Let G be a connected noncompact simple Hermitian symmetric group with nite center. Let H() denote the geometric realization of an irreducible unitary highest weight representation with highest weight. Then H() consists of vector-valued holomorphic functions on G=K and the action of G on H() is given in terms of a factor of automorphy. For highest weights corresponding to ladder representations,...

In this paper, we study irreducible unitary representations of GLn(R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, thereby generalizing results of Kostant, Matumoto and others. Our second result relates the annihilator to the sequence of adduced representations,...

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to ot...

Abstract. We propose the fundamental and two dimensional representation of the Lorentz groups on a (3+1)-dimensional hypercubic lattice, from which representations of higher dimensions can be constructed. For the unitary representation of the discrete translation group we use the kernel of the Fourier transform. From the Dirac representation of the Lorentz group (including reflections) we deriv...

0.1. Let Γ be a finite subgroup of SU(2). The question we will deal with in this paper is how an arbitrary (unitary) irreducible representation of SU(2) decomposes under the action of Γ. The theory of McKay assigns to Γ a complex simple Lie algebra g of type A−D−E. The assignment is such that if Γ̃ is the unitary dual of Γ we may parameterize Γ̃ by the nodes (or vertices) of the extended Coxeter-...

Mazur’s principle gives a criterion under which an irreducible mod l Galois representation arising from a classical modular form of level Np (with p prime to N) also arises from a classical modular form of level N . We consider the analogous question for Galois representations arising from certain unitary Shimura varieties. In particular, we prove an analogue of Mazur’s principle for U(2, 1) Sh...

In this paper we study the length of the longest induced cycle in the unitary Cayley graph Xn = Cay(Zn;Un), where Un is the group of units in Zn. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of eac...

Theta correspondence y over R is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that y preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show...

Harish-Chandra Collected Papers Edited by V. S. Varadarajan Volume 1: 1944–54 pp. 566 Volume 2: 1955–58 pp. 539 Volume 3: 1959–68 pp. 670 Volume 4: 1970–83 pp. 461 Group-representation theory is a broad topic which impinges on many domains of mathematics. The characters of abelian groups are of course central to Fourier analysis and entered at an early stage the theory of numbers, one of severa...