Let M be a Harish-Chandra module associated to a finite length, admissible representation of real reductive Lie group G 0 . Suppose that p is a parabolic subalgebra of the complexified Lie algebra of G 0 and let n ⊂ p be the nil radical of p. In this paper, motivated by some recent work in the study of zeta functions on locally symmetric spaces, we make a comparison between homological properties of M and homological properties of the minimal globalization of M . In particular, if p has a real Levi factor, we are able to show that, after conjugating by an element from G 0 , then the n-homology groups of the minimal globalization of M are, in a natural way, the minimal globalizations of the n-homology groups of M .

Arrangement graphs were introduced for their connection to computational networks and have since generated considerable interest in the literature. In a pair of recent articles by Chen, Ghorbani and Wong, the eigenvalues for the adjacency matrix of an (n,k)-arrangement graph are studied and shown to be integers. In this manuscript, we consider the adjaceny matrix directly in terms of the representation theory for the symmetric group. Our point of view yields a simple proof for an explicit fomula of the associated spectrum in terms of the characters of irreducibile representations evaluated on a transposition. As an application we prove a conjecture raised by Chen, Ghorbani and Wong.

In a recent preprint Kodiyalam and Verma give a particularly simple Gelfand model for the symmetric group that is built naturally on the space of involutions. In this manuscript we give a natural extension of Kodiyalam and Verma's model to a Gelfand model for Weyl groups of type Bn and D2n+1. Then we define an explicit isomorphism between this Gelfand model and the polynomial model using a technique we call telescopic decomposition.

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In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).