Induced sign representations and characters of Hecke algebras

introduction

Many combinatorial formulas for computations in the symmetric group Sn can be modified appropriately to describe computations in the Hecke algebra Hn(q), a deformation of C[Sn].

Unfortunately, the known formulas for induced sign characters of Sn are not among these.
For induced sign characters of Hn(q), we conjecture formulas which specialize at q=1 to formulas for induced sign characters of Sn.
We will discuss evidence in favor of the conjecture, and relations to the chromatic quasi-symmetric functions of Shareshian and Wachs.

Given a partition \lambda=(\lambda_1,\cdots, \lambda_n) of n

1 define W_{\lambda}=S_{\lambda_1}\times S_{\lambda_2} \cdots \times S_{\lambda_k}

2 For each coset of the form wW_{\lambda},

define T_{wW_{\lambda}}=\sum_{v\in wW_{\lambda}}(-q)^{\ell(v)}T_{v}

If we set q=1, we get a sum looks like (\sum_{w\in W} w_{\lambda} sgn(v)v)

3 Let H_n(q) act by lefy multiplication on coset sums T_{D} where D is of the form wW_{\lambda}

4 this left multiplication can be expressed as matrix multiplication

Let \rho_{q}^{\lambda}(T_v)=matrix that correspondes to left multiplication by T_v.

Let \rho^{\lambda}(v)=matrix corresponding to left multiplication by v.

the trace/character associated to representation \rho_{q}^{\lambda} are usually denoted by \epsilon_{q}^{\lambda}

Q. What is a nice formula for \epsilon_{q}^{\lambda}(T_{v}) ? (open)