arXiv subjects
Publications and source records attributed to Kei Yuen Chan.
Let $k_0$ be a $p$-adic field of odd residual characteristic, and $G$ a special orthogonal group defined as acting on a split $2n+1$-dimensional orthogonal space $V$ over $k_0$. Let $H$ be the Iwahori Hecke algebra of $G$. A purpose of this short article is to compute the Iwahori component of a Bessel model space and identify it with an explicit projective $H$-module.
This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch-Ciubotaru-Trapa and Ciubotaru. We apply our result to compute the Dirac cohomology of ladder representations for type $A_n$. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.
Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.
In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between discrete series vanish.
Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $ψ$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}ψ)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reductions, we show that the Bernstein-Zelevinsky derivatives can be determined using graded Hecke algebras. We give two applications of our study. Firstly, we compute the Bernstein-Zelevinsky derivatives of generalized Speh modules, which recovers a result of Lapid-Mínguez and Tadić. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representations of $GL(n+1,F)$, restricted to $GL(n,F)$, which is further used to verify a conjecture on an Ext-branching problem of D. Prasad for a class of examples.
We define an analogue of the Dirac operator for the degenerate affine Hecke-Clifford algebra. A main result is to relate the central characters of the degenerate affine Hecke-Clifford algebra with the central characters of the Sergeev algebra via Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.
We study a twisted Euler-Poincaré pairing for graded affine Hecke algebras, and give a precise connection to the twisted elliptic pairing of Weyl groups defined by Ciubotaru-He. The Ext-groups for an interesting class of parabolically induced modules are also studied in a connection with the twisted Euler-Poincaré pairing. We also study a certain space of graded Hecke algebra modules which equips with the twisted Euler-Poincaré pairing as an inner product.
A uniform parametrization for the irreducible spin representations of Weyl groups in terms of nilpotent orbits is recently achieved by Ciubotaru (2011). This paper is a generalization of this result to other real reflection groups. Let $(V_0, R, V_0^{\vee}, R^{\vee})$ be a root system with the real reflection group $W$. We define a special subset of points in $V_0^{\vee}$ which will be called solvable points. Those solvable points, in the case $R$ crystallographic, correspond to the nilpotent orbits whose elements have a solvable centralizer in the corresponding Lie algebra. Then a connection between the irreducible spin representations of $W$ and those solvable points in $V_0^{\vee}$ is established.
For a connected complex semi-simple Lie group $G$ and a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we determine when the intersection of a conjugacy class $C$ in $G$ and a double coset $BwB^-$ is non-empty, where $w$ is in the Weyl group $W$ of $G$. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on $W$ and an involution $\mc \in W$ associated to $C$. We study properties of the elements $\mc$. For $G = SL(n+1, \Cset)$, we describe $\mc$ explicitly for every conjugacy class $C$, and for the case when $w \in W$ is an involution, we also give an explicit answer to when $C \cap (BwB)$ is non-empty.