# A generalization of cyclic shift classes

Xuhua He  The Institute of Mathematical Sciences and Department of
Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR,
China<EMAIL_ADDRESS>
###### Abstract.

Motivated by Lusztig’s $G$-stable pieces, we consider the combinatorial
pieces: the pairs $(w,K)$ for elements $w$ in the Weyl group and subsets $K$
of simple reflections that are normalized by $w$. We generalize the notion of
cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We
show that the partial cyclic shift classes of combinatorial pieces associated
with minimal-length elements have nice representatives. As applications, we
prove the left-right symmetry and the compatibility of the induction functors
of the parabolic character sheaves.

###### Key words and phrases:

Weyl groups, cyclic shifts, parabolic character sheaves

###### 2010 Mathematics Subject Classification:

20F55, 20G40, 14F05

## Introduction

###  0.1. Cyclic shifts of Weyl group elements

Let $G$ be a connected reductive group over an algebraically closed field. Let
$B$ be a Borel subgroup of $G$ and $W$ be the Weyl group of $G$. Then we have
the Bruhat decomposition $G=\sqcup_{w\in W}B\dot{w}B$. A central theme in
geometric representation theory is understanding the (many) mysterious
relationships between the reductive group $G$ and its Weyl group $W$ (see,
e.g., the Kazhdan-Lusztig theory and the Springer theory).

We are particularly interested in the conjugation action on $W$. Note that $W$
is a Coxeter group and is generated by simple reflections. In particular, any
two elements in the same conjugacy class of $W$ are conjugated by a sequence
of simple reflections. We say that two elements are in the same cyclic shift
class if they are conjugated by a sequence of simple reflection, and each
conjugation preserves the length. This length-preserving condition is crucial
in the connection to the conjugation problems on $G$. If $w$ and $w^{\prime}$
are in the same cyclic shift class, then we have an isomorphism between the
quotient stacks

$\frac{B\dot{w}B}{\Delta(B)}\cong\frac{B\dot{w}^{\prime}B}{\Delta(B)}.$

Here $\Delta(B)$ denotes the conjugation action. Such an isomorphism plays an
important role in the interaction between the conjugacy classes of $W$ and
unipotent conjugacy classes of $G$ (see [10]).

###  0.2. Cyclic shifts of combinatorial pieces

In this paper, we consider an analogy of (a) for parabolic subgroups. Such an
isomorphism will play a role in the study of Lusztig’s theory of parabolic
character sheaves, which we will discuss in §0.4.

The object we consider is the quotient stack $\frac{P\dot{w}P}{\Delta(P)}$,
where $P=LU_{P}$ is a standard parabolic subgroup of $G$, and $w$ is an
element in the Weyl group which normalizes the standard Levi subgroup $L$. In
fact, for the applications, the object we consider will be more technical than
$\frac{P\dot{w}P}{\Delta(P)}$, but we ignore such complexity in the
introduction.

Our first goal is to establish isomorphisms between
$\frac{P\dot{w}P}{\Delta(P)}$ for various pairs $(w,P)$. Note that the
conjugation by simple reflections does not send standard Levi subgroups to
Levi subgroups. To overcome the difficulty, we introduce combinatorial pieces.
A combinatorial piece is a pair $(w,K)$, where $w$ is an element in the Weyl
group, and $K$ is a subset of simple reflections (which represents a standard
parabolic subgroup $P_{K}$ of $G$) such that $w$ normalizes $K$. We then
introduce the cyclic shift classes of the combinatorial pieces in §3.2 and
prove in Proposition 3.1 that if $(w,K)$ and $(w^{\prime},K^{\prime})$ are in
the same cyclic shift class, then we have an isomorphism between the quotient
stacks

$\frac{P_{K}\dot{w}P_{K}}{\Delta(P_{K})}\cong\frac{P_{K^{\prime}}\dot{w}^{\prime}P_{K^{\prime}}}{\Delta(P_{K^{\prime}})}.$

Note that we may regard the Weyl group $W$ as a subset of the combinatorial
pieces via the map $w(w,\emptyset)$. The combinatorial pieces $(w,\emptyset)$
and $(w^{\prime},\emptyset)$ are cyclic shift if and only if $w$ and
$w^{\prime}$ are cyclic shift in the usual sense. In this case,
$P_{\emptyset}=B$. Thus (b) can be regarded as a generalization of (a).

###  0.3. Representatives

Our next goal is to obtain nice representatives for certain cyclic shift
classes of combinatorial pieces.

Let $J$ be a subset of the simple reflections and $W_{J}$ be the standard
parabolic subgroup of $W$ generated by $J$. We consider the partial conjugacy
classes on $W$, i.e., the orbits of the conjugation by $W_{J}$ on $W$. By [5],
we have the decomposition

$\frac{W}{\Delta(W_{J})}=\bigsqcup_{(w,K)}\frac{W_{K}w}{\Delta(W_{K})},$

where $\frac{W}{\Delta(W_{J})}$ denotes the quotient stack of $W$ by the
conjugation action of $W_{J}$, and the right hand side runs over the pairs
$(w,K)$, where $w$ are minimal length representatives in the right cosets
$W_{J}\backslash W$ and $K$ is the maximal subset of $J$ such that $(w,K)$ is
a combinatorial piece. Moreover, if $w$ is a minimal length element in its
partial conjugacy class, then $w$ is cyclic shift (via partial conjugation,
i.e., a sequence of conjugations in $W_{J}$) to an element in $W_{K}w$ for
some $(w,K)$ occurring in the right side of (c).

We prove an “upgraded version” of the above statement in Theorem 3.2. Namely,
if $w_{1}$ is a minimal length element in its partial conjugacy class and
$(w_{1},K_{1})$ is a combinatorial piece with $K_{1}\subset J$, then

$\text{$(w_{1},K_{1})$ is cyclic shift (via partial conjugation) to
$(w_{2},K_{2})$},$

where $w_{2}$ is an element in $W_{K}w$ for some $(w,K)$ occurring in the
right side of (c).

###  0.4. Applications to parabolic character sheaves

Now we discuss some applications. Lusztig introduced the parabolic character
sheaves in [8]. Roughly speaking, the parabolic character sheaves are certain
simple perverse sheaves on the quotient stack
$\mathcal{Y}_{J}=\frac{U_{P_{J}}\backslash G/U_{P_{J}}}{\Delta(L_{J})}$. The
parabolic character sheaves rely on the following decomposition of
$\mathcal{Y}_{J}$ into $G$-stable pieces:

$\mathcal{Y}_{J}=\bigsqcup_{w}\mathcal{Y}_{J,w},\text{ where
}\mathcal{Y}_{J,w}``="\frac{P_{K}\dot{w}P_{K}}{\Delta(P_{K})},$

Here $w$ runs over the minimal length representatives in the right coset
$W_{J}\backslash W$. And the quotation mark means we ignore the gerbs for
unipotent groups. We refer to §2.2 for the precise statement.

It is worth pointing out that if $(w,K)$ is conjugate to
$(w^{\prime},K^{\prime})$, then it is easy to construct an isomorphism
$\frac{L_{K}\dot{w}}{\Delta(L_{K})}\cong\frac{L_{K^{\prime}}\dot{w}^{\prime}}{\Delta(L_{K^{\prime}})}$.
However, it is more involved to relate
$\frac{P_{K}\dot{w}P_{K}}{\Delta(P_{K})}$ for different combinatorial pieces
$(w,K)$. The results (b) and (d) on the cyclic shifts of combinatorial pieces
allow us to understand certain behavior of the sheaves associated with an
arbitrary combinatorial piece with respect to the decomposition (e).

We give two applications in §4.

(1) The left-right symmetry.

Note that in the decompositions (c) and (d), we use the right cosets
$W_{J}\backslash W$. One may define similar decompositions using the left
cosets $W/W_{J}$. A priori, the decompositions obtained using the left and
right cosets might differ. However, using the cyclic shifts of combinatorial
pieces, we show that there exists a symmetry between the left cosets and the
right cosets such that the corresponding substacks in
$\frac{W}{\Delta(W_{J})}$ and $\mathcal{Y}_{J}$ coincide.

(2) Induction functors.

Let $J\subset J^{\prime}$. Lusztig defined the functor
$f:\mathrm{Sh}(\mathcal{Y}_{J})\mathrm{Sh}(\mathcal{Y}_{J^{\prime}})$. Suppose
that $w$ (resp. $w^{\prime}$) is the minimal length representative in its
right coset $W_{J}\backslash W$ (resp. $W_{J^{\prime}}\backslash W$), $w$ and
$w^{\prime}$ are in the same $W_{J^{\prime}}$-conjugacy class and
$\ell(w)=\ell(w^{\prime})$. Then we have

$\mathcal{Y}_{J,w}``="\frac{P_{K}\dot{w}P_{K}}{\Delta(P_{K})}\text{ and
}\mathcal{Y}_{J^{\prime},w^{\prime}}``="\frac{P_{K^{\prime}}\dot{w}^{\prime}P_{K^{\prime}}}{\Delta(P_{K^{\prime}})}.$

In Theorem 4.3, we show that the restriction of the functor $f$ to
$\mathcal{Y}_{J,w}$ and $\mathcal{Y}_{J^{\prime},w^{\prime}}$ is essentially
the Harish-Chandra induction functor associated to the reductive group
$L_{K^{\prime}}$ and its Levi subgroups. This result is used in the work of
Li, Nadler, and Yun [11].

Acknowledgement:  XH is partially supported by funds connected with Choh-Ming
Chair at CUHK, by Hong Kong RGC grant 14300221, and by the Xplorer prize. This
paper is motivated by a question of Zhiwei Yun on the parabolic character
sheaves, which is used in [11]. We thank him for explaining to me the question
and related materials in [11]. We also thank George Lusztig for the helpful
discussions.

##  1\. Cyclic shift classes

###  1.1. Conjugacy graph

Let $W$ be a Weyl group and $S$ be the set of simple reflections. Let $\ell$
be the length function on $W$ and $\leqslant$ be the Bruhat order on $W$. Let
$\delta$ be a group automorphism on $W$, which preserves $S$. We consider the
$\delta$-conjugation action on $W$ defined by
$w\cdot_{\delta}w^{\prime}=ww^{\prime}\delta(w)^{-1}$.

Following [3, §3.2], we define the $\delta$-conjugacy graph as follows. It is
the direct graph with vertices in $W$ and the edges are of the form
$w\xrightarrow{s}_{\delta}w^{\prime}$, where $w,w^{\prime}\in W$, $s\in S$
such that $w^{\prime}=sw\delta(s)$ and $\ell(w^{\prime})\leqslant\ell(w)$.

We denote by δ the pre-order relation induced on $W$ by the $\delta$-conjugacy
graph. In other words, for $w,w^{\prime}\in W$, $w_{\delta}w^{\prime}$ if
there exists a sequence of elements $w=w_{1},w_{2},\ldots,w_{n}=w$ and
$s_{1},\ldots,s_{n-1}\in S$ such that
$w_{i}\xrightarrow{s_{i}}_{\delta}w_{i+1}$ for $1\leqslant i\leqslant n-1$.

We write $w\approx_{\delta}w^{\prime}$ if $w_{\delta}w^{\prime}$ and
$w^{\prime}_{\delta}w$. It is easy to see that $\approx_{\delta}$ gives an
equivalence relation on $W$. For $w\in W$, we write
$\text{Cyc}_{\delta}(w)=\\{w^{\prime}\in W;w\approx_{\delta}w^{\prime}\\}$ and
call it the $\delta$-cyclic shift class  of $w$.

If $\delta$ is the identity map, we may ignore the subscript $\delta$.

Now we give an example of the conjugacy graph.

######  Example 1.1.

Let $W=S_{4}$ be the finite Weyl group of type $A_{3}$ with the set of simple
reflections $S=\\{s_{1},s_{2},s_{3}\\}$. We simply write $s_{abc\cdots}$
instead of $s_{a}s_{b}s_{c}\cdots$. It is easy to see that the conjugacy class
of $s_{1}s_{2}s_{3}$ forms a connected component of the conjugacy graph. This
connected component is listed below.

$\textstyle{s_{12132}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{2}}$$\scriptstyle{s_{1},s_{3}}$$\textstyle{s_{23212}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{2}}$$\textstyle{s_{123}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\scriptstyle{s_{3}}$$\textstyle{s_{213}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{3}}$$\textstyle{s_{132}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\textstyle{s_{321}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###  1.2. Geometric cyclic shifts

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed
field $\mathbf{k}$. Let $G=\mathbf{G}(\mathbf{k})$ and $B$ be a Borel subgroup
of $G$ with maximal torus $T$ and unipotent radical $U$. Let $\mathcal{R}$ be
the root datum of $G$ and $\delta$ be an automorphism of $\mathcal{R}$. In
particular, $\delta(B)=B$. We still denote by $\delta$ for the corresponding
automorphism on $G$ and on its Weyl group $W=N_{G}(T)/T$. For any $w\in W$, we
choose a representative $\dot{w}$ in $N_{G}(T)$.

Consider the $\delta$-conjugation action of $G$ defined by
$g\cdot_{\delta}g^{\prime}=gg^{\prime}\delta(g)^{-1}$. Let
$\frac{G}{{\mathrm{Ad}}_{\delta}(G)}$ be the quotient stack. We have the
Bruhat decomposition $G=\sqcup_{w\in W}B\dot{w}B$. Each Bruhat cell
$B\dot{w}B$ is stable under the $\delta$-conjugation action by $B$. Let
$\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}$ be the corresponding quotient
stack and

$\pi_{w}:\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\frac{G}{{\mathrm{Ad}}_{\delta}(G)}$

be the natural morphism.

On the other hand, we have the automorphism on $T$ defined by
$t\dot{w}\delta(t)\dot{w}^{-1}$. Let
$\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}$ be the quotient stack and

$p_{w}:\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}$

be the natural morphism.

We have the following (upgraded) geometric version of cyclic shifts.

######  Proposition 1.2.

Let $w,w^{\prime}\in W$. Suppose that $w\approx_{\delta}w^{\prime}$. Then
there exist isomorphisms
$\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\cong\frac{B\dot{w}^{\prime}B}{{\mathrm{Ad}}_{\delta}(B)}$
and
$\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}\cong\frac{T}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(T)}$
such that the following diagram commutes

$\textstyle{\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{w}}$$\scriptstyle{\pi_{w}}$$\scriptstyle{\cong}$$\textstyle{\frac{G}{{\mathrm{Ad}}_{\delta}(G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{T}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(T)}}$$\textstyle{\frac{B\dot{w}^{\prime}B}{{\mathrm{Ad}}_{\delta}(B)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{w^{\prime}}}$$\scriptstyle{\pi_{w^{\prime}}}$$\textstyle{\frac{G}{{\mathrm{Ad}}_{\delta}(G)}.}$

######  Remark 1.3.

(1) An isomorphism
$\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\frac{B\dot{w}^{\prime}B}{{\mathrm{Ad}}_{\delta}(B)}$
was constructed in [6, §3.4]. We follow the proof in loc.cit.

(2) The isomorphisms
$\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\cong\frac{B\dot{w}^{\prime}B}{{\mathrm{Ad}}_{\delta}(B)}$
and
$\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}\cong\frac{T}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(T)}$
we construct can be regarded as the geometric lifting of the path of cyclic
shifts
$w=w_{1}\xrightarrow{s_{1}}\cdots\xrightarrow{s_{n-1}}w_{n}=w^{\prime}$. Such
isomorphisms depend on the choice of such paths, and there seem to be no
canonical choices of isomorphisms.

###### Proof.

It suffices to consider the case where $w\xrightarrow{s}w^{\prime}$. In this
case, $\ell(w^{\prime})=\ell(w)$. It is easy to see that if $sw>w$ and
$w\displaystyle(s)>w$, then $w^{\prime}=w$. Without loss of generality, we may
assume furthermore that $sw<w$. Set $w_{1}=sw$. Then
$w_{1}\delta(s)=w^{\prime}$. We have

$B\dot{s}B\times^{B}B\dot{w}_{1}B\cong B\dot{w}B,$

where $B$ acts on $B\dot{s}B\times B\dot{w}_{1}B$ via
$b\cdot(g_{1},g_{2})=(g_{1}b^{-1},bg_{2})$ and
$B\dot{s}B\times^{B}B\dot{w}_{1}B$ is the quotient space. Similarly, we have

$B\dot{w}_{1}B\times^{B}B\delta(\dot{s})B\cong B\dot{w}^{\prime}B.$

We have a natural isomorphism

$B\dot{s}B\times B\dot{w}_{1}B\cong B\dot{w}_{1}B\times
B\delta(\dot{s})B,\qquad(g_{1},g_{2})(g_{2},\delta(g_{1})).$

By definition, $g_{1}g_{2}$ and $g_{2}\delta(g_{1})$ are in the same
$\delta$-conjugacy class of $G$. This isomorphism induces the desired
isomorphism
$\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\cong\frac{B\dot{w}^{\prime}B}{{\mathrm{Ad}}_{\delta}(B)}$.

Note that
${\mathrm{Ad}}(\dot{s})\circ{\mathrm{Ad}}(\dot{w})\circ\delta={\mathrm{Ad}}(\dot{w}^{\prime})\circ\delta\circ{\mathrm{Ad}}(\dot{s})$
on $T$. The desired isomorphism
$\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}\cong\frac{T}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(T)}$
is induced from the conjugation action by $\dot{s}$. ∎

######  Corollary 1.4.

Let $w,w^{\prime}\in W$ with $w\approx_{\delta}w^{\prime}$. Let
$f:\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}\cong\frac{T}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(T)}$
be an isomorphism in Proposition 1.2. Then we have

$(\pi_{w})_{!}(p_{w})^{*}=(\pi_{w^{\prime}})_{!}(p_{w^{\prime}})^{*}f:\text{Sh}(\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)})\text{Sh}(\frac{G}{{\mathrm{Ad}}_{\delta}(G)}).$

Here $\text{Sh}(-)$ denotes the derived category of complexes of sheaves.

##  2\. Parabolic character sheaves

###  2.1. Partial conjugation on $W$

Let $J\subset S$. Recall that $W_{J}$ is the subgroup of $W$ generated by the
simple reflections in $J$. Let $W^{J}$ (resp. ${}^{J}W$) be the set of minimal
coset representatives in $W/W_{J}$ (resp. $W_{J}\backslash W$). For
$J,K\subset S$, we simply write ${}^{J}W^{K}$ for ${}^{J}W\cap W^{K}$.

Consider the $\delta$-conjugation action of $W_{J}$ on $W$ defined by
$w\cdot_{\delta}w^{\prime}=ww^{\prime}\delta(w)^{-1}$ for $w\in W_{J}$ and
$w^{\prime}\in W$. For $w\in W$, we write ${\mathrm{Ad}}(w)\delta(J)=J$ if for
any simple reflection $s\in J$, there exists a simple reflection
$s^{\prime}\in J$ such that $w\delta(s)w^{-1}=s^{\prime}$. In this case,
$w\in{}^{J}W$ if and only if $w\in W^{\delta(J)}$.

For any $J\subset S$ and $w\in W$, we set

$I(J,w,\delta)=\max\\{K\subset J;{\mathrm{Ad}}(w)\delta(K)=K\\}.$

Now we recall Bédard’s description [1] of ${}^{J}W$.

Let $\mathcal{T}(J,\delta)$ be the set of sequences $(J_{n},w_{n})_{n\geqslant
0}$ with $J_{n}\subset J$ and $w_{n}\in W$ such that

  1. (1)

$J_{0}=J$;

  2. (2)

$J_{n}=J_{n-1}\cap{\mathrm{Ad}}(w_{n-1})(\delta(J_{n-1}))$ for $n\geqslant 1$;

  3. (3)

$w_{n}\in{}^{J_{n}}W^{\delta(J_{n})}$ for $n\geqslant 0$;

  4. (4)

$w_{n}\in W_{J_{n}}w_{n-1}W_{\delta(J_{n-1})}$ for $n\geqslant 1$.

Then for any sequence $(J_{n},w_{n})_{n\geqslant 0}$, we have that
$w_{m}=w_{m+1}=\cdots$ and $J_{m}=J_{m+1}=\cdots$ for $m\geqslant 0$. By [8,
Proposition 2.5], the assignment $(J_{n},w_{n})_{n\geqslant 0}w_{m}$ for $m\gg
0$ defines a bijection $\mathcal{T}(J,\delta){}^{J}W$. Moreover, $w_{n}\in
wW_{\delta(J_{n})}$ for all $n\geqslant 0$ and $I(J,w,\delta)=J_{m}$ for $m\gg
0$.

We have the following description of $W_{I(J,w,\delta)}$.

######  Lemma 2.1.

Let $w\in{}^{J}W$. Then
$W_{I(J,w,\delta)}=\bigcap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})$.

###### Proof.

By definition,
${\mathrm{Ad}}(w)\circ\delta(W_{I(J,w,\delta)})=W_{I(J,w,\delta)}\subset
W_{J}$.

Let $(J_{n},w_{n})_{n\geqslant 0}$ be the sequence in $\mathcal{T}(J,\delta)$
corresponding to $w$. Now we prove that
$\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\subset
W_{J_{n}}$ for all $n$.

By definition
$\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\subset
W_{J}=W_{J_{0}}$. Assume that we have proved
$\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\subset
W_{J_{i}}$ for some $i$. By definition, $w=w_{i}\delta(a)$ for some $a\in
W_{J_{i}}$. Hence

$\displaystyle\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})$
$\displaystyle=\bigl{(}\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\bigr{)}\cap{\mathrm{Ad}}(w)\circ\delta\bigl{(}\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\bigr{)}$
$\displaystyle\subset
W_{J_{i}}\cap{\mathrm{Ad}}(w)\circ\delta(W_{J_{i}})=W_{J_{i}}\cap{\mathrm{Ad}}(w_{i})\circ\delta(W_{J_{i}})$
$\displaystyle=W_{J_{i+1}}.$

This finishes the proof. ∎

We have the following classification of the
${\mathrm{Ad}}_{\delta}(W_{J})$-orbits on $W$.

######  Proposition 2.2.

Let $J\subset S$. Then

(1) $W=\bigsqcup_{w\in{}^{J}W}W_{J}\cdot_{\delta}(W_{I(J,w,\delta)}w)$;

(2) For any $w\in{}^{J}W$, the embedding $W_{I(J,w,\delta)}W$, $uuw$ induces
the bijection between the quotient stacks

$\frac{W_{I(J,w,\delta)}}{{\mathrm{Ad}}_{w\circ\delta}(W_{I(J,w,\delta)})}\cong\frac{W_{J}\cdot_{\delta}(W_{I(J,w,\delta)}w)}{{\mathrm{Ad}}_{\delta}(W_{J})}.$

###### Proof.

Part (1) and the bijection of orbits in part (2) were proved in [5, Corollary
2.6]. It remains to show that the bijection on the orbits also gives an
isomorphism between the isotropy groups. In other words, it remains to show
that if $(a,b)\in W_{J}\times W_{I(J,w,\delta)}w$ with $ab\delta(a)^{-1}\in
W_{I(J,w,\delta)}w$, then $a\in W_{I(J,w,\delta)}$.

Let $(J_{n},w_{n})_{n\geqslant 0}$ be the element in $\mathcal{T}(J,\delta)$
corresponding to $w$. We argue by induction that $a\in W_{J_{n}}$ for all $n$.

By definition, $a\in W_{J}=W_{J_{0}}$. Assume that $a\in W_{J_{i}}$ for some
$i$. Then $W_{I(J,w,\delta)}w\subset w_{i}W_{\delta(J_{i})}$ and
$ab\delta(a)^{-1}\in aw_{i}W_{\delta(J_{i})}$. Thus

$a\in W_{J_{i}}\cap w_{i}W_{\delta(J_{i})}w_{i}^{-1}=W_{J_{i+1}}.$

Hence $a\in W_{J_{n}}$ for all $n\geqslant 0$. In particular, $a\in
W_{I(J,w,\delta)}$. ∎

###  2.2. Lusztig’s variety $Z_{J,\delta}$

For any $J\subset S$, let $P_{J}\supset B$ be the standard parabolic subgroup
of type $J$. Let $\mathcal{P}_{J}=G/P_{J}$ be the partial flag variety. We may
identify $\mathcal{P}_{J}$ with the set of parabolic subgroups of $G$ that are
conjugate to $P_{J}$. For any parabolic subgroup $P$ of $G$, we denote by
$U_{P}$ its unipotent radical. For $g\in G$ and $H\subset G$, we simply write
${}^{g}H$ for $gHg^{-1}$. Following [8], we set

$Z_{J,\delta}=\\{(P,P^{\prime},g\delta(U_{P}));P\in\mathcal{P}_{J},P^{\prime}\in\mathcal{P}_{\delta(J)},g\in
G,{}^{g}\delta(P)=P^{\prime}\\}.$

Define the action of $G\times G$ on $Z_{J,\delta}$ by

$(g_{1},g_{2})\cdot(P,P^{\prime},g\delta(U_{P}))=({}^{g_{2}}P,{}^{g_{1}}P^{\prime},g_{1}g\delta(g_{2})^{-1}\,\delta(U_{{}^{g_{2}}P})).$

Then $G\times G$ acts transitively on $Z_{J,\delta}$. Let
$h_{J,\delta}=(P_{J},P_{\delta(J)},U_{P_{\delta(J)}})$ be the base point. Then
the isotropy group of $h_{J,\delta}$ is $\\{(\delta(l)u^{\prime},lu);l\in
L_{J},u\in U_{P_{J}},u^{\prime}\in U_{P_{\delta(J)}}\\}$. The map $G\times
GG,(g,g^{\prime})(g^{\prime})^{-1}g$ induces an isomorphism of stacks

$\frac{Z_{J,\delta}}{\Delta(G)}\cong\mathcal{Y}_{J,\delta}.$

Here $\frac{Z_{J,\delta}}{\Delta(G)}$ is the quotient stack for the diagonal
$G$-action on $Z_{J,\delta}$ and
$\mathcal{Y}_{J,\delta}=\frac{U_{P_{J}}\backslash
G/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\delta}(L_{J})}$ is the quotient stack for
the $\delta$-conjugation action of $L_{J}$ on $U_{P_{J}}\backslash
G/U_{P_{\delta(J)}}$.

Now we recall the decomposition of $Z_{J,\delta}$ into the $G$-stable pieces,
introduced by Lusztig in [8]. For $w\in{}^{J}W$, we set

$Z_{J,\delta;w}=G_{\Delta}\cdot(B\dot{w},B)h_{J,\delta}.$

The following result is established by Lusztig in [8].

######  Proposition 2.3.

Let $J\subset S$. Then

(1) $Z_{J,\delta}=\bigsqcup_{w\in{}^{J}W}Z_{J,\delta;w}$ is a decomposition
into smooth, locally closed subvarieties.

(2) For any $w\in{}^{J}W$, there is a canonical map

$\pi_{J,\delta;w}:\frac{Z_{J,\delta;w}}{\Delta(G)}\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})}$

which is an iterated gerbe for unipotent groups.

Under the isomorphism
$\frac{Z_{J,\delta}}{\Delta(G)}\cong\mathcal{Y}_{J,\delta}$, we may
reformulate the above decomposition as follows. For any $w\in W$, let
$\mathcal{Y}_{J,\delta;w}$ be the image of $B\dot{w}B$ in
$\mathcal{Y}_{J,\delta}$. Then for any $w\in{}^{J}W$,
$\frac{Z_{J,\delta;w}}{\Delta(G)}\cong\mathcal{Y}_{J,\delta;w}$ and
$\mathcal{Y}_{J,\delta}=\sqcup_{w\in{}^{J}W}\mathcal{Y}_{J,\delta;w}$.

By [4, Proposition 1.10 (1)],

$Z_{J,\delta;w}\cong
G\times^{P_{I(J,w,\delta)}}(P_{I(J,w,\delta)}\dot{w},P_{I(J,w,\delta)})\cdot
h_{J,\delta},$

where $P_{I(J,w,\delta)}$ acts on
$G\times(P_{I(J,w,\delta)}\dot{w},P_{I(J,w,\delta)})\cdot h_{J,\delta}$ by
$p\cdot(g,z)=(gp^{-1},(p,p)\cdot z)$ and
$G\times^{P_{I(J,w,\delta)}}(P_{I(J,w,\delta)}\dot{w},P_{I(J,w,\delta)})\cdot
h_{J,\delta}$ is the quotient space. We may reformulate it as follows.

######  Proposition 2.4.

Let $w\in{}^{J}W$. The embedding
$P_{I(J,w,\delta)}\dot{w}P_{\delta(I(J,w,\delta)}G$ induces an isomorphism

$\frac{U_{P_{J}}\backslash
P_{I(J,w,\delta)}\dot{w}P_{\delta(I(J,w,\delta)}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\delta}(L_{J}\cap
P_{I(J,w,\delta)})}\cong\mathcal{Y}_{J,\delta;w}.$

###  2.3. Parabolic character sheaves

We follow [8]. For any $w\in W$, we consider the following diagram

$\textstyle{\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)}}$$\textstyle{\frac{B\dot{w}B}{{\mathrm{Ad}}_{\delta}(B)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{w}}$$\scriptstyle{\pi_{J,w}}$$\textstyle{\mathcal{Y}_{J,\delta}.}$

A parabolic character sheaf on $\mathcal{Y}_{J,\delta}$ is a simple perverse
sheaf that is a composition factor of
${}^{p}H^{i}((\pi_{J,w})_{!}p_{w}^{*}\mathcal{L})$ for some $w\in W$,
$i\in\mathbb{Z}$ and
$\mathcal{L}\in\mathrm{Sh}(\frac{T}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(T)})$.

On the other hand, by Proposition 2.3, for any $w\in{}^{J}W$, the map
$\pi_{J,w}$ induces an equivalence of categories

$\pi_{J,\delta;w}^{*}:\text{Sh}(\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})})\cong\text{Sh}(\frac{Z_{J,\delta;w}}{\Delta(G)})=\mathrm{Sh}(\mathcal{Y}_{J,\delta;w}).$

Note that $L_{I(J,w,\delta)}$ is a connected reductive group. The character
sheaves on
$\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})}$
is defined by Lusztig in [7].

It is proved by Lusztig in [8] that

######  Theorem 2.5.

Let $J\subset S$. Then

(1) Any parabolic character sheaf on $\mathcal{Y}_{J,\delta}$ is the
intermediate extension of $\pi^{*}_{J,\delta;w}(A)$ for a unique $w\in{}^{J}W$
and a character sheaf $A$ on
$\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})}$.

(2) If $B$ is a parabolic character sheaf on $\mathcal{Y}_{J,\delta}$ and
$w\in{}^{J}W$, then any composition factor of
${}^{p}H^{i}(B\mid_{\mathcal{Y}_{J,\delta;w}})$ is of the form
$\pi^{*}_{J,\delta;w}(A)$ for some character sheaf $A$ on
$\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})}$.

###  2.4. Partial conjugation graph

Let $J\subset S$. We consider the ${\mathrm{Ad}}_{\delta}(W_{J})$-orbits on
$W$. The $(J,\delta)$-conjugacy graph, by definition, is the direct graph with
vertices in $W$ and the edges are of the form
$w\xrightarrow{s}_{\delta}w^{\prime}$ for $w,w^{\prime}\in W$, $s\in J$.

We denote by J,δ the pre-order relation induced by $\xrightarrow{s}_{\delta}$
for $s\in J$. We write $w\approx_{J,\delta}w^{\prime}$ if
$w_{J,\delta}w^{\prime}$ and $w^{\prime}_{J,\delta}w$. We call the equivalece
class $\approx_{J,\delta}$ the $(J,\delta)$-cyclic shift classes on $W$.

We have the following result.

######  Proposition 2.6.

[5, Proposition 3.4] For any $w\in W$, there exists $w^{\prime}\in{}^{J}W$ and
$u\in W_{I(J,w^{\prime},\delta)}$ such that $w_{J,\delta}uw^{\prime}$.

Moreover, if $w$ is of minimal length in $W_{J}\cdot_{\delta}w$, then $u$ is
of minimal length in $W_{I(J,w^{\prime},\delta)}\cdot_{\delta^{\prime}}u$ and
$w\approx_{J,\delta}uw^{\prime}$, where
$\delta^{\prime}={\mathrm{Ad}}(w^{\prime})\circ\delta$.

######  Remark 2.7.

By Proposition 2.2 (1), $w^{\prime}$ is uniquely determined by $w$. However,
$u$ is not unique in general.

###  2.5. Partial order

For any $w\in W$ and $w^{\prime}\in{}^{J}W$, we write
$w^{\prime}\leqslant_{J,\delta}w$ if there exists $u\in W_{J}$ such that
$uw^{\prime}\delta(u)^{-1}\leqslant w$. By [5, Corollary 4.6],

(a) The restriction of $\leqslant_{J,\delta}$ to ${}^{J}W$ gives a partial
order on ${}^{J}W$.

It is easy to see that for $w,w^{\prime}\in{}^{J}W$, $w^{\prime}\leqslant w$
implies that $w^{\prime}\leqslant_{J,\delta}w$ and
$w^{\prime}\leqslant_{J,\delta}w$ implies that
$\ell(w^{\prime})\leqslant\ell(w)$. However, the converse directions do not
hold in general.

######  Example 2.8.

Let $W=S_{4}$ and $J=\\{3\\}$. The simple reflections of $W$ are
$s_{1},s_{2},s_{3}$. We simply write $s_{abc\cdots}$ instead of
$s_{a}s_{b}s_{c}\cdots$. In Figure 1, we draw the Hasse diagram of ${}^{J}W$,
with respect to the usual Bruhat order and the partial order
$\leqslant_{J,\mathrm{id}}$ (the extra relation is in dotted line).

Figure 1. Hasse diagram for the partial orders on ${}^{J}W$
$\textstyle{s_{12132}}$$\textstyle{s_{2132}}$$\textstyle{s_{1213}}$$\textstyle{s_{123}}$$\textstyle{s_{121}}$$\textstyle{s_{213}}$$\textstyle{s_{12}}$$\textstyle{s_{21}}$$\textstyle{s_{23}}$$\textstyle{s_{1}}$$\textstyle{s_{2}}$$\textstyle{1}$

By [5, Proposition 5.8], we have

(a) Let $w\in W$. Then
$\overline{\mathcal{Y}_{J,\delta;w}}=\bigsqcup_{w^{\prime}\in{}^{J}W;w^{\prime}\leqslant_{J,\delta}w}\mathcal{Y}_{J,\delta;w^{\prime}}$.

Since $\mathcal{Y}_{J,\delta;w}$ is irreducible, there exists a unique
geometric piece $\mathcal{Y}_{J,\delta;w^{\prime}}$ which is dense in
$\overline{\mathcal{Y}_{J,\delta;w}}$. Therefore we have

(b) For any $w\in W$, the set $\\{w^{\prime}\in
W^{J};w^{\prime}\leqslant_{J,\delta}w\\}$ contains a unique maximal element
with respect to $\leqslant_{J,\delta}$.

We also would like to point out the special case of (a), which will be used in
[11].

(c) Let $w\in W$. If $W_{J}\cdot_{\delta}w\cap{}^{J}W=\emptyset$ or $w$ is not
of minimal length in $W_{J}\cdot_{\delta}w$, then
$\mathcal{Y}_{J,\delta;w}\subset\bigsqcup_{w^{\prime}\in{}^{J}W;\ell(w^{\prime})<\ell(w)}\mathcal{Y}_{J,\delta;w^{\prime}}$.

##  3\. Cyclic shifts of pieces

###  3.1. Combinatorial pieces

A $\delta$-combinatorial piece is a pair $(w,K)$, where $w\in W$ and $K\subset
S$ with $w\in{}^{K}W$ and ${\mathrm{Ad}}(w)\delta(K)=K$ (and hence $w\in
W^{\delta(K)}$). To each $\delta$-combinatorial piece $(w,K)$, we associate a
subset $W_{K}w$ of $W$. In particular, if $K=\emptyset$, then we naturally
identify the $\delta$-combinatorial piece $(w,\emptyset)$ with the element $w$
of $W$. In this way, we identify $W$ as a subset of the set of
$\delta$-combinatorial pieces.

We say that two $\delta$-combinatorial pieces $(w,K)$ and
$(w^{\prime},K^{\prime})$ are $\delta$-conjugated by an element $x\in W$ if
$w^{\prime}=x^{-1}w\delta(x)$, and ${\mathrm{Ad}}(x)^{-1}(K)=K^{\prime}$ (and
hence $x\in W^{K}$). In this case,
$x^{-1}(W_{K}w)\delta(x)=W_{K^{\prime}}w^{\prime}$.

It is worth pointing out that the whole Weyl group $W$ does not act on the set
of $\delta$-combinatorial pieces. The element $x$ acts on $(w,K)$ only if
${\mathrm{Ad}}(x)^{-1}(K)\subset S$.

###  3.2. Cyclic shifts of pieces

The definition of cyclic shifts on $W$ in §1.1 does not generalize to the set
of $\delta$-combinatorial pieces.

However, there is an equivalent definition of cyclic shifts on $W$, due to
Broué and Michel in [2]. The definition is as follows. Let $w,w^{\prime}\in
W$. Then $w\approx_{\delta}w^{\prime}$ if there exists a sequence of elements
$w=w_{1},\ldots,w_{n}=w^{\prime}$ and elements $x_{i},y_{i}\in W$ such that
$w_{i}=x_{i}y_{i}$, $w_{i+1}=y_{i}\delta(x_{i})$ and
$\ell(w_{i})=\ell(x_{i})+\ell(y_{i})=\ell(w_{i+1})$ for all $1\leqslant
i\leqslant n-1$. See [3, Exercise 3.7].

Now we define cyclic shifts of the combinatorial pieces.

Let $(w,K)$ and $(w^{\prime},K^{\prime})$ be $\delta$-combinatorial pieces. We
write $(w,K)\stackrel{{\scriptstyle
x}}{{\approx}}_{\delta}(w^{\prime},K^{\prime})$ if $(w,K)$ and
$(w^{\prime},K^{\prime})$ are $\delta$-conjugated by $x$, and
$\ell(w)=\ell(x)+\ell(x^{-1}w)=\ell(w^{\prime})$. Let $\approx_{\delta}$ be
the equivalence relation on the set of $\delta$-combinatorial pieces generated
by $\stackrel{{\scriptstyle x}}{{\approx}}_{\delta}$ for $x\in W$. We call it
the cyclic shift relation of the $\delta$-combinatorial pieces. When
restricting to $W$, this definition coincides with the definition of Broué and
Michel.

Similarly, for any $J\subset S$, let $\approx_{J,\delta}$ be the equivalence
relation on the set of $\delta$-combinatorial pieces generated by
$\stackrel{{\scriptstyle x}}{{\approx}}_{\delta}$ for $x\in W_{J}$. In
geometric applications, we usually consider the $\approx_{J,\delta}$ on the
set of $\delta$-combinatorial pieces $(w,K)$ with the additional assumption
that $K\subset J$.

######  Proposition 3.1.

Let $(w,K)$ and $(w^{\prime},K^{\prime})$ be $\delta$-combinatorial pieces
with $K,K^{\prime}\subset J$. Suppose that
$(w,K)\approx_{J,\delta}(w^{\prime},K^{\prime})$. Then there exists
isomorphisms $\frac{U_{P_{J}}\backslash
P_{K}\dot{w}P_{\delta(K)}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{J}\cap
P_{K})}\cong\frac{U_{P_{J}}\backslash
P_{K^{\prime}}\dot{w}^{\prime}P_{\delta(K^{\prime})}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{J}\cap
P_{K^{\prime}})}$ and
$\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})}\cong\frac{L_{K^{\prime}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{K^{\prime}})}$
such that the following diagram commutes

$\textstyle{\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{U_{P_{J}}\backslash
P_{K}\dot{w}P_{\delta(K)}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{J}\cap
P_{K})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{Y}_{J,\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{L_{K^{\prime}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{K^{\prime}})}}$$\textstyle{\frac{U_{P_{J}}\backslash
P_{K^{\prime}}\dot{w}^{\prime}P_{\delta(K^{\prime})}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{J}\cap
P_{K^{\prime}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}_{J,\delta}.}$

###### Proof.

It suffices to consider the case where $(w,K)\stackrel{{\scriptstyle
x}}{{\approx}}_{\delta}(w^{\prime},K^{\prime})$ for some $x\in W_{J}$. Set
$y=x^{-1}w$. Then $w^{\prime}=y\delta(x)$. We have

$P_{K}\dot{w}P_{\delta(K)}=U_{P_{K}}\dot{w}P_{\delta(K)}\cong(U_{P_{K}}\cap\dot{w}U^{-}\dot{w}^{-1})\times
P_{\delta(K)}.$

Moreover,
$U_{P_{K}}\cap\dot{w}U^{-}\dot{w}^{-1}=U\cap\dot{w}U^{-}\dot{w}^{-1}$. Note
that
${\mathrm{Ad}}(y)\delta(K)={\mathrm{Ad}}(x^{-1}w)\delta(K)={\mathrm{Ad}}(x)^{-1}(K)=K^{\prime}$.
Similarly, we have

$\displaystyle
P_{K}\dot{x}P_{K^{\prime}}\cong(U\cap\dot{x}U^{-}\dot{x}^{-1})\times
P_{K^{\prime}};$ $\displaystyle
P_{K^{\prime}}\dot{y}P_{\delta(K)}\cong(U\cap\dot{y}U^{-}\dot{y}^{-1})\times
P_{\delta(K)}.$

Since $\ell(w)=\ell(x)+\ell(y)$, we have
$U\cap\dot{w}U^{-}\dot{w}^{-1}\cong(U\cap\dot{x}U^{-}\dot{x}^{-1})\times(U\cap\dot{y}U^{-}\dot{y}^{-1})$.
Hence

$\displaystyle P_{K}\dot{w}P_{\delta(K)}$ $\displaystyle\cong
U\cap\dot{w}U^{-}\dot{w}^{-1}\times P_{\delta(K)}$
$\displaystyle\cong(U\cap\dot{x}U^{-}\dot{x}^{-1})\times(U\cap\dot{y}U^{-}\dot{y}^{-1})\times
P_{\delta(K)}$ $\displaystyle\cong(U\cap\dot{x}U^{-}\dot{x}^{-1})\times
P_{K^{\prime}}\dot{y}P_{\delta(K)}$
$\displaystyle\cong\bigl{(}(U\cap\dot{x}U^{-}\dot{x}^{-1})\times
P_{K^{\prime}}\bigr{)}\times^{P_{K^{\prime}}}P_{K^{\prime}}\dot{y}P_{\delta(K)}$
$\displaystyle\cong
P_{K}\dot{x}P_{K^{\prime}}\times^{P_{K^{\prime}}}P_{K^{\prime}}\dot{y}P_{\delta(K)}.$

The map $(g_{1},g_{2})(g_{2},\delta(g_{1}))$ gives a natural isomorphism

$f:\frac{P_{K}\dot{x}P_{K^{\prime}}\times^{P_{K^{\prime}}}P_{K^{\prime}}\dot{y}P_{\delta(K)}}{{\mathrm{Ad}}_{\delta}(P_{K})}\cong\frac{P_{K^{\prime}}\dot{y}P_{\delta(K)}\times^{P_{\delta(K)}}P_{\delta(K)}\delta(\dot{x})P_{\delta(K^{\prime})}}{{\mathrm{Ad}}_{\delta}(P_{K^{\prime}})}.$

Since $K,K^{\prime}\subset J$ and $x\in W_{J}$, we have
$P_{K}\dot{x}P_{K^{\prime}}\subset P_{K}$ and hence the conjugation action of
$P_{K}\dot{x}P_{K^{\prime}}$ normalizes $U_{P_{J}}$. Thus $f$ induces the
desired isomorphism $\frac{U_{P_{J}}\backslash
P_{K}\dot{w}P_{\delta(K)}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\delta}(L_{J}\cap
P_{K})}\cong\frac{U_{P_{J}}\backslash
P_{K^{\prime}}\dot{w}^{\prime}P_{\delta(K^{\prime})}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\delta}(L_{J}\cap
P_{K^{\prime}})}$.

Note that the representative of $w^{\prime}$ in $N_{G}(T)$ is unique up to
right multiplication by $T$. As we consider the morphisms on the quotient
stacks, we may assume furthermore that $\dot{w}^{\prime}$ is chosen so that
$\dot{w}^{\prime}=\dot{x}^{-1}\dot{w}\delta(\dot{x})$. Then
${\mathrm{Ad}}(\dot{x})^{-1}\circ{\mathrm{Ad}}(\dot{w})\circ\delta={\mathrm{Ad}}(\dot{w}^{\prime})\circ\delta\circ{\mathrm{Ad}}(\dot{x})^{-1}$
on $L_{K}$. The desired isomorphism
$\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})}\cong\frac{L_{K^{\prime}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{K^{\prime}})}$
is induced from the conjugation action by $\dot{x}^{-1}$. ∎

######  Theorem 3.2.

Let $J\subset S$ and $(w,K)$ be a $\delta$-combinatorial piece. Suppose that
$K\subset J$ and $w$ is of minimal length in $W_{J}\cdot_{\delta}w$. Then
there exists unique $w^{\prime}\in{}^{J}W$, $x\in W_{J}\cap
W^{I(J,w^{\prime},\delta)}$, $u\in W_{I(J,w^{\prime},\delta)}$ such that
$x^{-1}w\delta(x)=uw^{\prime}$, ${\mathrm{Ad}}(x)^{-1}(K)\subset J$ and
$(w,K)\approx_{J,\delta}(uw^{\prime},{\mathrm{Ad}}(x)^{-1}(K))$.

###### Proof.

By Proposition 2.2 (1), there exists a unique $w^{\prime}\in{}^{J}W$ such that
$w\in W_{J}\cdot_{\delta}W_{I(J,w^{\prime},\delta)}w^{\prime}$. By Proposition
2.2 (2), there exists a unique $x\in W_{J}\cap W^{I(J,w^{\prime},\delta)}$
such that $x^{-1}w\delta(x)\in W_{I(J,w^{\prime},\delta)}w^{\prime}$. Let
$u\in W_{I(J,w^{\prime},\delta)}$ with $x^{-1}w\delta(x)=uw^{\prime}$.

By Lemma 2.1,

$\displaystyle W_{I(J,w^{\prime},\delta)}$
$\displaystyle=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w^{\prime})\circ\delta)^{n}(W_{J})=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(uw^{\prime})\circ\delta)^{n}(W_{J})$
$\displaystyle=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(x)^{-1}{\mathrm{Ad}}(w)\circ\delta\circ{\mathrm{Ad}}(x))^{n}(W_{J})$
$\displaystyle=\cap_{n\in\mathbb{Z}}{\mathrm{Ad}}(x)^{-1}({\mathrm{Ad}}(w)\circ\delta)^{n}{\mathrm{Ad}}(x)(W_{J})$
$\displaystyle={\mathrm{Ad}}(x)^{-1}\bigl{(}\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})\bigr{)}$
$\displaystyle\supset{\mathrm{Ad}}(x)^{-1}(W_{K}).$

The second equality uses the fact that
$({\mathrm{Ad}}(uw^{\prime})\circ\delta)^{n}=({\mathrm{Ad}}(w^{\prime})\circ\delta)^{n}{\mathrm{Ad}}(u^{\prime})$
for some $u^{\prime}\in W_{I(J,w^{\prime},\delta)}$.

Therefore $x^{-1}$ sends any simple root in $K$ to a root spanned by
$I(J,w^{\prime},\delta)$. Since $x\in W^{I(J,w^{\prime},\delta)}$, we have
${\mathrm{Ad}}(x)^{-1}(K)\subset I(J,w^{\prime},\delta)$. Let
$K^{\prime}={\mathrm{Ad}}(x)^{-1}(K)$. Then

$\displaystyle{\mathrm{Ad}}(uw^{\prime})\delta(K^{\prime})$
$\displaystyle={\mathrm{Ad}}(x)^{-1}{\mathrm{Ad}}(w)\delta{\mathrm{Ad}}(x)(K^{\prime})={\mathrm{Ad}}(x)^{-1}{\mathrm{Ad}}(w)\delta(K)$
$\displaystyle={\mathrm{Ad}}(x)^{-1}(K)=K^{\prime}.$

Thus $(uw^{\prime},K^{\prime})$ is a $\delta$-combinatorial piece. It remains
to prove that $(w,K)\approx_{J,\delta}(uw^{\prime},K^{\prime})$.

We associate a quadruple $(J_{n},w_{n},x_{n},y_{n})_{n\geqslant 0}$ to $w$.
Here $J_{n}\subset J$, $w_{n}\in{}^{J_{n}}W^{\delta(J_{n})}$, $x_{n}\in
W_{J_{n}}\cap W^{J_{n+1}}$ and $y_{n}\in W_{J_{n+1}}$ for all $n$. The
quadruple is constructed as follows.

Set $J_{0}=J$. Suppose that $n\geqslant 0$ and $(J_{m},x_{m})$ are defined for
$0\leqslant m<n$. We set $w^{\prime}_{n}=(x_{0}\cdots
x_{n-1})^{-1}w\delta(x_{0}\cdots x_{n-1})$. We may write $w^{\prime}_{n}$ as
$w^{\prime}_{n}=z_{n}w^{\prime\prime}_{n}$, where
$w^{\prime\prime}_{n}\in{}^{J_{n}}W$ and $z_{n}\in W_{J_{n}}$. Set
$w_{n}=\min(w^{\prime\prime}_{n}W_{\delta(J_{n})})$ and
$J_{n+1}=J_{n}\cap{\mathrm{Ad}}(w_{n})\delta(J_{n})$. We write $z_{n}$ as
$z_{n}=x_{n}y_{n}$, where $x_{n}\in W_{J_{n}}\cap W^{J_{n+1}}$ and $y_{n}\in
W_{J_{n+1}}$. The quadruple $(J_{n},w_{n},x_{n},y_{n})_{n\geqslant 0}$ is
defined inductively.

By definition, $(J_{n},w_{n})_{n\geqslant 0}\in\mathcal{T}(J,\delta)$. Then
there exists $m$ such that $J_{m}=J_{m+1}=\cdots$, $w_{m}=w_{m+1}=\cdots$. We
then have $x_{m}=x_{m+1}=\cdots=1$ and $y_{m}=y_{m+1}=\cdots$. Moreover,
$J_{m}=I(J,w_{m},\delta)$. Set $x^{\prime}=x_{0}x_{1}\cdots x_{m}\in W_{J}\cap
W^{I(J,w_{m},\delta)}$. Then $(x^{\prime})^{-1}wx^{\prime}\in
W_{J_{m}}w_{m}=W_{I(J,w_{m},\delta)}w_{m}$. By Proposition 2.2, we have
$w_{m}=w^{\prime}$ and $x^{\prime}=x$.

Moreover, we have
$\ell(w^{\prime}_{n})=\ell(w^{\prime\prime}_{n})+\ell(z_{n})=\ell(w^{\prime\prime}_{n})+\ell(x_{n})+\ell(y_{n})=\ell(y_{n}w^{\prime\prime}_{n})+\ell(x_{n})$.
Thus

$\displaystyle\ell(w^{\prime}_{n+1})$
$\displaystyle=\ell(x_{n}^{-1}w^{\prime}_{n}\delta(x_{n}))\leqslant\ell(x_{n}^{-1}w^{\prime}_{n})+\ell(x_{n})=\ell(y_{n}w^{\prime\prime}_{n})+\ell(x_{n})$
$\displaystyle=\ell(w^{\prime}_{n}).$

In particular, we have
$\ell(w)=\ell(w^{\prime}_{0})\geqslant\ell(w^{\prime}_{1})\geqslant\cdots\geqslant\ell(w^{\prime}_{m})=\ell(x^{-1}w\delta(x))$.
By our assumption, $w$ is of minimal length in $W_{J}\cdot_{\delta}w$. Hence
$\ell(w)=\ell(w^{\prime}_{1})=\cdots=\ell(w^{\prime}_{m})=\ell(x^{-1}w\delta(x))$.
Moreover, we have
$w=w^{\prime}_{0}\approx_{J,\delta}w^{\prime}_{1}\approx_{J,\delta}\cdots\approx_{J,\delta}w^{\prime}_{m}=x^{-1}w\delta(x)$.

Note that ${\mathrm{Ad}}(x_{0})^{-1}(W_{K})={\mathrm{Ad}}(x_{1}\cdots
x_{m})W_{K^{\prime}}\subset W_{J_{1}}$. Since $x_{0}\in W^{J_{1}}$, we have
${\mathrm{Ad}}(x_{0})^{-1}(K)\subset J_{1}$. By the same argument, we may show
that ${\mathrm{Ad}}(x_{0}\cdots x_{i})^{-1}(K)\subset J_{i+1}$ for all
$i\geqslant 0$.

Hence we have $(w,K)\stackrel{{\scriptstyle
x_{0}}}{{\approx}}_{\delta}(w^{\prime}_{1},{\mathrm{Ad}}(x_{0})^{-1}(K))\stackrel{{\scriptstyle
x_{1}}}{{\approx}}_{\delta}\cdots\stackrel{{\scriptstyle
x_{m}}}{{\approx}}_{\delta}(uw^{\prime},K^{\prime})$. ∎

##  4\. Applications

###  4.1. Left-right symmetry

Recall that

$W=\bigsqcup_{w\in{}^{J}W}W_{J}\cdot_{\delta}(W_{I(J,w,\delta)}w).$

By [5, §2], we have the following decomposition of $W$ indexed by
$W^{\delta(J)}$ instead of ${}^{J}W$:

$W=\bigsqcup_{w\in W^{\delta(J)}}W_{J}\cdot_{\delta}(W_{I(J,w,\delta)}w).$

Now we show that there exists a natural bijection between ${}^{J}W$ and
$W^{\delta(J)}$ so that the corresponding subsets of $W$ coincide.

######  Proposition 4.1.

Let $J\subset S$. Then there exists a unique bijection
$\iota:W^{\delta(J)}{}^{J}W$ such that

$\bigl{(}w,I(J,w,\delta)\bigr{)}\approx_{J,\delta}\bigl{(}\iota(w),I(J,\iota(w),\delta)\bigr{)}.$

In particular,
$W_{J}\cdot_{\delta}(W_{I(J,w,\delta)}w)=W_{J}\cdot_{\delta}(W_{I(J,\iota(w),\delta)}\iota(w))$.

###### Proof.

Let $w\in W^{\delta(J)}$ and $K=I(J,w,\delta)$. By definition, $(w,K)$ is a
$\delta$-combinatorial piece. For any $u\in W_{J}$, we have

$\ell(uw\delta(u)^{-1})\geqslant\ell(w\delta(u)^{-1})-\ell(u)=\ell(w)+\ell(\delta(u))-\ell(u)=\ell(w).$

Therefore $w$ is of minimal length in $W_{J}\cdot_{\delta}w$. By Theorem 3.2,
there exists unique $w^{\prime}\in{}^{J}W$, $x\in W_{J}\cap
W^{I(J,w^{\prime},\delta)}$, $u\in W_{I(J,w^{\prime},\delta)}$ such that
$x^{-1}w\delta(x)=uw^{\prime}$, ${\mathrm{Ad}}(x)^{-1}(K)\subset J$ and
$(w,K)\approx_{J,\delta}(uw^{\prime},{\mathrm{Ad}}(x)^{-1}(K))$.

Set $K^{\prime}={\mathrm{Ad}}(x)^{-1}(K)$. By Lemma 2.1,
$W_{K}=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})$. Hence

$\displaystyle W_{K^{\prime}}$
$\displaystyle={\mathrm{Ad}}(x)^{-1}(W_{K})=\cap_{n\in\mathbb{Z}}{\mathrm{Ad}}(x)^{-1}({\mathrm{Ad}}(w)\circ\delta)^{n}(W_{J})$
$\displaystyle=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(uw^{\prime})\circ\delta)^{n}{\mathrm{Ad}}(x)^{-1}(W_{J})$
$\displaystyle=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(uw^{\prime})\circ\delta)^{n}(W_{J})$
$\displaystyle=\cap_{n\in\mathbb{Z}}({\mathrm{Ad}}(w^{\prime})\circ\delta)^{n}(W_{J})$
$\displaystyle=W_{I(J,w^{\prime},\delta)}.$

The second to last equality uses the fact that
$({\mathrm{Ad}}(uw^{\prime})\circ\delta)^{n}=({\mathrm{Ad}}(w^{\prime})\circ\delta)^{n}{\mathrm{Ad}}(u^{\prime})$
for some $u^{\prime}\in W_{I(J,w^{\prime},\delta)}$.

Therefore $K^{\prime}=I(J,w^{\prime},\delta)$. Hence
$xuw^{\prime}=w\delta(x)\in W^{\delta(I(J,w^{\prime},\delta)}$. This implies
that $u=1$. So $(w,K)\approx_{J,\delta}(w^{\prime},K^{\prime})$. By the
definition of cyclic shifts of combinatorial pieces, we also have
$W_{J}\cdot_{\delta}(W_{K}w)=W_{J}\cdot_{\delta}(W_{K^{\prime}}w^{\prime})$. ∎

Let $w\in W^{\delta(J)}$ and $w^{\prime}=\iota(w)\in{}^{J}W$. Since
$w\approx_{J,\delta}w^{\prime}$, we have
$\mathcal{Y}_{J,\delta;w}=\mathcal{Y}_{J,\delta;w^{\prime}}$. Moreover, by
Proposition 3.1, we have the following commutative diagram

$\textstyle{\mathrm{Sh}(\frac{L_{I(J,w,\delta)}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{I(J,w,\delta)})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{Ad}}(\dot{x})^{-1}}$$\scriptstyle{\cong}$$\textstyle{\mathrm{Sh}(\mathcal{Y}_{J,\delta;w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sh}(\frac{L_{I(J,w^{\prime},\delta)}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{I(J,w^{\prime},\delta)})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{Sh}(\mathcal{Y}_{J,\delta;w^{\prime}}),}$

where $x$ is the unique element in $W_{J}\cap W^{I(J,w^{\prime},\delta)}$ such
that $w^{\prime}=x^{-1}w\delta(x)$.

Now we provide a nontrivial example of the map $\iota$.

######  Example 4.2.

Let $W=S_{5}$, $J=\\{1,3\\}$ and $\delta$ is the identity map. Then one may
check that $\iota(s_{121324})=s_{213243}$. In particular, the map $\iota$ is
different from taking inverse, and different from the one-step operation
$xyyx$ for $x\in W_{J}$ and $y\in{}^{J}W$.

###  4.2. The functors $f^{J}_{J^{\prime}}$

We follow [8, §6]. Let $J\subset J^{\prime}\subset S$. Let
$\pi:\mathcal{P}_{J}\mathcal{P}_{J^{\prime}}$ be the projection map. Set

$Z_{J,J^{\prime},\delta}=\\{(P,P^{\prime},g\delta(U_{\pi(P)}));P\in\mathcal{P}_{J},P^{\prime}\in\mathcal{P}_{\delta(J)},{}^{g}\delta(P)=P^{\prime}\\}.$

Define the action of $G\times G$ on $Z_{J,J^{\prime},\delta}$ by
$(g_{1},g_{2})\cdot(P,P^{\prime},gU_{\pi(P)})=({}^{g_{2}}P,{}^{g_{1}}P^{\prime},g_{1}g\delta(g_{2})^{-1}\delta(U_{\pi({}^{g_{2}}P)}))$.
Then $G\times G$ acts transitively on $Z_{J,J^{\prime},\delta}$. Let
$h_{J,J^{\prime},\delta}=(P_{J},P_{\delta(J)},U_{P_{\delta(J^{\prime})}})$ be
the base point. Then we may identify $Z_{J,J^{\prime},\delta}$ with $(G\times
G)/(U_{P_{J^{\prime}}}\times
U_{P_{\delta(J^{\prime})}}){\mathrm{Ad}}_{\delta}(L_{J^{\prime}}\cap P_{J})$,
where $L_{J^{\prime}}\cap P_{J}$ is a standard parabolic subgroup of
$L_{J^{\prime}}$.

We consider the following diagram

$\textstyle{Z_{J,\delta}}$$\textstyle{Z_{J,J^{\prime},\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{d}$$\textstyle{Z_{J^{\prime},\delta},}$

where

$\displaystyle
c(P,P^{\prime},g\delta(U_{\pi(P)}))=(P,P^{\prime},g\delta(U_{P})),$
$\displaystyle
d(P,P^{\prime},g\delta(U_{\pi(P)}))=(\pi(P),\pi(P^{\prime}),g\delta(U_{\pi(P)})).$

It is easy to see that $c$ is a locally trivial fibration with fibers
isomorphic to an affine space $U_{P_{J}}/U_{P_{J^{\prime}}}$ and $d$ is a
locally trivial fibration with fibers isomorphic to the flag variety
$L_{J^{\prime}}/(L_{J^{\prime}}\cap P_{J})$. Set

$\displaystyle
f^{J}_{J^{\prime}}=d_{!}c^{*}:\mathrm{Sh}(\frac{Z_{J,\delta}}{\Delta
G})\mathrm{Sh}(\frac{Z_{J^{\prime},\delta}}{\Delta G}).$

We may reformulate the functors as follows. Consider the following diagram of
stacks

$\textstyle{\mathcal{Y}_{J,\delta}}$$\textstyle{\mathcal{Y}_{J,J^{\prime},\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{\mathcal{Y}_{J^{\prime},\delta}}$

where $\mathcal{Y}_{J,J^{\prime},\delta}=\frac{U_{P_{J^{\prime}}}\backslash
G/U_{P_{\delta(J^{\prime})}}}{{\mathrm{Ad}}_{\delta}(L_{J^{\prime}}\cap
P_{J})}$ and $q$ and $p$ are natural projection maps. Under the isomorphism
$\frac{Z_{J,\delta}}{\Delta(G)}\cong\mathcal{Y}_{J,\delta}$ and
$\frac{Z_{J^{\prime},\delta}}{\Delta(G)}\cong\mathcal{Y}_{J^{\prime},\delta}$,
we may rewrite the functor $f^{J}_{J^{\prime}}$ as

$\displaystyle
f^{J}_{J^{\prime}}=p_{!}q^{*}:\mathrm{Sh}(\mathcal{Y}_{J,\delta})\mathrm{Sh}(\mathcal{Y}_{J^{\prime},\delta}).$

Let $w\in{}^{J}W$ and $w^{\prime}\in{}^{J^{\prime}}W$. Let
$i_{J,\delta;w}:\mathcal{Y}_{J,\delta;w}\mathcal{Y}_{J,\delta}$ and
$i_{J^{\prime},\delta;w^{\prime}}:\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}\mathcal{Y}_{J^{\prime},\delta}$
be the embeddings. We define

$\displaystyle
f^{J,w}_{J^{\prime},w^{\prime}}=i_{J^{\prime},\delta;w^{\prime}}^{*}\,p_{!}q^{*}(i_{J,\delta;w})_{!}:\mathrm{Sh}(\mathcal{Y}_{J,\delta;w})\mathrm{Sh}(\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}).$

By definition, for any $w\in{}^{J}W$,
$p(q^{-1}(\mathcal{Y}_{J,\delta;w}))=\mathcal{Y}_{J^{\prime},\delta;w}$. Hence
by §2.5 (c), we have

(a) Let $w\in{}^{J}W$ and $A$ be a parabolic character sheaf on
$\mathcal{Y}_{J,\delta}$ with support in
$\overline{\mathcal{Y}_{J,\delta;w}}$. Assume that
$W_{J^{\prime}}\cdot_{\delta}w\cap{}^{J^{\prime}}W=\emptyset$ or $w$ is not of
minimal length in $W_{J^{\prime}}\cdot_{\delta}w$. Then the support of
$f^{J}_{J^{\prime}}(A)$ is contained in
$\sqcup_{w^{\prime}\in{}^{J^{\prime}}W;\ell(w^{\prime})<\ell(w)}\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}$.

###  4.3. Induction functor

Let $(w,K)$ be a $\delta$-combinatorial piece. Consider the following diagram

$\textstyle{\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})}\cong\frac{L_{K}\dot{w}}{{\mathrm{Ad}}_{\delta}(L_{K})}}$$\textstyle{\frac{P_{K}\dot{w}P_{\delta(K)}}{{\mathrm{Ad}}_{\delta}(P_{K})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{\frac{G}{{\mathrm{Ad}}_{\delta}(G)},}$

where $a$ is induced from the projection map
$P_{K}\dot{w}P_{\delta(K)}L_{K}\dot{w}$ and $b$ is induced from the embedding
$P_{K}\dot{w}P_{\delta(K)}G$.

Following [12, §4.1], we define

$\displaystyle{\mathrm{Ind}}^{(G,\delta)}_{(L_{K},\dot{w})}=b_{!}a^{*}:\mathrm{Sh}(\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})})\mathrm{Sh}(\frac{G}{{\mathrm{Ad}}_{\delta}(G)}).$

If $w=1$, then $K=\delta(K)$ and $P_{K}$ is a $\delta$-stable standard
parabolic subgroup of $G$. In this case, ${\mathrm{Ind}}$ is the Harish-
Chandra induction functor. If $K=\emptyset$, then $P_{K}=B$ and
${\mathrm{Ind}}$ is the Deligne-Lusztig induction functor.

Now we prove that

######  Theorem 4.3.

Let $J\subset J^{\prime}\subset S$. Let $w\in{}^{J}W$ such that $w$ is of
minimal length in $W_{J}\cdot_{\delta}w$. Let $w^{\prime}\in{}^{J^{\prime}}W$,
$u\in W_{I(J^{\prime},w^{\prime},\delta)}$ and $x\in W_{J^{\prime}}\cap
W^{I(J^{\prime},w^{\prime},\delta)}$ such that $x^{-1}w\delta(x)=uw^{\prime}$.
Set $K=I(J,w,\delta)$, $K_{1}={\mathrm{Ad}}(x)^{-1}(K)$,
$K^{\prime}=I(J^{\prime},w^{\prime},\delta)$ and
$\delta^{\prime}={\mathrm{Ad}}(\dot{w}^{\prime})\circ\delta$. Then we have the
following commutative diagram

$\textstyle{\mathrm{Sh}(\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{Ind}}^{(L_{K^{\prime}},\delta^{\prime})}_{(L_{K_{1}},\dot{u})}\circ{\mathrm{Ad}}(\dot{x})^{-1}}$$\scriptstyle{\pi^{*}_{J,\delta;w}}$$\scriptstyle{\cong}$$\textstyle{\mathrm{Sh}(\mathcal{Y}_{J,\delta;w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{J,w}_{J^{\prime},w^{\prime}}}$$\textstyle{\mathrm{Sh}(\frac{L_{K^{\prime}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{K^{\prime}})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}_{J^{\prime},\delta;w^{\prime}}}$$\scriptstyle{\cong}$$\textstyle{\mathrm{Sh}(\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}).}$

###### Proof.

We have the following commutative diagram

$\textstyle{\frac{L_{K}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{K})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{Ad}}(\dot{x})^{-1}}$$\scriptstyle{\cong}$$\textstyle{\mathcal{Y}_{J,\delta;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{J,\delta;w}}$$\scriptstyle{i_{J,\delta;w}}$$\scriptstyle{\square}$$\textstyle{\mathcal{Y}_{J,\delta}}$$\textstyle{q^{-1}(\mathcal{Y}_{J,\delta;w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{f}$$\scriptstyle{\cong}$$\scriptstyle{i}$$\textstyle{\mathcal{Y}_{J,J^{\prime},\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{\frac{L_{K_{1}}}{{\mathrm{Ad}}_{\dot{u}\circ\delta^{\prime}}(L_{K_{1}})}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\frac{(L_{K^{\prime}}\cap
P_{K_{1}})\dot{u}\delta^{\prime}(L_{K^{\prime}}\cap
P_{K_{1}})}{{\mathrm{Ad}}_{\delta^{\prime}}(L_{K^{\prime}}\cap
P_{K_{1}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{\square}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{b^{\prime}}$$\textstyle{\frac{L_{K^{\prime}}}{{\mathrm{Ad}}_{\delta^{\prime}}(L_{K^{\prime}})}}$$\textstyle{\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{J^{\prime},\delta;w^{\prime}}}$$\scriptstyle{i_{J^{\prime},\delta;w^{\prime}}}$$\textstyle{\mathcal{Y}_{J^{\prime},\delta}.}$

Here $X=\frac{U_{P_{J^{\prime}}}\backslash
P_{K_{1}}\dot{u}\dot{w}^{\prime}P_{\delta(K_{1})}/U_{P_{\delta(J^{\prime})}}}{{\mathrm{Ad}}_{\dot{u}\dot{w}^{\prime}\circ\delta}(L_{J^{\prime}}\cap
P_{K_{1}})}$.

By Proposition 2.4, $\mathcal{Y}_{J,\delta;w}\cong\frac{U_{P_{J}}\backslash
P_{K}\dot{w}P_{\delta(K)}/U_{P_{\delta(J)}}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{J}\cap
P_{K})}$. Thus by definition
$q^{-1}(\mathcal{Y}_{J,\delta;w})\cong\frac{U_{P_{J^{\prime}}}\backslash
P_{K}\dot{w}P_{\delta(K)}/U_{P_{\delta(J^{\prime})}}}{{\mathrm{Ad}}_{\dot{w}\circ\delta}(L_{J^{\prime}}\cap
P_{K})}$. By Theorem 3.2, $(w,K)\approx_{J,\delta}(uw^{\prime},K_{1})$. The
isomorphism $f:q^{-1}(\mathcal{Y}_{J,\delta;w})X_{1}$ is given in Proposition
3.1. The map
$\pi_{1}:X\frac{L_{K_{1}}}{{\mathrm{Ad}}_{\dot{u}\circ\delta^{\prime}}(L_{K_{1}})}$
is induced from the projection map
$P_{K_{1}}\dot{u}\dot{w}^{\prime}P_{\delta(K_{1})}L_{K_{1}}\dot{u}\dot{w}^{\prime}$.
The map $\pi_{2}:X\frac{(L_{K^{\prime}}\cap
P_{K_{1}})\dot{u}\delta^{\prime}(L_{K^{\prime}}\cap
P_{K_{1}})}{{\mathrm{Ad}}_{\delta^{\prime}}(L_{K^{\prime}}\cap P_{K_{1}})}$ is
induced from the projection

$P_{K_{1}}\dot{u}\dot{w}^{\prime}P_{\delta(K_{1})}P_{K_{1}}\dot{u}\dot{w}^{\prime}P_{\delta(K_{1})}\cap
L_{K^{\prime}}\dot{w}^{\prime}=(L_{K^{\prime}}\cap
P_{K_{1}})\dot{u}\delta^{\prime}(L_{K^{\prime}}\cap P_{K_{1}}).$

By Proposition 2.4,
$\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}\cong\frac{U_{P_{J^{\prime}}}\backslash
P_{K^{\prime}}\dot{w}^{\prime}P_{\delta(K^{\prime})}/U_{P_{\delta(J^{\prime})}}}{{\mathrm{Ad}}_{\dot{w}^{\prime}\circ\delta}(L_{J^{\prime}}\cap
P_{K^{\prime}})}$. The map
$b^{\prime}:X\mathcal{Y}_{J^{\prime},\delta;w^{\prime}}$ is induced from the
embedding $P_{K_{1}}\dot{u}\dot{w}^{\prime}P_{\delta(K_{1})}\subset
P_{K^{\prime}}\dot{w}^{\prime}P_{\delta(K^{\prime})}$.

Thus

$\displaystyle f^{J,w}_{J^{\prime},w^{\prime}}\pi^{*}_{J,\delta;w}$
$\displaystyle=i_{J^{\prime},\delta;w^{\prime}}^{*}p_{!}q^{*}(i_{J,\delta;w})_{!}\pi^{*}_{J,w,\delta}=i_{J^{\prime},\delta;w^{\prime}}^{*}p_{!}i_{!}q^{*}\pi^{*}_{J,w,\delta}$
$\displaystyle=(b^{\prime})_{!}f_{!}q^{*}\pi^{*}_{J,w,\delta}=(b^{\prime})_{!}\pi_{1}^{*}{\mathrm{Ad}}(\dot{x})^{-1}$
$\displaystyle=(b^{\prime})_{!}\pi_{2}^{*}a^{*}{\mathrm{Ad}}(\dot{x})^{-1}=\pi_{J^{\prime},\delta;w^{\prime}}^{*}b_{!}a^{*}{\mathrm{Ad}}(\dot{x})^{-1}$
$\displaystyle=\pi_{J^{\prime},\delta;w^{\prime}}^{*}{\mathrm{Ind}}^{(L_{K^{\prime}},\delta^{\prime})}_{(L_{K_{1}},\dot{u})}\circ{\mathrm{Ad}}(\dot{x})^{-1}.$

The proof is finished. ∎

###  4.4. A special case

Now we discuss a special case of Theorem 4.3.

Let $J\subset J^{\prime}\subset S$. Let $w\in{}^{J}W$ and
$w^{\prime}\in{}^{J^{\prime}}W$ such that $w\in
W_{J^{\prime}}\cdot_{\delta}w^{\prime}$ and $\ell(w)=\ell(w^{\prime})$. Set
$K=I(J,w,\delta)$ and $K^{\prime}=I(J^{\prime},w^{\prime},\delta)$. By Theorem
3.2, there exists a unique $x\in W_{J^{\prime}}\cap W^{K^{\prime}}$ such that
$x^{-1}w\delta(x)=w^{\prime}$ and $(w,K)\approx_{J,\delta}(w^{\prime},K_{1})$,
where $K_{1}={\mathrm{Ad}}(x)^{-1}(K)\subset K^{\prime}$. Since
${\mathrm{Ad}}(w)\delta(K)=K$, we have
${\mathrm{Ad}}(w^{\prime})\delta(K_{1})=K_{1}$. Set
$\delta^{\prime}={\mathrm{Ad}}(\dot{w}^{\prime})\circ\delta$. Then $L_{K_{1}}$
is a $\delta^{\prime}$-stable Levi subgroup and $L_{K^{\prime}}\cap P_{K_{1}}$
is a $\delta^{\prime}$-stable parabolic subgroup of $L_{K^{\prime}}$. The
functor
${\mathrm{Ind}}^{(L_{K^{\prime}},\delta^{\prime})}_{(L_{K_{1}},\dot{u})}$ in
Theorem 4.3 in this case is the Harish-Chandra induction functor.

In this case, the functors $f^{J,w}_{J^{\prime},w^{\prime}}$ is just the
Harish-Chandra induction functor, composed with the conjugation by
$\dot{x}^{-1}$.

###  4.5. Generalization to loop groups

Theorem 3.2 holds not only for Weyl groups but also for arbitrary Coxeter
groups. The same proof works in such generality. In particular, Theorem 3.2
holds for any affine Weyl group together with a length-preserving automorphism
on it.

Let $\mathbf{k}$ be an algebraically closed field and
$F=\mathbf{k}((\epsilon))$ be the field of the formal Laurent series. Let
$\mathbf{G}$ be a connected reductive group over $F$ and $G=\mathbf{G}(F)$.
Let $I$ be an Iwahori subgroup of $G$ and $\tilde{W}$ be the Iwahori-Weyl
group of $G$. Let $P\supset I$ be a parahoric subgroup. It is a pro-algebraic
group. Let $U_{P}$ be the pro-unipotent radical of $P$ and $L\cong P/U_{P}$ be
its reductive quotient. In [9], Lusztig introduced the affine analogy of the
parabolic character sheaves. They are certain simple perverse sheaves on the
ind-stack $\frac{U_{P}\backslash G/U_{P}}{\Delta(L)}$.

Note that the Iwahori-Weyl group $\tilde{W}$ is not a Coxeter group in
general, but a quasi-Coxeter group. Namely, let $G^{0}$ be the subgroup of $G$
generated by all the parahoric subgroups. The Iwahori-Weyl group of $G^{0}$ is
an affine Weyl group $W_{a}$. And $\tilde{W}=W_{a}\rtimes\Omega$, where
$\Omega$ is the subgroup of length-zero elements of $\tilde{W}$. The
conjugation action of any element in $\Omega$ on $W_{a}$ is a length-
preserving automorphism. We have

$\frac{U_{P}\backslash
G/U_{P}}{\Delta(L)}=\bigsqcup_{\tau\in\Omega}\frac{U_{P}\backslash
G^{0}\dot{\tau}/U_{P}}{\Delta(L)}\cong\bigsqcup_{\tau\in\Omega}\frac{U_{P}\backslash
G^{0}/U_{\delta_{\dot{\tau}}(P)}}{{\mathrm{Ad}}_{\delta_{\dot{\tau}}}(L)}.$

Here $\delta_{\dot{\tau}}$ is the automorphism on $G^{0}$ given by the
conjugation action of $\dot{\tau}$. Now Theorem 4.3 may be applied to
$\frac{U_{P}\backslash
G^{0}/U_{\delta_{\dot{\tau}}(P)}}{{\mathrm{Ad}}_{\delta_{\dot{\tau}}}(L)}$
using Theorem 3.2 for the pair $(W_{a},{\mathrm{Ad}}(\tau))$.

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