text:
Public Safety Canada
Connecting with Canadian Communities
Counter Proliferation
National Security – More
Border Strategies
Beyond the Border
Border Law Enforcement
Immigration Enforcement
Preclearance in Canada and the United States
Canada-US Cross Border Crime Forum
Border Strategies – More
Countering Crime
Contraband Tobacco
Child Sexual Exploitation on the Internet
Human Smuggling
Gun and Gang Violence
Memorial Grant Program for First Responders
Countering Crime – More
Disaster Prevention and Mitigation
Natural Hazards of Canada
Responding to Emergency Events
Recovery from Disasters
Emergency Management – More
Canadian Disaster Database
Service Standards for Transfer Payment Programs
Resources – More
Research Summary: Civilianization of Police in Canada
Research Summary: Civilianization of Police in Canada PDF Version (97 Kb)
While civilianization may result in limiting budget increases, the overall cost of policing may not be reduced.
According to Statistics Canada, after many decades of stability, the proportion of civilian personnel working in Canadian police services increased from 26 percent of all employees in 2001 to 29 percent in 2014, or 2.4 sworn police officers for every civilian worker, down from 2.9 in 2001. While in the past many civilian employees held clerical positions, they now also carry out duties previously performed only by sworn officers including administration and management, specialized functions such as crime analysis, sophisticated investigations (e.g., forensic accountants), and public relations. This study of civilianization of police in Canada focuses on the costs and benefits, along with organizational challenges, for police services.
A survey was sent to a purposive sample of small, medium, and large-sized police services throughout the country and follow-up interviews were conducted with police executives from responding organizations to further explore issues of interest uncovered in the literature.
The survey results and interviews reveal that many respondents believe that civilianization has reduced policing costs as the salaries and benefits paid to civilians are typically less than those paid to sworn officers. These respondents contend that training costs in particular can be reduced significantly over the long term by hiring civilians who already possess specialized knowledge rather than by providing additional training for sworn officers to carry out specialized roles such as crime analysis. This is thought to be the case because a sworn officer's tenure in a specialized role such as crime analyst may be relatively short and their replacement by other sworn personnel would require repeated training to develop the appropriate skill set. This contrasts with civilians, who may occupy these positions either for their entire careers or on an "as needed" basis.
Findings on the issue of turnover, however, are mixed. While some police executives report that civilian job continuity is the norm, the turnover of employees with highly specialized or expert skills is sometimes reported to be high. Other reported benefits of hiring civilians are difficult to quantify. These include building relationships with ethno-cultural or professional communities not represented in the sworn officer contingent, offering a different perspective on problem-solving than that provided by sworn officers, and enabling sworn officers to return to traditional policing roles for which they are specifically qualified.
Although introducing civilian personnel into police services is being promoted as an effective cost‑savings measure, it is neither well-supported by evidence nor without challenges. Perhaps the most significant of challenges are the difficulties civilians experience adapting to a police services' organizational culture. Interviews with police executives reveal that their civilian employees do not always feel accepted, especially when they work alongside sworn police officers or in roles that had previously been carried out by sworn officers. Many police officers, for example, do not believe that civilian personnel share their values or entirely understand front-line policing. These findings are consistent with the review of the literature on civilianization which shows that the morale of civilian employees is lower than that of sworn officers. Lower morale among civilian personnel might also be a consequence of fewer opportunities for movement in police organizations.
There is some opposition to increasing the occupational roles and the proportion of civilian personnel working in police services, particularly in certain areas. Police associations generally oppose hiring civilians for positions that had previously been filled by sworn officers on the basis that this restricts opportunities for sworn officers' career growth. Police executives are sometimes reluctant to extend the roles assigned to civilian employees as this reduces the flexibility of the organization in exercising their duty of accommodation. This could include placing officers into meaningful jobs when they are unable to work "on the streets" because they are disabled, injured, or under investigation. Many respondents also referred to the challenges of supervising civilians when their duties fall under different collective agreements than those governing sworn officers.
Some challenges of civilianization were unforeseen. For example, once a civilian employee gains a foothold in a job role previously carried out by sworn officers, their professional associations may file grievances if similar vacant positions are not filled by civilians. Moreover, though not opposing civilianization, many police governance boards prefer to hire sworn officers on the premise that their visible uniformed presence is thought to increase public support.
Pressures from provincial and municipal governments to reduce policing costs are likely to continue given the rapid increases that have characterized the past decade, the revenue constraints most police services are now confronting, and the demand to get the best value for tax dollars. In addition, an increasing number of legislative requirements have swollen the administrative workload of police organizations. It is likely that the pressure to hire more civilians will be greatest in larger police services, given the variety of demands placed on them by the complex environments in which they serve. The largest police services in this study had the highest proportion of civilian personnel.
Altogether, the results of this project suggest that decisions about increasing the proportion of civilian personnel working within a police service should be based not only on careful identification of the overall direct short-term financial implications for the organization, but also on indirect costs and benefits over the long-term. A key finding of both the literature review and empirical research is that the overall costs of policing may not be declining as the number of sworn officers does not appear to decline commensurate with increased numbers of civilian employees. Indeed, the overall number of police (both sworn and civilian) continues to rise and costs to increase. There seems to be reluctance, for a variety of reasons, on the part of police executives, police boards and other governance bodies to reduce the number of uniformed sworn police officers, the most publicly visible component of police personnel.
In sum, the growth in numbers of both sworn officer positions and civilians and the continuing rise in policing costs is a significant phenomenon that merits thorough research and analysis.
Kiedrowski, J., Melchers, R. F., Ruddell, R., and Petrunik, M. (2016). The Civilianization of Police in Canada. Ottawa, Ontario: Public Safety Canada.
For more information, or to get a copy of the full research report, or to be placed on our distribution list, please contact:
Ottawa, Ontario, K1A 0P8
PS.CPBResearch-RechercheSPC.SP@ps-sp.gc.ca
Research Summaries are produced for the Community Safety and Countering Crime Branch, Public Safety Canada. The summary herein reflects interpretations of the report authors' findings and do not necessarily reflect those of Public Safety Canada.
Associate Deputy Minister
About Us – More
News – More
Contact Us – More
Stay Connected – More


text:
Cheiron Songwriters and Max Martin fan-forum and music discussion board www.swedishsongs.de - All about Swedish songwriters and music
The Songwriters => New Songs => Topic started by: Vanth on August 07, 2007, 12:39:43 AM
Title: Martin involved in the new BSB album?
Post by: Vanth on August 07, 2007, 12:39:43 AM
I'm new to this forum and i'm from germany.
I wrote an email to cheiron studios because i wanted to know if max martin is producing any songs in the upcoming BSB album.
All i got was: ,,You'll never know..Regards". Well, i was surprised they've even written back. ^^ Kind of honor..
Well, i wonder if anybody of you guys got some news to the new BSB album..The songs circulating in this forum are never cheiron-style..They're even kind of crappy in my eyes.
Would be very sad if the new BSB album got no songs bei cheiron, because their voices are brilliant!
Well so far..
PS: Sorry for my fucked up english. :D
Post by: Rebecca on August 07, 2007, 02:06:22 PM
Hi Vanthor, welcome to the forum
wow, you got a reply, do you mean from Maratone Studios?
great they replied, even though they didn't give you much of an answer
I don't have much news of the new album, it won't be long until it's released though, I think October.
I know Dan Muckala has worked with them, not too sure of the other songwriters.
Post by: turnaround on August 07, 2007, 05:48:25 PM
Hallo Vanthor, yes welcome to the forum!
Hope you like it, another German user, I am glad ;)
Concerning the album, the last information we had was, that there will be not a Max Martin song on it.
There are some information in this thread
http://songwriter.so.ohost.de/viewtopic.php?t=74
I still don't understand why, but who knows what will happen!
Post by: Vanth on August 07, 2007, 06:18:23 PM
Yes i meant from Maratone studios..i just wrote an email to info@maratone.se :D...
And well..they said ,,You'll never know.." That doesn't mean no. :)
The behavior of max and his studio ist very ridiculous and childish..okay, they're very good producers but very arrogant like they're the popstars not the bsb :D..
And the whole secret crap..maybe it's kind'a trick to dube the fans..
i don't know..i'm bargain for the worst..:(
Post by: LJ on August 07, 2007, 06:40:33 PM
Quote from: "Vanth0r"
Honestly... you're really unknowledge if you call Max Martin ridiculous and childish and he is far from arrogant. What grounds do you even base such ignorant statement from? You seriously upset me by saying that, because it just proves you have no idea what you are talking about. Secondly, it's not Max Martin that reply to your email, and the person that does, has probably no clue. BSB haven't worked with Max Martin. That is 99,99% certainty.
Very probably it was just a webmaster or someone else who replied, and he doesn't give out any concrete information about anything business related ;)
Still I am pretty surprised they answered anyway, and the answer is basically a "No", that's how I'd interprete it.
I know that Max didn't answer to my email..It was a guy called Mark..
And not to reply to calls and emails from the BSB who actually made him so famous is kind of childish and arrogant...
And all those secretivenesses about him and his studio is sort of weird in my eyes..
Maybe he isn't arrogant and stuff, but he seems to me, i'm sorry..Official statements from his site would be adequate and fair.
He have to get straigth about his fans. Even when he gets interviewed his answer are very short, mysterious and lazy like he is beware of secrets..
This is my impression..but he'll be the greatest producer to me ever!
Maybe all my impressions are the excrescences of my anger about the new bsb-album without him. :(
Yeah, I can't understand it either, maybe there will be another interview soon which will clear this up.
I haven't seen Max in an interview yet, or read one, so it's hard for me to judge, but I think there is a reason why these guys are in the background of the music scene.
I would still love to hear from them more, Savan luckily writes on boards, it's VERY interesting to hear from their work and the latest news.
Well, I can just tell you that you are completely wrong about him. He is not the person you make of him... but I don't feel like sitting here to try to prove you wrong. It just made me sad to see how someone who doesn't know him put an image of him that is so completely wrong.
Vanthor, I think you have a wrong view of Max too I'm sorry. I was disappointed too to hear that Max didn't work with BSB (I think it's pretty certain he didn't), but times change and people move on, Max has already given BSB a lot of hits.
In an interview with Max, I can't remember which one, he is asked why he doesn't do much press and he replies that it's 'not his job'. Which I agree with, it's the job of the artist to promote themselves and the songwriter can choose to stay in the background.
I think it's great that somebody, even the receptionist replied to you. Max and Rami can't reply to everyone personally - they are busy producing.
Post by: Tiiger on August 08, 2007, 03:16:34 PM
Maybe they realised they can't write any hits of their own.
I mean, who do they think they are?
Post by: NIKLAS"NiCkY" on August 11, 2007, 11:23:49 AM
I think it´s sad if not Martin,Rami or any of the swedish producers are involved in BSB next album...For me BSB are Max/Rami....they did all the hits and who out there dosen´t want a new shape of my heart or climbing the walls, get another boyfriend, don´t wane lose you???.. I am not sure if i will buy the album if any of the ex cheiron guys not are involved..
just want to say this.....Backstreet boys are Max Martin/Rami, or Max Martin/Rami are BSB
BSB new single is incosolable
That's right!
The other tracks on the album were often just fillers. The Cheiron tracks made them successful!
Post by: Linda on August 11, 2007, 06:16:06 PM
yea I don't think I will be buying the new album because of this ......
its quite sad.. not to have a max/rami song on the new album... I mean after all these years....
it is sad not to have a Max or Rami song on BSB's new album. I would love more than anything to have a new Shape of my Heart or Climbing The Walls or I Still or The Call, but i guess it's not to be.
I will still buy the new BSB album no matter what! i am still a diehard fan of theirs, they are still my favourite singers.
Let's hope that by Max and Rami not writing for BSB, they are saving some awesome songs for other singers.
Post by: ANJ600 on September 02, 2007, 10:00:43 AM
Although it is not yet 100% confirmed, it seems Max/Maratone is not involved in the writing nor the producing of BSB's new album "Unbreakable".
Title: Re: Martin involved in the new BSB album?
Post by: ahmed on September 10, 2007, 11:29:37 PM
why max is not working with backstreet or britney ,its really dissapointing cuz he had his best production with the both,i was on wikipedia looking if max will work with backstreet and found that one of the member was in an interview and said they r not working with max for that album
Post by: Tiiger on September 11, 2007, 12:09:30 AM
I really don't think it's up to Max if artists want to work with him. They come to him, not the other way around.
Britney will be running back into his arms sooner or later, of that I'm sure.
Post by: ahmed on September 11, 2007, 02:29:47 AM
i hope u r right about britney cuz i really believe that she needs max martin songs , and do u think there will be later for her ,cuz they said her performance for her new song gimme more slammed by many people even her fans,and she was seen crying backstage and that her career could be over soon
Britney has destroyed her image all by herself. Even if she approaches Max to "save her", I hope Max tells her to get lost. She doesn't deserve anymore of his talent. She did not want Max to work on her last album "In the Zone" and when she saw how awful her sales and songs were for that album, (compared with her other albums) she saw she needs Max and Maratone to dig herself out of the pile of crap she has buried herself under. She is taking advantage of Maratone by "wanting them only when she needs them"
In fact Max should just focus on other upcoming stars like Marion Raven. Brintey doesn't want to do "pop" music anymore and why the heck she has approached Max, I don't know. Because if Britney even thinks about singing rock (the music Max is now producing) her career will be over faster than ever and she will be gone from the spotlight.
i agree with u that britney dstroyed her image by herself but if u mean her personal life sorry but its not our business and everybody can live the life the way he \ she want but all i'm talking about is her music an i never enjoyed any artist's music like i enjoy her music especially with max ,and i think any other star will never be like britney especially when she was under max martin controll as a style ,music or charisma she was really amazing ,and i hope she sooner or later get musical help from max in the case that her career lasted and i hate her new song gimme more and i hate all the producers she is working with for that album except max if he is on the album haha , and i wish someday she go to max into max martin arms to bring the old britney back .
p.s: i think britney career will be over soon especially after the last vma performance which was slammed by many people even her fans and they say she was seen crying backstage.
Post by: Rebecca on September 11, 2007, 12:34:11 PM
I agree that Britney has destroyed her image all by herself. It will be very interesting to see if she does work with Max, it wouldn't surprise me if she doesn't.
Post by: Linda on September 11, 2007, 09:15:57 PM
Britney hasn't destoyed her image.. by herself...
the media helped, a LOT!
Post by: Tiiger on September 11, 2007, 11:49:36 PM
America just isn't a good place to live in. I wonder why almost all young American singers/actors do drugs and flaunt their vaginas?
This would never, NEVER, happen in Sweden.
there r many young stars does things worse than britney look at TATU girls and many others every where but because the american media is strong and have the upper hand so its effective,i wish britney was european.
Yes, the media has helped to destroy Britney's image - they are so cruel to her. Though she hasn't done herself any favours. I thought she looked really pretty and did some good dancing at the MTV performance but I only saw a little bit of it on the news. I'm feeling a bit sad about it all, I hope she can get it together and make a proper comeback. Though good for her for giving it a go - sometimes I get tired of all the perfect celebrities with their flawless performances.

redpajama_set_name: RedPajamaCommonCrawl


text:
22 of 58 for 58: Take Yourself (and Your Work) Seriously
I hadn't given a lot of thought to the career of Ethan Hawke. Then I read this morning's profile in the New York Times.
It seems Hawke's work had been routinely and harshly criticized for years. Now, it seems he's receiving more than his share of respect and even acclaim.
How? By doing the work he felt he needed to do. By shunning the words of critics. By persisting in the face of all obstacles. By recognizing that his actions themselves were just reward. By admitting that he might not receive recognition in this lifetime.
These lessons are hard to reconcile with the current obsession with fast and easy fame. At best, they offer some salve in the face of unrelenting online attacks against those who dare to think and act differently.
His story in particular challenges my own obsession with criticism. I may receive overwhelmingly positive feedback on a project. Still, I latch on to negative comments. I do so with a ferocity that seems irrational. That tendency clouds the inner voice that inspires my action. It gets in the way of my personal and professional growth.
What a great ideal Ethan Hawke's career offers. He has taken his life and work seriously, what others say be damned. I don't know how he did it; he just did. And he's a role model for any of us grappling with doubts and a world seemingly hostile to our best efforts.
Take yourself seriously. Just do the work you were meant to do.


Usually, each time you install a new application it is registering it's own filetype. But not all of them remove it from the registry when you are uninstalling it. Also you can simply remove the application from the hard disk but registry remains unchanged. This tab helps you delete such "lost" filetypes from the registry. Find title of filetypes in the list that are belongs to non-existent application (usually it has no icon) and remove it from the list.


Once a wheat plant starts to move from vegetative growth to reproductive growth in early spring it can be thought of as an upright cylinder.
All leaves are approximately the same size and live for approximately the same length of time. This means that in theory they could deliver the same yield for the plant. And they would if it wasn't that the lower leaves, those below final leaf two (counting down from the flag leaf), receive far less light than the upper leaves due to canopy shading.
The result is that the top two leaves of a wheat plant deliver around 70% of the yield, while the ear contributes a further 20%. Therefore these leaves and the ear must be kept disease free for as long as possible, to fully express a crop's yield potential.
How well do you know your farm?
Below is an editable PDF designed so that you can record simple but powerful data which will allow you to make informed decisions in future years.
What data do we suggest you record?
We suggest keeping it simple, focusing on plating date, crop growth stage and fungicide application dates. The aim is to strengthen your position in future years about the date GS39 will occur as this is a critical fungicide application timing for wheat. Once you know when this will occur, the other fungicide timings tend to fall into place. As your data set grows with time, it will be an invaluable record for you from your farm.
Would you be prepared to share?
Recording data on your farm in really important, but being able to benchmark how your crops perform alongside your neighbours and others in your region is invaluable. Use the downloadable PDF below to keep your own records or share your data by clicking 'submit' on the PDF and Bayer will collate the data and distribute findings from across the region.
There are three critical timings for fungicide application to a wheat crop and these should be adhered to almost without exception. These are GS32, GS39 and GS65. This programme has stood the test of time being the approach taken to control disease in wheat since the 1970s.
The first critical fungicide application should be made at GS32. At this growth stage final leaf 3 has emerged and importantly part of final leaf 2 is visible.
While leaf 3 can contribute 10% of the final yield the real importance of protecting this leaf from disease is that it prevents disease epidemics from getting established.
An effective fungicide application at GS32 will give 25-28 days protection to leaf 3 and to part of leaf 2, allowing the crop unhindered growth as it lays down the foundations for yield.
GS39 is the most critical, the "must-do", fungicide application for wheat.
At GS39 both the flag leaf and leaf 2 have fully emerged and are vulnerable to disease attack.
With these leaves contributing 70% of yield that can't be allowed to happen.
Applying a robust fungicide application at GS39 will give 25‑28 days protection to the flag leaf.
It will also help to reduce disease development on leaf 2 even though this leaf has been present for a while.
A fungicide applied at full ear emergence serves three main purposes.
It protects the ear from disease infection thereby protecting 20% of the final yield, it protects against diseases reducing grain quality and it tops up the fungicide protection of the flag leaf.
Then there are some other, less critical timings which deserve mention. They are described here because at times they are necessary and also they are discussed in a European disease control articles.
This application should be considered when crops emerge from winter with a heavy burden of Septoria leaf blotch or stripe rust that needs to be controlled urgently.
This application is in addition to a GS32 application.
Last year Bayer was faced with a situation in its wheat fungicide trial on the lower North Island where the crop grew slower than normal due to cold, cloudy weather. Here the interval between the GS32 and GS39 stretched to 35 days and fungicide control began to break down.
Consultations with Bayer experts in the UK showed that in some seasons they experience similar problems.
The solution is to apply an additional fungicide at GS33. To adopt a GS32, GS33 and GS39 programme.
Clearly this isn't ideal and it can be made worse when a GS30 spray has also been applied. The most effective way of minimsing this happening is to ensure that the GS32 spray isn't applied too early. Wait until GS32 and don't be tempted to spray early at GS31.
In NZ a mixture of two fungicides, Folicur® plus a strobilurin fungicide are sometimes applied at this late growth stage, especially on irrigated crops, to prolong green leaf area.
This approach protects against most of the late season diseases while providing a sound resistance management approach. Ensure the withholding period of both fungicides is adhered to.
Illustrated above is a high performing but cost effective fungicide programme to protect an autumn or winter planted wheat crop. This programme will provide disease protection throughout the season.


Designed in house to fulfill all your bike jersey needs.
Outtabounds custom house coaches jacket.
Designed in house and printed in Saskatoon.
Locally designed, using imagery of the Broadway bridge in Saskatoon.
We love Corduroy on the slopes and on our hats.


text:
\section{Introduction} \label{intro}
In the representation theory of $p$-adic groups, in particular, in the framework of the local Langlands correspondence for a connected reductive algebraic group $\bold G$ over a $p$-adic field $F,$ it is of great importance to study the reducibility of parabolically induced representations.
This study yields information on constructing tempered $L$-packets of $\bold G$ from discrete $L$-packets of its $F$-Levi subgroup.
The determination of reducibility has been developing over decades via several approaches, for example, by means of harmonic analysis from investigation of poles or zeros of residues of intertwining operators, Plancherel measures, and local $L$-functions, \cite{sh90, sh95, sil79}.
The method we address here is in terms of the Knapp-Stein $R$-group, which provides a combinatorial description of the tempered dual of $\bold G(F)$ as well as its elliptic tempered spectrum.
Further, as conjectured by Arthur, the isomorphism of the Knapp-Stein and Langlands-Arthur $R$-groups, via the endoscopic $R$-group, plays a significant role in the comparison of trace formulas and the endoscopic classification of automorphic representations \cite{art12, kmsw14, mok13}.

While there has been a great deal of progress on the theory of Knapp-Stein $R$-groups for $F$-quasi-split groups $\bold G,$ little is known for non $F$-quasi-split groups $\bold G'.$
In \cite{chgo12}, we investigated the behavior of $R$-groups between $F$-inner forms of $\SL_n,$ and determined the Knapp-Stein $R$-groups for the non quasi-split inner form from those of the split $\SL_n.$
We also proved the Knapp-Stein $R$-group for $\bold G'(F)$ embeds as a subgroup of the $R$-group for $\SL_n,$ and we characterized the quotient.
Another approach to this case was carried out in \cite{chaoli} and an example was discovered for which the Knapp-Stein $R$-group for $\bold G'(F)$ is strictly smaller than that of $\SL_n.$
In \cite{cgclassical}, we further showed the invariance of $R$-groups between $F$-inner forms of quasi-split classical groups $\SO_{2n+1},$ $\Sp_{2n},$ $\SO_{2n},$ or $\SO_{2n}^*,$ and transferred all known, relevant facts developed by the second-named author in the quasi-split classical groups  to their non quasi-split inner forms.
Non quasi-split inner forms of $\Sp_{4n}$ and $\SO_{4n}$ are also treated in \cite{han04}.

To study the Knapp-Stein $R$-group for a non quasi-split group $\bold G'$ in general, as one may notice from our previous works \cite{chgo12, cgclassical}, it is natural that we investigate its behavior between $\bold G$ and $\bold G'$ and transfer the developed theories regarding the Knapp-Stein $R$-group from $\bold G$ to $\bold G'.$
This is due to the notion of Langlands functoriality.
Since $\bold G'$ is induced from a new Galois action twisted by a Galois $1$-cocycle in the inner automorphisms of $\bold G,$ their $L$-groups are isomorphic over the Galois group and the set of $L$-parameters for $\bold G'$ is contained in that for $\bold G.$
In this paper, combining this strategy and the restriction method, we study the Knapp-Stein $R$-group for non quasi-split $F$-inner forms of quasi-split special unitary groups.

More precisely,
we fix a quadratic extension $E$ of a $p$-adic field $F$ of characteristic zero.
Let $\bold G_n = \SU_n$ be a quasi-split special unitary group over $F$ with respect to $E/F$ and let $\bold G'_n$ be its non quasi-split inner form over $F$.
A simple consequence from the Satake classification or a computation of Galois cohomology reduces our study to the case when $n$ is even. In fact, there is a unique non quasi-split $F$-inner form $\bold G'_n,$ up to $F$-isomorphism (see Section \ref{structure}).
For the rest of the introduction, we assume that $n$ is even, unless otherwise stated. Let $\bold M'$ be an $F$-Levi subgroup of $\bold G'_n,$ which is an $F$-inner form of an $F$-Levi subgroup $\bold M$ of $\bold G_n.$ Then, $\bold M = \boldsymbol{\mathcal{M}} \cap \bold G_n$ and $\bold M' = \boldsymbol{\mathcal{M}}' \cap \bold G'_n,$ where $\boldsymbol{\mathcal{M}}$ is an $F$-Levi subgroup of a quasi-split unitary group $\boldsymbol{\mathcal{G}}_n = \U_n$ over $F$ with respect to $E/F$ and $\boldsymbol{\mathcal{M}}'$ is an $F$-Levi subgroup of a non quasi-split $F$-inner form $\boldsymbol{\mathcal{G}}'_n$ of $\boldsymbol{\mathcal{G}}_n.$
We shall use $G$ for the group  $\bold G(F)$ of $F$-points of any connected reductive algebraic group $\bold G$ over $F.$

Given an elliptic tempered $L$-parameter $\phi \in \Phi_{\disc}(M),$
by \cite[Th\'{e}or\`{e}m 8.1]{la85}, we have a lifting $\varphi \in \Phi_{\disc}(\mathcal{M})$ commuting with the natural projection $\widehat{\mathcal{M}} \twoheadrightarrow \widehat{M},$
where $\widehat{\mathcal{M}}$ and $\widehat{M}$ respectively denote the connected components of the $L$-groups of $\boldsymbol{\mathcal{M}}$ and $\bold M$ (see Section \ref{pre} for the details).
Restricting the $L$-packet $\Pi_{\varphi}(\mathcal{M})$ constructed by Rogawski \cite{rog90} and Mok \cite{mok13}
(see Section \ref{LLC for unitary}),
we construct an $L$-packet  $\Pi_{\phi}(M)$ as the set of isomorphism classes of irreducible constituents in the restriction from $\mathcal{M}$ to $M.$
All the arguments used for $\bold M$ can be applied to  $\bold M',$ as in Kaletha-Minguez-Shin-White \cite{kmsw14}
and the details are described in Section \ref{L-packets}.
For any $\sigma \in \Pi_{\phi}(M),$ and $\sigma' \in \Pi_{\phi}(M'),$ we prove
\[
R_{\sigma} \simeq R_{\phi, \sigma} ~~ \text{and} ~~R_{\sigma'} \simeq R_{\phi, \sigma'}.
\]
In each of the above isomorphisms, the left side is the Knapp-Stein $R$-group, and and the right side is the Langlands-Arthur $R$-group, defined in Section \ref{section for def of R}.
This is known as Arthur's conjecture, predicted in \cite{art89ast}, for $\bold G$ and $\bold G'$ (See Theorem \ref{arthur conj}).
In the course of the proofs, we apply some known results about $R$-groups for
$\boldsymbol{\mathcal{G}}_n$ and $\boldsymbol{\mathcal{G}}'_n$
in \cite{go95, kmsw14, mok13}, which are recalled in Section \ref{R-groups for U(n) and its inner forms}.
We also investigate and utilize some relationships between identity components of the centralizers of the images of $\varphi$ and $\phi$ in $\widehat G$ and $\widehat \mathcal{G}$ (see Section \ref{L-groups} and Lemma \ref{lm for identity comp}).

We further study the invariance of the Knapp-Stein $R$-groups for $\bold G_n$ and $\bold G'_n.$
Namely, given $\sigma_1, ~ \sigma_2 \in \Pi_{\phi}(M),$ and $\sigma'_1, ~ \sigma'_2 \in \Pi_{\phi}(M'),$ we prove
$
R_{\sigma_1} \simeq R_{\sigma_2}, ~~ \text{and}~~
R_{\sigma'_1} \simeq R_{\sigma'_2}
$
(Theorem \ref{invariance within L-packet}).
Moreover, given $\sigma \in \Pi_{\phi}(M)$ and $\sigma' \in \Pi_{\phi}(M),$ we prove
$
R_{\sigma} \simeq R_{\sigma'}
$
(Theorem \ref{invariance 2 within L-packet}).
The crucial idea is to study the stabilizers $W(\sigma)$ and $W(\sigma').$
Theorem \ref{equality on W} shows
\begin{equation} \label{iso intro}
W(\sigma) \simeq
\{
w \in W_M : {^w}\Sigma \simeq \Sigma \lambda ~~ \text{ for some}~ \lambda \in (\mathcal{M}/M)^D
\},
\end{equation}
where $W_M$ is the Weyl group of $\bold M$ in $\bold G,$
$\Sigma \in \Pi_{\varphi}(\mathcal{M})$ is a lifting of $\sigma,$ and $(\mathcal{M}/M)^D$ is the group of continuous characters on $\mathcal{M}$ which are trivial on $M.$
The same is true for $W(\sigma').$
The isomorphism \eqref{iso intro} stems from the group structure of $M$ and $M'$ and the description of irreducible representations of $M$ and $M',$ which are discovered in \cite[Section 2]{go95} and explained in Section \ref{behavior of Rgps}.

As an application, a combination of Theorem \ref{invariance 2 within L-packet} and the second-named author's earlier result \cite[Theorem 3.7]{go95} shows that the Knapp-Stein $R$-group, $R_{\sigma'},$ for non quasi-split inner forms $\bold G'$ can be expressed in terms of a subgroup in $R_{\sigma'}$ and $\mathbb{Z}^d$ for some integer $d.$
It follows that $R_{\sigma'}$ is in general non-abelian (see Remark \ref{final remark}).

In Section \ref{pre}, we recall basic notation and background, provide the detailed group structure of $\boldsymbol{\mathcal{G}}_n,$ $\bold G_n,$ $\boldsymbol{\mathcal{G}}'_n,$ $\bold G'_n,$ and their $F$-Levi subgroups, and study relations in their $L$-groups.
In Section \ref{weyl group actions}, we describe Weyl group actions on Levi subgroups and their representations. In Section  \ref{R-groups for U(n) and its inner forms}, we revisit the theory of $R$--groups for $\boldsymbol{\mathcal{G}}_n$ and $\boldsymbol{\mathcal{G}}_n'$ based on \cite{go95, kmsw14, mok13}.
In Section \ref{R-groups for SU(n) and its inner forms}, we prove the Arthur's conjecture for $\bold G_n$ and $\bold G_n'$ and the invariance of their $R$-groups.

\subsection*{Acknowledgements}
K. Choiy was partially supported by an AMS-Simons Travel Grant.
D. Goldberg was partially supported by  Simons Foundation Collaboration Grant 279153.

\section{Preliminaries} \label{pre}
\subsection{Basic notation and background} \label{basic notation}
Let $p$ be a prime.  We let $F$ be a $p$-adic field of characteristic $0,$ i.e., a finite extension of $\mathbb{Q}_{p}.$
Fix an algebraic closure $\bar{F}$ of $F.$
Given a connected reductive algebraic group $\bold G$ defined over $F,$  we write $\bold G(F)$ for the group of $F$-points. Unless otherwise stated, we shall use $G$ for $\bold G(F)$ and likewise for other algebraic groups defined over $F.$

Fix a minimal $F$-parabolic subgroup $\bold P_0$ of $\bold G$ with Levi decomposition $\bold P_0=\bold M_0 \bold N_0,$ where $\bold M_0$ denotes a Levi factor and $\bold N_0$ denotes the unipotent radical.
We denote by $\bold A_0$ the split component of $\bold M_0,$ that is, the maximal $F$-split torus in the center of $\bold M_0,$ and by $\Delta$ the set of simple roots of $\bold A_0$ in $\bold N_0.$
We say an $F$-parabolic subgroup $\bold P$ of $\bold G$ to be standard if it contains $\bold P_0.$

Given an $F$-parabolic subgroup $\bold P$ with Levi decomposition $\bold P=\bold M\bold N,$ there exist a subset $\Theta \subseteq \Delta$ such that $\bold M$ equals $\bold M_{\Theta},$ the Levi subgroup generated by $\Theta.$
Note that $\bold M \supseteq \bold M_0$ and $\bold N \subseteq \bold N_0.$
We write  $\bold A_{\bold M_{\Theta}}$ for the split component $\bold A_{\bold M}$ of $\bold M=\bold M_{\Theta}.$
It follows that $\bold A_{\bold M}$ equals the identity component
$(\cap_{\alpha \in \Theta} \ker \alpha
)^{\circ}$ in $\bold A_0,$
so that $\bold M = Z_\bold G(\bold A_\bold M),$
where $Z_\bold G(\bold A_\bold M)$ is the centralizer of $\bold A_\bold M$ in $\bold G.$
We refer the reader to \cite[Proposition 20.4]{borel91} and \cite[Section 15.1]{springer98} for the details.

We let $\Phi(P, A_M)$ denote the set of reduced roots of $\bold P$ with respect to $\bold A_\bold M.$
Denote by $W_M = W(\bold G, \bold A_\bold M) := N_\bold G(\bold A_\bold M) / Z_\bold G(\bold A_\bold M)$ the Weyl group of $\bold A_\bold M$ in $\bold G,$
where $N_\bold G(\bold A_\bold M)$ is the normalizer of $\bold A_\bold M$ in $\bold G.$
For simplicity, we write $\bold A_0 = \bold A_{\bold M_0}.$

For any topological group $H,$ we write $Z(H)$ for the center of $H.$ Denote by $H^\circ$ the identity component of $H$ and by $\pi_0(H)$ the group $H/H^\circ$ of connected components of $H.$
We write $H^D$ for the group, $\Hom(H , \mathbb{C}^{\times}),$ of all continuous characters and by $\mathbbm{1}$ the trivial character.
We say a character is unitary if its image is in the unit circle $S^1 \subset \mathbb{C}^{\times}.$
For any Galois module $J,$ we denote by $H^i(F, J) := H^i(\Gal (\bar{F} / F), J(\bar{F}))$
the Galois cohomology of $J$ for $i \in \mathbb{N}.$

Let $\bold G$ be a connected reductive algebraic group  over $F.$
We denote by $\Irr(G)$ the set of isomorphism classes of irreducible admissible complex representations of $G.$
If there is no confusion, we do not make a distinction between each isomorphism class and its representative. For any $\sigma \in \Irr(M),$ we write $\mathbf{\textit{i}}_{G,M} (\sigma)$ for the normalized (twisted by $\delta_{P}^{1/2}$) induced representation, where $\delta_P$ denotes the modulus character of $P.$
We denote by $\sigma^{\vee}$ the contragredient of $\sigma.$

Denote by $\Pi_{\disc}(G)$ and $\Pi_{\temp}(G)$  the subsets of $\Irr(G)$ which respectively consist of discrete series and tempered representations,
where, a discrete series representation is an irreducible, admissible, unitary representation whose matrix coefficients are square-integrable modulo the center of $G,$ that is, in $L^2(G/Z(G)),$
and a tempered representation is an irreducible, admissible, unitary representation whose matrix coefficients are in $L^{2+\epsilon}(G/Z(G))$ for all $\epsilon > 0.$
It is clear that $\Pi_{\disc}(G) \subset \Pi_{\temp}(G).$

We denote by $W_F$ the Weil group of $F$ and by $\Gamma$ the absolute Galois group $\Gal(\bar{F} / F).$
By fixing $\Gamma$-invariant splitting data,
the $L$-group of $G$ is defined as a semi-direct product $^{L}G := \widehat{G} \rtimes W_F$ (see \cite[Section 2]{bo79}).
Following \cite[Section 8.2]{bo79}, an $L$-parameter for $G$ is an admissible homomorphism
\[
\phi: W_F \times SL_2(\mathbb{C}) \rightarrow {^L}G.
\]
Two $L$-parameters are said to be equivalent if they are conjugate by $\widehat{G}.$
We denote by $\Phi(G)$ the set of equivalence
classes of $L$-parameters for $G.$

We denote by $C_{\phi}(\widehat{G})$ the centralizer of the image of $\phi$ in $\widehat{G}.$
The center of $^{L}G$ is the $\Gamma$-invariant group $Z(\widehat{G})^{\Gamma}.$
Note that $C_{\phi} \supset Z(\widehat{G})^{\Gamma}.$
We say that $\phi$ is elliptic if the quotient group
$C_{\phi}(\widehat G) / Z(\widehat{G})^{\Gamma}
$
is finite, and $\phi$ is tempered if $\phi(W_F)$ is bounded.
We denote by $\Phi_{\el}(G)$ and $\Phi_{\temp}(G)$ the subset of $\Phi(G)$ which respectively consist of elliptic and tempered $L$-parameters of $G.$
We set $\Phi_{\disc}(G) = \Phi_{\el}(G) \cap \Phi_{\temp}(G).$

The local Langlands conjecture for $G$ predicts that there is a surjective finite-to-one map from $\Irr(G)$ to $\Phi(G).$
Given $\phi \in \Phi(G),$
we write $\Pi_{\phi}(G)$ for the $L$-packet attached to $\phi,$ and then the local Langlands conjecture implies that
\[
\Irr(G) = \bigsqcup_{\phi \in \Phi(G)}\Pi_{\phi}(G).
\]
It is expected that $\Phi_{\disc}(G)$ and $\Phi_{\temp}(G)$ respectively parameterize $\Pi_{\disc}(G)$ and $\Pi_{\temp}(G).$

Given two connected reductive algebraic groups $\bold G$ and $\bold G'$ over $F,$
 $\bold G'$ is said to be an \textit{$F$-inner form} of $\bold G$ with respect to an $\bar{F}$-isomorphism $\varphi: \bold G' \overset{\sim}{\rightarrow} \bold G$ if $\varphi \circ \tau(\varphi)^{-1}$ is an inner automorphism ($g \mapsto xgx^{-1}$) defined over $\bar{F}$ for all $\tau \in \Gal (\bar{F} / F)$ (see \cite[2.4(3)]{bo79} or \cite[p.280]{kot97}).
If there is no confusion, we often omit the references to $F$ and $\varphi.$
It is well known from \cite[p.280]{kot97} that there is a bijection between $H^1(F, \bold G_{\ad})$ and the set of isomorphism classes of $F$-inner forms of $\bold G,$ where $\bold G_{\ad}:=\bold G / Z(\bold G).$
We note that, when $\bold G$ and $\bold G'$ are inner forms of each other, we have $^L{G} \simeq {^LG'}$ \cite[Section 2.4(3)]{bo79}.
\subsection{$R$-groups} \label{section for def of R}
We review the definitions of Knapp-Stein, Langlands-Arthur, and endoscopic $R$--groups.
Let $\bold G$ be a connected reductive algebraic group over $F,$ and let $\bold M$ be an $F$-Levi subgroup of $\bold G.$
Given $\sigma \in \Irr(M)$ and $w \in W_M,$ we write ${^w}\sigma$ for the representation given by ${^w}\sigma(x)=\sigma(w^{-1}xw).$
Note that, for the purpose of studying $R$--groups, we do not distinguish the representative of $w,$ since the isomorphism class of ${^w}\sigma$ is independent of the choices of representatives in $G$ of $w \in W_M.$
Assume that $\sigma$ lies in $\Pi_{\disc}(M),$ we define the stabilizer of $\sigma$ in $W_M$
\[ W(\sigma) := \{ w \in W_M : {^w}\sigma \simeq \sigma \}.
\]
Write $\Delta'_\sigma$ for $\{ \alpha \in \Phi(P, A_M) : \mu_{\alpha} (\sigma) = 0 \},$ where $\mu_{\alpha} (\sigma)$ is the rank one Plancherel measure for $\sigma$ attached to $\alpha$ \cite[p.1108]{goldberg-class}.
The \textit{Knapp-Stein $R$-group} is defined by
\[
R_{\sigma} := \{ w \in W(\sigma) : w \alpha > 0, \; \forall \alpha \in \Delta'_\sigma \}.
\]
We denote by $W^{\circ}_{\sigma}$ the subgroup of $W(\sigma),$ generated by the reflections in the roots of $\Delta'_\sigma.$
Then, for any $\sigma \in \Pi_{\disc}(M),$ we have
\[
W(\sigma) = R(\sigma) \ltimes W^{\circ}_{\sigma},
\]
which yields another description of the Knapp-Stein $R$-group
\[
R(\sigma) \simeq  W(\sigma)/W^{\circ}_{\sigma}.
\]
We refer to \cite{ks72, sil78, sil78cor} for details.

Given an $L$-parameter $\phi \in \Phi(M),$
we also consider $\phi$ as an $L$-parameter for $G$ via the inclusion $\widehat M \hookrightarrow \widehat G.$
Fix a maximal torus $T_{\phi}$ in $C_{\phi}(\widehat{G})^{\circ}.$
We set
\[
W_{\phi}^{\circ} := N_{C_{\phi}(\widehat{G})^{\circ}} (T_{\phi}) /  Z_{C_{\phi}(\widehat{G})^{\circ}} (T_{\phi}), \quad W_{\phi} := N_{C_{\phi}(\widehat{G})} (T_{\phi}) /  Z_{C_{\phi}(\widehat{G})} (T_{\phi}).
\]
The \textit{endoscopic $R$-group} $R_{\phi}$ is defined as follows
\[
R_{\phi}:=W_{\phi}/W_{\phi}^{\circ}.
\]
We identify $W_{\phi}$ with a subgroup of $W_M$ (see \cite[p.45]{art89ast}).
For $\sigma \in \Pi_{\phi}(M),$ we set
\begin{equation*} \label{def of W_phi, sigma}
W_{\phi, \sigma}^{\circ} :=  W_{\phi}^{\circ} \cap W(\sigma), \quad  W_{\phi, \sigma} :=  W_{\phi} \cap W(\sigma).
\end{equation*}
The \textit{Langlands-Arthur $R$-group} $R_{\phi, \sigma}$ is defined as follows
\[
R_{\phi, \sigma}:=W_{\phi, \sigma}/W_{\phi, \sigma}^{\circ}.
\]

\section{Structure theory of Levi subgroups} \label{structure}
We discuss the group structure of $\U_n,$ $\SU_n,$ their $F$-inner forms, and their $F$-Levi subgroups.
We mainly refer to \cite{go95, go06, kmsw14, mok13, sa71}.
Fix a quadratic extension $E/F$ with $\bar{~~}$ the non-trivial Galois element.
For any positive integer $n,$ we let
\[
J_n :=
\left( \begin{array}{ccccc}
 0& & & & 1\\
 & & & -1 &\\
 & & 1 & &\\
 & \iddots & & &\\
(-1)^n & & & &0 \\
 \end{array} \right) \in {\Mat}_{n \times n}(\mathbb{Z}).
\]
We denote by $\Res_{E/F}$ the Weil restriction of scalars of $E/F$ (see \cite[Chapter 1]{weil82} and \cite[2.1.2]{pr94}).
For $g=(g_{ij}) \in \Res_{E/F} \GL_n,$ we let $\bar{g}=(\bar g_{ij})$ and set $\varepsilon(g) = J_n {^t\bar{g}}^{-1} J_n^{-1},$ where $g \mapsto  {^t}{g}$ is the transpose.
\subsection{$\U_n$ and its inner forms} \label{structure for Un}
Let $\boldsymbol{\mathcal{G}}=\boldsymbol{\mathcal{G}}_n$ denote the quasi-split unitary group $\U_{n}$ with respect to $E/F$ and $J_{n}.$ Thus,
\[
\boldsymbol{\mathcal{G}} = \{
g \in {\Res}_{E/F} {\GL}_n : gJ_n {^t\bar{g}} = J_n
\}
\]
(see \cite[Section 1]{go06}; our $J_n$ is the inverse of $u_n$ therein).
We denote by $\boldsymbol{\mathcal{M}}$ an $F$-Levi subgroup of $\boldsymbol{\mathcal{G}}.$ Then, $\boldsymbol{\mathcal{M}}$ is of the form
\begin{equation} \label{form of tbM}
{\Res}_{E/F} {\GL}_{n_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n_k} \times {\boldsymbol{\mathcal{G}}}_m,
\end{equation}
where $\sum_{i=1}^k 2n_i + m = n$ with $n_i \geq 0$ and $m \geq 0.$
By convention, we note that, ${\boldsymbol{\mathcal{G}}}_0 = 1$ for $n$ even, ${\boldsymbol{\mathcal{G}}}_1 = \U_1$ for $n$ odd, and ${\GL}_0=1.$
The group of $F$-points $,A_{\boldsymbol{\mathcal{M}}},$ of the split component $\bold A_{\boldsymbol{\mathcal{M}}}$ is of the form
\begin{equation} \label{type of AtM}
\{
{\diag} (x_1I_{n_1}, \cdots, x_k I_{n_k}, I_m, {x}_k^{-1}I_{n_k}, \cdots, {x}_1^{-1}I_{n_1}) : x_i \in F^{\times}
\}.
\end{equation}

We let $\boldsymbol{\mathcal{G}}'=\boldsymbol{\mathcal{G}}'_n$ denote an $F$-inner form of $\boldsymbol{\mathcal{G}}.$
By the Satake classification in \cite[Section 3.3]{sa71},
for $n$ \textbf{odd}, there is \textit{no non quasi-split} $F$-inner form of $\boldsymbol{\mathcal{G}}_n.$
On the other hand, for $n$ \textbf{even},
there is a \textit{unique} non quasi-split $F$-inner form $\boldsymbol{\mathcal{G}}'_n,$ up to $F$-isomorphism.
The $\Gamma$-diagram of the connected, simply-connected, semi-simple type of such $\boldsymbol{\mathcal{G}}'$ is
\begin{equation} \label{diagram}
\xy
\POS (10,0) *\cir<2pt>{} ="a" ,
\POS (20,0) *\cir<2pt>{} ="b" ,
\POS (30,0) *\cir<0pt>{} ="e",
\POS (40,0) *\cir<0pt>{} ="f",
\POS (50,0) *\cir<2pt>{} ="h",
\POS (60,0) *\cir<2pt>{} ="i",
\POS (70,-5) *{\bullet} ="j",

\POS (10,-10) *\cir<2pt>{} ="a1" ,
\POS (20,-10) *\cir<2pt>{} ="b1" ,
\POS (30,-10) *\cir<0pt>{} ="e1",
\POS (40,-10) *\cir<0pt>{} ="f1",
\POS (50,-10) *\cir<2pt>{} ="h1",
\POS (60,-10) *\cir<2pt>{} ="i1",

\POS (10,-2) *\cir<0pt>{} ="a2" ,
\POS (20,-2) *\cir<0pt>{} ="b2" ,
\POS (50,-2) *\cir<0pt>{} ="h2",
\POS (60,-2) *\cir<0pt>{} ="i2",

\POS (10,-8) *\cir<0pt>{} ="a3" ,
\POS (20,-8) *\cir<0pt>{} ="b3" ,
\POS (50,-8) *\cir<0pt>{} ="h3",
\POS (60,-8) *\cir<0pt>{} ="i3",

\POS "a" \ar@{-}^<<<{}_<<{}  "b",
\POS "b" \ar@{-}^<<<{}_<<{}  "e",
\POS "e" \ar@{.}^<<<{}_<<{}  "f",
\POS "f" \ar@{-}^<<<{}_<<{} "h",
\POS "h" \ar@{-}^<{}_<<<{} "i",
\POS "i" \ar@{-}^<<<{}_<<{} "j",

\POS "a1" \ar@{-}^<<<{}_<<{}  "b1",
\POS "b1" \ar@{-}^<<<{}_<<{}  "e1",
\POS "e1" \ar@{.}^<<<{}_<<{}  "f1",
\POS "f1" \ar@{-}^<<<{}_<<{} "h1",
\POS "h1" \ar@{-}^<{}_<<<{} "i1",
\POS "i1" \ar@{-}^<<<{}_<<{} "j",

\POS "a2" \ar@{<->} "a3",
\POS "b2" \ar@{<->} "b3",
\POS "h2" \ar@{<->} "h3",
\POS "i2" \ar@{<->} "i3",
\endxy
\end{equation}
(see the table in p.119 of \cite{sa71}).
In the diagram above, the arrow indicates the non-trivial Galois action.
Furthermore, the black vertex indicates a root in the set of simple roots of a fixed minimal $F$-Levi subgroup $\boldsymbol{\mathcal{M}}'_0$ of $\boldsymbol{\mathcal{G}}'.$
So, we remove only a subset (denoted by $\vartheta$) of pairs ($\Gal(E/F)$-orbits) of white vertices to obtain an $F$-Levi subgroup $\boldsymbol{\mathcal{M}}'$ (see \cite[Section 2.2]{sa71} and \cite[Section I.3]{bo79}).
As discussed in Section \ref{basic notation}, the $F$-Levi subgroup $\boldsymbol{\mathcal{M}}',$ corresponding to $\Theta = \Delta \setminus \vartheta,$
is the centralizer in $\boldsymbol{\mathcal{G}}'$ of the split component $\bold A_{\boldsymbol{\mathcal{M}}'}=(\cap_{\alpha \in \Theta} \ker \alpha)^{\circ}.$
Then, $\boldsymbol{\mathcal{M}}'$ is of the form
\begin{equation} \label{form of tbM'}
{\Res}_{E/F} {\GL}_{n'_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n'_{k'}} \times {\boldsymbol{\mathcal{G}}'}_{m'},
\end{equation}
where $\sum_{i=1}^{k'} 2n'_i + m' = n$ with $n'_i \geq 0$ and $m' \geq 2.$ Notice here that $m'$ is always even.
Considering the forms \eqref{form of tbM} and \eqref{form of tbM'}, it is obvious that, if $n_i=n'_i$ for all $i,$ $k=k',$ and $m=m',$ then $\boldsymbol{\mathcal{M}}'$ is an $F$-inner form of $\boldsymbol{\mathcal{M}}.$
In this case, furthermore,
two split components $\bold A_{\boldsymbol{\mathcal{M}}}$ and $\bold A_{\boldsymbol{\mathcal{M}}'} $ are isomorphic over $F.$

\begin{rem}
Since there is a bijection between $H^1(F, (\boldsymbol{\mathcal{G}}_n)_{\ad})$ and the set of isomorphism classes of $F$-inner forms of $\boldsymbol{\mathcal{G}}_n$ (see Section \ref{basic notation}), using the fact (\cite[Lemma 1.2.1(ii) and p.657]{hala04}) that
\[
H^1(F, (\boldsymbol{\mathcal{G}}_n)_{\ad}) = \left\{ \begin{array}{ll}
         1, & \mbox{if $n$ is odd},\\
        \mathbb{Z}/2\mathbb{Z}, & \mbox{if $n$ is even},\end{array} \right .
\]
it follows that, when $n$ is odd, there is no non quasi-split $F$-inner form of $\boldsymbol{\mathcal{G}}_n,$ and when  $n$ is even, there is a unique non quasi-split $F$-inner form, $\boldsymbol{\mathcal{G}}'_n,$  of $\boldsymbol{\mathcal{G}}_n$ up to $F$-isomorphism, as we discussed above.
\end{rem}
\subsection{$\SU_n$ and its inner forms} \label{structure for SUn}
We let $\bold G=\bold G_n$ denote the quasi-split special unitary group $\SU_n$ with respect to $E/F$ and $J_n.$ Thus,
\[
\bold G=\boldsymbol{\mathcal{G}} \cap {\Res}_{E/F} {\SL}_n =
\{
g \in {\Res}_{E/F} {\SL}_n : gJ_n {^t\bar{g}} = J_n
\}.
\]
We denote by $\bold M$ an $F$-Levi subgroup of $\bold G.$
Then, $\bold M$ is of the form $\boldsymbol{\mathcal{M}} \cap \bold G$ and $\bold A_{\bold M} $ is of the form $\bold A_{\boldsymbol{\mathcal{M}}} \cap \bold G.$
{

Due to \eqref{type of AtM}, the $F$-points $A_{\bold M}$ of the split component $\bold A_{\bold M}$ thus equals
\begin{equation*} \label{type of AM}
\{
{\diag} (x_1I_{n_1}, \cdots, x_k I_{n_k}, I_m, {x}_k^{-1}I_{n_k}, \cdots, {x}_1^{-1}I_{n_1}) : x_i \in F^{\times}\}.
\end{equation*}
}
Especially, when $m \geq 2,$ from \eqref{form of tbM} and \cite[Lemma 2.8]{go06} we have a useful isomorphism
\begin{equation} \label{form of bM}
M \simeq ({\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)) \rtimes G_m,
\end{equation}
where $\sum_{i=1}^k 2n_i + m = n$ with $n_i \geq 0.$
More precisely, for $(g, h)$ with $g \in {\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)$ and $h \in G_m,$ the isomorphism \eqref{form of bM} from $({\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)) \rtimes G_m$ to $M$ is given by
\begin{equation} \label{imp iso}
(g, h) \longmapsto
\left( \begin{array}{ccc}
 g& & \\
 & \alpha_m(g)^{-1}h & \\
 & & \varepsilon(g) \\
 \end{array} \right) \in \mathcal{M} \cap {\SU}_n(F) = M,
\end{equation}
where $\alpha_m(g) = \alpha_m(\det g)$ and for $a \in E^{\times},$
\begin{equation} \label{def of alpha}
\alpha_m(a) :=
\left( \begin{array}{ccc}
 a& & \\
 & I_{m-2} & \\
 & & \bar{a}^{-1}\\
 \end{array} \right) \in {\GL}_{m}(E).
\end{equation}
By convention, we note that ${\bold G}_0={\bold G}_1 = 1.$
Given $(g, h)$ with $g \in {\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)$ and $h \in G_m,$ the action of  $g$ on $h$ is given by
\[
g \circ h = \alpha_m(g) \cdot h \cdot \alpha_m(g)^{-1}.
\]

Next, we let $\bold G'=\bold G'_n$ denote an $F$--inner form of $\bold G.$
As discussed in Section \ref{structure for Un},
for $n$ \textbf{odd}, there is \textit{no non quasi-split} $F$-inner form of $\bold G_n.$
On the other hand, for $n$ \textbf{even},
there is a \textit{unique} non quasi-split $F$-inner form $\bold G'_n,$ up to $F$-isomorphism, whose $\Gamma$-diagram is \eqref{diagram}.
Any $F$-Levi subgroup $\bold M'$ of $\bold G'$ is of the form  $\boldsymbol{\mathcal{M}}' \cap \bold G'.$
Thus, from \eqref{form of tbM'} and \eqref{form of bM}, we have an isomorphism
\begin{equation} \label{form of bM'}
M' \simeq
({\GL}_{n'_1}(E) \times \cdots \times {\GL}_{n'_{k'}}(E)) \rtimes {G'}_{m'},
\end{equation}
where $\sum_{i=1}^{k'} 2n'_i + m' = n$ with $n'_i \geq  0$ and $m' \geq 2.$ Notice as before that $m'$ is always even.
To explain \eqref{form of bM'} more precisely,
for $(g', h')$ with $g' \in {\GL}_{n'_1}(E) \times \cdots \times {\GL}_{n'_k}(E)$ and $h' \in {G'}_{m'},$ the isomorphism \eqref{form of bM'} from $({\GL}_{n'_1}(E) \times \cdots \times {\GL}_{n'_{k'}}(F)) \rtimes {G'}_{m'}$ to $M'$ is given by
\[
(g', h') \longmapsto
\left( \begin{array}{ccc}
 g'& & \\
 & \alpha_{m'}(g')^{-1}h' & \\
 & & \varepsilon(g') \\
 \end{array} \right) \in \mathcal{M}' \cap {G'}_n = M',
\]
where $\alpha_{m'}(g') = \alpha_{m'}(\det g')$ and for $a \in E^{\times},$
\[
\alpha_{m'}(a) :=
\left( \begin{array}{ccc}
 a& & \\
 & I_{m'-2} & \\
 & & \bar{a}^{-1}\\
 \end{array} \right) \in {\GL}_{m'}(E).
\]
By convention, we note that, ${\bold G'}_0 = 1.$
Given $(g', h') \in M'$ with  $g' \in {\GL}_{n'_1}(E) \times \cdots \times {\GL}_{n'_k}(E)$ and $h' \in G'_{m'},$ the action of $g'$ on $h'$ is given by
\[
g' \circ h' = \alpha_{m'}(g') \cdot h' \cdot \alpha_{m'}(g')^{-1}.
\]
Like the unitary case in Section \ref{structure for Un}, it is obvious from \eqref{form of bM} and \eqref{form of bM'} that, if $n_i=n'_i$ for all $i,$ $k=k',$ and $m=m',$ then $\bold M'$ is an $F$-inner form of $\bold M.$
In this case,
there is an $F$--isomorphism $\bold A_{\bold M} \simeq \bold A_{\bold M'}$ between two split components.

We have following exact sequences of algebraic groups
\[
1 \longrightarrow {\bold G} \longrightarrow {\boldsymbol{\mathcal{G}}}  \overset{\det}{\longrightarrow}  {\U}_1 \longrightarrow 1,
\]
and
\[
1 \longrightarrow {\bold G'} \longrightarrow {\boldsymbol{\mathcal{G}}'} \overset{\det}{\longrightarrow}  {\U}_1 \longrightarrow 1.
\]
Applying Galois cohomology, since $H^1(F, \bold G)=H^1(F, \bold G')=1$  due to \cite[Theorem 6.4]{pr94}, we have following exact sequences of their $F$-points
\[
1 \longrightarrow {G} \longrightarrow {\mathcal{G}} \overset{\det}{\longrightarrow}  E^{1} \longrightarrow 1,
\]
and
\[
1 \longrightarrow {G'} \longrightarrow {\mathcal{G}'} \overset{\det}{\longrightarrow}  E^{1} \longrightarrow 1,
\]
where $E^1 = \{ x \in E : N_{E/K} (x) = x \bar x = 1 \}.$
All above exact sequences are true for $F$-Levi subgroups.
In particular, $H^1(F, \bold M) =H^1(F, \bold M')= 1$ since $H^1(F, \bold M) \hookrightarrow H^1(F, \bold G)=1$ and $H^1(F, \bold M') \hookrightarrow H^1(F, \bold G')=1$
(see \cite[p.95, footnote]{morel}, \cite[p.270]{kot97}, and \cite[Remark 2.5]{chgo12}).
We further have
\[
{\boldsymbol{\mathcal{G}}}_{\der} = {\bold G}_{\der}=\bold G, ~~{\boldsymbol{\mathcal{G}}'}_{\der} = {\bold G'}_{\der}=\bold G', ~~{\boldsymbol{\mathcal{M}}}_{\der} = {\bold M}_{\der} \subset \bold M, ~~{\boldsymbol{\mathcal{M}}'}_{\der} = {\bold M'}_{\der} \subset  \bold M'.
\]

\subsection{$L$-groups} \label{L-groups}
We describe $L$-groups of $\boldsymbol{\mathcal{G}}=\U_n, \bold G=\SU_n,$ their inner forms $\boldsymbol{\mathcal{G}}', \bold G',$ and the $L$--groups of  their $F$-Levi subgroups $\boldsymbol{\mathcal{M}}, \boldsymbol{\mathcal{M}}', \bold M,$ and  $\bold M'.$
Furthermore, we explain a relationship between $L$-groups of $\boldsymbol{\mathcal{M}}$ and $\bold M,$ and investigate a connection between two $\Gamma$-split components ${\bold A}_{\widehat{\mathcal{M}}}$ and ${\bold A}_{\widehat{M}} $ of $\widehat{\mathcal{M}}$ and $\widehat{M},$ defined as follows
\[
{\bold A}_{\widehat{\mathcal{M}}} := (Z(\widehat{\mathcal{M}})^{\Gamma})^\circ \quad \text{and} \quad {\bold A}_{\widehat{M}} := (Z(\widehat{M})^{\Gamma})^\circ.
\]
These will be used in Section \ref{R-groups for SU(n) and its inner forms}.
Based on \cite{mok13}, we set
\[
{^L}{\mathcal{G}}={^L}{\mathcal{G}'}={\GL}_n(\mathbb{C})\rtimes W_F,
\]
where $W_E$ acts trivially on ${\GL}_n(\mathbb{C}),$ and the action of $w_c \in W_F \setminus W_E$ on $ \hat g \in \GL_n(\mathbb{C})$ is given by
\begin{equation} \label{galois action on L-gp}
w_c( \hat g) = J_n^{-1} {^t}\hat g^{-1} J_n.
\end{equation}
For $\bold G$ and $\bold G',$ the $L$-group is
\[
{^L}{G}={^L}{G'}={\PGL}_n(\mathbb{C})\rtimes W_F,
\]
where $W_E$ acts trivially on ${\PGL}_n(\mathbb{C}),$ and the action of $w_c \in W_F \setminus W_E$ on $\hat g \in \PGL_n(\mathbb{C})$ is given by \eqref{galois action on L-gp}.

Let $\boldsymbol{\mathcal{M}},$ $\boldsymbol{\mathcal{M}}',$ $\bold M,$ and $\bold M'$ be $F$--Levi subgroups of $\boldsymbol{\mathcal{G}},$ $\boldsymbol{\mathcal{G}}',$ $\bold G,$ and $\bold G',$ respectively, such that $\boldsymbol{\mathcal{M}}'$ is an $F$--inner form of $\boldsymbol{\mathcal{M}},$ and $\bold M'$ is an $F$--inner form of $\bold M.$
We have
\begin{equation*}  \label{iso between L-groups}
\widehat \mathcal{M} = \widehat{\mathcal{M}'} \subset \widehat \mathcal{G} = \widehat{\mathcal{G}'}, \quad {^L}{\mathcal{M}}={^L}{\mathcal{M}'}=\widehat \mathcal{M} \rtimes W_F, \quad \widehat M = \widehat{M'}\subset\widehat G = \widehat{G'}, \quad {^L}{M}={^L}{M'}=\widehat M\rtimes W_F.
\end{equation*}
Considering \eqref{form of tbM} and \eqref{form of tbM'}, we set
\[
\boldsymbol{\mathcal{M}} = {\Res}_{E/F} {\GL}_{n_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n_k} \times \boldsymbol{\mathcal{G}}_m,
\]
\[
\boldsymbol{\mathcal{M}}' = {\Res}_{E/F} {\GL}_{n_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n_k} \times \boldsymbol{\mathcal{G}}'_m,
\]
where $\boldsymbol{\mathcal{G}}'_m$ is an $F$-inner form of $\boldsymbol{\mathcal{G}}_m,$ $\sum_{i=1}^k 2n_i + m = n$ with $n_i \geq  0,$ and $m \geq 0.$
Then, we have
\[
\widehat{\boldsymbol{\mathcal{M}}}=\widehat{\boldsymbol{\mathcal{M}}'} = ({\GL}_{n_1}(\mathbb{C})\times{\GL}_{n_1}(\mathbb{C})) \times \cdots \times ({\GL}_{n_k}(\mathbb{C})\times{\GL}_{n_k}(\mathbb{C}))\times {\GL}_m(\mathbb{C}).
\]
$W_E$ acts trivially on $\widehat{\boldsymbol{\mathcal{M}}},$ and the action of $w_c \in W_F \setminus W_E$ on $((\hat g_{11}, \hat g_{12}), (\hat g_{21}, \hat g_{22}), \cdots, (\hat g_{k1}, \hat g_{k2}), \hat h) \in \widehat{\boldsymbol{\mathcal{M}}}$ is given by
\begin{equation} \label{galois action on L-gp of tM}
w_c((\hat g_{11}, \hat g_{12}), (\hat g_{21}, \hat g_{22}), \cdots, (\hat g_{k1}, \hat g_{k2}), \hat h)
=
((\hat g_{12}, \hat g_{11}), (\hat g_{22}, \hat g_{21}), \cdots, (\hat g_{k2}, \hat g_{k1}),
 J_m^{-1} {^t}\hat h^{-1} J_m).
\end{equation}

Next, we note that, since $\boldsymbol{\mathcal{M}}_{\der}$ and $\bold M_{\der}$ are simply connected, we have
\[
\widehat{\mathcal{M}_{\der}} =
 {(\widehat{\mathcal{M}})}_{\ad} =
  \widehat{\mathcal{M}}/Z(\widehat{\mathcal{M}}), ~~
   \widehat{M_{\der}} = {\widehat{(M)}}_{\ad} = \widehat{M}/Z(\widehat{M}).
\]
Further, from \cite[(1.8.1) p.616]{kot84}, the exact sequence of algebraic groups
\[
1 \longrightarrow \bold M \longrightarrow \boldsymbol{\mathcal{M}} \longrightarrow {\U}_1 \longrightarrow 1
\]
yields an exact sequence
\begin{equation} \label{exact of L centers}
1 \longrightarrow \widehat{{\U}_1}=\mathbb{C}^{\times} \longrightarrow Z({\widehat{\mathcal{M}}}) \simeq (\mathbb{C}^{\times})^{2k+1} \longrightarrow Z({\widehat{M}}) \longrightarrow 1.
\end{equation}
We thus have the following commutative diagram of $L$-groups (cf., \cite[Remark 2.4]{chgo12})
\[
\begin{CD}
@.  1 @. 1 @.  @.
\\
@.      @VVV        @VVV   @. @.\\
1 @>>> \widehat{{\U}_1} = \mathbb{C}^{\times} @>{\simeq}>> \ker @>>> 1 @. @.
\\
@.      @VVV        @VVV   @VVV  @.\\
1 @>>> Z(\widehat{\mathcal{M}}) = (\mathbb{C}^{\times})^{2k+1} @>>> \widehat{\mathcal{M}} @>>> \widehat{\mathcal{M}_{\der}} @>>> 1
\\
@.      @VVV          @VVV   @|  @.
\\
1 @>>> Z(\widehat{M}) = (\mathbb{C}^{\times})^{2k+1} / \mathbb{C}^{\times} @>>> \widehat{M} @>>> \widehat{M_{\der}} @>>> 1 \\
@.      @VVV          @VVV   @VVV  @.\\
@.  1 @. 1 @. 1 @.
\end{CD}
\]
The middle vertical exact sequence becomes
\begin{equation} \label{exact of L-gp of M tM}
1 \longrightarrow \widehat{{\U}_1}=\mathbb{C}^{\times} \longrightarrow \widehat{\mathcal{M}}=\widehat{\mathcal{M}'} \longrightarrow \widehat{M}=\widehat{M'} \longrightarrow 1.
\end{equation}
We also have
\begin{equation} \label{exact of L-gp of G tG}
1 \longrightarrow \widehat{{\U}_1}=\mathbb{C}^{\times} \longrightarrow \widehat{\mathcal{G}}=\widehat{\mathcal{G}'} \longrightarrow  \widehat{G}=\widehat{G'} \longrightarrow 1.
\end{equation}

Note that $\widehat{\U_1}$ in \eqref{exact of L-gp of M tM} and \eqref{exact of L-gp of G tG} equals $ Z(\widehat{\mathcal{G}})=Z(\widehat{\mathcal{G}'}).$
Furthermore, $\widehat{\U_1}$ is diagonally embedded into $\widehat{\mathcal{G}}$ in \eqref{exact of L-gp of G tG} and the action of $W_F$ on $\widehat{\U_1} = \mathbb{C}^{\times}$ is obtained from \eqref{galois action on L-gp}.
The action of $W_F$ can be also obtained from \eqref{galois action on L-gp of tM}, since it is diagonally embedded into  $Z(\widehat{\mathcal{M}})$ in \eqref{exact of L centers} as well.
It thus follows that the subgroups of $\Gamma=\Gal(\bar F/F)$-invariants satisfy
\begin{equation} \label{gamma invariants}
Z(\widehat{\mathcal{G}})^{\Gamma}=Z(\widehat{\mathcal{G}'})^{\Gamma} = \widehat{{\U}_1}^{\Gamma} = \{ \pm 1 \}.
\end{equation}
Moreover, using the action of $W_F$ on $\widehat{\mathcal{M}}=\widehat{\mathcal{M}'}$ in \eqref{galois action on L-gp of tM} and the surjective map $\widehat{\mathcal{M}}=\widehat{\mathcal{M}'} \twoheadrightarrow \widehat{M}=\widehat{M'}$ in \eqref{exact of L-gp of M tM}, the action of $W_F$ on $\widehat M = \widehat{M'}$ can be obtained.

From \eqref{galois action on L-gp of tM}, we have
\[
{\bold A}_{\widehat{\mathcal{M}}} := (Z(\widehat{\mathcal{M}})^{\Gamma})^\circ \simeq ((\mathbb{C}^{\times})^k \times \{ \pm 1 \})^\circ = (\mathbb{C}^{\times})^k.
\]
We note that
\[
{\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma} \simeq ({\bold A}_{\widehat{\mathcal{M}}} \cdot Z(\widehat{\mathcal{G}}))/Z(\widehat{\mathcal{G}}) \subset \widehat{G}.
\]
Then, we have the following lemmas.
\begin{lm} \label{before gamma-inv of centers}
The quotient ${\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}$ is connected.
\end{lm}
\begin{proof}
Since ${\bold A}_{\widehat{\mathcal{M}}} = (Z(\widehat{\mathcal{M}})^{\Gamma})^\circ$ by definition and since $Z(\widehat{\mathcal{G}}) \subset {\bold A}_{\widehat{\mathcal{M}}},$ we have
\[
{\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}  = (Z(\widehat{\mathcal{M}})^{\Gamma})^\circ / (Z(\widehat{\mathcal{G}})^{\Gamma} \cap (Z(\widehat{\mathcal{M}})^{\Gamma})^\circ)
\]
which is isomorphic to $Z(\widehat{\mathcal{M}})^{\Gamma}/Z(\widehat{\mathcal{G}})^{\Gamma} $ by \cite[Lemma 1.1]{art99}.
Note that $Z(\widehat{\mathcal{M}})^{\Gamma}/Z(\widehat{\mathcal{G}})^{\Gamma} $ is connected due to the proof of \cite[Lemma 1.1]{art99}.
\end{proof}
From \eqref{exact of L centers} and \eqref{gamma invariants}, we have the following exact sequence
\[
1 \longrightarrow  {\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma} \longrightarrow Z(\widehat{M})^{\Gamma} \longrightarrow  H^1(F, Z(\widehat{\mathcal{G}})).
\]
The following lemma proves that the embedding ${\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}  \hookrightarrow Z(\widehat{M})^{\Gamma}$ is in fact an equality.
\begin{lm} \label{gamma-inv of centers}
With the notation above, we have
\[
{\bold A}_{\widehat M} = Z(\widehat{M})^{\Gamma}  ~~\text{and}~~
{\bold A}_{\widehat{M}} = {\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}.
\]
\end{lm}
\begin{proof}
We note from \cite[Lemma 1.1]{art99} that
\[
Z(\widehat{M})^{\Gamma} = Z(\widehat{G})^{\Gamma} \cdot (Z(\widehat{M})^{\Gamma})^{\circ}.
\]
Since $Z(\widehat{G}) = 1$ and $A_{\widehat M} = (Z(\widehat{M})^{\Gamma})^{\circ},$ the first equality is verified (hence, $Z(\widehat{M})^{\Gamma}$ is connected).
We note that
\begin{equation*} \label{an inclusion}
{\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma} \subset Z(\widehat{M})^{\Gamma} = {\bold A}_{\widehat{M}}
\end{equation*}
and ${\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}$ is connected by Lemma \ref{before gamma-inv of centers}.
Since ${\bold A}_{\widehat{M}}$ is a maximal torus in $C_{\phi}^{\circ},$ and since ${\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma}$ is also a torus having the same dimension with ${\bold A}_{\widehat{M}}$ (cf., Lemma \ref{lm for identity comp} in Section \ref{arthur conj}),
we have ${\bold A}_{\widehat{\mathcal{M}}}/Z(\widehat{\mathcal{G}})^{\Gamma} = {\bold A}_{\widehat{M}}.$
\end{proof}
\section{Weyl group actions} \label{weyl group actions}
For an $F$-Levi subgroup $\bold M$ of a connected reductive algebraic group $\bold G,$ we recall from \ref{basic notation} that
the Weyl groups are
$W_M = W(\bold G, \bold A_\bold M) := N_\bold G(\bold A_\bold M) / Z_\bold G(\bold A_\bold M),$
$W_{M'} = W(\bold G', \bold A_{\bold M'}) := N_{\bold G'}(\bold A_{\bold M'}) / Z_{\bold G'}(\bold A_{\bold M'}),$ and
$W_{\widehat M} = W(\widehat G, A_{\widehat M}) := N_{\widehat G}(A_{\widehat M}) / Z_{\widehat G}(A_{\widehat M}).$
Through the duality
\begin{equation*} \label{duality}
s_{\alpha} \mapsto s_{\alpha^{\vee}}
\end{equation*}
between simple reflections for $\alpha \in \Delta,$ we have
\begin{equation*}  \label{two isos}
W_M \simeq W_{\widehat M} \quad W_{\widehat{M}'} \simeq W_{M'}.
\end{equation*}
We thus identify
\begin{equation}  \label{identity for Weyls}
W_M = W_{\widehat M}=W_{\widehat{M}'} =W_{M'}.
\end{equation}
\subsection{On Levi subgroups} \label{WM-actions on Levis}
For simplicity, in Sections \ref{WM-actions on Levis} and \ref{WM-actions on rep of Levis}, we will write $\boldsymbol{\mathcal{G}}$ for \textit{both} quasi-split unitary groups $\U_n$ and its non quasi-split $F$--inner forms $\boldsymbol{\mathcal{G}}',$ and $\bold G$ for \textit{both} quasi-split special unitary groups $\SU_n$ and its non quasi-split $F$--inner forms $\bold G'.$

Let $\boldsymbol{\mathcal{M}}$ and $\bold M$ be $F$-Levi subgroups of $\boldsymbol{\mathcal{G}}$ and $\bold G,$ respectively. We recall from Section \ref{structure} that
\[
\mathcal{M} \simeq
{\GL}_{n_1}(E) \times \cdots \times  {\GL}_{n_{k}}(E) \times \mathcal{G}_m,
\]
\[
M \simeq
({\GL}_{n_1}(E) \times \cdots \times  {\GL}_{n_{k}}(E)) \rtimes G_m,
\]
where $\sum_{i=1}^{k} 2n_i + m = n$ with $n_i \geq  0$ and $m \geq 0.$ Notice here that $m\geq 2$ and is always even for non quasi-split inner forms.
We describe the action of Weyl group on Levi subgroups $\boldsymbol{\mathcal{M}}$ and $\bold M$ as well as irreducible representations of $\mathcal{M}$ and $M,$ based on the results in \cite{go95, go06}.
We denote by $S_k$ the symmetric group in $k$ letters.
From \cite{go95, go06}, we have
\begin{equation} \label{desc WM}
W_{\mathcal{M}} = W_{M} = \subset S_k \ltimes \mathbb{Z}_2^k.
\end{equation}

More precisely, $W_{\mathcal{M}}\simeq S\ltimes\mathcal C,$
where  $S=\langle (ij)|n_i=n_j \rangle,$ and $\mathcal C = \mathbb{Z}_2^k.$
For $g \in \mathcal{M},$ write
\[
g=(g_1,\dots g_i,\dots,g_j,\dots, g_k,h) \in GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E) \times \mathcal{G}_m.
\]
The permutation $(ij)$ acts on $g \in \mathcal{M}$ by
\begin{equation}  \label{action ij}
(ij):(g_1,\dots g_i,\dots,g_j,\dots, g_k,h) \mapsto (g_1,\dots,g_j,\dots,g_i,\dots,g_k,h).
\end{equation}
The finite $2$-group $\mathbb{Z}_2^k$ is generated by ``block sign changes" $C_i$ which acts on $g \in \mathcal{M}$ by
\begin{equation}  \label{action Ci-s}
C_i:(g_1,\dots,g_i,\dots,g_k,h) \mapsto (g_1,\dots,\, \varepsilon(g_i),\dots,g_k,h),
\end{equation}
where $\varepsilon(g_i) = {^t}\bar g_i^{-1}.$

\subsection{On representations of Levi subgroups} \label{WM-actions on rep of Levis}
Set $\Sigma$ to be $\Sigma_1 \otimes \Sigma_2 \otimes \cdots \otimes \Sigma_k \otimes \Upsilon.$
From \eqref{action ij} and \eqref{action Ci-s}, we have
\begin{gather*}
(ij)\Sigma=\Sigma_1\otimes\cdots
\otimes\Sigma_j\otimes\cdots
\otimes\Sigma_i\otimes\cdots
\otimes\Sigma_k\otimes\Upsilon;\\
C_i\Sigma=\Sigma_1\otimes\cdots
\otimes\varepsilon(\Sigma_i)\otimes\cdots
\otimes\Sigma_k\otimes\Upsilon,
\end{gather*}
where $\varepsilon(\Sigma_i)(g_i)=\Sigma_i(\varepsilon(g_i)),$
and these describe the action of $W_{\mathcal{M}}$ on $\Sigma.$

Now we turn to the case of $\bold M.$
Note that, since $G_0=G_1=1,$ $W_M$ acts on $M$ and an irreducible representation of $M$ in the same manner for $m=0, 1.$
In particular,
when $m=0$ (thus, $n$ is even),
from the proof of \cite[Lemma 2.4]{go06},
$M$ is of the form
\[
M = \{
\left( \begin{array}{cc}
 g & \\
 & \varepsilon(g) \\
 \end{array} \right) : \det (g) \det(\varepsilon(g)) = 1, ~ g \in GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E) \},
\]
which implies $M \simeq \{ g \in GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E) : \det g \in F^{\times} \}.$
When $m=1$ (thus, $n$ is odd), since $\mathcal{M} \simeq GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E) \times {\U}_1(F),$ we have
\[
M \simeq GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E).
\]
Note $\U_1(F) = E^{1} = \{ x \in E : x \bar x=1 \}.$
Further, for any $\Sigma \simeq \Sigma_1 \otimes \Sigma_2 \otimes \cdots \otimes \Sigma_k \otimes \omega \in \Irr(\mathcal{M}),$ we have
\[
{\Res}_{G_n}^{\mathcal{G}_n} \simeq \Sigma_1 \otimes \Sigma_2 \otimes \cdots \otimes \Sigma_k \otimes \omega \omega^ \varepsilon,
\]
which is always irreducible
(see the proof of \cite[Lemma 2.5]{go06}).
Let us move to the case of $m\geq 2.$
For $g \in M,$ from \eqref{form of bM'}, we write
\[
g=(g_0, h)=(g_1,\dots g_i,\dots,g_j,\dots, g_k,h) \in (GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E)) \rtimes G_m.
\]
Denote $wgw^{-1}$ by $(wg_0, h)$ for $w \in W_{\mathcal{M}}.$
Following arguments in \cite[p.353]{go06},
the permutation $(ij)$ acts on $g \in M$
\begin{align} \label{action ij-s}
(ij):(g_1,\dots g_i,\dots,g_j,\dots, g_k,h) \mapsto
& (g_1,\dots,g_j,\dots,g_i,\dots,g_k,
\alpha_m(g_0)^{-1}\alpha_m((ij)g_0)h) \\
&= (g_1,\dots,g_j,\dots,g_i,\dots,g_k, \nonumber
h),
\end{align}
since $\alpha_m(g_0)^{-1}\alpha_m((ij)g_0) = I_m.$
That is, the permutation $(ij)$ acts trivially on $G_m.$

The finite $2$-group $\mathbb{Z}_2^k$ is generated by ``block sign changes" $C_i$ which acts on $g \in M$
\begin{equation}  \label{action Ci}
C_i:(g_1,\dots,g_i,\dots,g_k,h) \mapsto (g_1,\dots,\, \varepsilon(g_i),\dots,g_k,\alpha_m(g_0)^{-1}\alpha_m(C_ig_0)h).
\end{equation}
Note from the definition of $\alpha_m$ in \eqref{def of alpha} that
\[
\alpha_m(g_0)^{-1}\alpha_m(C_ig_0) :=
\left( \begin{array}{ccc}
 (\det(g_i)\det(\bar g_i))^{-1}& & \\
 & I_{m-2} & \\
 & & \det(g_i)\det(\bar g_i)\\
 \end{array} \right) \in {\SU}_{m}(F).
\]

Given $\sigma \in \Irr(M),$ fix a lift $\Sigma \in \Irr(\mathcal{M}).$
Set $\Sigma \in \Irr(\mathcal{M})$ to be $\sigma_1 \otimes \sigma_2 \otimes \cdots \otimes \sigma_k \otimes \Upsilon,$
and $\Upsilon$ to be a lift in $\Irr(\mathcal{G}_m)$ with $\tau \hookrightarrow \Res^{\mathcal{G}_m}_{G_m} {\Upsilon}.$
Write $V_{\sigma}, V_{\Sigma_i},$ for $i=1, \dots, k,$ $V_{\Upsilon},$ and $V_\tau$ for the corresponding representation $\mathbb{C}$-vector spaces.
Then, $V_\sigma$ is of the form
\[
V_{\Sigma_1} \otimes V_{\Sigma_2} \otimes\cdots \otimes V_{\Sigma_k} \otimes V_\tau.
\]
From \cite[p. 353]{go06}, for $(g_0, h)=(g_1,\dots g_i,\dots,g_j,\dots, g_k,h) \in (GL_{n_1}(E) \times GL_{n_2}(E) \times \cdots \times GL_{n_k}(E)) \rtimes G_m,$ the representation $\sigma$ acts on $(v_1, v_2, \cdots, v_k, v_0) \in V_\sigma$ as follows,
\begin{equation*} \label{any sigma equ}
\sigma(g_0,h)(v_1, v_2, \cdots, v_k, v_0) =  \sigma_1(g_1)(v_1) \otimes \sigma_2(g_2)(v_2) \otimes \cdots \otimes \sigma_k(g_k)(v_k) \otimes \Upsilon(\alpha_m(g_0)^{-1})\tau(h)(v_0).
\end{equation*}
From \eqref{action ij-s} and \cite[p. 353]{go06},
we have
\begin{align*} \label{action ij on rep}
(ij)\sigma(g_0,h) =
& \sigma_1(g_1) \otimes \cdots \otimes \sigma_j(g_i) \otimes \cdots \otimes \sigma_i(g_j)\cdots \otimes\sigma_k(g_k) \otimes \Upsilon(\alpha_m((ij)g_0)^{-1})\tau(\alpha_m(g_0)^{-1}\alpha_m((ij)g_0)h) \\
= & \sigma_1(g_1) \otimes \cdots \otimes \sigma_i(g_j) \otimes \cdots \otimes \sigma_j(g_i)\cdots \otimes\sigma_k(g_k) \otimes \Upsilon(\alpha_m(g_0)^{-1})\tau(h).
\end{align*}
Likewise, using \eqref{action Ci},
we have
\begin{align*}
C_i\sigma(g_0,h) =
& \sigma_1(g_1) \otimes \cdots \otimes \varepsilon(\sigma_i)(g_i) \otimes \cdots \otimes\sigma_k(g_k) \otimes \Upsilon(\alpha_m(C_ig_0)^{-1})\tau(\alpha_m(g_0)^{-1}\alpha_m(C_ig_0)h) \\
= & \sigma_1(g_1) \otimes \cdots \otimes \sigma_i(\varepsilon(g_i)) \otimes \cdots \otimes \sigma_k(g_k) \otimes \Upsilon(\alpha_m(g_0)^{-1})\tau(h).
\end{align*}
Therefore, $W_{M}$ acts non-trivially only on $\sigma_1,\sigma_2, \dots, \sigma_k,$ but trivially on $\tau.$
\section{Revisiting $R$--groups for $\U_n$ and their inner forms} \label{R-groups for U(n) and its inner forms}
Based on some known results in \cite{bangoldberg12, go95, kmsw14, mok13} regarding $R$-groups for $\U_n$ and its $F$-inner form, we discuss Arthur's conjecture for $\U_n$ and its inner forms, behavior of $R$-groups within $L$-packets of $\U_n$ and its inner forms, and behavior of $R$-groups between $\U_n$ and its inner forms.
\subsection{$L$-packets for $\U_n$ and its inner forms} \label{LLC for unitary}
Let $\boldsymbol{\mathcal{G}}=\boldsymbol{\mathcal{G}}_n$ denote the quasi-split unitary group $\U_n$ with respect to $E/F$ and $J_n,$ and $\boldsymbol{\mathcal{M}}$ an $F$-Levi subgroup of $\boldsymbol{\mathcal{G}}.$
For our purpose of studying $R$-groups, we focus on $\Phi_{\temp}(\mathcal{G}).$
In \cite[Theorem 2.5.1.(b)]{mok13}, Mok generalized Rogawski's results (\cite{rog90}) in the case of unitary groups in three variables as follows.
There is a surjective finite-to-one map
\[
\Pi_{\temp}(\mathcal{G}) \longrightarrow \Phi_{\temp}(\mathcal{G}),
\]
and for $\varphi \in \Phi_{\temp} (\mathcal{G}),$ the tempered $L$-packet $\Pi_{\varphi}(\mathcal{G})$ is constructed.
The same is true for an $F$-inner form $\boldsymbol{\mathcal{G}}'$ of $\boldsymbol{\mathcal{G}}$ by Kaletha-Minguez-Shin-White \cite[Section 1.6.1]{kmsw14}.

Let $\boldsymbol{\mathcal{M}}$ and $\boldsymbol{\mathcal{M}}'$ be $F$-Levi subgroups of $\boldsymbol{\mathcal{G}}$ and $\boldsymbol{\mathcal{G}}',$ respectively, which are $F$-inner forms of each other. Then, from \eqref{form of tbM} and \eqref{form of tbM'}, we recall
\[
\boldsymbol{\mathcal{M}} \simeq {\Res}_{E/F} {\GL}_{n_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n_k} \times \boldsymbol{\mathcal{G}}_m,
\]
\[
\boldsymbol{\mathcal{M}}' \simeq {\Res}_{E/F} {\GL}_{n_1} \times \cdots \times {\Res}_{E/F} {\GL}_{n_k} \times \boldsymbol{\mathcal{G}}'_m,
\]
where $\boldsymbol{\mathcal{G}}'_m$ is an $F$-inner form of $\boldsymbol{\mathcal{G}}_m,$ $\sum_{i=1}^k 2n_i + m = n$ with $n_i \geq  0,$ and $m \geq 0.$
Let $\varphi \in \Phi_{\disc}(\mathcal{M})$ be given.
Then, $\varphi$ is $\boldsymbol{\mathcal{M}}$-relevant (see \cite[Section 8.2]{bo79}) and lies in $\Phi_{\disc}(\mathcal{M}')$ as well.
We note that $\varphi$ is of the form $\varphi_1 \oplus \varphi_2 \oplus \cdots \oplus \varphi_k \oplus \varphi_-,$
where $\varphi_i \in \Phi_{\disc}(\GL_{n_i}(E))$
and $\varphi_- \in \Phi_{\disc}(\mathcal{G}_m) = \Phi_{\disc}(\mathcal{G}'_m).$
For each $\varphi_i,$ due to \cite{ht01, he00, scholze13}, we construct $L$-packets $\Pi_{\varphi_i}(\GL_{n_i}(E))$ consisting of discrete series representations of $\GL_{n_i}(E).$
Note that $\Pi_{\varphi_i}(\GL_{n_i}(E))$ is a singleton.
For $\varphi_-,$ due to \cite{kmsw14, mok13, rog90},
we construct $L$-packets $\Pi_{\varphi_-}(G_m)$ and $\Pi_{\varphi_-}(G'_m)$ consisting of discrete series representations of $G_m$ and $G'_m,$ respectively.
By taking the tensor product of members in packets for each $\varphi_i$ and $\varphi_-,$ we thus construct $L$-packets ${\Pi}_{\varphi}(\mathcal{M})$ of $\mathcal{M}$ and ${\Pi}_{\varphi}(\mathcal{M}')$ of $\mathcal{M}',$ associated to the elliptic tempered $L$-parameter $\varphi.$
\subsection{Invariance of $R$-groups between $\U_n$ and its inner forms} \label{invariance for unitary}
\begin{thm} (Goldberg, \cite[Theorem 3.4]{go95}) \label{r-groups for tG}
Let $\boldsymbol{\mathcal{M}}$ be an $F$-Levi subgroup of $\boldsymbol{\mathcal{G}}.$
Given $\Sigma \simeq \Sigma_1 \otimes \Sigma_2 \otimes \cdots \otimes \Sigma_k \otimes \Upsilon \in \Pi_{\disc}(\mathcal{M}),$ we have
\[
R_{\Sigma} \simeq \mathbb{Z}_2^d,
\]
where $d$ is the number of inequivalent $\Sigma
_i$ such that the induced representation
$\mathbf{\textit{i}}_{\mathcal{G}_{n_i + m}, ~{\GL}_{n_i}(E)\times \mathcal{G}_m} (\Sigma \otimes \Upsilon)$ is reducible.
\qed
\end{thm}

\begin{thm} \label{iso bw R for tG}
Let $\varphi \in \Phi_{\disc}(\mathcal{M})$ be given.
Under the identity \eqref{identity for Weyls}, for any $\Sigma \in \Pi_{\varphi}(\mathcal{M})$ and $\Sigma' \in \Pi_{\varphi}(\mathcal{M}'),$ we have
\begin{equation*} \label{identity of W}
W(\Sigma) = W_{\varphi} = W(\Sigma').
\end{equation*}
and
\begin{equation} \label{identity of W'}
W^{\circ}_{\Sigma} = W^{\circ}_{\varphi} = W^{\circ}_{\Sigma'}.
\end{equation}
Therefore, we have
\[
R_{\Sigma} \simeq R_{\varphi, \Sigma} \simeq R_{\varphi} \simeq R_{\varphi, \Sigma'} \simeq R_{\Sigma'}.
\]
\end{thm}
\begin{proof}
This is a consequence of \cite[Lemma 11]{bangoldberg12} for maximal Levi subgroups and \cite[Section 7.6]{mok13} and \cite[Section 4.6]{kmsw14} for general Levi subgroups.
\end{proof}
We have following corollaries to Theorems \ref{r-groups for tG} and \ref{iso bw R for tG}.
\begin{cor}
Let $\varphi \in \Phi_{\disc}(\mathcal{M})$ be given.
For $\Sigma \in \Pi_{\varphi}(\mathcal{M})$ and $\Sigma' \in \Pi_{\varphi}(\mathcal{M}'),$ we have $\mathbf{\textit{i}}_{\mathcal{G},\mathcal{M}}(\Sigma)$ is irreducible if and only if $\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'}(\Sigma')$ is irreducible.
\qed
\end{cor}
\begin{cor} \label{r-groups for G'=bc}
Let $\boldsymbol{\mathcal{M}}'$ be an $F$-Levi subgroup of $\boldsymbol{\mathcal{G}}'.$
Given $\Sigma' \simeq \Sigma'_1 \otimes \Sigma'_2 \otimes \cdots \otimes \Sigma'_k \otimes \Upsilon' \in \Pi_{\disc}(\mathcal{M}'),$ we have
\[
R_{\Sigma'} \simeq \mathbb{Z}_2^d,
\]
where $d$ is the number of inequivalent $\Sigma'
_i$ such that the induced representation
$\mathbf{\textit{i}}_{\mathcal{G}'_{n_i + m}, ~{\GL}_{n_i}(E)\times \mathcal{G}'_m} (\Sigma' \otimes \Upsilon')$ is reducible.
\qed
\end{cor}
\begin{cor} \label{trivial on w}
Suppose $w \in R_{\Sigma'}$ and $w=sc,$ with $s\in S$ and $c\in\mathcal C.$  Then $w=1.$
\end{cor}
\begin{proof}
This is a consequence of \cite[Lemma 3.2]{go95} and Theorem \ref{iso bw R for tG}.
\end{proof}

Let $\Sigma' \in \Pi_{\disc}(\mathcal{M}')$ be given. Denote by $\mathbb{C}[R(\Sigma')]_{\eta}$ the group algebra of $R(\Sigma')$ twisted by a $2$-cocycle $\eta,$ and by $C(\Sigma'),$ known as the commuting algebra of $\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'} (\Sigma'),$ the algebra ${\End}_{\mathcal{G}'}(\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'} (\Sigma'))$ of $\mathcal{G}'$-endomorphisms of $\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'} (\Sigma').$
Then, we have
\[
C(\Sigma') \simeq \mathbb{C}[R(\Sigma')]_{\eta}
\]
as group algebras (see \cite{ks72, sil78, sil78cor}).
\begin{pro} \label{split 2-cocycle for G' bc}
With the above notation, we have
\[
C(\Sigma') \simeq \mathbb{C}[R_{\Sigma'}].
\]
\end{pro}
\begin{proof}
The proof is similar to \cite[Proposition 4.1]{go95}, due to Theorem \ref{iso bw R for tG} and Corollaries \ref{r-groups for G'=bc} and \ref{trivial on w}.
\end{proof}
As consequence of Proposition \ref{split 2-cocycle for G' bc}, we have the following corollary.
\begin{cor}
Let $\Sigma' \in \Pi_{\disc}(\mathcal{M}')$ be given. Then, each constituent of $\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'}(\Sigma')$ appears with multiplicity one.
\qed
\end{cor}

\begin{pro} \label{analogue of go95 thm4.3}
Let $\Sigma' \in \Pi_{\disc}(\mathcal{M}')$ be given. Then, $\mathbf{\textit{i}}_{\mathcal{G}', \mathcal{M}'}(\Sigma')$ has an elliptic constituent if and only if all constituents of $\mathbf{\textit{i}}_{\mathcal{G}',\mathcal{M}'}(\Sigma')$ are elliptic if and only if
\[
R_{\Sigma'} \simeq \mathbb{Z}_2^k.
\]
\end{pro}
\begin{proof}
The proof is similar to \cite[Theorem 4.3]{go95}, due to Corollary \ref{trivial on w} and Proposition \ref{split 2-cocycle for G' bc}.
\end{proof}
\begin{cor}
Let $\varphi \in \Phi_{\disc}(\mathcal{M})$ be given.
For any $\Sigma \in \Pi_{\varphi}(\mathcal{M})$ and $\Sigma' \in \Pi_{\varphi}(\mathcal{M}'),$
there is an elliptic constituent in $\mathbf{\textit{i}}_{\mathcal{G}, \mathcal{M}}(\Sigma)$
if and only if
there is an elliptic constituent in $\mathbf{\textit{i}}_{\mathcal{G}', \mathcal{M}'}(\Sigma').$
\end{cor}
\begin{proof}
This is a consequence of \cite[Theorem 4.3]{go95} and Proposition \ref{analogue of go95 thm4.3}.
\end{proof}
\section{Behaviour of $R$--groups for $\SU_n$ and its inner forms} \label{R-groups for SU(n) and its inner forms}
In this section, we prove Arthur's conjecture for both quasi-split special unitary groups $\SU_n$ and its inner forms.
Furthermore, we study the behavior of $R$-groups within $L$-packets and between inner forms of $\SU_n.$
We will use the notation in Sections \ref{structure}, \ref{weyl group actions}, and \ref{R-groups for U(n) and its inner forms}.
\subsection{$L$-packets for $\SU_n$ and its inner forms} \label{L-packets}
We discuss tempered $L$-packets of $\bold G=\SU_n$ and its $F$-inner form $\bold G'.$
It is natural to construct $L$-packets for $\bold G$ and $\bold G',$
by restricting $L$-packets for $\boldsymbol{\mathcal{G}}=\U_n$ and its $F$-inner form $\boldsymbol{\mathcal{G}}'$
which has been done by Rogawski \cite{rog90}, Mok \cite{mok13}, and Kaletha-Minguez-Shin-White \cite{kmsw14} (see Section \ref{LLC for unitary}).
Thus, given $\phi \in \Phi(G),$ from \cite[Th\'{e}or\`{e}m 8.1]{la85}, there exists a lifting $\varphi \in \Phi(\mathcal{G})$ such that
\[
\phi = \varphi \circ pr,
\]
where $pr$ is the projection  $\widehat{\mathcal{G}} \twoheadrightarrow \widehat{G}$ in \eqref{exact of L-gp of G tG}.
Note that  the homomorphism $pr$ is compatible with $\Gamma$-actions on $\widehat{\mathcal{G}}$ and $\widehat{G}$ (see Section \ref{L-groups}) and the lifting $\varphi \in \Phi(\mathcal{G})$ is chosen uniquely up to a $1$-cocycle of $W_F$ in $\widehat{(\mathcal{G}/G)}$ (see \cite[Section 7]{la85} and \cite[Theorem 3.5.1]{chaoli}).

For our purpose, we are interested in $\phi \in \Phi_{\temp}(G)$ and the lifting $\varphi$ lies in $\Phi_{\temp}(\mathcal{G}).$
Using the $L$-packet $\Pi_{\varphi}(\mathcal{G})$ for $\varphi \in \Phi_{\temp}(\mathcal{G})$ in Section \ref{LLC for unitary}, we construct an $L$-packet  $\Pi_{\phi}(G)$ for $\phi \in \Phi_{\temp}(G)$ as the set of isomorphism classes of irreducible constituents in the restriction from $\mathcal{G}$ to $G$ as follows:
\[
\Pi_{\phi}(G) := \{ \sigma \hookrightarrow {\Res}^{\mathcal{G}}_{G}(\Sigma),~~ \Sigma \in \Pi_{\varphi}(\mathcal{G}) \} / \simeq.
\]
Likewise, given $\phi \in \Phi_{\disc}(M),$ we construct an $L$-packet $\Pi_{\phi}(M)$ for $\phi \in \Phi_{\disc}(M)$ as follows:
\[
\Pi_{\phi}(M) := \{ \sigma \hookrightarrow {\Res}^{\mathcal{M}}_{M}(\Sigma),~~ \Sigma \in \Pi_{\varphi}(\mathcal{M}) \} / \simeq,
\]
where $\varphi$ lies in $\Phi_{\disc}(\mathcal{M})$ such that $
\phi = \varphi \circ pr,$ with the projection $pr: \widehat{\mathcal{M}} \twoheadrightarrow \widehat{M}$ in \eqref{exact of L-gp of M tM}.

All the above arguments apply verbatim to $F$-inner forms $\boldsymbol{\mathcal{G}}'$ and their $F$-Levi subgroups $\boldsymbol{\mathcal{M}}'.$
\subsection{Arthur conjecture for $\SU_n$ and its inner forms} \label{arthur conj}
The purpose of the section is to prove Arthur's conjecture, predicted in \cite{art89ast}, for $\bold G$ and $\bold G'.$
\begin{thm}  \label{arthur conj}
Given $\phi \in \Phi_{\disc}(M),$ $\sigma \in \Pi_{\phi}(M),$ and $\sigma' \in \Pi_{\phi}(M'),$ we have
\[
R_{\sigma} \simeq R_{\phi, \sigma} ~~ \text{and} ~~R_{\sigma'} \simeq R_{\phi, \sigma'}.
\]
\end{thm}
The rest of the section is devoted to the proof of Theorem \ref{arthur conj}.
Since all the following techniques apply to both $\bold M$ and $\bold M',$ we shall state the proof for $\bold M.$

Let $\phi \in \Phi_{\disc}(M)$ and $\sigma \in \Pi_{\phi}(M)$ be given.
Identifying $W_\phi$ with a subgroup of $W_M$ (see Section \ref{section for def of R}), we note that $W(\sigma) \subset W_\phi,$ which implies that
\begin{equation} \label{reduction for W}
W(\sigma) = W_{\sigma, \phi} = W_\phi \cap W(\sigma).
\end{equation}

Denote by $\varphi \in \Phi_{\disc}(\mathcal{M})$ a lifting of $\phi$ as in Section \ref{L-packets}.
For any $\Sigma \in \Phi_{\varphi}(\mathcal{M}),$ we have
\[
W^{\circ}_{\Sigma} = W^{\circ}_{\sigma},
\]
since the Plancherel measure is compatible with restriction (see \cite[Proposition 2.4]{choiy1}, \cite[Lemma 2.3]{go06}, for example) and $\Phi(P, A_M) = \Phi(\mathcal{P}, A_{\mathcal{M}})$ (see Section \ref{basic notation}).
Using \eqref{identity of W'}, we have
\begin{equation} \label{reduction for W'}
W^{\circ}_{\sigma} = W^{\circ}_{\varphi}.
\end{equation}
To prove Theorem \ref{arthur conj}, from \eqref{reduction for W} and \eqref{reduction for W'}, it is thus enough to show that
\begin{equation} \label{last arg for Arthur conj}
W^{\circ}_{\varphi} = W^{\circ}_{\phi}.
\end{equation}
Indeed, if so, then we have
\[
W^\circ_{\sigma, \phi} \overset{\text{definition}}{=} W^\circ_{\phi} \cap W(\sigma) \overset{\eqref{last arg for Arthur conj}}{=} W^\circ_{\varphi} \cap W(\sigma) \overset{\eqref{reduction for W'}}{=} W^\circ_{\sigma} \cap W(\sigma) = W^\circ_\sigma.
\]

In what follows, we prove \eqref{last arg for Arthur conj}.
From \cite[Chapter 3.4]{mok13} we set a maximal torus $T_{\varphi}$ in $C_{\varphi}(\widehat{\mathcal{G}})^{\circ}$ to be the identity component
\[
A_{\widehat \mathcal{M}} = (Z(\widehat{\mathcal{M}})^{\Gamma})^{\circ}
\]
of the $\Gamma$--invariants of the center $Z(\widehat{\mathcal{M}}).$
So, we have $\widehat \mathcal{M} = Z_{\widehat \mathcal{G}}(T_{\varphi}).$
Likewise, we set $T_{\phi}= A_{\widehat M} \subset C_{\phi}(\widehat{G})^{\circ},$ and  $\widehat M = Z_{\widehat G}(T_{\phi}).$
Here, we put
\[
\bar C_{\varphi}^\circ := C_{\varphi}(\widehat{\mathcal{G}})^\circ/Z(\widehat \mathcal{G})^{\Gamma}, ~~ \bar T_{\varphi} := T_{\varphi}/Z(\widehat \mathcal{G})^{\Gamma}.
\]
Note that $\bar C_{\varphi}^\circ \subset C_{\phi}(\widehat{G})^\circ \subset \widehat G$ and $\bar T_{\varphi} \subset T_{\phi} \subset \widehat M.$
Using an equality in \cite[p.64]{mok13}, we have
\[
W_{\varphi}^\circ \overset{\text{definition}}{=} N_{C_{\varphi}(\widehat{\mathcal{G}})^\circ}(T_{\varphi}) = N_{\bar C_{\varphi}^\circ}(\bar T_{\varphi}).
\]
Thus, it suffices to show that
\[
N_{\bar C_{\varphi}^\circ}(\bar T_{\varphi}) = N_{C_{\phi}(\widehat{G})^\circ}(T_{\phi}).
\]

With above notation, the following lemma holds.
\begin{lm} \label{lm for identity comp}
\[
\bar C_{\varphi}^\circ =  C_{\phi}(\widehat{G})^\circ
\]
\end{lm}
\begin{proof}
We first consider the following commutative diagram
\[
\begin{CD}
@.   @.1  @.  1 @.
\\
@.      @.        @VVV   @VVV  @.\\
1 @>>> Z(\widehat{\mathcal{G}})^\Gamma  @>>> C_{\varphi}(\widehat{\mathcal{G}})^\circ @>>> \bar C_{\varphi}^\circ := C_{\varphi}(\widehat{\mathcal{G}})^\circ/ Z(\widehat{\mathcal{G}})^\Gamma @>>> 1
\\
@.      @|          @VVV   @VVV  @.
\\
1 @>>> Z(\widehat{\mathcal{G}})^\Gamma @>>> C_{\varphi}(\widehat{\mathcal{G}}) @>>> C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma @>>> 1 \\
@.      @.          @VVV   @VVV  @.\\
@.   @. S_{\varphi}:=\pi_0(C_{\varphi}(\widehat{\mathcal{G}})) @= S_{\varphi}:=\pi_0(C_{\varphi}(\widehat{\mathcal{G}})) @>>> 1 .
\end{CD}
\]
It follows from the right vertical exact sequence above that
\[
\bar C_{\varphi}^\circ = C_{\varphi}(\widehat{\mathcal{G}})^\circ/ Z(\widehat{\mathcal{G}})^\Gamma \subset (C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma)^\circ \subset C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma.
\]
Note that the index
$
[\bar C_{\varphi}^\circ : C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma]
$
is finite and so is $
[\bar C_{\varphi}^\circ : (C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma)^\circ].
$ Since $\bar C_{\varphi}^\circ$ is connected due to the isomorphism $C_{\varphi}(\widehat{\mathcal{G}})^\circ/ Z(\widehat{\mathcal{G}})^\Gamma \simeq (C_{\varphi}(\widehat{\mathcal{G}})^\circ \cdot Z(\widehat{\mathcal{G}}))/ Z(\widehat{\mathcal{G}}),$ we have
\begin{equation} \label{an eqaulity in iendtity comp}
\bar C_{\varphi}^\circ = (C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma)^\circ.
\end{equation}
From the proof of \cite[Lemma 5.3.4]{chaoli} and the fact that $C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma \subset C_{\phi}(\widehat{G}),$ we have the exact sequence
\[
1 \longrightarrow C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma \longrightarrow C_{\phi}(\widehat{G}) \longrightarrow X^{\mathcal{G}}(\varphi) \longrightarrow 1,
\]
where $X^{\mathcal{G}}(\varphi) := \{ \bold a \in H^1(W_F,\widehat{\U_1}) : \bold a \varphi \simeq \varphi  \}.$ Since $X^{\mathcal{G}}(\varphi)$ is finite, we have
\[
(C_{\varphi}(\widehat{\mathcal{G}})/Z(\widehat{\mathcal{G}})^\Gamma)^\circ = C_{\phi}(\widehat{G})^\circ.
\]
Therefore, from \eqref{an eqaulity in iendtity comp}, Lemma \ref{lm for identity comp} is proved.
\end{proof}

Combining Lemmas \ref{gamma-inv of centers} and \ref{lm for identity comp}, we have proved \eqref{last arg for Arthur conj}.
This completes the proof of Theorem \ref{arthur conj}.
\subsection{Invariance of $R$-groups within $L$-packets and between inner forms} \label{behavior of Rgps}
Let $\phi \in \Phi_{\disc}(M)$ be given. In this section, we will discuss the behavior of Knapp-Stein $R$-groups within $L$-packets and between inner forms.
We first provide the following theorem that is crucial to the invariance (Theorems \ref{invariance within L-packet} and \ref{invariance 2 within L-packet}).
\begin{thm} \label{equality on W}
Fix a lift $\varphi \in \Phi_{\disc}(\mathcal{M})$ and $\Sigma \in \Pi_{\varphi}(\mathcal{M}).$ Then, we have
\[
W(\sigma) \simeq
\{
w \in W_M : {^w}\Sigma \simeq \Sigma \lambda ~~ \text{for some }~ \lambda \in (\mathcal{M}/M)^D
\}.
\]
\end{thm}
\begin{proof}
It suffice to consider the case of $m \geq 2,$ since the equality is already true for the case  $m =1$ due to \cite[Lemma 2.5]{go06}.
Let $m \geq 2.$ We then recall from \eqref{form of bM} that
\[
M \simeq ({\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)) \rtimes {\SU}_m(F).
\]
Since $\Sigma$ is of the form
\[
\Sigma_1 \otimes \Sigma_2 \otimes \cdots \otimes \Sigma_k \otimes \Upsilon,
\]
we set $\Sigma = \Sigma_0 \otimes \Upsilon.$
Denote by $\tau$ is an irreducible constituent in $\Res^{\U_m}_{\SU_m}(\Upsilon).$
Write $V_{\Sigma},$ $V_{\Sigma_0}=\otimes_{i=1}^kV_{\Sigma_i},$ $V_{\Upsilon},$ $V_\tau$ for the corresponding representation $\mathbb{C}$-vector spaces.

Given $(g,h)\in M$ with $g = (g_1, \cdots, g_k) \in {\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E)$ and $h \in {\SU}_m(F),$ $v'_0 = v_1 \otimes \cdots \otimes v_k \in V_{\Sigma_0},$ and $v_0 \in V_\tau,$
we define a representation $\sigma^*$ of $M$
\begin{equation} \label{def sigma*}
\sigma^*((g, h))(v'_0 \otimes v_0) =\Sigma_0(g)v'_0 \otimes \Upsilon(\alpha_m(g)^{-1})\tau(h)v_0.
\end{equation}
It turns out from \cite[p.353]{go06} that
$\sigma^*$ is an irreducible constituent in $\Res^{\U_n}_{\SU_n}(\Sigma).$

In what follows, using the idea in the proof of \cite[Proposition 2.9]{go06}, we will show
\begin{equation} \label{W*=}
W(\sigma^*) = \{
w \in W_M : {^w}\Sigma \simeq \Sigma \lambda ~~ \text{for some }~ \lambda \in (\mathcal{M}/M)^D
\}.
\end{equation}
The inclusion $\subset$ is obvious.
Suppose that ${^w}\Sigma \simeq \Sigma \lambda$ for some $\lambda \in (\mathcal{M}/M)^D.$
From \eqref{desc WM}, we set $w=sc$ with $s\in S_k$ and $c \in \mathbb{Z}_2^k.$
From Sections \ref{WM-actions on Levis} and \ref{WM-actions on rep of Levis}, we have
\[
{^w}\Sigma \simeq \bigotimes_{i=1}^k\varepsilon_i(\Sigma_{s(i)}) \otimes \Upsilon,
\]
where $\varepsilon_i$ is either $\varepsilon$ or trivial.
Since ${^w}\Sigma \simeq \Sigma \lambda,$ we have
$\varepsilon_i(\Sigma_{s(i)})  \simeq \Sigma_i \lambda$
for each $i,$ and $\Upsilon \simeq \Upsilon \lambda.$
We fix intertwining maps $T_i:V_{\Sigma_{s(i)}}\rightarrow V_{\Sigma_i}$ with $\Sigma_i \lambda T_i=T_i \varepsilon_i(\Sigma_{s(i)}),$
and $T_\lambda:V_{\Upsilon} \rightarrow V_{\Upsilon}$ with $\Upsilon \lambda T_{\lambda}=T_{\lambda} \Upsilon.$
Write
\[
T=\bigotimes_{i=1}^k T_i \otimes T_{\lambda}.
\]
Note that $\Sigma T = T {^w}\Sigma.$
Then, from the definition \eqref{def sigma*} of $\sigma^*,$ we have
\begin{align*}
T{^w}\sigma^*(g,h)(v'_0 \otimes v_0)
&= \bigotimes_{i=1}^k T_i \varepsilon_i(\Sigma_{s(i)})(g_i)v_i \otimes T_{\lambda} \Upsilon (\alpha_m({^w}g)^{-1})\tau(\alpha_m(g)^{-1}\alpha_m({^w}g)h)v_0 \\
&= \bigotimes_{i=1}^k T_i \varepsilon_i(\Sigma_{s(i)})(g_i)v_i \otimes T_{\lambda} \Upsilon (\alpha_m(g)^{-1})\tau(h)v_0 \\
&= \bigotimes_{i=1}^k
\Sigma_i(g_i)\lambda(g_i)T_i(v_i) \otimes  \Upsilon(\alpha_m(g)^{-1}) \lambda(\alpha_m(g)^{-1}) \tau(h)T_{\lambda}(v_0) \\
&= \sigma^*\lambda(g,h)T(v_0'\otimes v_0)\\
&=\sigma^*T(v_0'\otimes v_0),
\end{align*}
since
\[
\lambda(g,h) \overset{\eqref{imp iso}}{=} \lambda(
\left( \begin{array}{ccc}
 g& & \\
 & \alpha_m(g)^{-1}h & \\
 & & \varepsilon(g) \\
 \end{array} \right)) = 1.
\]
Thus, we have $w \in W(\sigma^*),$ which implies
\eqref{W*=}.
From the proof of \cite[Theorem 4.19]{chgo12}, we note
\begin{equation*} \label{W*}
W(\sigma) \simeq W(\sigma^*),
\end{equation*}
since both are in the same restriction $\Res^{\U_n}_{\SU_n}(\Sigma).$
Therefore, we proved the theorem.
\end{proof}
\begin{rem}
The difference between $W(\sigma)$ and $W(\Sigma)$ turns out to be whether the twisting by a character in $(\mathcal{M}/M)^D$ is allowed or not.
We also refer to \cite[Theorem 3.7]{go06} for another description of this difference.
\end{rem}
\begin{rem}
We note that Theorem \ref{equality on W} holds for $F$-inner forms $\bold M'$ as well.
To be precise, given $\sigma' \in \Pi_{\phi}(M'),$ we fix $\Sigma' \in \Pi_{\varphi}(\mathcal{M}').$
Then, we have
\[
W(\sigma') \simeq
\{
w \in W_{M'} : {^w}\Sigma' \simeq \Sigma' \lambda ~~ \text{for some }~ \lambda \in (\mathcal{M}'/M')^D
\}.
\]
\end{rem}
\begin{thm} \label{invariance within L-packet}
Given $\sigma_1, ~ \sigma_2 \in \Pi_{\phi}(M),$ we have
\[
R_{\sigma_1} \simeq R_{\sigma_2}.
\]
Furthermore, given $\sigma'_1, ~ \sigma'_2 \in \Pi_{\phi}(M'),$ we have
\[
R_{\sigma'_1} \simeq R_{\sigma'_2}.
\]
\end{thm}
\begin{proof}
Since $W^{\circ}(\sigma_1) = W^{\circ}(\sigma_2),$ it is enough to show that
\[
W(\sigma_1) \simeq W(\sigma_2).
\]
Fix $\Sigma^1$ and $\Sigma^2$  in $\Pi_{\varphi}(\mathcal{M})$ such that $\sigma_1 \hookrightarrow \Res^{\U_n}_{\SU_n}(\Sigma^1)$ and $\sigma_2 \hookrightarrow \Res^{\U_n}_{\SU_n}(\Sigma^2).$
Suppose that ${^w}\Sigma^1 \simeq \Sigma^1 \lambda$ for some $\lambda \in (\mathcal{M}/M)^D.$
From \eqref{desc WM}, we write $w=sc$ with $s\in S_k$ and $c \in \mathbb{Z}_2^k.$
Set $\Sigma^1 \simeq \Sigma^1_{1} \otimes \Sigma^1_{2} \otimes \cdots \otimes \Sigma^1_{k} \otimes \Upsilon_1.$
Then $\Sigma^1_{i} \simeq \varepsilon_i(\Sigma^1_{s(i)}) \lambda$
for each $i,$ and $\Upsilon_1 \simeq \Upsilon_1 \lambda.$
Set
\[
\lambda' := \left\{ \begin{array}{ll}
         \lambda, & \mbox{on}~~{\GL}_{n_1}(E) \times \cdots \times {\GL}_{n_k}(E),\\
        \mathbbm{1}, & \mbox{on} ~~{\U}_m(F),
        \end{array} \right.
\]
which is a character on $\mathcal{M}/M.$
Moreover, $\lambda'$ satisfies
\[
{^w}\Sigma^2 \simeq \Sigma^2 \lambda',
\]
which implies, from Theorem \ref{equality on W}, that
$W(\sigma_1) \hookrightarrow W(\sigma_2).$
In the same manner, one can verify $W(\sigma_2) \hookrightarrow W(\sigma_1).$
Thus, we have $R_{\sigma_1} \simeq R_{\sigma_2}.$
Since the method is the same, we omit the proof for $R_{\sigma'_1} \simeq R_{\sigma'_2}.$
\end{proof}
\begin{rem}
The trivial character $\mathbbm{1}$ in the definition of $\lambda'$ can be replaced by
a character $\lambda_0$ on $\U_m(F)$ such that $\Upsilon_2 \simeq \Upsilon_2 \lambda_0,$ where $\Upsilon_2$ is the representation of $\U_m(F)$ in the decomposition $\Sigma^2 \simeq \Sigma^2_{1} \otimes \Sigma^2_{2} \otimes \cdots \otimes \Sigma^2_{k} \otimes \Upsilon_2.$
\end{rem}
\begin{thm} \label{invariance 2 within L-packet}
Given $\sigma \in \Pi_{\phi}(M)$ and $\sigma' \in \Pi_{\phi}(M),$ we have
\[
R_{\sigma} \simeq R_{\sigma'}.
\]
\end{thm}
\begin{proof}
The proof is similar to Theorem \ref{invariance within L-packet}.
\end{proof}
\begin{rem} \label{final remark}
Due to \cite[Theorem 3.7]{go95}, Theorem \ref{invariance 2 within L-packet} shows that the Knapp-Stein $R$-group $R_{\sigma'}$ is of the form
\[
\Gamma_{\sigma'} \ltimes \mathbb{Z}^d,
\]
for some subgroup $\Gamma_{\sigma'}$ in $R_{\sigma'}$ defined in \textit{loc. cit} and some integer $d$ in Corollary \ref{r-groups for G'=bc}.
This implies that $R_{\sigma'}$ is in general non-abelian from the argument in \cite[Remark p.359]{go95}.
\end{rem}

redpajama_set_name: RedPajamaArXiv