Her appearance — with her father, former Vice President Dick Cheney, as well as former Sen. Alan Simpson and the children of Norman Mineta, a Democratic congressman turned transportation secretary who was sent to the camp when he was 10 — was part of a novelty for the new Mineta-Simpson Institute; Mrs. Cheney was moved, she said, by the presence of the survivors and their enduring commitment to the country that imprisoned them during World War II. There was something else, however, that got to the MP during the bipartisan ceremony with the elders of the party she was brought up to respect. “It was just a whole mix of emotions,” she recalled in a recent interview. With Mrs. Cheney facing almost certain defeat Tuesday in the House primary, it is the likely end of the two-generation Cheney dynasty and the death of a less racial and more partisan and bottom-line brand of politics. “We were a very strong agency and we worked with the other side, that was the key, because you couldn’t function if you didn’t,” Mr. Simpson, now 90, recalled after being awarded the Presidential Medal of Freedom. and as ragged as ever for his ancestral party. “My dad was a senator and a governor, and if I were to run again today as a Republican, I’d be kicking my ass — it’s not about legacy.” Elected to the Senate in 1978, the same year Mr. Cheney won Wyoming’s House seat, they worked closely together, two Republicans fighting on behalf of the nation’s least populous state at a time when Democrats always controlled at least one part of it. Congress. It’s no mere influence, however, that Wyoming’s traditional Republicans deplore as they look back on their golden past and ponder the state’s less certain political and economic future. Ahead of Tuesday’s election, which is likely to push Harriet Hagman, who is endorsed by former President Donald J. Trump, in the House, nostalgia in the state runs deeper than Buffalo Bill’s reservoir. Mr. Cheney and Mr. Simpson were not alone in the leadership of their respective chambers in the 1980s. Along with Senator Malcolm Wallop, a Yale-educated cold warrior whose grandfather served in both the British House of Lords and in the Wyoming legislature, they got along well and often appeared together as a delegation in a kind of road show across the vast state (“A small town with big roads,” as the Wyoming saying goes). Even more dangerous was the administration of President George W. Bush. Mr. Cheney became defense secretary and his wife, Lynne, served as chairman of the National Endowment for the Humanities, while Mr. Simpson was both the second-ranking Republican in the Senate and one of the president’s closest friends. In addition, then-Secretary of State James A. Baker III spent summers at his ranch in Wyoming, meaning two of the nation’s top national security officials could be found doing unofficial promotional work for the state’s tourism industry . “You’d have army helicopters grabbing Cheney and Baker from the fishing holes,” recalled Rob Wallace, who was Mr. Wallop’s chief of staff. As conservative as the state has been nationally — Lyndon B. Johnson is the only Democrat to carry Wyoming in 70 years — Wyoming’s Republican delegation worked effectively with two respectable Democratic governors during the same period, Ed Herschler and Mike Sullivan. Now, Ms. Cheney hardly even speaks to the two other Wyomingites in Congress — Senators John Barrasso and Cynthia Lummis, both Republicans — and has little contact with Gov. Mark Gordon. Ms. Lummis has endorsed Ms. Hageman. But Mr. Barrasso and Mr. Gordon, who are the main Republicans in the Cheney tradition, have tried to remain neutral in the hope of avoiding Mr. Trump’s wrath.

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“They have to make their own choices and live with the choices they make,” Ms. Cheney said of the two men, before adding: “There are too many people who think that somebody else is going to fix the problem, that we can fix it. stay on the sidelines and Trump will fade.”

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Asked about Cheney’s legacy in Wyoming, Mr. Barrasso and Mr. Gordon declined to comment. Mr. Gordon, however, certainly hasn’t forgotten that when he ran for governor in 2018, in a primary that included Ms. Hageman, Ms. Cheney did not endorse him. “Everything is political in Wyoming except politics, which is personal,” Mr. Simpson likes to say. What worries some longtime Wyoming Republicans more than the tensions of the campaign season is what the combination of Trump fear and loyalty will ultimately mean for the strength of the delegation — and for the future of the state. Wyoming has neither a state income tax nor a corporate tax, having long relied on severance taxes on oil, gas and coal. With growing national support for clean energy — and a major federal bill pushing the climate-friendly mandate soon to become law — this mainstay of Wyoming’s economy could face an uncertain future. “Wyoming has to fight to redefine itself with the decline of fossil energy,” said Mr. Wallace, who also served as a senior official in the Interior Department under Mr. Trump. “We need a strategy at the state and federal level to figure out how Wyoming will grow and prosper for generations to come.” Ms. Cheney’s critics, however, believe that her overwhelming focus on Mr. Trump and her alienation from other House Republicans after being ousted from the party leadership would make her ineffective, particularly if Republicans claim the majority. of Parliament in November. “She is not a true Republican in the sense of our Republican values ​​here in Wyoming,” said Gina Krohn, who works in Casper for the federal Department of Agriculture, arguing that Ms. Cheney should be less consumed with the former president and “all about fossil energy.” For those with deep roots in Wyoming’s institutions, however, Ms. Cheney’s apparent death symbolizes something as troubling as any debate about the future of energy. “There are a lot of good, well-intentioned people on both sides of the aisle who want nothing to do with politics today, and that’s a scary fact,” said Marilyn Kite, a Laramie native who was Wyoming’s first lady. Court. Like many multigenerational Wyomingites, Ms. Kite blames the influx of “a lot of unpleasant people” into the state for her change in politics. Attracted by the lack of income tax, conservative politics and famous “high elevations and low crowds,” transplants, he said, changed the perception of the individualistic Cowboy State, well chronicled by John Gunther in his famous mid- century. Inside the US” “There is no boss in Wyoming, no machine rule,” Mr. Gunter wrote, adding: “Here we are still in the open spaces where a man tries, at any rate, to think for himself.” What Ms. Kite failed to mention is that Ms. Hageman grew up on a ranch near Fort Laramie, and that a significant number of Ms. Cheney’s constituents will be transplants themselves, whether it’s a liberal college town here or the community around the University of Wyoming in Laramie, or amid the rapid rise of the Tetons and rising home prices near Jackson Hole. Still, there’s an undeniable sense that Tuesday represents an end-of-an-era moment in Wyoming politics. Even after Mr. Cheney left the vice presidency, Mr. former Sen. Michael Enzi still led the Wyoming delegation, said Cat Urbigkit, a writer and self-proclaimed “shepherd” of the sheep near Pinedale. “It was a different kind of leadership in the Republican Party at that point,” Ms. Urbigkit said. This election is especially poignant for Cheney fans. Dick and Lynn Cheney both grew up in Casper, high school sweethearts at Natrona County High, where the football field is now called Cheney Alumni Field. Living in a cook tent and reading Churchill’s story of World War II from a Coleman lantern, Mr. Cheney worked for a power-line crew across the state after he was kicked out of Yale twice – and before claiming two degrees at Wyoming’s top university. That he would rise to the vice presidency, Wyoming’s closest to the Oval Office, and that his daughter would eventually succeed him in the House is a “point of pride” for the entire state, Ms. Kite said. The Simpsons, however, aren’t sure Cheney’s story is quite complete. After the ceremony in July at Heart Mountain — where Mr. Simpson, who grew up in Cody, first befriended Mr. Mineta, his fellow Boy Scout — Ann Simpson, the former senator’s wife, approached Mr. Cheney. He said he thought Mrs. Cheney should run for president. “Dick just nodded at it,” Mr. Simpson later recalled his wife telling him. “He just said, ‘I’m so proud of her.’