# Iwahori Spherical Whittaker Functions for Steinberg Representations

Markos Karameris

Let $G(F)$ be a split reductive group over a $p$-adic field $F$ and let
$(\pi_{St},V)$ be a (generalized) Steinberg representation of $G(F)$. It is
known that the space of Iwahori fixed vectors in $V$ is one dimensional. The
Iwahori Hecke algebra acts on this space via a character. We determine this
fixed vector and use the Hecke algebra action on it to determine in full the
Whittaker function associated with this Iwahori fixed vector generalizing our
previous result [1] for $GL_{n}(F)$.

##  1 Introduction and Notations

In [7], Reeder considers two eigenvectors of the Iwahori Hecke algebra
$\mathcal{H}(G(F),J)$ corresponding to the spherical and the Steinberg
representation. The spherical Whittaker function is given by the well known
Casselman Shalika formula [8]. For a split reductive group, Reeder computes
the Whittaker function of the non-spherical vector for a general parahoric on
diagonal values and states that computing the values for a general element can
be difficult. In [6], Bump, Brubaker, Buciumas Henrik and Gustaffson give a
formula that can be used to compute the values of each summand $f_{w}$ that
appears in the Iwahori fixed vector. This, however, does not immediately
provide a simple formula for the Whittaker function in question. We provide an
explicit formula for the Whittaker function of $\pi_{St}$ on $G(F)$ and
compute all values for the case of the Steinberg fixed vector.

Let $F$ be a non-archimedean local field and $|.|$ the standard absolute value
on $F$. We denote by $\mathfrak{p}$ the maximal ideal, generated by the
uniformizer $\varpi$, $\mathcal{O}$ the ring of integers and
$\mathfrak{f}=\mathcal{O}/\mathfrak{p}$ the residue field with $q$ elements.
We also consider the connected, split reductive group of $F$-rational points
$G(F)$ where $G$ is the corresponding group valued functor. We furthermore let
$\mathcal{Z}$ denote the center of $G$ and also consider the following
subgroups inside $G(F)$: the maximal compact subgroup $K$, the Borel subgroup
$B$, the maximal unipotent $N$, the restriction $N_{\mathcal{O}}=N\cap K$, the
maximal torus $T$, the Weyl group of permutations $\mathbf{W}=N_{G}(T)/T$
111Here we pick a representative in $K$ for each $w\in\mathbf{W}$. generated
by the simple reflections $s_{\alpha}$ for the simple roots $\alpha$ and
$w_{0}$ the longest Weyl element and the Iwahori subgroup $J$ which is the
preimage of the Borel under the canonical projection $\mod\mathfrak{p}$.
Furthermore let $\lambda(d)$ be the weight in the group $X_{*}(T)$ of rational
cocharacters of $T$ corresponding to $d\in T$ under the regular isomorphism
$X_{*}(T)\to T/T^{0}$, where $T^{0}=T\cap K$ and denote by $\langle.,.\rangle$
the perfect pairing between $X_{*}(T)$ and the group of characters $X^{*}(T)$.
Denote by $\Phi$ the associated root system with positive simple roots
$\Delta$. We also let $x_{\alpha}(t)$ be the one parameter subgroup
corresponding to $\alpha\in\Phi$. Then
$N=\prod\limits_{\alpha\in\Phi^{+}}x_{\alpha}(F)$ and moreover the elements
$d\in T$ act on the roots by conjugation as
$dx_{\alpha}(t)d^{-1}=x_{\alpha}(\varpi^{\langle\alpha,\lambda(d)\rangle}t)$
and similarly for the Weyl elements: $wx_{\alpha}(t)w^{-1}=x_{w\alpha}(t)$.

##  2 The Iwahori-Bruhat decomposition

Let $\mathcal{H}(G(F),J)$ be the Hecke algebra of $J$ bi-invariant and
compactly supported functions on $G(F)$. $\mathcal{H}(G(F),J)$ acts on spaces
of $J$ invariant vectors in a representation of $G(F)$ and in particular on
right $J$ invariant functions on $G(F)$. We denote by
$X_{g}\in\mathcal{H}(G(F),J)$ the characteristic function of $JgJ$. Then the
left action of the Hecke algebra on a $J$ invariant function
$F:G(F)\to\mathbbm{C}$ is given by $X_{g}(F)(x)=\int_{JgJ}F(xh)dh$ (see 1.2 of
[3]). In fact we can compute this action explicitly:

###### Lemma 1

For any $g\in G(F)$ with coset decomposition
$JgJ=\overset{m}{\underset{i=1}{\bigcup}}\gamma_{i}J$ and $f$ a $J$ invariant
function. Then

$X_{g}(f)(x)=\sum\limits_{i=1}^{m}f(x\gamma_{i}).$

The results on coset decompositions below are partially known from Iwahori-
Matsumoto [2]:

###### Lemma 2

For any $w\in\mathbf{W}$ with $\ell(w)=k$:

$JwJ=\underset{(t_{\alpha})\in(\mathcal{O}/\mathfrak{p})^{k}}{\bigcup}\prod\limits_{\alpha\in\Phi|w^{-1}\alpha\in\Phi^{-}}x_{\alpha}(t_{\alpha})wJ.$

Proof  By the Iwahori factorization for $J$:
$J=N_{\mathcal{O}}T^{0}N_{\mathfrak{p}^{-}}$ and
$N_{\mathfrak{p}}^{-}=\prod\limits_{\beta\in\Phi^{-}}x_{\beta}(\varpi
s_{\beta})$ where $s_{\beta}\in\mathcal{O}$. But
$w^{-1}T^{0}N_{\mathfrak{p}^{-}}w\subset J$. It thus remains to check the
terms in $N_{\mathcal{O}}=\prod_{\alpha\in\Phi^{+}}x_{\alpha}(s_{\alpha})$. We
now have that $w^{-1}x_{\alpha}(s_{\alpha})w=x_{w^{-1}\alpha}(s_{\alpha})\in
J$ if and only if $w^{-1}\alpha\in\Phi^{+}$ or $w^{-1}\alpha\in\Phi^{-}$ and
$s_{\alpha}\in\mathfrak{p}$. Finally observing that
$\\#\\{\alpha\in\Phi|w^{-1}\alpha\in\Phi^{-}\\}=\ell(w^{-1})=\ell(w)$
completes the proof. $\Box$
Using Lemma 2 and the Iwasawa decomposition we obtain the refined Bruhat
decomposition of $G(F)$ relative to the Iwahori subgroup:

######  Lemma 3 (Bruhat-Iwahori decomposition [2])

There exists a double coset decomposition of $G(F)$ given by:

$G(F)=\underset{d\in T}{\bigcup}\underset{w\in\mathbf{W}}{\bigcup}NdwJ$

.

##  3 Iwahori spherical vectors of Steinberg Representations

We let $\delta_{B}$ be the modular character of $B$ and denote by
$I(\chi)=Ind_{B}^{G(F)}(\delta_{B}^{1/2}\chi)$ the principal series
representation formed by normalized induction from the smooth multiplicative
character $\chi$ of $T$. By definition then for an unramified trivial central
character $\chi$ (see also section 1. of [4]):

###### Definition 1

The Steinberg representation corresponding to an unramified character $\chi$
is the unique irreducible quotient of $I(\bar{\chi}\delta_{B}^{-1/2})$ that
contains the $J$ fixed vector that affords the sign character of
$\mathcal{H}(G(F),J)$. We denote this representation by ${St}^{\chi}$.

For any split reductive group $G(F)$, the Casselman basis of
$I(\chi\delta_{B}^{1/2})^{J}$ which consists of characteristic functions of
the form:

$f_{w}(g)=\begin{cases}\chi(b)\delta_{B}(b)&\text{if }g=bwj,b\in B,j\in J,\\\
0&\text{otherwise}\end{cases}.$

We write again the Iwahori factorization of the Iwahori subgroup
$J=N_{\mathcal{O}}T^{0}N^{-}_{\mathfrak{p}}$. Then due to Borel (see 5.7, [3])
we have the well known result that the characteristic functions corresponding
to the simple reflections $X_{s_{\alpha}}$ act on $(St^{\chi})^{J}$ with the
scalar $-1$. This enables us to determine the corresponding eigenvector.

###### Proposition 1

The Iwahori fixed part of the principal series representation
$I(\chi\delta_{B}^{1/2})^{J}$ contains up to scalar multiplication two
eigenvectors of $\mathcal{H}(G(F),J)$ given by
$\phi_{G}^{-}(g)=\sum\limits_{w\in W}(-q)^{-\ell(w)}f_{w}(g)$ and
$\phi_{G}^{+}(g)=\sum\limits_{w\in W}f_{w}(g)$ (see also p.23 of Reeder [7]).

Proof  Consider the two complex characters of the affine part of
$\mathcal{H}(G(F),J)$ given by
$\rho^{-}:X_{s_{\alpha}}\to-1,\forall\alpha\in\Delta$ and
$\rho^{+}:X_{s_{\alpha}}\to q,\forall\alpha\in\Delta$. For these two specific
characters we identify the corresponding fixed vectors
$\phi_{G}^{-}=\sum\limits_{w\in W}\lambda_{w}f_{w}(g)$ and
$\phi_{G}^{+}=\sum\limits_{w\in W}\lambda^{\prime}_{w}f_{w}(g)$. From Lemma 2
now for any $w\in\bf{W}$:
$X_{w}(\phi_{G}^{-})(1)=\rho^{-}(X_{w})\phi_{G}^{-}(1)=(-1)^{\ell(w)}\lambda_{1}$
and
$X_{w}(\phi_{G}^{+})(1)=\rho^{+}(X_{w})\phi_{G}^{+}(1)=q^{\ell(w)}\lambda^{\prime}_{1}$.
We observe that for any $w^{\prime}\in\bf{W}$:

$X_{w}(f_{w^{\prime}})(1)=\sum\limits_{(t_{\alpha})\in(\mathcal{O}_{F}/\mathfrak{p})^{\ell(w)}}f_{w^{\prime}}(\prod\limits_{\alpha\in\Phi|w^{-1}\alpha\in\Phi^{-}}x_{\alpha}(t_{\alpha})w)=q^{\ell(w)}\delta_{w,w^{\prime}}$

which implies

$X_{w}(\phi_{G}^{-})(1)=q^{\ell(w)}\lambda_{w}\text{ and
}X_{w}(\phi_{G}^{+})(1)=q^{\ell(w)}\lambda^{\prime}_{w}$

and thus we obtain:

$\lambda_{w}=(-q)^{-\ell(w)}\lambda_{1}\text{ and
}\lambda^{\prime}_{w}=\lambda_{1}$

respectively. It is easy to verify that these are eigenvectors of
$\mathcal{H}(G(F),J)$ indeed. $\Box$

Since $\phi_{G}^{-}$ is the unique eigenvector inside the induced space
affording the $-1$ eigenvalue for every simple root, it’s eigenvalues
$\rho(g):\rho(g)\phi_{G}^{-}=X_{g}(\phi_{G}^{-})$ fully determine the action
of $\mathcal{H}(G(F),J)$. The uniqueness of the character is also stated in
section 1. of [4].

##  4 Iwahori fixed Whittaker functions

We compute the Whittaker function associated to a Steinberg representation.
For an unramified character $\psi:F\to\mathbbm{C}^{\times}$ we extend it to a
character $\psi:N\rightarrow\mathbbm{C}^{\times}$ by
$\psi(\prod\limits_{\alpha\in\Phi^{+}}x_{\alpha}(t_{\alpha}))=\sum\limits_{\alpha\in\Delta^{+}}\psi(t_{\alpha})$.
Let $\mathcal{W}(\pi_{St^{\chi}},\psi)$ be the Whittaker model of $St^{\chi}$
with character $\psi$. Then there exists a unique up to scalar Whittaker
functional $\mathcal{W}_{\psi}\in\mathcal{W}(\pi_{St^{\chi}},\psi)$ which we
take to be $\mathcal{W}_{\psi}(f)(g)=\int_{N}f(w_{0}ng)\bar{\psi}(n)dn$. We
are interested in $\mathcal{W}(\pi_{St^{\chi}},\psi)^{J}$ which, by the
linearity of the Whittaker functional, has dimension $1$. Let
$W\in\mathcal{W}(\pi_{St^{\chi}},\psi)^{J}$ with $W\not\equiv 0$, then
$W(gj)=W(g),\forall g\in G(F)$ and $j\in J$ and $(F*W)(g)=\rho_{\chi}(F)W(g)$
with $\rho_{\chi}$ the character of $\mathcal{H}(G(F),J)$ as above. From Lemma
3 it is clear that the values of $W$ are completely determined by the elements
in $TW$ and more specifically by the elements of $T/T^{0}W$ since the units
can be conjugated inside $J$. This means that we can restrict ousrelves to
only computing the value of $W$ on $\varpi^{\lambda(d)}w$. In what follows we
always assume $W$ is as above.

Observe that the values of $W$ depend only on $G(F)/\mathcal{Z}(F)$. However
this is not necessarily a scheme but sheafifies to $(G/\mathcal{Z})(F)$ which
is a reductive group of adjoint type (since $\mathcal{Z}(G/\mathcal{Z})=1$ for
any reductive group) and in fact there is an injection induced by the central
isogeny $G\to G^{ad}$:
$(G(F)/\mathcal{Z}(F))\hookrightarrow(G/\mathcal{Z})(F)$. This isogeny from
the group $G$ to $G/\mathcal{Z}$ results in a bijection of the corresponding
root spaces of $G(F)$ and $(G/\mathcal{Z})(F)$ which have the same root
groups. In practice if $G^{\prime}=(G/\mathcal{Z})(F)$ then identifying
$G(F)/\mathcal{Z}(F)$ with its image in $G^{\prime}$ we obtain
$\mathcal{W}_{\psi}(\phi_{G}^{-})(g)=\int_{N}\phi_{G}^{-}(w_{0}ng)\bar{\psi}(n)dn=\int_{N}\phi_{G^{\prime}}^{-}(w_{0}ng)\bar{\psi}(n)dn=\mathcal{W}_{\psi}(\phi_{G^{\prime}}^{-})(g)$
for all $g\in G(F)/\mathcal{Z}(F)$ and in fact for every $g\in G(F)$, since
the center acts trivially on both groups. This proves that:

###### Proposition 2

Let $G(F)$ be a reductive group and $G^{ad}(F)$ it’s adjoint. Then the unique
Iwahori spherical Whittaker corresponding to the eigenfunction of the sign
character of $\mathcal{H}(G(F),J)$ is the restriction of the unique Iwahori
spherical Whittaker function corresponding to the eigenfunction of the sign
character of $\mathcal{H}(G^{ad}(F),J)$.

From the above proposition we see that it suffices to specify the Whittaker
function of a reductive group of adjoint type.

A generalization of the set $T^{+}$ is the following result a variant of which
appears in [6]:

###### Lemma 4

Let $d\in T$, then $W(dw)=0$ unless
$\langle\alpha,\lambda(d)\rangle\geq\begin{cases}0&\text{if
$w^{-1}\alpha\in\Phi^{+}$}\\\ -1&\text{if
$w^{-1}\alpha\in\Phi^{-}$}\end{cases}$ for all $\alpha\in\Delta$. A weight
$\lambda(d)$ satisfying this condition will be called $w$-dominant.

Proof Assume first that there exist $\alpha\in\Delta$ such that
$w^{-1}(\alpha)\in\Phi^{+}$ and $\langle\alpha,\lambda(d)\rangle<0$ or that
there exist $\alpha\in\Delta$ such that $w^{-1}(\alpha)\in\Phi^{-}$ and
$\langle\alpha,\lambda(d)\rangle<-1$. Let $s\in F$ be such that $|s|=q$ and
$\psi(x_{\alpha}(s))\neq 1$. Then:
$\psi(x_{\alpha}(s))W(dw)=W(x_{\alpha}(s)dw)=W(dd^{-1}x_{\alpha}(s)dw)=W(dww^{-1}x_{\alpha}(\varpi^{-\langle\alpha,\lambda(d)\rangle}s)w)=W(dwx_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\lambda(d)\rangle}s))$.
Since $x_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\lambda(d)\rangle}s)\in J$ in
each case, we then have that
$\psi(x_{\alpha}(s))W(dw)=W(dwx_{w^{-1}\alpha}(\varpi^{-\langle\alpha,\lambda(d)\rangle}s))=W(dw)$
and thus it follows that $W(dw)=0$. $\Box$

###### Corollary 1

For any $d\in T$ we have $W(dw)=0$ unless $d\in T_{w}^{+}T^{0}$.

###### Definition 2

We denote the semigroup of $w$-dominant matrices in $T$ by $T_{w}^{+}$ and
define the element $d_{w}=\\{d\in
T/\mathcal{Z}(F):\langle\alpha,\lambda(d)\rangle=\begin{cases}0&\text{if
$w^{-1}\alpha\in\Phi^{+}$}\\\ -1&\text{if
$w^{-1}\alpha\in\Phi^{-}$}\end{cases},\forall\alpha\in\Delta\\}$.

The existence of $d_{w}$ for any $w$ inside $G$ follows from the adjoint
property of $G$ i.e. the root lattice is equal to it’s weight lattice. For a
simply connected group this is not the case (ex. $SL_{n}(F)$ does not contain
any of the $s_{\alpha}$-dominant weights). It is easy to see that
$T_{w}^{+}=d_{w}T^{+}$, so in particular $T^{+}\subseteq T_{w}^{+},\forall
w\in\mathbf{W}$.

###### Lemma 5

For a Weyl element $w$ and $d\in T^{+}$ the following relation is satisfied:

$W(dw)=(-q)^{-\ell(w)}W(d).$

Proof  Inductively it follows that: $X_{w}(W)(g)=(-1)^{\ell(w)}W(g)$. From
Lemma 2 however we have: $JwJ=\underset{(t_{1},\dots
t_{\ell(w)})\in\mathcal{(}{O}_{F}/\mathfrak{p})^{\ell(w)}}{\bigcup}\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})wJ$
with $\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})\in N_{\mathcal{O}}$
and $\mu(JwJ)=|JwJ/J|=q^{\ell(w)}$. From Lemma 1 now
$(-1)^{\ell(w)}W(d)=\sum\limits_{(t_{1},\dots
t_{\ell(w)})\in\mathcal{(}{O}_{F}/\mathfrak{p})^{\ell(w)}}W(d\prod\limits_{j=1}^{\ell(w)}x_{\alpha_{j}}(t_{j})w)=q^{\ell(w)}W(dw)\implies
W(dw)=(-q)^{-\ell(w)}W(d)$. $\Box$

###### Lemma 6

For any $d\in T^{+}$ we have $JdJ\subseteq N_{\mathcal{O}}^{+}dJ$ i.e.
$JdJ=\bigcup\limits_{\ell}(n^{+}_{\mathcal{O}})_{\ell}dJ$.

Proof  For any $d\in T^{+}$ and any $\alpha\in\Phi^{+}$ write $a$ as a finite
sum of simple roots $\alpha=\sum\limits_{i}\alpha_{i}$. By definition now
$\langle\alpha_{i},\lambda(d)\rangle\geq 0$ for every
$\alpha_{i}\in\Delta^{+}$ and thus $\langle\alpha,\lambda(d)\rangle\geq 0$.
But observe that

$d^{-1}x_{-\alpha}(s\varpi)d=x_{-\alpha}(s\varpi^{1+\langle\alpha,\lambda(d)\rangle})$

where $s\in\mathcal{O}$ and thus $d^{-1}x_{-\alpha}(s\varpi)d\in J$. From the
Iwahori factorization finally it follows that
$JdJ=N_{\mathcal{O}}^{+}dd^{-1}T^{0}N_{\mathfrak{p}}^{-}dJ\subseteq
N_{\mathcal{O}}^{+}dJ$. $\Box$

We are now ready to show that:

######  Theorem 1 (Diagonal Whittaker values)

Let $W\in\mathcal{W}(\pi,\psi)^{J}$ with $W(1)=1$ be such that

$W(zg)=W(g),\;z\in Z,g\in G(F)$

and $W$ is an eigenfunction of $\mathcal{H}(G(F),J)$ with

$(F*W)(g)=\rho_{\chi}(F)W(g).$

Then $W$ has the following diagonal values:

$W(d)=\begin{cases}(\chi\delta_{B})(d)&\text{if $d\in T^{+}$}\\\
0&\text{otherwise}\end{cases}.$

Proof  Let $\epsilon_{g}=\rho_{\chi}(X_{g})\in\mathbb{C}$ for any $g\in G(F)$.
Since $\phi_{G}^{-}$ is an eigenfunction it holds that

$\epsilon_{g}\phi_{G}^{-}(g)=X_{h}(\phi_{G}^{-})(g),\text{for all }h\in G(F)$

and thus setting $h=d$ and $g=1$:

$\epsilon_{d}\phi_{G}^{-}(1)=X_{d}(\phi_{G}^{-})(1).$

We compute $\phi_{G}^{-}(1)=(-q)^{0}\phi_{1}(1)=1$ so it remains to compute
$X_{d}(\phi_{G}^{-})(1)$. For this we notice that
$X_{d}(\phi_{w})(1)=\int_{JdJ}\phi_{w}(j)dj$ which from Lemma 1 and Lemma 6
equals $\sum\limits_{\ell}\phi_{w}((n^{+}_{\mathcal{O}})_{\ell}d)$ which means
that $X_{d}(\phi_{w})(1)\neq 0$ if and only if $w=1$ and in that case
$X_{d}(\phi_{1})(1)=\sum\limits_{\ell}(\chi\delta_{B})(d)=\mu(JdJ)(\chi\delta_{B})(d)$.
Substituting in the original euation gives us:

$\frac{\epsilon_{d}}{\mu(JdJ)}=(\chi\delta_{B})(d).$

Now using Lemma 1 again yields:

$\epsilon_{p}W(g)=T_{d}(W)(g)=\sum\limits_{\ell}W(g(n^{+}_{\mathcal{O}})_{\ell}d)$

and thus since $W(1)=1$, for $g=1$ we obtain:

$\epsilon_{p}=\epsilon_{p}W(1)=\sum\limits_{\ell}W((n^{+}_{\mathcal{O}})_{\ell}d)=\mu(JdJ)W(d).$

Solving for $W(d)$ yields:

$W(d)=\frac{\epsilon_{p}}{\mu(JdJ)}=(\chi\delta_{B})(d),\text{ for any }d\in
T^{+}.$

$\Box$

###### Lemma 7

For any Weyl element $w$:
$Jd_{w}wJ=\underset{\bar{s}=(s_{\alpha})}{\bigcup}\prod\limits_{\alpha\in\Phi^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w}wJ$ where $s_{\alpha}\in\mathcal{O}/\mathfrak{p}^{n_{\alpha}}$
for some $n_{\alpha}\in\mathbbm{N}\cup\\{0\\}$ and each $\alpha\in\Phi^{-}$.

Proof  We write $J=N^{-}_{\mathfrak{p}}T^{0}N_{\mathcal{O}}$ as before. For
any $n\in N_{\mathcal{O}}$ observe that
$n=\prod\limits_{i=1}^{r}x_{a_{i}}(s_{a_{i}})$ for some $r\in\mathbb{N}$ with
$a_{i}\in\Delta^{+}$ and $s_{a_{i}}\in\mathcal{O}$ for all indices. Since
$w^{-1}d_{w}^{-1}nd_{w}w=\prod\limits_{i=1}^{r}(w^{-1}d_{w}^{-1}x_{a_{i}}(s_{a_{i}})d_{w}w)$
it suffices to show that
$x_{w^{-1}(\beta)}{(\varpi^{-\langle\beta,\lambda(d_{w})\rangle}s_{\beta})}\in
J,\forall\beta\in\Delta^{+}$ and any $s_{\beta}\in\mathcal{O}$. However,
taking cases, we see that if $\beta\in\Delta^{+}:w^{-1}\beta\in\Phi^{+}$ then
$\langle\beta,\lambda(d_{w})\rangle=0$ and similarly if
$w^{-1}\beta\in\Phi^{-}$ then $\langle\beta,\lambda(d_{w})\rangle=-1$. Thus in
every case
$w^{-1}d_{w}^{-1}x_{\beta}(s_{\beta})d_{w}w=x_{w^{-1}(\beta)}{(\varpi^{-\langle\beta,\lambda(d_{w})\rangle}s_{\beta})}\in
J$. $\Box$

###### Lemma 8

For every Weyl element $w\in\mathbf{W}$ we have
$d_{w_{0}w}=w_{0}d_{w}w_{0}d_{w_{0}}z_{w}$ for some $z_{w}\in\mathcal{Z}(F)$.

Proof  For all $\alpha\in\Delta^{+}$ we observe that
$w_{0}\alpha\in\Delta^{-}$ and compute:
$\langle\alpha,\lambda(d_{w_{0}})+(\lambda(w_{0}d_{w}w_{0}))\rangle=\langle\alpha,\lambda(d_{w_{0}})\rangle+\langle\alpha,(\lambda(w_{0}d_{w}w_{0}))\rangle=-1+\langle
w_{0}\alpha,\lambda(d_{w})\rangle=-1-\langle-
w_{0}\alpha,\lambda(d_{w})\rangle$. But for $w^{-1}w_{0}\alpha\in\Phi^{-}$
i.e. $w^{-1}(-w_{0}\alpha)\in\Phi^{+}$: $\langle-
w_{0}\alpha,\lambda(d_{w})\rangle=0$ and similarly when
$w^{-1}w_{0}\alpha\in\Phi^{+}$ i.e. $w^{-1}(-w_{0}\alpha)\in\Phi^{-}$:
$\langle-w_{0}\alpha,\lambda(d_{w})\rangle=-1$ thus
$\langle\alpha,\lambda(d_{w_{0}})w_{0}\lambda(d_{w})w_{0}\rangle=-\langle\alpha,\lambda(d_{w})\rangle-1=0\text{
if }w^{-1}w_{0}\alpha\in\Phi^{+}$ and
$\langle\alpha,\lambda(d_{w_{0}})w_{0}\lambda(d_{w})w_{0}\rangle=-1\text{ if
}w^{-1}w_{0}\alpha\in\Phi^{-}$ which is exactly
$\langle\alpha,\lambda(d_{w_{0}w})\rangle$ by definition. This implies that
$w_{0}d_{w}w_{0}d_{w_{0}}$ and $d_{w_{0}w}$ are equal in $T/\mathcal{Z}(F)$.
$\Box$

######  Theorem 2 (Whittaker values at every cell $Tw$)

Under the assumptions of Theorem 1

$W(dw)=\begin{cases}(\chi\delta_{B})(d)(-q)^{-\ell(w)}&\text{if $\lambda(d)$
is $w$-dominant}\\\ 0&\text{otherwise}\end{cases}$

Proof  We use again the relation $X_{d_{w}w}(W)(g)=\epsilon_{d_{w}w}W(g)$. By
Lemma 7 we have
$\epsilon_{d_{w}w}W(g)=\sum\limits_{\bar{s}}W(g\prod\limits_{\alpha\in\Phi^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w}w)$. Setting $g=dd_{w_{0}}w_{0}(=dw_{0}d_{w_{0}}^{-1})$ we
obtain $w_{0}d_{w_{0}}^{-1}\prod\limits_{\alpha\in\Delta^{-}}x_{\alpha}(\varpi
s_{\alpha})d_{w_{0}}w_{0}=\prod\limits_{\alpha\in\Delta^{-}}x_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,\lambda(d_{w_{0}}^{-1})\rangle})\in
N_{\mathcal{O}}$ since $\langle\alpha,-\lambda(d_{w_{0}})\rangle={-1}$ and
$\psi(dx_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,\lambda(d_{w_{0}}^{-1})\rangle})d^{-1})=1$ for
any $\alpha\in\Phi^{-}\backslash\Delta^{-}$ and $d\in T$. Thus we can
conjugate $\prod\limits_{\alpha\in\Phi^{-}}x_{w_{0}(\alpha)}(\varpi
s_{\alpha}\varpi^{\langle\alpha,\lambda(d_{w_{0}}^{-1})\rangle})$ with any
$d\in T^{+}$. We now get for any $d\in T^{+}$:
$\epsilon_{d_{w}w}W(dd_{w_{0}}w_{0})=\mu(Jd_{w}wJ)W(dd_{w_{0}}w_{0}d_{w}w_{0}w_{0}w)$.
From Lemma 8 now this becomes:

$\epsilon_{d_{w}w}W(dd_{w_{0}}w_{0})=\mu(Jd_{w}wJ)W(dd_{w_{0}w}w_{0}w),\forall
d\in T.$

Setting $w=w_{0}$ and comparing scalar terms with Lemma 5 for
$d=d_{w_{0}}^{-1}$ we get that
$\epsilon_{d_{w_{0}}w_{0}}=(-q)^{\ell(w_{0})}\mu(Jd_{w_{0}}w_{0}J)W(d_{w_{0}}^{-1})$.
Substituting the value of $\epsilon_{d_{w_{0}}w_{0}}$ and setting $d\to
dd_{w_{0}}^{-1}$ leads to:

$W(dw_{0})=(-q)^{-\ell(w_{0})}W(d_{w_{0}}^{-1})^{-1}W(dd_{w_{0}}^{-1}),\forall
d\in T$

Similarly we set $w\to w_{0}w$ and compute
$\epsilon_{d_{w_{0}w}w_{0}w}=(-q)^{\ell(w_{0})-\ell(w)}\mu(Jd_{ww_{0}}ww_{0}J)W(d_{w_{0}}^{-1}d_{w})$.
Substituting this value of $\epsilon_{d_{w_{0}w}w_{0}w}$ and using the above
relation for the $W(dw_{0})$ term implies:

$W(dw)=(-q)^{-\ell(w)}W(d_{w}^{-1})^{-1}W(dd_{w}^{-1}),\forall d\in T\;\;(**)$

and for any $d\in T_{w}^{+}$, i.e. $dd_{w}^{-1}\in T^{+}$, a direct
application of Theorem 1 proves the claim. $\Box$

###### Corollary 2

If $W\in\mathcal{W}(\pi_{St^{\chi}},\psi)^{J}$ and $W(1)=0$ then relations (*)
imply that $W(d)=0,\forall d\in T^{+}\iff W(d)=0,\forall d\in T$. Relation
(**) then implies that $W(dw)\equiv 0$ in every cell thus contradicting the
assumption $W\not\equiv 0$. This strengthens a more general non-vanishing
result in [7].

##  5 Whittaker Functions and New-vectors

From the relation $(F*W)(g)=\rho_{\chi}(F)W(g)$ and the uniqueness in Theorem
2 we conclude that:

###### Theorem 3

The function described in Theorem 2

$W(dw)=\begin{cases}(\chi\delta_{B})(d)(-q)^{-\ell(w)}&\text{if $\lambda(d)$
is $w$-dominant}\\\ 0&\text{otherwise}\end{cases}$

is the unique Whittaker function that corresponds to the Iwahori spherical
vector $\phi_{G}^{-}\in(St^{\chi})^{J}$.

We now use this explicit description of the Whittaker function associated to
$v\in St^{\chi}$ to show that $v$ is not fixed by any other parahoric subgroup
of $G(F)$ for all generalized Steinberg representations.
For $S=\\{s_{\alpha_{1}},...,s_{\alpha_{r}}\\}$ a subset of the simple
permutations generating $\mathbf{W}$, let $\mathbf{W}_{S}$ be the subset of
$\mathbf{W}$ generated by the elements of $S$.

###### Theorem 4

For any parahoric $K_{S}=J\mathbf{W}_{S}J$ with $S\neq\\{1\\}$ it holds that
$(St^{\chi})^{K_{S}}=\\{0\\}$.

Proof  Assume $v\in(St^{\chi})^{K_{S}}$ is non-zero, then
$v\in(St^{\chi})^{J}$ and thus $W_{v}$ is of the form we compute in Theorem 2.
Then $v\in(St^{\chi})^{K_{S}}\implies W_{v}(gk_{S})=W(g),\forall k_{S}\in
K_{S}$ so picking $k_{S}=s_{\alpha}$ for some $s_{\alpha}\in S$ we see that
$W(gs_{\alpha})=W(g),\forall g\in G(F)$. Setting $g=d_{s_{\alpha}}$ yields:
$W(d_{s_{\alpha}}s_{\alpha})=W(d_{s_{\alpha}})=0$ since $d_{s_{\alpha}}\not\in
T^{+}$. This however contradicts the formula in Theorem 2 according to which
$W(d_{s_{\alpha}}s_{\alpha})\neq 0$. $\Box$

## References

  * [1] E.M. Baruch, M. Karameris Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field, 2023, arXiv: 2311.02706 [math.RT]
  * [2] N. Iwahori, H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $p$-adic Chevalley groups, Publications Mathématiques de l’I.H.É.S., tome 25 (1965), p. 5-48
  * [3] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Inv. Math., 35, 233- 259 (1976)
  * [4] J. A. Shalika, On the space of cusp forms of a P-adic Chevalley group, Ann. Math. , Second Series, 92 (2) p. 262–278 (1970)
  * [5] W. Casselman, An introduction to the theory of admissible representations of reductive p-adic groups, Unpublished notes
  * [6] B. Brubaker, V. Buciumas, D. Bump, Henrik P. A. Gustafsson, Colored Vertex Models and Iwahori Whittaker Functions, 2019, arXiv: 1906.04140 [math.RT]
  * [7] M. Reeder, $p$-adic Whittaker functions and vector bundles on flag manifolds, Comp. Math. 85, No 1 (1980), p. 9 - 36
  * [8] T. Shintani, On an Explicit Formula for Class-$1$ ”Whittaker Functions” on $GL_{n}$ over $p$-adic Fields, Proc. Japan Acad. 52, (1976), p. 180 - 182
  * [9] W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Comp. Math. 40, No 3 (1980), p. 387-406