text:
Hasse Olsson är en organist, keyboardist och producent som varit verksam i Sverige sedan slutet av 1970-talet. Han är organist i bandet Stockholm Stoner med Mats Ronander.

Biografi
Olsson första framträdanden var då han deltog på Peter Lundblads soundtrack till Tove Janssons Vem ska trösta knyttet, som spelades in 1977 och kom ut 1978, och live med "Wellander Ronander Band" 1978. Under 1980-talet var han på många turnéer då han var kapellmästare i Ulf Lundells band från 1980 till 1987 både i studio och på scen och samtidigt var medlem i Dan Hylanders och Py Bäckmans Raj Montana Band. Han spelade även hammondorgel i Low budget blues band med bland andra Mats Ronander och Lasse Wellander. Under stor del av 80-talet spelade han med, och producerade, Tove Naess. Med henne var han även verksam som låtskrivare. Hasse Olsson var med och skrev musiken till filmatiseringen av Ulf Lundells roman Sömnen 1983. Han spelade med många av artisterna som deltog under ANC-galan i Göteborg 1985.

Olsson medverkade även på turnén "Klubbrock" 1992, med både svenska och internationella artister. 1995 gav Hasse Olsson, under namnet "Cloudscape", ut en meditationsplatta som hette "Ambiens Volym 1" och där han själv var upphovsman. Bland andra artister som Olsson spelat med under sin karriär finns;

Ted Gärdestad / Eva Dahlgren / Pugh Rogefeldt /  Mikael Rickfors / Lis Sørensen / Totta Näslund / Sanne Salomonsen / Åge Aleksandersen / Kal P. Dal / Lisa Nilsson / Mick Ronson / Tommy Nilsson / Ola Magnell / Lasse Tennander / Idolerna (Lalla Hansson, Svenne Hedlund, Tommy Blom, Lennart Grahn) / Marie Bergman / Anne-Lie Rydé / Louise Hoffsten  / Ole Paus / Plura Jonsson / Ola Bjurman / Börge Ring / Ingemar Olsson / Dan Tillberg / Carola Häggkvist / Lena Philipsson / Marie Fredriksson / Sven Ingvars / Harpo / Anna Lotta Larsson

Videosessioner från olika delar av Olssons karriär

1982 Ulf Lundell - Odysseus - Live från Konserthuset 1982
1983 Ola Magnell - Tomma tunnor - Klipp från Tv-framträdande []
1985 Ulf Lundell - Bandpresentation/Fri man i stan - Live från Hovet/Stockholm
1985 Tottas bluesband - Going down - Live från Anc-galan
1985 Tottas bluesband - Sail away - Live från Anc-galan
1985 Dan Hylander & Raj Montana Band - Hymn till en snubblad vandrare - Live från Anc-galan
1990 Grymlings - Mitt bästa för dig - Officiella musikvideon
2009 Stockholm Stoner - Broken wing angel & The bridge - Live från Nyhetsmorgon

Album producerade/medproducerade av Hasse Olsson

Tove Naess Fighting for love 1985, Shine on 1987, The Album 1988, Time of trust 1990
Big Deal
Ola Magnell Blå neon 1986
Björn Holm Man över bord 1987
Lennart Grahn Den nakna sanningen 1996
Peter Lundblad Allt betyder allt 1997
The Funk Gang 1997
Charlotte Perrellis hyllnings-skiva till Monica Zetterlund I din röst 2006.

Diskografi (ett urval av album där Hasse Olsson medverkat)

1978 Peter Lundblad - Vem ska trösta knyttet
1978 Lasse Tennander - På jakt
1979 Lasse Tennander - Längst därinne är himlen ändå röd
1979 Peter Lundblad - Club Öken
1980 Ulf Lundell - Längre inåt landet
1980 Lasse Lindbom Band - Tärningen är kastad
1980 Py Bäckman - Hard wind blows
1980 Kal P Dal - Svarta fåret
1981 Peter Lundblad - Jag tänker på dom ibland
1981 Dan Hylander - September
1981 Ola Magnell - Europaväg 66
1981 Niklas Strömstedt - Skjut inte… det är bara jag
1982 Ulf Lundell - Kär och galen
1982 Low Budget Blues Band - Low Budget Blues band
1982 Dan Hylander - Bella notte
1982 Py Bäckman - Belle de jour
1982 Mats Ronander - God bok
1983 Ulf Lundell - Sweethearts
1983 Ola Magnell - Gaia
1983 Mikael Rickfors - Blue Fun
1983 Niklas Strömstedt – Andra äventyr
1983 Dan Hylander - Calypso
1983 Py Bäckman - Sista föreställningen
1983 Eric Andersen - Tight in the night
1984 Dan Hylander - Om änglar och sjakaler
1984 Py Bäckman - Kvinna från Tellus
1984 Mats Ronander - 50/50
1984 Ratata - Paradis
1984 Marie Fredriksson - Het vind
1985 Ulf Lundell - Den vassa eggen
1985 Mats Ronander - Tokig
1985 Tove Naess - Fighting for love
1985 Dan Hylander & Py Bäckman - Telegram långt farväl / Live
1985 ANC - galan (Raj Montana Band/Tottas Bluesband/Hansson De Wolfe United)
1986 Mikael Rickfors - Rickfors
1986 Ola Magnell - Blå neon
1987 Ulf Lundell - Det goda livet
1987 Tove Naess - Shine on
1987 Björn Holm - Man över bord
1988 Tove Naess - The Album
1990 Tove Naess - Time of trust
1990 Grymlings - Grymlings
1990 Low Budget Blues Band - Low Budget Blues band II
1990 Lisa Nilsson - Indestructible
1990 Carola - Much more
1991 Pugh Rogefeldt - Människors hantverk
1994 Lasse Lindbom - Vägen söderut
1994 Low Budget Blues Band - Country File
1996 Lennart Grahn - Den nakna sanningen
1997 Peter Lundblad - Allt betyder allt
1998 Magnus Johansson - Simplistico
1999 Totta Näslund - En dåre som jag
2000 Idolerna - Idolerna
2001 Jan Malmsjö - Välkommen till min jul
2006 Charlotte Perrelli - I din röst

Referenser

Externa länkar
Hasse Olsson på Discogs

Svenska musiker
Män
Levande personer
Födelseår saknas
Svenska keyboardister


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Kim Davis – Let me show you what persecution looks like
Tuesday, September 22nd, 2015
ON US TV today (Sept. 22), the clerk who refused to issue marriage licenses to ANY couple because she didn't believe in same sex marriage has said what hurts her most about how she has been treated since she defied the Supreme Court ruling on marriage in June – being called 'a hypocrite'.
Kim Davis of Rowan County, Kentucky, has been jailed, received a 'Religious Liberty' award for fighting 'legal tyranny', been called 'the bravest woman in America' and compared to Abraham Lincoln by her lawyer. She has also been called a martyr and a victim of anti-Christian persecution. She even admitted denying marriage licences to friends, and receiving death threats.
All because she believes in her Christian duty to break the oath she took as a US Government official to uphold the law, in a country whose constitution explicitly forbids the primacy of any religion. The First Amendment clearly states that the US government cannot make any law 'respecting an establishment of religion.' In the words of the Supreme Court, this generally means that the government can't "pass laws which aid one religion, aid all religions, or prefer one religion over another," or otherwise become entangled in religious affairs.
Click to continue »
Posted in Bi-National Couples, Commentary, Discrimination, England, Immigration, Issues, Kentucky, LGBT Rights, Marriage Equality/Gay Marriage, Marriage License, Opponents, People, Religion, Religious Liberty, Sexual Orientation, USA, Washington State | 1 Response »
Tags: gay, gay couple, Gay Marriage, gay rights, kentucky, kim davis, marriage equality, opponents, religious liberty, same sex marriage, supreme court, united kingdom, usa, washington, washington state
'Rainbow Profile Pictures for Sex Offenders' – Homophobia NOT Satire
Facebook founder Mark Zuckerberg was one of 26 million people who used the rainbow filter on his profile picture. SOURCE: Facebook
IT'S LATE – my phone rings. It's someone I haven't heard from in a while. I answer.
His greeting is warm, but it sounds like he's been drinking.
He chats about Facebook – he says he doesn't use it much but he's logged in and noticed the rainbow filter on my profile picture, the only one among his list of friends on the site.
He says he is concerned because he has just read an article claiming that sex offenders have been ordered to use rainbow profile pictures on social media so others can easily identify them as potential predators. Click to continue »
Posted in Bullying, Coming Out, Commentary, Discrimination, Employment, England, Facebook, LGBT Rights, Marriage Equality/Gay Marriage, Opponents, Sexual Orientation, US Supreme Court, USA | No Responses »
Tags: facebook, Gay Marriage, homophobia, marriage equality, same sex marriage, satire, sex offenders, sexual offences, supreme court, usa
Anti-LGBT Bills Become Law in Michigan, North Carolina
Two states enshrined anti-LGBT discrimination into law yesterday, legalizing dangerously broad "religious freedom" aka right to discriminate laws.
The first one was North Carolina, where the House rushed through a veto override with no time for discussion. The Washington Blade reports:
The North Carolina legislature has succeeded in overriding Gov. Pat McCrory's veto of controversial legislation that would enable magistrates to opt out of performing marriages to which they have religious objection, including a same-sex union. The override succeeded in the House on Thursday by a vote of 69-41, which is three votes more than the three-fifths vote needed for the legislative maneuver. The Senate already succeeded in overriding the veto last week by a vote of 32-16. The legislation, Senate Bill 2, would enable magistrates and registers of deeds to opt out of issuing marriage licenses for a period of at least six months to same-sex couples — or any couple — based on a religious objection.
Up north in Michigan, Governor Rick Snyder wasted no time signing his own state's bills. The Washington Blade reports:
"Michigan Gov. Rick Snyder signed into law on late Thursday morning a package of religious freedom bills seen to enable anti-LGBT legislation in the state's adoption agencies. Snyder announced he had signed the legislation — House Bills 4188, 4189, 4190 — just one day after the Michigan Senate approved the bills. The Michigan House already passed the bills in March. In a statement, Snyder said he signed the legislation as a means to ensure children without homes in Michigan have access to families to care for them. The new laws are set to go into effect in 90 days."
Let's be clear that the laws in both states allow someone to refuse service to anyone, not just gays and lesbians, based on their own personal religious beliefs.
Posted in Adoption, Bills, Discrimination, Marriage License, Michigan, North Carolina, Opponents, Politicians, Republicans, USA | 8 Responses »
Tags: anti gay, laws, michigan, north carolina
Five Reasons Why Marriage Equality Isn't The Be All And End All Of The Fight
Monday, June 1st, 2015
Two years after striking down DOMA, the U.S. Supreme Court is once again set to issue a ruling on same-sex marriage. If everything goes as expected and the Court lifts the remaining state bans on same-sex marriage, it will make history. But it won't be the end of the fight for equality.
In fact, here are five reasons why achieving marriage equality is only the beginning of the end…
Nondiscrimination laws
More than half (29 total) of the states lack laws banning sexual orientation discrimination, and even more (32) don't prohibit it when it comes to gender identity. And (thanks, Republican Congress!) there is no federal statute either. What does this mean exactly? In a nutshell: That there some states where gay people can both get married and be fired.
Authored By Graham Gremore – See the Full Story at Queerty
Find more articles and gay wedding resources.
Posted in Commentary, Discrimination, LGBT Rights, Marriage Equality/Gay Marriage | No Responses »
Tags: discrimination, Gay Marriage, lgbt rights, marriage equality, same sex marriage
Nicaragua Outlaws Marriage Equality, Same Sex Adoption
Nicaragua has rejected same-sex marriage and adoption by gay single people or couples. The Family Code establishes that marriage is defined as 'between a man and a woman'. It also ensures only opposite-sex couples, foreign or Nicaraguan, can adopt.
Gay couples will also be banned from using fertility treatments to get pregnant and will not be entitled to social security protection or inheritance in the case of loss of a spouse.
This is one of the biggest step backs for a Central American country in recent years, taking effect yesterday (8 April).
Authored By Joe Morgan – See the Full Story at Gay Star News
Image via Google Maps
Posted in Adoption, Central America, Discrimination, Marriage Equality/Gay Marriage, Nicaragua | 1 Response »
Tags: adoption, ban, Gay Marriage, marriage equality, nicaragua, same sex marriage
Springfield, Missouri Repeals LGBT Rights With Heartbreaking 51.4% Vote
The religious right has it figured out – put LGBT rights repeals on the ballot in special elections, and they can turn-out enough people to get them off the books. It happened again in Springfield, Missouri yesterday:
Voters in Springfield, Missouri today repealed protections for LGBT citizens which were added by the City Council to the city's non-discrimination ordinance last October, KY3 reports: With all the votes tallied, about 51.4 percent of the voters on Tuesday said they wanted to repeal the City Council's action, while 48.6 percent wanted to retain it. The raw numbers are 15,347 people voted "yes" to repeal the anti-discrimination ordinance and 14,493 people voted "no" to keep it on the books. For a brief time on Tuesday night, when about 60 percent of the precincts were counted, the "no repeal" votes had a slim lead over the "yes, repeal" votes. As more precincts were counted, however, the "yes" votes went back on top.
They did it with the same pack of lies that has worked elsewhere, using scary lines about men using women's bathrooms. It just makes me crazy to see these hateful lies used again and again, and even more to see that they work.
Posted in Discrimination, Initiatives, LGBT Rights | 2 Responses »
Tags: lgbt rights, lies, missouri, repeal, springfield, vote
Arkansas Governor Signs Updated Bill
Friday, April 3rd, 2015
We also have some updates out of Arkansas, after the Governor there unexpectedly declined to sign that state's right to discriminate bill, asking for changes.
Former AG Opposes Bill: Former Arkansas Attorney General Dustin McDaniel wrote an op-ed in the Arkansas Democrat-Gazette condemning the "religious freedom" bill passed this week by the legislature but that Gov. Asa Hutchinson has refused to sign. full story
Rights Initiative Can Collect Signatures: Arkansas Attorney General Leslie Rutledge approved a proposed referendum to add sexual orientation to a list of categories protected from discrimination full story
Governor Signs Amended Bill: Arkansas Gov. Asa Hutchinson signed a religious freedom measure into law on Thursday after state lawmakers overhauled their proposal so that it mirrors the federal law. full story
Governor Considers Anti Discrimination Order: Hutchinson said he's considering signing an executive order that bars discrimination among the state's workforce. full story
ACLU Praises Governor: The ACLU issued a statement praising the Governor's actions. full story
HRC Calls for LGBT Protections: "The fact remains that the only way to ensure LGBT Arkansans are treated equally under state law is to add explicit protections for them. Moving forward, Arkansas should explicitly clarify that the RFRA cannot be used to undermine non-discrimination protections at any level." full story
State Supremes Create New Marriage Equality Case: In related news, the state Supreme Court created a new marriage equality case to determine if a justice sworn in in January should help decide whether gay marriage is legal in Arkansas. full story
Posted in Arkansas, Bills, Discrimination, LGBT Rights, Politicians, Religious Liberty, Republicans, USA | No Responses »
Tags: arkansas, bill, governor asa hutchinson, RFRA, signed
Indiana Governor Signs "Fix" for Right to Discriminate Law
Governor Pence has signed a "fix" to the state's right to discriminate law, but the debate is still swirling.
Here are the general updates around the law and the "fix":
Legislature Passes the "Fix": The Indiana House voted 66-30 to alter the divisive "religious freedom" law to remove fears that it would allow businesses to discriminate against gays and lesbians. The Indiana Senate voted 34-16 to OK changes to the "religious freedom" law. full story
Governor Pence Signs the "Fix": Indiana Gov. Mike Pence has signed into law a measure preventing the state's new "religious freedom" law from being used to discriminate against LGBT individuals. full story
Commerce Groups Praise "Fix": Indiana Chamber of Commerce President Kevin Brinegar said the proposed changes that would prohibit service providers from discriminating against gays and lesbians are "what businesses and individuals from around the state, country and world needed to hear" while Indy Chamber President and CEO Michael Huber called the proposed changes "a major step in the right direction in efforts to protect the rights of all Hoosiers." full story
"Fix" Doesn't Go Far Enough, Groups Say: Lambda Legal and the Human Rights Campaign say the "fix" doesn't go far enough to solve the state's lack of protections against LGBT discrimination. full story
Angie's List Calls "Fix" Insufficient: Angie's List calls the revision to the "religious freedom" law "insufficient," becoming the first major local company to reject the deal hammered out by Indiana legislators, the business community and others. full story
GOP Quiet About Law: The Washington Blade notes that few in the GOP have spoken out against the law. full story
55% Oppose Law: A HuffPost/YouGov poll found that 55 percent of people think businesses should be required to provide the same services to LGBT customers, while 31 percent said businesses should be allowed to refuse on religious grounds. The poll was taken before the "fix". full story
Nix it Now Campaign Launches: In the wake of the Indiana experience, the National LGBTQ Task Force launched the "Nix it Now" campaign to discourage similar laws in the future. full story
Anti Gay Pizza Parlor Gets Cash Bonanza: The anti-gay pizzeria that spoke out blatantly about its plans to discriminate will net almost $500K for their bigotry, thanks to an online crowdfunding page. full story
We also have some analysis and opinions about the law and the "fix":
What the "Fix" Does: According to Ian Millhiser at Think Progress, the "fix" basically prevents the law from usurping local LGBT rights laws, but does not apply to religious groups, and gives no new rights to LGBT citizens. full story
Dem's Alternative Leaves Out Trans Citizens: Autumn Sandeen takes apart the law, and notices that the Democrats' alternative leaves out transgender folks. full story
Gay Couple Speaks Out: A married Indiana gay couple speaks out about the law from inside the state. full story
How Hoosiers Feel About Boycot: After calls to boycott Indiana over the state's 'religious freedom' law, Skylar Baker-Jordan looks at what LGBT people in the state actually think. full story
Fixes Are a Good First Step: Zack Ford at Think Progress says the "fix" is a good first step, but much more needs to be done to secure LGBT rights. full story
Law Passed Because Liberals Stayed Home: How did this law get passed in the first place? The Dallas Voice's Hardy Haberman has the answer – liberals stayed home. full story
More states and localities consider banning travel to the state:
Maryland Governor – No Travel Ban: Maryland Gov. Larry Hogan on Wednesday rejected a gay lawmaker's request to ban state-funded travel to Indiana over its religious freedom law. full story
Colorado Governor Considers Ban: DENVER – Colorado Governor John Hickenlooper's (D-Colorado) office says the governor has not ruled out the option of banning or discouraging official travel to Indiana by Colorado state employees, but is waiting to see the "fix" being worked on by Gov. Mike Pence (R-Indiana) and his state legislature. full story
Opponents of the las continue to speak out:
Drag Queen Argues Against Boycott: Drag Queen Lt. Moxie Anne Magnus argues for not boycotting the state, although she opposes the law. full story
Salesforce to Help Employees Relocate: Salesforce.com Inc. employees in Indiana who are uncomfortable with the state's recently adopted Religious Freedom Restoration Act can relocate with help from the company, CEO Marc Benioff told CNN Wednesday. full story
LGBT Weekly Calls for Ban to Continue: Joel Trambley calls for continuing the boycott against the state until it repeals the law completely. full story
Ten Best Quotes: Queerty rounds up the ten best quotes opposing the law. full story
Mellancamp Speaks Out: Singer John Mellancamp, an Indiana native, speaks out against the law. full story
Boy George Calls kLaw Outdated: Out singer Boy George called the law outdated, but said he would perform in Indiana despite the state's so-called religious freedom law. full story
Supporters of the law are still sounding off too:
Christie Defends Pence: NJ Governor Chris Christie, in his first remarks on the subject in an interview with The Wall Street Journal on Wednesday, focused on defending Pence: "I know Mike Pence, and he's got nothing but love in his heart for people." full story
Professors Support Marriage Equality, RFRA Laws: The Christian Post profiles three law professors who support both marriage equality and RFRA laws. full story
Ohio Group Jumps Into Fray: Citizens for Community Values, the Ohio group that successfully pushed an Ohio constitutional ban 11 years ago, is urging people to contact GOP lawmakers to preserve the right to discriminate law. full story
Senator Cotton – Be Glad You're Not Being Hanged: Arkansas Senator Tom Cotton says gays should get some perspective on the Indiana bill and be glad they aren't being hanged, like in Iran. full story
Jeb Bush Wavers on Support: After endorsing the law, Bush has now backtracked, saying a "better approach" would have been a "consensus-oriented" process that brought both people of faith and members of the LGBT community to the table. full story
Tweets from Supporters Who Hate "Fix": Joe.My.God rounds up tweets from supporters of the law who hate the "fix". full story
Perkins Says Veto the "Fix": Tony Perkins urged the Governor to veto the "fix". full story
NOM Urges Veto Too: The National Organization for Marriage's Brian Brown also urged the Governor to veto the "fix". full story
Bryan Fischer is Sad: Bryan Fischer is upset about the "fix" and says Pence can forget about any Presidential aspirations in 2016. full story
Tony Perkins is Sad, Too: In the wake of the Governor's signing of the "fix", Perkins says "If the government punishes people for living their faith, there are no limits to what government can control." full story
Wolf in Sheep's Rainbow Clothing: Thomas Peters at The American Papist vividly illustrated "Christian" fears. full story
And finally, a little comedy to lighten your day:
Funny or Die Spoofs Law: James Van Der Beek and Anna Camp play a pair of bigoted home shopping network hosts who, thanks to Indiana's "religious freedom" bill, are now free to refuse to do business with icky gay customers. full story
Posted in Bills, Discrimination, Indiana, LGBT Rights, Religious Liberty, USA | No Responses »
Tags: fix, governor mike pence, indiana, law, right to discriminate
Indiana Mess Continues As Legislature Struggles to "Fix" Law
The Indiana legislature started on a "fix" for the state's new, reviled right to discriminate law. So many more updates for you – first the general stufff:
New Bill Might Have Some LGBT Protections: Indiana GOP leaders are vetting a proposed RFRA change that would clarify that private businesses may not discriminate against LGBT citizens on the basis of religious beliefs. full story
Text of Clarification: The clarification would say that the new "religious freedom" law does not authorize a provider – including businesses or individuals – to refuse to offer or provide its services, facilities, goods, or public accommodation to any member of the public based on sexual orientation or gender identity, in addition to race, color, religion, ancestry, age, national origin, disability, sex, or military service. full story
Law Could Cost State $250 Million: Backlash over the passage of the so-called "Religious Freedom Restoration Act," signed into law in Indiana last week by Gov. Mike Pence, could cost the Hoosier state as much as $250 million in economic activity — and counting. full story
We also have some analyses of the new law and its impact and effects:
Law Exposes GOP Divisions: The fight over the law has exposed deep divisions in the GOP ahead of the 2016 elections. full story
Waldman Analyzes Law: Towleroad.com's Ari Ezra Waldman takes a close look at what's in the right to discriminate law. full story
Ramifications of the Law: Zack Ford at Think Progress looks at how the law empowers businesses to discriminate. full story
Should We Boycott: Matt Baume explores the good a boycott does, and the way to get these bills off the table before they are passed. full story
Law is Consolation Prize: The law is as a direct response to the failed attempt to place a constitutional ban on same-sex marriage, a consolation prize for anti-gay activists. full story
People and Corporations are still lining up to oppose the new law:
Evan Bayh Opposes Law: Retired Sen. Evan Bayh (D-Ind.) sharply criticized his home state governor on Tuesday for signing a controversial "religious freedom" law that he said would unambiguously invite discrimination against gays and lesbians. full story
Celebrities Tweet Opposition: Stars of stage, screen, sports, and television continue to lash out against Indiana Gov. Mike Pence and the state's Religious Freedom Restoration Act as they call for their fans to #BoycottIndiana. full story
More Corporations Oppose Law: Human Rights Campaign (HRC) yesterday launched a statement supported by a number of corporations calling on public officials to defeat or abandon efforts to enact harmful anti-LGBT legislation being considered around the United States, including American Airlines, Apple, Levi Strauss, Microsoft, Orbitz, Replacements, Starwood Hotels, Symantec and Wells Fargo. full story
CNN Reporter Opposes Law: Out CNN anchor Don Lemon on Tuesday criticized an Indiana bill that supporters say protects religious freedom but critics argue targets the LGBT community. full story
Michael Stipe Has Suggestion for Governor: The former R.E.M. front man, who came out 20 years ago, shared his feelings in a brief video uploaded to his private Tumblr page: "Indiana and people of Indiana, I love you. Indiana Governor Mike Pence, no! Go fuck yourself." full story
Leagues Should Condemn Law: College Sports writer Dick Weiss is calling on the NCAA, NFL, NBA and WNBA to condemn the state's right to discriminate law full story
Meanwhile supporters of the law are growing more frustrated:
Rightwing Headlines Target Pence: Joe.My.God rounds up some of the angry headlines against Governor Pence's call to "fix" the law. full story
USA: Micah Clark, executive director of the Indiana branch of the American Family Association and one of the people who helped draft the law, said that Christians involved in "spiritual warfare" are the real victims in the state's "right to discriminate" fight. full story
Pat Robertson is Pissed: The anti-gay commentator says "They're not liberal, they're totalitarian dictators!" full story
Pete LaBarbera Also Pissed: Anti-gay activist Peter LaBarbera is also pissed that the state appears to be backing down. full story
Christians Will Be Slaves: Bryan Fischer is bringing back the "gays are making slaves of Christians" argument. full story
Gingrich Invokes "Lynch Mob": Former Speaker of the House Newt Gingrich slammed the opposition to the recently passed religious freedom bills in Indiana and Arkansas, calling resistance to the legislation is part of a liberal "lynch mob" mentality and is an example of "selective outrage." full story
Fiorina Endorses Law: Republican presidential hopeful Carly Fiorina defends Indiana's "religious freedom" law, and also suggested she would support civil unions for gays and lesbians. full story
Ted Cruz Takes Up Religious Liberty Cry: Texas Sen. Ted Cruz says concerns about religious freedom could unite Republicans and rally conservative Democrats to the GOP in 2016. full story
Stossel Calls Gays Totalitarians: Fox Business' John Stossel backs gay marriage too, but told Bill O'Reilly tonight, "This movement has moved from tolerance to totalitarianism. It's the totalitarianism of the left." full story
USA, Indiana: Memories Pizza, an anti-gay pizzeria in Walkerton, has temporarily closed its doors due to the backlash it has received since publicly stating they would not cater a same-sex couple's wedding. full story
Pizzeria Gets $33K in Crowdfunding: But don't feel too badly for them – online supporters donated more than $33,000 in three hours to Memories Pizza after the pie maker's controversial pledge not to cater same-sex weddings spawned a wave of criticism. full story
Meanwhile the fight still rages over travel bans to the state:
NYC Bans Travel: In addition to the state, NYC's Mayor Bill DeBlasio has also banned official travel to the state. full story
Illinois Governor Won't Ban Travel: Illinois Gov. Bruce Rauner will not be issuing a directive banning official travel to the state. full story
Minneapolis Firefighters Cancel Trip: Members of the Minneapolis Fire Department won't make a planned trip for a conference in Indianapolis, after Mayor Betsy Hodges asked for a moratorium on city funded travel to Indiana. full story
Others are trying to poach the state's businesses:
Dayton, Ohio Goes After State Businesses: Dayton officials are working to attract companies and individuals looking to leave Indiana because they oppose the new law, touting the southwestern Ohio city's recognition for diversity and inclusiveness. full story
Virginia Wants Indiana Businesses: Virginia Governor Terry McAuliffe says his state would also welcome Indiana businesses. full story
Ontario Wants State Businesses, Too: Kathleen Wynne, Ontario's openly lesbian premier, urged state businesses upset with the law to move to Canada's most populous province. full story
And last, a little comedy:
Conan Spoofs State's Homophobia: Conan O'Brien had a little fun at the state's expense, inviting Indiana's "Religious Freedom Czar" on the show for a spoof segment. full story
Cartoon Spoofs Governor: A new The Washington Blade cartoon spoofs the state and Governor. full story
Posted in Bills, Discrimination, Indiana, LGBT Rights, Politicians, Religious Liberty, Republicans, USA | No Responses »
Tags: bill, fix, indiana, law, right to discriminate
Arkansas Governor Won't Sign Bill Without Changes
Lots of news out of Arkansas yesterday after the legislature sent a right to discriminate bill to the Governor's desk:
Governor Refuses to Sign Bill: Gov. Asa Hutchinson (R) announced Wednesday that he would not sign the "Religious Freedom Restoration Act" (RFRA) as it was presented to him by the legislature on Tuesday, urging the legislature to recall the bill and tweak it so that it better mirrors the narrower federal bill. full story
Governor's Son Publically Opposed Bill: The Governor revealed that his son Seth Hutchinson publicly denounced his plans to sign the bill, signing a petition. full story
Little Rock Nine Opposed: In a symbolic move, two members of the Little Rock Nine — Carlotta Walls Lanier and Ernie Green have voiced their opposition to Arkansas' H.B. 1228. full story
Activists Rally for Changes: Hundreds of LGBT rights activists rallied for changes to the bill. full story
Starnes Hates on Walmart: Anti gay commentator Todd Starnes tweeted that Walmart, which opposed the law, is "the home of low prices and no values" and called for a boycott. full story
Senate Approves Amended Bill: The Arkansas Senate on Wednesday night approved transmitting two amended existing bills to the House that add language that mirrors the federal Religious Freedom Restoration Act. full story
Jake Tapper Interviews Bill Author: Watch this guy babble and deflect in some truly Pence-ian doublespeak and note how he repeatedly says "homosexual" rather than "gay." full story
Tags: arkansas, bill, governor asa hutchinson, religious freedom, right to discriminate

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

Let $\fg = \mathfrak{sl}_{n+1}(\C)$ with index set $I=\{1,\ldots,n\}$, and fix a triangular decomposition
$\fg = \fn_+ \oplus \fh \oplus \fn_-$.
Denote by $\varpi_i$ ($i \in I$) the fundamental weights.
Let $\fg[t]=\fg\otimes \C[t]$ be the associated current algebra.
For $m \in I$ and a sequence $\bm{\ell}=(\ell_1, \ell_2,\ldots,\ell_p)$ of nonnegative integers,
we define a $\fg[t]$-module $V_m(\bm{\ell})$ by
\[ V_m(\bm{\ell}) = V(\ell_1 \varpi_m) *V(\ell_2\varpi_m) * \cdots * V(\ell_p\varpi_m).
\]
Here $V(\gl)$ is the simple $\fg$-module with highest weight $\gl$,
and $*$ denotes the fusion product defined by Feigin and Loktev in \cite{MR1729359}.
We may assume without loss of generality that $\ell_1 \ge \ell_2 \ge \cdots \ge \ell_p$, that is, $\bm{\ell}$ is a partition.

In \cite{MR3296163}, Chari and Venkatesh have introduced a large family of indecomposable $\fg[t]$-modules
(with $\fg$ a general simple Lie algebra) indexed by a sequence of partitions, in terms of generators and relations.
In this note, we will show that the fusion product $V_m(\bm{\ell})$ is isomorphic to a module belonging to their family.
More explicitly, we show the following defining relations of $V_m(\bm{\ell})$.\\[-5pt]

\par
\noindent
{\bfseries Theorem.}\ \
{\itshape
 Let $m \in I$ and $\bm{\ell} = (\ell_1\ge\cdots\ge\ell_p)$ be a partition.
 Set $L_i = \ell_i +\cdots +\ell_{p-1}+\ell_p$ for $1 \le i \le p$ and $L_i = 0$ for $i>p$.
 Then $V_m(\bm{\ell})$ is isomorphic to the
 $\fg[t]$-module generated by a vector $v$ with relations
 \begin{align*
  \fn_+[t]v=0, \ \ \ (h\otimes t^s) v&= \gd_{s0}L_1\langle h,\varpi_m\rangle v \text{ for }
  h \in \fh, \ s \in \Z_{\ge 0},\nonumber\\
  (f_\ga\otimes \C[t])v&= 0 \text{ for } \ga \in \gD_+ \text{ with } \langle h_\ga,\varpi_m\rangle = 0,\nonumber\\
  f_\ga^{L_1+1}v&=0 \text{ for } \ga \in \gD_+ \text{ with } \langle h_\ga, \varpi_m\rangle =1, \nonumber\\
  (e_\ga\otimes t)^{s}f_\ga^{r+s}v&=0 \text{ for } \ga \in \gD_+, r,s \in \Z_{>0} \text{ with } \langle h_\ga, \varpi_m
   \rangle = 1,\nonumber\\
  &\hspace{40pt} r+s \ge 1+kr + L_{k+1} \text{ for some } k\in\Z_{>0}.
 \end{align*}
 Here $\gD_+$ is the set of positive roots, $ h_\ga$ is the coroot corresponding to $\ga$, and $e_\ga$ and $f_\ga$ are
 root vectors corresponding to $\ga$ and $-\ga$ respectively.}
\par

This theorem for $\fg = \mathfrak{sl}_2$ has been proved in \cite{MR1988973} and \cite{MR3296163}.
In the case $p=2$, this can be found in \cite{MR3336341} and \cite{Fourier} (see also \cite{CSVW}).

Let us introduce a motivation of the theorem.
For that we consider the case $\fg = \mathfrak{sl}_2(\C)$ for a moment.
Let $(\ell_1\ge \ell_2), (r_1\ge r_2)$ be partitions of a positive integer $\ell$.
By the well-known Clebsch-Gordan formula
\[ V(\ell\varpi_1)\otimes V(r\varpi_1) = V\big(|\ell-r|\varpi_1\big) \oplus \cdots \oplus
   V\big((\ell+r-2)\varpi_1\big) \oplus V\big((\ell+r)\varpi_1\big),
\]
we see that there exists a surjective $\fg$-module homomorphism
\[ V(\ell_1\varpi_1)\otimes V(\ell_2\varpi_1) \twoheadrightarrow V(r_1\varpi_1) \otimes V(r_2\varpi_1)
\]
if and only if $\ell_2 \ge r_2$.
This surjection implies that the difference of their characters can be written as a sum of characters of simple $\fg$-modules.
Since the characters of simple $\fg$-modules are known as Schur functions,
this property is called \textit{Schur positivity}.
Generalization of the surjection to a more general $\fg$ and more general $\fg$-modules has been studied in
\cite{MR2358610, MR2369890, MR3248990,MR3226992}.
In particular when $\fg = \mathfrak{sl}_{n+1}$, it follows from \cite{MR3248990} (see also \cite{MR2369890})
that for $m \in I$ and two partitions $(\ell_1\ge\cdots\ge \ell_p)$, $(r_1\ge\cdots \ge r_p)$ of a positive integer $\ell$,
there exists a surjective $\fg$-module homomorphism
\begin{equation}\label{eq:surjection}
 V(\ell_1\varpi_m) \otimes \cdots \otimes V(\ell_p\varpi_m) \twoheadrightarrow V(r_1\varpi_m) \otimes \cdots \otimes
 V(r_p\varpi_m)
\end{equation}
if $\ell_i + \cdots +\ell_p \geq r_i + \cdots +r_p$ holds for each $1\le i \le p$.

Fourier and Hernandez have raised the following question in the introduction of \cite{MR3226992}: Can surjections
such as (\ref{eq:surjection}) be obtained from surjective $\fg[t]$-module homomorphisms between the
corresponding fusion products?
(Recall that the $\fg$-module structures of a tensor product and a fusion product are the same.)
By inspecting the defining relations of the theorem we obtain the following corollary,
which gives a positive answer to their question in our setting.

\begin{Cor}
 Let $m \in I$, and $\bm{\ell}=(\ell_1 \geq \cdots \geq \ell_p)$, $\bm{r}=(r_1 \geq \cdots \geq r_p)$ be two partitions of a
 positive integer $\ell$.
 We assume that $\ell_i + \cdots +\ell_p \geq r_i + \cdots +r_p$ holds for each $1 \le i \le p$.
 Then there exists a surjective $\fg[t]$-module homomorphism
 from $V_m(\bm{\ell})$ onto $V_m(\bm{r})$.
\end{Cor}

It would be an interesting problem to generalize the theorem to a more general $\fg$ or more general modules.
These will be studied elsewhere.

The organization of this paper is as follows.
We fix basic notations in Subsection \ref{Subsection:preliminary},
and recall the definition of fusion products in Subsection \ref{Subsection:Fusion_product}.
By \cite{MR2964614} $V_m(\bm{\ell})$ can be realized as a $\fg[t]$-submodule of a module over the affine Lie algebra $\hfg$,
which is recalled in Subsection \ref{Subsection:Realization}.
In Subsection \ref{Subsection:Chari-Venkatesh}, we recall some results in \cite{MR3296163} needed for the proof of
the main theorem, and show one technical lemma.
Then we prove the theorem in Section \ref{Section:proof} by determining the defining relations recursively
using the realization, in which we apply the method used in \cite{MR3120578}.

\section{Preliminaries}

\subsection{Simple Lie algebra, current algebra, and affine Kac-Moody Lie algebra of type
{\boldmath$A$}}\label{Subsection:preliminary}

Let $\fg = \mathfrak{sl}_{n+1}(\C)$ with index set $I=\{1,\ldots,n\}$.
We fix a triangular decomposition $\fg = \fn_+ \oplus \fh \oplus \fn_-$.
Let $\ga_i$ and $\varpi_i$ ($i\in I$) be simple roots and fundamental weights respectively.
We use the labeling in \cite[Section 4.8]{MR1104219}.
For convenience we set $\varpi_0=0$.
Let $\gD$ be the root system, $\gD_+$ the set of positive roots, $W$ the Weyl group with simple reflections
$\{s_i\mid i\in I\}$ and longest element $w_0$,
and $(\ , \ )$ the unique non-degenerate
$W$-invariant symmetric bilinear form on $\fh^*$ satisfying $(\ga,\ga) = 2$ for all $\ga \in \gD$.
Let
\[ \gt = \ga_1 + \cdots + \ga_{n-1}+\ga_n
\]
be the highest root.
For each $\ga \in \gD$, let $ h_\ga$ be its coroot, $\fg_\ga$ the corresponding root space,
and $e_{\ga} \in \fg_\ga$ a root vector satisfying $[e_\ga,e_{-\ga}] = h_\ga$.
We also use the notations $f_\ga = e_{-\ga}$ for $\ga \in \gD_+$, $h_i = h_{\ga_i}$, $e_i = e_{\ga_i}$ and $f_i = f_{\ga_i}$.
Denote by $P$ the weight lattice, by $P_+$ the set of dominant integral weights,
and by $V(\gl)$ ($\gl \in P_+$) the simple $\fg$-module with highest weight $\gl$.
For $i \in I$, set
\[ i^* = n+1-i \in I.
\]
Note that $w_0(\varpi_i) = -\varpi_{i^*}$ holds.

Given a Lie algebra $\fa$, its \textit{current algebra} $\fa[t]$ is defined by the tensor product $\fa \otimes \C[t]$
equipped with the Lie algebra structure given by
\[ [x\otimes f(t), y \otimes g(t)] = [x,y] \otimes f(t)g(t).
\]
For $k\in \Z_{> 0}$, let $t^k\fa[t]$ denote the ideal $\fa \otimes t^k\C[t] \subseteq \fa \otimes \C[t]$.

Let $\hfg = \fg\otimes \C[t,t^{-1}] \oplus \C c \oplus \C d$ be the nontwisted affine Lie algebra associated with $\fg$.
Here $c$ is the canonical central element and $d$ is the degree operator.
Note that $\fg$ and $\fg[t]$ are naturally considered as Lie subalgebras of $\hfg$.
Let $\hI = I \sqcup \{0\}$, and define Lie subalgebras $\hfh$, $\hfn_+$, and $\hfb$ as follows:
\[ \hfh = \fh \oplus \C c \oplus \C d, \ \ \hfn_+ = \fn_+ \oplus t\fg[t], \ \
   \hfb = \hfh \oplus \hfn_+.
\]
We also define $\hfh_{\mathrm{cl}} = \fh \oplus \C c$.
We often consider $\fh^*$ (resp.\ $\hfh_{\mathrm{cl}}^*$) as a subspace of $\hfh^*$ by setting
\[ \langle c,\gl\rangle = \langle d,\gl\rangle = 0 \text{ for }  \gl \in \fh^* \ \ \ \big(\text{resp.}\
   \langle d,\gl\rangle = 0 \text{ for }  \gl \in \hfh_{\mathrm{cl}}^*\big).
\]
Let $\hat{\gD}$ be the root system of $\hfg$, $\hP$ the weight lattice, $\hP_+$ the set of dominant integral weights,
and $\hat{W}$ the Weyl group with simple reflections $\{s_i \mid i \in \hI\}$.
Denote by $\gd\in \hP$ the null root, and by $\gL_0 \in \hP$ the unique element satisfying
\[ \langle \fh,\gL_0\rangle = \langle d, \gL_0 \rangle = 0, \ \ \ \langle c, \gL_0 \rangle = 1.
\]
Let $\ga_0 = \gd -\gt$, $e_0 = f_{\gt} \otimes t$ and $f_0 = e_{\gt} \otimes t^{-1}$.
Given an integrable $\hfg$-module $M$ and $i \in \hI$, define a linear automorphism $\Phi_i^M$ on $M$ by
\[ \Phi^M_i = \mathrm{exp}(f_i)\mathrm{exp}(-e_i)\mathrm{exp}(f_i),
\]
see \cite[Lemma 3.8]{MR1104219}.
For each $w\in \hW$ fix a reduced expression $w=s_{i_k}\cdots s_{i_1}$, and set
$\Phi^M_w=\Phi^M_{i_k}\cdots \Phi^M_{i_1}$.
Then $\Phi^M_w$ satisfies
\begin{equation*
 \Phi^M_w(M_\mu) = M_{w(\mu)} \ \text{ for } \mu \in \hP, \ \ \ \Phi_w^M \hfg_\ga (\Phi_w^M)^{-1}
  = \hfg_{w(\ga)} \text{ for } \ga \in \hat{\gD}.
\end{equation*}
In particular by considering the adjoint representation, an algebra automorphism on $U(\hfg)$
is defined for each $w \in \hW$, which is denoted by $\Phi_w$.
Note that $\Phi^M_w$ for $w \in W$ is also defined on a finite-dimensional $\fg$-module $M$.

Define $t_\gl \in \mathrm{GL}(\hfh^*)$ for $\gl \in P$ by
\begin{equation*}
  t_{\gl}(\mu) = \mu + \langle c,\mu \rangle \gl - \big((\mu, \gl) +
  \frac{1}{2}(\gl, \gl)\langle  c, \mu \rangle\big)\gd,
\end{equation*}
see \cite[Chapter 6]{MR1104219}.
Let $T(P) = \{t_\gl \mid \gl \in P\}$ and $\widetilde{W} = W \ltimes T(P)$,
which is called the \textit{extended affine Weyl group}.
Here $w \in W$ and $t_\gl \in T(P)$ satisfy $wt_\gl w^{-1}=t_{w(\gl)}$.
For $i \in \hI$, let
\begin{equation*}
 \tau_i = t_{\varpi_i}w_{i,0}w_0 \in \widetilde{W}
\end{equation*}
where $w_{i,0}$ is the longest element of $W_{\varpi_i}$,
the stabilizer of $\varpi_i$ in $W$.
We have
\begin{equation}\label{eq:rule_of_tau}
 \tau_i(\gd) = \gd, \ \ \ \tau_i(\ga_j) = \ga_{\ol{i+j}}, \ \text{ and } \ \tau_i(\varpi_j+\gL_0) \equiv \varpi_{\ol{i+j}}+
  \gL_0 \ \text{ mod } \Q\gd \ \text{ for } j \in \hI
\end{equation}
where $\ol{i+j} \equiv i+j$ mod $n+1$.
Set $\gS = \{\tau_i \mid i\in \hI\}$.
It is known that $\widetilde{W} = \hW\rtimes \gS $.
We define an action of $\gS$ on $\hfg$ by letting $\tau_i$ act as a Lie algebra automorphism determined by
\begin{align*
 \tau_i(e_j) = e_{\ol{i+j}}, \ \ \tau_i(f_j) = f_{\ol{i+j}} \text{ for }  j \in \hI, \ \
 \langle \tau_i(h), \tau_i(\gl)\rangle = \langle h,\gl\rangle \text{ for } h \in \hfh, \gl \in \hfh^*.
\end{align*}

\subsection{Fusion product}\label{Subsection:Fusion_product}

Let us recall the definition of the fusion product introduced in \cite{MR1729359}.
Note that $U(\fg[t])$ has a natural $\Z_{\ge 0}$-grading defined by
\[ U(\fg[t])^k = \{X \in U(\fg[t]) \mid [d,X] = kX\}.
\]
Let $\gl_1,\ldots,\gl_p$ be a sequence of elements of $P_+$, and $c_1,\ldots,c_p$ pairwise distinct complex numbers.
We define a $\fg[t]$-module structure on $V(\gl_i)$ as follows:
\[ \big(x\otimes f(t)\big) v = f(c_i)xv \ \text{ for } x \in \fg, f(t) \in \C[t], v \in V(\gl_i).
\]
Denote this $\fg[t]$-module by $V(\gl_i)_{c_i}$.
Let $v_i$ be a highest weight vector of $V(\gl_i)$.
Then the $\fg[t]$-module $V(\gl_1)_{c_1} \otimes \cdots \otimes V(\gl_p)_{c_p}$ is
generated by $v_1 \otimes \cdots \otimes v_p$ (see \cite{MR1729359}),
and the grading on $U(\fg[t])$ induces a filtration on $V(\gl_1)_{c_1}\otimes \cdots \otimes V(\gl_p)_{c_p}$ by
\[ \Big(V(\gl_1)_{c_1}\otimes \cdots \otimes V(\gl_p)_{c_p}\Big)^{\leq k} = \sum_{r \leq k}
   U(\fg[t])^r(v_1\otimes \cdots \otimes v_p).
\]
Now the $\fg[t]$-module obtained by taking the associated graded is denoted by
\[ V(\gl_1) * \cdots * V(\gl_p),
\]
and called the \textit{fusion product} of $V(\gl_1),\ldots,V(\gl_p)$.
Though the definition depends on the parameters $c_i$, we omit them for the notational convenience.
All fusion products appearing in this paper do not depend on the parameters up to isomorphism.
Note that, by definition, we have
\[ V(\gl_1) * \cdots * V(\gl_p) \cong V(\gl_1) \otimes \cdots \otimes V(\gl_p)
\]
as a $\fg$-module.

\subsection{Another realization of fusion products}\label{Subsection:Realization}

Kirillov-Reshetikhin modules for $\fg[t]$ are $\fg[t]$-modules defined in terms of generators and relations,
which have been introduced in \cite{MR2238884}.
In \cite{MR2964614} the fusion products of Kirillov-Reshetikhin modules for $\fg[t]$
were studied when $\fg$ is of type $ADE$, and a new realization of these modules using Joseph functors was given.
When $\fg$ is of type $A$, a Kirillov-Reshetikhin module is just the evaluation module at $t=0$ of $V(k\varpi_i)$
with $k\in \Z_{> 0}$ and $i \in I$,
and hence their fusion products are what we are studying in this note.
In this subsection we will reformulate the result of \cite{MR2964614} in type $A$ in a different way
(see Remark \ref{Rem}).
This formulation has previously been used in \cite{MR3120578}, and is more suitable for later use since we can apply Lemma
\ref{Lem:proceed} stated below.

First we introduce several notations.
Assume that $V$ is a $\hfg$-module and $D$ is a $\hfb$-submodule of $V$.
For $\tau \in \gS$, denote by $F_\tau V$ the pull-back $(\tau^{-1})^*V$ with respect to the Lie algebra
automorphism $\tau^{-1}$ on $\hfg$, and define a $\hfb$-submodule $F_\tau D \subseteq F_\tau V$ in the obvious way.
For $i \in \hI$ let $\hfp_i$ denote the parabolic subalgebra $\hfb \oplus \C f_i \subseteq \hfg$,
and set $F_iD= U(\hfp_i)D \subseteq V$ to be the $\hfp_i$-submodule generated by $D$.
Finally we define $F_w D$ for $w \in \widetilde{W}$ as follows:
let $\tau \in \gS$ and $w' \in \hat{W}$ be the elements such that $w=w'\tau$, and choose a reduced
expression $w'=s_{i_k} \cdots s_{i_1}$.
Then we set
\[ F_{w}D = F_{i_k}\cdots F_{i_1} F_\tau D\subseteq F_\tau V.
\]
Though the definition depends on the choice of the expression of $w'$, we use $F_w$ by an abuse of notation.

For $\gL \in \hP_+$ let $\hV(\gL)$ be the simple highest weight $\hfg$-module with highest weight $\gL$.
Denote by $\C_{\gL}$ the $1$-dimensional $\hfb$-submodule of $\hV(\gL)$ spanned
by a highest weight vector.
Note that $F_\tau\hV(\gL) \cong \hV(\tau\gL)$ and $F_\tau \C_{\gL} \cong \C_{\tau\gL}$ for $\tau \in \gS$.
Let
\begin{equation*
 \hfb' = \hfb\cap \fg[t]= \fh \oplus \hfn_+.
\end{equation*}
Now \cite[Theorem 6.1]{MR2964614} is reformulated as follows.

\begin{Thm}\label{Thm:realization}
 Let $\bm{\ell} = (\ell_1 \ge \cdots\ge\ell_p)$ be a partition, and $m_1,\ldots,m_p$ a sequence of elements
 of $I$.
 As a $\hfb'$-module, we have
 \begin{align*}
  V(\ell_1\varpi_{m_1}) * \cdots& * V(\ell_p\varpi_{m_p})\\
  &\cong F_{t_{-\varpi_{m_1^*}}}\Big(\C_{(\ell_1-\ell_2)\gL_0} \otimes \cdots
  \otimes F_{t_{-\varpi_{m_{p-1}^*}}}\big(\C_{(\ell_{p-1}-\ell_p)\gL_0} \otimes F_{t_{-\varpi_{m_p^*}}}\C_{\ell_p\gL_0}\big)\!
  \cdots \!\Big).
 \end{align*}
\end{Thm}

\begin{Rem}\label{Rem}\normalfont
 In \cite[Theorem 6.1]{MR2964614} the right-hand side is defined in terms of Joseph functors, but it can easily be proved to
 be isomorphic to the right-hand side of Theorem \ref{Thm:realization} as follows.
 By the universality of Joseph functors, there exists a surjection between two modules.
 Moreover their characters coincide by \cite[Corollary 6.2]{MR2964614} and \cite[Theorem 5]{MR1887117},
 and hence they are isomorphic. (See \cite[a paragraph below Lemma 5.2]{MR3120578} for more detail,
in which a similar argument is given.)
\end{Rem}

For $i \in \hI$, let $\hfn_i$ be the nilradical of $\hfp_i$.
More explicitly, $\hfn_i$ is defined as follows:
\[ \hfn_i = \bigoplus_{\ga \in \gD_+\setminus \{\ga_i\}} \C e_\ga \oplus t\fg[t] \ (i \in I), \
   \hfn_0 = \fn_+ \oplus \bigoplus_{\ga \in \gD\setminus\{-\theta\}} \C (e_\ga \otimes t) \oplus (\fh\otimes t)\oplus
   t^2\fg[t].
\]
The following lemma is useful to determine defining relations of modules constructed using $F_w$'s.
For the proof, see \cite[Lemma 5.3]{MR3120578}.

\begin{Lem}\label{Lem:proceed}
  Let $V$ be an integrable $\hfg$-module, $T$ a finite-dimensional $\hfb$-submodule of $V$,
  $i \in \hI$ and $\xi \in \hP$ such that $\langle h_i, \xi \rangle \ge 0$.
  Assume that the following conditions hold:\\
  {\normalfont(i)}
    $T$ is generated by a weight vector $v \in T_\xi$ satisfying $e_i v=0$.\\
  {\normalfont(ii)}
    There is an $\ad(e_i)$-invariant left $U(\hfn_i)$-ideal $\mathcal{I}$ such that
    \[ \mathrm{Ann}_{U(\hfn_+)}v = U(\hfn_+)e_i + U(\hfn_+)\mathcal{I}.
    \]
  {\normalfont(iii)} We have $\ch F_i T = \mathcal{D}_i \ch T$, where $\mathrm{ch}$ denotes the character with respect to
   $\hfh$, and $\mathcal{D}_i$ is the Demazure operator defined by
   \[ \mathcal{D}_i(f) = \frac{f-e^{-\ga_i}s_i(f)}{1-e^{-\ga_i}}.
   \]
  Let $v' = f_i^{\langle h_i, \xi \rangle} v$. Then we have
  \[ \mathrm{Ann}_{U(\hfn_+)} v'= U(\hfn_+)e_i^{\langle h_i, \xi \rangle+1} + U(\hfn_+)\Phi_i( \mathcal{I}).
  \]
\end{Lem}

\subsection{Presentation by Chari and Venkatesh}\label{Subsection:Chari-Venkatesh}

Following \cite{MR3296163}, we introduce some notations.
For $r,s \in\Z_{\ge 0}$, let
\[ \mathbf{S}(r,s)=\Big\{(b_j)_{j\ge 0}\Bigm| b_j \in \Z_{\ge 0}, \ \sum_j b_j = r, \ \sum_j jb_j = s\Big\}.
\]
Note that $\mathbf{S}(0,s) = \emptyset$ if $s > 0$, and if $(b_j)_{j\ge 0} \in \mathbf{S}(r,s)$ then
$b_j = 0$ for $j>s$.
For $x \in \fg$ and $r,s\in \Z_{\ge 0}$, define a vector $x(r,s) \in U(\fg[t])$ by
\[ x(r,s) = \sum_{(b_j)_{j\ge 0}\in \mathbf{S}(r,s)} (x\otimes 1)^{(b_0)}(x\otimes t)^{(b_1)} \cdots
   (x\otimes t^s)^{(b_s)},
\]
where for $X \in \fg[t]$, $X^{(b)}$ denotes the divided power $X^b/b!$.
We understand $x(r,s) = 0$ if $\mathbf{S}(r,s)=\emptyset$.
For $\ga \in \gD_+$, define Lie subalgebras $\mathfrak{sl}_{2,\ga}$ and $\fb_{\ga}$ of $\fg$ by
\[ \msl_{2,\ga}= \C e_\ga \oplus \C  h_\ga \oplus \C f_{\ga},\ \ \ \fb_{\ga}=\C e_\ga\oplus \C h_\ga.
\]
We also define a Lie subalgebra $\hfm_\ga$ of $\msl_{2,\ga}[t]$ by
\[ \hfm_\ga = t\msl_{2,\ga}[t] \oplus \C f_\ga.
\]
By \cite{MR502647} (see also \cite[Lemma 1.3]{MR1850556}), we have
\begin{align}\label{eq:Garland}
 (e_\ga\otimes t)^{(s)}f_{\ga}^{(r+s)} - &(-1)^sf_{\ga}(r,s)\in U(\hfm_\ga)t\fb_\ga[t].
\end{align}
For $k \in \Z_{\ge 0}$, let $\mathbf{S}(r,s)_k$ (resp.\ ${}_k\mathbf{S}(r,s)$) be the subset of $\mathbf{S}(r,s)$ consisting
of elements $(b_j)_{j\ge0}$, satisfying
\[ b_j =0 \ \text{ for } \ j \ge k \ \ \ (\text{resp.\ } b_j = 0 \ \text{ for } \ j<k).
\]
For $x\in \fg$, define a vector $x(r,s)_k$ and ${}_kx(r,s)$ by
\begin{align*}
 x(r,s)_k &= \sum_{(b_j)_{j\ge 0}\in \mathbf{S}(r,s)_k} (x\otimes 1)^{(b_0)}(x\otimes t)^{(b_1)} \cdots
  (x\otimes t^{k-1})^{(b_{k-1})},\\
 {}_kx(r,s)&= \sum_{(b_j)_{j\ge0}\in {}_k\mathbf{S}(r,s)} (x\otimes t^k)^{(b_k)}(x\otimes t^{k+1})^{(b_{k+1})} \cdots
  (x\otimes t^{s})^{(b_s)}.
\end{align*}
The following was proved in \cite{MR3296163}.

\begin{Lem}\label{Lem:Chari-Venkatesh}
 {\normalfont(i)} Let $x \in \fg$. If $r,s,k\in \Z_{>0}$ and $K \in \Z_{\ge 0}$ satisfy $r+s\ge kr+K$, then
 \[ x(r,s)={}_kx(r,s) + \sum_{(r',s')} x(r-r',s-s')_k\cdot {}_kx(r',s'),
 \]
 where the sum is over all pairs $r',s' \in \Z_{\ge 0}$ such that $r' < r, s'\le s$ and $r'+s' \ge kr' + K$.\\
 \noindent{\normalfont(ii)} Let $\ga \in \gD_+$, $V$ be an $\mathfrak{sl}_{2,\ga}[t]$-module, $v \in V$
  and $K \in \Z_{\ge 0}$.
  Assume that $e_\ga\otimes \C[t]$ and $h_\ga\otimes t\C[t]$ act trivially on $v \in V$.
  Then,
  \[ (e_\ga \otimes t)^{s}f_{\ga}^{r+s}v = 0 \text{ for all } r,s \in \Z_{>0} \text{ with }
      r+s \ge 1 + kr + K \text{ for some } k\in \Z_{>0}
  \]
  if and only if
  \[ {}_kf_{\ga}(r,s)v=0 \ \text{ for all } r,s,k \in \Z_{>0} \text{ with } r+s \ge 1 + kr + K.
  \]
\end{Lem}

The following proposition plays an important roll in the next section.

\begin{Prop}\label{Prop2}
 Let $\ga \in \gD_+$. If $r,s,k \in \Z_{>0}$ and $K \in \Z_{\ge 0}$ satisfy $r+s \ge kr + K$,
 then we have
 \[ \big[ e_\ga, \ {}_kf_{\ga}(r,s)\big] \in \sum_{(r',s')}U(t\mathfrak{sl}_{2,\ga}[t]){}_k\bfa(r',s') +
    U(t\msl_{2,\ga}[t])t\fb_\ga[t],
 \]
 where the sum is over all pairs $r',s' \in \Z_{> 0}$ such that $r'+s'\ge kr'+K$.
\end{Prop}

\begin{proof}
 First we introduce some notation.
 We write $\mathfrak{f}_\ga = \C f_\ga$ here.
 Define Lie subalgebras $\hfm_\ga^h$, $\mathfrak{f}_\ga[t]_{< k}$ and $\ff_\ga[t]_{<k}^h$ by
 \[ \hfm_\ga^h = \hfm_\ga \oplus \C h_\ga, \ \ \ \ff_\ga[t]_{<k} = \bigoplus_{j=0}^{k-1} \C (f_\ga \otimes t^j), \ \ \
    \ff_{\ga}[t]_{<k}^h = \ff_{\ga}[t]_{<k} \oplus \C h_\ga.
 \]
 Since
 \[ \hfm_\ga^h = \ff_{\ga}[t]_{<k}^h \oplus t^k\ff_{\ga}[t] \oplus t\fb_\ga[t],
 \]
 we have by the PBW theorem that
 \[ U(\hfm_\ga^h) = \ff_{\ga}[t]_{<k}^hU(\hfm_\ga^h) \oplus U\big(t^k\ff_{\ga}[t]\oplus t\fb_\ga[t]\big).
 \]
 Denote by $p$ the projection
 \[ U(\hfm_\ga^h) \twoheadrightarrow U\big(t^k\ff_\ga[t] \oplus t\fb_\ga[t]\big)
 \]
 with respect to this decomposition.
 It follows from Lemma \ref{Lem:Chari-Venkatesh} (i) that
 \begin{equation}\label{eq:image_of_p}
  p\big(\bfa(r',s')\big) = {}_k\bfa(r',s').
 \end{equation}
 Denote by $\mathcal{I}$ the left $U(t\msl_{2,\ga}[t])$-ideal in the assertion.

 Now we begin the proof of the proposition.
 By (\ref{eq:Garland}), it follows that
 \[ \big[e_\ga,\ (e_\ga\otimes t)^{(s)}f_\ga^{(r+s)}\big] -
    (-1)^s\big[e_\ga, \ \bfa(r,s)\big] \in U(\hfm_\ga^h)t\fb_\ga[t].
 \]
 By applying $p$ to this, we have
\begin{align}\label{eq:containment2}
 p\Big(\big[e_\ga,\ (e_\ga\otimes t)^{(s)}f_\ga^{(r+s)}\big]\Big) -
    (-1)^sp&\Big(\big[e_\ga, \ \bfa(r,s)\big]\Big)\in U\big(t^k\ff_\ga[t]\oplus t\fb_\ga[t]\big)t\fb_\ga[t]\subseteq \mathcal{I}.
\end{align}
 The following calculation is elementary:
\begin{align*
 \big[e_\ga, \ (e_\ga\otimes t)^{(s)}f_\ga^{(r+s)}\big] = (e_\ga\otimes t)^{(s)}\big[e_\ga, \, f_\ga^{(r+s)}\big]
   = (h_\ga+r-s-1)(e_\ga\otimes t)^{(s)}f_\ga^{(r+s-1)}.
\end{align*}
Note that the pair $(r-1,s)$ satisfies the condition $(r-1)+s \ge k(r-1)+K$ since $k \in \Z_{>0}$.
By (\ref{eq:Garland}), the above equality implies
\begin{align}\label{eq:containment}
 p\Big(\big[e_\ga, \ (e_\ga\otimes t)^{(s)}f_\ga^{(r+s)}\big]\Big) \in&\ p\Big(\C(e_\ga\otimes t)^{(s)}
  f_\ga^{(r+s-1)}\Big)\nonumber\\
 \subseteq&\ p\Big(\C \bfa(r-1,s) + U(\hfm_\ga)t\fb_\ga[t]\Big)\\ =&\ \C\, {}_k\bfa(r-1,s) +
   U\big(t^k\ff_\ga[t] \oplus t\fb_\ga[t]\big)t\fb_\ga[t] \subseteq \mathcal{I},\nonumber
\end{align}
where the equality holds by (\ref{eq:image_of_p}).
On the other hand, we have by Lemma \ref{Lem:Chari-Venkatesh} (i) that
\begin{align}\label{eq:pro_equal}
 p\Big(\big[e_\ga, \ \bfa(r,s)\big]\Big) = p \Big( \big[e_\ga,\ {}_k\bfa(r,s)\big]\Big) + \sum_{(r',s')}
 p\Big(\big[ e_\ga, \ \bfa(r-r',s-s')_k\cdot {}_k\bfa(r',s')\big]\Big).
\end{align}
Since $\big[e_\ga,\ {}_k\bfa(r,s)\big] \in U(t^k\mathfrak{sl}_{2,\ga}[t])$,
it follows that $p\Big(\big[e_\ga, \ {}_k\bfa(r,s)\big]\Big) = \big[e_\ga, \ {}_k\bfa(r,s)\big]$, and
\begin{align*}
 p\Big(\big[ e_\ga, \ \bfa(r&-r',s-s')_k\cdot {}_k\bfa(r',s')\big]\Big)\\
 \hspace{-5pt}=&\, p\Big(\big[e_\ga, \ \bfa(r-r',s-s')_k\big]{}_k\bfa(r',s')\Big) + p\Big(\bfa(r-r',s-s')_k
 \big[e_\ga, \ {}_k\bfa(r',s')\big]\Big)\\
 \hspace{-5pt}\in&\, p\Big(U(\hfm_\ga^h){}_k\bfa(r',s')\Big) + 0 = U\big(t^k\ff_\ga[t] \oplus t\fb_\ga[t]\big){}_k\bfa(r',s')
 \subseteq \mathcal{I}.
\end{align*}
Hence (\ref{eq:pro_equal}) implies
\begin{align*
 p\Big(\big[e_\ga, \ \bfa(r,s)\big]\Big) - \big[e_\ga, \ {}_k\bfa(r,s)\big]
 \in \mathcal{I}.
\end{align*}
Now $\big[e_\ga, \ {}_k\bfa(r,s)\big] \in \mathcal{I}$ follows from this, together with (\ref{eq:containment2}) and
(\ref{eq:containment}).
The proof is complete.
\end{proof}

\section{Main theorem and proof}\label{Section:proof}

Let $m \in I$ and $\bm{\ell} = (\ell_1\ge\cdots\ge\ell_p)$ be a partition,
and denote by $V_m(\bm{\ell})$ the fusion product $V(\ell_1\varpi_m) * \cdots * V(\ell_p\varpi_m)$.
Set $L_i = \ell_i + \cdots +\ell_p$ for $1\le i \le p$, and $L_i = 0$ for $i>p$.
As mentioned in the introduction, the main theorem of this note is the following.

\begin{Thm}\label{Thm:Main2}
 The fusion product $V_m(\bm{\ell})$ is isomorphic to the
 $\fg[t]$-module generated by a vector $v$ with relations
 \begin{align*
  \fn_+[t]v=0, \ \ \ (h\otimes t^s) v&= \gd_{s0}L_1\langle h,\varpi_m\rangle v \text{ for }
  h \in \fh, \ s \in \Z_{\ge 0},\nonumber\\
  \big(f_\ga\otimes \C[t]\big)v&= 0 \text{ for } \ga \in \gD_+ \text{ with } \langle h_\ga,\varpi_m\rangle = 0,\nonumber\\
  f_\ga^{L_1+1}v&=0 \text{ for } \ga \in \gD_+ \text{ with } \langle h_\ga, \varpi_m\rangle =1, \nonumber\\
  (e_\ga\otimes t)^{s}f_\ga^{r+s}v&=0 \text{ for } \ga \in \gD_+, r,s \in \Z_{>0} \text{ with } \langle h_\ga, \varpi_m
   \rangle = 1,\nonumber\\
  &\hspace{40pt} r+s \ge 1+kr + L_{k+1} \text{ for some } k\in\Z_{>0}.
 \end{align*}
\end{Thm}

\begin{Rem}\normalfont
 In \cite{MR3296163}, the authors have introduced a collection of $\fg[t]$-modules $V(\bm{\xi})$
 (with $\fg$ a general simple Lie algebra) indexed by
 a $|\gD_+|$-tuple of partitions $\bm{\xi} = (\xi^\ga)_{\ga \in \gD_+}$ satisfying $|\xi^\ga|=\langle h_\ga,\gl\rangle$
 for some $\gl \in P_+$.
 In their terminology, the theorem says that $V_m(\bm{\ell})$ is isomorphic to $V(\bm{\xi})$ where
 $\bm{\xi} = (\xi^\ga)_{\ga\in \gD_+}$ with
 \[ \xi^\ga = \begin{cases} \bm{\ell} & \text{if }\langle h_\ga,\varpi_m\rangle = 1,\\
                            0         & \text{if }\langle h_\ga,\varpi_m\rangle = 0.
              \end{cases}
 \]
\end{Rem}

The proof of the theorem will occupy the rest of this paper.
Fix $m \in I$ and $\bm{\ell}$ from now on.
By Theorem \ref{Thm:realization}, we have
\begin{align}\label{eq:isom}
  V_m(\bm{\ell})
  \cong F_{t_{-\varpi_{m^*}}}\Big(\C_{(\ell_1-\ell_2)\gL_0}\otimes \cdots \otimes F_{t_{-\varpi_{m^*}}}
  \big(\C_{(\ell_{p-1}-\ell_p)\gL_0}
  \otimes F_{t_{-\varpi_{m^*}}}\C_{\ell_p\gL_0}\big)\!\cdots\!\Big)
\end{align}
as $\hfb'$-modules.
We shall determine defining relations of the right-hand side recursively.
In the sequel, we write $\tau= \tau_m$ and $\gs = w_0w_{m,0}$ (see Subsection \ref{Subsection:preliminary}).
Note that
\[ \gs(\varpi_m) = w_0(\varpi_m) = -\varpi_{m^*} \ \text{ and } \ t_{-\varpi_{m^*}} = \gs t_{\varpi_m}\gs^{-1} = \gs\tau
\]
hold.
Let $\gs=s_{i_{\ell(\gs)}}\cdots s_{i_2}s_{i_1}$ be a reduced expression of $\gs$, and set $\gs_j =
s_{i_j}\cdots s_{i_2}s_{i_1}$ for $0\le j \le \ell(\gs)$.
For $a \in \{0,\pm 1\}$, define a subset $\gD[a] \subseteq \gD$ by
\[ \gD[a] = \{ \ga \in \gD\mid \langle h_\ga,\varpi_m\rangle = a\}.
\]
Note that $\gD[\pm 1] \subseteq \pm \gD_+$, and
\begin{equation}\label{eq:small_remark}
  \ga \in \gD[a] \text{ if and only if } \langle \gs(h_\ga),\varpi_{m^*}\rangle
  = -a.
\end{equation}
We also write $\gD[\ge\! 0] = \gD[0] \sqcup \gD[1]$, etc.
It should be noted that, since $\gs$ is the shortest element such that $\gs(\varpi_m) = -\varpi_{m^*}$,
for every $1 \le j \le \ell(\gs)$ we have
\begin{equation}\label{eq:positive2}
 \langle h_{i_j},\gs_{j-1}(\varpi_m)\rangle = 1 \text{ and } \gs_{j-1}^{-1}(\ga_{i_j}) \in \gD[1].
\end{equation}
Define a parabolic subalgebra $\fp_{\varpi_m}$ of $\fg$ by
\[ \fp_{\varpi_m} = \bigoplus_{\ga \in \gD[\ge 0]} \C e_\ga \oplus \fh = \bigoplus_{\ga \in \gD[0] \cap \gD_+} \C f_\ga
   \oplus \fb.
\]
For $1\le q \le p$ and $0\le j \le \ell(\gs)$, let $V(q,j)$ be the $\hfb$-module
\[ F_{\gs_j\tau}\Big(\C_{(\ell_q-\ell_{q+1})\gL_0} \otimes F_{t_{-\varpi_{m^*}}}\Big(\cdots\otimes F_{t_{-\varpi_{m^*}}}
   \big(\C_{(\ell_{p-1}-\ell_p)\gL_0}
   \otimes F_{t_{-\varpi_{m^*}}}\C_{\ell_p\gL_0}\big)\!\cdots\!\Big)\!\Big).
\]

\begin{Prop}\label{Prop*}
 For every $q$ and $j$, there exists a nonzero vector $v_{q,j}$ in $V(q,j)$ whose $\hfh_{\mathrm{cl}}$-weight is
 $L_q\gs_j(\varpi_m)+\ell_q\gL_0$, such that $V(q,j)$ is generated by $v_{q,j}$ as a $\hfb'$-module and
 \begin{align}\label{eq:annihilators}
  \mathrm{Ann}_{U(\hfn_+)}v_{q,j}=  \sum_{\begin{smallmatrix}\ga \in \gD[-1] \\ \gs_j(\ga) \in \gD_+\end{smallmatrix} }
   U(\hfn_+)& e_{\gs_j(\ga)}^{L_q+1} + \sum_{\begin{smallmatrix}\ga \in \gD[\ge0] \\ \gs_j(\ga) \in \gD_+\end{smallmatrix} }
   U(\hfn_+) e_{\gs_j(\ga)} \\&+\sum_{\ga\in\gD[-1]}\sum_{(r,s,k)}
   U(\hfn_+){}_ke_{\gs_j(\ga)}(r,s) +U(\hfn_+)\Phi_{\gs_j}\big(t\fp_{\varpi_m}[t]\big),  \nonumber
 \end{align}
 where the sum for $(r,s,k)$ is over all $r,s,k \in \Z_{>0}$ such that $r+s \ge 1+ kr + L_{k+q}$.
\end{Prop}

For a while we assume this proposition, and give a proof to Theorem \ref{Thm:Main2}.
Denote by $T_q$ the running set of $(r,s,k)$ in (\ref{eq:annihilators}), that is,
\[ T_q=\big\{(r,s,k)\in \Z_{>0}^3\bigm| r+s \ge 1 + kr + L_{k+q}\big\}.
\]
Since $\langle h_\ga, \varpi_{m^*}\rangle = -\langle h_{\gs^{-1}(\ga)}, \varpi_m\rangle$, we see that
for $\ga \in \gD_+$, $\gs^{-1}(\ga) \in \gD[-1]$ is equivalent to $\langle h_\ga,\varpi_{m^*}\rangle = 1$.
Hence (\ref{eq:isom}) and Proposition \ref{Prop*} with $q = 1$ and $j = \ell(\gs)$ imply that
there exists a nonzero vector $v'$ in $V_m(\bm{\ell})$ whose $\fh$-weight is $-L_1\varpi_{m^*}$, such that $V_m(\bm{\ell})$ is
generated by $v'$ and
\begin{align}\label{eq:annihilators_of_1}
 \mathrm{Ann}_{U(\hfn_+)}v' =  \sum_{\ga \in \gD_+}
   U(\hfn_+)& e_{\ga}^{L_1\langle h_\ga, \varpi_{m^*}\rangle+1}\\
  &+ \sum_{\ga \in \gD[-1]}\sum_{(r,s,k) \in T_1}U(\hfn_+){}_ke_{\gs (\ga)}(r,s)
   +U(\hfn_+)\Phi_{\gs}(t\fp_{\varpi_m}[t]),\nonumber
\end{align}
where the first summation in the right-hand side is obtained using (\ref{eq:small_remark}).
$V_m(\bm{\ell})$ being a finite-dimensional $\fg$-module, $\Phi_{w_0}^{V_m(\bm{\ell})}$ is defined.
Set $v''= \Phi^{V_m(\bm{\ell})}_{w_0}(v') \in V_m(\bm{\ell})_{L_1\varpi_m}$,
and $\hfm_+=\Phi_{w_0}(\hfn_+) = \fn_-[t] \oplus t\fb[t]$.
Since each $\gD[a]$ is stable by $w_0\gs  = w_{m,0}$ and $\gD[-1] = -\gD[1]$, it follows that
\begin{align*
 \mathrm{Ann}_{U(\hfm_+)}v'' =  \sum_{\ga \in \gD_+} U(\hfm_+)f_\ga^{L_1\langle h_\ga,\varpi_m\rangle+1}
  + \sum_{\ga \in \gD[1]}\sum_{(r,s,k)\in T_1}U(\hfm_+){}_kf_{\ga}(r,s)
   +U(\hfm_+)t\fp_{\varpi_m}[t].
\end{align*}
Let $M$ be the $\fg[t]$-module generated by a vector $v$ with relations in Theorem \ref{Thm:Main2}.
By Lemma \ref{Lem:Chari-Venkatesh} (ii), $v$ satisfies
\[ {}_kf_\ga(r,s)v=0 \ \text{ for } \ga \in \gD[1], (r,s,k) \in T_1.
\]
Then we see from the above description of $\mathrm{Ann}_{U(\hfm_+)}v''$ that
there exists a surjective $\hfm_+$-module homomorphism
from $V_m(\bm{\ell})$ to $M$ mapping $v''$ to $v$.
On the other hand, since
\[ V_m(\bm{\ell}) \cong V(\ell_1\varpi_m) \otimes \cdots \otimes V(\ell_p\varpi_m)
\]
as a $\fg$-module,
we have $V_m(\bm{\ell})_{\mu} = 0$ if $\mu > L_1\varpi_{m}$, which implies $\fn_+v''=0$.
Then again by Lemma \ref{Lem:Chari-Venkatesh} (ii), $v''$ satisfies $(e_\ga \otimes t)^sf_\ga^{r+s}v'' = 0$ for
$\ga \in \gD[1]$ and $r,s$ with $(r,s,k) \in T_1$ for some $k \in \Z_{>0}$,
and we also see that there exists a surjective $\fg[t]$-module homomorphism from $M$ to $V_m(\bm{\ell})$ mapping $v$ to $v''$.
Hence $V_m(\bm{\ell}) \cong M$ holds, and the theorem is proved.

The rest of this paper is devoted to prove Proposition \ref{Prop*}.
Define a left $U(\hfn_+)$-ideal $\mathcal{I}(q,j)$ by the right-hand side of (\ref{eq:annihilators}).
We prove the assertion by the induction on $(q,j)$.
When $q=p$ and $j = 0$,
\[ V(p,0) = F_\tau\C_{\ell_p\gL_0} \cong \C_{\ell_p(\varpi_m+\gL_0)}
\]
is a $1$-dimensional module with $\hfh_{\mathrm{cl}}$-weight $\ell_p(\varpi_m+\gL_0)$ on which $\hfn_+$ acts trivially.
Hence in order to verify the assertion in this case, it suffices to show that $\mathcal{I}(p,0) = U(\hfn_+)$.
The containment $\mathcal{I}(p,0) \subseteq U(\hfn_+)$ is obvious,
and $\fn_+ + t\fp_{\varpi_m}[t] \subseteq \mathcal{I}(p,0)$ is easily seen.
Moreover since $L_{1+p} = 0$, $(1,s,s) \in T_p$ for every $s\in \Z_{>0}$,
and hence we have
\[ {}_se_\ga(1,s) = e_\ga \otimes t^s \in \mathcal{I}(p,0) \ \text{ for } \ga \in \gD[-1], \ s \in \Z_{>0}.
\]
Hence $U(\hfn_+) \subseteq \mathcal{I}(p,0)$ holds.

Next we shall prove that, if the assertion for $(q,j-1)$ holds, then that for
$(q,j)$ also holds.
We write $i = i_j$ for short.
We have $V(q,j) = F_{i}V(q,j-1)$,
and the $\hfh_{\mathrm{cl}}$-weight of $v_{q,j-1}$ is $L_q\gs_{j-1}(\varpi_m)+\ell_q\gL_0$.
Moreover $e_iv_{q,j-1}=0$ holds by (\ref{eq:positive2}).
Set $v_{q,j} = f_i^{L_q}v_{q,j-1}$.
Since $V(q,j)$ is a submodule of an integrable module,
it follows from the representation theory of $\mathfrak{sl}_2$ that
\[ v_{q,j} \neq 0, \ \ \ f_iv_{q,j} = 0,\ \text{ and } \ e_i^{L_q}v_{q,j}
   \in \C^\times v_{q,j-1}.
\]
Hence we have
\[ V(q,j)=F_iV(q,j-1) = U(\hfp_i)v_{q,j-1} = U(\hfp_i)v_{q,j} = U(\hfb')v_{q,j},
\]
and the cyclicity of $V(q,j)$ is proved.
Moreover it is obvious that the $\hfh_{\mathrm{cl}}$-weight of $v_{q,j}$ is $L_{q}\gs_j(\varpi_m)+\ell_q\gL_0$.
It remains to prove $\mathrm{Ann}_{U(\hfn_+)}(v_{q,j}) = \mathcal{I}(q,j)$.
Let $\mathcal{J}$ be the left $U(\hfn_{i})$-ideal defined by
\begin{align*}
  \mathcal{J}=  \sum_{\begin{smallmatrix}\ga \in \gD[-1]\\ \gs_{j-1}(\ga) \in \gD_+\end{smallmatrix}}
   U(\hfn_i)&e_{\gs_{j-1}(\ga)}^{L_q+1} + \sum_{\begin{smallmatrix}\ga \in \gD[\ge0]\\ \gs_{j-1}(\ga) \in
   \gD_+\setminus \{\ga_i\}
   \end{smallmatrix}}
   U(\hfn_i)e_{\gs_{j-1}(\ga)}\\&+\sum_{\ga\in\gD[-1]}\sum_{(r,s,k)\in T_q}
   U(\hfn_i){}_ke_{\gs_{j-1}(\ga)}(r,s) +U(\hfn_i)\Phi_{\gs_{j-1}}\big(t\fp_{\varpi_m}[t]).
\end{align*}
By the induction hypothesis we have
\begin{align*}
 \mathrm{Ann}_{U(\hfn_+)}v_{q,j-1} =\mathcal{I}(q,j-1)= U(\hfn_+)e_{i}+U(\hfn_+)\mathcal{J}.
\end{align*}
It suffices to show that $V(q,j-1)$, $v_{q,j-1}$ and $\mathcal{J}$ satisfy the conditions (i)--(iii) in Lemma
\ref{Lem:proceed}.
Indeed if they satisfy the conditions, it follows from the lemma that
\begin{align*
 \mathrm{Ann}_{U(\hfn_+)}v_{q,j} = U(\hfn_+)e_i^{L_{q}+1} + U(\hfn_+)\Phi_i(\mathcal{J})= \mathcal{I}(q,j),
\end{align*}
as required.
The condition (i) follows from the induction hypothesis, and the condition (iii) is proved by \cite[Theorem 5]{MR1887117},
or \cite[Corollary 2.13 and Lemma 3.2(ii)]{MR2964614}.
In order to show the condition (ii) we need to prove that $\mathcal{J}$ is $\mathrm{ad}(e_i)$-invariant.
For that, we first verify for $\ga \in \gD \setminus \{-\ga_i\}$ that
\[ \ad(e_i)U\big(t\fg_\ga[t]\big) \subseteq \mathcal{J}.
\]
If $\ga + \ga_i \notin \gD$, this is obvious.
Assume that $\ga + \ga_i \in \gD$.
Then it follows from (\ref{eq:positive2}) that
\[ \ga + \ga_i = \gs_{j-1}\big(\gs_{j-1}^{-1}(\ga) + \gs_{j-1}^{-1}(\ga_i)\big) \in \gs_{j-1}\big(\gD[\ge\!0]\big).
\]
Since $[e_{\ga+\ga_i},e_\ga] = 0$ holds, this implies
\[ \ad(e_i)U\big(t\fg_\ga[t]\big) \subseteq U(t\fg_\ga[t])t\fg_{\ga+\ga_i}[t] \subseteq \mathcal{J},
\]
as required.
In a similar manner, $\ad(e_i)U\big(\C e_\ga\big) \subseteq \mathcal{J}$ for $\ga \in \gD_+$ is also proved.
In addition, $\ad(e_i)\big(t\fh[t]\big)= t\fg_{\ga_i}[t] \subseteq \mathcal{J}$ follows from (\ref{eq:positive2}).
Now combining these facts with Proposition \ref{Prop2}, $\ad(e_i)\big(\mathcal{J}\big) \subseteq \mathcal{J}$ is proved.

Finally it remains to prove that the assertion for $\big(q+1,\ell(\gs)\big)$ implies that for $(q,0)$.
Note that
\begin{align*}
 V(q,0) = F_\tau \Big(\C_{(\ell_q-\ell_{q+1})\gL_0} \otimes V\big(q+1,\ell(\gs)\big)\Big)
  \cong \C_{(\ell_q-\ell_{q+1})(\varpi_m+\gL_0)} \otimes (\tau^{-1})^*V\big(q+1,\ell(\gs)\big).
\end{align*}
Let $z$ be a basis of $\C_{(\ell_q-\ell_{q+1})(\varpi_m+\gL_0)}$ and set
$v_{q,0} = z \otimes (\tau^{-1})^*v_{q+1,\ell(\gs)}$,
where $(\tau^{-1})^*v_{q+1,\ell(\gs)}$ is the image of $v_{q+1,\ell(\gs)}$ under the linear isomorphism
$V\big(q+1,\ell(\gs)\big) \to (\tau^{-1})^*V\big(q+1,\ell(\gs)\big)$.
By (\ref{eq:rule_of_tau}), the $\hfh_{\mathrm{cl}}$-weight of $v_{q,0}$ is
\begin{align*}
 (\ell_q-\ell_{q+1})(\varpi_m+\gL_0&) + \tau(-L_{q+1}\varpi_{m^*}+\ell_{q+1}\gL_0)\\ &= (\ell_q-\ell_{q+1})(\varpi_m+\gL_0)
   + \tau\big(-L_{q+1}(\varpi_{m^*}+\gL_0) + (\ell_{q+1}+L_{q+1})\gL_0\big)\\
  &= (\ell_q-\ell_{q+1})(\varpi_m+\gL_0) + \big(-L_{q+1}\gL_0+ (\ell_{q+1}+L_{q+1})(\varpi_m+\gL_0)\big)\\
  &= L_q\varpi_m + \ell_q\gL_0.
\end{align*}
Moreover since $\hfn_+$ acts trivially on $z$, we have
\begin{align}\label{eq:final_ann}
 \mathrm{Ann}_{U(\hfn_+)}(v_{q,0}) = \tau\Big(\mathrm{Ann}_{U(\hfn_+)}(v_{q+1,\ell(\gs)})\Big)
 =\tau\Big(\mathcal{I}\big(q+1,\ell(\gs)\big)\Big).
\end{align}
By noting $\tau \gs = t_{\varpi_m}$, we see that
\begin{equation}\label{eq:tau2}
 \tau(e_{\gs(\ga)}\otimes t^s) = e_\ga \otimes t^{s-\langle h_\ga,\varpi_m\rangle} \ \text{ for } \ga \in \gD,
 s \in \Z_{\ge 0}.
\end{equation}

\begin{Lem}\label{Lem:elementary_lemma}
 For $\ga \in \gD[-1]$ and $r,s,k \in \Z_{>0}$ we have
 \[ \tau\big({}_ke_{\gs(\ga)}(r,s)\big) = {}_{k+1}e_{\ga}(r,s+r).
 \]
\end{Lem}

\begin{proof}
 It is easily seen that the map
 \[ {}_k\mathbf{S}(r,s) \ni (b_j)_{j\ge 0} \to (b_j')_{j\ge0}\in{}_{k+1}\mathbf{S}(r,s+r)
 \]
 defined by $b_j' = b_{j-1}$ is bijective.
 Then the assertion is proved from (\ref{eq:tau2}).
\end{proof}

Using (\ref{eq:tau2}) and the above lemma, we see that
\begin{align}\label{eq:tauI}
 \tau\Big(\mathcal{I}\big(q+1,\ell(\gs)\big)\Big) =
 \sum_{\ga \in \gD[1]} U(\hfn_+&)\Big\{\C(f_\ga\otimes t)^{L_{q+1}+1}
 \\ +&\sum_{(r,s,k)\in T_{q+1}}\C\, {}_{k+1}f_\ga(r,s+r)\Big\}
 + U(\hfn_+)\big(\fn_+ + t\fp_{\varpi_m}[t]\big).\nonumber
\end{align}
On the other hand, we have
\begin{align}\label{eq:Iq0}
 \mathcal{I}(q,0) = \sum_{\ga \in \gD[1]} \sum_{(r,s,k)\in T_q}U(\hfn_+){}_{k}f_\ga(r,s)
 + U(\hfn_+)\big(\fn_+ + t\fp_{\varpi_m}[t]\big).
\end{align}
It is easily checked that
\[ \big\{(r,s+r,k+1)\bigm| (r,s,k) \in T_{q+1}\big\} = \{(r,s,k) \in T_q\mid k>1\},
\]
which implies
\begin{equation}\label{eq:easy_equal}
  \sum_{(r,s,k)\in T_{q+1}}U(\hfn_+) {}_{k+1}f_\ga(r,s+r) = \sum_{\begin{smallmatrix} (r,s,k)\in T_q \\ k > 1
  \end{smallmatrix}}U(\hfn_+) {}_{k}f_\ga(r,s).
\end{equation}
We shall prove for each $\ga \in \gD[1]$ that
\begin{align}\label{eq:final1}
 (f_\ga\otimes \,t)^{L_{q+1}+1}&\in \sum_{(r,s,k)\in T_q}U(\hfn_+){}_{k}f_\ga(r,s), \ \text{ and}\\
 \label{eq:final2}
 {}_1f_\ga(r,s) &\in U(\hfn_+)\Big(\C(f_\ga\otimes t)^{L_{q+1}+1} +  \fn_+[t] + t\fh[t]\Big)\ \text{ for } r,s
 \text{ with } (r,s,1) \in T_q.
\end{align}
By comparing (\ref{eq:tauI}) and (\ref{eq:Iq0}) and using (\ref{eq:easy_equal}), we see that these imply
\[ \tau\Big(\mathcal{I}\big(q+1,\ell(\gs)\big)\Big) = \mathcal{I}(q,0),
\]
and then by (\ref{eq:final_ann}) we have $\mathrm{Ann}_{U(\hfn_+)}(v_{q,0}) = \mathcal{I}(q,0)$,
which completes the proof.
Setting $r=s=L_{q+1}+1$, we have
\[ {}_1f_\ga(r,s) = (f_\ga\otimes t)^{L_{q+1}+1},
\]
and hence (\ref{eq:final1}) follows.
Assume that $r,s$ satisfy $(r,s,1) \in T_q$, which implies $s \ge L_{q+1}+1$.
Since ${}_1f_\ga(r,s) = 0$ if $r >s$, we may assume $r\le s$.
If we apply the Lie algebra automorphism $\tau\circ\Phi_{\gs}$ to (\ref{eq:Garland}) with $s$ replaced by $s-r$, we have
from Lemma \ref{Lem:elementary_lemma} that
\[ e_{\ga}^{(s-r)}(f_\ga\otimes t)^{(s)} - (-1)^{s-r}{}_1f_\ga(r,s) \in U(\hfn_+)\big(\fn_+[t] \oplus t\fh[t]\big).
\]
Hence (\ref{eq:final2}) also holds.

\section*{Acknowledgement}
The author is supported by JSPS Grant-in-Aid for Young Scientists (B) No.\ 25800006.

\def\cprime{$'$} \def\cprime{$'$}

redpajama_set_name: RedPajamaArXiv


text:
Михайло Борисович Гіліс (20 червня 1907, с. Вербка-Мурована —14 жовтня 1989, м. Львів) — радянський учений-агроном; доктор сільськогосподарських наук, професор.

Життєпис
Народився 20 червня 1907 р. в с. Вербка-Мурована на Хмельниччині. Вищу освіту здобув в Уманському садоогородньому інституті в 1931 році. Починаючи з 1934 року, працював у Київському сільськогосподарському інституті, пізніше в Львівському політехнічному інституті.
У 1947—1978 рр. завідував кафедрою агрохімії та ґрунтознавства Львівського сільськогосподарського інституту.
З 1961 року — доктор сільськогосподарських наук, а з 1962 — професор.

Нагороджений державними нагородами СРСР. Розробляв раціональні способи внесення добрив під сільськогосподарські культури. Вивів високоврожайний сорт помідорів «Львівський».

Його найбільш відомі праці: «Локальне внесення добрив» (1962), «Рациональные способы внесения удобрений» (1975).

Помер 14 жовтня 1989 року в м. Львові. Похований на Янівському цвинтарі.

Джерела

Поховані на Янівському цвинтарі

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Equipment-driven rental and services companies are themselves driven by field service requirements. Therefore, mobility has become essential to working with remote employees. There are many areas to explore in mobile apps for field service.
As a result, we focus on the importance of choosing an app with workflows that guide employees through service processes precisely. How important is it to choose an app and platform which lets you configure those workflows for all your tasks? With DynaRent PowerApps for Microsoft Dynamics 365 Finance and Operations you can provide both mobile staff and your back office with what they need.
DynaRent Mobile includes PowerApps and out-of-the-box configured mobile apps made for employees who work on a depot to directly view equipment, its status, and other required information. In this article, we focus on the Equipment Overview PowerApp that can be made available for every employee working on a depot.
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text:
Preparing the future, winning now
Stregspiller September 20, 2016 Sascha Staat Unplugged, News Leave a Comment
I've seen a lot of handball games on location, especially in recent times. Finals of all major tournaments like the EURO, the World Championship or even the Olympic games. Final matches of European club competitions like the EHF Cup or the Cup Winners Cup. But none of them did come anywhere close to the dramatic ending of this year's Champions League final.
The game between Hungarian side Veszprem and Polish powerhouse Kielce was over midway through the second half – it was a done deal. But for some reason it didn't end the way it should have. Kielce came back in a way I've never seen before, capturing the title in a penalty shoot-out. Back in May, the team of Talant Dushebajev had come into the final weekend of the season as the big underdog. So can they defend their title in 2017?
Taking a look at the players who left the club only two of them come to mind right away, Croatian right-wing Ivan Cupic and his compatriot Denis Buntic (right-back). While Cupic shared time on the court with Tobias Reichmann – latter one has been performing on a world-class level in the past few months – the Croatian national played less and less because of Reichmann's outstanding form, especially during crunch time.
Buntic was only a back-up option to Krzysztof Lijewski, and rightfully so. Often it's no fun to watch Buntic in action anymore, as he has his best days behind him. Cutting a long story short, following the victory in Cologne, Cupic joined Vardar Skopje, while Buntic moved on to Pick Szeged. Goalkeeper Marin Sego also left Kielce, but he was replaced by another Croatian, 24-year old Filip Ivic, one of the top talents in Europe.
Serbian winger Darko Djukic replaced Cupic. He recently played for Turkish outfit Besiktas and scored 87 goals during last year's Champions League campaign. With Reichmann as a starter the new tandem on right-wing offers as much quality as it did last season.
So who will step in for Buntic? Kielce fans will have to wait one more summer to enjoy the skills of Alex Dushebajev, who will join his father in Poland. Pawel Paczkowski and especially young Macedonian, Branko Vujovic could get some playing time behind veteran Lijewski. And Reichmann is also a capable of playing right-back in an emergency situation.
The main acquisition over the summer surely is Dean Bombac. The Slovenian wizard is one of the most entertaining players to watch in all of handball. He was the main reason that Pick Szeged established itself as one of the notable clubs in Europe, winning the EHF Cup in 2014. He will help a lot to reduce the workload of injured Michal Jurecki, as well Uros Zorman and Karol Bielecki.
So what's the main weakness of the team that has gotten better on paper? It probably is the average age of some of the core players, such as Jurecki, Zorman and Bielecki. But also legendary goalkeeper Slawomir Szmal is inching closer to the end of his career. His descending form curve has become more and more obvious in recent years. But the club signed Ivic to make sure it will have a proper replacement once Szmal retires.
Kielce has one major advantage, or two, to be honest. They have only one major rival in the Polish league – Orlen Wisla Plock. And Group B of the Champions League also offers a chance to rest some key players if needed. Rhein-Neckar Löwen – although the club favors the Bundesliga compared to success in international competitions – and Vardar Skopje are the biggest contenders for first place, while Zagreb has an outside shot.
My bet is that Kielce will find a way to cruise through the group stage and finish first or second. Then it depends on the draw and what other team the Polish champion will face in the quarterfinals. We all remember quite well that Vive was one decision away from going out against Flensburg. But I think they will, once again, qualify for the Final 4. And as we all know, anyhing can happen in Cologne.


Today I'd like to discuss a bit about the upcoming fun day we all refer to as Halloween! To a lot of us, this one day out of the year signals one of a handful of holidays we have which entails lots of candy. The thing that is great about Halloween is the wonderful creative element with costumes and makeup. Now, some people love the traditional scary aspects of the holiday. (And remember if you believe in any kind of religion whatever "evil" you associate with Halloween originates from your perspective religion and should not be transferred to others who do not share your specific beliefs.) Others love the cute things that have popped up in the last 20 years or the TV and Movie characters that have trickled down to this popular holiday or candy day. The one thing that should be on all of our minds as this day quickly approaches is that no matter what we each believe and value, children are children from zero to nineteen. So, if you see a group of teenagers dressed up in costumes and trick-or-treating, that is great! We want the teens out there to be safe and enjoy this day along with all the other children.
Of course, not every family out there believes trick-or-treating is something their children should participate in. This is especially true if you don't live in an area that is safe or a community that just doesn't feel the need to doll out cavity-causing candy. So, parties are the norm with a lot of families. Again, if you know where your children are as responsible adult parents or guardians then your children can enjoy a fun and safe Halloween. Whether it be trick-or-treating or at a party, children of all ages should be allowed this one day of the year to have a lot of fun in their communities. This is a way for communities to come together and all be a positive part of every child out there having a huge smile on their faces and the pure elated joy of the whole experience! And, why deny older children the opportunity to see such creativity with the many ways people love to decorate the outsides of their homes for this special evening of the year.
One more quick reminder. Please remember if you see a teal or aqua colored pumpkin while out trick-or-treating, that means allergy safe candy is being dolled out. Nut, dairy, and gluten allergies should fall under the teal pumpkin. If you want to provide for both children with allergies and children without please make sure the candies are completely separate and never stored in the same area of your home. Also, make sure the bowl you serve the cany in has not housed candies containing allergy-inducing ingredients. Even though the candy comes individually wrapped we can never be too careful.
Please wear light reflective clothing, bring flashlights or other glowing devices along with your candy bags. Parents and guardians please drive or walk with your children to make sure they are safe every step of the way. The last thing any child needs to think about or worry about is their safety when engaged in so much fun. Also, avoid houses or apartments with the porch light turned off. Most likely those people do not participate in this holiday or they are off doing what you all are doing.
Be safe, have fun and include teenagers because they are still children and deserve to enjoy this holiday until they are unable to due to being an adult. Let kids be kids until they have to grow up. Being an adult will last a lifetime, being a kid does not.
Tips from The Marigold School of Early Learning: A Reggio Emilia inspired preschool.


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