RIGHT THERE in the middle of a dramatic transition from one mayor to quite another for the District of Columbia is the serious financial matter of a budget deficit that needs urgent attention -- but is producing political posturing from three directions. Unlike other lingering business that mayor-elect Sharon Pratt Dixon wants Mayor Marion Barry to drop so she can take charge -- such as contracts and new appointments to boards, commissions and other agencies -- budget-balancing should be left totally in the lap of Mr. Barry, she says, to be cleaned up by the time she moves into the office Jan. 2.

Easy for her to say -- but hardly consistent with her repeated campaign pledge to get off to a fast start on the city's financial plight. Enter John Wilson, D.C. Council chairman-elect and current chairman of the council's finance and revenue committee, with a self-serving yet pertinent poke at both the mayor and the mayor-to-be: "If the council is going to deal with any tax increase or budget cuts, they want to know what the mayor-elect's position is. What she's saying is, 'You all raise taxes, you all balance the budget, and I'll stand here and say nothing.' It is not that simple." At the same time, Mr. Wilson charged, Mr. Barry's fiscal plan doesn't address the "total financial picture" but instead deals only with the city's cash-flow difficulties.

So for now you have a political tweak-fest: There is Mr. Barry, inviting his successor to join in the budget work, knowing full well that she has pledged to cut waste first before supporting anything that looks like a tax. There is Mrs. Dixon, dumping the matter entirely back on Mr. Barry, even if anything he would propose includes some new taxes and thus will be shredded when she takes office. And there is Mr. Wilson, seizing the moment to bawl them both out and repeat his five-point plan for recovery -- new taxes, a hiring freeze, an early retirement program, cuts in programs and a 20 percent increase in the annual federal payment to the city.

It is not unusual during any change of administrations for the lame duck to grind out one last budget proposal, only to see it chucked for something quite different by the next executive. Mrs. Dixon has said that a trimming of the bureaucracy and an overhaul of programs must come before any arguments for more taxes or a higher federal payment. Clearly the voters saw the logic in this plan and will await the new mayor's first sweeps of that broom in January.