A manager unless has a core competency to have their teams and people fully engaged, supporting, productive and efficient: give their best, they will end up over burdened with undue responsibility and drain themselves of energy.
Coach approach imparts Managers an insight to keep their team members make self-discoveries, be driven by purpose and strive for excellence. They help them to champion their own development plan and generate rewarding choices. This happens as Managers use their power to interact targeting Shift at the InnerMost level.
Focused Coaching Sessions : This requires them to phase weekly, bi-weekly or monthly sessions. They take special time out and encourage their people to bring their agenda and work out very best solutions in focus coaching sessions.
Making Coaching Conversations : This does not require them to take time out and have exclusive coaching sessions. They adopt coach approach in their everyday work interactions and become popular for their out of the box behaviors.
Either ways Managers as Coaches become enablers, facilitators and catalysts for creating aappreciative, resource empowering coaching culture in their organizations.
They create such a profound culture because they get specially trained in ICF laid coaching competencies and proven model of coaching focused to create InnerMost Shift as follow.
Purpose : Decide what purpose will it serve what they are about to talk or ask.
Intention : Discover and clarify intention; align behaviors with purposeful interaction.
Outcome : Clearly know what would they accomplish and get delivered.
Listen deeply : Completely listen, give full ears and attention to non-verbal expressions.
Question : Ask wise and thoughtful and focused questions.
Navigate : Keep communication on purpose and does not deviate.
Encourage : Assist to come out with the very best ideas and the best solutions.
Action Commitment : Have others commit to take action and accountable.
Track Progress : Ensure continuous action in the right direction and take responsibility.
Deliver Results : Bring their people to whole heartedly deliver results.
All this happens as Managers, while becoming coaches, are well trained in all of the above competencies. Because of Mangers orientation as coaches, they are able to help make their people InnerMost shift in the following core areas necessary for one's growth and development.
Planning : Draw practical and executable plan: decide steps and mile stones.
Performance : Help improving quality and quantum of performance.
Productivity : Assist in employees to exceed their limits to enhance productivity.
Progress : Encourage teams to grow, keeping personal goals aligned, with business goals.
What makes Managers as Coaches more in demand and very useful for the organisations is their skills and competency to optimise following area.
Innovate : Continuously innovate to upgrade company's product, services and support.
Time wise : Allocate and keep track of time spent to be wise and efficient.
Manage Conflicts : Immediately help people resolve differences and synergise.
Get Going : Going often gets tough; they keep people sailing through turbulences.
Stress Free : They are artful and competent to keep people at ease.
Irrespective of customers, suppliers, colleagues,bossed, team members or their own Managers, may be CEO's they are Emotionally wise and Intelligence to help people remain vibrant and full of energy.
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text:
EU welcomes normalisation of relations between Israel and Gulf States
New trams to run in Antwerp from Monday
The ceremony at the White House, 15 September, credit: Avi Ohayon (GPO)
A historic event took place at the White House in Washington on Tuesday when Israel signed a peace agreement with the United Arab Emirates (UAE) and a declaration on establishing diplomatic relations with Bahrain.
The agreements, signed between Israel's Prime Minister Benyamin Netanyahu and the foreign ministers of UAE and Bahrain with President Trump as witness, break Israel's isolation in the Middle East and opens the door for diplomatic relations with more Arab countries. They may even pave the way for the elusive peace between Israel and the Palestinians who for the time being feel betrayed by the two Gulf States.
In fact, this is the third peace treaty between Israel and an Arab state, following the treaties with Egypt in 1979 and Jordan in 1984. Israel and the Palestinian Authority were close to a comprehensive peace deal in 1999 but then Palestinian President Yasser Arafat did not get the support he needed from the Arab League to finalise the agreement on Jerusalem.
For the Israeli prime minister, the peace agreement with UAE vindicated his formula "peace for peace" but that is because Israel was never in war with UAE and Bahrein. To achieve peace with the Palestinians, Israel will have to apply the formula "peace for territories" and withdraw from most of the occupied territories.
This however did not cloud the ceremony at the White House. President Trump and Prime Minister Netanyahu took all credit for the agreements - that also serve their political agendas - but the agreements are steps in the right direction. Normalisation of relations between Israel and Arab countries is in the interest of the peoples concerned and serves the cause of peace in the Middle East.
A historic turning point
The Israeli-Emirati treaty is a full-fledged agreement covering a long list of cooperation areas. It will enter into force after it has been ratified by both parties. For Israel, it implies that the treaty will have to be approved by the government and ratified by the Israeli parliament (Knesset). Until the signature of the agreement, the text was kept a secret even for the government.
"This day is a pivot of history. It heralds a new dawn of peace," Netanyahu said at the ceremony and thanked President Trump. "He is already lining up more and more countries. This was unimaginable a few years ago, but with resolve, determination, a fresh look at the way peace is done, this is being achieved."
"The blessings of the peace we make today will be enormous," Netanyahu promised and listed three reasons for his optimism.
"First, because this peace will eventually expand to include other Arab states, and ultimately it can end the Arab-Israeli conflict once and for all. Second, because the great economic benefits of our partnership will be felt throughout our region, and they will reach every one of our citizens. And third, because this is not only a peace between leaders, it's a peace between peoples."
Both agreements mention the need for overall peace in the Middle East and a solution to the Israeli-Palestinian conflict. But such a solution can hardly be based on Trump's one-sided peace plan which has been rejected by the Palestinians.
The Israeli-Emirati peace agreement states in general terms that the two countries are "committed to working together to realize a negotiated solution to the Israeli-Palestinian conflict that meets the legitimate needs and aspirations of both peoples, and to advance comprehensive Middle East peace, stability and prosperity."
The joint declaration by Israel and Bahrain states that the two countries will continue "the efforts to achieve a just, comprehensive, and enduring resolution of the Israeli- Palestinian conflict."
Common future
Under the radar for years, Israel has developed relations with both UAE and Bahrain based on shared interests. Summarizing the economic importance of the agreements, Professor Efraim Inbar, President of the Jerusalem Institute for Strategy and Security, told The Brussels Times that, "The Emirates could become Israel's gate to Asia and Israel could attract investments from the UAE and Bahrain."
Politically, Inbar did not exclude that Israel might have to pay a price for the agreements in the form of American sales of advanced combat aircraft to UAE. Netanyahu denies it but Trump has already hinted that this is his intention. "Possibly the UAE will end up with F-35s but Israel will be compensated to keep its qualitative military edge."
Another issue was the Israeli government's plan for unilateral annexation – partly based on Trump's peace plan – of up to 30 % of the West Bank. A condition for the Gulf States signing the agreements was that Israel cancelled the plan. But according to Netanyahu the plan has only been suspended. Professor Inbar commented that annexation is still not definitely off the table.
A future Israeli-Palestinian peace settlement will have to be a mutually agreed compromise. Will the Palestinians accept that they cannot block the peace process between Israel and the Arab world and reconsider their own policy? "They'll have to adopt a more mature policy and understand that no side can realize all its dreams," Professor Inbar, the expert on Middle Eastern strategic issues, replied.
EU position
In a declaration on behalf of the EU (15 September), High Representative Josep Borell welcomed the normalisation of relations between Israel and the UAE, and acknowledged the "constructive role played by the US in this respect." The EU has for many years promoted the development of relations between Israel and the countries of the region, the declaration says.
"A normalisation of their bilateral relations will be beneficial to both countries and a fundamental step for the stabilisation of the region as a whole." EU also considers that Israel's commitment to suspend plans to unilaterally annex areas of the occupied Palestinian territory is a positive step and reiterated its commitment to a negotiated and viable two-state solution.
EU has kept a low profile in the normalisation process and was represented at the signing ceremony by the head of its delegation, ambassadors of some member states and the foreign minister of Hungary. Will EU use the momentum to resume the Israeli-Palestinian process?
"EU has consistently argued for the resumption of the talks and been stressing that recent developments are creating a momentum to restart the process," Peter Stano, lead spokesperson for EU foreign affairs, told The Brussels Times.
"We continue to be ready to help in any way we can," he said and mentioned planned contacts between High Representative Borrell and his counterparts on the Israeli and Palestinian sides.
M.Apelblat
The Brussels Times

redpajama_set_name: RedPajamaCommonCrawl


We'll work with you so you don't have to move, just improve. We can handle everything from design to delivery so you can sit back as we change your home to your dream home.
Have a growing family? Thinking of moving because you don't have enough space? Home additions are the perfect solution.
Whether an addition or a remodel, kitchens are the heart of the home and bathrooms are a home's sanctuary. We enjoy focusing on these high-impact areas to boost your satisfaction with your home and increase its value for you.


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text:
Home Community Joe Biden Attends This Year's "Blue Dream" Gala
Joe Biden Attends This Year's "Blue Dream" Gala
US Presidential hopeful Joe Biden addressing the audience at the Blue Dream event. Credit: https://www.bluedreamhamptons.org/
This year's "Blue Dream" charity event in Southampton, New York, took place on August 24, with hundreds of guests once again helping raise money for the Greek Orthodox Church of the Hamptons and a charitable cause of its choosing.
Distinguished members of the Greek-American community attended the gala as well as the newly-elected Greek Orthodox Archbishop of America, Elpidophoros.
Among the guests was also the former Vice President of the US, and the front-runner for the Democratic nomination for the 2020 election, Joe Biden of Delaware.
Biden has had a long-standing relationship with the Greek-American community; he is even a member of the distinguished "Order of the Lords" ("Tagma ton Archonton" in Greek) of the Ecumenical Patriarchate.
According to social media reports, Biden also told the guests jokingly that he would like to change his surname to "Bidenopoulos," in order to make it sound more Greek!
Organized by the Greek Orthodox Church of the Hamptons and the Greek-American community, the "Blue Dream" charity gala this year was held in support of Stony Brook Southampton Hospital.
As the organziers noted, "the Parish will donate 70% of this year's proceeds to the Stony Brook Southampton Hospital. All proceeds from the event will be shared between the Stony Brook Southampton Hospital and Greek Orthodox Church of the Hamptons.''
The event was completely sold out, with many thousands of dollars being raised for these worthy causes.
Calamos


"Real Housewives of New York" star Bethenny Frankel is speaking out about the sudden passing of her on-again/off-again boyfriend, Dennis Shields.
Frankel took to Instagram on Monday morning to share a photo of Shields laying beside her dog, Cookie, who passed away last year.
Frankel attended Shields' funeral service earlier on Monday in Long Island.
As The Blast previously reported, 51-year-old Shields died last Friday inside his apartment at Trump Tower in New York City. He suffered an apparent overdose and his death is currently being investigated by the medical examiner.


I run out of white nail polish really quickly because I use it so often in nail art, both as a base and accent/detail work. A good white polish is surprisingly hard to come by - OPI Alpine Snow and Sephora Formula X White Matter are my absolute favourites, but I'm not too keen on spending $10+ every couple months for a new one! Shoppers Drug Mart had Ceramic Glaze on sale for $3, so I snapped one up in Pure.
It's not quite on par with Alpine Snow or White Matter, but it's a pretty good white polish for the price. The brush is a little stiff and uneven though, making application not as effortless as the other two. However, the formula is smooth, opaque in two coats, and not too thick or gloppy - you'll be seeing this polish often in future posts!


The Lost Swimmer (Ann Turner) has been optioned by Film Art Media Pty Ltd to be made into a film. Ann Turner will write the screenplay and co-produce the film based on her critically acclaimed first novel published by Simon and Schuster in 2015.
"Ann Turner's debut novel has all the hallmarks for a brilliant suspense thriller and we are delighted to be working with her to adapt The Lost Swimmer into a film with international audience appeal," said producer Sue Maslin.
Ann and Sue have previously worked together on Irresistible starring Susan Sarandon, Sam Neill and Emily Blunt – where Sue was executive producer and Ann was writer/director. "It's a real coup to have Sue come on board, as she is such a talented and highly experienced producer," said Ann.


Your profile houses all the important information you will need on your journey through Suitest. If you are a member of an organization you will have at least two separate profiles; your personal account and organization account. You can switch in between them by clicking your profile and choosing your personal or organization account (organization name will be displayed here).
Suitest has two types of accounts; Personal and Organization. Everyone who uses Suitest has a personal account, which is purely your own. An Organization account is only created if you are invited to an organization or when you create an organization. Organization accounts are used mainly for collaboration, they allow multiple team members to work on applications or tests. Personal and organization accounts do not share subscriptions plans or applications (Tests, Elements, Test packs), therefore if you need to use both of the accounts then you would need to have two separate plans.
The summary page is a review of your current plan and especially the number of minutes that have been used within the 30 day period. Minutes are separated into Interactive and Automation.
If you have created or are a member of an organization then the team page will be available to you depending on your team's permissions. If you have created an organization or have the appropriate permissions, you can invite team members via email based on your plan's team size limit. You can view the team member list of your organization and individually set permissions.
Admin - has the right to read/write API Tokens, receives email notifications from Suitest billing.
Manage Devices - gives the ability to add, remove, enable, disable and share devices.
Manage Users - gives access to the Team tab for inviting and removing team members from the organization.
Access - permitting team members to be included in the organization.
We call CandyBoxes and SuitestDrives control units, as they are used to interface and control other devices.
The available control units page contains a list of all CandyBoxes and SuitestDrives belonging to a selected organization or personal account. Selecting a control unit allows you to view the list of devices which are configured with that particular control unit. The device list contains the status and assigned devices, you can also quickly edit a devices details if needed.
Important: If you delete a SuitestDrive unit, all devices attached will be removed as well. It is worth mentioning that CandyBoxes cannot be deleted from control units.
The get a control unit page is where the download to SuitestDrive, for all the operating systems that are supported. You can order a CandyBox here as well.
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To order a CandyBox simply fill out your billing information and press Order CandyBox, or upgrade to a plan (Super Sweet or The Sweetest) which includes a CandyBox. The shipping costs are automatically calculated based on the shipping address. Our friendly sales team will contact you to verify the shipping address.
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Suitest provides access to the Continuous Integration API, which allows you to interact directly with Suitest based on your needs. To access the API, create an API Token, name it and then save the secret token password given to a secure place. Then head over to the CI API docs.


text:
Arabibarbus is a genus of Cyprinidae. They are medium-small to very large freshwater carps found in the Western Asia.

Species
The taxonomic position of these species has historically caused considerable confusion and they were formerly placed in Barbus or Tor, although the species described in 2014 was placed in Arabibarbus from the beginning.

There are currently three recognized species of this genus:

Arabibarbus arabicus (Trewavas, 1941)
Arabibarbus hadhrami (Borkenhagen, 2014)
Arabibarbus grypus (Heckel, 1843) (Shirbot)

References

Vertebrates of the Arabian Peninsula

redpajama_set_name: RedPajamaWikipedia


text:
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\head Introduction\endhead
\subhead 0.1\endsubhead
Let $G$ be a connected reductive algebraic group over $\CC$, with Weyl group $W$.
In \cite{L15} we have defined a partition of $G$ into finitely many strata. The set of strata
of $G$ is in natural bijection with the image of a map $cl(W)@>>>\Irr(W)$ where
$cl(W)$ is the set  of conjugacy classes in $W$ and $\Irr(W)$ is a set of representatives
for the isomorphism classes of irreducible representations of $W$ over $\QQ$.
We define a partition of $cl(W)$ into subsets (called the {\it strata of $cl(W)$}): a stratum of $cl(W)$ is by definition a nonempty fibre of the map $cl(W)@>>>\Irr(W)$.
We define a partition of $W$ into subsets (called the {\it strata of $W$}): a stratum of $W$
is by definition the inverse image of a stratum of $cl(W)$ under the surjective
map $W@>>>cl(W)$ which takes an element of $W$ to its conjugacy class. The set of strata of
$W$ is in obvious bijection with the set of strata of $cl(W)$ which is in bijection
with the set of strata of $G$.

In this paper we are interested in two problems:

(i) How to parametrize the strata of $cl(W)$ (or $W$)?

(ii) How to describe explicitly each individual stratum of $cl(W)$.
\nl
These problems are solved in \cite{L15}, but we would like to get a simpler are more direct
approach. We shall reduce these problems to the same problems
restricted to a much smaller set of cases.

The set of strata can be viewed as an enlargement of the set of unipotent classes of $G$
(a unipotent class of $G$ is contained in exactly one stratum); this enlargement is built
from the sets of unipotent classes in groups like $G$ but in all characteristics.
According to \cite{BC76}, the classification of unipotent classes of $G$ can be reduced to
the classification of a smaller
set of unipotent classes, the distinguished ones. We will show that, similarly, the
classification of strata of $G$ can be reduced to the classification of a much
smaller set, that of distinguished strata. It turns out that the distinguished strata of $G$
are indexed by a subset $CL_{dist}(W)$ of the set of elliptic conjugacy classes of $W$
which may be called distinguished conjugacy classes. (It would be interesting to find
a description of distinguished conjugacy classes which is purely in terms of $W$ and is
not case by case.) One of our results is that

(a) the set of strata of $G$ (or $W$) has a simple parametrization in terms of the sets
$CL_{dist}(W)$ where $W$ is replaced by various parabolic subgroups of $W$ (see 0.6(a))
to which ``parabolic inclusion'' (see 0.6) is applied.
\nl
Another result of this paper is that

(b) the strata of $cl(W)$ are precisely the connected components of an (oriented) graph with
set of vertices $cl(W)$ (the graph structure on $cl(W)$ is defined in \S2); the
connected components of this graph are remarkably simple: they are products of
Coxeter graphs of type $A$.
\nl
Our definition of the edges of the graph is by first defining (case by case) the edges
for which one end is an elliptic conjugacy class and then applying ``parabolic inclusion''
to such elementary edges where $W$ is replaced by various parabolic subgroups of $W$.
It would be interesting to find a description of these elementary edges which is not case by
case.

In \S3 we show how

(c) $CL_{dist}(W)$ can be parametrized in terms of certain reflection subgroups of $W$.
\nl
Note that by combining (a),(b),(c) one can hope to understand the classification of conjugacy
classes in $W$ in different terms than those in the classification of Carter \cite{C72}.
Namely, the subset $CL_{dist}(W)$ is classified by (c); next, the set of
strata of $cl(W)$ is obtained from (a) by parabolic inclusion and finally the objects
of $cl(W)$ should be described by their position in the graph (product of graphs of type $A$
in (b)) associated to a stratum of $cl(W)$.

We note that the set $CL_{dist}(W)$ and (a),(b) above depend only on $W$ as a Coxeter group
(not on $G$); however, this is not so for (c).

\subhead 0.2\endsubhead
Let $\cp=\{2,3,5,\do\}$ be the set of prime numbers. For any $r\in\cp$ let $G^{(r)}$
be a connected reductive group over an algebraically closed field of characteristic $r$
of the same type as $G$. We set $G^{(0)}=G$. For $r\in\{0\}\cup\cp$ let
$\cu^{(r)}$ be the set of unipotent classes of $G^{(r)}$. By the Springer correspondence
(extended in \cite{L84} to small characteristic) there is a natural imbedding
$\io^{(r)}:\cu^{(r)}@>>>\Irr(W)$
whose image is denoted by $\SS^{(r)}(W)$; it is known
that $\SS^{(0)}(W)\sub\SS^{(r)}(W)$.
Let
$$\SS(W)=\cup_{r\in\{0\}\cup\cp}\SS^{(r)}(W)=\cup_{r\in\cp}\SS^{(r)}(W)\sub\Irr(W).$$

In \cite{L15} (where the notation $\cs_2(W)$ is used instead of $\SS(W)$)
it is shown that $\SS(W)$ depends only on $W$
as a Coxeter group, not on the underlying root datum (but it is not clear whether $\SS(W)$ makes sense
for a finite non-crystallographic Coxeter group).

In \cite{L15} we have defined for any $r\in\{0\}\cup\cp$ a surjective map $\k^{(r)}:G^{(r)}@>>>\SS(W)$
whose fibres are called the {\it strata} of $G^{(r)}$; each stratum is a union of conjugacy classes of the
same dimension, independent of $r$ and, according to \cite{C20}, is locally closed in $G^{(r)}$.
If $g\in G^{(r)}$ is unipotent, then $\k^{(r)}(g)$ is the same as the image of the conjugacy class of $g$
under $\io^{(r)}$. It follows that for any $E\in\SS(W)$, there exists $r\in\{0\}\cup\cp$ such that
the stratum $(\k^{(r)})\i(E)$ contains some unipotent element.

\subhead 0.3\endsubhead
An element of $G^{(r)}$ is said to be distinguished
if it is not contained in a Levi subgroup of a proper parabolic subgroup of $G^{(r)}$ (see \cite{BC76}).
Let $\cu^{(r)}_{dist}$ be the set of unipotent classes in $G^{(r)}$ in which some/any element is
distinguished. Let $\SS^{(r)}_{dist}(W)=\io^{(r)}(\cu^{(r)}_{dist})$.

We say that $E\in\SS(W)$ is distinguished if $E\in\cup_{r\in\{0\}\cup\cp}\SS^{(r)}_{dist}(W)$
or equivalently if there exists $r\in\{0\}\cup\cp$ such that
the stratum $(\k^{(r)})\i(E)$ contains some distinguished unipotent element.
Let $\SS_{dist}(W)$ be the set of distinguished elements of $\SS(W)$.

In \S1 we will show:

(a) {\it $E\in\SS(W)$ is distinguished if and only if there exists $r\in\{0\}\cup\cp$ such that the
stratum $(\k^{(r)})\i(E)$ contains some distinguished (not necessarily unipotent) element of $G^{(r)}$.}
\nl
A stratum of $G^{(r)}$ (with $r\in\{0\}\cup\cp$)
is said to be distinguished if it is of the form $(\k^{(r)})\i(E)$ where
$E\in\SS_{dist}(W)$ (such a stratum need not contain a distinguished unipotent element).

\subhead 0.4\endsubhead
For $C\in cl(W)$ let $m(C)$ be the dimension of the $1$-eigenspace of some/any $w\in C$ on the
reflection representation of $W$.
We shall write $\Ph:cl(W)@>>>\SS(W)$ for what in \cite{L15} is denoted by ${}'\Ph$ (a surjective
map).
In \cite{L15} it is shown that

(a) {\it for any $E\in\SS(W)$ there is a unique $C_E\in\Ph\i(E)$ which is as elliptic as
possible, that is $m(C_E)\le m(C)$ for any $C\in\Ph\i(E)$;}
\nl
thus $E\m C_E$ is a cross section of the surjective map $\Ph$. The following variant
of (a) will be verified in \S1.

(b) {\it for any $E\in\SS(W)$ there is a unique $C'_E\in\Ph\i(E)$ which is as non-elliptic as possible,
that is $m(C'_E)\ge m(C)$ for any $C\in\Ph\i(E)$.}
\nl
Let $CL(W)$ be the image of the map $E\m C'_E$, $\SS(W)@>>>cl(W)$. Note that $\Ph$ restricts to a bijection
$CL(W)@>\si>>\SS(W)$. Under this bijection, the subset $\SS_{disc}(W)$ of $\SS(W)$ corresponds to
a subset $CL_{dist}(W)$ of $CL(W)$. The conjugacy classes of $W$ contained in $CL_{dist}(W)$
are said to be
distinguished. The following result will be proved in \S1.

(c) {\it Let $C\in CL(W)$. We have $C\in CL_{dist}(W)$ if and only if $C$ is elliptic (that is, $m(C)=0$).}

\subhead 0.5 \endsubhead
For $r\in\{0\}\cup\cp$ let
$\ps^{(r)}:cl(W)@>>>\cu^{(r)}$ be the surjective map
defined in \cite{L11a}. (In the case $r=0$, an alternative definition of this map is
given in \cite{Y20}.) Let $\Ph^{(r)}:cl(W)@>>>\Irr(W)$ be the composition of $\ps^{(r)}$
with $\io^{(r)}:\cu^{(r)}@>>>\Irr(W)$.
From the explicit description of $\ps^{(r)}$ in \cite{L12} and the explicit description
of $CL_{dist}(W)$ given in this paper we see that

(a) {\it if $C\in CL_{dist}(W)$ then $\Ph^{(r)}(C)$ is independent of $r$. Hence, by the definition
of $\Ph$ in \cite{L15, 4.1}, we have $\Ph(C)=\Ph^{(r)}(C)$ for all $r$.}

\subhead 0.6 \endsubhead
Let $\{s_i;i\in I\}$ be the set of simple reflections of $W$. For $J\sub I$ let $W_J$
be the subgroup of $W$ generated by $\{s_i;i\in J\}$; this is the Weyl group of a Levi
subgroup of a parabolic subgroup of $G$. Hence $CL(W_J)$ and its subset $CL_{dist}(W_J)$
are defined. For $C_1\in cl(W_J)$ we define $\r_J(C_1)\in cl(W)$ by the condition
$C_1\sub\r_J(C_1)$. Now $C_1\m\r_J(C_1)$ is an injective map $cl(W_J)@>>>cl(W)$.
(We call it {\it parabolic inclusion}.)
If $J\sub I,J'\sub I$, we say that $J,J'$ are equivalent if
$W_J,W_{J'}$ are conjugate under an element of $W$. Let $\cx$ be a set of representatives
for the equivalence classes of subsets $J\sub I$ for the equivalence relation above.
The following result can be deduced from the explicit description of $CL(W)$ given in
\cite{L15} and that of $CL_{dist}(W)$ given in this paper.

(a) $CL(W)=\sqc_{J\in\cx}\r_J(CL_{dist}(W_J))$.

\subhead 0.7. Notation \endsubhead
A bipartition is a sequence $\l_*=(\l_1,\l_2,\l_3,\do)$ in $\NN$ such that $\l_i=0$ for
$i$ large and $\l_1\ge\l_3\ge\l_5\ge\do$, $\l_2\ge\l_4\ge\l_6\ge\do$. We write
$|\l_*|=\l_1+\l_2+\l_3+\do$. Let $BP$ be the set of bipartitions.

Let $T$ be the set of all $\l_*\in BP$ such that $\l_1\ge\l_2\ge\l_3\ge\l_4\ge\do$.
Let $R$ be the set of all $\l_*\in T$ such that $\l_i$ is even for any $i$.
Let $P$ be the set of all $\l_*\in T$ such that $\l_1=\l_2,\l_3=\l_4,\l_5=\l_6,\do$,
For $\l_*\in T$ and $j>0$ we set $\mu_j(\l_*)=\sha(\{k\ge1;\l_k=j\})$.
For $m\in\NN$ let $BP^m=\{\l_*\in BP,|\l_*|=m\}$, $T^m=T\cap BP^m$.
Let $T_{ev}$ (resp. $R_{ev}$) be the subset of $T$ (resp. $R$) consisting of the $\l_*$
with an even number of $>0$ terms.
For $a,b$ in $\ZZ$ we write $a\gg b$ instead $a-b\ge2$.

\head 1. Proof of 0.3(a), 0.4(b), 0.4(c)\endhead
\subhead 1.1\endsubhead
In this section we prove 0.3(a), 0.4(b), 0.4(c). To do this we can assume that $G$ is almost simple.
It is also enough to consider only one such $G$ in each isogeny class.
The case where $G$ is of classical (resp. exceptional) type is considered in 1.2-1.9 (resp.
1.10-1.15).

\subhead 1.2\endsubhead
Assume first that $G=SL_n(\CC), n\ge2$. In this case $\SS^{(r)}(W)=\Irr(W)=\SS(W)$ for any $r$
and the map $\Ph$ is a bijection $cl(W)@>\si>>\SS(W)$. In this case 0.4(b) is obvious and we have
$CL(W)=cl(W)$. Also 0.3(a) is immediate (an element
is distinguished if and only if is regular unipotent
times a central element).
Note that $CL_{dist}(W)$ consists of a single element:
the class of the Coxeter element; thus 0.4(c) holds.

\subhead 1.3\endsubhead
Until the end of 1.5 we assume that $G=Sp_{2n}(\CC),n\ge2$.
Then $\SS(W)=\SS^{(2)}(W)$ and $\Ph$ becomes a map $cl(W)@>>>\SS^{(2)}(W)$.
By \cite{L12} we have bijections

(a) $\cu^{(2)}\lra$ (set of all pairs $(c_*,\e)$ where $c_*\in T^{2n}$ is such that
$\mu_j(c_*)\in2\NN$ for any odd $j$ and
$\e:\{j\in2\NN+2;\mu_j(c_*)\in2\NN+2\}\}@>>>\{0,1\}$),

(b) $cl(W)\lra(R\T P)^{2n}:=\{(r_*,p_*)\in R\T P;|r_*|+|p_*|=2n\}$.

Via (a),(b), $(\io^{(2)})\i\Ph:cl(W)@>>>\cu^{(2)}$ becomes the map

(c) $(r_*,p_*)\m(c_*,\e)$ where $\mu_j(c_*)=\mu_j(r_*)+\mu_j(p_*)$ for $j>0$
and for any $j\in2\NN+2$ such that $\mu_j(c_*)\in2\NN+2$, we have
$\e(j)=1$ if $j=r_i$ for some $i$ and $\e(j)=0$, otherwise.
\nl
Via (b), the map $cl(W)@>>>\NN, C\m m(C)$ becomes the map which to any
$(r_*,p_*)\in(R\T P)^{2n}$ associates $|p_*|/2$.
To prove 0.4(b) we must show that, if $(c_*,\e)$ (as in (a)) is given, then
there is a unique $(r_*,p_*)\in(R\T P)^{2n}$ which maps to it (as in (c))
and has $|p_*|$ maximum possible. Thus,

for $j\in2\NN+2$ such that $\mu_j(c_*)\in2\NN+2$, we must have that
$\mu_j(r_*)=2,\mu_j(p_*)=\mu_j(c_*)-2$ (if $\e(j)=1$) and $\mu_j(r_*)=0,\mu_j(p_*)=\mu_j(c_*)$
(if $\e(j)=0$); for $j\in2\NN+2$ such that $\mu_j(c_*)\in2\NN+1$, we must have that
$\mu_j(r_*)=1,\mu_j(p_*)=\mu_j(c_*)-1$; for $j\in2\NN+2$ such that $\mu_j(c_*)=0$, we must have
that $\mu_j(r_*)=0,\mu_j(p_*)=0$; for $j\in2\NN+1$, we must have that $\mu_j(r_*)=0,\mu_j(p_*)=\mu_j(c_*)$.
\nl
This proves 0.4(b) in our case.

Note that:

(e) {\it $CL(W)$ is (via (b)) the set of all $(r_*,p_*)\in(R\T P)^{2n}$
such that $\mu_j(r_*)\le2$ for any $j>0$.}

\subhead 1.4\endsubhead
As in \cite{L15} we have a bijection

(a) $\Irr(W)\lra BP^n$.
\nl
Using \cite{L15, \S3}, we see that

(b) {\it when $r\ne2$, the subset $\io^{(r)}(\cu^{(r)}_{dist})$ of $\Irr(W)$
becomes via (a) the subset of $BP^n$ consisting of sequences of the form
$(a_1>a_2>\do>a_s>0,0,0,\do)$.}
\nl
By \cite{W63} (see also \cite{LS12, 6.2}),
the set $\cu^{(2)}_{dist}$ can be identified via 1.3(a) with
the subset of $\cu^{(2)}$ consisting of

(c) {\it all $(c_*,\e)$ (as in 1.3(a)) such that $\mu_j(c_*)=0$ for odd $j$, $\mu_j(c_*)\le2$ for
even $j$ and $\e(j)=1$ whenever $j$ is even and $\mu_j(c_*)=2$.}
\nl
Using \cite{L15, \S3} we see that the subset $\io^{(2)}(\cu^{(2)}_{dist})$ of $\Irr(W)$ becomes
via (a) the subset of $BP^n$ formed by the sequences $(c_1/2,c_2/2,c_3/2,\do)$ for various
$(c_*,\e)$ as in (c). This is the same as the set of all $(a_1\ge a_2\ge a_3\ge\do)\in BP^n$
such that there are no consecutive equalities between the non zero $a_i$.
This set contains the set (b). It follows that

(d) $\SS^{(r)}_{dist}(W)\sub\SS^{(2)}_{dist}(W)$ for any $r$.
\nl
Under our bijection $CL_{dist}(W)\lra \SS_{dist}(W)$,
the set of $(c_*,\e)$ as in (c) corresponds to the set of $(r_*,p_*)\in(R\T P)^{2n}$
such that $\mu_j(r_*)\le2$ for $j>0$ and $p_*=(0,0,0,\do)$; this is the same as the set of
all $(r_*,p_*)\in(R\T P)^{2n}$ which under 1.3(b) correspond to
elliptic conjugacy classes in $W$ which are in $CL(W)$.
This implies (by (d))
that 0.4(c) holds in our case.

\subhead 1.5\endsubhead
Let $g\in G^{(r)}$ be a distinguished element. To prove 0.3(a) it is enough to show
that $\k^{(r)}(g)\in\SS_{dist}(W)$. If $r=2$ then $g$ is unipotent and the result is clear.
Thus we can assume that $r\ne2$. Using \cite{L15, \S3} we see that under the bijection 1.4(a),
$\k^{(r)}(g)$ corresponds to a bipartition of the form

(a) $((a_1+b_1)/2,(a_2+b_2)/2,(a_3+b_3)/2,\do)$
\nl
where
$$a_1=\nu_1/2,a_2=\nu_2/2,a_3=\nu_3/2,\do,a_s=\nu_s/2,a_{s+1}=0,a_{s+2}=0,\do,$$
$$b_1=\nu'_1/2,b_2=\nu'_2/2,b_3=\nu'_3/2,\do,b_t=\nu'_t/2,b_{t+1}=0,b_{t+2}=0,\do,$$
and $\nu_1>\nu_2>\nu_3>\do>\nu_s$, $\nu'_1>\nu'_2>\nu'_3>\do>\nu'_t$ are even integers $\ge2$
with $\sum_k\nu_k+\sum_k\nu'_k=2n$.

Clearly, $(a_1+b_1)/2\ge(a_2+b_2)/2\ge(a_3+b_3)/2\ge\do$. If
$(a_i+b_i)/2=(a_{i+2}+b_{i+2})/2$
then $a_i=a_{i+2},b_i=b_{i+2}$ hence $a_{i+2}=0,b_{i+2}=0$ and $(a_{i+2}+b_{i+2})/2=0$.
Thus (a) corresponds to an element of $\SS^{(2)}_{dist}(W)$. This proves 0.3(a) in our case.

\subhead 1.6\endsubhead
We now assume that $G=SO_{2n+1}(\CC), n\ge3$.
Then the arguments in 1.3, 1.4 can be used word by word in the present case except that
1.4(b) must be replaced by the following statement:

(a) When $r\ne2$, the subset $\io^{(r)}(\cu^{(r)}_{dist})$ of $\Irr(W)$
becomes via 1.4(a) the subset of $BP^n$ consisting of sequences $(c_1,c_2,c_3,\do)$ where
$$\align&c_1=(\nu_1-1)/2,c_2=(\nu_2+1)/2,c_3=(\nu_3-1)/2,\\&
c_4=(\nu_4+1)/2,\do,c_{2s+1}=(\nu_{2s+1}-1)/2,c_{2s+2}=0, c_{2s+3}=0,\do\endalign$$
and $\nu_1>\nu_2>\nu_3>\do>\nu_{2s+1}$ are odd integers $\ge1$ with sum $2n+1$.

(Note that $c_1,c_2,\do)\in P$, with no two successive equalities between its nonzero terms
hence it corresponds to an element in $\SS^{(2)}_{dist}(W)$.)

The argument in 1.5 also continues to hold except that 1.5(a) must be replaced by

(b) $((a_1+b_1)/2,(a_2+b_2)/2,(a_3+b_3)/2,\do)$
\nl
where
$$\align&a_1=(\nu_1-1)/2,a_2=(\nu_2+1)/2,a_3=(\nu_3-1)/2,\\&
a_4=(\nu_4+1)/2,\do,a_{2s+1}=(\nu_{2s+1}-1)/2,a_{2s+2}=0, a_{2s+3}=0,\do,\endalign$$
$$\align&b_1=(\nu'_1+1)/2,b_2=(\nu'_2-1)/2,b_3=(\nu'_3+1)/2,\\&
b_4=(\nu'_4-1)/2,\do,b_{2t}=(\nu'_{2t}-1)/2,b_{2t+1}=0, b_{2t+2}=0,\do,\endalign$$
and $\nu_1>\nu_2>\nu_3>\do>\nu_{2s+1}$, $\nu'_1>\nu'_2>\nu'_3>\do>\nu'_{2t}$ are odd integers $\ge1$
with $\sum_k\nu_k+\sum_k\nu'_k=2n+1$.

We have $a_1\ge a_2\ge a_3\ge\do$ and $b_1\ge b_2\ge b_3\ge\do$ hence
$$(a_1+b_1)/2\ge(a_2+b_2)/2\ge(a_3+b_3)/2\ge\do.$$
If $(a_i+b_i)/2=(a_{i+2}+b_{i+2})/2$ then $a_i=a_{i+2},b_i=b_{i+2}$ hence $a_i=b_i=0$.
\nl
Thus (a) corresponds to an element of $\SS^{(2)}_{dist}(W)$. This proves 0.3(a) in our case.

\subhead 1.7\endsubhead
Until the end of 1.9 we assume that $G=SO_{2n}(\CC), n\ge4$.
We have $\SS(W)=\SS^{(2)}(W)$ and $\Ph$ becomes a map $cl(W)@>>>\SS^{(2)}(W)$.
Now each of
$$cl(W),\Irr(W),\SS^{(2)}(W),\cu^{(2)}$$ has a natural involution
induced by conjugation by an element in the non-identity component of the
full orthogonal group. We then have partitions
$$cl(W)=cl(W)'\sqc cl(W)'',\Irr(W)=\Irr(W)'\sqc\Irr(W)'',$$
$$\SS^{(2)}(W)=\SS^{(2)}(W)'\sqc\SS^{(2)}(W)'',\cu^{(2)}=\cu^{(2)}{}'\sqc\cu^{(2)}{}'',$$
where $()'$ denotes the set of fixed points of the involution and $()''$ denotes its
complement.

By \cite{L12} we have bijections

(a)  (set of orbits of the involution on $\cu^{(2)}{}''$) $\lra$
(set of all pairs $(c_*,\e)$ as in 1.3(a) such that $\e=0$ and
$\mu_j(c_*)=0$ for $j$ odd, $\mu_j(c_*)$ even for $j$ even),

(b) (set of orbits of the involution on $cl(W)''$) $\lra$
(set of all pairs $(r_*,p_*)\in(R\T P)^{2n}$
such that $r_*=(0,0,\do)$ and
$\mu_j(p_*)=0$ for $j$ odd),

(c) $\cu^{(2)}{}'\lra$
(set of all pairs $(c_*,\e)$ as in 1.3(a) which are not as in (a) and are such that
$c_*\in T_{ev}$),

(d) $cl(W)'\lra$ (set of all pairs $(r_*,p_*)\in(R\T P)^{2n}$ which are not as in (b)
and are such that $r_*\in R_{ev}$).
\nl
Now $(\io^{(2)})\i\Ph$ induces the (surjective) map $cl(W)'@>>>\cu^{(2)}{}'$ which by (c),(d)
becomes the map $(r_*,p_*)\m(c_*,\e)$ given by the same rule as in 1.3(c).
The same proof as in 1.3 shows that if $(c_*,\e)$ (as in (c)) is given, then
there is a unique $(r_*,p_*)$ (as in (d)) which maps to it and has $|p_*|$ maximum possible.
This implies that 0.4(b) holds for any $E\in\SS(W)=\SS^{(2)}(W)$ that is contained in
$\SS^{(2)}(W)'$. If $E\in\SS^{(2)}(W)''$ then $\Ph\i(E)$ consists of a single element so
that 0.4(b) holds automatically for such $E$. This proves 0.4(b) in our case.

In our case

(e) the set $CL(W)$ becomes the set of pairs $(r_*,p_*)$ as in 1.3(e)
such that $r_*\in R_{ev}$ and with each pair as in (b) repeated twice.

\subhead 1.8\endsubhead
As in 1.4(a) we have a bijection

(a) $\Irr(W)'\lra$ set of all $(a_1,a_2,a_3,\do)\in BP^n$ such that $a_1-a_2,a_3-a_4,\do$ are
not all zero.

Using \cite{L15, \S3} we see that

(b) when $r\ne2$, the subset $\io^{(r)}(\cu^{(r)}_{dist})$ of $\Irr(W)'$
becomes via (a) the subset of $BP^n$ consisting of sequences of the form
$$((\nu_1+1)/2,(\nu_2-1)/2,(\nu_3+1)/2,\do,(\nu_{2s}-1)/2,0,0,0,\do)$$
where $\nu_1>\nu_2>\do>\nu_{2s}$ are odd $\ge1$.

By \cite{W63} (see also \cite{LS12, 6.2}),
the set $\cu^{(2)}_{dist}$ can be identified via 1.7(a) with
the set consisting of

(c) all $(c_*,\e)$ (as in 1.3(a)) where $c_*\in T_{ev}$ and
such that $\mu_j(c_*)=0$ for odd $j$, $\mu_j(c_*)\le2$ for
even $j$ and $\e(j)=1$ whenever $j$ is even and $\mu_j(c_*)=2$.

Using \cite{L15, \S3} we see that the subset $\io^{(2)}(\cu^{(2)}_{dist})$ of $\Irr(W)'$ becomes
via (a) the subset of $BP^n$ formed by the sequences
$$((c_1+2)/2,(c_2-2)/2,(c_3+2)/2,(c_4-2)/2,\do,(c_{2s}-2)/2,0,0,0,\do)$$
for various $(c_*,\e)$ as in (c) with $c_*=(c_1\ge c_2\ge\do\ge c_{2s}>0,0,0,\do)$.

If $\nu_1>\nu_2>\do>\nu_{2s}$ are odd $\ge1$ then
$$\align&((\nu_1+1)/2,(\nu_2-1)/2,(\nu_3+1)/2,\do,(\nu_{2s}-1)/2,0,0,0,\do)\\&=
((c_1+2)/2,(c_2-2)/2,(c_3+2)/2,(c_4-2)/2,\do,(c_{2s}-2)/2,0,0,0,\do)\endalign$$
where
$c_1=\nu_1-1, c_2=\nu_2+1,c_3=\nu_3-1,\do, c_{2s}=\nu_{2s}+1$
are even and $c_1\ge c_2\gg c_3\ge c_4\gg\do\ge c_{2s}\gg0$.
From this we see that $\io^{(r)}(\cu^{(r)}_{dist})\sub\io^{(2)}(\cu^{(2)}_{dist})$ for any $r$, that is

(d) $\SS^{(r)}_{dist}(W)\sub\SS^{(2)}_{dist}(W)$ for any $r$.
\nl
Under our bijection $CL_{dist}(W)\lra \SS_{dist}(W)$,
the set of $(c_*,\e)$ as in (c) corresponds to the set of $(r_*,p_*)$
as in 1.7(d) such that $\mu_j(r_*)\le2$ for $j>0$ and $p_*=(0,0,0,\do)$; this is the same as the set of
all $(r_*,p_*)$ in 1.7(e) which correspond to elliptic conjugacy classes in $W$. This implies (by (d))
that 0.4(c) holds in our case.

\subhead 1.9 \endsubhead
Let $g\in G^{(r)}$ be a distinguished element. To prove 0.3(a) it is enough to show
that $\k^{(r)}(g)\in\SS_{dist}(W)$. If $r=2$ then $g$ is unipotent and the result is clear.
Thus we can assume that $r\ne2$. Using \cite{L15, \S3} we see that under the bijection 1.8(a),
$\k^{(r)}(g)$ corresponds to a bipartition of the form

(a) $((a_1+b_1)/2,(a_2+b_2)/2,(a_3+b_3)/2,\do)$
\nl
where
$$\align &
a_1=(\nu_1+1)/2,a_2=(\nu_2-1)/2,a_3=(\nu_3+1)/2,a_4=(\nu_4-1)/2,\do,\\&a_{2s}=(\nu_{2s}-1)/2,
a_{2s+1}=0, a_{2t+2}=0,\do,\endalign$$
$$\align&b_1=(\nu'_1+1)/2,b_2=(\nu'_2-1)/2,
b_3=(\nu'_3+1)/2,\\&b_4=(\nu'_4-1)/2,\do,b_{2t}=(\nu'_{2t}-1)/2,
b_{2t+1}=0, b_{2t+2}=0,\do,\endalign$$
and
$$\nu_1>\nu_2>\nu_3>\do>\nu_{2s}, \qua \nu'_1>\nu'_2>\nu'_3>\do>\nu'_{2t}$$
are odd integers $\ge1$
with $\sum_k\nu_k+\sum_k\nu'_k=2n$. We can assume that $t\le s$. We have
$$\align&((a_1+b_1)/2,(a_2+b_2)/2,(a_3+b_3)/2,\do)\\&
=((c_1+2)/2,(c_2-2)/2,(c_3+2)/2,(c_4-2)/2,\do,(c_{2s}-2)/2,0,0,0,\do)\endalign$$
where
$$\align&c_1=\nu_1+\nu'_1,c_2=\nu_2+\nu'_2,\do,c_{2t}=\nu_{2t}+\nu'_{2t}, \\&
c_{2t+1}=\nu_{2t+1}-1,c_{2t+2}=\nu_{2t+2}+1,\do, c_{2s}=\nu_{2s}+1.\endalign$$

(The terms $c_{2t+1},\do,c_{2s}$ are missing if $t=s$.)
Note that $c_1,c_2,\do,c_{2s}$ are even, nonzero and
$c_1\ge c_2\ge c_3\ge c_4\ge\do\ge c_{2s}$ with no consecutive equalities.
Thus (a) corresponds to an element of $\SS^{(2)}_{dist}(W)$. This proves 0.3(a) in our case.

\subhead 1.10\endsubhead
In the remainder of this section we assume that $G$ is simple of exceptional type.
In this case 0.4(b) can be deduced from tables in \cite{L15}.
In 1.11-1.15 we describe in each case, using notation of
Carter \cite{C72} and that of \cite{L15},
the bijection between $CL(W)$ and $\SS(W)$ in the form $C\lra E$.
Here we also use the description of distinguished unipotent elements in \cite{M80}, \cite{LS12}.
Now 0.3(a),0.4(c)  can be verified in each case using the definitions.

\subhead 1.11. Type $G_2$\endsubhead

$[G_2]\lra1_0$; $[A_2]\lra2_1$; $[A_1+\tA_1]\lra2_2$;

$[\tA_1]\lra1_3$;

$[A_1]\lra1_3$;

$[A_0]\lra1_6$.

Here the first three items are in $CL_{dist}(W)$.
Note that $2_2\in\SS^{(r)}_{dist}(W)$ for a single $r$ namely $r=3$.

\subhead 1.12. Type $F_4$ \endsubhead

$[F_4]\lra1_0$; $[B_4]\lra4_1$; $[F_4(a_1)]\lra9_2$; $[D_4(a_1)]\lra12_4$;
$[A_3+\tA_1]\lra16_5$; $[\tA_2+\tA_2]\lra6_6$;

$[B_3]\lra8_3$;

$[C_3]\lra8_3$;

$[A_3]\lra9_6$;

$[B_2+A_1]\lra9_6$;

$[A_2+\tA_1]\lra4_7$;

$[\tA_2+A_1]\lra4_7$;

$[B_2]\lra 4_8$;

$[\tA_2]\lra 8_9$;

$[A_2]\lra 8_9$;

$[A_1+\tA_1]\lra9_{10}$;

$[2A_1]\lra 4_{13}$;

$[A_1]\lra 2_{16}$;

$[\tA_1]\lra 2_{16}$;

$[A_0]\lra 1_{24}$.

Here the first six items are in $CL_{dist}(W)$.

Note that $16_5,6_6$ are in $\SS^{(r)}_{dist}(W)$ for a single $r$ namely $r=2$.

\subhead 1.13. Type $E_6$\endsubhead

$[E_6]\lra1_0$; $[E_6(a_1)]\lra6_1$; $[E_6(a_2)]\lra 30_3$,

$[D_5]\lra20_2$; $[D_5(a_1)]\lra64_4$

$[A_5]\lra15_4$

$[A_4+A_1]\lra60_5$

$[2A_2+A_1]\lra 10_9$

$[D_4]\lra 24_6$; $[D_4(a_1)]\lra 80_7$

$[A_4]\lra81_6$

$[A_3+A_1]\lra60_8$

$[A_2+2A_1]\lra 60_{11}$

$[2A_2]\lra 24_{12}$

$[A_3]\lra 81_{10}$

$[A_2+A_1]\lra 64_{13}$

$[3A_1]\lra 15_{16}$

$[A_2]\lra 30_{15}$

$[2A_1]\lra 20_{20}$

$[A_1]\lra 6_{25}$

$[A_0]\lra 1_{36}$

Here the first three items are in $CL_{dist}(W)$.

\subhead 1.14. Type $E_7$\endsubhead

$[E_7]\lra 1_0$; $[E_7(a_1)]\lra 7_1$; $[E_7(a_2)]\lra 27_2$;
$[E_7(a_3)]\lra 56_3$; $[A_7]\lra189_5$; $[E_7(a_4)]\lra315_7$;

$[E_6]\lra 21_3$; $[E_6(a_1)]\lra120_4$; $[E_6(a_2)]\lra405_8$;

$[D_6]\lra35_4$; $[D_6(a_1)]\lra210_6$; $[D_6(a_2)]\lra280_8$; $[2A_3]\lra378_{14}$;

$[A_6]\lra105_6$;

$[D_5+A_1]\lra168_6$; $[D_5(a_1)+A_1]\lra378_9$;

$[(A_5+A_1)']\lra70_9$;

$[A_4+A_2]\lra210_{10}$;

$[A_3+A_2+A_1]\lra210_{13}$;

$[D_5]\lra189_7$; $[D_5(a_1)]\lra420_{10}$;

$[A''_5]\lra216_9$;

$[A_4+A_1]\lra512_{11}$;

$[A'_5]\lra105_{12}$;

$[D_4+A_1]\lra84_{12}$; $[D_4(a_1)+A_1]\lra405_{15}$;

$[A_3+A_2]\lra84_{15}$;

$[(A_3+2A_1)']\lra216_{16}$;

$[2A_2+A_1]\lra70_{18}$;

$[A_2+3A_1]\lra105_{21}$;

$[A_4]\lra420_{13}$;

$[D_4]\lra105_{15}$; $[D_4(a_1)]\lra315_{16}$;

$[(A_3+A_1)'']\lra280_{17}$;

$[(A_3+A_1)']\lra189_{20}$;

$[2A_2]\lra168_{21}$;

$[A_2+2A_1]\lra189_{22}$;

$[(4A_1)']\lra15_{28}$;

$[A_3]\lra210_{21}$;

$[A_2+A_1]\lra120_{25}$;

$[(3A_1)'']\lra35_{31}$;

$[(3A_1)']\lra21_{36}$;

$[A_2]\lra56_{30}$;

$[2A_1]\lra27_{37}$;

$[A_1]\lra7_{46}$;

$[A_0]\lra1_{63}$.

Here the first six items are in $CL_{dist}(W)$.

There are two conjugacy classes $4A_1$; one of them, $(4A_1)''$, comes from $W_J$ of type
$D_4$.
There are two conjugacy classes $A_5+A_1$; one of them, $(A_5+A_1)''$, comes from $W_J$ of type
$E_6$.

\subhead 1.15. Type $E_8$\endsubhead

$[E_8]\lra 1_0$; $[E_8(a_1)]\lra 8_1$; $[E_8(a_2)]\lra 35_2$; $[E_8(a_4)]\lra 112_3$;

$[E_8(a_5)]\lra 210_4$; $[D_8]\lra 560_5$; $[E_8(a_3)]\lra 700_6$; $[E_8(a_7)]\lra1400_7$;

$[E_8(a_6)]\lra1400_8$; $[D_8(a_2)]\lra 3240_9$; $[A_8]\lra 2240_{10}$;

$[D_8(a_3)]\lra 1400_{11}$; $[A_7+A_1]\lra 4536_{13}$; $[E_8(a_8)]\lra 4480_{16}$;

$[E_7]\lra 84_4$; $[E_7(a_1)]\lra 567_6$; $[E_7(a_2)]\lra1344_8$; $[E_7(a_3)]\lra 2268_{10}$;

$[E_7(a_4)]\lra7168_{17}$;  $[A'_7]\lra 6075_{14}$;

$[A''_7]\lra175_{12}$;

$[A_6+A_1]\lra 2835_{14}$;

$[D_5+A_2]\lra 840_{14}$; $[D_5(a_1)+A_2]\lra1344_{19}$;

$[D_7]\lra 400_7$; $[D_7(a_1)]\lra 1050_{10}$; $[D_7(a_2)]\lra 4200_{12}$; $[D_4+A_3]\lra4200_{21}$;

$[E_6+A_1]\lra 448_9$; $[E_6(a_1)+A_1]\lra 4096_{11}$; $[E_6(a_2)+A_1]\lra3150_{18}$;

$[A_4+A_3]\lra420_{20}$;

$[A_4+A_2+A_1]\lra2835_{22}$;

$[E_6]\lra 525_{12}$; $[E_6(a_1)]\lra 2800_{13}$; $[E_6(a_2)]\lra5600_{21}$;

$[A_6]\lra4200_{15}$;

$[D_6]\lra 972_{12}$; $[D_6(a_1)]\lra5600_{15}$; $[D_6(a_2)]\lra4200_{18}$; $[(2A_3)']\lra3240_{31}$;

$[D_5+A_1]\lra3200_{16}$; $[D_5(a_1)+A_1]\lra6075_{22}$;

$[(A_5+A_1)'']\lra2016_{19}$;

$[A_4+A_2]\lra4536_{23}$;

$[A_4+2A_1]\lra4200_{24}$;

$[D_4+A_2]\lra168_{24}$; $[D_4(a_1)+A_2]\lra2240_{28}$;

$[(2A_3)'']\lra840_{26}$;

$[A_3+A_2+A_1]\lra1400_{29}$;

$[2A_2+2A_1]\lra175_{36}$;

$[A_5]\lra 3200_{22}$

$[D_5]\lra2100_{20}$; $[D_5(a_1)]\lra2800_{25}$;

$[A_4+A_1]\lra4096_{26}$;

$[D_4+A_1]\lra700_{28}$; $[D_4(a_1)+A_1]\lra1400_{32}$;

$[A_3+A_2]\lra972_{32}$;

$[(A_3+2A_1)'']\lra1050_{34}$;

$[2A_2+A_1]\lra448_{39}$;

$[A_2+3A_1]\lra400_{43}$;

$[D_4]\lra525_{36}$; $[D_4(a_1)]\lra1400_{37}$;

$[A_4]\lra2268_{30}$;

$[A_3+A_1]\lra1344_{38}$;

$[2A_2]\lra700_{42}$;

$[A_2+2A_1]\lra560_{47}$;

$[(4A_1)'']\lra 50_{56}$;

$[A_3]\lra567_{46}$;

$[A_2+A_1]\lra210_{52}$;

$[3A_1]\lra84_{64}$;

$[A_2]\lra 112_{63}$;

$[2A_1]\lra 35_{74}$;

$[A_1]\lra8_{91}$;

$[A_0]\lra1_{120}$.

Here the first fourteen items are in $CL_{dist}(W)$.
There are two conjugacy classes $A_7$; one of them, $A_7'$, comes from $W_J$ of type $E_7$.
There are two conjugacy classes $2A_3$; one of them, $2A_3'$, comes from $W_J$ of type $D_6$.
There are two conjugacy classes $A_5+A_1$; one of them, $(A_5+A_1)'$, comes from $W_J$ of
type $E_6$.
There are two conjugacy classes $A_3+2A_1$; one of them, $(A_3+2A_1)'$, comes from $W_J$ of
type $D_5$.
There are two conjugacy classes $4A_1$; one of them, $(4A_1)'$, comes from $W_J$ of
type $D_4$.

Note that $3240_9,4536_{11}$ are in $\SS^{(r)}_{dist}(W)$ for a single $r$ namely $r=2$;
$1400_{11}$ is in $\SS^{(r)}_{dist}(W)$ for a single $r$ namely $r=3$.

\head 2. A graph structure on $cl(W)$\endhead
\subhead 2.1 \endsubhead
Let $cl_{ell}(W)=\{C\in cl(W);m(C)=0\}$ (elliptic conjugacy classes). Let
$cl_{s-ell}(W)=\{C\in cl(W);m(C)=1\}$ (sub-elliptic conjugacy classes).
We shall now define a subset $\ce_{ell}(W)$ of $cl_{ell}(W)\T cl_{s-ell}(W)$.
If $W$ is a product $W_1\T W_2$ of two Weyl group and if $\ce_{ell}(W_1),\ce_{ell}(W_2)$
are already defined, then $\ce_{ell}(W)$ consists of $(C_1\T C_2,C'_1\T C'_2)$ where either
$(C_1,C'_1)\in\ce_{ell}(W_1),C_2=C'_2\in cl_{ell}(W_2)$ or
$(C_2,C'_2)\in\ce_{ell}(W_2),C_1=C'_1\in cl_{ell}(W_1)$.
In this way we see that it is enough to define $\ce_{ell}(W)$ when $W$ is irreducible.

When $W$ is of type $A$, we set $\ce_{ell}(W)=\emp$.

Assume that $W$ is of type $B_n,n\ge2$. With the identification 1.3(b), we define
$\ce_{ell}(W)$ to be the set of all
$((r_*,p_*),(r'_*,p'_*))\in(R\T P)^{2n}\T(R\T P)^{2n}$
such that for some $2t\in2\NN+2$ the following holds:

(a) $r'_*$ is obtained from $r_*$ by removing two consecutive terms equal to $2t$
and $2t$ appears at least once in $r'_*$;

(b) $p_*=(0,0,\do)$, $p'_*=(2t,2t,0,0,\do)$.

Assume that $W$ is of type $D_n,n\ge4$. With the identification 1.7(d), we define
$\ce_{ell}(W)$ to be the set of all
$((r_*,p_*),(r'_*,p'_*))\in (R_{ev}\T P)^{2n}\T(R_{ev}\T P)^{2n}$
such that for some $2t\in2\NN+2$, (a) and (b) hold.

For conjugacy classes in $W$ of exceptional type
we use the notation of Carter \cite{C72} except that
we sometimes write $D_4+A_3$ for what Carter denotes by $D_6(a_2)+A_1$.

\subhead 2.2 \endsubhead
If $W$ is of type $G_2$ we set $\ce_{ell}(W)=\emp$.

\subhead 2.3 \endsubhead
If $W$ is of type $F_4$, $\ce_{ell}(W)$ consists of:

$([D_4],[B_3])$; $([C_3+A_1],[C_3])$; $([4A_1],[3A_1])$.

\subhead 2.4 \endsubhead
If $W$ is of type $E_6$, $\ce_{ell}(W)$ consists of:

$([A_5+A_1],[A_5])$; $([3A_2],[2A_2+A_1])$.

\subhead 2.5 \endsubhead
If $W$ is of type $E_7$, $\ce_{ell}(W)$ consists of:

$([D_6+A_1],[D_6])$; $([D_6(a_2)+A_1],[D_6(a_2)])$; $([D_4+3A_1],[D_4+2A_1])$;

$([A_5+A_2],[(A_5+A_1)'])$; $([2A_3+A_1],[A_3+A_2+A_1])$; $([7A_1],[6A_1])$.

\subhead 2.6 \endsubhead
If $W$ is of type $E_8$, $\ce_{ell}(W)$ consists of:

$([E_7+A_1],[E_7])$; $([E_7(a_2)+A_1],[E_7(a_2)])$; $([E_7(a_4)+A_1],[E_7(a_4)])$;

$([E_6+A_2],[E_6+A_1])$; $([E_6(a_2)+A_2],[E_6(a_2)+A_1])$;

$([D_8(a_1)],[D_7])$; $([D_6+2A_1],[D_6+A_1])$;

$([D_5(a_1)+A_3],[D_5(a_1)+A_2])$; $([2D_4],[D_4+A_3])$;

$([2D_4(a_1)],[D_4(a_1)+A_3])$; $([D_4+4A_1],[D_4+3A_1])$;

$([A_5+A_2+A_1],[A_5+A_2])$; $([A_5+A_2+A_1],[A_5+2A_1])$;

$([2A_4],[A_4+A_3])$;

$([2A_3+2A_2],[A_3+A_2+2A_1])$;

$([2A_3+2A_2],[2A_3+A_1])$;

$([4A_2],[3A_2+A_1])$;

$([8A_1],[7A_1])$.

\subhead 2.7 \endsubhead
We define a subset $\ce(W)$ of $cl(W)\T cl(W)$ as follows.
We say that $(C,C')\in cl(W)\T cl(W)$ is in $\ce(W)$ if there exists $J\sub I$
and $(C_1,C'_1)\in \ce_{ell}(W_J)$ such that $C=\r_J(C_1),C'=\r_J(C'_1)$, ($\r_J$ as in 0.6).
Note that $\ce_{ell}(W)\sub\ce(W)$ and that if $(C,C')\in \ce(W)$ then $m(C')=m(C)+1$.
We can regard $\ce(W)$ as the set of edges of a graph with vertices $cl(W)$.
This graph is oriented: the edge $(C,C')$ is oriented from $C$ to $C'$.
From the results of \cite{L15} one can verify that:

(a) {\it the strata of $cl(W)$ (or $W$) are exactly the connected components of this graph.}
\nl
Now each stratum of $cl(W)$ (or $W$) can be viewed as the set of vertices of an oriented graph
(restriction of the graph above to the stratum). From the results of \cite{L15} one can
verify the following strengthening of 0.4(a) and 0.4(b):

(b) {\it this oriented graph is a product of finitely many Coxeter graphs
of type $A$ (with the usual orientation).}
\nl
Recall that an oriented Coxeter graph of type $A$ is of the form
$$\bul\to\bul\to\do\to\bul.$$

For example, if $W$ is of type $E_8$, then the graph attached to a stratum of $cl(W)$ (or $W$)
is of one of the types $A_5,A_4,A_3,A_2\T A_2,A_2,A_1$. (Type $A_5$ appears for a unique
stratum; type $A_2\T A_2$ appears for two strata.)
If $W$ is of type $E_7$, then the graph attached to a stratum of $cl(W)$ (or $W$)
is of one of the types $A_4,A_3,A_2,A_1$. (Type $A_4$ appears for a unique
stratum.) If $W$ is of type $E_6$, then the graph attached to a stratum of $cl(W)$ (or $W$)
is of one of the types $A_2,A_1$.
If $W$ is of type $F_4$, then the graph attached to a stratum of $cl(W)$ (or $W$)
is of one of the types $A_4,A_2,A_1$. (Type $A_4$ appears for a unique stratum.)
If $W$ is of type $G_2$. then the graph attached to a stratum of $cl(W)$ (or $W$)
is of type $A_1$.

\head 3. Complements\endhead
\subhead 3.1 \endsubhead
Let $r\in\{0\}\cup\cp$. We state two properties similar to 0.4(a),(b).

(a) For any $E\in\SS^{(r)}(W)$  there is a unique $C^{(r)}_E\in(\Ph^{(r)})\i(E)$ which is as elliptic as
possible, that is $m(C_E^{(r)})\le m(C)$ for any $C\in(\Ph^{(r)})\i(E)$;

(b) For any $E\in\SS^{(r)}(W)$  there is a unique $C'{}^{(r)}_E\in(\Ph^{(r)})\i(E)$ which is as non-elliptic as
possible, that is $m(C'{}^{(r)}_E)\le m(C)$ for any $C\in (\Ph^{(r)})\i(E)$.
\nl
Now (a) is proved in \cite{L12}; the proof of (b) is similar.
Let $CL^{(r)}(W)$ be the image of the map $E\m C'_E$, $\SS^{(r)}(W)@>>>cl(W)$. Note that
$\Ph^{(r)}$ restricts to a bijection $CL^{(r)}(W)@>\si>>\SS^{(r)}(W)$.
Let $CL_{dist}^{(r)}(W)$ be the subset of $CL^{(r)}(W)$ corresponding to $\SS^{(r)}_{disc}\sub\SS^{(r)}(W)$
under this bijection. We have the following analogue of 0.4(c).

(c) {\it Let $C\in CL^{(r)}(W)$. We have $C\in CL_{dist}^{(r)}(W)$ if and only if $C$ is elliptic.}
\nl
Note that $CL_{dist}(W)=\cup_rCL_{dist}^{(r)}(W)$.

\subhead 3.2 \endsubhead
For any semisimple element $s\in G=G^{(0)}$ let $E_s=j_{W_s}^W(\text{sign})$ ($j$-induction)
where $W_s$ is the Weyl group of the connected centralizer of $s$ viewed as a subgroup of $W$.
We have $E_s\in\SS(W)$; the subset of $\SS(W)$ formed by the $E_s$ for various $s$ as above is
denoted by $\Irr_{ss}(W)$.

Now let $E\in\SS_{dist}(W)$ and let $C\in CL_{dist}(W)$ be the corresponding conjugacy class.
According to 0.5(a) we have $E=\Ph^{(0)}(C)$
hence $E=\io^{(0)}(\g)$ where $\g=\ps^{(0)}(C)\in\cu^{(0)}$.
(The elements of $\g$ need not be distinguished.)
From \cite{L11b} there exists a semisimple element $s\in G^{(0)}$ such that
$E=E_s$; thus, we have $\SS_{dist}(W)\sub\Irr_{ss}(W)$.
(We can assume that $G$ is almost simple; then
the statement in the previous sentence holds for $E=\Ph^{(0)}(C)$ for any
elliptic $C\in cl(W)$ with a single exception
in type $E_8$; but that exception is not distinguished, see 1.15.)
Note that $s$ belongs to the stratum of $G^{(0)}$ that contains $\g$.

Thus, to $E\in\SS_{dist}(W)$ (or $C\in CL_{dist}(W)$)
one can associate a collection of reflection subgroups $W_s$
(for various $s$ as above).

Now, \cite{L11a, 4.4(b)} implies that the minimum length of
an element in $C\in CL_{dist}(W)$ is equal to the dimension of the centralizer of $s$ (as above)
in $G$ modulo the centre of $G$.

\subhead 3.3\endsubhead
In the 3.4-3.10 we describe explicitly a correspondence

(a) $C\m W_s$
\nl
which to any $C\in CL_{dist}(W)$ associates a reflection subgroup $W_s$ of $W$ (up to
conjugacy) as in 3.2 (assuming that $W$ is irreducible of type $\ne A$); if $W$ is of type
$A$, then $C$ is the Coxeter class and the corresponding $W_s$ is $\{1\}$.

\subhead 3.4\endsubhead
We now assume that $G=Sp_{2n}(\CC),n\ge2$ or that $G=SO_{2n+1}(\CC),n\ge2$.
According to 1.4, 1.6, $CL_{dist}(W)$
can be identified with the set of pairs $(r_*,p_*)\in(R\T P)^{2n}$ such that $p_*=(0,0,\do)$
and $r_*$ is a sequence $r_1\ge r_2\ge r_3\ge\do\ge r_\s$ of even integers $>0$ without
two consecutive equalities. For such $(r_*,p_*)$ we define
$k=r_1/2$. We define $\br_1\ge\br_2\ge\do\ge\br_k>0$
by $\br_i=\sha(t\in[1,\s];r_t/2\ge i)$ for $i\in[1,k]$; note that $\br_1=\s$ and
$\br_1+\br_2+\do+\br_k=n$.
Using 0.5(a) and \cite{L11b},
we see that the reflection subgroup $W_s$ corresponding to $(r_*,p_*)$
is a product of symmetric groups
$$S_{\br_k}\T S_{\br_{k-1}}\T\do\T S_{\br_1}$$
(a subgroup of $S_n$ which is itself naturally a subgroup of $W$ of the form $W_J$ for some
$J$).

We see that $CL_{dist}(W)$ is in natural bijection with the set of sequences
$\br_1\ge\br_2\ge\br_3\ge\do$ of integers $\ge0$ such that
$\br_1+\br_2+\do=n$ and $\br_i-\br_{i+1}\le2$ for all $i\ge1$.

\subhead 3.5\endsubhead
We now assume that $G=S0_{2n}(\CC),n\ge4$. According to 1.8, $CL_{dist}(W)$
can be identified with the set of pairs $(r_*,p_*)\in(R\T P)^{2n}$ such that $p_*=(0,0,\do)$
and $r_*$ is a sequence $r_1\ge r_2\ge r_3\ge\do\ge r_\s$ of even integers $>0$ without
two consecutive equalities with $\s$ even. For such $(r_*,p_*)$ we define
$\br_1\ge\br_2\ge\do\ge\br_k>0$ as in 3.4. Note that $\br_1$ is even and $\ge2$.
Using 0.5(a) and \cite{L11b},
we see that the reflection subgroup $W_s$ corresponding to $(r_*,p_*)$ is
of the form
$$S_{\br_k}\T S_{\br_{k-1}}\T\do\T S_{\br_2}\T W_{D_{\br_1/2}}\T W_{D_{\br_1/2}}.$$
Here $W_{D_m}$ denotes a Weyl group of type $D_m$ (for $m=1$ this is taken to be $\{1\}$;
for $m=2$ this taken to be $S_2\T S_2$). We view
$$S_{\br_k}\T S_{\br_{k-1}}\T\do\T S_{\br_2}\T W_{D_{\br_1}}$$
as a subgroup of $W$ of the form $W_J$ for some $J$ and $W_{D_m}\T W_{D_m}$ as a
 reflection subgroup of $W_{D_{2m}}$ in the standard way.

We see that $CL_{dist}(W)$ is in natural bijection with the set of sequences
$\br_1\ge\br_2\ge\br_3\ge\do$ of integers $\ge0$ such that
$\br_1+\br_2+\do=n$, $\br_1$ is even and $\br_i-\br_{i+1}\le2$ for all $i\ge1$.

\subhead 3.6\endsubhead
Until the end of 3.10, $W$ is of exceptional type.
In each case the reflection subgroup $W_s$ attached by 3.3(a)
to $C\in CL_{dist}(W)$ is specified by its type. (We use 0.5(a) and \cite{L11b}.)

If $W$ is of type $G_2$, the correspondence 3.3(a) is:

$[G_2]\m A_0$; $[A_2]\m A_1$; $[A_1+\tA_1]\m 2A_1$.

Note that the group $W_s$ is of the form $W_J$ for some $J\sub I$
except for the last case.

\subhead 3.7\endsubhead
If $W$ is of type $F_4$, the correspondence 3.3(a) is:

$[F_4]\m A_0$; $[B_4]\m A_1$; $[F_4(a_1)]\m 2A_1$; $[D_4(a_1)]\m A_2+A_1$;

$[A_3+\tA_1]\m B_2+A_1$; $[\tA_2+\tA_2]\m A_2+A_2$.

Note that the group $W_s$ is of the form $W_J$ for some $J\sub I$
except for the last two cases.

\subhead 3.8\endsubhead
If $W$ is of type $E_6$, the correspondence 3.3(a) is:

$[E_6]\m A_0$; $[E_6(a_1)]\m A_1$; $[E_6(a_2)]\m 3A_1$.

Note that the group $W_s$ is of the form $W_J$ for some $J\sub I$.

\subhead 3.9\endsubhead
If $W$ is of type $E_7$, the correspondence 3.3(a) is:

$[E_7]\m A_0$; $[E_7(a_1)]\m A_1$; $[E_7(a_2)]\m 2A_1$; $[E_7(a_3)]\m 3A_1$;

$[A_7]\m A_2+2A_1$; $[E_7(a_4)]\m 2A_2+A_1$.

Note that the group $W_s$ is of the form $W_J$ for some $J\sub I$.

\subhead 3.10\endsubhead
If $W$ is of type $E_8$, the correspondence 3.3(a) is:

$[E_8]\m A_0$; $[E_8(a_1)]\m A_1$; $[E_8(a_2)]\m 2A_1$; $[E_8(a_4)]\m 3A_1$;

$[E_8(a_5)]\m 4A_1$; $[D_8]\m A_2+2A_1$; $[E_8(a_3)]\m A_2+3A_1$;

$[E_8(a_7)]\m 2A_2+A_1$; $[E_8(a_6)]\m 2A_2+2A_1$;

$[A_8]\m A_3+A_2+A_1$; $[E_8(a_8)]\m A_4+A_3$;

$[D_8(a_2)]\m A_3+3A_1$; $[D_8(a_3)]\m A_3+A_2+2A_1$; $[A_7+A_1]\m 2A_3+A_1$.

Note that the group $W_s$ is of the form $W_J$ for some $J\sub I$
except for the last three cases.

\subhead 3.11\endsubhead
Let $K$ be a maximal compact subgroup of $G^{(0)}$. The following result was stated in \cite{L21, 5.2}:

(a) {\it Let $X$ be a stratum  of $G^{(0)}$ and let $E$ be
the corresponding element of $\SS(W)$. We have $X\cap K\ne\emp$ if and only if $E\in\Irr_{ss}(W)$.}
\nl
By the results in 3.2, we have $E\in \Irr_{ss}(W)$ if and only if $X$
contains a semisimple element of $G^{(0)}$. This last condition is clearly satisfied when
$X\cap K\ne\emp$. Conversely, assume that $X$ contains a semisimple element $s$ of $G^{(0)}$. It is
well known that we can find $s'\in K$ such that the connected centralizers of $s$ and $s'$ are
conjugate. It follows that $E_s=E_{s'}$ hence $s,s'$ belongs to the same stratum. Since $s\in X$ we
have $s'\in X$ so that $X\cap K\ne\emp$. This proves (a).

We show:

(b) {\it Let $X,E$ be as in (a). If $E$ is distinguished then $X\cap K\ne\emp$.}
\nl
By (a) it is enough to show that $E\in\Irr_{ss}(W)$.
This follows from $\SS_{dist}(W)\sub\Irr_{ss}(W)$, see 3.2.

{\it Errata to \cite{L15}}.

p.355, line  containing $378_{14}$: replace $[A_3+A_2]$ by $[2A_3]$.

p.356, line containing $3200_{22}$: replace $[(A_5+A_1)']$ by $[(A_5+A_1)',A_5]$.

\widestnumber\key{ABCD}
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\publaddr Providence, R.I.\yr2012\endref
\ref\key{L84}\by G.Lusztig\paper Intersection cohomology complexes on a reductive group
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\ref\key{L11a}\by G.Lusztig\paper From conjugacy classes in the Weyl group to unipotent classes
\jour Represent.Th.\vol15\yr2011\pages494-530\endref
\ref\key{L11b}\by G.Lusztig\paper On C-small conjugacy classes in a reductive group
\jour Transfor.Groups\vol16\yr2011\pages807-825\endref
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From conjugacy classes in the Weyl group to unipotent classes II\jour
Represent.Th.\vol16\yr2012\pages189-211\endref
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arxiv:2010.13642\endref

\endRefs
\enddocument

redpajama_set_name: RedPajamaArXiv


text:
Hillsborough Park's new Snake Slide with Councillors Bob Johnson and Mary Lea alongside Friends of Hillsborough Park's Christine Welburn. (Photo: Sheffield City Council)
New attractions now open at popular Sheffield park playground.
Snakes Head Slide at Hillsborough Park Playground
The popular playground in Sheffield's Hillsborough Park has reopened complete with a number of improvements.
The first phase of a bigger redevelopment of the playground has completed with the opening of a brand new snake-themed slide and obstacle course.
Damaged playground apparatus has been removed and replaced also as part of this initial phase of improvements.
"I think we can all agree that this first phase of improvements has already made a huge difference to the playground and I know that local people are very excited to start using the new equipment.
"Parks have been a lifeline for people during this pandemic, for leisure, fun, exercise and fresh air and we understand now more than ever just how important it is that we continue to invest in our fantastic green spaces." Councillor Mary Lea, Cabinet Member for Culture, Parks and Leisure
Playground improvements made with the help of the Friends of Hillsborough Park
The £50,000 worth of improvements came with the collaboration of Sheffield City Council's Parks and Leisure department and the Friends of Hillsborough Park community group.
Residents and visitors were provided with three different options at an event in Hillsborough Park's Walled Garden and the popular choice was adopted.
Funding was raised from a number of different avenues, including Public Health funding and a donation from the Tramlines Festival, which is held in Hillsborough Park.
"The Friends of Hillsborough Park are excited that the first stage of the playground renovation is complete. The new equipment lifts the whole park and the snakes head is stunning. Local families and children deserve to have such an innovative play area. We look forward to the communities' support for fundraising towards the next phase of development." Christine Welburn, Friends of Hillsborough Park
Further improvements coming to Hillsborough Park
Hillsborough Park is increasingly growing in its designation as a 'destination park' with further improvements to come.
An all-wheeled bike track is set to be built, whilst the Hillsborough Library's old Coach House is in the process of being converted to a dementia-friendly cafe.
"Hillsborough Park has always been one of the city's most loved parks, and with the new improved playground, plus our plans for phase 2, the All Wheeled bike track and Coach House community café next year, it is going to be a real destination for recreation in 2021." Councillor Mary Lea, Cabinet Member for Culture, Parks and Leisure
Visitors are reminded to utilise Hillsborough Park Playground space and equipment in line with current coronavirus guidance and restrictions.
"Hillsborough Park is real local gem, meeting the needs of so many people of all ages. With the first phase of playground improvements now complete, including the great new snake's head slide, it's brilliant that we can reopen and welcome families back on to the play equipment.
"This was designed in partnership with local people, so we're really pleased that they can now enjoy the benefits of this investment and look forward to developing phase two with the local community firmly at the heart of it." Councillor Bob Johnson, Cabinet Member for Transport and Development and local councillor Hillsborough ward
"We have been very fortunate with phase one to be able to use some of the Public Health money committed in this final year of investment, as well as a generous donation from Tramlines and other funding projects, which we are extremely grateful for. We can't wait to get started on plans for phase two which will provide an outstanding play facility for thousands of children across the city." Councillor Mary Lea, Cabinet Member for Culture, Parks and Leisure
In this article:Bob Johnson, Cafe, Children's Playground, Christine Welburn, Coronavirus, Friends of Hillsborough Park, Hillsborough All-Wheeled Bike Park, Hillsborough Library, Hillsborough Park, Hillsborough Park Walled Garden, Mary Lea, Old Coach House, Public Heath Funding, Sheffield, Sheffield City Council, Tramlines Festival
Great to see that projects like this are still going on in our current situation! 😀


text:
What the Next President Faces
November 2, 2020 Winter Watch Around the Web, Politics, US News 0
PHOTO: Buchanan.org
Given the cards he has been dealt in 2020, and the hatred of the media he daily confronts, it is astonishing that Trump retains his energy and enthusiasm for the battle. Most presidents would have long ago been broken… Should Joe Biden win, he would be, on Jan. 20, 2021, the oldest and most visibly enfeebled leader to win the presidency in the history of the republic …
By Patrick J. Buchanan | 30 October 2020
BUCHANAN — Of the presidents in the modern era, many have been dealt a difficult hand by history, but perhaps none more so than Donald Trump.
In 1952, Harry Truman was in his third year of a stalemated war in Korea that was costing 200 American lives every week. He lost the New Hampshire primary to Sen. Estes Kefauver and decided to pack it in.
In 1968, Lyndon B. Johnson had also been challenged in New Hampshire, by Sen. Eugene McCarthy. And, he, too, had on his hands a seemingly endless Asian war if he was not prepared to escalate militarily and add hundreds of thousands more troops to the 500,000 already in Vietnam.
Like Truman before him, LBJ stood down.
In 1980, Jimmy Carter also had a challenge from within his party — Edward Kennedy. And for the entire last year of his presidency, 52 U.S. hostages were held in Teheran while Carter presided over an economy where the interest rates had hit 21% and inflation 13%.
Trump had no primary challenger. He had not taken us into any new wars. And he had begun 2020 with the U.S. economy firing on all cylinders. But it all crashed in March and April in the worst pandemic in a century, which destroyed his economy and has since consumed a quarter of a million American lives.
October of Trump's reelection year saw a new wave of COVID-19 infections and the bottom fall out of the stock market. […]
Operation Keelhaul: Britain's Repatriation of Anti-Communist Russians and Ukrainians to Stalin for Certain Death
The FBI's Increasingly Odd Silence on Boston Bombing
CIA Drug Smuggling and Dealing: The Birth of the Dark Alliance
US Secretary of State Antony Blinken is Yet Another in a Long Line of Made Men


text:
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The World Bank's $12B COVID-19 Fund Can Have Lasting Impact: Here's how
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To say that COVID-19 is affecting everything, not just healthcare, is no hyperbole. So far, it's responsible for more than 395,000…
Gambia's Third COVID-19 Case Recorded in URR
The governor's office of the Upper Region (URR) has confirmed a positive Coronavirus (COVID-19) case in the region on Tuesday,…
How Gambians React to the Coronavirus Outbreak in the Country
The Coronavirus outbreak has caused a great concern among Gambians and the world in general. With most countries in Asia, Europe,…
Gambia's second COVID-19 Patient Dies
The Gambia's health minister, Amadou Samateh has confirmed Sunday the death of a 70-year-old Bangladeshi national in the country…
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The Gambian authorities have confirmed the first Coronavirus case (COVID-19) on Tuesday, but the country still maintains its…
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Feed-In Tariffs were introduced in April 2010 to provide financial incentives for individuals and businesses to install renewable energy generating technologies such as solar electricity panels and wind turbines. It does this by guaranteeing a minimum payment for the electricity generated, as well as any electricity exported to the grid.
What are the options for installation?
It's worth noting that there are two types of panel; Solar Photo Voltaic and Solar Thermal. Solar PV generates electricity and solar thermal heats up water. The Feed-in Tariff relates only to solar PVs and there are two ways to go about it.
You can fund the installation of the panels yourself. This involves upfront capital and you are guaranteed a Feed-in tariff for 20 years. The cost of installation will depend on the size of the installation and can vary from £8-£25,000. The payback is usually around 8-10 years (usually dependent on the orientation of the roof) after which you should make a healthy return on investment until the 20 years is up.
Export tariff: your energy supplier will pay you a further rate for each unit you export back to the electricity grid, so you can sell any electricity you generate but don't use yourself. At some stage smart meters will be installed to measure what you export, but until then it is estimated as being 50 per cent of the electricity you generate (only systems above 30kWp need to have an export meter fitted, and a domestic system is unlikely to be that big).
The second way of installing Solar PV is to work with a company who will entirely fund the installation – essentially the club is leasing its roof to the company. They will then sell you the electricity you need for about half the price you usually pay. The company then gets the Feed-in Tariff.
What other eligibility criteria is there?
What is changing with the Feed-in Tariff scheme?
The generous initial rates of FIT meant that the uptake of solar panels was faster than expected so the government is planning to reduce FIT rates from 1 January 2016 by as much as 87%. For example, for the 4kWh band, the generation tariff will reduce from 12.92p to 1.63p p/kWh.