y_{i}^{*}=&\prod_{j=1}^{2}\prod_{k=1}^{n-2}(1+x_{j,k}^{2})^{\eta_{n-2+j,i,k}/2},&\delta^{\prime}=&(\delta_{1},\ldots,\delta_{n-2}),\\\
\mu^{\prime}=&(\mu_{1},\ldots,\mu_{n-2})+\tfrac{\mu_{n-1}+\mu_{n}}{n-2}(1,\ldots,1),&\mu^{\prime\prime}=&(\mu_{n-1},\mu_{n})-\tfrac{\mu_{n-1}+\mu_{n}}{2}(1,1).\end{aligned}$

In a neighborhood of $i\mathfrak{a}^{*}_{0}(\Lambda)$, the first integral
representation is best in the case $\Lambda_{n}=0$, while the second is better
for $\Lambda_{n}\neq 0$ since then $\Lambda_{n-1}+\Lambda_{n}=0$ so
${\mathop{\symoperators Re}}(\mu_{n-1}+\mu_{n})$ is close to $0$. Note that
Stade and Ishii have given decompositions in [Stade03, Theorem 2.1] and
[IshiiStade, Theorem 14] in the spherical case (in the latter paper by passing
through the Mellin transform); it would be interesting to see if one can
arrive at their formula by starting with this one. This also recovers the
integral representations of [HWI, (3.22), (3.24)].

We use these expressions, starting with (LABEL:eq:WhitInd1), to obtain
expressions for the Mellin transform of the Whittaker function. Since we can
conjugate by an element of the Weyl group to transform $\hat{k}_{n,i}(x)$ back
to an element of $SO(2)$, our assumptions on $\sigma$ (see section 6.3) imply
that the entries of $\sigma(\hat{k}_{n,i}(x))$ are trigonometric polynomials
in $\arg\frac{1-ix}{\sqrt{1+x^{2}}}$. So we replace each
$\sigma(\hat{k}_{n,n-j}(x_{j}))$ with
$\left(\frac{1-ix_{j}}{\sqrt{1+x_{j}^{2}}}\right)^{a_{j}}$ for some
$a_{j}\in\mathbb{Z}$ and replace the Whittaker function $W^{*}_{n-1,\sigma}$
with its entry $W^{*}_{n-1,\sigma_{b_{1}},b_{2},b_{3}}$ where $\sigma_{b_{1}}$
is an irreducible component of $\sigma|_{O(n-1)}$ and $1\leq
b_{2},b_{3}\leq\dim\sigma_{b_{1}}$. Next, Mellin expand the $GL(n-1)$
Whittaker function and apply [MeThesis, (6.1)]

$\displaystyle e\left(x\right)$
$\displaystyle=\lim_{\theta\to\frac{\pi}{2}^{-}}\exp\left(-\left|2\pi
x\right|e^{-i\theta{\mathop{\symoperators\,sgn}}(x)}\right)=\lim_{\theta\to\frac{\pi}{2}^{-}}\int_{{\mathop{\symoperators
Re}}(s)=1}\left|2\pi
x\right|^{-s}e^{is\theta{\mathop{\symoperators\,sgn}}(x)}\Gamma\left(s\right)\,\frac{ds}{2\pi
i},\qquad x\neq 0,$

to each term in the complex exponential, then use [HWI, Section 2.3.2]

$\displaystyle\int_{\mathbb{R}}{\mathop{\symoperators\,sgn}}(x)^{d}\left|x\right|^{a-1}(1+x^{2})^{-\frac{a+b}{2}}\left(\frac{1-ix}{\sqrt{1+x^{2}}}\right)^{c}dx=B_{d,c}(a,b),\qquad
d\in\left\\{0,1\right\\}$ $\displaystyle\begin{aligned}
B_{d,c}(a,b)=&(i{\mathop{\symoperators\,sgn}}(c))^{d}\sum_{j=0}^{(\left|c\right|-d)/2}\binom{\left|c\right|}{2j+d}(-1)^{j}B\left(\frac{d+a}{2}+j,\frac{\left|c\right|-d+b}{2}-j\right),\\\
B(a,b)=&\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},\qquad
e^{it\theta{\mathop{\symoperators\,sgn}}(x)}=\sum_{d\in\left\\{0,1\right\\}}\cos\left(d\frac{\pi}{2}-\theta
t\right){\mathop{\symoperators\,sgn}}(x)^{d},\end{aligned}$

(which is the Mellin transform of the classical Whittaker function, compare
(24)) and since the resulting integral converges absolutely, we may replace
$\theta$ with $\frac{\pi}{2}$ in the integrand. We conclude that the entries
of the Mellin transform $\widehat{W}_{n,\sigma}(s,\mu,\delta)$ of the
Whittaker function (as in (20)) are linear combinations (with coefficients of
modulus at most 1) of at most $(\dim\sigma)^{2n-1}$ terms of the form

(49)
$\displaystyle\sum_{d\in\left\\{0,1\right\\}^{n-1}}\int_{{\mathop{\symoperators
Re}}(q)=\mathfrak{q}}\widehat{W}_{n-1,\sigma_{b_{1}},b_{2},b_{3}}(q,\mu^{\prime},\delta^{\prime})\prod_{j=1}^{n-1}A_{d_{j},+}\left(s_{j}-q_{j-1}+\tfrac{n-j}{n-1}\mu_{n}\right)$
$\displaystyle\times
B_{\tilde{d}_{j},a_{j}}\left(1-s_{j}-s_{j+1}+q_{j-1}+q_{j}-\tfrac{2n-2j-1}{n-1}\mu_{n},s_{j}-q_{j}-\tfrac{j}{n-1}\mu_{n}\right)\frac{dq}{(2\pi
i)^{n-2}},$

where

$A_{d,\pm}(u):=(2\pi)^{-u}\Gamma(u)\cos\frac{\pi}{2}(d\pm u),$

$\left\\{0,1\right\\}\ni\tilde{d}_{j}\equiv d_{j}+d_{j+1}\pmod{2}$ using
$s_{n}=q_{0}=q_{n-1}=d_{n}=0$.

By Stirling’s formula, the beta function $B(u+v,u-v)$ has exponential decay in
$\left|{\mathop{\symoperators Im}}(v)\right|$ unless
$\left|{\mathop{\symoperators Im}}(v)\right|<\left|{\mathop{\symoperators
Im}}(u)\right|$, and the same holds for $B_{d,c}(u+v,u-v)$. Suppose the Mellin
transform of the lower-rank Whittaker function forces
$\left\|{\mathop{\symoperators
Im}}(q)\right\|\ll_{n}\left\|{\mathop{\symoperators
Im}}(\mu^{\prime})\right\|$, then the beta functions force
$\left\|{\mathop{\symoperators
Im}}(s)\right\|\ll_{n}\left\|{\mathop{\symoperators Im}}(\mu)\right\|$ and by
induction this holds for all $n$ since it holds for $n=2$.

In (49), consider shifting the $q$ contours far to the right, starting with
$q_{1}$ and ending with $q_{n-2}$, we see that
$\widehat{W}_{n,\sigma}(s,\mu,\delta)$ is holomorphic in $\mu$ and $s$ for
${\mathop{\symoperators Re}}(s)$ large compared to ${\mathop{\symoperators
Re}}(\mu)$, so $W^{*}_{n\sigma}(t,\mu,\delta)$ has super-polynomial decay as
the coordinates of $t$ tend to infinity. Furthermore, the potential poles of
$\widehat{W}_{n,\sigma}(s,\mu,\delta)$ occur when $s=-\hat{\mu}^{w}-m$ for
some $w\in W$, $m\in\mathbb{N}_{0}^{n-1}$ (imagine shifting the $q$ contours
far to the right, picking up residues at
$q_{j-1}=s_{j}+\mu_{n}-\frac{j-1}{n-1}\mu_{n}+m_{j}$ and
$q_{j}=s_{j}-\frac{j}{n-1}\mu_{n}+m_{j}$, $m_{j}\in\mathbb{N}_{0}$), and by
contour shifting this implies

######  Theorem 12 (Inverse Whittaker Paley-Wiener).

If $F(\mu)$ is holomorphic and has rapid decay in $\left\|\mu\right\|$ on a
tube domain containing $i\mathfrak{a}^{*}_{0}(\Lambda)$, then there exists
$\eta,\eta^{\prime},A,B>0$ so that

$\int_{i\mathfrak{a}^{*}_{0}(\Lambda)}F(\mu)W_{\sigma}(y,\mu,\delta)d_{\text{spec}}\mu\ll_{\eta,\eta^{\prime},A,B}p_{(1+\eta^{\prime})\rho}(y)(\dim\sigma)^{B}\left\|F\right\|_{\eta,A},$

where

$\left\|F\right\|_{\eta,A}:=\sup_{\left\|\beta\right\|<\eta}\int_{{\mathop{\symoperators
Re}}(\mu)=\beta}\left|F(\mu)\right|(1+\left\|\mu\right\|)^{A}\left|d\mu\right|.$

Applying the same process to (LABEL:eq:WhitInd2) yields

(50)
$\displaystyle\sum_{d\in\left\\{0,1\right\\}^{n-2}}\sum_{d^{\prime}\in\left\\{0,1\right\\}^{n-3}}\sum_{d^{\prime\prime}\in\left\\{0,1\right\\}^{n-2}}\int_{{\mathop{\symoperators
Re}}(u)=\epsilon}\widehat{W}_{n-2,\sigma_{b_{1}},b_{2},b_{3}}(u,\mu^{\prime},\delta^{\prime})\int_{{\mathop{\symoperators
Re}}(u^{\prime})=\epsilon}\widehat{W}_{2,\sigma_{b^{\prime}}}(u^{\prime},\mu^{\prime\prime},(\delta_{n-1},\delta_{n}))$
$\displaystyle\int_{{\mathop{\symoperators
Re}}(q)=\epsilon(n-3,\ldots,1)}\int_{{\mathop{\symoperators
Re}}(q^{\prime})=\epsilon}A_{d_{1},-}\left(s_{1}-q_{1}-u^{\prime}+\tfrac{\mu_{n-1}+\mu_{n}}{2}\right)A_{d^{\prime\prime}_{1},+}(s_{2}+\mu_{n-1}+\mu_{n})$
$\displaystyle
B_{\tilde{d}_{1,1},a_{1,1}}\bigl{(}1-s_{1}-s_{3}+q_{1}+q^{\prime}_{1}+u_{1}+u^{\prime}-\tfrac{3n-8}{2(n-2)}(\mu_{n-1}+\mu_{n}),s_{1}-u_{1}-\tfrac{1}{n-2}(\mu_{n-1}+\mu_{n})\bigr{)}$
$\displaystyle
B_{\tilde{d}_{2,1},a_{2,1}}\bigl{(}1-s_{1}-s_{2}+q_{1}-q^{\prime}_{1}+u^{\prime}-\tfrac{3}{2}(\mu_{n-1}+\mu_{n}),s_{2}-s_{3}-q_{1}+q^{\prime}_{1}-u^{\prime}-\tfrac{1}{2}(\mu_{n-1}+\mu_{n})\bigr{)}$
$\displaystyle\prod_{j=1}^{n-3}A_{d_{j+1},-}(q_{j}-q_{j+1})A_{d^{\prime\prime}_{j+1},+}(q^{\prime}_{j})A_{d^{\prime}_{j},+}\left(s_{j+2}-q^{\prime}_{j}-u_{j}+\tfrac{n-2-j}{n-2}(\mu_{n-1}+\mu_{n})\right)$
$\displaystyle\prod_{j=2}^{n-2}B_{\tilde{d}_{1,j},a_{1,j}}\bigl{(}1-s_{j+1}-s_{j+2}-q_{j-1}+q_{j}+q^{\prime}_{j-1}+q^{\prime}_{j}+u_{j-1}+u_{j}-\tfrac{2n-3-2j}{n-2}(\mu_{n-1}+\mu_{n}),$
$\displaystyle\qquad\qquad\qquad
s_{j+1}+q_{j-1}-q^{\prime}_{j-1}-u_{j}+u^{\prime}+\tfrac{n-2j-2}{2(n-2)}(\mu_{n-1}+\mu_{n})\bigr{)}$
$\displaystyle
B_{\tilde{d}_{2,j},a_{2,j}}\bigl{(}1-q_{j-1}+q_{j}-q^{\prime}_{j-1}-q^{\prime}_{j},s_{j+1}-s_{j+2}-q_{j}+q^{\prime}_{j}-u^{\prime}-\tfrac{1}{2}(\mu_{n-1}+\mu_{n})\bigr{)}$
$\displaystyle\frac{dq^{\prime}}{(2\pi i)^{n-3}}\frac{dq}{(2\pi
i)^{n-3}}\frac{du^{\prime}}{2\pi i}\frac{du}{(2\pi i)^{n-3}},$

where $\left\\{0,1\right\\}\ni\tilde{d}_{1,j}\equiv
d_{j}+d^{\prime}_{j}+d^{\prime}_{j-1}\pmod{2}$,
$\left\\{0,1\right\\}\ni\tilde{d}_{2,j}\equiv
d_{j}+d^{\prime\prime}_{j}+d^{\prime\prime}_{j+1}\pmod{2}$ using
$q_{n-2}=q^{\prime}_{n-2}=d^{\prime}_{0}=d^{\prime\prime}_{n-1}=0$.
Convergence of this integral is somewhat more difficult as the exponential
decay factors coming from the beta functions and Mellin transforms of the
Whittaker functions are not sufficient, but the polynomial part of Stirling’s
formula does give enough decay.

###  9.1. Jacquet integrals and limits

For the Asymptotics Theorem, we will need to evaluate the Jacquet integral of
a limit of a Whittaker function, and we perform that computation here.
Throughout this section, the letters $a,b$ as subscripts are used to
distinguish matrices of a similar type, not as indices.

For $y\in Y_{w}$, define $\alpha_{w}\in\mathbb{R}^{n-1}$ by
$\psi_{\alpha_{w}}=\lim_{y\to 0}\psi_{y}$, where the limit is taken along the
non-constant entries of $y$, so $\alpha_{w,i}$ is 0 if $i=\hat{r}_{j}$ for
some $j$ and 1 otherwise.

Suppose ${\mathop{\symoperators Re}}(\mu_{i})>{\mathop{\symoperators
Re}}(\mu_{j})$ for all $i<j$, then for $y\in Y_{w}$,

(51) $\displaystyle W_{\sigma}(yg,\mu,\delta)=$ $\displaystyle
p_{\rho+\mu^{w_{l}}}(y)\chi_{\delta^{w_{l}}}(y)W_{\sigma}(g,w_{l},\mu,\delta,\psi_{y})$
$\displaystyle\sim$ $\displaystyle
p_{\rho+\mu^{w_{l}}}(y)\chi_{\delta^{w_{l}}}(y)W_{\sigma}(g,w_{l},\mu,\delta,\psi_{\alpha_{w}}),\qquad\text{
as }y\to 0.$

Then we decompose $U=\overline{U}_{w^{-1}}U_{w^{-1}}$ so that

$\displaystyle W_{\sigma}(wg,w_{l},\mu,\delta,\psi_{\alpha_{w}})=$
$\displaystyle\int_{U(\mathbb{R})}I_{\mu,\delta,\sigma}(w_{l}uwg)\overline{\psi_{\alpha_{w}}(u)}du$
$\displaystyle=$
$\displaystyle\int_{U_{w^{-1}}(\mathbb{R})}\int_{\overline{U}_{w^{-1}}(\mathbb{R})}I_{\mu,\delta,\sigma}(w_{l}u_{a}u_{b}wg)\overline{\psi_{I}(u_{b})}du_{a}\,du_{b},$

since $\psi_{\alpha_{w}}(u_{a}u_{b})=\psi_{I}(u_{b})$. Then conjugating
$w^{-1}u_{b}w\mapsto u_{b}$ with $\psi_{I}(w^{-1}u_{b}w)=\psi_{I}(u_{b})$
gives

$\displaystyle W_{\sigma}(wg,w_{l},\mu,\delta,\psi_{\alpha_{w}})=$
$\displaystyle\int_{U_{w}(\mathbb{R})}\int_{\overline{U}_{w^{-1}}(\mathbb{R})}I_{\mu,\delta,\sigma}(w_{l}u_{a}wu_{b}g)\overline{\psi_{I}(u_{b})}du_{a}\,du_{b}.$

Now write $w_{l}u_{b}g=u^{*}y^{*}k^{*}$, then

$\displaystyle W_{\sigma}(wg,w_{l},\mu,\delta,\psi_{\alpha_{w}})=$
$\displaystyle\int_{U_{w}(\mathbb{R})}\int_{\overline{U}_{w^{-1}}(\mathbb{R})}I_{\mu,\delta,\sigma}(w^{-1}u_{a}u^{*}y^{*})\overline{\psi_{I}(u_{b})}\sigma(k^{*})du_{a}\,du_{b}$
$\displaystyle=$
$\displaystyle\int_{U_{w}(\mathbb{R})}W_{\sigma}(u^{*}y^{*},w^{-1},\mu,\delta,\psi_{0})\overline{\psi_{I}(u_{b})}\sigma(k^{*})du_{b},$

using $w_{l}w^{-1}\overline{U}_{w^{-1}}ww_{l}=\overline{U}_{w^{-1}}$ and
$w_{l}ww_{l}=w^{-1}$. Using the known properties, e.g. (17), of the degenerate
Whittaker function,

$\displaystyle W_{\sigma}(wg,w_{l},\mu,\delta,\psi_{\alpha_{w}})=$
$\displaystyle
W_{\sigma}(I,w^{-1},\mu,\delta,\psi_{0})\int_{U_{w}(\mathbb{R})}I_{\mu^{w^{-1}},\delta^{w^{-1}},\sigma}(w_{l}u_{b}g)\overline{\psi_{I}(u_{b})}du_{b}.$

Since the original integral converged absolutely, we know the $u_{b}$ integral
converges absolutely on ${\mathop{\symoperators Re}}(\mu)=\epsilon\rho$, even
at $\mu^{w^{-1}}$, and by analytic continuation, this equality holds wherever
the $u_{b}$ integral converges. In fact, the $u_{b}$ integral can be expressed
in terms of lower-rank, non-degenerate Whittaker functions, and this is entire
in $\mu$. As in section 8, the fully degenerate Whittaker function can be
evaluated and, up to matrix factors of the form $\sigma(w),w\in W$, is given
by a product of sums of quotients of gamma functions

$\int_{\mathbb{R}}(1+x^{2})^{-\frac{1+\mu_{i}-\mu_{j}}{2}}\left(\frac{1-ix}{\sqrt{1+x^{2}}}\right)^{d}dx=\mathcal{W}(0,\mu_{i}-\mu_{j},d),$

as in (24), and the possible poles occur at
$\mu_{i}-\mu_{j}\in-\mathbb{N}_{0}$ for some $i<j$.

Hence

(52)
$\displaystyle\int_{\overline{U}_{w}(\mathbb{R})}W_{\sigma}(wxt,w_{l},\mu,\delta,\psi_{\alpha_{w}})\overline{\psi_{I}(x)}dx=$
$\displaystyle
W_{\sigma}(I,w^{-1},\mu,\delta,\psi_{0})W_{\sigma}(t,\mu^{w^{-1}},\delta^{w^{-1}}),$

and this holds wherever the $x$ integral converges, in particular, on the
region ${\mathop{\symoperators Re}}(\mu^{w^{-1}}_{i})>{\mathop{\symoperators
Re}}(\mu^{w^{-1}}_{j})$ for all $i<j$.

##  10\. The Strong Interchange of Integrals

For the Strong Interchange of Integrals Conjecture, we start by taking

$f_{1}(g)=\int_{i\mathfrak{a}_{0}^{*}(\Lambda)}f(\mu)W_{\sigma}(yg,\mu,\delta)d_{\text{spec}}\mu,$

with $f(\mu)$ as in the Interchange of Integrals Conjecture and
$f_{2}(x)=\overline{\psi_{t}(x)}$ in (46), which is effectively a choice of
coordinates on $\overline{U}_{w}(\mathbb{R})$ where
$w=w_{r_{1},\ldots,r_{\ell}}$. Note that compared to the Interchange of
Integrals Conjecture, we have conjugated $x\mapsto txt^{-1}$ and substituted
$ywtw^{-1}\mapsto y$.

We can also address the convergence using (46): The final argument of the
Whittaker function is

$Y^{+}(ywx)=y\prod_{k=1}^{\ell-1}\prod_{m=\hat{r}_{k}+1}^{\hat{r}_{k+1}}\prod_{i=1}^{\hat{r}_{k}}\hat{y}_{m,i^{\hat{w}_{k}}}(u_{m,i}),\qquad\hat{w}_{k}=w_{r_{1},\ldots,r_{k}},$

and modelling $W_{\sigma}(y,\mu,\delta)$ as roughly $p_{\rho}(y)$ for small
$y$ (with, say, ${\mathop{\symoperators Re}}(\mu)=0$; the general case tends
to be worse) means that without the extra decay coming from the $\mu$
integral, each $u_{m,j}$ integral would converge like

$\int_{\mathbb{R}}p_{\rho}\left(\hat{y}_{m,j^{\hat{w}_{k}}}(u_{m,j})\right)\left(1+u_{m,j}^{2}\right)^{\frac{m-j^{\hat{w}_{k}}-1}{2}}du_{m,j}=\int_{\mathbb{R}}\left(1+u_{m,j}^{2}\right)^{-\frac{1}{2}}du_{m,j}=\infty.$

On the other hand, the slight decay given by Theorem 12 means the integrals
all converge absolutely.

Since the $\overline{U}_{w}(\mathbb{R})$ integral converges absolutely
(outside the $\mu$ integral), we can apply a smooth, dyadic partition of unity
as follows: Let $g_{1},g_{2}:\mathbb{R}\to[0,1]$ be even and smooth with
$g_{1}$ supported on $[-1,1]$ and $g_{2}$ supported on $[-4,-1]\cup[1,4]$ such
that

$g_{1}(x)+\sum_{i=1}^{\infty}g_{2}\left(x/2^{i}\right)=1$

identically on $\mathbb{R}$. We apply the partition of unity to each $u_{m,j}$
coordinate, taking $C_{m,j}=2^{i}$ for some $i\in\mathbb{N}$ or $C_{m,j}=1$
for the $g_{1}$ term. We set $\left\|C\right\|=\max_{m,j}C_{m,j}$ and pull the
(now compactly supported) $u_{m,j}$ integrals inside the $\mu$ integral.

From the matrix decompositions and in particular Lemma 11, we see that
$\sigma(K(wx))$ is a product of $\sigma$ at some fixed Weyl elements and
$\sigma\left(h_{1}\left(\arg\frac{1-iu_{m,j}}{\sqrt{1+u_{m,j}^{2}}}\right)\right)$,
which we have assumed are diagonalized, so that the entries are just powers
$\left(\frac{1-iu_{m,j}}{\sqrt{1+u_{m,j}^{2}}}\right)^{a_{m,j}}$,
$a_{m,j}\in\mathbb{Z}$ with $\left|a_{m,j}\right|\leq a_{\sigma}$ for some
$a_{\sigma}\in\mathbb{N}_{0}$, the highest weight character occurring in the
restriction of $\sigma$ to $SO(2)$. (For $n\geq 3$, we have
$a_{\sigma}\leq\dim\sigma$, but for $n=2$, $\dim\sigma\leq 2$ while
$a_{\sigma}$ can be arbitrarily large.)

Lastly, the Whittaker function has super-polynomial (actually exponential)
decay in $\left\|C\right\|$ unless each coordinate of $Y^{+}(wx)$ is
$\ll_{\mu}\left\|C\right\|^{\epsilon}$, with polynomial dependence on
$1+\left\|\mu\right\|$. We record this bound on the $C_{m,j}$ variables and
Mellin expand the Whittaker function. Recalling (33), the $i$-th
$Y$-coordinate of $\hat{y}_{m,j^{\hat{w}_{k}}}(u_{m,j})$ is
$\left(1+u_{m,j}^{2}\right)^{\eta_{n,i,m,j^{\hat{w}_{k}}}/2}$, and then

$b(s)_{m,j}=-\frac{1}{2}\sum_{i=1}^{n-1}\eta_{n,i,m,j^{\hat{w}_{k}}}s_{i}-\frac{1}{2}$

is the power of $(1+u_{m,j}^{2})$ resulting from the Mellin expansion of the
Whittaker function. To simplify notation we define

$g_{m,j,s}(u_{m,j})=\frac{1-iu_{m,j}}{\sqrt{1+u_{m,j}^{2}}}^{a_{m,j}}(1+u_{m,j}^{2})^{b(s)_{m,j}}\times\left\\{\begin{matrix}g_{1}(x)&\mbox{if
}C_{m,j}=1,\\\ g_{2}(x/C_{m,j})&\mbox{otherwise}.\end{matrix}\right.$

Note that

$\frac{d^{k}}{du_{m,j}^{k}}g_{m,j,s}(u_{m,j})\ll_{k}\left(a_{\sigma}+\left\|s\right\|\right)^{k}C_{m,j}^{-k}$

and $\sqrt{1+u_{m,j}^{2}}\asymp C_{m,j}$ on the support of $g_{m,j}(u_{m,j})$.

We inductively define $\mathcal{I}_{n}(r,a,s,C,y,t,\alpha)$ by
$\mathcal{I}_{n}((n),a,s,C,y,t,\alpha)=1$ and for $r=(r_{1},\ldots,r_{\ell})$
a non-trivial (i.e. $\ell>1$) partition of $n$,

(53) $\displaystyle\mathcal{I}_{n}(r,a,s,C,y,t,\alpha)=$
$\displaystyle\int_{\mathbb{R}^{n-r_{1}}}e\left(\sum_{i=1}^{n-r_{1}}\alpha_{i}u_{n,i}\prod_{j=1}^{i-1}\sqrt{1+u_{n,j}^{2}}-t_{n-r_{1}}u_{n,1}+\delta_{r_{1}=1}\frac{y_{r_{1}}u_{n,r_{2}}}{(1+u_{n,r_{2}}^{2})\prod_{i=1}^{r_{2}-1}\sqrt{1+u_{n,i}^{2}}}\right)$
$\displaystyle\qquad
e\left(-\sum_{i=2}^{\ell-1}t_{n-\hat{r}_{i}}\frac{u_{n,\hat{r}_{i}-r_{1}}u_{n,\hat{r}_{i}-r_{1}+1}}{\sqrt{1+u_{n,\hat{r}_{i}-r_{1}}^{2}}}-\sum_{i=2}^{\ell}\sum_{j=1}^{r_{i}-1}y_{\hat{r}_{i-1}+j}\frac{u_{n,\hat{r}_{i}-r_{1}-j}u_{n,\hat{r}_{i}-r_{1}-j+1}}{\sqrt{1+u_{n,\hat{r}_{i}-r_{1}-j}^{2}}}\right)$
$\displaystyle\qquad\mathcal{I}_{n-1}\left(\widetilde{r},a,s,C,\widetilde{y},\widetilde{t},\widetilde{\alpha}\right)\prod_{j=1}^{n-r_{1}}g_{n,j,s}(u_{n,j})du_{n,j},$

with $\widetilde{r},\widetilde{y},\widetilde{t},\widetilde{\alpha}$ as in
(40)-(44).

Finally, we have:

######  Conjecture (Strong Interchange of Integrals).

Fix $r$ and the values of $a_{m,j}\in\mathbb{Z}$ for
$j=1,\ldots,n-\hat{r}_{k}$, $n-\hat{r}_{k}+1\leq m\leq n-\hat{r}_{k-1}$,
$k=1,\ldots,\ell$ and ${\mathop{\symoperators Re}}(s_{i})$ for
$i=1,\ldots,n-1$. Then for tuples $(C_{m,j})_{m,j}$,
$\log_{2}C_{m,j}\in\mathbb{N}_{0}$ satisfying

$\prod_{k=1}^{\ell-1}\prod_{m=\hat{r}_{k}+1}^{\hat{r}_{k+1}}\prod_{j=1}^{\hat{r}_{k}}C_{m,j}^{\eta_{n,i,m,j^{\hat{w}_{k}}}}\ll\left\|C\right\|^{\epsilon},\qquad
i=1,\ldots,n-1,$

the integral $\mathcal{I}_{n}(r,a,s,C,y,t,0)$ has super-polynomial decay in
$\left\|C\right\|$ for $\left\|C\right\|$ larger than some fixed power of the
norm of $(y,t,y^{-1},t^{-1},s,a_{\sigma})$.

It is important to note that we are not conjecturing that the inner integral
$\mathcal{I}\left(\widetilde{r},a,s,C,\widetilde{y},\widetilde{t},\widetilde{\alpha}\right)$
has super-polynomial decay in the corresponding coordinates of $C$; having
nonzero $\alpha$ will interfere.

###### Proof of Proposition 3.

The Interchange of Integrals Conjecture follows by applying the decomposition
(of the $\overline{U}_{w}(\mathbb{R})$ integral into linear combinations of
$\mathcal{I}(r,a,s,C,y,t,\alpha)$, where $w=w_{r_{1},\ldots,r_{\ell}}$)
described above to get something of the form

$\displaystyle
p_{\rho-\rho^{w}}(t)\widetilde{K}_{w}(ywt^{-1}w^{-1},t,\Lambda,\mu,\delta,\sigma)_{i,j}=$
$\displaystyle\lim_{R\to\infty}\sum_{\left\|C\right\|\leq
R}\sum_{a,b,i^{\prime},j^{\prime}}d_{i,j,a,b,i^{\prime},j^{\prime}}\int_{{\mathop{\symoperators
Re}}(s)=\mathfrak{s}}\widehat{W}_{n,\sigma}(s,\mu,\delta)_{i^{\prime},j^{\prime}}\hat{p}_{-s}(y)\mathcal{I}_{n}(r,a,s,C,y,t,0)\frac{ds}{(2\pi
i)^{n-1}},$

for appropriate coefficients $d_{i,j,a,b,i^{\prime},j^{\prime}}\in\mathbb{C}$
and $\mathfrak{s}$ is sufficiently large compared to ${\mathop{\symoperators
Re}}(\mu)$. The Strong Interchange of Integrals Conjecture implies we can
freely interchange the $R$ limit with the $\mu$ in the Interchange of
Integrals Conjecture. Note that differentiating
$\mathcal{I}(r,a,s,C,y,t,\alpha)$ in $y$, $t$ and $s$ produces linear
combinations of integrals of the same form having different values for $a$,
$b$ and $d_{i,j,a,b,i^{\prime},j^{\prime}}$, while the trivial bound on

$\mathcal{I}_{n}(r,a,s,C,y,t,0)$

is clearly polynomial in $s$, hence
$\widetilde{K}_{w}(y,t,\Lambda,\mu,\delta,\sigma)$ is polynomially bounded in
$\left\|\mu\right\|$.

Furthermore, the Analytic Continuation Conjecture follows because the
Whittaker function is entire in $\mu$ and our construction of
$\widetilde{K}_{w}(y,t,\Lambda,\mu,\delta,\sigma)$ above is visibly
independent of $\Lambda$. ∎

In the integral for the Bessel function, taking $y=0$ (in the complex
exponential), $w=w_{l}$, and replacing the Whittaker function with the power
function gives the Jacquet integral for the Whittaker function, hence

######  Conjecture (Jacquet-Whittaker Direct Continuation).

Fix the values of $a_{m,j}\in\mathbb{Z}$ for $j=1,\ldots,n-\hat{r}_{k}$,
$n-\hat{r}_{k}+1\leq m\leq n-\hat{r}_{k-1}$, $k=1,\ldots,\ell$. Suppose
$\left|{\mathop{\symoperators Re}}(\mu_{i})\right|\leq\delta$ for
$i=1,\ldots,n-1$ and some fixed $\delta>0$. Then for tuples $(C_{m,j})_{m,j}$,
$\log_{2}C_{m,j}\in\mathbb{N}_{0}$ and some fixed $\epsilon>0$, we have

$\mathcal{I}_{n}((1,\ldots,1),a,\hat{\mu},C,0,t,0)\ll\left\|C\right\|^{-\epsilon}$

for $\left\|C\right\|$ larger than some fixed power of the norm of
$(t,t^{-1},\mu,a_{\sigma})$.

##  11\. The Asymptotic Analysis

Assume the Interchange of Integrals, Analytic Continuation and Differential
Equations and Power Series Conjectures; we want to prove the Asymptotics
Theorem. The Weyl invariant of part 1 follows by Proposition 2, the
$W_{w}$-invariance of $J_{w}(y,\mu,\delta)$ and the holomorphy of both
$K_{w}(y,\mu,\delta)$ and $J_{w}(y,\mu,\delta)$; the meromorphy follows from
part 2. For the remaining parts, we first prove the informal Asymptotics
Conjecture; i.e. we derive the relation between the asymptotics of the Bessel
functions and the Whittaker functions. This is primarily Lemma 13, below. Then
we determine the coefficients in terms of Shahidi’s local coefficients in
section 11.2. We generally view $GL(n-j)\subset GL(n)$ by the upper-left
embedding.

###  11.1. Proof of the Informal Asymptotics Conjecture

Let $w=w_{r_{1},\ldots,r_{\ell}}\in W^{\text{rel}}$ where
$r_{1}+\ldots+r_{\ell}=n$. For convenience of notation, we extend (for this
section only) by the Iwasawa decomposition for $x\in U(\mathbb{R})$, $y\in
Y^{+}$, $k\in K$, and any character $\psi$ of $U(\mathbb{R})$,

$\psi(xyk)=\psi(x),\quad
p_{S,\mu}(xyk)=p_{S,\mu}(y),\quad\hat{p}_{S^{c},-s_{S^{c}}}(xyk)=\hat{p}_{S^{c},-s_{S^{c}}}(y),\quad\sigma(xyk)=\sigma(k),$

recalling $p_{S,\mu}$ and $\hat{p}_{S^{c},-s_{S^{c}}}$ from section 6.4.1.

Define a test function $f:M_{n}\to\mathbb{C}$ by

$f(\mu)=f_{\mathcal{S}}(\mu)f_{\mathcal{S}^{c}}(\mu),\qquad
f_{\mathcal{S}^{c}}(\mu):=\prod_{i\in\mathcal{S}^{c}}f_{0}(\mu_{n-i}),\qquad
f_{\mathcal{S}}(\mu):=\prod_{i\in\mathcal{S}}f_{1}(\hat{\rho}_{i},\hat{\mu}_{i}),$

where $\mathcal{S}=\left\\{\hat{r}_{i}\,\middle|\,i=1,\ldots,\ell\right\\}$,

$f_{0}(t)=2\sqrt{\pi}R_{0}e^{R_{0}^{2}t^{2}},\qquad
f_{1}(u,t)=\frac{R_{1}e^{t^{2}}}{u-R_{1}t},\qquad R_{0}>1,\qquad
R_{1}>4+\frac{1}{\epsilon}.$

Let $\beta\in M_{n}^{0}$, then we apply the Interchange of Integrals
Conjecture to the test function $f(\mu-\beta)$, we have

$\displaystyle\mathcal{I}:=$
$\displaystyle\int_{\overline{U}_{w}(\mathbb{R})}\int_{M_{n}^{0}}f(\mu)W_{\sigma}(ywxy^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-1}}\overline{\psi_{I}(x)}dx$
$\displaystyle=\int_{M_{n}^{0}}f(\mu)K_{w}(y,\mu+\beta,\delta)W_{\sigma}(y^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-1}}.$

We will compare asymptotics as $R_{1}\to\infty$ with

$y_{j}={\mathop{\symoperators\,sgn}}(y_{j})e^{-R_{1}^{2}},\qquad
j\in\mathcal{S}$

in the two expressions for $\mathcal{I}$, and then take the limit as
$R_{0}\to\infty$.

######  Lemma 13.

Let

$T=p_{(1+R_{1}^{-1})\rho}(y)\prod_{i\in\mathcal{S}}y_{i}^{\hat{\beta}_{i}}{\mathop{\symoperators\,sgn}}(y_{i})^{\hat{\delta}_{i}},$

the leading term as $R_{1}\to\infty$ is proportional to $T$. More precisely,

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}(\mu)C_{w}^{*}(\mu+\beta,\delta)W_{\sigma}(y^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{-1}\right)$ $\displaystyle=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}(\mu)\widetilde{C}_{w}(\mu+\beta,\delta)W_{\sigma}(y^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{-1}\right),$

where $\widetilde{C}_{w}(\beta,\delta)$ is the scalar determined by

(54) $\displaystyle
W_{\sigma}(I,w^{-1},\mu^{w_{l}},\delta^{w_{l}},\psi_{0})=\widetilde{C}_{w}(\mu,\delta)M_{\sigma}(w^{-1},\mu^{w_{l}},\delta^{w_{l}}).$

The lemma implies the asymptotics as follows: Equating the main terms, we may
then send $R_{0}\to\infty$ and divide by $W_{\sigma}(y^{\prime},\beta,\delta)$
(which is not identically zero for $y^{\prime}$ sufficiently small) to
conclude

$\displaystyle C^{*}_{w}(\beta,\delta)=\widetilde{C}_{w}(\beta,\delta),$

for all $\delta$ and by analytic continuation all $\beta\in M_{n}$.

We will now proceed to prove the lemma, but first a simple lemma:

######  Lemma 14.

Let $x\in\overline{U}_{w}(\mathbb{R})$. If we write the Iwasawa decomposition
of $wx$ as $wx=x^{*}y^{*}k^{*}$ then

  1. 1.

for $j=1,\ldots,n-1$, we have $x^{*}_{j,j+1}\ll_{n}p_{-\rho}(y^{*})$,

  2. 2.

for any $\alpha\in[-1,1]^{n-1}$, we have $y_{1}^{*\alpha_{1}}\cdots
y_{n-1}^{*\alpha_{n-1}}\leq p_{-2\rho}(y^{*})$.

###### Proof.

We may use the coordinates on $x$ defined by the substitutions throughout
section 8, and we note that in those coordinates $y^{*}$ is a product of the
matrices $\hat{y}_{m,i}(x)$, $i<m\leq n$ of Lemma 11. So it’s enough to prove
part 2 for such a $\hat{y}_{m,i}(x)$, and we see that

$p_{-\rho}\left(\hat{y}_{m,i}(x)\right)=\left(1+x^{2}\right)^{\frac{\rho_{i}-\rho_{m}}{2}}=\left(1+x^{2}\right)^{\frac{m-i}{2}}\geq\sqrt{1+x^{2}},$

while (recall (33)) the $j$-th $Y$-coordinate of $\hat{y}_{m,i}(x)$ is
$\left(1+x^{2}\right)^{\eta_{n,j,m,i}/2}$ and $\left|\eta_{n,j,m,i}\right|\leq
2$.

Similarly, $x^{*}_{j,j+1}$ is a sum of terms appearing in the complex
exponential in (46) (or possibly deeper in the inductive definition). Since
each $x$ coordinate $u_{n,i}$ appears in a numerator as either $u_{n,i}$ or
$\sqrt{1+u_{n,i}^{2}}$ (possibly deeper in the inductive definition through
$\widetilde{y},\widetilde{t}$ or $\widetilde{\alpha}$) and only ever once in a
single summand, we immediately have part 1. ∎

####  11.1.1. The integral transform side

For $S\subset[n-1]$, define $\rho_{S}\in M_{n}$ and $\hat{\rho}_{S^{c}}$ as
follows: If $i_{1},i_{2}\in S\cup\left\\{0,n\right\\}$ with $i_{1}<i_{2}$ and
$i_{1}<i_{3}<i_{2}\Rightarrow i_{3}\notin S$, define

$\rho_{S,i_{3}}=\frac{\hat{\rho}_{i_{2}}-\hat{\rho}_{i_{1}}}{i_{2}-i_{1}}=\frac{n-i_{1}-i_{2}}{2}$

for all $i_{1}<i_{3}\leq i_{2}$; in particular, we have
$\hat{\rho}_{S,\hat{r}_{i}}=\hat{\rho}_{\hat{r}_{i}}$ for all
$i=1,\ldots,\ell-1$. Define
$\hat{\rho}_{S^{c}}=(\hat{\rho}_{i^{\prime}_{1}},\ldots,\hat{\rho}_{i^{\prime}_{n-1-q}})$,
where $S^{c}=\left\\{i^{\prime}_{1},\ldots,i^{\prime}_{n-1-q}\right\\}$ in
increasing order. Note that if ${\mathop{\symoperators
Re}}(\mu)=\alpha\rho_{S}$, then ${\mathop{\symoperators
Re}}(\hat{\mu}_{i})=\alpha\hat{\rho}_{i}$ for all $i\in S$ and any
$\alpha\in\mathbb{R}$, and if also ${\mathop{\symoperators
Re}}(s_{S^{c}})=-\alpha\hat{\rho}_{S^{c}}$, then

(55)
$\displaystyle\left|p_{S,\mu}(y)\hat{p}_{S^{c},-s_{S^{c}}}(y)\right|=p_{\alpha\rho}(y).$

Applying (21), we have

$\displaystyle\mathcal{I}=$ $\displaystyle\sum_{S\subset[n-1]}\sum_{w_{b}\in
W/W_{S}}\int_{\overline{U}_{w}(\mathbb{R})}\psi_{y}(wxy^{\prime})p_{\rho}(ywxy^{\prime})\int\limits_{\begin{subarray}{c}M_{n}\\\
{\mathop{\symoperators
Re}}(\mu)=\alpha\rho_{S}\end{subarray}}f(\mu^{w_{b}^{-1}})p_{S,\mu+\beta^{w_{b}}}(ywxy^{\prime})$
$\displaystyle\qquad\int\limits_{{\mathop{\symoperators
Re}}(s_{S^{c}})=-\alpha\hat{\rho}_{S^{c}}}\widehat{W}_{n,\sigma}(s_{S^{c}},\mu+\beta^{w_{b}},\delta)\hat{p}_{S^{c},-s_{S^{c}}}(ywxy^{\prime})\frac{ds_{S^{c}}}{(2\pi
i)^{\left|S^{c}\right|}}$
$\displaystyle\qquad\sigma(ywxy^{\prime})\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-1}}\overline{\psi_{I}(x)}dx,$

where $\alpha=n^{-1}R_{1}^{-1}$. (We will use $\alpha$ as a parameter in our
contour shifting arguments.)

####  11.1.2. Contour shifting

We pause to consider the contour shifting implicit in the above. Starting from
${\mathop{\symoperators Re}}(\mu)=0$ and the contours ${\mathop{\symoperators
Re}}(s_{S^{c}})=-\epsilon$ in (21), we have shifted first
${\mathop{\symoperators Re}}(\mu)$ leftward to $-\alpha\rho_{S}$ and then
${\mathop{\symoperators Re}}(s_{S^{c}})$ rightward to
$-\alpha\hat{\rho}_{S^{c}}$. The poles of
$\widehat{W}_{n,\sigma}(s,\mu,\delta)$ occur whenever $-s=\hat{\mu}^{w_{c}}+m$
for some $m\in\mathbb{N}_{0}^{n-1}$ and $w_{c}\in W$, and our contour shifting
in $\mu$ is certainly small enough that we won’t cross any poles, so when we
start shifting the $s_{S^{c}}$ contours rightward, we have
${\mathop{\symoperators Re}}(s_{i}+\hat{\mu}^{w_{c}}_{i})\in(-1,0)$ for all
$i\in S^{c}$, $w_{c}\in W_{S}$ and
$\widehat{W}_{n,\sigma}(s_{S^{c}},\mu,\delta)$ is holomorphic on this region.
Note that this condition forms a convex subset of the parameter space
$({\mathop{\symoperators Re}}(s_{S^{c}}),{\mathop{\symoperators Re}}(\mu))$.
Therefore, if the endpoint of the contour shifting also satisfies this
condition, the contour shifting may be performed by a (linear) substitution on
the $s_{S^{c}}$ and a contour shift on a single $s_{i}$ integral (which is
trivial to accomplish by the rapid decay of
$\widehat{W}_{n,\sigma}(s_{S^{c}},\mu,\delta)$), while remaining in the region
of holomorphy of $\widehat{W}_{n,\sigma}(s_{S^{c}},\mu,\delta)$. Again, it is
sufficient to check that ${\mathop{\symoperators
Re}}(s_{i}+\hat{\mu}^{w_{c}}_{i})<0$, since the shifts are small and in fact
we are shifting the $s_{i}$ contours in the positive direction.

Now the coordinates of $\mu^{w_{c}}$ for an arbitrary $w_{c}\in W_{S}$ are
determined as follows: if $i_{1},i_{2}\in S\cup\left\\{0,n\right\\}$ with
$i_{1}<i_{2}$ and $i_{1}<i_{3}<i_{2}\Rightarrow i_{3}\notin S$, then
$\left\\{\mu^{w_{c}}_{i_{1}+1},\ldots,\mu^{w_{c}}_{i_{2}}\right\\}=\left\\{\mu_{i_{1}+1},\ldots,\mu_{i_{2}}\right\\}$,
since the powers of $y_{i_{1}}$ and $y_{i_{2}}$ in the restricted power
function $p_{S,\mu^{w_{c}}}(y)$ must agree with $p_{S,\mu}(y)$. We see that

$\displaystyle\alpha^{-1}{\mathop{\symoperators
Re}}(\hat{\mu}^{w_{c}}_{i_{3}})=$
$\displaystyle\hat{\rho}_{i_{1}}+\rho_{S,i_{1}+1}+\ldots+\rho_{S,i_{3}}$
$\displaystyle=$
$\displaystyle\hat{\rho}_{i_{1}}+(i_{3}-i_{1})\frac{n-i_{1}-i_{2}}{2}$
$\displaystyle<$
$\displaystyle\hat{\rho}_{i_{1}}+(i_{3}-i_{1})\frac{n-i_{1}-i_{3}}{2}$
$\displaystyle=$
$\displaystyle\hat{\rho}_{i_{1}}+\rho_{i_{1}+1}+\ldots+\rho_{i_{3}}$
$\displaystyle=$ $\displaystyle\hat{\rho}_{i_{3}}$ $\displaystyle=$
$\displaystyle-\alpha^{-1}{\mathop{\symoperators Re}}(s_{i_{3}}),$

for all $i_{1}<i_{3}<i_{2}$, and in particular, ${\mathop{\symoperators
Re}}(s_{i_{3}}+\hat{\mu}^{w_{c}}_{i_{3}})<0$ holds whenever $i_{3}\in S^{c}$.

####  11.1.3. Returning to the main argument

For $j\in\mathcal{S}$, the total exponent on $y_{j}$ in our previous
expression for $\mathcal{I}$ has real part $(1+\alpha)\hat{\rho}_{j}$ and as
above, we have

$\frac{1}{\alpha}{\mathop{\symoperators
Re}}(\hat{\mu}^{w_{b}^{-1}}_{j})\leq\hat{\rho}^{w_{b}^{-1}}_{j}\leq\hat{\rho}_{j}.$

Here the first inequality is strict when
$\hat{\mu}^{w_{b}^{-1}}_{j}=\hat{\mu}_{i}$ with $i\in S^{c}$, and in that case

$\frac{1}{\alpha}{\mathop{\symoperators
Re}}(\hat{\mu}^{w_{b}^{-1}}_{j})\leq\hat{\rho}_{j}-\frac{1}{2}$

by the computation of the previous section. The second inequality is strict
when $w_{b}$ fails to fix the set $\left\\{\mu_{1},\ldots,\mu_{j}\right\\}$
and in that case

$\frac{1}{\alpha}{\mathop{\symoperators
Re}}(\hat{\mu}^{w_{b}^{-1}}_{j})\leq\hat{\rho}_{j}-2.$

So we shift $\alpha\mapsto\left(1+n^{-3}\right)R_{1}^{-1}$, possibly
encountering poles at $\hat{\mu}^{w_{b}^{-1}}_{j}=R_{1}^{-1}\hat{\rho}_{j}$.
By the above reasoning and recalling (55), the least power of $y$ (that is,
the least power of $e^{-R_{1}^{2}}$) occurring in the shift is
$p_{(1+R_{1}^{-1})\rho}(y)$ and this happens when $\mathcal{S}\subset S$ and
$w_{b}\in W_{w}$. (One might be confused, as the author sometimes is, why the
minimum power is not
$p_{\rho+R_{1}^{-1}\rho^{w_{l}}}(y)=p_{(1-R^{-1})\rho}(y)$ at $w_{b}=w_{l}$;
the answer is simply that when $w_{b}=w_{l}$, shifting $\alpha$ in the
positive direction means heading away from the poles, so the contour shifting
actually ends at power $p_{(1+(1+n^{-3})R_{1}^{-1})\rho}(y)$ without
encountering a single pole.) For the other terms, we gain at least a power
$\frac{1}{n^{3}R_{1}}$ of some $y_{j}$, $j\in\mathcal{S}$, and we have

$\left|p_{\rho}(wxy^{\prime})p_{S,\mu+\beta^{w_{b}}}(wxy^{\prime})\hat{p}_{S^{c},-s_{S^{c}}}(wxy^{\prime})\right|\ll_{y^{\prime}}p_{\left(1+\left(1-n^{-2}\right)R_{1}^{-1}\right)\rho}(wxy^{\prime}),$

by Lemma 14. So the $x$ integral converges with a pole of order at most
$\frac{n(n-1)}{2}$ at $R_{1}=\infty$. We handle the residues by substituting

$\mu^{w_{b}^{-1}}=\mu_{\mathcal{S}}+\sum_{i=1}^{\ell-1}t_{i}(e_{\hat{r}_{i}}-e_{\hat{r}_{i}+1}),$

where $\mu_{\mathcal{S}}\in M_{\mathcal{S}}$ and then taking the residues at
each

$t_{i}=R_{1}^{-1}\hat{\rho}_{\hat{r}_{i}}.$

Therefore,

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\left(\prod_{i\in\mathcal{S}}e^{\hat{\rho}_{i}^{2}/R_{1}^{2}}\right)\sum_{\mathcal{S}\subset
S\subset[n-1]}\sum_{w_{b}\in
W_{w}/W_{S}}\int_{\overline{U}_{w}(\mathbb{R})}\psi_{y}(wxy^{\prime})p_{\rho}(wxy^{\prime})$
$\displaystyle\qquad\int\limits_{\begin{subarray}{c}M_{\mathcal{S}}\\\
{\mathop{\symoperators
Re}}(\mu+R_{1}^{-1}\rho^{*})=R_{1}^{-1}\rho^{w_{b}^{-1}}_{S}\end{subarray}}f_{\mathcal{S}^{c}}^{*}(\mu+R_{1}^{-1}\rho^{*})p_{S,(\mu+R_{1}^{-1}\rho^{*}+\beta)^{w_{b}}}(wxy^{\prime})$
$\displaystyle\qquad\int\limits_{{\mathop{\symoperators
Re}}(s_{S^{c}})=-R_{1}^{-1}\hat{\rho}_{S^{c}}}\widehat{W}_{n,\sigma}(s_{S^{c}},(\mu+R_{1}^{-1}\rho^{*}+\beta)^{w_{b}},\delta)\hat{p}_{S^{c},-s_{S^{c}}}(wxy^{\prime})\frac{ds_{S^{c}}}{(2\pi
i)^{\left|S^{c}\right|}}$
$\displaystyle\qquad\sigma(wxy^{\prime})\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}\overline{\psi_{I}(x)}dx+O\left(R_{1}^{n^{2}}e^{-n^{-3}R_{1}}\right),$

where

$\rho^{*}=\sum_{i=1}^{\ell-1}\hat{\rho}_{\hat{r}_{i}}(e_{\hat{r}_{i}}-e_{\hat{r}_{i}+1}),$

using the standard basis elements $e_{i}$.

A reality check for the $\mu$ contour: To determine if this contour is
possible, it is sufficient to check that $\rho^{w_{b}^{-1}}_{S}-\rho^{*}\in
M_{\mathcal{S}}$, or in other words, that

$\hat{\rho}^{w_{b}^{-1}}_{S,\hat{r}_{i}}=\hat{\rho}_{S,\hat{r}_{i}}=\hat{\rho}_{\hat{r}_{i}}=\hat{\rho}^{*}_{\hat{r}_{i}},\qquad
i=1,\ldots,\ell-1,$

and this is true since $w_{b}^{-1}\in W_{w}$ and $\mathcal{S}\subset S$ and by
construction of $\rho^{*}$.

Now we may remove the factor $\psi_{y}(wxy^{\prime})$ by applying

$\left|e\left(y_{j}x^{*}_{j,j+1}\right)-1\right|\ll\left|y_{j}x^{*}_{j,j+1}\right|^{\frac{1}{n^{3}R_{1}}}\ll_{n,y^{\prime}}\left|y_{j}\right|^{\frac{1}{n^{3}R_{1}}}p_{-\frac{1}{n^{3}R_{1}}\rho}(wxy^{\prime}),$

for each $j\in\mathcal{S}$, by Lemma 14, where $x^{*}y^{*}k^{*}=wxy^{\prime}$.
Thus we have

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\left(\prod_{i\in\mathcal{S}}e^{\hat{\rho}_{i}^{2}/R_{1}^{2}}\right)\int_{\overline{U}_{w}(\mathbb{R})}\mathcal{I}_{1}(x)\overline{\psi_{I}(x)}dx+O\left(R_{1}^{n^{2}}e^{-n^{-3}R_{1}}\right),$
$\displaystyle\mathcal{I}_{1}(x):=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}^{*}(\mu+R_{1}^{-1}\rho^{*})\mathcal{I}_{2}(x,\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}$ $\displaystyle\mathcal{I}_{2}(x,\mu,\delta):=$
$\displaystyle\sum_{\mathcal{S}\subset S\subset[n-1]}\sum_{w_{b}\in
W_{w}/W_{S}}p_{\rho}(wxy^{\prime})p_{S,\mu^{w_{b}}}(wxy^{\prime})\int\limits_{{\mathop{\symoperators
Re}}(s_{S^{c}})=-\epsilon}\widehat{W}_{n,\sigma}(s_{S^{c}},\mu^{w_{b}},\delta)$
$\displaystyle\qquad\hat{p}_{S^{c},-s_{S^{c}}}(wxy^{\prime})\frac{ds_{S^{c}}}{(2\pi
i)^{\left|S^{c}\right|}}\sigma(wxy^{\prime}),$

after shifting the $\mu$ and $s_{S^{c}}$ contours in the interated integral
(as we may).

####  11.1.4. Asymptotics and functional equations of the Whittaker function

Suppose that

$0<{\mathop{\symoperators
Re}}(\mu_{j}-\mu_{i})<\frac{\epsilon}{n^{2}},\qquad\forall i<j,$

then by apply the exact same reasoning as the previous section directly to
(21), we see that $\mathcal{I}_{2}$ is the first term asymptotic of

$\displaystyle W_{\sigma}(ywxy^{\prime},\mu,\delta)=$ $\displaystyle
p_{\rho+\mu}(y)\chi_{\delta}(y)M_{\sigma}(w_{l},\mu,\delta)W_{\sigma}(wxy^{\prime},w_{l},\mu^{w_{l}},\delta^{w_{l}},\psi_{y})$
$\displaystyle\sim$ $\displaystyle
p_{\rho+\mu}(y)\chi_{\delta}(y)M_{\sigma}(w_{l},\mu,\delta)W_{\sigma}(wxy^{\prime},w_{l},\mu^{w_{l}},\delta^{w_{l}},\psi_{\alpha_{w}})$

as in (51). So we compute that from the Jacquet integral and then apply
analytic continuation in $\mu$. Note: The reversal from the region
${\mathop{\symoperators Re}}(\mu_{i}-\mu_{j})>0,\forall i<j$ where we did the
work of the previous section to the region ${\mathop{\symoperators
Re}}(\mu_{j}-\mu_{i})>0,\forall i<j$ used above has to do with the contour
shifting and the pole of $f_{1}$ – when ${\mathop{\symoperators Re}}(\mu)$ is
fixed, then the lead asymptotic in $y_{j}$ is the one encountered first while
shifting the $s$ contours to the left.

From (52), we have

(56)
$\displaystyle\int_{\overline{U}_{w}(\mathbb{R})}\mathcal{I}_{2}(x,\mu,\delta)\overline{\psi_{I}(x)}dx=$
$\displaystyle\qquad
p_{\rho+\mu}(y)\chi_{\delta}(y)M_{\sigma}(w_{l},\mu,\delta)W_{\sigma}(I,w^{-1},\mu^{w_{l}},\delta^{w_{l}},\psi_{0})W_{\sigma}(y^{\prime},\mu^{w_{l}w^{-1}},\delta^{w_{l}w^{-1}}),$

provided the $x$ integral converges, and ${\mathop{\symoperators
Re}}(\mu)=R_{1}^{-1}\rho^{ww_{l}}$ is sufficient for that.

Now we would like to shift the $\mu$ integral in $\mathcal{I}_{1}(x)$ to
${\mathop{\symoperators Re}}(\mu)=R_{1}^{-1}(\rho^{ww_{l}}-\rho^{*})$ in order
to accommodate that condition. To determine if this is possible, it is
sufficient to check that $\rho^{ww_{l}}-\rho^{*}\in M_{\mathcal{S}}$, or in
other words, that

$\hat{\rho}_{\hat{r}_{i}}^{ww_{l}}=\hat{\rho}^{*}_{\hat{r}_{i}}=\hat{\rho}_{\hat{r}_{i}},\qquad
i=1,\ldots,\ell-1,$

and since $ww_{l}=\operatorname{diag}(w_{l,r_{1}},\ldots,w_{l,r_{\ell}})$, it
is trivial that $ww_{l}$ fixes the required blocks of coordinates. Then we may
apply (56) and the end result is

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\left(\prod_{i\in\mathcal{S}}e^{\hat{\rho}_{i}^{2}/R_{1}^{2}}\right)\int\limits_{\begin{subarray}{c}M_{\mathcal{S}}\\\
{\mathop{\symoperators
Re}}(\mu)=R_{1}^{-1}(\rho^{ww_{l}}-\rho^{*})\end{subarray}}f_{\mathcal{S}^{c}}(\mu+R_{1}^{-1}\rho^{*})M_{\sigma}(w_{l},\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)$
$\displaystyle\qquad
W_{\sigma}(I,w^{-1},(\mu+R_{1}^{-1}\rho^{*}+\beta)^{w_{l}},\delta^{w_{l}},\psi_{0})$
$\displaystyle\qquad
W_{\sigma}(y^{\prime},(\mu+R_{1}^{-1}\rho^{*}+\beta)^{w_{l}w^{-1}},\delta^{w_{l}w^{-1}})\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{n^{2}}e^{-n^{-3}R_{1}}\right)$ $\displaystyle=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}(\mu)M_{\sigma}(w_{l},\mu+\beta,\delta)W_{\sigma}(I,w^{-1},(\mu+\beta)^{w_{l}},\delta^{w_{l}},\psi_{0})$
$\displaystyle\qquad
W_{\sigma}(y^{\prime},(\mu+\beta)^{w_{l}w^{-1}},\delta^{w_{l}w^{-1}})\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{-1}\right),$

by the Mean Value Theorem.

Applying (54), we have

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}(\mu)\widetilde{C}_{w}(\mu+\beta,\delta)W_{\sigma}(y^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{-1}\right).$

####  11.1.5. The power series side

From (12), we have

$\displaystyle\mathcal{I}=$ $\displaystyle\sum_{w_{b}\in
W/W_{w}}\int_{M_{n}^{0}}f(\mu^{w_{b}^{-1}})C_{w}^{*}(\mu+\beta^{w_{b}},\delta^{w_{b}})$
$\displaystyle\qquad
J_{w}^{*}(y,\mu+\beta^{w_{b}},\delta^{w_{b}})W_{\sigma}(y^{\prime},\mu^{w_{b}^{-1}}+\beta^{w_{a}},\delta^{w_{a}})\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-1}}.$

As before, we shift to ${\mathop{\symoperators Re}}(\mu)=\epsilon\rho$ and the
leading term occurs when $w_{b}\in W_{w}$, so

$\displaystyle\mathcal{I}=$
$\displaystyle\int\limits_{\begin{subarray}{c}M_{\mathcal{S}}\\\
{\mathop{\symoperators
Re}}(\mu)=R_{1}^{-1}(\rho-\rho^{*})\end{subarray}}f_{\mathcal{S}^{c}}(\mu)C_{w}^{*}(\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)J_{w}^{*}(y,\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)$
$\displaystyle\qquad
W_{\sigma}(y^{\prime},\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(Te^{-R_{1}}\right).$

Note that the presumed bound on the coefficients in part 6 of the Differential
Equations and Power Series Conjecture implies that
$J_{w}^{*}(y,\mu,\delta)\ll_{\mu,n}p_{\rho+{\mathop{\symoperators
Re}}(\mu)}(y)$ on the region with all $y_{i}\ll 1$.

Applying the power series expansion of $J_{w}^{*}$, we have

$\displaystyle T^{-1}\mathcal{I}=$
$\displaystyle\int\limits_{\begin{subarray}{c}M_{\mathcal{S}}\\\
{\mathop{\symoperators
Re}}(\mu)=R_{1}^{-1}(\rho-\rho^{*})\end{subarray}}f_{\mathcal{S}^{c}}(\mu+R_{1}^{-1}\rho^{*})C_{w}^{*}(\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)$
$\displaystyle\qquad
W_{\sigma}(y^{\prime},\mu+R_{1}^{-1}\rho^{*}+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(e^{-R_{1}}\right)$ $\displaystyle=$
$\displaystyle\int_{M_{\mathcal{S}}^{0}}f_{\mathcal{S}^{c}}(\mu)C_{w}^{*}(\mu+\beta,\delta)W_{\sigma}(y^{\prime},\mu+\beta,\delta)\frac{d_{\text{spec}}\mu}{(2\pi
i)^{n-\ell}}+O\left(R_{1}^{-1}\right).$

###  11.2. Computing the $C^{*}_{w}(\beta,\delta)$ coefficients

Since the degenerate Whittaker function is also the intertwining operator and
induces a map between Whittaker models, multiplicity one implies it acts as a
scalar multiple of the functional equation (18). So we can take
$C^{\dagger}_{w}(\beta,\delta)$ to be the scalar determined by

(57) $\displaystyle
W_{\sigma}(I,\mu,\delta,w,\psi_{0})=C^{\dagger}_{w}(\mu,\delta)M_{\sigma}(w,\mu,\delta).$

This definition relates to $\widetilde{C}_{w}(\mu,\delta)$ as in Lemma 13 by

(58) $\displaystyle C^{\dagger}_{w}(\mu,\delta)=$
$\displaystyle\widetilde{C}_{w^{-1}}(\mu^{w_{l}},\delta^{w_{l}})=C^{*}_{w^{-1}}(\mu^{w_{l}},\delta^{w_{l}}).$

These are the local coefficients in Shahidi’s chapter 5 [Shah01].

Using [Shah01, Theorem 4.2.2] and the definition of
$M_{\sigma}(w,\mu,\delta)$, we have the factorizations

$\displaystyle W_{\sigma}(I,w_{a}w_{b},\mu,\delta,\psi_{0})=$ $\displaystyle
W_{\sigma}(I,w_{a},\mu,\delta,\psi_{0})W_{\sigma}(I,w_{b},\mu^{w_{a}},\delta^{w_{a}},\psi_{0}),$
$\displaystyle M_{\sigma}(w_{a}w_{b},\mu,\delta)=$ $\displaystyle
M_{\sigma}(w_{a},\mu,\delta)M_{\sigma}(w_{b},\mu^{w_{a}},\delta^{w_{a}})$

and it follows that

$C^{\dagger}_{w_{a}w_{b}}(\mu,\delta)=C^{\dagger}_{w_{a}}(\mu,\delta)C^{\dagger}_{w_{b}}(\mu^{w_{a}},\delta^{w_{a}}).$

In [Shah01, Theorem 4.2.2], the notation is rather different and there are
conditions on $\mu$, but with analytic continuation, they are sufficient to
factor any given $w$ into a product of transpositions of adjacent indices.

By this factorization of the functional equations, it’s enough to consider $w$
a transposition of adjacent indices when computing
$C^{\dagger}_{w}(\beta,\delta)$. By conjugating $\sigma$ as necessary (using
the inductive formula for the Whittaker function (38)), it’s enough to
consider the Whittaker function on the upper left copy of $GL(2,\mathbb{R})$.
Comparing (26) to (25) at $y=0$ using (24), we have

$C^{\dagger}_{w}(\mu,\delta)=(-1)^{\delta_{1}}\pi^{\frac{1}{2}-\mu_{1}+\mu_{2}}i^{m}\frac{\Gamma\left(\frac{m+\mu_{1}-\mu_{2}}{2}\right)}{\Gamma\left(\frac{1+m-\mu_{1}+\mu_{2}}{2}\right)},$

where $\left\\{0,1\right\\}\ni m\equiv\delta_{1}+\delta_{2}\pmod{2}$.

Part 2 of the Asymptotics Theorem follows by induction over the factorization
$w=w_{(w(n)\,w(n)+1)}w^{\prime}$ where now $w^{\prime}(n)=w(n)+1$ and we use
the fact that $w^{-1}(j)=j^{w}$.

To prove the last piece of the Asymptotics Theorem, it’s enough to show four
things: For $u\in\mathbb{C}$, $\delta\in\left\\{0,1\right\\}^{n}$ and
$\left\\{0,1\right\\}\ni m\equiv\delta_{1}+\delta_{2}\pmod{2}$,

$\mathbf{sin}(\mu^{w},\delta^{w})=\mathbf{sin}(\mu,\delta){\mathop{\symoperators\,sgn}}(w),$
$\pi^{\frac{1}{2}+u}\frac{\Gamma\left(\frac{m-u}{2}\right)}{\Gamma\left(\frac{1+m+u}{2}\right)}\times(2\pi)^{-u}\Gamma\left(1+u\right)\times\sin\frac{\pi}{2}\left(m+u\right)=(-1)^{m}(-\pi),$
$(-1)^{\delta_{2}}i^{m}\times\frac{\sin\frac{\pi}{2}\left(u+\delta_{1}-\delta_{2}\right)}{\sin\frac{\pi}{2}\left(m+u\right)}\times(-1)^{m}=(-i)^{\delta_{1}+\delta_{2}}$
$\prod_{j<k}(-i)^{\delta_{j}+\delta_{k}}=i^{-(n-1)(\delta_{1}+\ldots+\delta_{n})}.$

The last one is $\\#\left\\{j:1\leq j<q\right\\}+\\#\left\\{k:q<k\leq
n\right\\}=n-1$, and the rest are fairly straightforward.

##  12\. The long Weyl element

###  12.1. The Differential Equations

We assume the Interchange of Integrals and Analytic Continuation Conjectures
and prove the Differential Equations and Power Series Conjecture for the long
Weyl element, outside of the exceptional cases $\mu_{i}-\mu_{j}\in\mathbb{Z}$,
$i\neq j$. Consider the function

$f(g):=p_{-\rho}(g)K_{w_{l}}(g{g}^{T},\mu,\delta),$

which satisfies

$f(xyk)=\psi_{I}^{2}(x)f(y).$

We wish to show that $\Delta f=\lambda_{2\mu}(\Delta)f$ for every $\Delta$ in
the center of the universal enveloping algebra (i.e. for the Casimir
operators), where

$\Delta p_{\rho+\mu}=\lambda_{\mu}(\Delta)p_{\rho+\mu}.$

Let ${(x^{*})}^{T}y^{*}k^{*}=w_{l}xt$ in (11), then

$\Delta
p_{-\rho}(g)W_{\sigma}(g{g}^{T}w_{l}xt,\mu,\delta)=\overline{\psi_{I}(x^{*})}p_{-\mu^{w_{l}}}(y^{*})\Delta
p_{-\rho}(y^{*}x^{*}g)W_{\sigma}(y^{*}x^{*}g{g}^{T}{(x^{*})}^{T}y^{*},w_{l},\mu,\delta,\psi_{(y^{*})^{-1}}),$

and by translation invariance, it suffices to determine

$\Delta p_{-\rho}(g)W_{\sigma}(g{g}^{T},w_{l},\mu,\delta,\psi).$

Now we assume $\mu$ is in the region of absolute convergence of the Whittaker
function, so we need to compute

$\Delta p_{-\rho}(g)I_{\mu,\sigma}(w_{l}ug{g}^{T}),$

and again we let $x^{\dagger}y^{\dagger}k^{\dagger}=w_{l}u$ so that the above
may be written

$p_{\rho+\mu}(y^{\dagger})\Delta
p_{-\rho}(g)I_{\mu,\sigma}(k^{\dagger}g{g}^{T}{(k^{\dagger})}^{T})\sigma(k^{\dagger}).$

Because $\Delta$ may be written as a linear combination of compositions of
first-order derivatives (coming from the Lie algebra), each of which is left-
translation invariant, it is sufficient to determine $\Delta f^{*}$ where
$f^{*}(g)=p_{-\rho}(g)p_{\rho+\mu}(g{g}^{T})$. Now
$\Delta_{G}f^{*}(g)=\Delta_{G/K}f^{*}(z)$ by $K$-invariance (i.e. using
$k\exp(tX)=\exp(tkXk^{-1})k$), and it is easy to see that
$\Delta_{G/K}f^{*}(z)=\lambda_{2\mu}(\Delta)f^{*}(z)$ since the equivalent
statement holds for every element of the Lie algebra. (For
$u=E_{i,j}\in{\mathfrak{n}}^{T}$, i.e. $i>j$, write $u={u}^{T}+(u-{u}^{T})$
and note that $u-{u}^{T}\in\mathfrak{k}$.)

We denote by $2\in Y^{+}$ the matrix such that
$\psi_{y}^{2}(x)=\psi_{I}(2yx(2y)^{-1})$. We now believe that $f(g)$ is the
spherical Whittaker function; that is,

$K_{w_{l}}(y^{2},\mu,\delta)=C(\mu)p_{\rho}(y)W_{1}(2y,2\mu,0)$

for $y\in Y^{+}$ and some constant $C(\mu)$. We check this by comparing
asymptotics as $y\to 0$: By the functional equations of the spherical
Whittaker function,

$\displaystyle\Lambda_{\text{JW}}(2\mu)W_{1}(2y,2\mu,0)\sim$
$\displaystyle\sum_{w\in
W}p_{\rho+2\mu^{ww_{l}}}(2y)\Lambda_{\text{JW}}(2\mu^{w})W_{1}(I,w_{l},2\mu^{w},0,\psi_{0}),$
$\displaystyle\Lambda_{\text{JW}}(\mu):=$ $\displaystyle\prod_{1\leq i<j\leq
n}\pi^{-\frac{1+\mu_{i}-\mu_{j}}{2}}\Gamma\left(\tfrac{1+\mu_{i}-\mu_{j}}{2}\right),$
$\displaystyle W_{1}(I,w_{l},\mu,0,\psi_{0})=$ $\displaystyle\prod_{1\leq
i<j\leq
n}\pi^{\frac{1}{2}}\frac{\Gamma\left(\frac{\mu_{i}-\mu_{j}}{2}\right)}{\Gamma\left(\frac{1+\mu_{i}-\mu_{j}}{2}\right)}$

and from the Asymptotics Theorem (for $y\in Y^{+}$),

(59) $\displaystyle\mathbf{sin}(\mu,\delta)K_{w_{l}}(y,\mu,\delta)\sim$
$\displaystyle(-\pi)^{\frac{n(n-1)}{2}}i^{-(n-1)(\delta_{1}+\ldots+\delta_{n})}\sum_{w\in
W}\frac{{\mathop{\symoperators\,sgn}}(w)}{\Lambda_{w_{l}}(\mu^{w})}p_{\rho+\mu^{w}}(y).$

The ratio

$\frac{\Lambda_{\text{JW}}(2\mu){\mathop{\symoperators\,sgn}}(w)}{\Lambda_{w_{l}}(\mu^{w})p_{\rho+2\mu^{w}}(2)\Lambda_{\text{JW}}(2\mu^{ww_{l}})W(I,2\mu^{ww_{l}},\psi_{0})}$

is invariant under $w$ (consider the case of a transposition of consecutive
indices), as required, and we have

$C(\mu)=\frac{i^{-(n-1)(\delta_{1}+\ldots+\delta_{n})}\mathbf{sin}(2\mu,0)\Lambda_{\text{JW}}(2\mu)}{p_{\rho}(2)\mathbf{sin}(\mu,\delta)},$

after some simplification. Note: It might be possible to prove instead that
$p_{-\rho}(y)K_{w_{l}}(y^{2},\mu,\delta)$ has moderate growth in $y$, but this
isn’t entirely obvious, a priori.

Michihiko Hashizume [Hashi] shows that the space of spherical Whittaker
functions (not necessarily of moderate growth in $y$) has a spanning set given
by power series of the form $J^{*}_{w_{l}}(y^{2},(\mu/2)^{w})$, $w\in W$, for
some coefficients $a_{w_{l},m}(\mu)$ determined recursively by the action of
the Casimir operators. It follows that for positive $y$,
$K_{w_{l}}(y,\mu,\delta)$ has the form (12). Now the restriction of the
Casimir operators to the space of long-element Bessel functions (i.e.
functions that transform by $\psi_{I}$ on both the left and right) is composed
of polynomials in $y_{i}$ and $y_{i}\partial_{y_{i}}$, $i=1,\ldots,n-1$ (for
such a function $f$, consider the Lie algebra action of $E_{i,j}$ on $f(y)$,
then $i>j$ multiplies by constants, $j>i$ multiplies by coordinates of $y$ and
$2\pi i$, and $j=i$ have the form in $y_{i}\partial_{y_{i}}$); these operators
act on $p_{\rho+\mu}(y)\hat{p}_{m}(y)$ by

$y_{i}^{k_{1}}(y_{i}\partial_{y_{i}})^{k_{2}}p_{\rho+\mu}(y)\hat{p}_{m}(y)=\left(\hat{\rho}_{i}+\hat{\mu}_{i}+m_{i}\right)^{k_{2}}p_{\rho+\mu}(y)\hat{p}_{m+k_{1}e_{i}}(y),$

(using the standard basis element $e_{i}$), and this holds independent of the
sign of $y$. In particular, (12) holds for every sign (at $w=w_{l}$), since
the asymptotics of $K_{w_{l}}(y,\mu,\delta)$ have the correct dependence on
${\mathop{\symoperators\,sgn}}(y)$ (that is, the coefficient of
$p_{\rho+\mu^{w}}(y)$ transforms by $\chi_{\delta^{w}}(y)$).

###  12.2. Integral Representations of the Long-Element Bessel Function

####  12.2.1. Ishii’s Theorem

Ishii [Ishii, Theorem 4] gave a useful recurrence relation for the
coefficients of the power series solutions to the Whittaker differential
equations on $GL(n)$, and we record it here as it applies to the long-element
Bessel functions, anticipating its utility in the construction of integral
representations.

For $k\in\mathbb{N}_{0}^{n-1}$ and $\mu\in\mathbb{C}^{n}$, set

$S(k)=\left\\{m\in\mathbb{N}_{0}^{n-3}\,\middle|\,0\leq m_{i}\leq
k_{i+1}\right\\},$
$P_{n,k}(\mu)=\prod_{j=1}^{n-1}\frac{\left(\mu_{1}-\mu_{n}+1\right)_{k_{j}+k_{j+1}}}{\left(\mu_{1}-\mu_{n}+1\right)_{k_{j}}\left(\mu_{1}-\mu_{j+1}+1\right)_{k_{j}}\left(\mu_{j+1}-\mu_{n}+1\right)_{k_{j+1}}},$

for $m\in S(k)$,

$\displaystyle Q_{n,k,m}(\mu)=$
$\displaystyle\prod_{j=1}^{n-1}\frac{\left(\mu_{j+1}-\mu_{n}+k_{j+1}+1-m_{j-1}\right)_{m_{j-1}}\left(\mu_{1}-\mu_{n}+k_{j}+1-m_{j-1}\right)_{m_{j-1}}}{(k_{j}-m_{j-1})!\,\left(\mu_{1}-\mu_{n}+k_{j}+k_{j+1}+1-m_{j-1}-m_{j}\right)_{m_{j-1}+m_{j}}}$
$\displaystyle\qquad\times\left(\mu_{1}-\mu_{j+1}+k_{j}+1-m_{j}\right)_{m_{j}},$
$R_{n,k}(\mu)=\sum_{j=1}^{n-1}\left((k_{j}-k_{j-1})^{2}+2k_{j}(\mu_{j}-\mu_{j+1})\right),$

using $m_{j}=0$ for $j\notin\left\\{1,\ldots,n-3\right\\}$ and $k_{j}=0$ for
$j\notin\left\\{1,\ldots,n-1\right\\}$. Define $G_{n,k}(\mu)$ by

$G_{n,0}(\mu)=1,\qquad
R_{n,k}(\mu)G_{n,k}(\mu)=\sum_{j=1}^{n-1}G_{n,k-e_{j}}(\mu).$

Let

$\Omega_{n}=\left\\{\mu\in M_{n}\,\middle|\,R_{n,k}(\mu^{w})\neq 0,\forall
k\in\mathbb{N}_{0}^{n-1}\setminus\left\\{0\right\\}\forall w\in
W,\mu_{i}-\mu_{j}\notin\mathbb{Z},i\neq j.\right\\}$

Then for $\mu\in\Omega_{n}$, define $\mu^{\prime}\in\mathbb{C}^{n-2}$ by
$\mu^{\prime}_{j}=\mu_{j+1}+\frac{\mu_{1}+\mu_{n}}{n-2}$ and suppose
$\mu^{\prime}\in\Omega_{n-2}$. Then Ishii’s recurrence relation is

(60) $\displaystyle G_{n,k}(\mu)=P_{n,k}(\mu)\sum_{m\in
S(k)}G_{n-2,m}(\mu^{\prime})Q_{n,k,m}(\mu),$

and these compare to the coefficients $a_{w_{l},m}(\mu)$ in the Differential
Equations and Power Series Conjecture as

$a_{w_{l},m}(\mu)=G_{n,m}(\mu)\prod_{j=1}^{n-1}\left(4\pi^{2}\right)^{m_{j}}.$

Then [Ishii, Theorem 3] for ${\mathop{\symoperators Re}}(\mu_{1}-\mu_{n})>-2$,
and $y_{j}>0$, becomes

$\displaystyle J_{n,w_{l}}^{\dagger}(I,y,\mu)=$
$\displaystyle\prod_{j=1}^{n-1}y_{j}^{\frac{n-2j}{2(n-2)}(\mu_{1}+\mu_{n})}\frac{1}{(2\pi
i)^{n-2}}\int_{\left|u_{1}\right|=1}\cdots\int_{\left|u_{n-2}\right|=1}\prod_{j=1}^{n-1}I_{\mu_{1}-\mu_{n}}\left(4\pi\sqrt{y_{j}(1+u_{j-1})(1+1/u_{j})}\right)$
$\displaystyle\qquad
J_{n-2,w_{l}}^{\dagger}\left(I,\left(y_{2}u_{1}/u_{2},\ldots,y_{n-2}u_{n-3}/u_{n-2}\right),\mu^{\prime}\right)\prod_{j=1}^{n-2}u_{j}^{-\frac{n}{2(n-2)}(\mu_{1}+\mu_{n})}\frac{du_{j}}{u_{j}},$

taking $u_{0}=1/u_{n-1}=0$. Note: We’ve expressed Ishii’s result in terms of
the Bessel function by replacing his $y$ with $2\sqrt{y}$ and $\mu\mapsto
2\mu$. To avoid confusion about branch cuts and convergence, it’s best to
imagine ${\mathop{\symoperators Im}}(\mu)=0$ with
$\mu_{1}>\mu_{2}>\ldots>\mu_{n}$ and use analytic continuation on the final
result.

####  12.2.2. The other signs

We now prove Proposition 6.

The result of the analytic continuation of $J_{n,w_{l}}(I,y,\mu)$ from
${\mathop{\symoperators\,sgn}}(y)=I$ to ${\mathop{\symoperators\,sgn}}(y)=v$
by rotating each coordinate $y_{j}$ (having $v_{j}=-1$) in the counter-
clockwise direction is given by

$J_{n,w_{l}}^{\dagger}(v,(\left|y_{1}\right|,\ldots,\left|y_{n-1}\right|),\mu)\exp\left(i\pi\sum_{v_{j}=-1}\hat{\mu}_{j}\right).$

Meanwhile, we have $I_{\alpha}(ix)=e^{i\frac{\pi}{2}\alpha}J_{\alpha}(x)$, so
the analytic continuation of the right-hand side of Ishii’s integral becomes

$\displaystyle\prod_{j=1}^{n-1}\left|y_{j}\right|^{\frac{n-2j}{2(n-2)}(\mu_{1}+\mu_{n})}\frac{1}{(2\pi
i)^{n-2}}\int_{\left|u_{1}\right|=1}\cdots\int_{\left|u_{n-2}\right|=1}\prod_{j=1}^{n-1}Z^{v_{j}}_{\mu_{1}-\mu_{n}}\left(4\pi\sqrt{\left|y_{j}\right|(1+u_{j-1})(1+1/u_{j})}\right)$
$\displaystyle\times
J_{n-2,w_{l}}^{\dagger}\left(v^{\prime},\left(\left|y_{2}\right|u_{1}/u_{2},\ldots,\left|y_{n-2}\right|u_{n-3}/u_{n-2}\right),\mu^{\prime}\right)\prod_{j=1}^{n-2}u_{j}^{-\frac{n}{2(n-2)}(\mu_{1}+\mu_{n})}\frac{du_{j}}{u_{j}}$
$\displaystyle\times\exp\left(i\frac{\pi}{2}\left(\sum_{v_{j}=-1}\left(\frac{n-2j}{(n-2)}(\mu_{1}+\mu_{n})+\mu_{1}-\mu_{n}\right)+2\sum_{\begin{subarray}{c}2\leq
j\leq n-2\\\
v_{j}=-1\end{subarray}}\sum_{k=2}^{j}\left(\mu_{j}+\frac{\mu_{1}+\mu_{n}}{n-2}\right)\right)\right),$

and the complex exponential is the same on both sides.

##  13\. Acknowledgments

The author would like to thank Tyrone Crisp and Peter Humphries for helpful
discussions on representation theory.

## Appendix A – Bessel functions on $GL(4)$

The eventual goal of this project is to treat the Bessel functions in a
systematic manner. However, detecting a pattern using a single non-trivial
example is rather difficult, so we include here the constructions (proofs) on
$GL(4)$ through the Differential Equations and Power Series Conjecture,
ignoring the degenerate cases $\mu_{i}-\mu_{j}\in\mathbb{Z},i\neq j$, and
assuming the Strong Interchange of Integrals Conjecture. The immediate goal
for this section is not a complete description of the Bessel functions on
$GL(4)$, but to provide evidence that the above conjectures should hold on
$GL(n)$. For brevity, we drop commas in subscripts where possible, i.e.
writing $E_{11}$ rather than $E_{1,1}$

We start by parameterizing $G_{w_{l}}=U(\mathbb{R})Y{U(\mathbb{R})}^{T}$ with
coordinates $xy{u}^{T}$, $x,u\in U(\mathbb{R}),y\in Y$ where

$x=\begin{pmatrix}1&x_{1}&x_{4}&x_{6}\\\ &1&x_{2}&x_{5}\\\ &&1&x_{3}\\\
&&&1\end{pmatrix},\qquad y=\begin{pmatrix}y_{1}y_{2}y_{3}\\\ &y_{2}y_{3}\\\
&&y_{3}\\\ &&&1\end{pmatrix}.$

Then the differential operators $E_{i,j}$ acting on functions of $G_{w_{l}}$
have the form

(61) $\displaystyle\begin{pmatrix}E_{11}&E_{12}&E_{13}&E_{14}\\\
E_{21}&E_{22}&E_{23}&E_{24}\\\ E_{31}&E_{32}&E_{33}&E_{34}\\\
E_{41}&E_{42}&E_{43}&E_{44}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}y_{1}\partial_{y_{1}}&y_{1}\partial_{x_{1}}&y_{2}y_{1}\partial_{x_{4}}&y_{3}y_{2}y_{1}\partial_{x_{6}}\\\
\partial_{u_{1}}&y_{2}\partial_{y_{2}}-y_{1}\partial_{y_{1}}&y_{2}\partial_{x_{2}}+x_{1}y_{2}\partial_{x_{4}}&y_{3}y_{2}\partial_{x_{5}}+x_{1}y_{3}y_{2}\partial_{x_{6}}\\\
\partial_{u_{4}}&\partial_{u_{2}}+u_{1}\partial_{u_{4}}&y_{3}\partial_{y_{3}}-y_{2}\partial_{y_{2}}&y_{3}\partial_{x_{3}}+x_{2}y_{3}\partial_{x_{5}}+x_{4}y_{3}\partial_{x_{6}}\\\
\partial_{u_{6}}&\partial_{u_{5}}+u_{1}\partial_{u_{6}}&\partial_{u_{3}}+u_{2}\partial_{u_{5}}+u_{4}\partial_{u_{6}}&-y_{3}\partial_{y_{3}}\end{pmatrix}.$

If we renormalize the Casimir operators slightly to define

$\displaystyle\Delta_{1}=$ $\displaystyle\sum_{1\leq i\leq 4}E_{ii},$
$\displaystyle-2\Delta_{2}=$ $\displaystyle\sum_{1\leq i,j\leq 4}E_{ij}\circ
E_{ji},$ $\displaystyle 3\Delta_{3}=$ $\displaystyle\sum_{1\leq i,j,k\leq
4}E_{ij}\circ E_{jk}\circ E_{ki}+4\Delta_{2},$ $\displaystyle-4\Delta_{4}=$
$\displaystyle\sum_{1\leq i,j,k,l\leq 4}E_{ij}\circ E_{jk}\circ E_{kl}\circ
E_{li}-12\Delta_{3}+\Delta_{2},$

then the eigenvalues of the power function
$\Delta_{i}p_{\rho+\mu}=\lambda_{i}(\mu)p_{\rho+\mu}$ are

$\displaystyle\lambda_{1}(\mu)=$ $\displaystyle 0,$
$\displaystyle\lambda_{2}(\mu)=$
$\displaystyle\frac{5}{2}-\frac{\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}+\mu_{4}^{2}}{2},$
$\displaystyle\lambda_{3}(\mu)=$
$\displaystyle\mu_{1}\mu_{2}\mu_{3}+\mu_{1}\mu_{2}\mu_{4}+\mu_{1}\mu_{3}\mu_{4}+\mu_{2}\mu_{3}\mu_{4},$
$\displaystyle\lambda_{4}(\mu)=$
$\displaystyle\frac{41}{16}-\frac{\mu_{1}^{4}+\mu_{2}^{4}+\mu_{3}^{4}+\mu_{4}^{4}}{4}.$

The relevant Weyl elements are

$I,w_{13},w_{31},w_{22},w_{112},w_{121},w_{211},w_{l},$

where, e.g. $w_{13}=w_{1,3}$ as in (2). The long Weyl element was treated in
section 12.1 and from Proposition 2, it suffices to consider the elements

$w_{31},w_{22},w_{121},w_{211},$

and we apply the algorithm discussed after the Differential Equations and
Power Series Conjecture.

For $w_{31}$, we have

$\overline{U}_{w_{31}}=\begin{pmatrix}1&*&*&*\\\ &1&0&0\\\ &&1&0\\\
&&&1\end{pmatrix},\qquad
W_{w_{31}}=\left\\{I,w_{(1\,2)},w_{(2\,3)},w_{(3\,2\,1)},w_{(1\,2\,3)},w_{(1\,3)}\right\\},$

and $Y_{w_{31}}$ is characterized by $y_{1}=y_{2}=1$. We consider a test
function of the form

$f(xy{u}^{T})=e\left(x_{1}+x_{2}+x_{3}+u_{6}\right)f(y_{1},y_{2},y_{3},u_{1},u_{2},u_{4}),$

and find that $J_{w_{31}}(y,\mu)$ is annihilated by the operator

(62)
$\displaystyle\bigl{(}-(\Delta_{4}-\lambda_{4}+2(3+2E_{44})\circ(\Delta_{3}-\lambda_{3})$
$\displaystyle\qquad\left.-(4E_{44}\circ(3+E_{44})+2\Delta_{2}+2\lambda_{2}-1)\circ(\Delta_{2}-\lambda_{2})\bigr{)}f\right|_{x=u=I}$
$\displaystyle=\bigl{(}4y_{3}^{4}\partial_{y_{3}}^{4}+4\lambda_{2}y_{3}^{2}\partial_{y_{3}}^{2}+4(\lambda_{3}-2\lambda_{2})y_{3}\partial_{y_{3}}$
$\displaystyle\qquad\left.+\left(4\left(16\pi^{4}y_{3}+\lambda_{4}\right)-6\lambda_{3}+2\lambda_{2}^{2}-\lambda_{2}\right)\bigr{)}f\right|_{x=u=I}$

whose kernel has dimension at most $4=\left|W/W_{w_{31}}\right|$. Comparing
the powers of $y_{3}$ not offset by a derivative $\partial_{y_{3}}$, it
becomes somewhat more convenient to work in the variable $t=16\pi^{4}y_{3}$,
and in that variable, the power series (Frobenius series) coefficients satisfy
a recurrence relation

$a_{w_{31},m-1}+m(\mu_{1}-\mu_{4}+m)(\mu_{2}-\mu_{4}+m)(\mu_{3}-\mu_{4}+m)a_{w_{31},m}=0,$

and hence for $y\in Y_{w_{31}}$,

(63) $\displaystyle J_{w_{31}}(y,\mu)=$
$\displaystyle\frac{\left|16\pi^{4}y_{3}\right|^{\frac{3}{2}-\mu_{4}}}{(2\pi)^{6}}\sum_{m=0}^{\infty}\frac{(-16\pi^{4}y_{3})^{m}}{m!\,\Gamma(1+\mu_{1}-\mu_{4}+m)\Gamma(1+\mu_{2}-\mu_{4}+m)\Gamma(1+\mu_{3}-\mu_{4}+m)}.$

For $w_{22}$, we have

$\overline{U}_{w_{22}}=\begin{pmatrix}1&0&*&*\\\ &1&*&*\\\ &&1&0\\\
&&&1\end{pmatrix},\qquad
W_{w_{22}}=\left\\{I,w_{(1\,2)},w_{(3\,4)},w_{(1\,2)(3\,4)}\right\\},$

and $Y_{w_{22}}$ is characterized by $y_{1}=y_{3}=1$. We consider a test
function of the form

$f(xy{u}^{T})=e\left(x_{1}+x_{2}+x_{3}+u_{6}-u_{3}u_{4}\right)f(y_{1},y_{2},y_{3},u_{1},u_{3}),$

and find that $J_{w_{22}}(y,\mu)$ is annihilated by the operator

(64)
$\displaystyle\bigl{(}4(4+A\circ(A-4))\circ(\Delta_{4}-\lambda_{4})+(\Delta_{3}+\lambda_{3})\circ(\Delta_{3}-\lambda_{3})$
$\displaystyle\qquad\left.-(A-2)\circ(A-2)\circ(2A\circ(A-4)-(\Delta_{2}+\lambda_{2}-13))\circ(\Delta_{2}-\lambda_{2})\bigr{)}f\right|_{x=u=I}$
$\displaystyle=\bigl{(}y_{2}^{6}\partial_{y_{2}}^{6}+3y_{2}^{5}\partial_{y_{2}}^{5}+2\lambda_{2}y_{2}^{4}\partial_{y_{2}}^{4}-4\lambda_{2}y_{2}^{3}\partial_{y_{2}}^{3}+(4\lambda_{4}+\lambda_{2}^{2}-19\lambda_{2}+64\pi^{4}y_{2})y_{2}^{2}\partial_{y_{2}}^{2}$
$\displaystyle\qquad\left.+3\left(4\lambda_{4}+\lambda_{2}^{2}-15\lambda_{2}+32\pi^{4}y_{2}\right)y_{2}\partial_{y_{2}}+\left(-16\lambda_{4}-\lambda_{3}^{2}-4\lambda_{2}^{2}+52\lambda_{2}-96\pi^{4}y_{2}\right)\bigr{)}f\right|_{x=u=I},$
$\displaystyle A:=E_{11}+E_{22},$

whose kernel has dimension at most $6=\left|W/W_{w_{22}}\right|$. In the
variable $t=16\pi^{4}y_{2}$, the power series coefficients satisfy a
recurrence relation

$\displaystyle
4(\mu_{1}+\mu_{2}-\tfrac{1}{2}+m)(\mu_{1}+\mu_{2}+m)a_{w_{22},m-1}=$
$\displaystyle
m(2\mu_{1}+2\mu_{2}+m)(\mu_{1}-\mu_{3}+m)(\mu_{1}-\mu_{4}+m)(\mu_{2}-\mu_{3}+m)(\mu_{2}-\mu_{4}+m)a_{w_{22},m},$

and hence for $y\in Y_{w_{22}}$,

(65) $\displaystyle J_{w_{22}}(y,\mu)=$
$\displaystyle\frac{\left|16\pi^{4}y_{2}\right|^{2+\mu_{1}+\mu_{2}}}{(2\pi)^{8}}\sum_{m=0}^{\infty}\frac{(16\pi^{4}y_{2})^{m}\Gamma(1+2\mu_{1}+2\mu_{2}+2m)}{m!\,\Gamma(1+2\mu_{1}+2\mu_{2}+m)\Gamma(1+\mu_{1}-\mu_{3}+m)}$
$\displaystyle\qquad\times\frac{1}{\Gamma(1+\mu_{1}-\mu_{4}+m)\Gamma(1+\mu_{2}-\mu_{3}+m)\Gamma(1+\mu_{2}-\mu_{3}+m)}.$

For $w_{121}$, we have

$\overline{U}_{w_{121}}=\begin{pmatrix}1&*&*&*\\\ &1&0&*\\\ &&1&*\\\
&&&1\end{pmatrix},\qquad W_{w_{121}}=\left\\{I,w_{(2\,3)}\right\\},$

and $Y_{w_{121}}$ is characterized by $y_{2}=1$. We consider a test function
of the form

$f(xy{u}^{T})=e\left(x_{1}+x_{2}+x_{3}+u_{4}+u_{5}-u_{2}u_{1}\right)f(y_{1},y_{2},y_{3},u_{2}),$

and find that $J_{w_{121}}(y,\mu)$ is annihilated by the operators

(66) $\displaystyle\begin{aligned}
&\left.\left((\Delta_{3}-\lambda_{3})+(E_{22}+E_{33})\circ(\Delta_{2}-\lambda_{2})\right)f\right|_{x=u=I}\\\
&=\bigl{(}y_{1}^{3}\partial_{y_{1}}^{3}-y_{3}^{3}\partial_{y_{3}}^{3}+y_{3}^{2}y_{1}\partial_{y_{3}}^{2}\partial_{y_{1}}-y_{3}y_{1}^{2}\partial_{y_{3}}\partial_{y_{1}}^{2}+\lambda_{2}y_{1}\partial_{y_{1}}-\lambda_{2}y_{3}\partial_{y_{3}}\\\
&\qquad\left.-\left(\lambda_{3}+8\pi^{3}(y_{3}+y_{1})\right)\bigr{)}f\right|_{x=u=I},\end{aligned}$
(67) $\displaystyle\begin{aligned}
&\bigl{(}-4(\Delta_{4}-\lambda_{4})+2(\Delta_{2}-\lambda_{2})\circ(\Delta_{2}-\lambda_{2})\\\
&\qquad\left.+(2E_{44}+3)\circ(2E_{11}-3)+4\Delta_{2}-10)\circ(\Delta_{2}-\lambda_{2})\bigr{)}f\right|_{x=u=I}\\\
&=\bigl{(}4y_{3}^{3}y_{1}\partial_{y_{3}}^{3}\partial_{y_{1}}-6y_{3}^{3}\partial_{y_{1}}^{3}-4y_{3}^{2}y_{1}^{2}\partial_{y_{3}}^{2}\partial_{y_{1}}^{2}+2y_{3}^{2}y_{1}\partial_{y_{3}}^{2}\partial_{y_{1}}+4y_{3}y_{1}^{3}\partial_{y_{3}}\partial_{y_{1}}^{3}+2y_{3}y_{1}^{2}\partial_{y_{3}}\partial_{y_{1}}^{2}\\\
&\qquad+4\lambda_{2}y_{3}y_{1}\partial_{y_{3}}\partial_{y_{1}}-2(3\lambda_{3}+16\pi^{3}iy_{1})y_{3}\partial_{y_{3}}-6y_{1}^{3}\partial_{y_{1}}^{3}-2(2\lambda_{2}-16\pi^{3}iy_{3})y_{1}\partial_{y_{1}}\\\
&\qquad\left.+\left(4\lambda_{4}+2\lambda_{2}^{2}-\lambda_{2}-48\pi^{3}i(y_{3}-y_{1})\right)\bigr{)}f\right|_{x=u=I}.\end{aligned}$

Switching to the variables $t_{1}=8\pi^{3}iy_{1}$, $t_{2}=8\pi^{3}iy_{3}$
simplifies the differential operators somewhat, and it can be shown that
$J_{w_{121}}(y,\mu)$ satisfies a (somewhat complicated) twelfth-order
differential equation in $t_{1}$ alone, whose solution then determines the
function in $t_{2}$ (up to a constant), giving (a basis of)
$12=\left|W/W_{w_{121}}\right|$ solutions. In the $t$ variables, the power
series coefficients satisfy the recurrence relations

$\displaystyle a_{w_{121},m_{1},m_{2}-1}+a_{w_{121},m_{1}-1,m_{2}}=$
$\displaystyle\bigl{(}m_{1}(\mu_{1}-\mu_{2}+m_{1})(\mu_{1}-\mu_{3}+m_{1})-m_{2}(\mu_{2}-\mu_{4}+m_{2})(\mu_{3}-\mu_{4}+m_{2})$
$\displaystyle\qquad+m_{2}m_{1}(2\mu_{2}+2\mu_{3}+m_{2}-m_{1})\bigr{)}a_{w_{22},m_{1},m_{2}},$
$\displaystyle(\mu_{1}+m_{1})a_{w_{121},m_{1},m_{2}-1}+(\mu_{4}-m_{2})a_{w_{121},m_{1}-1,m_{2}}=$
$\displaystyle\Bigl{(}\mu_{1}m_{2}(\mu_{2}-\mu_{4}+m_{2})(\mu_{3}-\mu_{4}+m_{2})-\mu_{4}m_{1}(\mu_{1}-\mu_{2}+m_{1})(\mu_{1}-\mu_{3}+m_{1})$
$\displaystyle\qquad+m_{2}m_{1}\bigl{(}(\mu_{2}+\mu_{3}+m_{1}+m_{2})(\mu_{1}+m_{1}+m_{2})+(\mu_{2}+\mu_{3}+m_{2})(2\mu_{3}-3m_{1})$
$\displaystyle\qquad+\mu_{2}(2\mu_{2}+3\mu_{3}+2m_{2})+\mu_{1}^{2}\bigr{)}\Bigr{)}a_{w_{22},m_{1},m_{2}},$

and taking $(\mu_{2}+m_{2})$ times the first plus the second gives

$\displaystyle(\mu_{2}-\mu_{4}+m_{1}+m_{2})a_{w_{121},m_{2},m_{1}-1}=m_{1}(\mu_{1}-\mu_{2}+m_{1})(\mu_{1}-\mu_{3}+m_{1})(\mu_{1}-\mu_{4}+m_{1})a_{w_{22},m_{2},m_{1}}.$

Substituting

$a_{w_{22},m_{1},m_{2}}=\frac{\Gamma(1+\mu_{1}-\mu_{4}+m_{1}+m_{2})}{m_{1}!\Gamma(1+\mu_{1}-\mu_{2}+m_{1})\Gamma(1+\mu_{1}-\mu_{3}+m_{1})\Gamma(1+\mu_{1}-\mu_{4}+m_{1})}b_{m_{2}}$

into the first recurrence gives

$b_{m_{2}-1}=m_{2}(\mu_{1}-\mu_{4}+m_{2})(\mu_{2}-\mu_{4}+m_{2})(\mu_{3}-\mu_{4}+m_{2})b_{m_{2}},$

and hence for $y\in Y_{w_{121}}$,

(68) $\displaystyle J_{w_{121}}(y,\mu)=$
$\displaystyle\frac{\left|8\pi^{3}y_{3}\right|^{\frac{3}{2}-\mu_{4}}\left|8\pi^{3}y_{1}\right|^{\frac{3}{2}+\mu_{1}}}{(2\pi)^{9}}\sum_{m_{2}=0}^{\infty}\sum_{m_{1}=0}^{\infty}\frac{(8\pi^{3}iy_{1})^{m_{1}}(8\pi^{3}iy_{3})^{m_{2}}}{m_{2}!\,m_{1}!\,\Gamma(1+\mu_{1}-\mu_{2}+m_{1})}$
$\displaystyle\qquad\times\frac{\Gamma(1+\mu_{1}-\mu_{4}+m_{1}+m_{2})}{\Gamma(1+\mu_{1}-\mu_{3}+m_{1})\Gamma(1+\mu_{1}-\mu_{4}+m_{1})}$
$\displaystyle\qquad\times\frac{1}{\Gamma(1+\mu_{1}-\mu_{4}+m_{2})\Gamma(1+\mu_{2}-\mu_{4}+m_{2})\Gamma(1+\mu_{3}-\mu_{4}+m_{2})}.$

The story for $w_{211}$ is a bit more complicated; we have

$\overline{U}_{w_{211}}=\begin{pmatrix}1&*&*&*\\\ &1&*&*\\\ &&1&0\\\
&&&1\end{pmatrix},\qquad W_{w_{211}}=\left\\{I,w_{(1\,2)}\right\\},$

and $Y_{w_{211}}$ is characterized by $y_{1}=1$. We consider a test function
of the form

$f(xy{u}^{T})=e\left(x_{1}+x_{2}+x_{3}+u_{3}+u_{4}\right)f(y_{1},y_{2},y_{3},u_{1}),$

and find that $J_{w_{211}}(y,\mu)$ is annihilated by the operators

(69) $\displaystyle\begin{aligned}
&\left.\left((\Delta_{3}-\lambda_{3})+(E_{11}+E_{22})\circ(\Delta_{2}-\lambda_{2})\right)f\right|_{x=u=I}\\\
&=\bigl{(}4y_{3}^{2}\partial_{y_{3}}^{2}-2y_{3}^{2}y_{2}\partial_{y_{3}}^{2}\partial_{y_{2}}+2y_{3}y_{2}^{2}\partial_{y_{3}}\partial_{y_{2}}^{2}-2y_{3}y_{2}\partial_{y_{3}}\partial_{y_{2}}-y_{2}^{3}\partial_{y_{2}}^{3}-(\lambda_{2}-8\pi^{2}y_{3})y_{2}\partial_{y_{2}}\\\
&\qquad\left.-\left(\lambda_{3}-\lambda_{2}+8\pi^{2}(2y_{3}+\pi
iy_{2})\right)\bigr{)}f\right|_{x=u=I},\end{aligned}$ (70)
$\displaystyle\begin{aligned}
&\bigl{(}4(\Delta_{4}-\lambda_{4})+2(\Delta_{2}-\lambda_{2})\circ(\Delta_{2}-\lambda_{2})\\\
&\qquad\left.-(2E_{44}+3)\circ(2E_{33}+1)-4\lambda_{2}+10+16\pi^{2}y_{3})\circ(\Delta_{2}-\lambda_{2})\bigr{)}f\right|_{x=u=I}\\\
&=\bigl{(}-4y_{3}^{4}\partial_{y_{3}}^{4}-16y_{3}^{3}\partial_{y_{3}}^{3}-6y_{2}^{3}\partial_{y_{2}}^{3}+8y_{3}^{3}y_{2}\partial_{y_{3}}^{3}\partial_{y_{2}}-8y_{3}^{2}y_{2}^{2}\partial_{y_{3}}^{2}\partial_{y_{2}}^{2}+12y_{3}^{2}y_{2}\partial_{y_{3}}^{2}\partial_{y_{2}}\\\
&\qquad+4y_{2}^{3}y_{3}\partial_{y_{3}}\partial_{y_{2}}^{3}+16\pi^{2}y_{3}y_{2}^{2}\partial_{y_{2}}^{2}+4y_{3}y_{2}^{2}\partial_{y_{3}}\partial_{y_{2}}^{2}+4\left(-\lambda_{2}+8\pi^{2}y_{3}-2\right)y_{3}^{2}\partial_{y_{3}}^{2}\\\
&\qquad-4\left(1-\lambda_{2}+8\pi^{2}y_{3}\right)y_{3}y_{2}\partial_{y_{3}}\partial_{y_{2}}-2\left(3\lambda_{2}+8\pi^{2}y_{3}\right)y_{2}\partial_{y_{2}}+32\pi^{2}\left(\pi
iy_{2}+y_{3}\right)y_{3}\partial_{y_{3}}\\\
&\qquad\left.+\lambda_{2}\left(13-2\lambda_{2}+16\pi^{2}y_{3}\right)-16\pi^{3}\left(4\pi
y_{3}^{2}+3iy_{2}\right)-4\lambda_{4}\bigr{)}f\right|_{x=u=I}.\end{aligned}$

Switching to the variables $t_{1}=8\pi^{3}iy_{2}$, $t_{2}=4\pi^{2}y_{3}$
simplifies the differential operators somewhat, and by the method of
Frobenius, any series solution of the form

$f(t)=\left|t_{1}\right|^{s_{1}}\left|t_{2}\right|^{s_{2}}\sum_{m_{2}=0}^{\infty}\sum_{m_{1}=0}^{\infty}a_{w_{211},m_{2},m_{1}}t_{1}^{m_{1}}t_{2}^{m_{2}}$

satisfies $s_{1}=2+\mu^{w}_{1}+\mu^{w}_{2}$, $s_{2}=\frac{3}{2}-\mu^{w}_{4}$
for some $w\in W$, hence there are $\left|W/W_{w_{211}}\right|=12$ Frobenius
series solutions (i.e. solutions that are well-behaved near $y_{3}=y_{2}=0$).
Taking $s_{1}=2+\mu_{1}+\mu_{2}$, $s_{2}=\frac{3}{2}-\mu_{4}$, the
coefficients in general satisfy the recurrence relations

$\displaystyle
a_{w_{211},m_{1}-1,m_{2}}-2\left(\mu_{1}+\mu_{2}+m_{1}\right)a_{w_{211},m_{1},m_{2}-1}=$
$\displaystyle\left(-m_{1}\left(\mu_{1}-\mu_{3}+m_{1}\right)\left(\mu_{2}-\mu_{3}+m_{1}\right)-2m_{2}\left(\mu_{1}+\mu_{2}+m_{1}\right)\left(\mu_{3}-\mu_{4}+m_{2}-m_{1}\right)\right)a_{w_{211},m_{1},m_{2}},$
$\displaystyle
a_{m_{1},m_{2}-1}-\left(1-\mu_{4}+m_{2}\right)a_{w_{211},m_{1}-1,m_{2}+1}=$
$\displaystyle\left(\left(\mu_{1}+\mu_{3}+1\right)\left(\mu_{2}+\mu_{3}+1\right)+2m_{2}\left(\mu_{3}-\mu_{4}+m_{2}+1\right)+m_{1}\left(-2\mu_{3}-2m_{2}+m_{1}-1\right)\right)a_{w_{211},m_{1},m_{2}}$
$\displaystyle\qquad-\Bigl{(}\left(m_{2}+1\right)\left(\mu_{1}+\mu_{3}+m_{2}+1\right)\left(\mu_{2}+\mu_{3}+m_{2}+1\right)\left(\mu_{3}-\mu_{4}+m_{2}+1\right)$
$\displaystyle\qquad+\left(1-\mu_{4}\right)m_{1}\left((\mu_{1}+\mu_{2}-\mu_{3}-4)\mu_{3}-\mu_{1}\mu_{2}-m_{1}\left(\mu_{1}+\mu_{2}-2\mu_{3}+m_{1}-2\right)-2\right)$
$\displaystyle\qquad-
m_{2}m_{1}\bigl{(}(2\mu_{3}-3\mu_{4}+12)\mu_{3}+4\mu_{1}+\mu_{1}\mu_{2}+4\mu_{2}+2m_{2}\left(2\mu_{3}-\mu_{4}+m_{2}-m_{1}+3\right)$
$\displaystyle\qquad+m_{1}\left(\mu_{4}-3\mu_{3}+m_{1}-4\right)+6\bigr{)}\Bigr{)}a_{w_{211},m_{1},m_{2}+1}$

Finally, taking $(1-\mu_{4}+m_{2})$ times the first at $m_{2}\mapsto m_{2}+1$
plus the second gives

$\displaystyle
m_{2}\left(\mu_{1}-\mu_{4}+m_{2}\right)\left(\mu_{2}-\mu_{4}+m_{2}\right)\left(\mu_{3}-\mu_{4}+m_{2}\right)a_{w_{211},m_{1},m_{2}}+a_{w_{211},m_{1},m_{2}-2}=$
$\displaystyle\bigr{(}2\mu_{1}^{2}+\mu_{1}+\mu_{2}\left(5\mu_{1}+2\mu_{2}+1\right)+\mu_{3}\left(-2\mu_{3}-3\mu_{4}+2\right)+\left(m_{1}+1\right)\left(2\mu_{1}+2\mu_{2}+m_{1}\right)$
$\displaystyle\qquad+2\left(m_{2}-1\right)\left(m_{2}-2\mu_{4}\right)+1\bigl{)}a_{w_{211},m_{1},m_{2}-1},$

and one might hope that the solution to this would be expressible in terms of
hypergeometric functions at one (but not a quotient of gamma functions). This
does not appear to be the case as both Mathematica and the “Hyper” algorithm
of Petkovšek [Petkov] fail to return a closed form solution. The author
expects the coefficients to be a two-variable hypergeometric function at one.

## Appendix B – Matrix decompositions in Mathematica

Here we give an implementation of the decompositions of section 8 in
Mathematica. The two main functions below are NiceIwaxb and NiceIwawx. Both
take a vector of parameters $rs=\left\\{r_{1},\ldots,r_{\ell}\right\\}$ and
return a tuple $\left\\{x,x^{*},y^{*},kss\right\\}$ where $kss$ is a vector of
matrices and we take $k^{*}=$ Dot@@$kss$ to be the product. For NiceIwaxb, $x$
is a choice of coordinates on
$w_{b}\overline{U}_{w_{b}}(\mathbb{R})w_{b}^{-1}$ and the Iwasawa
decomposition is $x=x^{*}y^{*}k^{*}$. For NiceIwawx, $x$ is a choice of
coordinates on $\overline{U}_{w}(\mathbb{R})$ and the Iwasawa decomposition is
$wx=X^{*}y^{*}k^{*}$.

DoSymbol[name_, index_] := Symbol[name <>ToString[index]]
DoSymbolList[name_, length_] := Module[{i},
Table[DoSymbol[name, i], {i, 1, length}]]
xmat[n_, name_] := Module[{vars = DoSymbolList[name, (n (n - 1))/2], i, j},
Table[If[i == j, 1, If[j >i, vars[[i + (j - 1) (j - 2)/2]], 0]],
{i, 1, n}, {j, 1, n}]]
wl[rs_] := Module[{n = Length[rs], i, j}, ArrayFlatten[Table[
If[i + j == n + 1, IdentityMatrix[rs[[i]]],
ConstantArray[0, {rs[[i]], rs[[n + 1 - j]]}]], {i, 1, n}, {j, 1, n}]]]
Embed[n_, mat_] := If[Length[mat] == n, mat, ArrayFlatten[{{mat,
ConstantArray[0, {Length[mat], n - Length[mat]}]},
{ConstantArray[0, {n - Length[mat], Length[mat]}],
IdentityMatrix[n - Length[mat]]}}]]
getws[w_] := Module[{row}, Table[Position[row, 1][[1, 1]], {row, w}]]
GetUw[w_, x_] := Module[{n = Length[w], ws = getws[Transpose[w]]},
Table[If[j <i, 0, If[j == i, 1, If[ws[[i]] <ws[[j]], x[[i, j]], 0]]],
{i, 1, n}, {j, 1, n}]]

MDhatr[rs_, i_] := Total[rs[[;; i]]]
MDhatr[rs_] := Table[MDhatr[rs, i], {i, 1, Length[rs]}]
MDnp[rs_, m_] := MDhatr[rs, Length[rs] - m]
MDwp[rs_, m_] := If[m <Length[rs], wl[rs[[;; -m - 1]]], Null]
MDwpp[rs_, m_] := wl[rs[[-m ;;]]]
MDwa[rs_, m_] := ArrayFlatten[{{MDwp[rs, m], 0}, {0, MDwpp[rs, m]}}]
MDwb[rs_, m_] := If[m == Length[rs], IdentityMatrix[Total[rs]],
ArrayFlatten[{{0, IdentityMatrix[MDhatr[rs, Length[rs] - m]]},
{IdentityMatrix[Total[rs] - MDhatr[rs, Length[rs] - m]], 0}}]]

MDDecomps[rs_, m_, xa_, xbs_] := Module[{n = Total[rs], wa = MDwa[rs, m],
wb = MDwb[rs, m], np = MDnp[rs, m], A, B, C, D, E, F, H},
A = xa[[;; np, ;; np]];
B = xbs[[np + 1 ;;, np + 1 ;;]];
C = xbs[[;; np, ;; np]];
D = xbs[[;; np, np + 1 ;;]];
E = xbs[[np + 1 ;;, np + 1 ;;]];
F = GetUw[MDwp[rs, m], C];
H = GetUw[MDwpp[rs, m], E];
{A, B, C, D, E, F, H}]

MDhaty[n_, m_, i_, x_] := DiagonalMatrix[Join[ConstantArray[1, i - 1],
{(1 + x^2)^(-1/2)}, ConstantArray[1, m - i - 1], {(1 + x^2)^(1/2)},
ConstantArray[1, n - m]]]
RevEmbed[n_, s_] := If[Length[s] >= n, s,
ArrayFlatten[{{IdentityMatrix[n - Length[s]], 0}, {0, s}}]]
MDhatk2[n_, x_] := If[n <= 2, {{(1 + x^2)^(-1/2), -x (1 + x^2)^(-1/2)},
{x (1 + x^2)^(-1/2), (1 + x^2)^(-1/2)}}, ArrayFlatten[{{(1 + x^2)^(-1/2),
0, -x (1 + x^2)^(-1/2)}, {0, IdentityMatrix[n - 2], 0},
{x (1 + x^2)^(-1/2), 0, (1 + x^2)^(-1/2)}}]]
MDhatk[n_, m_, i_, x_] := Embed[n, RevEmbed[m, MDhatk2[m - i + 1, x]]]

MDucstarentry[i_, j_, u_] := Module[{l},
-u[[i]] u[[j]] Product[(1 + u[[l]]^2)^(-1/2), {l, i, j - 1}]]
MDucstar[rs_, m_, u_] := Module[{n = Total[rs], np = MDnp[rs, m], i, j},
Table[If[j <i, 0, If[j == i, 1, MDucstarentry[i, j, u]]], {i, 1, np},
{j, 1, np}]]
MDucstar[rs_, m_] := MDucstar[rs, m, DoSymbolList["u", Total[rs]]]
MDudstar[rs_, m_, u_] := Module[{n = Total[rs], np = MDnp[rs, m], i, j},
Transpose[{Table[u[[i]] Product[(1 + u[[j]]^2)^(-1/2), {j, 1, np}]
Product[(1 + u[[j]]^2)^(-1/2), {j, i, np}], {i, 1, np}]}]]
MDudstar[rs_, m_] := MDudstar[rs, m, DoSymbolList["u", Total[rs]]]
MDuctildeentry[i_, j_] := Module[{u = DoSymbolList["u", j], l}, u[[i]] u[[j]]
(1 + u[[i]]^2)^(-1/2) Product[(1 + u[[l]]^2)^(1/2), {l, i + 1, j - 1}]]
MDuctilde[rs_, m_] := Module[{n = Total[rs], np = MDnp[rs, m], i, j},
Table[If[j <i, 0, If[j == i, 1, MDuctildeentry[i, j]]], {i, 1, np},
{j, 1, np}]]
MDudtilde[rs_, m_] := Module[{n = Total[rs], np = MDnp[rs, m], u, i, j},
u = DoSymbolList["u", n];Transpose[{Table[-u[[i]] (1 + u[[i]]^2)^(-1)
Product[(1 + u[[j]]^2)^(-1/2), {j, 1, i - 1}], {i, 1, np}]}]]
MDxdag[rs_, m_] := Module[{n = Total[rs], np = MDnp[rs, m]}, If[n - np <= 1,
ArrayFlatten[{{MDucstar[rs, m], MDudstar[rs, m]}, {0, 1}}],
ArrayFlatten[{{MDucstar[rs, m], 0, MDudstar[rs, m]},
{0, IdentityMatrix[n - np - 1], 0}, {0, 0, 1}}]]]
MDxdaginv[rs_, m_] := Module[{n = Total[rs], np = MDnp[rs, m]},
If[n - np <= 1, ArrayFlatten[{{MDuctilde[rs, m], MDudtilde[rs, m]}, {0, 1}}],
ArrayFlatten[{{MDuctilde[rs, m], 0, MDudtilde[rs, m]},
{0, IdentityMatrix[n - np - 1], 0}, {0, 0, 1}}]]]
MDydag[rs_, m_, u_] := Module[{n = Total[rs], np = MDnp[rs, m], i},
Dot @@ Table[MDhaty[n, n, i, u[[i]]], {i, 1, np}]]
MDydag[rs_, m_] := MDydag[rs, m, DoSymbolList["u", MDnp[rs, m]]]
MDkdagAll[rs_, m_, u_] := Module[{n = Total[rs], np = MDnp[rs, m], i},
Table[MDhatk[n, n, i, u[[i]]], {i, np, 1, -1}]]
MDxcdagentry[rs_, u_, xc_, i_] := Module[{np = MDnp[rs, 1], j, l},
Sum[xc[[np + i, j]] u[[j]]/Product[(1 + u[[l]]^2)^(1/2), {l, 1, j}],
{j, 1, MDnp[rs, 1]}]]
MDxcdag[rs_, u_, xc_] := Module[{i},
Table[{MDxcdagentry[rs, u, xc, i]}, {i, 1, Total[rs] - MDnp[rs, 1] - 1}]]

NiceIwaxb[rs_] := NiceIwaxb[rs, xmat[Total[rs], "x"]]
NiceIwaxb[rs_, origx_] := NiceIwaxb[rs, Last[rs], origx]
NiceIwaxb[rs_, 0, origx_] := Module[{n = Total[rs]},
{IdentityMatrix[n], IdentityMatrix[n], IdentityMatrix[n], {}}]
NiceIwaxb[rs_, ind_, origx_] := Module[{n = Total[rs], np = Total[rs[[;;
-2]]],
rl = rs[[-1]], rps, ydag, wb, kdag, u, umat, xc, xcs, ycs, kcs, ucs,
uds, xcdag, mat1, mat2, mat3, i, j, l},
{xc, xcs, ycs, kcs} = NiceIwaxb[rs, ind - 1, origx];
rps = Append[rs[[;; -2]], ind];
xc = Embed[n, xc];
xcs = Embed[n, xcs];
ycs = Embed[n, ycs];
u = origx[[ind, rl + 1 ;;]];
ucs = MDucstar[rps, 1, u];
uds = MDudstar[rps, 1, u];
xcdag = MDxcdag[rps, u, xc];
ydag = Embed[n, MDydag[rps, 1, u]];
wb = MDwb[rs, 1];
kdag = Map[Embed[n, #] &, MDkdagAll[rps, 1, u]];
mat1 = Embed[n, If[ind == 1, ArrayFlatten[{{ucs, uds}, {0, 1}}],
ArrayFlatten[{{ucs, 0, uds}, {0, IdentityMatrix[ind - 1], xcdag},
{0, 0, {{1}}}}]]];
mat2 = Embed[n, ucs];
umat = Table[If[i == ind, u[[j]] Product[(1 + u[[l]]^2)^(1/2),
{l, 1, j - 1}], 0], {i, 1, rl}, {j, 1, np}];
mat3 = ArrayFlatten[{{IdentityMatrix[np], 0}, {umat, IdentityMatrix[rl]}}];
{mat3 . mat2 . ydag . xc . Inverse[ydag] . Inverse[mat2],
mat1 . ydag . xcs . Inverse[ydag], ydag . ycs,
Join[Map[Embed[n, #] &, kcs], kdag]}]

NiceIwawx[rs_] := NiceIwawx[rs, xmat[Total[rs], "x"]]
NiceIwawx[rs_/; Length[rs] == 1, origx_] := Module[{n = Total[rs]},
{IdentityMatrix[n], IdentityMatrix[n], IdentityMatrix[n], {}}]
NiceIwawx[rs_, origx_] := Module[{n = Total[rs], rl = Last[rs],
wp = MDwp[rs, 1], wa = MDwa[rs, 1], wb = MDwb[rs, 1], xa, xb, xbs, ybs,
kbs, ybswa, Ap, Bp, C, D, E, F, H, xp, xs, ys, ks, mat1, mat2,
mat3, mat4},
{xa, xs, ys, ks} =
NiceIwawx[rs[[;; -2]], (wb.origx.Inverse[wb])[[;; n - rl, ;; n - rl]]];
{xb, xbs, ybs, kbs} = NiceIwaxb[rs, origx];
xa = Embed[n, xa];
xb = Inverse[wb] . Embed[n, xb] . wb;
{Ap, Bp, C, D, E, F, H} = MDDecomps[rs, 1, ybs . xa . Inverse[ybs], xbs];
mat1 = ArrayFlatten[{{IdentityMatrix[n - rl],
wp . F . Ap . Inverse[C] . D . Inverse[E]}, {0, IdentityMatrix[rl]}}];
mat2 = ArrayFlatten[{{wp . F . Inverse[wp], 0}, {0, E}}];
mat3 = ArrayFlatten[{{H, 0}, {0, F}}];
mat4 = ArrayFlatten[{{Inverse[E], 0}, {0, Inverse[C]}}];
ybswa = wa . ybs . Inverse[wa];
{mat3 . Inverse[wb] . ybs . xa . Inverse[ybs] . wb . mat4 . xb,
mat1 . mat2 . ybswa . Embed[n, xs] . Inverse[ybswa],
ybswa . Embed[n, ys], Join[Map[Embed[n, #] &, ks], kbs, {wb}]}]

## References


# Two particle correlations: a probe of the LHC QCD medium

Yaxian Mao1,2 for the ALICE Collaboration  1 Laboratoire de Physique
Subatomique et de Cosmologie, Grenoble 38026, France 2 Key Laboratory of Quark
& Lepton Physics (Huazhong Normal University), Ministry of Education, Wuhan
430079, China<EMAIL_ADDRESS>
###### Abstract

The properties of $\gamma$–jet pairs emitted in heavy-ion collisions provide
an accurate mean to perform a tomographic measurement of the medium created in
the collision through the study of the medium modified jet properties. The
idea is to measure the distribution of hadrons emitted on the opposite side of
the direct photon. The feasibility of such measurements is studied by applying
the approach on the simulation data, we have demonstrated that this method
allows us to measure, with a good approximation, both the jet fragmentation
and the back-to-back azimuthal alignment of the direct photon and the jet.
Comparing these two observables measured in pp collisions with the ones
measured in AA collisions reveals the modifications induced by the medium on
the jet structure and consequently allows us to infer the medium properties.
In this contribution, we discuss a first attempt of such measurements applied
to real proton-proton data from the ALICE experiment.

##  1 Introduction

Quantum ChromoDynamics (QCD) [1] is a theory of the strong interaction, the
fundamental force describing the interactions of quarks and gluons making up
hadrons. The QCD calculations performed on a lattice, indicate that a phase
transition from normal hadronic matter to partonic matter, the Quark-Gluon
Plasma (QGP), will occur beyond a critical temperature of $T_{\rm
c}\sim~{}170$ MeV [2]. By colliding heavy ions at ultra relativistic energies,
this new state of matter can be created and its properties, such as the
equation of state, the degrees of freedom and the transport properties can be
measured.

The phase diagram has been explored in various regions with heavy-ion
collisions at continuously increasing kinetic energies. Experiments at CERN’s
Super Proton Synchrotron (SPS) [3] concluded on the indirect evidence of a
”new state of matter”. Current experiments at Brookhaven National Laboratory’s
Relativistic Heavy Ion Collider (RHIC) [4] have found that matter does not
behave as an ideal gas of free quarks and gluons predicted by theory, but,
rather, as an almost perfect fluid. The new experiment ALICE [5] at CERN’s
Large Hadron Collider (LHC), will push further the study of the QCD medium.
Thanks to the huge step in collision energy ($\sqrt{s_{NN}}=5.5~{}TeV$ in Pb-
Pb collisions), LHC will open new avenues for the exploration of matter under
extreme conditions of temperature and density. Since the hot QCD medium will
be formed at higher temperatures than at RHIC, the deconfined phase will last
longer and more readily modify our experimental probes, allowing for a more
accurate study of this new state matter.

Hard scattered partons produced in initial stage of the collisions, have been
identified as a valuable probe of the medium. Indeed, medium properties can be
inferred from the modifications experienced by the partonic shower inside the
medium. Partons are only observed indirectly, as a collimated jet of hadrons
coming from the fragmentation of the partonic shower [6]. Comparing the
measurements of the jet fragmentation in proton-proton and heavy-ion
collisions will reveal the modifications produced by the medium on the hard
scattered partons. Ideally, one needs to know the 4-momentum of the parton
when it has been produced in the hard scattering and after it has been
modified by the medium. This can be achieved by selecting particular hard
processes in which there is a photon in the final state. Since the photon does
not interact with the medium, its 4-momentum is not modified and thus provides
a measure of the hard scattered parton emitted back-to-back with the photon.
Measuring the hadrons opposite to the photon is thus a promising way to
measure the jet fragmentation and misalignment between photon and hadrons to
quantify the modifications due to the medium.

##  2 Approach Validation with Monte-Carlo Data

The experimental technique consists in tagging events with a well identified
high energy direct photon and measuring the distribution of hadrons emitted
oppositely to the photon as a function of the parameter
$x_{E}=-\vec{p}_{T}^{h}\cdot\vec{p}_{T}^{\gamma}/\mid p_{T}^{\gamma}\mid^{2}$.
Such a measurement requires an excellent direct photon identification and the
measurement of charged and neutral hadrons with good $p_{T}$ resolution. In
ALICE, the electromagnetic calorimeters, PHOS ($|\Delta\eta|<0.12$ and
$\Delta\phi$ =100o) and EMCal ($|\Delta\eta|<0.7$ and $\Delta\phi$ =100o) [7,
8], are capable to measure photons with high efficiency and resolution [9].
The central tracking system (ITS and TPC), covers the pseudorapidity
$-0.9\leq\eta\leq+0.9$ and the full azimuth, is helpful for direct photon
extraction with the isolation technique.

We have first established the feasibility of $\gamma$–hadrons correlation
measurement with ALICE detectors using Monte-Carlo data. As a first result
[10] of this study, PYTHIA [11] generator is used to simulate $pp$ collisions
at $\sqrt{s}~{}=~{}14~{}$TeV containing a 2$\rightarrow$2 process with a
direct photon inside PHOS acceptance. we have demonstrated that this
measurement allows us to determine, both the jet fragmentation distribution
and the back-to-back azimuthal alignment of the direct photon and the jet.
However because of the limited acceptance covered by the calorimeters, the
measurement is restricted by statistics to photon with energies below 50 GeV.
This kinematic region is particularly interesting because jets of such low
energy loose a large fraction of their energy while traversing the medium,
rendering the medium modification most visible. In addition, because jets with
energy below 50 GeV can hardly be reconstructed in the heavy-ion environment,
the photon tagging technique provides a sensitive measurement of jets in this
kinematic range. Systematic errors due to the improper identification of
direct photons remain, within this kinematic range, lower than statistical
errors from our study [10].

To quantify the medium modification, the photon–hadrons correlation
distribution has been studied with events generated in $pp$ collisions at
$\sqrt{s}~{}=~{}5.5~{}$TeV containing a 2$\rightarrow$2 process with a direct
photon inside EMCal acceptance. They have been generated by PYTHIA [11]
generator and qPYTHIA, which includes a parton energy loss model [12] with the
medium transport parameter $\hat{q}$ =50 GeV2/fm for photon energies between 5
and 200 GeV. At this stage of the study, the heavy-ion collision background
has not yet been taken into account. Direct photons are identified with the
isolation technique requiring no hadronic activity around the direct photon
candidate inside a given cone size [13]. Hadrons detected in the azimuthal
range $\pi/2<\Delta\phi<3\pi/2$ relative to the photon were used to construct
the correlation function. The contribution of hadrons from the underlying
event was calculated from the hadrons emitted in the same azimuthal hemisphere
as the photon.

The relative azimuthal angle, $\Delta\phi=\phi_{\gamma}-\phi_{h}$, between the
direct photon and charged hadrons is strongly peaked at $\pi$ as expected for
the 2$\rightarrow$2 process (Fig. 2). When medium effects are simulated
(qPYTHIA), the $\Delta\phi$ distribution becomes broader. The broadening can
be related to the medium transport parameter $\hat{q}$. However, the effect is
quite small which will make the measurement in the heavy-ion environment quite
challenging. A stronger signal is expected to be observed in the
photon–hadrons distribution from heavy-ion collisions when compared to the
distribution from pp collisions. The resulting photon-triggered hadrons
distributions, after subtraction of underlying events, are shown in Fig. 2,
normalized to the number of trigger particles found in corresponding
generation. The statistical errors are estimated from the annual yield of
photon events with $p_{T}$ larger than 30 GeV we anticipate to collect during
one PbPb run at nominal luminosity [5]. The distribution exhibits the expected
suppression at high $x_{E}$, due to the enegy loss of the hard scatered parton
and the enhancement at low $x_{E}$ due to the fragmentation of soft gluons
radiated in the medium.

Figure 1: Relative azimuthal angle distribution
$\Delta\phi=\phi_{\gamma}-\phi_{hadron}$ for $\gamma$-jet events in pp
collisions at $\sqrt{s}$ = 5.5 TeV.

Figure 2: $\gamma$-hadron correlation distributions in quenched and unquenched
PYTHIA events as a function of $ln(1/x_{E})$.

##  3 Two Particle Correlations in pp@7TeV

Minimum bias data have been collected in pp collisions at center of mass
$\sqrt{s}~{}=7~{}$TeV. We have analyzed about 35 million events in a first
attempt to measure $\gamma$-hadrons correlations. The trigger particle is
selected as the one with the highest transverse momentum measured either in
the central tracking system or in the electromagnetic calorimeters. In the
calorimeters, electromagnetic particles are detected as clusters of hit
calorimeter cells. Roughly we have identified $\pi^{0}$ candidate as a pair of
clusters which invariant mass matches the $\pi^{0}$ mass range, $135\pm 15$
MeV, and single clusters (which do no pair with another cluster) as direct
photon candidates. No particle identification has been applied yet so that the
single cluster sample contains a sizable fraction of charged particles which
develop a shower in the calorimeters.

The azimuthal correlation between the trigger particle (charged particle,
$\pi^{0}$ candidate, single cluster) and the charged hadrons are shown in Fig.
4. The near side ($\Delta\phi=0$) and away side ($\Delta\phi=\pi$) peak are
clearly observed. Note, however, that these distributions have not been
corrected for efficiency. It is interesting to remark that at this very
preliminary stage of the analysis we find that the underlying event background
level, outside the peaks region, is independent of the type of trigger
particles, giving some confidence in the measurement. By applying an isolation
selection on the trigger candidate, where hadron activity carries less than 30
% transverse momentum of the trigger candidate inside a cone with size $R=0.4$
required, the probability of direct photon or single particle jets in the
sample enhances. Comparing the azimuthal correlation with and without
isolation selection, obviously, a suppression of the near side peak is
observed, but the away side peak is almost unaffected, as expected (Fig. 4).
However this preliminary analysis does not allow to draw any conclusion other
that these results indicate the expected behaviour. The isolation parameters
are not well adjusted and especially in our case only charged tracks are
considered in our isolation cone due to the limited calorimeter acceptance (40
% EMCAL and 60 % PHOS have been installed so far).

Figure 3: Relative azimuthal angle distribution
$\Delta\phi=\phi_{trigger}-\phi_{hadron}$ in pp collisions at $\sqrt{s}$ = 7
TeV.

Figure 4: Azimuthal correlation distributions before and after isolation cut
(IC) on the trigger particles with $p_{T}>5GeV/c$

##  4 Summary and outlook

The feasibility to measure $\gamma$–hadrons correlation in pp collisions and
medium modification effect in PbPb collisions with ALICE has been evaluated.
Such a measurement provides an exclusive observable sensitive to the
properties of the medium formed in heavy-ion collisions. So far only a
preliminary analysis has been performed on a small fraction of the data
collected by ALICE. Exciting physics will certainly come with the final
analysis of large statistics with well calibrated detectors.

## Acknowledgement

The work is partially supported by the NSFC (10875051, 10635020 and 10975061),
the Key Project of Chinese Ministry of Education (306022), the Program of
Introducing Talents of Discipline to Universities of China (QLPL200909, B08033
and CCNU09C01002).

##  5 References

## References

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