Poincaré series forSO(n, 1)

A theory of Poincaré series is developed for Lobachevsky space of arbitrary dimension. For a general non-uniform lattice a Selberg-Kloosterman zeta function is introduced. It has meromorphic continuation to the plane with poles at the corresponding automorphic spectrum. When the lattice is a unit group of a rational quadratic form, the Selberg-Kloosterman zeta function is computed explicitly in terms of exponential sums. In this way a non-trivial Ramanujan-like bound analogous to “Selberg’s 3/16 bound” is proved in general.     

Lagrangian construction of the enveloping algebraU(sln)

The Institute of the Earth Physics, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 1, pp. 64–66, January–March, 1992.     

On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1)

LetG _o be a non compact real semisimple Lie group with finite center, and letU U(g) ^K denote the centralizer inU U(g) of a maximal compact subgroupK _o ofG _o. To study the algebraU U(g) ^K , B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG _o=K _o A _o N _o ofG _o, adapted toK _o. WhenP is restricted toU U(g) ^K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g) ^K intoU(k) ^M ⊗U(a). HereU(k) ^M denotes the centralizer ofM _o inU(k),M _o being the centralizer ofA _o inK _o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g) ^K inU(k)^M×U(a). In this paper we describe such image whenG _o=SO(n,1)_e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g) ^K , and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG _o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case. Supported in part by Fundación Antorchas     

Contractions ofSU(1, n) andSU(n+1) via Berezin quantization

We establish contractions of discrete series representations ofSU(1,n) and of unitary irreducible representations ofSU(n+1) to the unitary irreducible representations of the (2n+1)-dimensional Heisenberg group by use of the Berezin calculus on the coadjoint orbits associated to these representations by the Kirillov-Kostant method of orbits.     

On strictly Deza graphs with parameters (n,k,k−1,a)

A non-empty $k$-regular graph $\Gamma$ on $n$ vertices is called a Deza graph if there exist constants $b$ and $a$ $(b\ge a)$ such that any pair of distinct vertices of $\Gamma$ has either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\Gamma$ and are written as the quadruple $(n,k,b,a)$. If a Deza graph has diameter 2 and is not strongly regular, then it is called a strictly Deza graph. In the present paper, we investigate strictly Deza graphs whose parameters $(n,k,b,a)$ satisfy the conditions $k=b+1$ and $\frac{k(k-1)-a(n-1)}{b-a}>1$.     

$A(n,k)$和$P(n,k)$的精确公式

设A(n,k)表示不定方程的非负整数解的个数,P(n,k)为整数n分为k个部分的无序分拆的个数,每个分部不小于1．本文给出了A(n,k)和P(n,k)的精确表达式．     

A property of (v,k, λ)-designs

It is shown that if the subsetsX ₁,...,X _v of a setX form a (v, k, λ)-design, then there does not exist another subsetX _v+1 ofX havingany cardinalityk ₁ and intersecting each of theX _j, 1≦j≦v, inany number λ₁ of elements, where 0<k ₁<v and 0<λ₁<k (in order to avoid uninteresting cases).     

On the Action of the Groups GL(n+1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) on PG(n, q t)

Let q be a prime power and let n ≥ 0, t ≥ 1 be integers. We determine the sizes of the point orbits of each of the groups GL(n + 1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) acting on PG(n, q ^t) and for each of these sizes (and groups) we determine the exact number of point orbits of this size.     

Second-order differential invariants of the rotation group O(n) and of its extensions: E(n), P(1, n), G(1, n)

Functional bases of second-order differential invariants of the Euclid, Poincaré, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant equations.     

Weak amenability of the universal covering group ofSU(1,n)

This work was completed, while the author had a research fellowship from the Mittag-Leffler Institute in Stockholm. He would like to thank the institute for its support and kind hospitality during the special year 1988/89 of operator algebras. The author would like to thank professor Uffe Haagerup for much useful and inspiring avice     

On the series ∑ k=1 ∞ ( k 3k )−1 k −n x k

In this paper we investigate the series ∑ _k=1 ^∞ ( _k ^3k )⁻¹ k ⁻ⁿ x ^k . Obtaining some integral representations of them, we evaluated the sum of them explicitly forn = 0, 1, 2.