Euler-Poincar\'e pairing, Dirac index and elliptic pairing for Harish-Chandra modules

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of equal rank, and let $\mathscr M$ be the category of Harish-Chandra modules for $G$. We relate three differentely defined pairings between two finite length modules $X$ and $Y$ in $\mathscr M$ : the Euler-Poincar\'e pairing, the natural pairing between the Dirac indices of $X$ and $Y$, and the elliptic pairing. (The Dirac index is a virtual finite dimensional representation of $\widetilde K$, the spin double cover of $K$.) Analogy with the case of Hecke algebras and a formal (but not rigorous) computation lead us to conjecture that the first two pairings coincide. In the second part of the paper, we show that they are both computed as the indices of Fredholm pairs (defined here in an algebraic sense) of operators acting on the same spaces. We construct index functions $f_X$ for any finite length Harish-Chandra module $X$. These functions are very cuspidal in the sense of Labesse, and their orbital integrals on elliptic elements coincide with the character of $X$. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. These results are the archimedean analog of results of Schneider-Stuhler for $p$-adic groups.Comment: 25 page

Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution.Comment: LaTeX2e, 51 pages; changed conten

Commuting families in Hecke and Temperley-Lieb algebras

Abstract We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.12