The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary r-fold atypical gl m|n-Kac-module and the set of composition factors of some r-fold atypical gl r|r-Kac-module. The result of...

LetO be the orbit of η ∈ g∗ under the coadjoint action of the compact Lie group G. We give two formulae for calculating the action integral along a closed Hamiltonian isotopy on O. The first one expresses this action in terms of a particular character of the isotropy subgroup of η. In the second one is involved the character of an irreducible representation of G.

We study the representation theory of the W -algebra Wk(ḡ) associated with a simple Lie algebra ḡ at level k. We show that the “−” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k ∈ C. Moreover, we show that the character of each irreducible highest weight representation of Wk(ḡ) is completely determined by that of the corresponding irre...

where St and S2 are projective representations of G. An irreducible projective representation is one that is not reducible. In this paper, we study the degrees of irreducible projective representations of a finite group G over an algebraically closed field K. In Theorem 2, we require further that char KJ(\G\. The degree of an irreducible projective representation of a finite group G, over the c...

For a finite group G, let cd(G) be the set of irreducible complex character degrees of G forgetting multiplicities and X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Suppose that p is a prime number. We prove that if G is a finite group such that |G| = |PGL(2, p)|, p ∈ cd(G) and max(cd(G)) = p+1, then G ∼= PGL(2, p), SL(2, p) or PSL(2, p) × A, where ...

We consider both local and global theta correspondences for GSp4 and GSO4,2. Because of the accidental isomorphism PGSO4,2 PGU2,2, these correspondences give rise to those between GSp4 and GU2,2 for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on GU(2, 2) by the...

Let K be a totally real number field, π an irreducible cuspidal representation of GL2(K)\GL2(AK) with unitary central character, and χ a Hecke character of conductor q. Then L(1/2, π ⊗ χ) ≪ (Nq) 1 2 − 1 8 , where 0 6 θ 6 1/2 is any exponent towards the Ramanujan– Petersson conjecture (θ = 1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a genera...

The number of shortest factorizations into reflections for a Singer cycle in GLn(Fq) is shown to be (qn − 1)n−1. Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and...

In this paper we obtain the orthogonality relations for the supergroup U(m|n), which are remarkably different from the ones for the U(N) case. We extend our results for ordinary representations, obtained some time ago, to the case of complex conjugated and mixed representations. Our results are expressed in terms of the Young tableaux notation for irreducible representations. We use the supersy...

where k (p) is the +1 (−1) eigenspace of θ on g. Let Z(g) be the center of the universal enveloping algebra of g, and fix a character χ of Z(g). Let HC be the category of Harish-Chandra modules for G (i.e. finitely generated (g,K)–modules), and let HCχ be the full subcategory of HC of the modules with generalized central character χ. We are interested in the distribution characters, say, of the...