n}\chi^{\lambda}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})V^{\lambda},$

in particular, taking the Frobenius character, we have

$\displaystyle\operatorname{ch}(IH^{*}(\mathcal{Y}_{w}(X)))$
$\displaystyle=\sum_{\lambda\vdash
n}\chi^{\lambda}(C^{\prime}_{w})\operatorname{ch}(V^{\lambda})$
$\displaystyle=\sum_{\lambda\vdash
n}\chi^{\lambda}(C^{\prime}_{w})s_{\lambda}$
$\displaystyle=\operatorname{ch}(C^{\prime}_{w}).$

##  5\. Induced characters of Hecke algebras

Let $W$ be a finite Coxeter group with set of simple reflections $S$ and set
of transpositions $\Upsilon$. Each element $w\in W$ can be written (possibly
non-uniquely) as a reduced word $w=s_{1}\ldots s_{n}$, and we define
$\ell(w):=n$. The Bruhat order on $W$ is given by $w^{\prime}\leq w$ if some
reduced word of $w^{\prime}$ is a subword of a reduced word of $w$. We also
define the right and left weak orders as

$\displaystyle w^{\prime}\leq_{R}w$ $\displaystyle\text{ if
}w=w^{\prime}u\text{ for some $u\in W$ with }\ell(w)=\ell(w^{\prime})+\ell(u)$
$\displaystyle w^{\prime}\leq_{L}w$ $\displaystyle\text{ if
}w=uw^{\prime}\text{ for some $u\in W$ with
}\ell(w)=\ell(w^{\prime})+\ell(u).$

We note that if $s\in S$ and $w\in W$, then either $sw<w$ or $sw>w$, and
either $sw<_{L}w$ or $sw>_{L}w$. Moreover $sw<w$ if and only if $sw<_{L}w$.
The same holds for $\leq_{R}$.

Define also the _left and right inversion set_ of $w$ by

$\displaystyle\Upsilon_{L}(w)$
$\displaystyle:=\\{t\in\Upsilon;\ell(tw)<\ell(w)\\},$
$\displaystyle\Upsilon_{R}(w)$
$\displaystyle:=\\{t\in\Upsilon;\ell(wt)<\ell(w)\\}.$

Since we will mostly consider the set of left inversions, we write
$\Upsilon(w):=\Upsilon_{L}(w)$. Similarly, the set of _left and right
descents_ of $w$ by:

$\displaystyle D_{L}(w)$ $\displaystyle:=\Upsilon_{L}(w)\cap S,$
$\displaystyle D_{R}(w)$ $\displaystyle:=\Upsilon_{R}(w)\cap S.$

We also recall a simple result on Coxeter groups:

######  Lemma 5.1.

Let $w\in W$ and $s,s^{\prime}\in S$ such that $s^{\prime}w>w$ and $ws>w$.
Then either

  1. (1)

$s^{\prime}ws=w$, or

  2. (2)

$s^{\prime}ws>ws$ and $s^{\prime}ws>s^{\prime}w$.

See [BB05, Corollary 2.2.8].

Given $J\subset S$ we denote by $W_{J}$ the induced parabolic subgroup, that
is, the subgroup generated by $J$. We also define ${}^{J}W$ as the set of
minimal representatives of the right cosets $W_{J}\backslash W$, that is, for
each coset $W_{j}w$ we choose its element with minimal length.

######  Example 5.2.

Let $W=S_{4}$, then $S=\\{(1,2),(2,3),(3,4)\\}$. If $J=\\{(1,2),(3,4)\\}$,
then ${}^{J}W$ is the set of permutations $w$ such that $w^{-1}(1)<w^{-1}(2)$
and $w^{-1}(3)<w^{-1}(4)$.

${}^{J}W=\\{1234,1324,1342,3124,3142,3412\\}.$

Analogously $W^{J}$ would be the set of permutations $W$ such that $w(1)<w(2)$
and $w(3)<w(4)$.

By Deodhar’s lemma [BB05], we have that if $s\in S$ and $w\in{}^{J}W$ then
either $ws\in{}^{J}W$ or $W_{J}w=W_{J}ws$. We define the right action
$\cdot_{J}$ of $W$ on ${}^{J}W$ given by

$w\cdot_{J}s=\begin{cases}ws&\text{ if }ws\in{}^{J}W\\\ w&\text{
otherwise}\end{cases}$

for every $w\in{}^{J}W$ and for every $s\in S$. This is equivalent to the
right action of $W$ on $W_{J}\backslash W$.

######  Example 5.3.

Keep the notation in Example 5.2 and take $w=1342\in{}^{J}W$. Then

$\displaystyle w\cdot_{J}(1,2)$ $\displaystyle=3142,$ $\displaystyle
w\cdot_{J}(2,3)$ $\displaystyle=1342,$ $\displaystyle w\cdot_{J}(3,4)$
$\displaystyle=1324.$

The Hecke algebra $H_{W}$ is the algebra over $\mathbb{C}(q^{\frac{1}{2}})$
with basis $\\{T_{w},w\in W\\}$ satisfying the following multiplication rules

$\begin{array}[]{rll}T_{u}\cdot T_{w}=&\\!\\!\\!\\!\\!T_{uw}&\text{ if
}\ell(uw)=\ell(u)+\ell(w),\\\ T_{s}^{2}=&\\!\\!\\!\\!\\!(q-1)T_{s}+q&\text{ if
}s\in S.\end{array}$

The Hecke algebra $H_{W}$ has an involution

$\displaystyle i\colon H_{W}$ $\displaystyle\to H_{W}$ $\displaystyle q$
$\displaystyle\mapsto q^{-1}$ $\displaystyle T_{w}$ $\displaystyle\mapsto
T_{w^{-1}}^{-1}.$

The Kazhdan-Lusztig elements $C^{\prime}_{w}$ of $H_{w}$ are defined by

$\displaystyle i(C^{\prime}_{w})=$ $\displaystyle C^{\prime}_{w},$
$\displaystyle q^{\frac{\ell(w)}{2}}C^{\prime}_{w}=$ $\displaystyle\sum_{u\leq
w}P_{u,w}T_{u},$

where $P_{u,w}\in\mathbb{Z}[q]$, $P_{w,w}(q)=1$ and
$\deg(P_{u,w})<(\ell(w)-\ell(u))/2$ for $u\neq w$. The existence of such
elements $C^{\prime}_{w}$ and polynomials $P_{u,w}$ is given by [KL79].

######  Example 5.4.

If $W=S_{4}$, we have

$\displaystyle q^{2}C^{\prime}_{3412}=$
$\displaystyle(1+q)T_{1234}+(1+q)T_{1324}+T_{2134}+T_{1243}+T_{1342}+T_{1423}+T_{2143}$
$\displaystyle+T_{3124}+T_{2314}+T_{1432}+T_{3142}+T_{2413}+T_{3214}+T_{3412},$

which means that $P_{1234,3412}(q)=1+q$.

Given a set $J\subset S$, we can define a right action (also called
$\cdot_{J}$) of $H_{W}$ on

$V_{J}:=\mathbb{C}(q^{\frac{1}{2}})^{{}^{J}W}=\bigoplus_{w\in{}^{J}W}\mathbb{C}(q^{\frac{1}{2}})w$

by

$w\cdot_{J}T_{s}:=q\max(w\cdot_{J}s,w)+\min(w\cdot_{J}s,w)-w.$

######  Example 5.5.

Keep the notation in examples 5.2 and 5.3. We have that

$\displaystyle w\cdot_{J}T_{(1,2)}$ $\displaystyle=[3142],$ $\displaystyle
w\cdot_{J}T_{(2,3)}$ $\displaystyle=q[1342],$ $\displaystyle
w\cdot_{J}T_{(3,4)}$ $\displaystyle=q[1324]+(q-1)[1342].$

This is indeed an action as can be seen in [GP00, Page 287]. More precisely,
the action in loc. cit. is defined as

$w\otimes_{J}T_{s}=\begin{cases}qw&\text{ if }ws\in W_{J}w\\\ ws&\text{ if
}ws\in{}^{J}W\text{ and }\ell(ws)>\ell(w)\\\ qws+(q-1)w&\text{ if
}ws\in{}^{J}W\text{ and }\ell(ws)<\ell(w)\end{cases}$

Our action can be obtained from $\otimes_{J}$ simply by a base change. If
$\overline{w}=\frac{w}{q^{\ell(w)}}$, then

$\overline{w}\otimes_{J}T_{s}=q\overline{\max(w\cdot_{J}s,w)}+\overline{\min(w\cdot_{J}s,w)}-\overline{w}.$

The action $\otimes_{J}$ is the induced representation $i_{H_{J}}^{H_{W}}$. In
particular, we have that

$\chi^{i_{H_{j}}^{H_{W}}}(a)=Tr((-)\otimes_{J}a)=Tr((-)\cdot_{J}a).$

For simplicity, we will write $c_{J}(a):=Tr((-)\cdot_{J}a)$.

######  Remark 5.6.

Let $W=S_{n}$ be the symmetric group, and $\chi^{\lambda}$ the irreducible
characters of $S_{n}$ and, abusing notation, of $H_{n}$. Recall that for every
$a\in H_{n}$ we have defined

$\operatorname{ch}(a)=\sum_{\lambda\vdash n}\chi^{\lambda}(a)s_{\lambda}.$

By [Ram91, Theorem 3.8], we have that

(5a) $\operatorname{ch}(a)=\sum_{\lambda\vdash n}c_{\lambda}(a)m_{\lambda}.$

More precisely, in [Ram91, Equation (3.2)] there is defined the action

$w\boxtimes_{J}T_{s}=\begin{cases}qw&\text{ if }ws\in W_{J}w\\\
q^{\frac{1}{2}}ws+(q-1)w&\text{ if }ws\in{}^{J}W\text{ and
}\ell(ws)>\ell(w)\\\ q^{\frac{1}{2}}ws&\text{ if }ws\in{}^{J}W\text{ and
}\ell(ws)<\ell(w).\end{cases}$

Again, we have to change basis. As usual we write $w_{0}$ for the maximal
element of $W$ and $w_{0}(J)$ for the maximal element of $W_{J}$. Let

$\overline{w}=\frac{w_{0}(J)ww_{0}}{q^{\frac{\ell(w)}{2}}}.$

By [BB05, Proposition 2.5.4] and the fact that $w_{0}s=sw_{0}$ for every $s\in
S$, we have that

$\overline{w}\boxtimes T_{s}=\begin{cases}q\overline{w}&\text{ if }ws\in
W_{J}w\\\ \overline{ws}+(q-1)\overline{w}&\text{ if }ws\in{}^{J}W\text{ and
}\ell(ws)<\ell(w)\\\ qws&\text{ if }ws\in{}^{J}W\text{ and
}\ell(ws)>\ell(w),\end{cases}$

which is precisely the action $\cdot_{J}$.

We are now interested in the elements
$(1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}})$, which correspond to the Bott-
Samelson resolution of the Schubert varieties (see Section 3.3). For each
sequence ${\underline{s}}=(s_{1},s_{2},\ldots,s_{n})$ we define

$c_{J}({\underline{s}})=c_{J}((1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}}))=Tr((-)\cdot_{J}(1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}})).$

Note that for each $w\in{}^{J}W$, we have

$w\cdot_{J}(1+T_{s})=q\max(w\cdot_{J}s,w)+\min(w\cdot_{J}s,w).$

Alternatively we can compute $c_{J}({\underline{s}})$ as follows. Define
$\phi_{J,s,i}\colon{}^{J}W\to{}^{J}W$ for $i=0,1$ as

$\displaystyle\phi_{J,s,1}(w)$ $\displaystyle=\max(w\cdot_{J}s,w)$
$\displaystyle\phi_{J,s,0}(w)$ $\displaystyle=\min(w\cdot_{J}s,w).$

Moreover, for each binary word $\underline{b}=(b_{1},\ldots,b_{n})$, define

(5b)
$\phi_{J,{\underline{s}},\underline{b}}=\phi_{J,s_{n},b_{n}}\circ\ldots\circ\phi_{J,s_{1},b_{1}}.$

Then we have

(5c)
$c_{J}({\underline{s}})=\sum_{w\in{}^{J}W}\sum_{\underline{b}\in\operatorname{Bin}_{J,{\underline{s}}}(w)}q^{|\underline{b}|}$

where $|\underline{b}|:=\sum b_{i}$ and
$\operatorname{Bin}_{J,{\underline{s}}}(w):=\\{\underline{b}\in\\{0,1\\}^{n};\phi_{J,{\underline{s}},\underline{b}}(w)=w\\}$.
When $J=\emptyset$, we will write $\phi_{{\underline{s}},\underline{b}}$
instead of $\phi_{\emptyset,{\underline{s}},\underline{b}}$.

######  Example 5.7.

Keep the notation in Examples 5.2 and 5.3, and consider
${\underline{s}}=((1,2),(2,3))$. Then

$\displaystyle[1234]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q+1)[1234]\cdot_{J}(1+T_{(2,3)})$ $\displaystyle=$
$\displaystyle q(q+1)[1324]+(q+1)[1234]),$
$\displaystyle[1324]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q[3124]+[1324])\cdot_{J}(1+T_{(2,3)})$ $\displaystyle=$
$\displaystyle q(q+1)[3124]+q[1324]+[1234],$
$\displaystyle[1342]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q[3142]+[1342])\cdot_{J}(1+T_{(2,3)})=$ $\displaystyle=$
$\displaystyle q^{2}[3412]+q[3142]+(q+1)[1342],$
$\displaystyle[3124]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q[3124]+[1324])\cdot_{J}(1+T_{(2,3)})=$ $\displaystyle=$
$\displaystyle q(q+1)[3124]+q[1324]+[1234],$
$\displaystyle[3142]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q[3142]+[1342])\cdot_{J}(1+T_{(2,3)})=$ $\displaystyle=$
$\displaystyle q^{2}[3412]+q[3142]+(q+1)[1342],$
$\displaystyle[3412]\cdot_{J}(1+T_{(1,2)})(1+T_{(2,3)})=$
$\displaystyle(q+1)[3412]\cdot_{J}(1+T_{(2,3)})$ $\displaystyle=$
$\displaystyle q(q+1)[3412]+(q+1)[3142].$

Hence

$c_{J}(L)=(q+1)+q+(q+1)+q(q+1)+q+q(q+1)=2q^{2}+6q+2.$

Alternatively, we have the following words $\underline{b}\in\\{0,1\\}^{2}$
satisfying $\phi_{J,{\underline{s}},\underline{b}}(w)=w$ for each
$w\in{}^{J}W$.

$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(1234)=$
$\displaystyle\\{(1,0),(0,0)\\}$
$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(1324)=$
$\displaystyle\\{(0,1)\\}$
$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(1342)=$
$\displaystyle\\{(0,0),(0,1)\\}$
$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(3124)=$
$\displaystyle\\{(1,1),(1,0)\\}$
$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(3142)=$
$\displaystyle\\{(1,0)\\}$
$\displaystyle\operatorname{Bin}_{J,{\underline{s}}}(3412)=$
$\displaystyle\\{(1,1),(0,1)\\}.$

which recovers $c_{J}(L)$.

######  Proposition 5.8.

If the sequence ${\underline{s}}=(s_{1},\ldots,s_{n})$ is such that for each
simple reflection $s\in S$ there exists $i=1,\ldots,n$ with $s_{i}=s$, then

$c_{J}({\underline{s}})=q^{n}+[\textnormal{lower degrees}].$

By Equation (5c) it suffices to prove that there exists exactly one element
$w\in{}^{J}W$ such that $\phi_{J,{\underline{s}},\underline{b}}(w)=w$ for
$\underline{b}=(1,1,\ldots,1)$. Let $w_{0}\in{}^{J}W$ be such that
$\phi_{J,{\underline{s}},\underline{b}}(w_{0})=w_{0}$, and define
$w_{i+1}:=\phi_{J,{\underline{s}}|_{i},\underline{b}|_{i}}(w_{i})$ for
$i=0,\ldots,n-1$. Since $w_{i+1}=\max(w_{i}\cdot_{J}s_{i+1},w_{i})$ and
$w_{0}=w_{n}$, we must have that $w_{i+1}=w_{i}$ for all $i=0,\ldots,n-1$.
This means that $w\geq w\cdot_{J}s_{i}$ for every $i=1,\ldots,n$, which
implies that $w\geq w\cdot_{j}s$ for every $s\in S$ (by the hypothesis that
each element $s\in S$ appears in the sequence ${\underline{s}}$). This happens
if and only if $w$ is the unique maximal element of ${}^{J}W$, which finishes
the proof.

######  Definition 5.9.

Fix $J\subset S$, and define
$D_{{\underline{s}},J}(w):=D_{{\underline{s}}}(w)\cap J$. Moreover, for
${\underline{s}}=(s_{1},\ldots,s_{n})$ and
$\underline{b}=(b_{1},\ldots,b_{n})$ let us define
${\underline{s}}|_{i}=(s_{1},\ldots,s_{i})$ and
$\underline{b}|_{i}=(b_{1},\ldots,b_{i})$. We say that a word $\underline{b}$
is $(L,J)$-_good_ for $w$ if the following holds:

  1. (1)

$\phi_{{\underline{s}},\underline{b}}(w)=w$, and

  2. (2)

for each $s\in D_{{\underline{s}},J}(w)$, there exists $i\in\\{1,\ldots,n\\}$
such that $\phi_{{\underline{s}}|_{i-1},\underline{b}|_{i-1}}(w)\cdot
s_{i}=s\cdot\phi_{{\underline{s}}|_{i-1},\underline{b}|_{i-1}}(w)$.

The intuition behind this definition is Theorem 8.2.

######  Example 5.10.

Consider $W=S_{4}$, $w=4213$,
${\underline{s}}=((2,3),(1,2),(3,4),(2,3),(1,2),(3,4))$, and $J=S$. Then
$\mathcal{D}_{\underline{s}}(w)=\\{(1,2),(3,4)\\}$, and we claim that the word
$\underline{b}=(1,0,1,1,1,0)$ is a $(L,J)$-good word for $w$. For simplicity
let $w_{i}:=\phi_{{\underline{s}}|_{i},\underline{b}|_{i}}(w)$.

  * •

Since $s_{1}=(2,3)$ and $b_{1}=1$, we have $ws_{1}=4123<w$ and
$w_{1}=\max(ws_{1},w)=w=4213$. Moreover, it holds that $(1,2)w=4123=ws_{1}$.

  * •

Since, $s_{2}=(1,2)$ and $b_{2}=0$, we have $w_{1}s_{2}=2413<w_{1}$ and
$w_{2}=\min(w_{1}s_{2},w_{1})=w_{1}s_{2}=2413$. Moreover, it holds that
$(2,4)w_{1}=2413=w_{1}s_{2}$.

  * •

Since, $s_{3}=(3,4)$ and $b_{3}=1$, we have $w_{2}s_{3}=2431>w_{2}$ and
$w_{3}=\max(w_{2}s_{3},w_{2})=w_{2}s_{3}=2431$. Moreover, it holds that
$(1,3)w_{2}=2413=w_{2}s_{3}$.

  * •

Since, $s_{4}=(2,3)$ and $b_{4}=1$, we have $w_{3}s_{4}=2341<w_{3}$ and
$w_{4}=\max(w_{3}s_{4},w_{3})=w_{3}=2431$. Moreover, it holds that
$(3,4)w_{3}=2341=w_{3}s_{4}$.

  * •

Since, $s_{5}=(1,2)$ and $b_{5}=1$, we have $w_{4}s_{5}=4231>w_{4}$ and
$w_{5}=\max(w_{4}s_{5},w_{4})=w_{4}s_{5}=4231$. Moreover, it holds that
$(2,4)w_{4}=4231=w_{4}s_{5}$.

  * •

Since, $s_{6}=(3,4)$ and $b_{6}=0$, we have $w_{5}s_{6}=4213<w_{5}$ and
$w_{6}=\min(w_{5}s_{6},w_{5})=w_{5}s_{6}=4213$. Moreover, it holds that
$(1,3)w_{5}=4213=w_{5}s_{6}$.

Then we have that the conditions of Definition 5.9 are satisfied. Indeed, we
have that $w_{6}=w$ and for each simple reflection
$s\in\\{(1,2),(3,4)\\}=D_{{\underline{s}},J}(w)$ there exists
$i\in\\{1,2,3,4\\}$ such that $w_{i-1}s_{i}=sw_{i}$. If $s=(1,2)$, then $i=1$,
and if $s=(3,4)$, then $i=4$.

Let $\operatorname{Good}(L,J,w)$ be the set of all $(L,J)$-good words for $w$
and define

$c^{\prime}_{J}(L)=\sum_{w\in W}\sum_{b\in Good(L,J,w)}q^{|b|}.$

The main goal of this section is to prove the following theorem.

######  Theorem 5.11.

Let $J$ be a subset of $S$ and ${\underline{s}}=(s_{1},\ldots,s_{n})$ a
sequence of simple reflections. We have that

$c^{\prime}_{J}(L)=c_{J}(L).$

See also [KLS19] for combinatorial characterizations of the induced sign
characters.

######  Example 5.12.

Keeping the notation in Examples 5.2, 5.3 and 5.7. We have that

$\displaystyle\operatorname{Good}(J,{\underline{s}},1234)=$
$\displaystyle\\{(0,0)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},1324)=$
$\displaystyle\\{(0,1)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},1342)=$
$\displaystyle\\{(0,0)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},1432)=$
$\displaystyle\\{(0,1)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},2134)=$
$\displaystyle\\{(1,0)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},3124)=$
$\displaystyle\\{(1,0)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},3142)=$
$\displaystyle\\{(1,0)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},3214)=$
$\displaystyle\\{(1,1)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},3412)=$
$\displaystyle\\{(0,1)\\},$
$\displaystyle\operatorname{Good}(J,{\underline{s}},4312)=$
$\displaystyle\\{(1,1)\\}.$

While the other good sets are empty. This tells us that
$c^{\prime}_{J}(L)=2q^{2}+6q+2$, which agrees with $c_{J}(L)$.

We will need a few results before we are able to prove Theorem 5.11. We begin
with the following lemma.

######  Lemma 5.13.

Let $s,s^{\prime}\in S$ and $b=0,1$. Then

$\phi_{s,b}(s^{\prime}w)=\begin{cases}\phi_{s,b}(w)&\text{if
}s^{\prime}w=ws\\\ s^{\prime}\phi_{s,b}(w)&\text{otherwise.}\\\ \end{cases}$

Moreover, if $s^{\prime}w>_{\underline{s}}w$, then
$\phi_{s,b}(s^{\prime}w)\geq_{\underline{s}}\phi_{s,b}(w)$. This means that
$\phi_{s,b}\colon W\to W$ is a weak-left order preserving map.

First we will assume that $s^{\prime}w>w$ and $ws>w$ (in particular
$s^{\prime}w>_{L}w$). By Lemma 5.1, we have two cases. In the first case we
have $s^{\prime}ws=w$, so that

$\phi_{s,1}(s^{\prime}w)=\max(s^{\prime}ws,s^{\prime}w)=\max(w,ws)=\phi_{s,1}(w).$

The same argument holds for $b=0$. In the second case, we have
$s^{\prime}ws>ws$ and $s^{\prime}ws>s^{\prime}w$, so

$\displaystyle\phi_{s,1}(s^{\prime}w)=\max(s^{\prime}ws,s^{\prime}w)=s^{\prime}ws=s^{\prime}\max(ws,w)=s^{\prime}\phi_{s,1}(w)$
$\displaystyle\phi_{s,0}(s^{\prime}w)=\min(s^{\prime}ws,s^{\prime}w)=s^{\prime}w=s^{\prime}\min(ws,w)=s^{\prime}\phi_{s,0}(w)$

Since $s^{\prime}ws>_{L}ws$, we have
$\phi_{s,1}(s^{\prime}w)>_{\underline{s}}\phi_{s,1}(w)$, and since
$s^{\prime}w>_{L}w$, we have
$\phi_{s,0}(s^{\prime}w)>_{\underline{s}}\phi_{s,0}(w)$.

The case $s^{\prime}w<w$ follows from the result above for
$w^{\prime}=s^{\prime}w$, and the case $ws<w$ from the fact that
$\phi_{s,b}(w)=\phi_{s,b}(ws)$.

######  Corollary 5.14.

For every ${\underline{s}}=(s_{1},\ldots,s_{k})$, $s^{\prime}\in S$ and binary
word $\underline{b}$, we have that

$\phi_{{\underline{s}},\underline{b}}(s^{\prime}w)=\begin{cases}\phi_{{\underline{s}},\underline{b}}(w)&\text{if
}s^{\prime}\phi_{{\underline{s}}|_{i-1},\underline{b}|_{i-1}}(w)=\phi_{{\underline{s}}|_{i-1},\underline{b}|_{i-1}}(w)s_{i}\text{
for some i}\\\
s^{\prime}\phi_{{\underline{s}},\underline{b}}(w)&\text{otherwise.}\\\
\end{cases}$

Moreover, $\phi_{{\underline{s}},\underline{b}}\colon W\to W$ is a weak-left
order preserving map of posets.

This follows from successive applications of Lemma 5.13.

######  Lemma 5.15.

Consider $J\subset S$, $\overline{w}\in{}^{J}W$,
${\underline{s}}=(s_{1},\ldots,s_{n})$ and $\underline{b}\in\\{0,1\\}^{n}$.
Then

$\phi_{{\underline{s}},\underline{b}}(W_{J}\overline{w})=W_{J}\phi_{J,{\underline{s}},\underline{b}}(\overline{w}).$

In particular, if
$\phi_{J,{\underline{s}},\underline{b}}(\overline{w})=\overline{w}$ we have
that

$\phi_{{\underline{s}},\underline{b}}(W_{J}\overline{w})=W_{J}\overline{w}.$

Let $w\in W_{J}\overline{w}$. That means that there exists $u\in W_{J}$ such
that $w=u\overline{w}$ with $u\in W_{J}$ and
$\ell(w)=\ell(u)+\ell(\overline{w})$. Let $s\in S$, then we have two cases:
Either $\overline{w}s\in{}^{J}W$ or $W_{J}\overline{w}=W_{J}\overline{w}s$, so
that either $ws=u\overline{w}s$ or $ws=u^{\prime}\overline{w}$ for
$u^{\prime}\in W_{J}$. In the first case we have

$\phi_{s,b}(w)=u\phi_{J,s,b}(\overline{w}),$

while in the second case we have $\phi_{J,s,b}(w)=w$, hence

$\phi_{s,b}(w)=\begin{cases}u\phi_{J,s,b}(\overline{w})\\\
u^{\prime}\phi_{J,s,b}(\overline{w}).\end{cases}$

This proves that

$\phi_{s,b}(W_{J}\overline{w})=W_{J}\phi_{J,s,b}(w).$

With successive applications we get the result of the Lemma. For each
$J\subset S$, consider the sets

$\displaystyle F_{1}:=$
$\displaystyle\\{(w,\underline{b});\underline{b}\in\operatorname{Good}(L,J,w)\\}\subset
W\times\\{0,1\\}^{n},$ $\displaystyle F_{2}:=$
$\displaystyle\\{(w,\underline{b});\phi_{J,{\underline{s}},\underline{b}}(w)=w\\}\subset{}^{J}W\times\\{0,1\\}^{n},$

and define the function

$\displaystyle\kappa_{J}\colon F_{1}$ $\displaystyle\to F_{2}$
$\displaystyle(w,\underline{b})$
$\displaystyle\mapsto(\overline{w},\underline{b}).$

where $\overline{w}$ is the minimal representative of $W$ in $W_{J}w$.

######  Example 5.16.

Keep the notation in Examples 5.2, 5.3, 5.7, and 5.12. Then

$F_{1}=\left\\{\begin{array}[]{l}(1234,(0,0)),(1324,(0,1)),(1342,(0,0)),(1432,(0,1)),(2134,(1,0)),\\\
(3124,(1,0)),(3142,(1,0)),(3214,(1,1)),(3412,(0,1)),(4312,(1,1))\end{array}\right\\}$

and

$F_{2}=\left\\{\begin{array}[]{l}(1234,(0,0)),(1234,(1,0)),(1324,(0,1)),(1342,(0,0)),(1342,(0,1)),\\\
(3124,(1,0)),(3124,(1,1)),(3142,(1,0)),(3412,(0,1)),(3412,(0,1))\end{array}\right\\}.$

Moreover, we can see that

$\displaystyle 1234,2134$ $\displaystyle\in W_{J}[1234],$ $\displaystyle 1324$
$\displaystyle\in W_{J}[1324]$ $\displaystyle 1342,1432$ $\displaystyle\in
W_{J}[1342]$ $\displaystyle 3124,3214$ $\displaystyle\in W_{J}[3124]$
$\displaystyle 3142$ $\displaystyle\in W_{J}[3142]$ $\displaystyle 3412,4312$
$\displaystyle\in W_{J}[3412].$

We can see that $\kappa_{J}$ is a bijection. Proposition 5.17 below proves
that this is always the case.

######  Proposition 5.17.

The function $\kappa_{J}$ is a bijection.

Fix $\overline{w}\in{}^{J}W$ and $\underline{b}\in\\{0,1\\}^{n}$ with
$\phi_{J,{\underline{s}},\underline{b}}(\overline{w})=\overline{w}$. Let us
prove that exists exactly one $w\in W_{J}\overline{w}$ such that
$\underline{b}\in Good(L,J,w)$. Consider
$\phi_{{\underline{s}},\underline{b}}\colon W\to W$. Since
$\phi_{J,{\underline{s}},\underline{b}}(\overline{w})=\overline{w}$, we have
by Lemma 5.15 that
$\phi_{{\underline{s}},\underline{b}}(W_{J}\overline{w})=W_{J}\overline{w}$.
Note that $\phi_{{\underline{s}},\underline{b}}|_{W_{J}\overline{w}}$ still is
weak-left order preserving on $W_{J}\overline{w}$. By Tarski’s fixed point
theorem ([Tar55]), the set of fixed points of
$\phi:=\phi_{{\underline{s}},\underline{b}}|_{W_{j}\overline{w}}$ is non-empty
and has a minimal element $w$. We will prove that this $w$ satisfies
$\underline{b}\in\operatorname{Good}(L,J,w)$ and is the unique element of
$W_{J}\overline{w}$ with this property.

Choosing $s^{\prime}\in D_{{\underline{s}},J}(w)$, we have
$s^{\prime}w<_{\underline{s}}w$ and $s^{\prime}w\in W_{J}\overline{w}$. Since
$w$ is minimal, we have $\phi(s^{\prime}w)\neq s^{\prime}w$. On the other
hand, by Corollary 5.14, we have that $\phi(s^{\prime}w)=\phi(w)=w$ or
$\phi(s^{\prime}w)=s^{\prime}\phi(w)=s^{\prime}w$, and therefore
$\phi(s^{\prime}w)=\phi(w)$. Again by Corollary 5.14, there exists $i$ such
that
$s^{\prime}\phi_{{\underline{s}}|_{i-1}},\underline{b}|_{i-1}(w)=\phi_{{\underline{s}}|_{i-1}},\underline{b}|_{i-1}(w)s_{i}$.
This proves that $\underline{b}\in\operatorname{Good}(L,J,w)$.

Choose a fixed point $\widehat{w}\in W_{J}\overline{w}$ that it is not
minimal. By Lemma 5.18 below there exists $s^{\prime}\in
D_{{\underline{s}},J}(\widehat{w})$ such that $s^{\prime}\widehat{w}$ is a
fixed point of $\phi$.

Repeating the argument above, we have that, in order for
$s^{\prime}\widehat{w}$ be a fixed point, it must hold that
$s^{\prime}\phi_{{\underline{s}}|_{i-1}},\underline{b}|_{i-1}(w)\neq\phi_{{\underline{s}}|_{i-1}},\underline{b}|_{i-1}(w)s_{i}$
for every $i$. This means that
$\underline{b}\notin\operatorname{Good}(L,J,\widehat{w})$.

######  Lemma 5.18.

Let $w$ be the minimal fixed point and let $\widehat{w}$ be a fixed point.
Choose a maximal chain
$w<_{L}w_{1}<_{\underline{s}}\ldots<_{\underline{s}}w_{k}<_{\underline{s}}\widehat{w}$
in $W_{J}\overline{w}$. Then $w_{i}$ is a fixed point for every $i$. In
particular, there exists $s^{\prime}\in D_{{\underline{s}},J}(\widehat{w})$
such that $s^{\prime}\widehat{w}$ is a fixed point.

Since $\phi$ is order preserving we have that
$w\leq_{\underline{s}}\phi(w_{1})\leq\ldots\leq\phi(w_{k})\leq\widehat{w}$.
Since $w_{i+1}=s^{\prime}_{i}w_{i}$, by Lemma 5.13 we have that
$\ell(\phi(w_{i+1}))=\phi(w_{i})$ or $\ell(\phi(w_{i+1}))=\phi(w_{i})+1$.
Since
$\ell(\phi(\widehat{w}))=\ell(\widehat{w})=\ell(w)+k+1=\ell(\phi(w))+k+1$, we
have that $\ell(\phi(w_{i+1}))=\phi(w_{i})+1$ holds for all $i=1,\ldots,k$. In
particular $\phi(w_{i+1})=s^{\prime}_{i}\phi(w_{i})$, which proves that
$\phi(w_{i})=w_{i}$ for all $i$. Moreover, $w_{k}=s^{\prime}\widehat{w}$ for
some $s^{\prime}\in D_{{\underline{s}},J}(w)$. This proves Theorem 5.11.

To finish this section we restrict ourselves to the case where $W$ is a Weyl
group. In this case, we have bijections between $S$ and $\Delta$ (as seen in
Section 3) and between $\Upsilon$ and $\Phi^{+}$. For a transposition
$t\in\Upsilon$ we write $\gamma_{t}$ for the induced positive root, and
conversely if $\gamma\in\Phi^{+}$ is a positive root, we write $t_{\gamma}$
for the induced transposition. This bijection between $\Upsilon$ and
$\Phi^{+}$ induces a bijection between $\Phi(w)$ and
$\Upsilon_{\underline{s}}(w)$ for every $w\in W$.

######  Definition 5.19.

Let $\underline{\delta}:=(\delta_{i})_{i=1,\ldots,m}$ be a sequence of
positive roots. We say that $\underline{\delta}$ is a _good sequence_ for $w$
if for every $\gamma\in\Phi(w)$ at least one of the following holds:

  1. (1)

There exists $i$ such that $\gamma=\delta_{i}$.

  2. (2)

For all $\alpha\in\Delta$ such that $\gamma-\alpha\in\Phi\setminus\Phi(w)$,
there exists $i$ such that $\gamma-\alpha=\delta_{i}$ and
$\delta_{j}\neq\alpha$ for every $j<i$.

######  Definition 5.20.

Let ${\underline{s}}=(s_{1},\ldots,s_{n})$ be a sequence of simple
reflections, $J\subset S$, $w\in W$ and
$\underline{b}\in\operatorname{Good}(L,J,w)$. Write $w=u\overline{w}$ with
$u\in W_{J}$ and $\overline{w}\in{}^{J}W$, and define
$w_{i}:=\phi_{{\underline{s}}_{i},\underline{b}|_{i}}(w)$. Let $j_{i}$ be the
unique integer such that $b_{j_{i}}=1$ and $\sum_{l=1}^{j_{i}}{b_{l}}=i$. Let
$t_{i}\in\Upsilon$ be the transposition such that
$t_{i}w_{j_{i}-1}=w_{j_{i}-1}s_{j_{i}}$. Define the sequence
$\underline{\delta}=\underline{\delta}(L,J,w,\underline{b})=(\delta_{1},\ldots,\delta_{|\underline{b}|})$
by $\delta_{i}=\gamma_{t_{i}}$.

######  Example 5.21.

Let $W$, $w$, ${\underline{s}}$ and $\underline{b}$ as in Example 5.10. Then
the sequence $\underline{\delta}$ defined in Proposition 5.22 is
$\gamma_{1,2},\gamma_{1,3},\gamma_{3,4},\gamma_{2,4}$ corresponding to the
transpositions $(1,2),(1,3),(3,4),(2,4)$.

######  Proposition 5.22.

Let ${\underline{s}}=(s_{1},\ldots,s_{n})$ be a sequence of simple reflections
(roots), $J\subset S$, $w\in W$ and
$\underline{b}\in\operatorname{Good}(L,J,w)$. Write $w=u\overline{w}$ with
$\overline{w}\in{}^{J}W$, then the sequence
$\underline{\delta}=\underline{\delta}(L,J,w,\underline{b})$ is a good
sequence for $u$ in $W_{J}$.

As in Definition 5.20, write
$w_{i}=\phi_{{\underline{s}}|_{i},\underline{b}|_{i}}(w)$. Choose
$\gamma\in\Phi(u)\subset\Phi_{J}$ such that $\gamma\neq\delta_{i}$ for every
$i=1,\ldots,|\underline{b}|$. We claim that $\gamma\in\Phi(w_{i})$ for every
$i$. Indeed, we have that $\gamma\in\Phi(w)$ because

$\ell(t_{\gamma}w)=\ell(t_{\gamma}u\overline{w})\leq\ell(t_{\gamma}u)+\ell(\overline{w})<\ell(w)+\ell(\overline{w})=\ell(w).$

Assume by contradiction that there exists $i$ such that
$\gamma\notin\Phi(w_{i})$ and choose $k:=\max\\{i;\gamma\notin\Phi(w_{i})\\}$.
Clearly, $k\neq n$ because $w_{n}=w$. Moreover, since
$\gamma\in\Phi(w_{k+1})\setminus\Phi(w_{k})$, we have $w_{k+1}=w_{k}s_{k+1}$.
By Lemma 5.23 below, $w_{k+1}>w_{k}$ and $t_{\gamma}w_{k}=w_{k+1}s_{k+1}$.
This means $b_{k+1}=1$ (because $w_{k+1}>w_{k}$) and hence
$\gamma=\delta_{k+1}$, a contradiction. This proves that
$\gamma\in\Phi(w_{i})$ for every $i$.

Let $s\in J$ be such that $\gamma-\alpha_{s}\in\Phi$ (and hence
$\gamma-\alpha_{s}\in\Phi_{J}$). Since both $\Phi(w)$ and
$\Phi^{+}\setminus\Phi(w)$ are closed by sums, and since $\gamma\in\Phi(w)$,
we have that if $\alpha_{s}\notin\Phi(w)$, then $\gamma-\alpha_{s}\in\Phi(w)$.

Assume that $\alpha_{s}\in\Phi(w)$ (equivalently,
$s\in\Upsilon_{\underline{s}}(w)$). By Lemma 5.23, either
$s\in\Upsilon_{\underline{s}}(w_{k})$ for every $k$ or there exists $k$ such
that
$s\in\Upsilon_{\underline{s}}(w_{k+1})\setminus\Upsilon_{\underline{s}}(w_{k})$.

Let us assume first that $s\in\Upsilon_{\underline{s}}(w_{k})$ for every $k$.
Since $\underline{b}$ is a $(L,J)$-good word for $w$, we have that there
exists $k$ such that $sw_{k}=w_{k}s_{k+1}$. Moreover, since
$s\in\Upsilon_{\underline{s}}(w_{k})$, we have that
$\ell(w_{k}s_{k+1})=\ell(sw_{k})<\ell(w_{k})$, so by Lemma 5.23 and the fact
that $s\in\Upsilon_{\underline{s}}(w_{k+1})$, we must have $b_{k+1}=1$.

In the second case, there exists $k$ such that
$\Phi(w_{k+1})=\Phi(w_{k})\setminus\\{\alpha_{s}\\}$. Since $w_{n}=w$, we must
have that there also exists some $k$ such that
$\Phi(w_{k+1})=\Phi(w_{k})\cup\\{\alpha_{s}\\}$. In that case, we have that
$w_{k+1}=\alpha w_{k}=w_{k}s_{k+1}$, $b_{k+1}=1$ and $w_{k+1}>w_{k}$.

Let $k$ be the minimum $i$ such that $b_{i+1}=1$ and $sw_{i}=w_{i+1}s_{i+1}$,
which exists by the discussion above. In particular, we have that
$\delta_{j}\neq\alpha_{s}$ for every $j<k+1$.

Let us proceed by contradiction and assume that
$\gamma-\alpha_{s}\notin\Phi(w)$ and that there does not exist $j\leq k$ with
$\gamma-\alpha_{s}=\delta_{j}$. By Lemma 5.23, this means that
$\gamma-\alpha_{s}\notin\Phi(w_{j})$ for every $j\leq k$. Since
$\gamma\in\Phi(w_{j})$ for every $j$, we must have that
$\alpha_{s}\in\Phi(w_{j})$ for every $j\leq k$. In particular, we have that
$\alpha_{s}\in\Phi(w_{k})$, $\gamma-\alpha_{s}\notin\Phi(w_{k})$,
$\gamma\in\Phi(w_{k})$ and that $w_{k}^{-1}sw_{k}=s_{k+1}$. This contradicts
Lemma 5.24

######  Lemma 5.23.

Let $w\in W$ and $s\in S$, and assume that $ws>w$. Then
$\Upsilon_{\underline{s}}(ws)=\Upsilon_{\underline{s}}(w)\cup\\{t\\}$, where
$t\in\Upsilon$ is such that $tw=ws$. Conversely, if $ws<w$ then
$\Upsilon_{\underline{s}}(ws)=\Upsilon_{\underline{s}}(w)\setminus\\{t\\}$.

We first prove that $t\in\Upsilon_{\underline{s}}(ws)$. Since $t(ws)=w<ws$,
then $t\in\Upsilon_{\underline{s}}(ws)$. Now let
$t^{\prime}\in\Upsilon_{\underline{s}}(w)$, so that
$\ell(t^{\prime}w)<\ell(w)$, which means

$\ell(t^{\prime}ws)\leq\ell(t^{\prime}w)+1\leq\ell(w)<\ell(ws).$

Then $t^{\prime}\in\Upsilon_{\underline{s}}(ws)$, hence
$\Upsilon_{\underline{s}}(w)\cup\\{t\\}\subset\Upsilon_{\underline{s}}(ws)$.
But these sets both have cardinality $\ell(w)+1$, so
$\Upsilon_{\underline{s}}(ws)=\Upsilon_{\underline{s}}(w)\cup\\{t\\}$ as
desired. The second statement follows directly from the first.

######  Lemma 5.24.

Let $w\in W$, $\alpha\in\Phi(w)$ and $\beta\notin\Phi(w)$ be such that
$\alpha+\beta\in\Phi(w)$. Then $w^{-1}(\alpha)\notin-\Delta$, or,
equivalently, $w^{-1}s_{\alpha}w$ is not a simple transposition.

By the hypothesis we have that

$w^{-1}(\alpha)\in\Phi^{-}\text{, }w^{-1}(\beta)\in\Phi^{+}\text{ and
}w^{-1}(\alpha+\beta)\in\Phi^{-}.$

Since $w^{-1}(\alpha)=w^{-1}(\alpha+\beta)-w^{-1}(\beta)$ we have that
$w^{-1}(\alpha)\notin-\Delta$.

##  6\. Paving by affines the Bott-Samelson variety

In this section we will describe an affine paving of the Bott-Samelson
variety. Usually, these affine pavings are described in terms of Tits
_buildings_ and _galleries_ , see [Tit13], [Gau01, Section 3] and [CC17,
Chapter 14]. We will avoid defining such objects and give instead an explicit
construction.

Our construction will be done inductively. Let
${\underline{s}}=(s_{1},\ldots,s_{n})$ be a word in $S$. When $n=0$, then
$\mathcal{BS}_{\underline{s}}=\mathcal{B}$ and the paving will simply be the
paving given by Schubert stratification, that is,
$\Omega_{w,e}^{\circ}=B\dot{w}B/B\subset\mathcal{B}$. We have the following
isomorphim (see Section 3)

$\displaystyle U^{w}$ $\displaystyle\to\Omega_{w,e}^{\circ}$ $\displaystyle u$
$\displaystyle\mapsto u\dot{w}B.$

######  Example 6.1.

Consider $w=4213\in S_{4}$. We saw in example 3.1 that

$U^{w}=\left\\{\left(\begin{array}[]{cccc}1&*&0&*\\\ 0&1&0&*\\\ 0&0&1&*\\\
0&0&0&1\end{array}\right)\right\\}.$

This means that $\Omega_{w,e}^{\circ}$ is the set of flags associated to
matrices of the form

$\left(\begin{array}[]{cccc}1&x_{1}&0&x_{2}\\\ 0&1&0&x_{3}\\\ 0&0&1&x_{4}\\\
0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\\\
1&0&0&0\end{array}\right)=\left(\begin{array}[]{cccc}x_{2}&x_{1}&1&0\\\
x_{3}&1&0&0\\\ x_{4}&0&0&1\\\
1&0&0&0\end{array}\right)=:(v_{1},v_{2},v_{3},v_{4}).$

The flag $V_{\bullet}$ can be recovered from the above matrix by setting
$V_{1}=\langle v_{1}\rangle$, $V_{2}=\langle v_{1},v_{2}\rangle$,
$V_{3}=\langle v_{1},v_{2},v_{3}\rangle$ and $V_{4}=\mathbb{C}^{4}$ (recall
Example 3.20). We will usually abuse notation and write
$V_{\bullet}=(v_{1},v_{2},v_{3},v_{4})$.

We generalize the above construction to any $n>0$. Let
${\underline{s}}=(s_{1},\ldots,s_{n})$ be a word in $S$, $w$ any element of
$W$ (not necessarily the one given by ${\underline{s}}$), and
$\underline{b}=(b_{1},\ldots,b_{n})$ a binary word of length $n$. Recall the
definition of the function $\phi_{{\underline{s}},\underline{b}}\colon W\to W$
in Equation 5b. We write
$w_{i}=\phi_{{\underline{s}}|_{i},\underline{b}|_{i}}(w)$ and, as usual,
$\dot{w}_{i}$ for the chosen representative of $w_{i}$ in $G$. Moreover, let
$\gamma_{i}$ be the positive root such that
$t_{\gamma_{i}}w_{i-1}=w_{i-1}s_{i}$.

For $\gamma\in\phi^{+}$ and $b=0,1$ define

$U_{\gamma,b}:=\begin{cases}U_{\gamma}&\text{ if }b=1\\\ \\{1\\}&\text{ if
}b=0,\end{cases}$

and consider the map $g_{{\underline{s}},\underline{b}}^{w}$ defined by

(6a) $\displaystyle U^{w}\times U_{\gamma_{1},b_{1}}\times\cdots\times
U_{\gamma_{n},b_{n}}$ $\displaystyle\to\mathcal{BS}_{{\underline{s}}}$
$\displaystyle(u_{0},u_{1},\ldots,u_{n})$
$\displaystyle\mapsto(u_{0}\dot{w}B,u_{0}u_{1}\dot{w}_{1}B,\ldots,u_{0}u_{1}\ldots
u_{n}\dot{w}_{n}B).$

To motivate this definition we give an example.

######  Example 6.2.

As in example 6.1, consider $w=4213\in S_{4}$ and let
${\underline{s}}=(s_{1})$, with $s_{1}=(2,3)$. Then

$\mathcal{BS}_{\underline{s}}=\\{(V_{\bullet}^{1},V_{\bullet}^{2});V_{j}^{1}=V_{j}^{2}\text{
for every }j=1,3,4\\}.$

Assuming that $V_{\bullet}^{1}\in\Omega_{w,e}^{\circ}$, we have
$V_{1}^{1}=\langle v_{1}\rangle$, $V_{2}^{1}=\langle v_{1},v_{2}\rangle$,
$V_{3}^{1}=\langle v_{1},v_{2},v_{3}\rangle$ and $V_{4}^{1}=\mathbb{C}^{4}$
for $v_{1},v_{2},v_{3},v_{4}$ as in Example 6.1.

If we fix $V_{\bullet}^{1}$, we have a $\mathbb{P}^{1}$ of choices for
$V_{\bullet}^{2}$, in fact, we just have to choose $V_{2}^{2}$ satisfying

$<v_{1}>=V_{1}^{1}=V_{1}^{2}\subset V_{2}^{2}\subset
V_{3}^{2}=V_{3}^{1}=\langle v_{1},v_{2},v_{3}\rangle.$

Which means that $V_{2}^{2}=\langle
v_{1},\lambda_{1}v_{2}+\lambda_{2}v_{3}\rangle$. In particular if

$\mathcal{BS}_{{\underline{s}}}^{w}:=\\{(V_{\bullet}^{1},V_{\bullet}^{2})\in\mathcal{BS}_{\underline{s}};V_{\bullet}^{1}\in\Omega_{w,e}^{\circ}\\},$

then
$\mathcal{BS}_{{\underline{s}}}^{w}=\Omega_{w,1}^{\circ}\times\mathbb{P}^{1}$.

The idea is to stratify $\mathbb{P}^{1}$ as
$\mathbb{P}^{1}=\mathbb{A}^{1}\cup\mathbb{A}^{0}$. There are two natural
choices for accomplishing this:

(6b) $\displaystyle\mathbb{P}^{1}$
$\displaystyle=\\{(1:\lambda_{2})\\}\cup\\{(0:1)\\},$ (6c)
$\displaystyle\mathbb{P}^{1}$
$\displaystyle=\\{(\lambda_{1}:1)\\}\cup\\{(1:0)\\}.$

Let us take a closer look at each of these options, defining in both cases
$\mathcal{BS}_{{\underline{s}},0}^{w}:=\Omega_{w,1}^{\circ}\times\mathbb{A}^{0}$
and
$\mathcal{BS}_{{\underline{s}},1}^{w}:=\Omega_{w,e}^{\circ}\times\mathbb{A}^{1}$.

Let us begin with the first possible stratification (6b). If we have
$(V_{\bullet}^{1},V_{\bullet}^{2})\in\mathcal{BS}_{{\underline{s}},1}^{w}$,
then $V_{2}^{2}=\langle v_{1},v_{2}+\lambda_{2}v_{3}\rangle$,
$V_{3}^{2}=\langle v_{1},v_{2}+\lambda_{2}v_{3},v_{3}\rangle$, and we can
write the matrix associated to $V_{\bullet}^{2}$ as

(6d)
$V_{\bullet}^{2}=\left(\begin{array}[]{cccc}x_{2}&x_{1}+\lambda_{2}&1&0\\\
x_{3}&1&0&0\\\ x_{4}&0&0&1\\\ 1&0&0&0\end{array}\right).$

If $(V_{\bullet}^{1},V_{\bullet}^{2})\in\mathcal{BS}_{{\underline{s}},0}^{w}$,
then $V_{2}^{2}=\langle v_{1},v_{3}\rangle$ and $V_{3}^{2}=\langle
v_{1},v_{3},v_{2}\rangle$, which means

(6e) $V_{\bullet}^{2}=\left(\begin{array}[]{cccc}x_{2}&1&x_{1}&0\\\
x_{3}&0&1&0\\\ x_{4}&0&0&1\\\ 1&0&0&0\end{array}\right).$

On the other hand, if we choose stratification (6c) and consider
$(V_{\bullet}^{1},V_{\bullet}^{2})\in\mathcal{BS}_{{\underline{s}},1}^{w}$
then, $V_{2}^{2}=\langle v_{1},\lambda_{1}v_{2}+v_{3}\rangle$ and
$V_{3}^{2}=\langle v_{1},\lambda_{1}v_{2}+v_{3},v_{2}\rangle$, so we can write

$V_{\bullet}^{2}=\left(\begin{array}[]{cccc}x_{2}&\lambda_{1}x_{1}+1&1&0\\\
x_{3}&\lambda_{1}&0&0\\\ x_{4}&0&0&1\\\ 1&0&0&0\end{array}\right).$

If we have
$(V_{\bullet}^{1},V_{\bullet}^{2})\in\mathcal{BS}_{{\underline{s}},0}^{w}$,
then $V_{2}^{2}=\langle v_{1},v_{2}\rangle$ and $V_{3}^{2}=\langle
v_{1},v_{2},v_{3}\rangle$, thus

$V_{\bullet}^{2}=\left(\begin{array}[]{cccc}x_{2}&x_{1}&1&0\\\ x_{3}&1&0&0\\\
x_{4}&0&0&1\\\ 1&0&0&0\end{array}\right).$

Among these two choices, note that only stratification (6b) has the property
that the matrix associated to $V_{\bullet}^{2}$ has a permutation of columns
which turns it into an upper triangular matrix with all diagonal entries equal
to $1$. That is to say, this matrix is of the form $u\dot{w}_{1}$ for some
$u\in U$ and $w_{1}\in W$. (In this case $w_{1}=w$, but that is not so in
general.) For this reason, this is the stratification we will choose.

More precisely, the matrices in Equations (6d) and (6e) can be rewritten as

(6f) $\left(\begin{array}[]{cccc}x_{2}&x_{1}+\lambda_{2}&1&0\\\ x_{3}&1&0&0\\\
x_{4}&0&0&1\\\
1&0&0&0\end{array}\right)=\left(\begin{array}[]{cccc}1&x_{1}&0&x_{2}\\\
0&1&0&x_{3}\\\ 0&0&1&x_{4}\\\
0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}1&\lambda_{2}&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 1&0&0&0\end{array}\right)$

and

$\left(\begin{array}[]{cccc}x_{2}&1&x_{1}&0\\\ x_{3}&0&1&0\\\ x_{4}&0&0&1\\\
1&0&0&0\end{array}\right)=\left(\begin{array}[]{cccc}1&x_{1}&0&x_{2}\\\
0&1&0&x_{3}\\\ 0&0&1&x_{4}\\\
0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\end{array}\right).$

Note that the permutation matrices appearing above are, respectively, the
matrices of $4213=w$ and $4123=ws_{1}=:w_{1}$ (see also the first item in
Example 5.10). Moreover, we have that $t_{\gamma_{1}}:=ws_{1}w^{-1}=(1,2)$, so

$U_{\gamma_{1},1}=\left\\{\left(\begin{array}[]{cccc}1&\lambda_{2}&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right)\right\\}\quad\text{ and }\quad
U_{\gamma_{1},0}=\left\\{\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right)\right\\}.$

This means that an element of $\mathcal{BS}_{{\underline{s}},1}^{w}$ can be
written as $(u_{0}\dot{w},u_{0}u_{1}\dot{w})$ for $u_{0}\in U^{w}$ and
$u_{1}\in U_{\gamma_{1}}$, while an element of
$\mathcal{BS}_{{\underline{s}},0}^{w}$ is of the form
$(u_{0}\dot{w},u_{0}\dot{w}_{1})$. It is also worth noting that $w_{1}<w$,
hence $w_{1}=\phi_{{\underline{s}},0}(w)$ and $w=\phi_{{\underline{s}},1}(w)$.
Finally, we have isomorphisms $U^{w}\times
U_{\gamma_{1},1}\to\mathcal{BS}_{{\underline{s}},1}^{w}$ and $U^{w}\times
U_{\gamma_{1},0}\to\mathcal{BS}_{{\underline{s}},0}^{w}$.

If ${\underline{s}}=(s_{1},\ldots,s_{n})$ is a longer word, we can repeat the
process above. Assuming that we have found matrices for
$V_{\bullet}^{1},\ldots,V_{\bullet}^{j}$, we start with the matrix associated
to $V_{\bullet}^{j}$ and find the possible matrices associated with
$V_{\bullet}^{j+1}$ and choose the stratification such that this matrix
associated with $V_{\bullet}^{n+1}$ can be made upper-triangular after a
permutation of the columns.

######  Remark 6.3.

It is also possible to write the map $g_{{\underline{s}},\underline{b}}^{w}$
is in terms of _braid matrices_ , that is, matrices of the form

$Br_{i}(\lambda)=\left(\begin{array}[]{cccccc}1&\dots&&&\dots&0\\\
\vdots&\ddots&&&\vdots\\\ 0&\dots&0&1&\dots&0\\\ 0&\dots&1&\lambda&\dots&0\\\
\vdots&\ddots&&&\vdots\\\ 0&\dots&&&\dots&1\\\ \end{array}\right).$

For instance, Equation (6f) can be written as

$\left(\begin{array}[]{cccc}x_{2}&x_{1}+\lambda_{2}&1&0\\\ x_{3}&1&0&0\\\
x_{4}&0&0&1\\\
1&0&0&0\end{array}\right)=\left(\begin{array}[]{cccc}x_{2}&x_{1}&1&0\\\
x_{3}&1&0&0\\\ x_{4}&0&0&1\\\ 1&0&0&0\end{array}\right)\cdot
Br_{2}(\lambda_{2})\cdot\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\
0&1&0&0\\\ 0&0&0&1\end{array}\right).$

######  Lemma 6.4.

The map $g_{{\underline{s}},\underline{b}}^{w}$ is well defined, that is,
$g_{{\underline{s}},\underline{b}}^{w}(u_{0},\ldots,u_{n})\in\mathcal{BS}_{\underline{s}}$
for every $(u_{0},\ldots,u_{n})\in U^{w}\times
U_{\gamma_{1},b_{1}}\times\cdots\times U_{\gamma_{n},b_{n}}$.

It is enough to check that

$(u_{0}u_{1}\ldots u_{i}\dot{w}_{i})^{-1}(u_{0}u_{1}\ldots
u_{i+1}\dot{w}_{i+1})\in P_{s_{i+1}}.$

However, we can simplify the expression to

$\dot{w}_{i}^{-1}u_{i+1}\dot{w}_{i+1}\in P_{s_{i+1}}.$

We have that either $w_{i+1}=w_{i}$ or $w_{i+1}=w_{i}s_{i+1}$. Since
$u_{i+1}\in U_{\gamma_{i+1}}$ and $t_{\gamma_{i+1}}=w_{i}s_{i}w_{i}^{-1}$, we
have $U_{\gamma_{i+1}}=w_{i}U_{\pm\alpha_{s_{i+1}}}w_{i}^{-1}$, where
$\alpha_{s_{i+1}}$ is the simple root associated to $s_{i+1}$. Since
$w_{i}U_{\pm\alpha_{s_{i+1}}}w_{i}^{-1}\subset w_{i}P_{s_{i+1}}w_{i}^{-1}$ and
$\dot{s}_{i+1}\in P_{s_{i+1}}$, then

$w_{i}^{-1}u_{i+1}w_{i+1}\in
w_{i}^{-1}w_{i}P_{s_{i+1}}w_{i}^{-1}w_{i+1}=P_{s_{i+1}},$

which finishes the proof.

We will now prove that $g_{{\underline{s}},\underline{b}}^{w}$ is an
immersion. Let $\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}$ be the image
of the map $g_{{\underline{s}},\underline{b}}^{w}$.

######  Proposition 6.5.

The maps $g_{{\underline{s}},\underline{b}}^{w}$ are isomorphisms onto their
images, and the variety $\mathcal{BS}_{\underline{s}}$ is paved by the
collection $\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}$ as
$\underline{b}$ varies in $\\{0,1\\}^{n}$ and $w$ in $W$.

We proceed by induction on $n$. The case $n=0$ follows from the Bruhat
decomposition. Assume that $n\geq 1$, let
${\underline{s}}^{\prime}:=(s_{1},\ldots,s_{n-1})$ be a truncation of
${\underline{s}}$, and suppose that
$g_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w}$ is an isomorphim
onto its image for each $w\in W$ and
$\underline{b}^{\prime}\in\\{0,1\\}^{n-1}$. Moreover assume that
$(X_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w})_{w,\underline{b}^{\prime}}$
pave $\mathcal{BS}_{{\underline{s}}^{\prime}}$. Fix $w$ and $\underline{b}$,
and define as before $w_{i}=\phi_{{\underline{s}}|_{i},\underline{b}|_{i}}$
and $\gamma_{i}w_{i-1}=w_{i-1}s_{i}$. Also, let
$\underline{b}^{\prime}=\underline{b}|_{n-1}$. Let
$p\colon\mathcal{BS}_{\underline{s}}\to\mathcal{BS}_{{\underline{s}}^{\prime}}$
and $q\colon U^{w}\times U_{\gamma_{1},b_{1}}\times\ldots\times
U_{\gamma_{n},b_{n}}\to U^{w}\times U_{\gamma_{1},b_{1}}\times\ldots\times
U_{\gamma_{n-1},b_{n-1}}$ be the projections. It is clear from the definition
that $p\circ
g_{{\underline{s}},\underline{b}}^{w}=g_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w}\circ
q$,

${U^{w}\times U_{\gamma_{1},b_{1}}\times\ldots\times
U_{\gamma_{n},b_{n}}}$${\mathcal{BS}_{\underline{s}}}$${U^{w}\times
U_{\gamma_{1},b_{1}}\times\ldots\times
U_{\gamma_{n-1},b_{n-1}}}$${\mathcal{BS}_{{\underline{s}}^{\prime}}.}$$\scriptstyle{q}$$\scriptstyle{g_{{\underline{s}},\underline{b}}^{w}}$$\scriptstyle{p}$$\scriptstyle{g_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w}}$

Let $\underline{b}_{0}:=(\underline{b}^{\prime},0)$ and
$\underline{b}_{1}:=(\underline{b}^{\prime},1)$. We have that
$\mathcal{BS}_{{\underline{s}},\underline{b}_{0}}^{w},\mathcal{BS}_{{\underline{s}},\underline{b}_{1}}^{w}\subset
p^{-1}(\mathcal{BS}_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w})$.
Define $w^{\prime}:=\phi_{{\underline{s}}^{\prime},\underline{b}^{\prime}}(w)$
and let
$x\in\mathcal{BS}_{{\underline{s}}^{\prime},\underline{b}^{\prime}}^{w}$ be a
point with last coordinate $u_{x}\dot{w}^{\prime}B$ for some $u_{x}\in U$.
Then

$p^{-1}(x)=\\{(x,gB);g\in u_{x}\dot{w}^{\prime}P_{s_{n}}\\}.$

Recall that, if $\alpha_{s_{n}}$ is the simple root associated to $s_{n}$,
then (see Equation 3b)

$P_{s_{n}}=U_{\alpha_{s_{n}}}\dot{s}_{n}B\cup B\quad\text{ and
}P_{s_{n}}=\dot{s}_{n}U_{\alpha_{s_{n}}}\dot{s}_{n}B\cup\dot{s}_{n}B.$

Now we consider two cases. The first is when $w^{\prime}s_{n}>w^{\prime}$. In
this case we have that
$\phi_{{\underline{s}},\underline{b}_{1}}(w)=w^{\prime}s_{n}$ and
$\phi_{{\underline{s}},\underline{b}_{0}}(w)=w^{\prime}$. Moreover
$\dot{w}^{\prime}U_{\alpha_{s_{n}}}\dot{w}^{\prime-1}=U_{\gamma_{n}}$, thus

$\displaystyle u_{x}\dot{w}^{\prime}P_{s_{n}}=$ $\displaystyle
u_{x}\dot{w}^{\prime}(U_{\alpha}\dot{s}_{n}B\cup B)$ $\displaystyle=$
$\displaystyle
u_{x}\dot{w}^{\prime}U_{\alpha}\dot{w}^{\prime-1}\dot{w}^{\prime}\dot{s}_{n}B\cup
u_{x}\dot{w}^{\prime}B$ $\displaystyle=$ $\displaystyle
u_{x}U_{\gamma_{n}}\dot{w}^{\prime}\dot{s}_{n}B\cup u_{x}\dot{w}^{\prime}B$
$\displaystyle=$ $\displaystyle
u_{x}U_{\gamma_{n}}\dot{w}^{\prime}\dot{s}_{n}B\cup u_{x}\dot{w}^{\prime}B$

The second case is when $w^{\prime}s_{n}<w^{\prime}$. In this case we have
$\phi_{{\underline{s}},\underline{b}_{1}}(w)=w^{\prime}$ and
$\phi_{{\underline{s}}.\underline{b}_{0}}(w)=w^{\prime}s_{n}$. Moreover
$\dot{w}^{\prime}\dot{s}_{n}U_{\alpha_{s_{n}}}\dot{s}_{n}\dot{w}^{\prime-1}=U_{\gamma_{n}}$,
so that

$\displaystyle u_{x}\dot{w}^{\prime}P_{s_{n}}=$ $\displaystyle
u_{x}\dot{w}^{\prime}(\dot{s}_{n}U_{\alpha}\dot{s}_{n}B\cup\dot{s}_{n}B)$
$\displaystyle=$ $\displaystyle
u_{x}\dot{w}^{\prime}\dot{s}_{n}U_{s_{n}}\dot{s}_{n}\dot{w}^{\prime-1}\dot{w}^{\prime}B\cup
u_{x}\dot{w}^{\prime}\dot{s}_{n}B$ $\displaystyle=$ $\displaystyle
u_{x}U_{\gamma_{n}}\dot{w}^{\prime}B\cup u_{x}\dot{w}^{\prime}\dot{s}_{n}B.$

The result follows.

##  7\. Basic properties of Lusztig varieties

In this section we give basic results about the varieties $\mathcal{Y}_{w}(X)$
and $\mathcal{Y}_{{\underline{s}}}(X)$ defined in Definition 1.21. We keep the
notation of Section 3 and consider the variety $\mathcal{Y}\subset
G\times\mathcal{B}\times\mathcal{B}$ given by

$\mathcal{Y}:=\\{(X,g_{1}B,g_{2}B);g_{1}^{-1}Xg_{2}\in
B\\}=\\{(X,g_{1}B,g_{2}B);X\in g_{1}Bg_{2}^{-1}\\}.$

Note that the projection $\mathcal{Y}\to\mathcal{B}\times\mathcal{B}$ is a
$B$-bundle (i.e., a smooth morphism with fibers isomorphic to $B$). Moreover,
$\mathcal{Y}$ is isomorphic to $G\times\mathcal{B}$ via the map

$\displaystyle G\times\mathcal{B}$ $\displaystyle\to\mathcal{Y}$
$\displaystyle(X,gB)$ $\displaystyle\mapsto(X,XgB,gB).$

Also we have that $G$ acts on $Y$ by conjugation on the first coordinate and
left multiplication on the other coordinates:

$\displaystyle G\times Y$ $\displaystyle\to Y$
$\displaystyle(g,(X,g_{1}B,g_{2}B))$
$\displaystyle\mapsto(gXg^{-1},gg_{1}B,gg_{2}B).$

Let $w\in W$ and ${\underline{s}}=(s_{1},\ldots,s_{\ell(w)})$ a reduced word
for $w$. We define the variety $\mathcal{Y}_{w}$ (and its associated strata
$\mathcal{Y}_{w}^{\circ})$ and the variety $\mathcal{Y}_{\underline{s}}$ as

$\displaystyle\mathcal{Y}_{w}$
$\displaystyle:=\mathcal{Y}\times_{\mathcal{B}\times\mathcal{B}}\Omega_{w}$
$\displaystyle\mathcal{Y}_{w}^{\circ}$
$\displaystyle:=\mathcal{Y}\times_{\mathcal{B}\times\mathcal{B}}\Omega_{w}^{\circ}$
$\displaystyle\mathcal{Y}_{\underline{s}}$
$\displaystyle:=\mathcal{Y}\times_{\mathcal{B}\times\mathcal{B}}\mathcal{BS}_{\underline{s}}.$

More explicitly we have

$\displaystyle\mathcal{Y}_{w}$ $\displaystyle:=\overline{\\{(X,gB);g^{-1}Xg\in
BwB\\}}\subset G\times\mathcal{B},$ $\displaystyle\mathcal{Y}_{w}^{\circ}$
$\displaystyle:=\\{(X,gB);g^{-1}Xg\in BwB\\}\subset G\times\mathcal{B},$
$\displaystyle\mathcal{Y}_{\underline{s}}$
$\displaystyle:=\\{(X,(g_{0}B,g_{1}B,\ldots,g_{\ell(w)}B));g_{0}^{-1}Xg_{\ell(w)}\in
B\\}\subset G\times\mathcal{BS}_{\underline{s}}.$

As before, $G$ acts on these $3$ varieties by conjugation on the first
coordinate and left multiplication on the other coordinates.

We have the following commutative diagram

${\mathcal{Y}_{\underline{s}}}$${\mathcal{BS}_{\underline{s}}}$${\mathcal{Y}_{w}}$${\Omega_{w}}$${\mathcal{Y}}$${\mathcal{B}\times\mathcal{B}.}$

Via the first projection $\mathcal{Y}\to G$, we have induced maps
$\mathcal{Y}_{w}\to G$ and $\mathcal{Y}_{\underline{s}}\to G$. These maps are
equivariant when we consider the adjoint action of $G$ on itself. For $X\in
G$, we denote by $\mathcal{Y}_{w}(X)$ and $\mathcal{Y}_{\underline{s}}(X)$ the
fibers of $\mathcal{Y}_{w}\to G$ and $\mathcal{Y}_{\underline{s}}\to G$ over
$X$. Set theoretically, these fibers can be described as (see Definition 1.21)

(7a) $\displaystyle\mathcal{Y}_{w}(X)$ $\displaystyle=\\{gB;g^{-1}Xg\in
B\\}\subset\mathcal{B}$ $\displaystyle\mathcal{Y}_{\underline{s}}(X)$
$\displaystyle=\\{(g_{0}B,g_{1}B,\ldots,g_{\ell(w)}B)\in\mathcal{BS}_{\underline{s}};g_{0}^{-1}Xg_{\ell(w)}\in
B\\}.$

######  Remark 7.1.

We can define $\mathcal{Y}_{\underline{s}}$ and
$\mathcal{Y}_{\underline{s}}(X)$ for every word ${\underline{s}}$, not just
the reduced ones.

Recall that $G^{rs}$ denotes the open subset of $G$ consisting of the regular
semisimple elements, and denote by $\mathcal{Y}_{w}^{rs}$,
$\mathcal{Y}_{w}^{rs,\circ}$, and $\mathcal{Y}_{\underline{s}}^{rs}$ the
restrictions of $\mathcal{Y}_{w}$, $\mathcal{Y}_{w}^{\circ}$ and
$\mathcal{Y}_{\underline{s}}$ to $G^{rs}$.

######  Example 7.2.

Let $G=GL(\mathbb{C},4)$ and take

$X_{1}:=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&2&0&0\\\ 0&0&3&0\\\
0&0&0&4\end{array}\right)\quad\text{ and }\quad
X_{2}:=\left(\begin{array}[]{cccc}1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&1\\\
0&0&0&1\end{array}\right).$

Also consider $w=2413$. Then

$\mathcal{Y}_{w}(X)=\\{V_{\bullet};XV_{1}\subset V_{2},V_{2}\subset
XV_{3}\\}.$

We try to give a more geometric description of $\mathcal{Y}_{w}(X_{1})$ and
$\mathcal{Y}_{w}(X_{2})$. First, we consider the forgetful map

$\mathcal{Y}_{w}(X)\to\operatorname{Gr}(2,4).$

Then it is clear that its image is given by
$\\{V_{2}\subset\mathbb{C}^{4};\dim(XV_{2}\cap V_{2})\geq 1\\}$.

Recall the Plücker coordinates of $\operatorname{Gr}(2,4)$: choose a basis
$v_{1},v_{2}$ of $V_{2}$, and let $M_{V_{2}}=(v_{1},v_{2})$ be the $4\times 2$
matrix with columns $v_{1}$ and $v_{2}$. Then the $2\times 2$ minors $p_{ij}$
given by lines $i$ and $j$ of $M_{V_{2}}$ define an embedding

$\displaystyle Gr(2,4)$ $\displaystyle\to\mathbb{P}^{5}$ $\displaystyle V_{2}$
$\displaystyle\mapsto(p_{12}:p_{13}:p_{14}:p_{23}:p_{24}:p_{34}).$

The image of $\operatorname{Gr}(2,4)$ has equation
$p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}=0$. The condition $\dim(XV_{2}\cap
V_{2})\geq 1$ is equivalent to the vanishing of the determinant
$\det(v_{1},v_{2},Xv_{1},Xv_{2})$. It is not hard to write this determinant in
terms of $p_{ij}$. For $X=X_{1}$, the ideal of the image of
$\mathcal{Y}_{w}(X)$ in $\mathbb{P}^{5}$ is

$I_{1}=(p_{13}p_{24}-4p_{12}p_{34},p_{14}p_{23}-3p_{12}p_{34}),$

while for $X=X_{2}$ it is

$I_{2}=(p_{24}^{2}-p_{14}p_{34}-p_{23}p_{34},p_{14}p_{23}-p_{13}p_{24}+p_{12}p_{34}).$

It is not hard to see that $V(I_{1})$ is singular precisely at the 6 points
$(1:0:\ldots:0),\ldots,(0:\ldots:0:1)$. These points correspond to the vector
spaces $V_{2}$ such that $X_{1}V_{2}=V_{2}$. For example at $(0:1:0:0:0:0)$,
the affine equation for $V(I_{1})$ becomes (in the affine chart $p_{13}=1$)

$(p_{24}-4p_{12}p_{34},p_{14}p_{23}-3p_{12}p_{34}).$

This means that $V(I_{1})\cap\\{p_{13}=1\\}$ is isomorphic to the cone
$\operatorname{Spec}(\frac{\mathbb{C}[x,y,z,w]}{(xy-zw)})$.

We claim that $\mathcal{Y}_{w}(X_{1})$ is $V(I_{1})$ blown up at the $6$
singular points. Indeed, the map $\mathcal{Y}_{w}(X_{1})\to V(I_{1})$ is one-
to-one outside the singular points, while if $XV_{2}\neq V_{2}$ the only
possible choices for $V_{1}$ and $V_{3}$ are $V_{1}=V_{2}\cap X_{1}^{-1}V_{2}$
and $V_{3}=V_{2}+X_{1}^{-1}V_{2}$. Over the points such that
$X_{1}V_{2}=V_{2}$, the fiber of $\mathcal{Y}_{w}(X_{1})\to V(I)$ is
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, since we are free to choose
$V_{1}\subseteq V_{2}$ and $V_{3}\supseteq V_{2}$. The preimages of the
singular points are Cartier divisors, and the blow-up of a cone on its
singular point has a $\mathbb{P}^{1}\times\mathbb{P}^{1}$ as its exceptional
divisor, so we have proved our claim.

Upon a rescaling of the variables $p_{ij}$, we could assume that

$I=(p_{13}p_{24}-p_{12}p_{34},p_{14}p_{23}-p_{12}p_{34}),$

and now $V(I)$ carries an action of $S_{4}$, where $wp_{ij}=p_{w(i)w(j)}$.
Since the singular points are invariant points in this action, we have that
this action of $S_{4}$ lifts to an action of $S_{4}$ on
$\mathcal{Y}_{w}(X_{1})$.

The unipotent case $X=X_{2}$ is more complicated (for instance, it is known
that regular nilpotent Hessenberg varieties are not even normal ). We have
that only the point $(1:0:0:0:0:0)$ (which corresponds to $V_{2}=\langle
e_{1},e_{2}\rangle$) corresponds to a point where $X_{2}V_{2}=V_{2}$. However,
the map $\mathcal{Y}_{w}(X_{2})\to V(I_{2})$ is not the blowup of this special
point, but the blowup of a more sophisticated subscheme.

##  8\. Paving by affines the variety $\mathcal{Y}_{{\underline{s}}}(X)$

In this section we will construct an affine paving of
$\mathcal{Y}_{\underline{s}}(X)$ for a regular element $X\in G^{r}$. The idea
is to prove that for each
$\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}\subset\mathcal{BS}_{\underline{s}}$
constructed in Section 6, we have that
$\mathcal{Y}_{\underline{s}}(X)\cap\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}$
is either empty or isomorphic to an affine space. In order to do that, we will
need a technical result.

Let $A$ be a $\mathbb{C}$-algebra and consider $A[x_{1},\ldots,x_{N}]$ the
polynomial ring in $N$ variables over $A$. Assume that the variable $x_{i}$
has weight $k_{i}\in\mathbb{Z}$. We consider a finite set of polynomials
$\mathcal{F}=\\{F_{k,j}\\}$ where $k\in\mathbb{Z}$ and
$j\in\\{1,\ldots,m_{k}\\}$ (note that $m_{k}=0$ except for a finite number of
values $k$).

######  Proposition 8.1.

Assume that we can write $F_{k,j}=G_{k,j}+H_{k,j}$ where $G_{k,j}$ is a linear
polynomial with complex coefficients in variables of weight $k$ and $H_{k,j}$
is a polynomial in variables of weight less than $k$. Moreover, assume that
for each $k$ the polynomials $\\{G_{k,j}\\}_{j\in\\{1,\ldots,m_{k}\\}}$ are
linearly independent. In that case, we have that
$V(F)=\operatorname{Spec}(\frac{A[x_{1},\ldots,x_{N}]}{\langle F\rangle})$ is
isomorphic to an affine space over $A$ of expected dimension
$N-\sum_{k\in\mathbb{Z}}m_{k}$ (over $spec(A)$).

The proof is done by induction. Consider
$X_{k}=\\{x_{i};\operatorname{wt}(x_{i})\leq k\\}$ and let $A_{k}=A[X_{k}]$
and $\mathcal{F}_{k}:=\\{F_{k^{\prime},j}\\}_{k^{\prime}\leq k}$. Clearly
$\mathcal{F}_{k}$ is a set of polynomials in $A_{k}$. For $k$ sufficiently
small, we have $X_{k}=\emptyset$ and $\mathcal{F}_{k}=\emptyset$, and in this
case the result holds trivially. Assume, by the induction hypothesis, that
$V(\mathcal{F}_{k})$ is isomorphic to an affine space of dimension
$|X_{k}|-\sum_{k^{\prime}\leq k}m_{k^{\prime}}$. Since
$G_{k+1,1},\ldots,G_{k+1,m_{k+1}}$ are linearly independent, upon change the
variables of weight $k$ with suitable linear combinations, we can assume that
$G_{k+1,j}=x_{j}$. Hence, there is an isomorphism
$V(\mathcal{F}_{k})\times\mathbb{A}^{\alpha_{k}-m_{k+1}}\to V(F_{k+1})$ given
by

$\displaystyle\mathbb{A}^{X_{k}}\times\mathbb{A}^{\alpha_{k}-m_{k+1}}$
$\displaystyle\to\mathbb{A}^{X_{k}}\times\mathbb{A}^{\alpha_{k}}$
$\displaystyle(y,(x_{m_{k}+1},\ldots,x_{\alpha_{k}}))$
$\displaystyle\mapsto(y,(-H_{k+1,1}(y),-H_{k+1,2}(y),\ldots,-H_{k+1,m_{k+1}}(y),x_{m_{k}+1},\ldots,x_{\alpha_{k}})$

where $\alpha_{k}=|X_{k+1}\setminus X_{k}|$. By the induction hypothesis, we
have our result.

Let $\exp\colon\mathfrak{g}\to G$ be the exponential map. As we saw in Section
3, the restriction $\exp:\mathfrak{u}\to U$ is an isomorphism of schemes (see
also Equation (3c)). Moreover, given $w\in W$, we have
$\exp(\mathfrak{u}^{w})=U^{w}$ and $\exp(\mathfrak{u}_{w})=U_{w}$. To fix
notation, for each $\gamma\in\Phi^{+}$ we will choose $E_{\gamma}$ a generator
of $\mathfrak{u}_{\gamma}$. Moreover, for $\beta,\gamma\in\Phi^{+}$, we define
constants $\lambda_{\beta,\gamma}$ by the formula

(8a) $[E_{\beta},E_{\gamma}]=\lambda_{\beta,\gamma}E_{\beta+\gamma}.$

The equation above only makes sense if $\beta+\gamma$ is a root, otherwise we
define $\lambda_{\beta,\gamma}:=0$. It is worth noting that
$\lambda_{\beta,\gamma}=-\lambda_{\gamma,\beta}$.

Let $X\in B$ a regular element. Recall that we can pave
$\mathcal{BS}_{\underline{s}}$ by affines isomorphic to $U^{w}\times
U_{\delta_{1},b_{1}}\times\ldots\times U_{\delta_{n},b_{n}}$ via the map
$g_{{\underline{s}},\underline{b}}^{w}$ defined in Equation (6a). Also recall
Equation (7a). Then we have that

(8b)
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))=\\{(u_{0},u_{1},\ldots,u_{n});\dot{w}^{-1}u_{0}^{-1}Xu_{0}u_{1}\ldots
u_{n}\phi_{{\underline{s}},\underline{b}}(\dot{w})\in B\\}.$

Or, equivalently,

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))=\\{(u_{0},u_{1},\ldots,u_{n});u_{0}^{-1}Xu_{0}u_{1}\ldots
u_{n}\phi_{{\underline{s}},\underline{b}}(\dot{w})\in\dot{w}B\\}.$

Since $u_{0}^{-1}Xu_{0}u_{1}\ldots u_{n}\in B$ for every
$(u_{0},\ldots,u_{n})\in U^{w}\times U_{\delta_{1},b_{1}}\times\ldots\times
U_{\delta_{n},b_{n}}$, we have that if
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))$
is non-empty, then $B\phi_{{\underline{s}},\underline{b}}(w)B\cap
BwB\neq\emptyset$. This implies that
$\phi_{{\underline{s}},\underline{b}}(w)=w$. In this case, we can rephrase
Equation (8b) as

(8c)
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))=\\{(u_{0},u_{1},\ldots,u_{n});u_{0}^{-1}Xu_{0}u_{1}\ldots
u_{n}\in B\cap\dot{w}B\dot{w}^{-1}\\}.$

For simplicity, we will write
$\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X):=\mathcal{Y}_{\underline{s}}(X)\cap\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}$.

######  Theorem 8.2.

Let $X\in B$ be a regular element such that $X=X_{s}X_{u}$ and
$Z_{G}(X_{s})=G_{J}$ for some $J\subset S$. Then
$\mathcal{Y}_{{\underline{s}}}(X)$ is paved by the affines
$\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X)$ where $w\in W$ and
$\underline{b}\in\operatorname{Good}(L,J,w)$, and
$\dim\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X)=|\underline{b}|$.

######  Corollary 8.3.

If ${\underline{s}}=(s_{1},\ldots,s_{n})$ is a word in $S$, we have the
equality

$Poin(H^{*}(\mathcal{Y}_{\underline{s}}(X_{J})))=\chi^{i_{W_{J}}^{W}}((1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}})).$

Since $\mathcal{Y}_{\underline{s}}(X_{J})$ is paved by affines, we have

$\displaystyle Poin(H^{*}(\mathcal{Y}_{\underline{s}}(X_{J})))$
$\displaystyle=\sum_{w\in
W}\sum_{\underline{b}\in\operatorname{Good}(L,J,w)}q^{\dim(\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X_{J}))}$
$\displaystyle=\sum_{w\in
W}\sum_{\underline{b}\in\operatorname{Good}(L,J,w)}q^{|\underline{b}|}$
$\displaystyle=\chi^{i_{W_{J}}^{W}}((1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}})).$

The first and second equalities follow from Theorem 8.2, the third equality
from Theorem 5.11.

######  Corollary 8.4.

The Betti numbers of $\mathcal{Y}_{\underline{s}}(X)$ are palindromic for
every regular element $X$.

This follows from Corollary 8.3 and the fact that
$\chi^{i_{W_{J}}^{W}}((1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}}))$ is
palindromic.

######  Corollary 8.5.

Consider $G=SL_{n}(\mathbb{C})$ and $W=S_{n}$. Let
${\underline{s}}=(s_{1},\ldots,s_{n})$ be a word in $S$, then we have the
following equality

$\operatorname{ch}((1+T_{s_{1}})(1+T_{s_{2}})\ldots(1+T_{s_{n}}))=\sum_{\lambda\vdash
n}Poin(H^{*}(\mathcal{Y}_{\underline{s}}(X_{\lambda})))m_{\lambda},$

where each $X_{\lambda}$ is a regular matrix with Jordan decomposition
$\lambda$.

This follows directly from Corollary 8.3 and Equation (5a).

######  Corollary 8.6.

Let ${\underline{s}}$ be a sequence such that every $s\in S$ appears in
${\underline{s}}$. Then the variety $\mathcal{Y}_{{\underline{s}}}(X)$ is
irreducible for every regular element $X$. In particular, $\mathcal{Y}_{w}(X)$
is irreducible for every $w\in W$.

The proof is the same as [Pre18, Corollary 14]. By proposition 5.8 and Theorem
8.2 we have that $\mathcal{Y}_{{\underline{s}}}(X)$ has only one stratum of
top dimension. By the Theorem of dimension of fibers applied to
$\mathcal{Y}_{{\underline{s}}}^{r}\to G^{r}$, the dimension of each
irreducible component of $\mathcal{Y}_{{\underline{s}}}(X)$ is at least $n$,
in particular $\mathcal{Y}_{{\underline{s}}}(X)$ is equidimensional and hence
irreducible (since it has only one stratum of top dimension).

The proof of Theorem 8.2 will take the rest of the section. We begin with an
example to fix the main ideas.

######  Example 8.7.

As in Examples 5.10, 5.21, 6.1 and 6.2, let $G=GL(4)$ then $W=S_{4}$. If
$w=4231$ then $\Upsilon(w)=\\{(1,2),(1,4),(2,4),(3,4)\\}$ and
$D_{\underline{s}}(w)=\\{(1,2),(3,4)\\}$. Let $\underline{b}=(1,0,1,1,1,0)$,
and ${\underline{s}}=((2,3),(1,2),(3,4),(2,3),(1,2),(3,4))$. We have that the
sequence $\underline{\delta}$ (defined in 5.20) is given by
$\delta_{1}=(1,2)$, $\delta_{2}=(1,3)$, $\delta_{3}=(3,4)$ and
$\delta_{4}=(2,4)$. This means that

$\displaystyle u_{0}$
$\displaystyle=\left(\begin{array}[]{cccc}1&x_{12}&0&x_{14}\\\ 0&1&0&x_{24}\\\
0&0&1&x_{34}\\\ 0&0&0&1\end{array}\right),$ $\displaystyle u_{1}$
$\displaystyle=\left(\begin{array}[]{cccc}1&y_{1}&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),$ $\displaystyle u_{2}$
$\displaystyle=\left(\begin{array}[]{cccc}1&0&y_{2}&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),$ $\displaystyle u_{3}$
$\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&y_{3}\\\
0&0&0&1\end{array}\right),$ $\displaystyle u_{4}$
$\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&y_{4}\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),$ $\displaystyle X$
$\displaystyle:=\left(\begin{array}[]{cccc}1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&1\\\
0&0&0&1\end{array}\right).$

Then

$u_{0}^{-1}Xu_{0}=\left(\begin{array}[]{cccc}1&1&-x_{12}&-x_{12}x_{34}+x_{24}\\\
0&1&1&x_{34}\\\ 0&0&1&1\\\ 0&0&0&1\end{array}\right).$

Moreover, we have that

$\displaystyle M:=$ $\displaystyle u_{0}^{-1}Xu_{0}u_{1}u_{2}u_{3}u_{4}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&y_{1}+1&-x_{12}+y_{2}&-x_{12}x_{34}-x_{12}y_{3}+y_{2}y_{3}+y_{1}y_{4}+x_{24}+y_{4}\\\
0&1&1&x_{34}+y_{3}+y_{4}\\\ 0&0&1&y_{3}+1\\\ 0&0&0&1\end{array}\right).$

The condition that $M\in\dot{w}B\dot{w}^{-1}$ says that $M_{12}=0$,
$M_{14}=0$, $M_{24}=0$ and $M_{34}=0$ (these are the same entries that are
non-zero in $u_{0}\in U^{w}$). Therefore, we have that the subscheme

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))\subset
U^{w}\times U_{\delta_{1}}\times U_{\delta_{2}}\times U_{\delta_{3}}\times
U_{\delta_{4}}=\operatorname{Spec}(\mathbb{C}[x_{12},x_{14},x_{24},x_{34},y_{1},y_{2},y_{3},y_{4}])$

is given by the following equations

$\displaystyle 0$ $\displaystyle=y_{1}+1,$ $\displaystyle 0$
$\displaystyle=-x_{12}x_{34}-x_{12}y_{3}+y_{2}y_{3}+y_{1}y_{4}+x_{24}+y_{4},$
$\displaystyle 0$ $\displaystyle=x_{34}+y_{3}+y_{4},$ $\displaystyle 0$
$\displaystyle=y_{3}+1.$

It is not hard to see that
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))\cong\mathbb{A}^{4}_{\mathbb{C}}$.

We note that if we had omitted $u_{1}$ in the product defining $M$, then
$M_{12}$ would be just $1$, in which case
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))$
would be empty. The same holds for $\delta_{3}$. This is the main motivation
behind the definition of a good sequence, to exclude that the matrix $M$ could
have an entry $M_{i,i+1}=1$ for some $(i,i+1)\in D_{\underline{s}}(w)$.

Fix ${\underline{s}}=(s_{1},\ldots,s_{n})$, $w\in W$ and $\underline{b}$. We
begin, restricting ourselves to the case that $X\in U$ is a regular unipotent
element. In this case, Equation 8c becomes

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))=\\{(u_{0},u_{1},\ldots,u_{n});u_{0}^{-1}Xu_{0}u_{1}\ldots
u_{n}\in U_{w}\\},$

hence we have the fiber diagram

(8d)
${(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))}$${U_{w}}$${U^{w}\times
U_{\delta_{1},b_{1}}\times\ldots\times U_{\delta_{n},b_{n}}}$${U,}$

where the bottom map is given by

$(u_{0},u_{1},\ldots,u_{n})\to u_{0}^{-1}Xu_{0}u_{1}\ldots u_{n}.$

Since $U_{\delta,0}=\\{1\\}$, we consider only the roots $\delta_{i}$ such
that $b_{i}=1$. Renaming then, we write
$\delta_{1},\delta_{2},\ldots,\delta_{n^{\prime}}$ for such roots. Then

$U^{w}\times U_{\delta_{1},b_{1}}\times\ldots\times U_{\delta_{n},b_{n}}$

becomes

$U^{w}\times U_{\delta_{1}}\times\ldots\times U_{\delta_{n^{\prime}}}.$

Of course, we have that $n^{\prime}=|\underline{b}|=\sum_{1\leq i\leq
n}b_{i}$. Alternatively, via BCH formula, the bottom map in Equation (8d) can
be viewed as the map

$\displaystyle\mathfrak{u}^{w}\times\mathfrak{u}_{\delta_{1}}\times\ldots\times\mathfrak{u}_{\delta_{n^{\prime}}}$
$\displaystyle\to\mathfrak{u}$ $\displaystyle(u_{0},\ldots,u_{n^{\prime}})$
$\displaystyle\mapsto(-u_{0})\otimes X\otimes u_{0}\otimes
u_{1}\otimes\ldots\otimes u_{n^{\prime}}.$

Via the (inverse of the) exponential map, we can assume that
$X\in\mathfrak{u}$, and since $X$ is regular, we can also assume that
$X=\sum_{\alpha\in\Delta}E_{\alpha}$ (up to conjugation).

######  Example 8.8.

We will repeat Example 8.7 using BCH formular. Everything stays the same
except that

$\displaystyle u_{0}$
$\displaystyle=\left(\begin{array}[]{cccc}0&x_{12}&0&x_{14}\\\ 0&0&0&x_{24}\\\
0&0&0&x_{34}\\\ 0&0&0&0\end{array}\right),$ $\displaystyle u_{1}$
$\displaystyle=\left(\begin{array}[]{cccc}0&y_{1}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),$ $\displaystyle u_{2}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&y_{2}&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),$ $\displaystyle u_{3}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&y_{3}\\\
0&0&0&0\end{array}\right),$ $\displaystyle u_{4}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&y_{4}\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right),$ $\displaystyle X$
$\displaystyle:=\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
0&0&0&0\end{array}\right).$

Then

$(-u_{0})\otimes X\otimes
u_{0}=\left(\begin{array}[]{cccc}0&1&-x_{12}&-x_{12}x_{34}+x_{24}\\\
0&0&1&x_{34}\\\ 0&0&0&1\\\ 0&0&0&0\end{array}\right),$

see Lemma 8.10 below. Moreover, we have that

$\displaystyle M:=$ $\displaystyle(-u_{0})\otimes X\otimes u_{0}\otimes
u_{1}\otimes u_{2}\otimes u_{3}\otimes
u_{4}=\left(\begin{array}[]{cccc}0&y_{1}+1&-x_{12}-\frac{1}{2}y_{1}+y_{2}&M_{14}\\\
0&0&1&x_{34}+\frac{1}{2}y_{3}+y_{4}\\\ 0&0&0&y_{3}+1\\\
0&0&0&0\end{array}\right),$

where

$M_{14}=-x_{12}x_{34}-\frac{1}{2}x_{34}y_{1}-\frac{1}{2}x_{12}y_{3}-\frac{1}{6}y_{1}y_{3}+\frac{1}{2}y_{2}y_{3}+\frac{1}{2}y_{1}y_{4}+x_{24}+\frac{1}{12}y_{1}-\frac{1}{2}y_{2}+\frac{1}{12}y_{3}+\frac{1}{2}y_{4}$

see Corollaries 8.13 and 8.14 below.

In view of Proposition 8.1, it is convenient to give weights to the variables
we are considering. Assume that we have variables $x_{\gamma,i}$ for each
$\gamma\in\Phi^{+}$ and each integer $i$. Recall that we have a partial
ordering $\leq$ in $\Phi^{+}$. For $\beta,\gamma\in\Phi^{+}$, we write
$\beta\lessdot\gamma$ if there exists $\alpha\in\Delta$ with
$\gamma=\beta+\alpha$ and write $\beta<<\gamma$ if $\beta\leq\gamma$ and
$\operatorname{ht}(\beta)\leq\operatorname{ht}(\gamma)-2$.

Given a property $P$ on the set $\Phi^{+}$ we will write
$g(x_{\beta,i},P(\beta))$ to mean a polynomial in the variables $x_{\beta,i}$
such that $\beta\in\Phi^{+}$ and $\beta$ satisfies $P$. Consider the elements
of $\mathbb{C}[x_{\gamma,i}]\otimes_{\mathbb{C}}\mathfrak{u}$ given by

(8e)
$Y=\sum_{\gamma\in\Phi^{+}}E_{\gamma}(f_{\gamma}(x_{\gamma,i})+g_{\gamma}(x_{\beta,i},(\beta\lessdot\gamma)\vee(\beta\in\Delta\cap\Upsilon_{\gamma}))+h_{\gamma}(x_{\beta,i},\beta<<\gamma)),$

where $g$ and $h$ have no constant terms. In particular, if $\gamma\in\Delta$,
then $g_{\gamma}=h_{\gamma}=0$.

######  Proposition 8.9.

If $Y^{1}$, $Y^{2}$ and $Y^{3}$ are elements as above with

$Y^{i}=\sum_{\gamma\in\Phi^{+}}E_{\gamma}(f^{i}_{\gamma}(x_{\gamma,i})+g^{i}_{\gamma}(x_{\beta,i},(\beta\lessdot\gamma)\vee(\beta\in\Delta\cap\Upsilon_{\gamma}))+h^{i}_{\gamma}(x_{\beta,i},\beta<<\gamma))$

and $Y^{3}=Y^{1}\otimes Y^{2}$, then

$\displaystyle f^{3}_{\gamma}$ $\displaystyle=f^{1}_{\gamma}+f^{2}_{\gamma},$
$\displaystyle g^{3}_{\gamma}$
$\displaystyle=g^{1}_{\gamma}+g^{2}_{\gamma}+\frac{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha,\gamma-\alpha}(f^{1}_{\alpha}f^{2}_{\gamma-\alpha}-f^{2}_{\alpha}f^{1}_{\gamma-\alpha})$
$\displaystyle\textnormal{if $\operatorname{ht}(\gamma)\geq 3$},$
$\displaystyle g^{3}_{\gamma}$
$\displaystyle=g^{1}_{\gamma}+g^{2}_{\gamma}+\frac{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha,\gamma-\alpha}(f^{1}_{\alpha}f^{2}_{\gamma-\alpha})$
$\displaystyle\textnormal{if $\operatorname{ht}(\gamma)=2$},$

where $\lambda_{\alpha,\gamma-\alpha}$ are the structure constants defined in
Equation (8a).

We have that

$\displaystyle Y^{1}\otimes
Y^{2}=Y^{1}+Y^{2}+\frac{1}{2}[Y_{1},Y_{2}]+[\text{higher commutators}].$

Since all variables in the coefficient of $E_{\gamma}$ are of the form
$x_{\beta,i}$, with $\beta\leq\gamma$, the higher commutators term will belong
entirely to $h^{3}_{\gamma}$ and can therefore be ignored. Continuing with the
computation, we have

(8f) $\displaystyle Y^{1}+Y^{2}$
$\displaystyle=\sum_{\gamma\in\phi^{+}}E_{\gamma}(f^{1}_{\gamma}+f^{2}_{\gamma}+g^{1}_{\gamma}+g^{2}_{\gamma}+h^{1}_{\gamma}+h^{2}_{\gamma}).$

and

(8g)
$\displaystyle[Y^{1},Y^{2}]=\sum_{\gamma}E_{\gamma}\bigg{(}\underset{\beta_{1}+\beta_{2}=\gamma}{\sum_{\beta_{1},\beta_{2}\in\phi^{+}}}\lambda_{\beta_{1},\beta_{2}}(f^{1}_{\beta_{1}}+g^{1}_{\beta_{1}}+h^{1}_{\beta_{1}})(f^{2}_{\beta_{1}}+g^{2}_{\beta_{2}}+h^{2}_{\beta_{2}})\bigg{)}.$

If neither $\beta_{1}$ nor $\beta_{2}$ is a simple root, then the coefficient

$\lambda_{\beta_{1},\beta_{2}}(f^{1}_{\beta_{1}}+g^{1}_{\beta_{1}}+h^{1}_{\beta_{1}})(f^{2}_{\beta_{2}}+g^{2}_{\beta_{2}}+h^{2}_{\beta_{2}})$

also belongs in $h^{3}_{\gamma}$, so we can consider only the summands where
either $\beta_{1}\in\Delta$ or $\beta_{2}\in\Delta$.

Now we are ready to prove the statements in the Proposition. The first follows
directly from Equations (8f) and (8g). Let us prove the second statement,
considering first the case where $\operatorname{ht}(\gamma)\geq 3$. In this
case, precisely one of the $\beta_{i}$ is in $\Delta$, and we assume without
loss of generality that $\beta_{1}\in\Delta$. In particular, we see that
$\beta_{1}<<\gamma$. Moreover $g^{1}_{\beta_{1}}=h^{1}_{\beta_{1}}=0$, and the
coefficient above becomes

$\lambda_{\beta_{1},\beta_{2}}(f^{1}_{\beta_{1}}f^{2}_{\beta_{2}}+f^{1}_{\beta_{1}}g^{2}_{\beta_{2}}+f^{1}_{\beta_{1}}h^{2}_{\beta_{2}}).$

It is easy to see that
$f^{1}_{\beta_{1}}g^{2}_{\beta_{2}}+f^{1}_{\beta_{1}}h^{2}_{\beta_{2}}$
belongs in $h^{3}_{\gamma}$ and $f^{1}_{\beta_{1}}f^{2}_{\beta_{2}}$ belongs
in $g^{3}_{\gamma}$. Since
$\lambda_{\beta_{1},\beta_{2}}=-\lambda_{\beta_{2},\beta_{1}}$, the case
$\beta_{2}\in\Delta$ is analogous and we get the second statement.

If $\operatorname{ht}(\gamma)=2$, then both $\beta_{1},\beta_{2}\in\Delta$, so
$g^{i}_{\beta_{i}}=h^{i}_{\beta_{i}}=0$ for $i=1,2$ and the coefficient above
becomes

$\lambda_{\beta_{1},\beta_{2}}f^{1}_{\beta_{1}}f^{2}_{\beta_{2}}.$

The result follows.

Each of the spaces $\mathfrak{u}^{w}$ and $\mathfrak{u}_{\delta_{i}}$ are
affine, and we write $u_{0}=\sum_{\beta\in\Upsilon(w)}x_{\beta}E_{\beta}$ and
$u_{i}=y_{i}E_{\delta_{i}}$. As before, we will have a root associated to each
variable. From now on, we will write the elements of
$\mathbb{C}[x_{\beta},y_{i}]\otimes_{\mathbb{C}}\mathfrak{u}$ as in Equation
8e, abusing notation, omit the variables and write simply
$f_{\gamma},g_{\gamma},h_{\gamma}$. Recall that
$[E_{\alpha},E_{\beta}]=\lambda_{\alpha,\beta}E_{\alpha+\beta}$, where
$\lambda_{\alpha,\beta}\neq 0$ if and only if $\alpha+\beta\in\Phi^{+}$.

######  Lemma 8.10.

We can write

$(-u_{0})\otimes
X\otimes(u_{0})=\sum_{\gamma\in\Phi^{+}}E_{\gamma}(f^{0}_{\gamma}+g^{0}_{\gamma}+h^{0}_{\gamma})$

where

$f^{0}_{\gamma}=\begin{cases}1&\text{ if }\gamma\in\Delta\\\
0&\text{otherwise},\end{cases}$

and

$\displaystyle g^{0}_{\gamma}$
$\displaystyle=\underset{\gamma-\alpha\in\phi_{w}}{\sum_{\alpha\in\Delta}}\lambda_{\alpha,\gamma-\alpha}x_{\gamma-\alpha}$

for every $\gamma\in\phi^{+}\setminus\Delta$.

We could use Proposition 8.9, but it is more informative to compute directly:

$\displaystyle(-u_{0})\otimes X\otimes u_{0}=$
$\displaystyle(-u_{0})\otimes(X+u_{0}+\frac{1}{2}[X,u_{0}]+[\text{higher
commutators}])$ $\displaystyle=$
$\displaystyle(-u_{0})+X+u_{0}+\frac{1}{2}[X,u_{0}]+[\text{higher
commutators}]$
$\displaystyle+\frac{1}{2}\big{[}-u_{0},X+u_{0}+\frac{1}{2}[X,u_{0}]+[\text{higher
commutators}]\big{]}+[\text{higher commutators}]$ $\displaystyle=$
$\displaystyle X+\frac{1}{2}[X,u_{0}]+\frac{1}{2}[-u_{0},X]+[\text{higher
commutators}]$ $\displaystyle=$ $\displaystyle X+[X,u_{0}]+[\text{higher
commutators}]$ $\displaystyle=$
$\displaystyle\sum_{\alpha\in\Delta}E_{\alpha}+\sum_{\gamma\notin\Delta}E_{\gamma}\bigg{(}\underset{\alpha+\beta=\gamma}{\sum_{\alpha\in\Delta,\beta\in\Upsilon(w)}}\lambda_{\alpha,\beta}x_{\beta}+h^{0}_{\gamma}\bigg{)}.$

This finishes the proof.

######  Example 8.11.

In the notation of Example 8.8, we have $g^{0}_{13}=-x_{12}$,
$g_{24}^{0}=x_{34}$, $g^{0}_{14}=x_{24}$, and $h_{24}^{0}=0$, $h_{13}^{0}=0$,
$h_{14}^{0}=-x_{12}x_{34}$.

In the proposition we have defined polynomials $f_{\gamma}^{0}$,
$g_{\gamma}^{0}$, $h_{\gamma}^{0}$ in the variables $x_{\beta}$ for
$\beta\in\Upsilon_{w}$. Define inductively
$f_{\gamma}^{i},g_{\gamma}^{i},h_{\gamma}^{i}$ by the formula (recall Equation
(8e))

$\sum_{\gamma\in\Phi^{+}}E_{\gamma}(f_{\gamma}^{i+1}+g_{\gamma}^{i+1}+h_{\gamma}^{i+1})=\bigg{(}\sum_{\gamma\in\Phi^{+}}E_{\gamma}(f_{\gamma}^{i}+g_{\gamma}^{i}+h_{\gamma}^{i})\bigg{)}\otimes
y_{i+1}E_{\delta_{i+1}}.$

Moreover, we set $f_{\gamma}:=f_{\gamma}^{n^{\prime}}$,
$g_{\gamma}:=g_{\gamma}^{n^{\prime}}$ and
$h_{\gamma}:=h_{\gamma}^{n^{\prime}}$ for the final polynomials. The next
result follows directly from Proposition 8.9.

######  Proposition 8.12.

We have the following relations.

  1. (1)

If $\gamma\neq\delta_{i+1}$, $\delta_{i+1}\notin\Delta$ and
$\gamma-\delta_{i+1}\notin\Delta$, then

$\displaystyle f_{\gamma}^{i+1}$ $\displaystyle=f_{\gamma}^{i},$
$\displaystyle g_{\gamma}^{i+1}$ $\displaystyle=g_{\gamma}^{i}.$

  2. (2)

If $\gamma\neq\delta_{i+1}$, $\delta_{i+1}\in\Delta$ or
$\gamma-\delta_{i+1}\in\Delta$, then

$\displaystyle f_{\gamma}^{i+1}$ $\displaystyle=f_{\gamma}^{i},$
$\displaystyle g_{\gamma}^{i+1}$
$\displaystyle=g_{\gamma}^{i}+\frac{1}{2}\lambda_{\gamma-\delta_{i+1},\delta_{i+1}}y_{i+1}f_{\gamma-\delta_{i+1}}^{i}.$

  3. (3)

If $\gamma=\delta_{i+1}$, then

$\displaystyle f_{\gamma}^{i+1}$ $\displaystyle=f_{\gamma}^{i}+y_{i+1},$
$\displaystyle g_{\gamma}^{i+1}$ $\displaystyle=g_{\gamma}^{i}.$

Write
$y_{i+1}E_{\delta_{i+1}}=\sum_{\gamma\in\Phi^{+}}E_{\gamma}(\widetilde{f}_{\gamma})$.
By Proposition 8.9, we have that

$\displaystyle f_{\gamma}^{i+1}=$ $\displaystyle
f_{\gamma}^{i}+\widetilde{f}_{\gamma}$ $\displaystyle g_{\gamma}^{i+1}=$
$\displaystyle
g_{\gamma}^{i}+\frac{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha,\gamma-\alpha}(f_{\alpha}^{i}\widetilde{f}_{\gamma-\alpha}-\widetilde{f}_{\alpha}f_{\gamma-\alpha}^{i}).$
if $\operatorname{ht}(\gamma)\geq 3$ $\displaystyle g^{i+1}_{\gamma}=$
$\displaystyle
g^{i}_{\gamma}+\frac{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha,\gamma-\alpha}(f^{i}_{\alpha}\widetilde{f}_{\gamma-\alpha}).$
if $\operatorname{ht}(\gamma)=2$

Since $\widetilde{f}_{\gamma}=0$, unless $\gamma=\delta_{i+1}$, we get our
results.

######  Corollary 8.13.

For every $\alpha\in\Delta$, we have

$\displaystyle f_{\alpha}^{i}=$ $\displaystyle
1+\underset{\delta_{i}=\alpha}{\sum_{j\in\\{1,\ldots,i\\}}}y_{j},$

while for every $\gamma\notin\Delta$,

$\displaystyle f_{\gamma}^{i}=$
$\displaystyle\underset{\delta_{i}=\gamma}{\sum_{j\in\\{1,\ldots,i\\}}}y_{j}.$

This follows directly from Proposition 8.12.

######  Corollary 8.14.

It holds that

(8h)
$g_{\gamma}=g_{\gamma}^{n^{\prime}}=g_{\gamma}^{0}+\frac{1}{2}\underset{\gamma-\delta_{j}\in\Delta}{\sum_{1\leq
j\leq
n^{\prime}}}\lambda_{\gamma-\delta_{j},\delta_{j}}y_{j}+\frac{1}{2}\underset{(\delta_{i}\in\Delta)\vee(\gamma-\delta_{i}\in\Delta)}{\sum_{1\leq
i\leq
n^{\prime}-1}}\lambda_{\delta_{i},\gamma-\delta_{i}}y_{i}(\underset{\delta_{j}=\gamma-\delta_{i}}{\sum_{i+1\leq
j\leq n}}y_{j})$

By Proposition 8.12, we see that

$\displaystyle g_{\gamma}^{n^{\prime}}=$ $\displaystyle
g_{\gamma}^{0}+\frac{1}{2}\underset{(\gamma-\delta_{j}\in\Delta)\vee(\delta_{j}\in\Delta)}{\sum_{1\leq
j\leq
n^{\prime}}}\lambda_{\gamma-\delta_{j},\delta_{j}}y_{j}f_{\gamma-\delta_{j}}^{j-1}$
$\displaystyle=$ $\displaystyle
g_{\gamma}^{0}+\frac{1}{2}\underset{(\gamma-\delta_{j}\in\Delta)\vee(\delta_{j}\in\Delta)}{\sum_{1\leq
j\leq
n^{\prime}}}\lambda_{\gamma-\delta_{j},\delta_{j}}y_{j}\big{(}[\gamma-\delta_{j}\in\Delta]+\underset{\delta_{i}=\gamma-\delta_{j}}{\sum_{1\leq
i\leq j-1}}y_{i}\big{)}$ $\displaystyle=$ $\displaystyle
g_{\gamma}^{0}+\frac{1}{2}\underset{\gamma-\delta_{j}\in\Delta}{\sum_{1\leq
j\leq
n^{\prime}}}\lambda_{\gamma-\delta_{j},\delta_{j}}y_{j}+\frac{1}{2}\underset{(\delta_{i}\in\Delta)\vee(\delta_{j}\in\Delta)}{\underset{\delta_{i}+\delta_{j}=\gamma}{\sum_{1\leq
i<j\leq n^{\prime}}}}\lambda_{\delta_{i},\delta_{j}}y_{j}y_{i}$
$\displaystyle=$ $\displaystyle
g_{\gamma}^{0}+\frac{1}{2}\underset{\gamma-\delta_{j}\in\Delta}{\sum_{1\leq
j\leq
n^{\prime}}}\lambda_{\gamma-\delta_{j},\delta_{j}}y_{j}+\frac{1}{2}\underset{(\delta_{i}\in\Delta)\vee(\gamma-\delta_{i}\in\Delta)}{\sum_{1\leq
i\leq
n^{\prime}-1}}\lambda_{\delta_{i},\gamma-\delta_{i}}y_{i}(\underset{\delta_{j}=\gamma-\delta_{i}}{\sum_{i+1\leq
j\leq n}}y_{j})$

######  Example 8.15.

Using the notation of Example 8.7 and restricting attention to
$\gamma\in\Upsilon(w)=\\{(1,2),(1,4),(2,3),(2,4)\\}$, we have

$\displaystyle f_{12}$ $\displaystyle=y_{1}+1$ $\displaystyle f_{13}$
$\displaystyle=y_{2}$ $\displaystyle f_{34}$ $\displaystyle=y_{3}+1$
$\displaystyle f_{24}$ $\displaystyle=y_{4}$ $\displaystyle f_{14}$
$\displaystyle=0$ $\displaystyle g_{24}$
$\displaystyle=x_{34}+\frac{1}{2}y_{3}$ $\displaystyle g_{13}$
$\displaystyle=-x_{12}-\frac{1}{2}y_{1}$ $\displaystyle g_{14}$
$\displaystyle=x_{24}-\frac{1}{2}y_{2}+\frac{1}{2}y_{4}+\frac{1}{2}y_{1}y_{4}+\frac{1}{2}y_{2}y_{3}$
$\displaystyle h_{14}$
$\displaystyle=x_{12}x_{34}-\frac{1}{2}x_{34}y_{1}-\frac{1}{2}x_{12}y_{3}-\frac{1}{6}y_{1}y_{3}+\frac{1}{12}y_{1}+\frac{1}{12}y_{3}$

in agreement with Corollaries 8.13 and 8.14. Indeed, let us compute $g_{14}$
via Corollary 8.14: We have $\gamma=(1,4)$, $\delta_{1}=(1,2)$,
$\delta_{2}=(1,3)$, $\delta_{3}=(3,4)$, and $\delta_{4}=(2,4)$, so
$\gamma-\delta_{i}\in\Delta$ only for $i=2,4$. Moreover, only for the pairs
$(i,j)=(1,4),(2,3)$ do we have that $\delta_{i}+\delta_{j}=\gamma$ and either
$\delta_{i}\in\Delta$ or $\delta_{j}\in\Delta$, so

$g_{14}=x_{24}+\frac{1}{2}(\lambda_{(3,4),(1,3)}y_{2}+\lambda_{(1,2),(2,4)}y_{4})+\frac{1}{2}(\lambda_{(1,2),(2,4)}y_{1}y_{4}+\lambda_{(1,3),(3,4)}y_{2}y_{3}).$

Since $\lambda_{(i,j),(j,k)}=1$ for $i<j<k$, we have

$g_{14}=x_{24}-\frac{1}{2}y_{2}+\frac{1}{2}y_{4}+\frac{1}{2}y_{1}y_{4}+\frac{1}{2}y_{2}y_{3}$

as expected.

We are interested in the space
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X)):=V(f_{\gamma}+g_{\gamma}+h_{\gamma})_{\gamma\in\phi(w)}$.
First note that if there exists $\alpha\in\Delta\cap\phi_{w}$ such that
$\delta_{j}\neq\alpha$ for every $j$, then
$\mathcal{Y}_{{\underline{s}},X}^{w,\underline{b}}=\emptyset$. This is because

$f_{\alpha}+g_{\alpha}+h_{\alpha}=1$

by Corollary 8.13 and the fact that $g_{\alpha}=h_{\alpha}=0$. In particular,
if $\underline{b}$ is not a good word for $w$, then
$\mathcal{Y}_{{\underline{s}},X}^{w,\underline{b}}=\emptyset$.

We now assume that $\underline{b}$ is a good word for $w$. In particular we
can apply Proposition 5.22, and see that for every
$\gamma\in\Upsilon_{\underline{s}}(w)$, one of the following conditions hold

  1. (1)

There exists $i$ such that $\gamma=\delta_{i}$.

  2. (2)

For all $\alpha\in\Delta$ such that
$\gamma-\alpha\in\phi\setminus\Upsilon(w)$, there exists $i$ such that
$\gamma-\alpha=\delta_{i}$ and $\delta_{j}\neq\alpha$ for every $j<i$.

For every $\beta\in\\{\delta_{1},\ldots,\delta_{n^{\prime}}\\}$, let
$y_{\beta}:=y_{k_{\beta}}$, where $k_{\beta}=\min\\{i,\delta_{i}=\beta\\}$.
Let us give weights to the variables according to
$\operatorname{wt}(x_{\beta}):=\operatorname{ht}(\beta)+1$ and
$\operatorname{wt}(y_{\beta}):=\operatorname{ht}(\beta)+\frac{1}{2}$ if
$\beta\in\Upsilon(w)$, or
$\operatorname{wt}(y_{\beta})=\operatorname{ht}(\beta)+1$ otherwise. For every
other $y_{i}$ we define $\operatorname{wt}(y_{i}):=0$.

######  Proposition 8.16.

Let $\gamma\in\Upsilon(w)$ be a root such that $\gamma\neq\delta_{i}$ for
every $i$. Then

$\displaystyle g_{\gamma}=$
$\displaystyle\sum_{\alpha\in\Delta,\gamma-\alpha\in\Upsilon(w)}\lambda_{\alpha,\gamma-\alpha}x_{\gamma-\alpha}+\sum_{\alpha\in\Delta,\gamma-\alpha\notin\Upsilon(w)}\lambda_{\alpha,\gamma-\alpha}y_{\gamma-\alpha}\frac{1}{2}(2-f_{\alpha})+h_{\gamma}^{\prime}$

where $h_{\gamma}^{\prime}$ has only variables of weight less than
$\operatorname{ht}(\gamma)$.

Since $\gamma\neq\delta_{i}$ for every $i$, condition (2) above holds. Fix
$\alpha\in\Delta$ such that $\gamma-\alpha\in\Phi^{+}$. If
$\gamma-\alpha\notin\Upsilon(w)$, then $\delta_{j}\neq\alpha$ for every
$j<k_{\gamma-\alpha}$. In particular the variable
$y_{\gamma-\alpha}=y_{k_{\gamma-\alpha}}$ only appears in the following
summand in Equation 8h

$\frac{1}{2}\lambda_{\alpha,\gamma-\alpha}y_{\gamma-\alpha}+\frac{1}{2}\lambda_{\gamma-\alpha,\alpha}y_{\gamma-\alpha}(\underset{\delta_{j}=\alpha}{\sum_{k_{\beta}+1\leq
j\leq n^{\prime}}}y_{j})$

Now we check that every other variable $y_{i}$ appearing in $g_{\gamma}$ has
weight less than $\operatorname{ht}(\gamma)$. If
$\gamma-\alpha\in\Upsilon(w)$, then all variables $y_{i}$ with
$\delta_{i}=\gamma-\alpha$ either have weight $0$,
$\operatorname{ht}(\gamma)-1$, or $\operatorname{ht}(\gamma)-\frac{1}{2}$. If
$\alpha\in\Upsilon(w)$, then all variables $y_{i}$ with $\delta_{i}=\alpha$
have weight either $0$, $1$, or $\frac{3}{2}$, and if
$\alpha\notin\Upsilon(w)$, then either $\alpha=\gamma-\alpha^{\prime}$ for
some $\alpha^{\prime}\in\Delta$ and we are in the case of the previous
paragraph, or $\operatorname{ht}(\gamma)\geq 3$ and all variables $y_{i}$ with
$\delta_{i}=\alpha$ have weight either $0$ or $2$.

We finish by noting that

$\underset{\delta_{j}=\alpha}{\sum_{i+1\leq j\leq
n^{\prime}}}y_{j}=f_{\alpha}-1,$

since there is no $j<k_{\beta}$ with $\delta_{j}=\alpha$. Since
$\lambda_{\alpha,\gamma-\alpha}=-\lambda_{\gamma-\alpha,\alpha}$ the result
follows.

######  Example 8.17.

In Example 8.15 we saw that

$f_{12}=1+y_{1}\quad,f_{34}=y_{3}+1\text{ and
}g_{14}=x_{24}-\frac{1}{2}y_{2}+\frac{1}{2}y_{4}+\frac{1}{2}y_{1}y_{4}+\frac{1}{2}y_{2}y_{3}.$

This means that

$\displaystyle g_{14}=$ $\displaystyle
x_{24}+(-1)y_{2}\frac{1}{2}(2-(y_{3}+1))+y_{4}+y_{1}y_{4}$ $\displaystyle=$
$\displaystyle
x_{24}+\lambda_{(3,4),(1,3)}y_{2}\frac{1}{2}(2-(y_{3}+1))+(y_{4}+y_{1}y_{4})$

which agrees with Proposition 8.16, since
$(1,4)-(3,4)=(1,3)=\delta_{2}\notin\Upsilon(w)$ and
$(1,4)-(1,2)=(2,4)\in\Upsilon(w)$.

We are finally in a position to use Proposition 8.1 and prove Theorem 8.2.

[Proof of Theorem 8.2] Assume first that $X$ is regular unipotent, so that
$J=S$.

We have three cases

  1. (1)

If $\alpha\in\Delta\cap\Upsilon(w)$, then $g_{\alpha}=h_{\alpha}=0$ and the
coefficient of $E_{\alpha}$ is

$f_{\alpha}=1+\sum_{\delta_{j}=\gamma}y_{j}.$

  2. (2)

If $\gamma\in\Upsilon(w)\setminus\Delta$ and there exists $j$ such that
$\gamma=\delta_{j}$, then the coefficient of $E_{\gamma}$ is

$f_{\gamma}+g_{\gamma}+h_{\gamma}=\sum_{\delta_{j}=\gamma}y_{j}+g_{\gamma}+h_{\gamma}.$

  3. (3)

If $\gamma\in\Upsilon(w)\setminus\Delta$ and there does not exists $j$ such
that $\gamma=\delta_{j}$, then the coefficient of $E_{\gamma}$ is

$g_{\gamma}+h_{\gamma}=\sum_{\alpha\in\Delta,\gamma-\alpha\in\Upsilon(w)}\lambda_{\alpha,\gamma-\alpha}x_{\gamma-\alpha}+\sum_{\alpha\in\Delta,\gamma-\alpha\notin\Upsilon(w)}\lambda_{\alpha,\gamma-\alpha}y_{\gamma-\alpha}\frac{1}{2}(2-f_{\alpha})+h_{\gamma}^{\prime}+h_{\gamma}.$

The variety $\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X)$ is
isomorphic, via $g_{{\underline{s}},\underline{b}}^{w}$ to
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))$,
the latter being the locus where all the polynomials in items (1), (2), and
(3) above vanish. It is straightforward to see that we can change the
polynomials in case (3) to

  1. (3’)

If $\gamma\in\Upsilon(w)\setminus\Delta$ we consider the polynomial

$\sum_{\alpha\in\Delta,\gamma-\alpha\in\phi(w)}\lambda_{\alpha,\gamma-\alpha}x_{\gamma-\alpha}+\sum_{\alpha\in\Delta,\gamma-\alpha\notin\phi(w)}\lambda_{\alpha,\gamma-\alpha}y_{\gamma-\alpha}+h_{\gamma}^{\prime}+h_{\gamma}.$

This is because $\alpha+(\gamma-\alpha)\in\Upsilon(w)$ and
$\gamma-\alpha\notin\Upsilon(w)$, hence $\alpha\in\Upsilon(w)$, which will
force $f_{\alpha}$ to vanish.

The only thing left is to see that the linear polynomials

$\sum_{\alpha\in\Delta,\gamma-\alpha\in\phi(w)}\lambda_{\alpha,\gamma-\alpha}x_{\gamma-\alpha}+\sum_{\alpha\in\Delta,\gamma-\alpha\notin\phi(w)}\lambda_{\alpha,\gamma-\alpha}y_{\gamma-\alpha}$

for
$\gamma\in\Upsilon(w)\setminus(\Delta\cup\\{\delta_{1},\ldots,\delta_{n^{\prime}}\\})$
are linearly independent, but this follows from Corollary 3.6.

Since we need a slightly more general result for the non-unipotent case, we
state the following Proposition, which is what we actually prove above.

######  Proposition 8.18.

Let $X\in U$ be a regular unipotent element, $w\in W$, and
$\underline{\delta}=(\delta_{1},\ldots,\delta_{n^{\prime}})$ a good sequence
for $w$ of positive roots. Then the locus

$U_{\underline{\delta}}^{w}(X):=\\{(u_{0},u_{1},\ldots,u_{n^{\prime}})\in
U^{w}\times U_{\delta_{1}}\times\ldots\times
U_{\delta_{n^{\prime}}};u_{0}^{-1}Xu_{0}u_{1}\ldots u_{n^{\prime}}\in
U_{w}\\}$

is isomorphic to an affine space of dimension $n^{\prime}$.

[Proof of Theorem 8.2, the general case]

Let $X=X_{s}X_{u}$, where $X_{s}$ is semisimple and $X_{u}$ is unipotent. We
know (by [Ste65]) that $X$ is regular if and only if $X_{u}$ is regular in
$Z_{G}(X_{s})$. Up to conjugation, we can assume that there exists
$J\subset\Delta$ such that $Z_{G}(X_{s})=G_{J}$ and $X_{u}\in U_{j}$ is
regular in $G_{J}$.

The main idea is to reduce to the regular unipotent case and use Propostion
8.1. As before, for each $\underline{b}\in\\{0,1\\}^{n}$ and $w\in W$ we
define
$\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X)=\mathcal{BS}_{{\underline{s}},\underline{b}}^{w}\cap\mathcal{Y}_{\underline{s}}(X)$.
Via the isomorphism $g_{{\underline{s}},\underline{b}}^{w}$, we have that
$\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}(X)$ is isomorphic to
$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))\subset
U^{w}\times U_{\delta_{1}}\times\ldots\times U_{\delta_{n^{\prime}}}$.
Moreover, recalling Equation (8c) we have

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))=\\{(u_{0},u_{1},\ldots,u_{n^{\prime}});u_{0}^{-1}Xu_{0}u_{1}\ldots
u_{n^{\prime}}\in\dot{w}B\dot{w}^{-1}\\}.$

Since $X_{s}\in T\subset\dot{w}B\dot{w}^{-1}$, we have that

$u_{0}^{-1}Xu_{0}u_{1}\ldots u_{m}\in\dot{w}B\dot{w}^{-1}\text{ if and only if
}X_{s}^{-1}u_{0}^{-1}Xu_{0}u_{1}\ldots u_{m}\in\dot{w}B\dot{w}^{-1}.$

By Lemma 3.2 the projection $p_{J}\colon U\to U_{J}$ is a group homomorphism.
Define

$U_{\delta_{i},J}:=U_{\delta_{i}}\cap U_{J}=\begin{cases}\\{1\\}&\text{ if
}\delta_{i}\notin\Phi_{J}\\\ U_{\delta_{i}}&\text{ if
}\delta_{i}\in\Phi_{J},\end{cases}$

and let $w_{J}\in W_{J}$ and ${}^{J}w\in{}^{J}W$ be such that
$w=w_{J}\cdot{}^{J}w$. We let $q_{J}$ be the projection map

$\displaystyle q_{J}\colon U^{w}\times U_{\delta_{1}}\times\ldots\times
U_{\delta_{n^{\prime}}}$ $\displaystyle\to U^{w_{J}}\times
U_{\delta_{1},J}\times\ldots\times U_{\delta_{n^{\prime}},J}$
$\displaystyle(u_{0},u_{1},\ldots,u_{n^{\prime}})$
$\displaystyle\mapsto(p_{J}(u_{0}),p_{J}(u_{1}),\ldots,p_{J}(u_{n^{\prime}})).$

Then by Lemma 3.2, we have that $p_{J}$ is a homomorphism,
$p_{J}(tut^{-1})=tp_{J}(u)t^{-1}$, and $p_{J}(u)\in U_{J}\subset Z_{G}(X_{s})$
for every $u\in U$. Thus

$\displaystyle p_{J}(X_{s}^{-1}u_{0}^{-1}X_{s}X_{u}u_{0}u_{1}\ldots u_{n})=$
$\displaystyle
p_{J}(X_{s}^{-1}u_{0}^{-1}X_{s})X_{u}p_{J}(u_{0})p_{J}(u_{1})\ldots
p_{J}(u_{n})$ $\displaystyle=$ $\displaystyle
X_{s}^{-1}p_{J}(u_{0})^{-1}X_{s}X_{u}p_{J}(u_{0})p_{J}(u_{1})\ldots
p_{J}(u_{n})$ $\displaystyle=$ $\displaystyle
p_{J}(u_{0})^{-1}X_{u}p_{J}(u_{0})p_{J}(u_{1})\ldots p_{J}(u_{n}),$

and in particular we have

$q_{J}((g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{{\underline{s}}}(X))=U^{w_{J}}_{\underline{\delta}_{J}}(X_{u}),$

where $\underline{\delta}_{J}$ is the sequence of positive roots obtained from
$\underline{\delta}$ by removing the entries $\delta_{i}$ such that
$\delta_{i}\not\in\Phi_{J}$. By Propositions 8.18 and 5.22 we have that
$U^{w_{J}}_{\underline{\delta}_{J}}(X_{U})$ is isomorphic to an affine space
of dimension $|\underline{\delta}_{J}|$. Moreover,

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))\subset
U_{\underline{\delta}_{J}}^{w_{J}}(X_{u})\times_{(U^{w_{J}}\times
U_{\delta_{1},J}\times\ldots U_{\delta_{n^{\prime}},J})}(U^{w}\times
U_{\delta_{1}}\times\ldots\times U_{\delta_{n^{\prime}},J}),$

and the right-hand side is isomorphic to

$U_{\underline{\delta}_{J}}^{w_{J}}(X_{u})\times\prod_{\gamma\in\Upsilon(w)\setminus\Phi_{J}}U_{\gamma}\times\prod_{i,\delta_{i}\notin\Phi_{J}}U_{\delta_{i}}.$

Let $A$ be the $\mathbb{C}$-algebra such that
$U_{\underline{\delta}_{J}}^{w_{J}}(X_{u})=\operatorname{Spec}(A)$. Then

$U_{\underline{\delta}_{J}}^{w_{J}}(X_{u})\times_{(U^{w_{J}}\times
U_{\delta_{1},J}\times\ldots U_{\delta_{n^{\prime}},J})}(U^{w}\times
U_{\delta_{1}}\times\ldots\times
U_{\delta_{n^{\prime}},J})=\operatorname{Spec}(A[x_{\gamma},y_{i}]_{\gamma\in\Upsilon(w)\setminus\Phi_{J},\delta_{i}\notin\Phi_{J}}).$

Via the exponential map, we can preform the computation in
$(\mathfrak{u},\otimes)$, since $X_{s}^{-1}u_{0}^{-1}X_{s}\in U$, $X_{u}\in
U$, and
$\log(X_{s}^{-1}u_{0}^{-1}X_{s})=\operatorname{Ad}(X_{s})(-\log(u_{0}))$.
Therefore we have

$(g_{{\underline{s}},\underline{b}}^{w})^{-1}(\mathcal{Y}_{\underline{s}}(X))=\\{(u_{0},u_{1},\ldots,u_{n^{\prime}});(\operatorname{Ad}(X_{s})(-u_{0}))\otimes
X_{n}\otimes u_{0}\otimes u_{1}\otimes\ldots\otimes
u_{n^{\prime}}\in\mathfrak{u}_{w}.\\}$

As before, write $u_{0}=\sum_{\gamma\in\Upsilon(w)}x_{\gamma}E_{\gamma}$ and
$u_{i}=y_{i}E_{\delta_{i}}$. However, if $\gamma,\delta_{i}\in\phi_{J}$, we
consider $x_{\gamma}$ and $y_{i}$ as elements of $A$. Then only $x_{\gamma}$
for $\gamma\in\Upsilon(w)\setminus\Phi_{J}$ and $y_{i}$ for
$\delta_{i}\notin\phi_{J}$ are variables in
$A[x_{\gamma},y_{i}]_{\gamma\in\Upsilon(w)\setminus\Phi_{J},\delta_{i}\notin\Phi_{J}}$.
If $\gamma\notin\Phi_{J}$ then $\gamma+\beta\notin\Phi_{J}$ for every $\beta$,
so by Proposition 8.12 we have that the coefficient of $E_{\gamma}$ in

$\operatorname{Ad}(X_{s})(-u_{0}))\otimes X_{n}\otimes u_{0}\otimes
u_{1}\otimes\ldots\otimes u_{n^{\prime}}$

is constant for $\gamma\in\Phi_{J}$. Thus we can restrict attention to
coefficients of $E_{\gamma}$ for $\gamma\notin\Phi_{J}$. As in Equation 8e, we
write some elements of
$A[x_{\gamma},y_{i}]_{\gamma\in\Upsilon(w)\setminus\Phi_{J},\delta_{i}\notin\Phi_{J}}\otimes_{\mathbb{C}}\mathfrak{u}$
as

$Y=\sum_{\gamma\in\Phi^{+}\setminus\Phi_{J}}E_{\gamma}(f_{\gamma}(x_{\gamma,i})+g_{\gamma}(x_{\beta},\beta<\gamma,y_{i})).$

By Lemma 8.19 below and Proposition 8.12, we can write the coefficient of each
$E_{\gamma}$ for $\gamma\in\Upsilon(w)\setminus\phi_{J}$ in
$(\operatorname{Ad}(X_{s})(-u_{0}))\otimes X_{n}\otimes
u_{0}\otimes\ldots\otimes u_{n}$ as

$f_{\gamma}+g_{\gamma}=(1-\gamma(X_{s}))x_{\gamma}+\sum_{\delta_{j}=\gamma}y_{j}+g_{\gamma}^{0}$

where $g_{\gamma}$ is a polynomial in variables $x_{\beta},y_{\beta,i}$ with
$\beta<\gamma$. Since $\gamma\notin\Phi_{J}$, we have $(1-\gamma(X_{s}))\neq
0$, so applying Proposition 8.1, we see that
$\mathcal{Y}_{{\underline{s}},\underline{b}}^{w}$ is an affine space over
$U_{\underline{\delta}_{J}}^{w_{J}}(X_{u})$ of the expected dimension, and
hence it is an affine space of the expected dimension
$n^{\prime}=|\underline{b}|$.

######  Lemma 8.19.

We can write

$(\operatorname{Ad}(X_{s})(-u_{0}))\otimes
X_{u}\otimes(u_{0})=\sum_{\gamma\in\Upsilon(w)\setminus\Phi_{J}}E_{\gamma}(f^{0}_{\gamma}+g^{0}_{\gamma})+\sum_{\gamma\notin(\Upsilon(w)\setminus\Phi_{J})}E_{\gamma}(g_{\gamma}^{0})$

where $f^{0}_{\gamma}=(1-\gamma(X_{s}))x_{\gamma}$ for every
$\gamma\in\Upsilon(w)\setminus\Phi_{J}$ and $g_{\gamma}^{0}$ is a polynomial
in variables $x_{\beta}$ with $\beta<\gamma$.

Recall that
$\operatorname{Ad}(X_{s})u_{0}=\sum_{\beta\in\phi(w)}\beta(X_{s})x_{\beta}E_{\beta}$.
Moreover, we have that if $\gamma\in\Phi_{J}$ and $\alpha+\beta=\gamma$, then
$\alpha,\beta\in\Phi_{J}$ which means that $\alpha(X_{s})=\beta(X_{s})=1$.

$\displaystyle(\operatorname{Ad}(X_{s})(-u_{0}))\otimes X_{u}\otimes u_{0}=$
$\displaystyle(\operatorname{Ad}(X_{s})(-u_{0}))\otimes(X_{n}+u_{0}+[\text{higher
commutators}])$ $\displaystyle=$
$\displaystyle(\operatorname{Ad}(X_{s})(-u_{0}))+X_{u}+u_{0}+[\text{higher
commutators}]$ $\displaystyle=$
$\displaystyle(I-\operatorname{Ad}(X_{s}))(u_{0})+X_{u}+[\text{higher
commutators}]$ $\displaystyle=$
$\displaystyle\sum_{\gamma\in\Upsilon(w)\setminus\Phi_{J}}E_{\gamma}((1-\gamma(X_{s}))x_{\gamma}+g_{\gamma}^{0})+\sum_{\gamma\notin(\Upsilon(w)\setminus\phi_{J})}E_{\gamma}(g_{\gamma}^{0})$

##  9\. The monodromy action on $\mathcal{Y}_{w}$ and
$\mathcal{Y}_{\underline{s}}$

The goal of this section is twofold. First, we use using Ehresmann’s lemma and
Thom’s first isotopy lemma to prove that both $\mathcal{Y}_{w}^{rs}\to G^{rs}$
and $\mathcal{Y}_{\underline{s}}^{rs}\to G^{rs}$ are topological fibrations.
Second, we study the monodromy action of $\mathcal{Y}_{w}$ and
$\mathcal{Y}_{{\underline{s}}}$. We begin with the following proposition.

######  Proposition 9.1.

The map $\mathcal{Y}_{w}^{rs,\circ}\to G^{rs}$ is a submersion.

First, consider $\widetilde{\mathcal{Y}}_{w}^{\circ}=\\{(X,g);g^{-1}Xg\in
BwB\\}\subset G\times G$. We have a map

$\displaystyle\widetilde{\mathcal{Y}}_{w}^{\circ}$
$\displaystyle\to\mathcal{Y}_{w}^{\circ}$ $\displaystyle(X,g)$
$\displaystyle\mapsto(X,XgB,gB),$

which is smooth with fibers isomorphic to $B$. Since we are interested in the
projection onto the first factor, it is enough to prove that

(9a)
$\mathbf{T}_{(X_{0},g_{0})}\widetilde{\mathcal{Y}}_{w}^{\circ}\to\mathbf{T}_{X_{0}}G$

is surjective for every pair $(X_{0},g_{0})\in G^{rs}\times G$. Since
$\widetilde{\mathcal{Y}}_{w}^{\circ}\to G$ is $G$-equivariant, we can assume
that $g_{0}=e$. Consider the composition

${G\times G}$${G\times
G}$${G}$${G}$${(g^{\prime},g)}$${(X_{0}g^{\prime},g)}$${g^{-1}X_{0}g^{\prime}g}$${X_{0}^{-1}g^{-1}X_{0}g^{\prime}g.}$

It is clear that $(e,e)$ is taken to $(e,e)$ and that the differential map is
given by

$\displaystyle\mathfrak{g}\times\mathfrak{g}$
$\displaystyle\to\mathfrak{g}\times\mathfrak{g}$ $\displaystyle(Y,Z)$
$\displaystyle\mapsto\operatorname{Ad}(X_{0})^{-1}(-Z)+Y+Z.$

Hence

$\mathbf{T}_{(X_{0},1)}\widetilde{H}_{w}=\\{(Y,Z);\operatorname{Ad}(X_{0})^{-1}(-Z)+Y+Z\in\mathbf{T}_{X_{0}}BwB\\}.$

Thus the surjectivity of map (9a) is equivalent to the fact that for every
$Y\in\mathfrak{g}$, there exists $Z\in\mathfrak{g}$ such that
$\operatorname{Ad}(X_{0})^{-1}(-Z)+Y+Z\in\mathbf{T}_{X_{0}}BwB$. In turn, this
is equivalent to the surjectivity of the map

$\displaystyle\mathfrak{g}$
$\displaystyle\to\frac{\mathfrak{g}}{\mathbf{T}_{X_{0}}BwB}$ $\displaystyle Z$
$\displaystyle\mapsto(1-\operatorname{Ad}(X_{0}^{-1}))(Z)+\mathbf{T}_{x_{0}}BwB,$

which we will now establish.

Since $X_{0}$ is regular semisimple, it lies inside a maximal torus
$T^{\prime}\subset G$ (with associated Cartan subalgebra
$\mathfrak{h}^{\prime}$). If $\Phi^{\prime}$ is the set of roots associated to
$T^{\prime}$, then
$\mathfrak{g}=\mathfrak{h}^{\prime}\oplus\bigoplus_{\gamma\in\Phi^{\prime}}\mathfrak{g}_{\gamma}$,
and if $E_{\gamma}$ is a generator of $\mathfrak{g}_{\gamma}$, and since
$X_{0}$ is regular, we have
$(1-\operatorname{Ad}(X_{0}^{-1}))(E_{\gamma})=(1-\gamma(X_{0}^{-1}))E_{\gamma}\neq
0$ (see [Ste65, Proposition 2.11]). Hence, the image of the map

$\displaystyle f\colon\mathfrak{g}$ $\displaystyle\to\mathfrak{g}$
$\displaystyle Z$ $\displaystyle\mapsto(1-\operatorname{Ad}(X_{0})^{-1}))(Z)$

is
$\mathfrak{g}_{\mathfrak{h}^{\prime}}:=\bigoplus_{\gamma\in\Phi^{\prime}}\mathfrak{g}_{\gamma}$.

Now, each map below is surjective

$\mathfrak{g}\xrightarrow{f}\mathfrak{g}_{\mathfrak{h}^{\prime}}\to\frac{\mathfrak{g}}{\mathfrak{b}}\to\frac{\mathfrak{g}}{\mathbf{T}_{x_{0}}BwB},$

the second by Lemma 3.4 and the third by Lemma 3.8. This finishes the proof.

######  Proposition 9.2.

The map $\mathcal{Y}_{w}^{rs}\to G^{rs}$ is a topological fibration.

For $z\leq w$, $\mathcal{Y}_{z}^{rs,\circ}$ is a Whitney stratification of
$\mathcal{Y}_{w}^{rs}$. Indeed for $z\leq w$, $\Omega_{z}$ is a Whitney
stratification of $\Omega_{w}$, and since $\mathcal{Y}_{w}\to\Omega_{w}$ is a
smooth morphism, we have that $\mathcal{Y}_{z}^{\circ}$ is a Whitney
stratification of $\mathcal{Y}_{w}$. Since $\mathcal{Y}_{w}^{rs}$ is an open
subset of $\mathcal{Y}_{w}$, $\mathcal{Y}_{z}^{rs,\circ}$ is a Whitney
stratification of $\mathcal{Y}_{w}^{rs}$. By Proposition 9.1, the restriction
$\mathcal{Y}_{u}^{rs,\circ}\to G$ is a submersion for every $u$, so the result
follows by Thom’s first isotopy lemma.

######  Proposition 9.3.

Let $w\in W$, and let ${\underline{s}}$ be a reduced word for $w$. Then the
map $\mathcal{Y}_{{\underline{s}}}^{rs}\to G^{rs}$ is a topological fibration.

Since $\mathcal{Y}_{\underline{s}}$ is smooth, it is sufficient, by
Ehresmann’s lemma, to prove that $\mathcal{Y}^{rs}_{\underline{s}}\to G^{rs}$
is a submersion. Take $Q$ a closed point in $\mathcal{Y}_{\underline{s}}^{rs}$
and denote by
$\beta_{\underline{s}}\colon\mathcal{Y}_{\underline{s}}\to\mathcal{Y}_{w}$ the
natural map. Then $Q$ belongs in some strata
$\beta_{\underline{s}}^{-1}(\mathcal{Y}_{u}^{\circ})$. Moreover,
$\beta_{\underline{s}}^{-1}(\mathcal{Y}_{u}^{\circ})\to\mathcal{Y}_{u}^{\circ}$
is a topological fibration since
$\alpha_{\underline{s}}^{-1}(\Omega_{u})\to\Omega_{u}$ is a topological
fibration, hence
$\mathbf{T}_{Q}\beta_{\underline{s}}^{-1}(\mathcal{Y}_{u}^{\circ})\to\mathbf{T}_{\beta_{\underline{s}}(Q)}\mathcal{Y}_{u}^{\circ}$
is surjective. However, we already know that
$\mathbf{T}_{\beta_{\underline{s}}(Q)}\mathcal{Y}_{u}^{\circ}\to\mathbf{T}_{f(\beta_{\underline{s}}(Q))}G$
is surjective, so the map
$\mathbf{T}_{Q}\beta_{\underline{s}}^{-1}(\mathcal{Y}_{u}^{\circ})\to\mathbf{T}_{f(g(Q))}G$
is surjective, which proves that
$\mathbf{T}_{Q}\mathcal{Y}_{\underline{s}}\to\mathbf{T}_{f(\beta_{\underline{s}}(Q))}G$
is surjective.

${\mathcal{Y}_{\underline{s}}}$${\mathcal{BS}_{\underline{s}}}$${\mathcal{Y}_{w}}$${\Omega_{w}}$${\mathcal{Y}}$${\mathcal{B}\times\mathcal{B}}$

The remainder of this section is devoted to proving the following result.

######  Proposition 9.4.

Let $X\in T^{r}$ be a regular element of $T$, $w$ an element of $W$, and
${\underline{s}}=(s_{1},\ldots,s_{\ell(w)})$ a reduced word for $w$. Then

  1. (1)

The action of $\pi_{1}(G^{rs},X)$ on $H^{*}(\mathcal{Y}_{{\underline{s}}}(X))$
factors through an action of $W$ on $H^{*}(\mathcal{Y}_{\underline{s}}(X))$.

  2. (2)

The action of $\pi_{1}(G^{rs},X)$ on $IH^{*}(\mathcal{Y}_{u}(X))$ factors
through an action of $W$ on $IH^{*}(\mathcal{Y}_{u}(X))$ for every $u\in W$.

  3. (3)

The actions of $W$ are compatible with the isomorphism

$H^{*}(\mathcal{Y}_{\underline{s}}(X))\cong
IH^{*}(\mathcal{Y}_{w}(X))\oplus\bigoplus IH^{*}(\mathcal{Y}_{u}(X))\otimes
L_{u}[-\ell(w)+\ell(u)]$

given by the decomposition theorem, where $W$ acts on $L_{u}$ trivially.

As in [BC18], the idea is to prove the results for the equivariant cohomology
and then restrict to the usual cohomology. We begin with a few definitions.

Let ${\underline{s}}=(s_{1},\ldots,s_{\ell(w)})$ be a reduced word of an
element $w\in W$. Recall the torus fiber bundle $\mathcal{T}$ defined in
Section 3.2. There is an action of $\mathcal{T}$ on $G^{rs}\times\mathcal{B}$,
given by

$\displaystyle m\colon\mathcal{T}\times_{G^{rs}}(G^{rs}\times\mathcal{B})$
$\displaystyle\to G^{rs}\times\mathcal{B}$
$\displaystyle((X,g^{\prime}),(X,gB))$ $\displaystyle\to(X,g^{\prime}gB).$

Since $g^{\prime}\in Z_{G}(X)$ we have that $m$ preserves each stratum
$\mathcal{Y}_{w}^{rs,\circ}\subset G^{rs}\times\mathcal{B}$. Hence $m$ will
also be considered as an action of $\mathcal{T}$ on $\mathcal{Y}_{w}^{rs}$ for
every $w\in W$.

Since
$\mathcal{BS}_{\underline{s}}=\mathcal{B}\times_{\mathcal{P}_{s_{1}}}\mathcal{B}\times\ldots\times\times_{\mathcal{P}_{s_{n}}}\mathcal{B}$,
we have an action of $\mathcal{T}$ in
$G^{rs}\times\mathcal{BS}_{\underline{s}}$ given by the action of $m$ on each
of the factors of $\mathcal{BS}_{\underline{s}}$, that is,

$\displaystyle
m^{\prime}\colon\mathcal{T}\times_{G^{rs}}(G^{rs}\times\mathcal{BS}_{\underline{s}})$
$\displaystyle\to G^{rs}\times\mathcal{BS}_{\underline{s}}$
$\displaystyle((X,g^{\prime}),(X,g_{0}B,g_{1}B,\ldots,g_{n}B))$
$\displaystyle\mapsto(X,(g^{\prime}g_{0}B,g^{\prime}g_{1}B,\ldots,g^{\prime}g_{n}B)).$

Again, since $g^{\prime}\in Z_{G}(X)$, we have that this action stabilizes
$\mathcal{Y}_{\underline{s}}$. Clearly the map
$\mathcal{Y}_{\underline{s}}^{rs}\to\mathcal{Y}_{w}^{rs}$ is
$\mathcal{T}$-equivariant.

We now turn our attention to the fixed locus of $m$. A point $(X,gB)\in
G^{rs}\times\mathcal{B}$ is a fixed point of $m$ if and only if
$gB=g^{\prime}gB$ for every $g^{\prime}\in Z_{G}(X)$. The latter occurs if and
only if $g^{-1}Z_{G}(X)g\subset B$, which is the same as
$Z_{G}(g^{-1}Xg)\subset B$. Since $X$ is regular semisimple, then so is
$g^{-1}Xg$. In particular, we have that $Z_{G}(g^{-1}Xg)\subset B$ if and only
$g^{-1}Xg\in B$. This proves that the fixed locus of the action $m$ is
precisely $\mathcal{Y}_{e}^{rs}$ (defined in Definition 1.21, Section 7 and
Equation (3e)).

The fixed locus of $m^{\prime}$ is little more involved. Since the action
$m^{\prime}$ is simply the action $m$ on the factors of
$\mathcal{BS}_{\underline{s}}$, we have that fixed locus of $m^{\prime}$ is

(9b)
$(G^{rs}\times\mathcal{BS}_{{\underline{s}}})^{\mathcal{T}}=\mathcal{Y}_{e}^{rs}\times_{G^{rs}\times
P_{s_{1}}}\mathcal{Y}_{e}^{rs}\times_{G^{rs}\times
P_{s_{2}}}\ldots\times_{G^{rs}\times P_{s_{n}}}\mathcal{Y}_{e}^{rs}.$

It is easier to see this fixed locus on a fiber over $X\in T^{r}$. We know
that $\mathcal{Y}_{e}^{rs}(X)=\\{\dot{w}B;w\in W\\}$, in particular, we have

$\mathcal{BS}_{\underline{s}}^{\mathcal{T}_{X}}=\\{(\dot{w}_{0}B,\dot{w}_{1}B,\ldots,\dot{w}_{n}B);w_{i}^{-1}w_{i+1}\in<s_{i}>\text{
for every }i=0,\ldots,n-1\\}.$

This gives a bijection between the set of fixed points
$\mathcal{BS}_{{\underline{s}}}^{\mathcal{T}_{X}}$ and the set
$W\times\\{0,1\\}^{\ell(w)}$. Explicitly, this bijection is given by

$\displaystyle W\times\\{0,1\\}^{\ell(w)}$
$\displaystyle\to\mathcal{BS}_{{\underline{s}}}^{\mathcal{T}_{X}}$
$\displaystyle(w_{0},\underline{b})$
$\displaystyle\mapsto(\dot{w}_{0}B,\dot{w}_{1}B,\ldots,\dot{w}_{\ell(w)}B),$

where $w_{i}=\phi_{{\underline{s}}|_{i},\underline{b}|_{i}}(w_{0})$. We will
abuse notation and say that a pair $(w_{0},\underline{b})$ is a fixed point in
$\mathcal{BS}_{\underline{s}}^{\mathcal{T}_{X}}$. Moreover, we have that a
fixed point
$(w_{0},\underline{b})\in\mathcal{BS}_{{\underline{s}}}^{\mathcal{T}_{X}}$ is
in $\mathcal{Y}_{\underline{s}}(X)$ if and only if
$\phi_{{\underline{s}},\underline{b}}(w_{0})=w_{0}$. Meanwhile, the Weyl group
$W$ acts on $\mathcal{BS}_{\underline{s}}^{\mathcal{T}_{X}}$ by

$\displaystyle W\times\mathcal{BS}_{\underline{s}}^{\mathcal{T}_{X}}$
$\displaystyle\to\mathcal{BS}_{\underline{s}}^{\mathcal{T}_{X}}$
$\displaystyle(w^{\prime},(\dot{w}_{0}B,\ldots,\dot{w}_{\ell(w)}B))$
$\displaystyle\mapsto(\dot{w}^{\prime}\dot{w}_{0}B,\dot{w}^{\prime}\dot{w}_{1}B,\ldots,\dot{w}^{\prime}\dot{w}_{\ell(w)}B)$

which can be restricted to $\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}}$.
Since $\mathcal{T}\to G^{rs}$ and $\mathcal{Y}_{\underline{s}}^{rs}\to G^{rs}$
are fibrations, we have that $\pi_{1}(G^{rs},X)$ acts on
$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X))$ and
$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}})$. By localization
we obtain an injective map

(9c) $H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X))\hookrightarrow
H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}})$

that is $\pi_{1}(G^{rs},X)$-equivariant.

######  Lemma 9.5.

The action of $\pi_{1}(G^{rs},X)$ on
$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}})$ factors through
an action of $W$ on
$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}})$.

We have that $\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}}$ is a union of
points, and the the action of $\pi_{1}(G^{rs},X)$ on these points factors
through $W$. This follows from Equation (9b) and the fact that
$\mathcal{Y}_{e}^{rs}\to G^{rs}$ is a $W$-Galois cover.

On the other hand, we have that the action of $\pi_{1}(G^{rs},X)$ on the Lie
algebra $\mathfrak{h}$ of $T=\mathcal{T}_{X}$ factors through $W$, which is
the content of Corollary 3.19. Since

$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}})=\bigoplus_{pt\in\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}}}H_{T}^{*}(pt)=\bigoplus_{pt\in\mathcal{Y}_{\underline{s}}(X)^{\mathcal{T}_{X}}}\mathbb{C}[\mathfrak{h}]$

we have our result.

######  Corollary 9.6.

The action of $\pi_{1}(G^{rs},X)$ on
$H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X))$ factors through an action of $W$
on $H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X))$.

This is a direct consequence of Lemma 9.5 and Equation (9c).

[Proof of Proposition 9.4] We begin by proving item (1). The result follows
from Corollary 9.6 and the fact that $H^{*}(\mathcal{Y}_{\underline{s}}(X))$
is a quotient of $H_{T}^{*}(\mathcal{Y}_{\underline{s}}(X))$. Since the fiber
diagram

${\mathcal{Y}_{\underline{s}}^{rs}}$${\mathcal{BS}_{\underline{s}}}$${\mathcal{Y}_{w}^{rs}}$${\Omega_{w}}$$\scriptstyle{\beta_{\underline{s}}}$$\scriptstyle{\alpha_{{\underline{s}}}}$

is a smooth base change, we have that the decomposition theorem for
$\mathcal{Y}_{\underline{s}}^{rs}\to\mathcal{Y}_{w}^{rs}$ reads

$(\beta_{\underline{s}})_{*}(IC_{\mathcal{Y}_{\underline{s}}(X)})=IC_{\mathcal{Y}_{w}}\oplus\bigoplus_{z\leq
w}IC_{\mathcal{Y}_{z}}\otimes L_{z},$

where $L_{u}$ are the same graded vector spaces appearing in Equation (3f).

Recalling that
$IH^{i}(\mathcal{X})=\mathcal{H}^{i-\dim_{\mathbb{C}}(\mathcal{X})}(IC_{\mathcal{X}})$,
$\dim(\mathcal{Y}_{\underline{s}})=\dim(G)+\ell(w)$, and
$\dim(\mathcal{Y}_{z})=\dim(G)+\ell(z)$, we have that (see also Proposition
2.5)

(9d) $H^{*}(\mathcal{Y}_{\underline{s}}(X))\cong
IH^{*}(\mathcal{Y}_{w}(X))\oplus\bigoplus_{z\leq
w}IH^{*}(\mathcal{Y}_{z}(X))\otimes L_{z}[-\ell(w)+\ell(z)].$

By Propositions 2.5, 9.2 and 9.3, the isomorphism in Equation (9d) is
compatible with the action of $\pi_{1}(G^{rs},X)$. Since the action of
$\pi_{1}(G^{rs},X)$ on the left-hand side factors through $W$, we have that
the action on the right-hand side will also factor through $W$. This proves
items (2) and (3).

##  10\. Proof of the main theorems

Now we have all the ingredients to prove Theorems 1.22 and 1.25.

[Proof of Theorem 1.25] Items (1) and (2) follow from Theorem 8.2 and
Corollary 8.3.

Item (3) follows from items (1) and (2), Theorems 2.7 and 2.8, and
Propositions 3.15 and 9.4.

Now we prove item (3). By Proposition 3.15, Theorems 2.7 and 2.8 and Corollary
8.4 we have that

$H^{*}(\mathcal{Y}_{\underline{s}}(X))^{W_{J}}\cong
H^{*}(\mathcal{Y}_{\underline{s}}(X_{J})),$

for some $X_{J}=X_{s}X_{u}$ with $Z_{G}(X_{s})=G_{J}$. In particular

(10a)
$\operatorname{Poin}(H^{*}(\mathcal{Y}_{\underline{s}}(X))^{W_{J}})=\operatorname{Poin}(H^{*}(\mathcal{Y}_{\underline{s}}(X_{J}))).$

By Corollary 8.3,

(10b)
$\operatorname{Poin}(H^{*}(\mathcal{Y}_{\underline{s}}(X_{J})))=\chi^{i_{W_{J}}^{W}}(\prod_{i=1}^{n}(1+T_{s_{i}}))),$

proving the first equality in item (3). To finish the proof, we just notice
that, in the case of $G=SL_{n}(\mathbb{C})$ and $W=S_{n}$, we have

$\displaystyle\operatorname{ch}(\prod_{i=1}^{n}(1+T_{s_{i}})$
$\displaystyle=\sum_{\lambda\vdash
n}\chi^{i_{S_{\lambda}}^{S_{n}}}(\prod_{i=1}^{n}(1+T_{s_{i}})))m_{\lambda},$
$\displaystyle\operatorname{ch}(H^{*}(\mathcal{Y}_{\underline{s}}(X))^{S_{\lambda}})$
$\displaystyle=\sum_{\lambda\vdash
n}\operatorname{Poin}(H^{*}(\mathcal{Y}_{\underline{s}}(X))^{S_{\lambda}})m_{\lambda}$

[Proof of Theorem 1.22]

Let ${\underline{s}}$ be a reduced word for $w$. The first statement of the
theorem follows directly from Proposition 9.4 item (2). By Proposition 9.4
item (3), we have that

$H^{*}(\mathcal{Y}_{\underline{s}}(X))^{W_{J}}\cong
IH^{*}(\mathcal{Y}_{w}(X))^{W_{J}}\oplus\bigoplus_{z<w}IH^{*}(\mathcal{Y}_{z}(X))^{W_{J}}\otimes
L_{z}[-\ell(w)+\ell(u)].$

In particular,

(10c)
$\operatorname{Poin}(H^{*}(\mathcal{Y}_{\underline{s}}(X))^{W_{J}})=\operatorname{Poin}(IH^{*}(\mathcal{Y}_{w}(X))^{W_{J}})+\sum_{z\leq
w}\operatorname{Poin}(IH^{*}(\mathcal{Y}_{z}(X))^{W_{J}})q^{\frac{\ell(w)-\ell(z)}{2}}P_{{\underline{s}},z}(q^{\frac{1}{2}}),$

where $P_{{\underline{s}},z}(q^{\frac{1}{2}})$ are the polynomials in
Proposition 3.21. By Proposition 3.21,

$\displaystyle\prod_{i=1}^{\ell(w)}(1+T_{s_{i}})=$ $\displaystyle
q^{\frac{\ell(w)}{2}}C^{\prime}_{w}+\sum_{z<w}q^{\frac{\ell(w)}{2}}P_{{\underline{s}},z}(q^{\frac{1}{2}})C^{\prime}_{z}.$
(10d) $\displaystyle=$ $\displaystyle
q^{\frac{\ell(w)}{2}}C^{\prime}_{w}+\sum_{z<w}q^{\frac{\ell(w)-\ell(z)}{2}}P_{{\underline{s}},z}(q^{\frac{1}{2}})q^{\frac{\ell(z)}{2}}C^{\prime}_{z}.$

Combining Equations (10c), (10a), (10b), and (10d), with an inductive argument
yields

$\operatorname{Poin}(IH^{*}(\mathcal{Y}_{w}(X))^{W_{J}})=\chi^{i_{W_{J}}^{W}}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w}),$

as desired. The last statement of the theorem follows from the observation
that

$\displaystyle\operatorname{ch}(IH^{*}(\mathcal{Y}_{w}(X)))$
$\displaystyle=\sum_{\lambda\vdash
n}\operatorname{Poin}(IH^{*}(\mathcal{Y}_{w}(X))^{S_{\lambda}})m_{\lambda},$
$\displaystyle\operatorname{ch}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})$
$\displaystyle=\sum_{\lambda\vdash
n}\chi^{i_{S_{\lambda}}^{S_{n}}}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})m_{\lambda},$

which finishes the proof of Theorem 1.22.

##  11\. Further directions

###  11.1. Rationally smoothness

One class of varieties that has palindromic Betti numbers and whose
intersection cohomology is isomorphic to its singular cohomology are
_rationally smooth varieties_ , that is, varieties $\mathcal{X}$ such that
$IC_{\mathcal{X}}=\mathbb{\mathbb{C}}_{\mathcal{X}}[\dim(\mathcal{X})]$. In
[Pre18], Precup asks if regular Hessenberg varieties are rationally smooth. We
conjecture the same should hold for the varieties
$\mathcal{Y}_{{\underline{s}}}(X)$.

######  Conjecture 11.1.

When $X$ is regular the varieties $\mathcal{Y}_{{\underline{s}}}(X)$ are
rationally smooth.

Notice that if $X$ is regular semisimple, then the varieties
$\mathcal{Y}_{{\underline{s}}}(X)$ are smooth and hence rationally smooth.

We note that $\mathcal{Y}_{w}(X_{\lambda})$ is not rationally smooth in
general, since already the Schubert variety $\Omega_{w}$ is not rationally
smooth. Nevertheless it seems reasonable to expect that the intersection
cohomology sheaf of $\mathcal{Y}_{w}(X_{\lambda})$ is not so different from
the intersection cohomology sheaf of the Schubert variety $\Omega_{w}$. More
precisely, we conjecture that:

######  Conjecture 11.2.

For $X\in G$ a regular element and $w\in W$, we have that
$IC_{\mathcal{Y}_{w}(X)}=i^{!}(IC_{\Omega_{w}})$, where
$i\colon\mathcal{Y}_{w}(X)\to\Omega_{w}$ is the natural map $i(gB)=(XgB,gB)$.

###  11.2. Cohomology of Hessenberg varieties

Since the varieties $\mathcal{Y}_{w}(X)$ are generalizations of Hessenberg
varieties, we believe that these varieties may help to understand the
cohomology of Hessenberg varieties (see [HP19], [HHM+21] and [HPT22] for
results about the cohomology ring of Hessenberg varieties). More precisely,
since $\mathcal{Y}_{w}(X)$ is a subvariety of $\mathcal{Y}_{\mathbf{m}}(X)$
for every $w\leq\mathbf{m}$, we have natural geometric classes
$[\mathcal{Y}_{w}(X)]\in H^{*}(\mathcal{Y}_{\mathbf{m}(X)})$. In the case that
$X$ is unipotent, we have that the number of codimension $i$ subvarieties that
appear in this way agrees with the known dimension of
$H^{2i}(\mathcal{Y}_{\mathbf{m}(X)})$. We therefore conjecture:

######  Conjecture 11.3.

Let $w\in S_{n}$ be a smooth permutation and $X\in SL(n)$ a regular element.
Then the classes $[\mathcal{Y}_{u}(X)]\in H^{*}(\mathcal{Y}_{w}(X))$ (for
$u\leq w$) are linearly independent. If $X$ is semisimple, then these classes
are fixed by the action of $S_{n}$. If $X$ is regular unipotent, then the
$[\mathcal{Y}_{u}(X)]$ will form a basis of $H^{*}(\mathcal{Y}_{w}(X))$.

When $X$ is regular semisimple and $w\in S_{n}$ is a smooth permutation, then
$\mathcal{Y}_{w}(X)$ is a GKM-space and the equivariant cohomology ring
$H_{T}^{*}(\mathcal{Y}_{w}(X))$ has an explicit description in terms of the
moment graph of $\mathcal{Y}_{w}(X)$ (see [Tym08]).

######  Question 11.4.

For each $u\in S_{n},u\leq w$, compute the class $[\mathcal{Y}_{u}(X)]$ in
terms of the moment graph of $\mathcal{Y}_{w}(X)$.

###  11.3. Relation with $\operatorname{LLT}$-polynomials

Given a Hessenberg function $\mathbf{m}$, besides the chromatic quasisymmetric
function $\operatorname{csf}_{q}(G_{\mathbf{m}})$ there are other interesting
symmetric functions associated to it. For example, the unicellular
$\operatorname{LLT}$-polynomial111111The $\operatorname{LLT}$-polynomials are
defined in a more general setting, see [LLT97]. The definition we are giving
was introduced in [CM18] and [AP18]. associated to $\mathbf{m}$ is defined as

(11a)
$\operatorname{LLT}(\mathbf{m})=\sum_{\kappa}q^{\operatorname{asc}(\kappa)}x_{\kappa},$

where the sum runs through all vertex colorings of $G_{\mathbf{m}}$ (not
necessarily proper). The chromatic quasisymmetric function and the
$\operatorname{LLT}$-polynomial are related via a plethystic relation (see
[HHL05, Section 5.1], [CM18, Proposition 3.5] and [Ale20, Lemma 12]):

$\operatorname{csf}_{q}(\mathbf{m}))=\frac{LLT(\mathbf{m})[\mathbf{x}(q-1)]}{(q-1)^{n}}.$

Or, equivalently

$\operatorname{LLT}(\mathbf{m})=(q-1)^{n}\operatorname{csf}_{q}(\mathbf{m})\Big{[}\frac{\mathbf{x}}{q-1}\Big{]}.$

We could ask:

######  Question 11.5.

We can define $\operatorname{LLT}$-like polynomials for non-codominant
permutations $w$ via

$\operatorname{LLT}(w):=(q-1)^{n}(\omega(\operatorname{ch}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})))\Big{[}\frac{\mathbf{x}}{q-1}\Big{]}.$

Do these symmetric function have an interesting combinatorial definition?

For example, we have that

$\operatorname{LLT}(3412)=(q^{4}+q^{3})s_{1,1,1,1}+(3q^{3}+3q^{2})s_{2,1,1}+(q^{3}+2q^{2}+q)s_{2,2}+(3q^{2}+3q)s_{3,1}+(q+1)s_{4}.$

Since for codominant permutations $w$ we have that $\operatorname{LLT}(w)$ is
Schur-positive (see [HG09] and [GP16]) and shifted $e$-positive (see [D’A20],
[AS22] and [AN21b]), the same properties would be enjoyed by
$\operatorname{LLT}(w)$ for every permutations $w\in S_{n}$, if (some
modification of) Conjecture 1.12 was true.

######  Conjecture 11.6.

For every $w\in S_{n}$ we have that $\operatorname{LLT}(w)$ is Schur-positive
and shifted $e$-positive.

In [AN21b], for each Hessenberg function $\mathbf{m}$, a weight function
$\operatorname{wt}_{\mathbf{m}}\colon\\{u\in
S_{n};u\leq\mathbf{m}\\}\to\mathbb{Z}_{\geq 0}$ was defined such that

$\operatorname{LLT}(\mathbf{m})=\sum_{u\leq\mathbf{m}}(q-1)^{n-\ell(\lambda(u))}q^{\operatorname{wt}_{\mathbf{m}}(u)}e_{\lambda(u)}.$

######  Question 11.7.

For each permutation $w\in S_{n}$, we may ask if there is a function
$\operatorname{wt}_{w}\colon\\{u\in S_{n};u\leq w\\}\to\mathbb{Z}_{\geq 0}$
such that

(11b) $\operatorname{LLT}(w)=\sum_{u\leq
w}(q-1)^{n-\ell(\lambda(u))}q^{\operatorname{wt}_{w}(u)}P_{u,w}(q)e_{\lambda(u)},$

where $P_{u,w}$ is the Kazhdan-Lusztig polnomial.

The above equation is equivalent to

$\operatorname{ch}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})=\sum_{u\leq
w}q^{\operatorname{wt}_{w}(u)}P_{u,w}(q)\rho_{\lambda(u)},$

where $\rho_{\lambda}$ is defined by $\rho_{i}:=q^{j-1}P_{i}(q^{-1})$,
$\rho_{\lambda}:=\prod\rho_{\lambda_{i}}$ and $P_{j}$ is the Hall-Littlewood
polynomial (see [AN21b, Proposition 2.1]).

###  11.4. Hopf algebra of Dyck paths

In [GP16] a proof of the Shareshian-Wachs conjecture was given by means of a
Hopf algebra of Dyck paths. Since each Dyck path corresponds to a codominant
permutation, we ask:

######  Question 11.8.

Does there exist a Hopf algebra $\mathcal{S}$ of permutations extending the
Hopf algebra of Dyck paths considered in [GP16]? Does there exist a
multiplicative map $\zeta\colon S\to\mathbb{C}(q)$ inducing a Hopf algebra
morphism $\Psi_{\zeta}\colon\mathcal{S}\to\mathcal{Q}Sym$ such that
$\operatorname{ch}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})=\Psi_{\zeta}(w)$ (or
for some other power of $q$). Can we give a proof of Theorem 1.22 using this
Hopf algebra as in [GP16]? Are the $\operatorname{LLT}$-polynomials
$\operatorname{LLT}(w)$ induced from a different multiplicative map
$\zeta^{\prime}$ as is the case in [GP16]?

###  11.5. Log-concavity of $h$-coefficients

Finally, motivated by computational results for the codominant case (see
[AN21a, Section 5]), we make one last conjecture:

######  Conjecture 11.9.

The $h$-coefficients of
$\operatorname{ch}(q^{\frac{\ell(w)}{2}}C^{\prime}_{w})$ are log-concave
polynomials in $q$ for every $w\in S_{n}$.

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