Let $G$ be the group $GL_n(F)$, $F$ a finite extension of $Q_p$.

Let $n = ab$. Then we consider the Speh representation $Sp(b, St_a)$, which is the unique quotient of the induced representation from the standard parabolic subgroup $P$ of type $(a,a,a,..a)$ of Steinberg_M tensor $\delta_P^{-1/2}$.

I was wondering if there exists a character formula for such representations in terms of (full) induced representations from parabolic subgroups.

For example, for the Steinberg representation we have the formula

$$ St_G = \sum_P \epsilon_P Ind_P^G(trivial) $$

in the Grothendieck group of G. Here $\epsilon_P$ is equal to $-1$ if the number of blocks of $P$ is even, and it is equal to $1$ otherwise.

I can find such a formula for all irreducible subquotients of $Ind_B^G(trivial)$. This is done via the filtration induced by the $Ind_P^G(trivial)$, and this is probably very classical, see for example Borel Walach Continuous Cohomology, Discrete Subgroups and representations of reductive groups chapter X.4.6.

However, the Speh representation occurs in $Ind_B^G(\delta_P^{-1/2})$. The problem is that when you induce from a non-trivial character $\chi$, then the representations $Ind_P^G(\chi)$ do not make sense ($\chi$ is not defined on the std Levi of $P$), and so you do not have the above filtration, or "only part of it".