fatou limit theorem was extended to the case of positive harmonic functions on a hyperbolic Riemann surface $R$ by Constantinescu-Cornea. They used extensively the notions of Martin's boundary and fine limit following the filter generated by the base of the subsets of $R$ whose complements are closed and thin at a minimal boundary point of $R$. We shall consider such a problem for positive solutions of the Schrödinger equation on a hyperbolic Riemann surface.

We proved $L_q-L_r$ type estimates of the Stokes semigroup in a two dimensional exterior domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

We classify complete conformally flat three dimensional Riemannian manifolds with constant scalar curvature and constant squared norm of Ricci curvature tensor by applying the Generalized Maximum Principle due to H. Omori.

We consider the group of foliation preserving homeomorphisms of afoliated manifold. We compute the first homologies of the groups for codimension one foliations. Especially, we show that the group for the Reeb foliation on the 3-sphere is perfect and the groups for irrational linear foliations on the torus are not perfect.

We prove an asymptotic formula for ${{\displaystyle \sum }}_{n\le N}r\left(n\right)r(n+m)$ using the spectral theory of automorphic forms and we specially study the uniformity of the error term in the asymptotic approximation when $m$ varies. The best results are obtained under a natural conjecture about the size of a certain spectral mean of the Maass forms. We also employ large sieve type inequalities for Fourier coefficients of cusp forms to estimate some averages (over $m$) of the error term.