$\DeclareMathOperator\ch{ch}$Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. I shall use $\kappa(F)$ to denote the residue field of $F$.
Let further $G = \mathrm{GL}_n$ denote the standard algebraic group scheme of the general linear group. Further I shall denote by
- $T$ : The standard split torus of $G$ consisting of diagonal matrices,
- $B$ : the standard Borel subgroup of $G$ of upper-triangular matrices with non-zero elements on the diagonal,
- $U$ : the standard unipotent subgroup of $G$ of upper-triangular matrices with $1$'s on the diagonal principal. Hence $B = TU$. I will also use $U^{-}$ for the subgroup of lower-diagonal matrices with $1$'s on its diagonal principal.
The roots and the Bruhat order are chosen in a standard way, i.e. all roots $\alpha_{ij}$ shall be positive iff $i > j$ and negative otherwise. For my needs, I realize the Weyl-Group $W = W(G(F),T(F))$ inside $G(\mathcal{O})$ by dropping the modulo $T(\mathcal{O})$-condition (this won't have any effect in my setting).
The Iwahori subgroup J of $G(F)$ is defined as the preimage of $B(\kappa(F))$ under the canonical projection $G(\mathcal{O}) \to G(\kappa(F))$. In other words $$ J = \begin{pmatrix} \mathcal{O}^{\times} & \mathcal{O} & \ldots & \mathcal{O} \\ \mathfrak{p} & \mathcal{O}^{\times} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \mathcal{O} \\ \mathfrak{p} & \ldots & \mathfrak{p} & \mathcal{O}^{\times} \end{pmatrix}. $$ The Iwahori subgroup may be decomposed as the direct product $$ J = U^{-}(\mathfrak{p})T(\mathcal{O})U(\mathcal{O}). $$ The (Iwahori-)Hecke algebra $\mathcal{H}$ is the space of $\mathbb{C}$-valued functions on $G(F)$, that are
- Iwahori-bi-invariant, i.e. $f(jgj') = f(g)$ for $j,j' \in J$ and $g \in G(F)$.
- compactly supported
- smooth, i.e. locally constant.
It is a $\mathbb{C}$-algebra under the convolution product given by $$ (f \ast g)(x) = \int_{G(F)} f(xy^{-1})g(y)dy, $$ where the Haar measure on $G(F)$ is chosen such that $vol(J,dg)=1$.
It is known, that as complex vector spaces, $$ \mathcal{H} = \mathcal{H}_W \otimes X_{*}(T(F)), $$ where $\mathcal{H}_W$ is the Hecke-algebra of $W$ (not necessary here) and $X_{*}(T(F))$ is the (commutative) group of rational cocharacters of $F$ (these are rational homomorphisms $F^{\times} \to T(F)/T(\mathcal{O})$. The cocharacter group $X_{*}(T(F))$ is known to be a free abelian group of rank n, and since I can drop the 'modulo $T(\mathcal{O})$'-condition, I will simply write $$ \pi^{\lambda} = \begin{pmatrix} \pi^{\lambda_1} & & \\ & \ddots & \\ & & \pi^{\lambda_n} \end{pmatrix} $$ for the image of $\pi$ under some $\lambda \in X_{*}(T(F))$.
The group of cocharacters can be realized inside $\mathcal{H}$ as follows: for a dominant $\lambda\in X_{*}(T(F))$ one defines the Hecke operator $$ T_{\lambda} := \ch(J\pi^{\lambda}J), $$ where by $\ch(\cdot)$ I mean the characteristic function of $\cdot$. As far as I understand it, if $\lambda\in X_{*}(T(F))$ is any (!) cocharacter, there are dominant $\lambda', \lambda'' \in X_{*}(T(F))$ s.t. $\lambda = \lambda' - \lambda''$ and one defines $$ \widetilde{T}_{\lambda} := T_{\lambda'} T^{-1}_{\lambda''}. $$ The definition should be independent on the choice of $\lambda'$ and $\lambda''$.
Question: I was wondering if there is any nice description of these $T^{-1}_{\lambda''}$, maybe any reference?
Example: I computed it in the simplest case when $n=2$ and $\lambda = (1,0)$ is the fundamental coweight. I got something like $$ T^{-1}_{(1,0)} = \frac{1}{q} \ch(J\pi^{-(1,0)}J) - \frac{1}{q(q-1)} \ch(J w_2J), $$ where $w_2 = \begin{pmatrix} & 1\\ 1 & \end{pmatrix}$ is the (only) simple root.
