Ladder representations of GL(n,Q_p)

In this paper, we recover certain known results about the ladder representations of GL(n, Q_p) defined and studied by Lapid, Minguez, and Tadic. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa-Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal formula proved by Lapid-Minguez and Tadic is a direct consequence of the BGG resolution of finite dimensional simple gl(n)-modules. We make a connection between the semisimplicity of Hecke algebra modules, unitarity with respect to a certain hermitian form, and ladder representations.Comment: 14 page

The Dunkl-Cherednik Deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{W}\otimes\mathcal{C}$ deforms to a Lie superalgebra inside $H_c\otimes\mathcal{C}$ and compute its centraliser algebra. For a generic parameter $c$ such that the standard $H_c$-module is unitary, we compute the joint $((H_c)^{\mathfrak{s}},\mathfrak{s})$- and $((H_c\otimes\mathcal{C})^{\mathfrak{g}},\mathfrak{g})$-decompositions of the relevant modules.Comment: 29 pages; version that was accepted for publication. In this revised version we shortened the discussion about Drinfeld orbifold algebras, added a List of symbols and made minor corrections throughou

Hermitian forms for affine Hecke algebras

We study star operations for Iwahori-Hecke algebras and invariant hermitian forms for finite dimensional modules over (graded) affine Hecke algebras with a view towards a unitarity algorithm.Comment: 29 pages, preliminary version. v2: the classification of star operations for the graded Hecke algebras and the construction of hermitian forms in the Iwahori case via Bernstein's projectives have been removed from this preprint and they will make the basis of a new pape

Dirac induction for rational Cherednik algebras

We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler-Poincar\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.Comment: 32 page