President Carter told a congressional delegation yesterday that he wants District of Columbia political and community leaders to form a special task force to recommend ways the federal government can help solve the capital region's nagging problems.

Del. Walter E. Fauntroy (D-D.C.) said after a White House meeting that the President requested one document containing "a comprehensive assessment of what the problems are, and what can be done in the (Current) 95th Congress."

The delegation, which included 10 other lawmakers who deal with District legislation, met for 30 minutes with Vice President Mondale before being joined by the President in the Cabinet room for another 20 minutes.

Problems for which answers will be sought range from several aspects of city finances to completion of the Metrorail system. They include Congressional voting representation for the city.

Carter "obviously knew what he was talking about" from the questions he asked and comments he made about area matters, Rep Herbert E. Harris (D-Va.), a member of the House District Committee, said.

Yesterday's meeting, the first of its kind since President Lyndon B. Johnson called in lawmakers a decade ago to demand adoption of D.C. home rule legislation, was requested by Rep. Charles C. Diggs Jr. (D-Mich.), chairman of the House District Committee.

Diggs said it was he who suggested the idea to the President of a special commission, or task force, to be composed of high-ranking federal, congressional and District government officials and community spokesmen. Diggs said members of the task force would be chosen soon.

Carter, agreeing with that approach, told the group he would see that the task force included representation from the Office of Management and Budget and the Treasury and Commerce Departments, and perhaps other agencies.

The President sat on one side of the long, oval-shaped Cabinet table, flanked by Mondale and by Martha (Bunny) Mitchell, his adviser on District matters, and facing most of the lawmakers. Frank Moore, the White House's chief congressional liaison officer, sat behind the President.

During the few minutes that news media personnel were admitted for picture-taking, Fauntroy was making a pitch for a proposed constitutional amendment that would grant the District two voting members of the House and two senators. Fauntroy, as the city's congressional delegate, can speak but cannot vote on the House floor.

"I hope with your leadership and the help of the Vice President we can get it," Fauntroy told the President.

Carter, who has endorsed the principle of District representation while leaving the details uncertain, put this on the list of things he wants the task force to study.

Talking to reporters later, Fauntroy said, "Obviously, I would have preferred a strong statement supporting full voting rights for the District . . . (but) he was not prepared today to go the full route . . ."

Fauntroy and others said matters to be studied, perhaps over a six-month period, would include a possible formula for setting the annual federal payment to the city development (now authorized to be $300 million), the need for dealing with the city's $1.4 billion in pension obligations, the impending payoff of $19.8 million of bonds sold to build the Robert F. Kennedy Stadium, completion of the Metrorail system and the Pennsylvania Avenue redevelopment.

When a reporter asked whether a "commuter tax" -- a controversial levy by the city on the earnings of suburbanites -- was on the list, Harris, a staunch foe of the idea, called out: "It was not brought up!"

Harris, a former member of the Metro board, returned to his Capitol Hill office and dictated a letter inviting the President to go for a ride --possibly to National Airport -- on the Metro subway.

That he wrote Carter, would provide "a first-hand look at what can be accomplished when the local, state and federal governments work together at regional cooperation."

Other lawmakers at the White House meeting were Reps. William H. Natcher (D-Ky.) and Clair W. Burgener (R-Calif.) of the House District Appropriations Subcommittee; Sen. Patrick J. Leahy (D-Vt.), of the Senate District Appropriations Subcommittee; Sens. Thomas F. Eagleton (D-.) and Charles McC. Mathias Jr. (R-Md.) of the newly created District Subcommittee of the Senate Government Affairs Committee, and three other members of the House District Committee -- Reps. Stewart B. McKinney (R-Conn.), Romano L. Maz- -- who are not, he said, the District's (D-Calif.).

At Washington's city hall, the meeting drew criticism from Council member Julius Hobson Sr., who said in a telegram to the President that instead of meeting with members of Congress -- who are not, he said, the District's spokesperson -- the President should meet with elected city officials.